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https://datascience.stackexchange.com/tags/categorical-data/new | # Tag Info
0
You may want to tokenize the strings, e.g., “Mustermann GmbH” tokenizes into "Mustermann" and "GmbH". Allow for spaces and commas certainly, perhaps also hyphens and other punctuation. You may want to look into Natural Language Processing (NLP) if you're classifying text, but whatever method you choose should have better luck sniffing out business vs. non-...
1
LabelEncoder is meant for the labels (target, dependent variable), not for the features. OrdinalEncoder can be used for features, and so can take a 2d array rather than the 1d array LabelEncoder requires, and so you can use a single transformer for all your categorical columns. (You can use a ColumnTransformer to select those categorical columns, if you ...
1
PCA is not recommended for categorical features. There are equivalent algorithms for categorical features like CATPCA and MCA.
2
Here is nice implementation of mixed type data in R- https://dpmartin42.github.io/posts/r/cluster-mixed-types This question right here- K-Means clustering for mixed numeric and categorical data and a Discussion Thread of Kaggle- https://www.kaggle.com/general/19741 There are ways, to either map your categorical data to numeric type and then you can go ...
1
You will need some way of converting categorical data to numerical, or numerical to categorical. One way to do this (convert categorical to numerical) is with one-hot encoding, where you look at the number of categories you have and make a vector of that size. Then, you can map each datapoint to a vector with 0 everywhere except for the location for the ...
0
I hope this is not a late answer but actually you can use category_encoders library, it follows sklearn's style. Example: import category_encoders as ce #I'll pretend that you've already split your data into train/test #your categorical features cat_features = ['cat_feature1', 'cat_feature2'] #count encoder count_encoder = ce.CountEncoder(cols=...
0
If the categorical variable is binary(like e.g sex) you try Point biserial correlation coefficient. Or recode levels of var(woman->1, man->0) and use pearson correlation. Recoding it's a risky way because of you indicate order(woman>man). You should be aware of that. Also $\chi^{2}$ test is used to determine whether an association (or relationship) between ...
1
If you don't encode numerical categories with dummy variables, some models will end up being trained to use an ordering of numbers (e.g. 1 < 2 < 3 < 4 < 5 < ...) in their predictions. Whether or not this is desirable or useful depends on the context, and in particular on the meaning of the numerical categories and the model and implementation ...
Top 50 recent answers are included | 2020-05-25 10:06:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3302874267101288, "perplexity": 1758.4439465990479}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347388427.15/warc/CC-MAIN-20200525095005-20200525125005-00153.warc.gz"} |
https://fourms.wiki.ifi.uio.no/Video_recording | # Video recording
If you want to work with video for analysis there are several things to think about:
• Permission: Remember to get a permission to record the performance
• Lighting: The better light, the better image result and the better analysis result. This is not so easy to control in the concert situation, unless you are able to talk to the light technician beforehand (if there is one).
## Recording tips
Before starting the discussion of various types of visualisation techniques, it might be useful to quickly review some things that may be worth thinking about when preparing video material for this type of analysis. Recording and editing video for analytical purposes is often quite different than when the video material is mainly going to be used for visual inspection. Here are some hints that may help improve the recordings:
Background: Try to place the camera so that there are as few distractions as possible in the background. This makes it easier to separate the foreground (e.g. a musician) from the background, something which is of vital importance for quantitative analysis but also very useful for qualitative analysis. In a laboratory setting this can be achieved easily by using a simple single-coloured backdrop. Outside the lab it might not be possible to change the background, but often it is possible to improve the result by looking for camera angles that give a better visual result.
Lighting: Having good and even lighting will always result in better video recordings, which again will lead to better analysis results. In a concert situation where the lighting is changing (colours and/or luminosity) it may be worth thinking about recording with night mode on the camera, or use a filter on the camera that will only let infrared light through.
Motion: Since most of the visualisation techniques that will be presented here are based on creating a \emph{motion image} (see Section~\ref{sect:motion-image}), it is important that there are no external moving elements in the image (people, curtains, etc.) in the background. The computer won't be able to differentiate meaningful from non-meaningful movements, so any external movements will also be reflected in the visualisation.
Stand: Camera motion will also influence the analysis, so it is best to use a camera stand and leave the camera untouched while recording. This includes any type of panning and zooming, since this will also show up in the final analysis. To get both detail and an overview (e.g. a single musician vs. a group of musicians), it is often better to use multiple cameras than trying to get both with one camera.
Microphones: It is often easier to place the camera(s) at a distance (e.g. in concert halls), and use the zoom to frame the image correctly. This is seldom the best placement in terms of getting a good sound quality, so it might be worth trying to record sound through external microphones placed closer to the sound source.
Thinking about some of these points will result in recordings that are better suited for visualisation and analysis.
## Recording with multiple cameras
Often it is best to record with two cameras. Place one camera in the back that will record everything, and use one camera to record closeups. In the editing software you can add both video streams to the timeline, and then use the best shot for the final video.
Synchronization of the two (or more) recordings is always a challenge. The classic method of doing this is by clapping in front of the camera, or taking a photo with a camera firing a flash. These markers can then be used to synchronize the video afterward. An alternative method, if the cameras allow for it, is to use something like SMPTE time code to handle the syncing.
## Selecting camera
There are lots of cameras out there. Which one to choose? A few things that are important from a musical point of view:
• Microphone input: many cameras do not have the ability to connect an external microphone, and should be avoided. Most cameras that do have a connector only offers a minijack, while some of the larger professional cameras also include an expander with XLR and phantom power.
• 3 CCD: lighting conditions often differ greatly during concerts, so having 3 CCD is important for getting the best light conditions
• Lens: in a typical concert situation you will either have to place the camera at the back or at the front of the room. Somewhere in between will most likely lead to problems with people getting the camera in the sightline, people bumping into the camera etc. If you are at the back you probably need a zoom lens, if you are in the front you need a wide angle lens or converter.
• HD: this is the future of digital video, and it does not make sense to buy a SD camera nowadays.
Some possible cameras:
Reviews:
## Editing
There is a multitude of video editing programs available. OSX ships with iMovie, Windows ships with Windows MovieMaker.
Many people don't know that QuickTime Pro is also a really quick and easy to use video editor.
## Converting
Converting files is often a challenge.
OSX:
• MPEG Streamclip
• HandBrake
• WonderShare Video Converter (shareware)
Windows: | 2017-09-26 01:51:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2858242690563202, "perplexity": 877.3614995582071}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818693940.84/warc/CC-MAIN-20170926013935-20170926033935-00576.warc.gz"} |
http://mathhelpforum.com/number-theory/206765-prime-numbers-print.html | Prime numbers
• November 4th 2012, 04:22 PM
makramer
Prime numbers
Quote:
Find all primes $p$ such that the number of $(p^6)+6$ is also a prime number.
How to begin to solve this problem? http://www.mymathforum.com/images/smilies/icon_wink.gif
• November 4th 2012, 05:02 PM
Petek
Re: Prime numbers
The way I solved this problem was to use Wolfram Alpha: Plug in expressions of the form
Factor p^6 + 6
where p = 2, 3, 5, 7, 11, 13, 17, 19 and so on until you see a pattern. Then use what you know (perhaps Fermat's Little Theorem) to prove your guess. Others might be able to see the solution more quickly, but that's how I did it.
• November 4th 2012, 06:36 PM
richard1234
Re: Prime numbers
Hint: Consider $p^6 + 6$ modulo some other prime q. Without much computation, I was able to find a q such that $p^6 + 6 \equiv 0 (\mod q)$ for all p. | 2016-08-27 07:23:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6960569024085999, "perplexity": 351.90959474936994}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982298551.3/warc/CC-MAIN-20160823195818-00037-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://kerodon.net/tag/00WN | # Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
Proposition 3.2.5.2. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes and let $n \geq 0$ be an integer. Then the sequence of pointed sets
$\pi _{n}(X_ s, x) \rightarrow \pi _{n}(X,x) \rightarrow \pi _{n}(S,s)$
is exact.
Proof of Proposition 3.2.5.2. Fix an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X$ such that $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ is the constant map carrying $\operatorname{\partial \Delta }^{n}$ to the base point $x \in X$. We wish to show that the homotopy class $[\sigma ]$ belongs to the image of the map $\pi _{n}(X_ s, x) \rightarrow \pi _{n}(X,x)$ if and only if the image $[f(\sigma )]$ is equal to the base point of $\pi _{n}(S,s)$. The “only if” direction is clear, since the composite map $X_{s} \hookrightarrow X \xrightarrow {f} S$ is equal to the constant map taking the value $s$. For the converse, suppose that $[ f(\sigma ) ]$ is the base point of $\pi _{n}(S,s)$. Then there exists a homotopy $h: \Delta ^{1} \times \Delta ^{n} \rightarrow S$ from $f(\sigma )$ to the constant map $\sigma '_0: \Delta ^{n} \rightarrow \{ s\} \subseteq S$, which is constant when restricted to the boundary $\operatorname{\partial \Delta }^ n$. Since $f$ is a Kan fibration, we can lift $h$ to a homotopy $\widetilde{h}: \Delta ^{1} \times \Delta ^{n} \rightarrow X$ from $\sigma$ to another $n$-simplex $\sigma ': \Delta ^{n} \rightarrow X$, where $\widetilde{h}$ is constant along the boundary $\operatorname{\partial \Delta }^{n}$ and $f( \sigma ') = \sigma '_0$ (Remark 3.1.4.3). Then $\sigma '$ represents a homotopy class $[\sigma '] \in \pi _{n}(X_ s, x)$, and the homotopy $\widetilde{h}$ witnesses that $[\sigma ]$ is equal to the image of $[\sigma ']$ in $\pi _{n}(X,x)$. $\square$ | 2020-07-06 09:29:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9803763031959534, "perplexity": 38.166871434887355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655890157.10/warc/CC-MAIN-20200706073443-20200706103443-00008.warc.gz"} |
https://economics.stackexchange.com/questions/26987/simple-econometric-question/27033 | # Simple Econometric question
From LLN, we have $$\text{plim} Y_n=\mu$$. Now you have $$W_n=Y^3_n$$, what is the probability limit of $$W_n$$?
Please help me answer this and thank you!
• Take a look at the continuous mapping theorem. – Matthew Gunn Feb 22 '19 at 23:48
• If you want to only find the probability limit, you can replace "plim" with "lim". Proving it involves probability theory. – chan1142 Feb 23 '19 at 8:35
## 1 Answer
The comments have the answer. Let $$\{X_n\}_{n=1}^{\infty}$$ be a sequence of iid random variables with $$E(X_1)=\mu$$ and $$Var(X_1)=\sigma^2$$ with $$\mu,\sigma^2<\infty$$. Define $$Y_n=\frac{1}{n}\sum_{i=1}^n X_i.$$ By WLLN, $$\text{plim}Y_n=\mu.$$ Since $$f(x)=x^3$$ is a continuous function, by the continuous mapping theorem, $$\text{plim}f(Y_n)=\text{plim}Y_n^3=f(\mu)=\mu^3.$$ | 2021-01-19 04:55:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9617291688919067, "perplexity": 298.87339334430527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703517966.39/warc/CC-MAIN-20210119042046-20210119072046-00378.warc.gz"} |
https://cs.stackexchange.com/questions/134059/finding-optimal-separating-value | # Finding optimal separating value
## Problem description
We are given two sorted arrays of even numbers: A and B. Values of A are generally supposed to be smaller than values of B. So we are asked to find a value X where X is an odd number to predict which set an element belongs to. If an element is smaller than X, it will be assigned to A,and if it is bigger - to B. The value should be optimal, meaning the number of improperly matched values should be minimal. For example, if A = [2,4,6] and B = [6, 10, 12]. Separating value 3 would cause 2 mismatched values 4 and 6. Optimal would be 5 or 7, both causing one improper matched element. The algorithm is supposed to return an optimal separating value (only one of there are many) and the number of mismatched items.
## My ideas
At first I thought about using Binary Search for each element of A to find a number of elements of B smaller than it. Then repeat for B.
A better idea would be to generate candidates for such separating numbers. For example, if we only have values 10, 50, 100 we don't have to check all odd numbers, we can just check 9, 11, 51, 101. We could then calculate prefix sums, indicating how many numbers from A are bigger than the checked value and how many elements from B are smart than that value. The sum of the two sums for a number would be the total number of errors. The only thing left is to find the value with minimal errors.
Is this approach even near optimal, maybe we can find a better way. Also, calculating such prefix sums can be tricky because if we want not to care about the range of numbers we would have to find a way to skip some numbers and that makes our iteration harder, although possible.
What are your ideas? Some pseudocode or code in Python is very welcome.
Initialize an array of pairs $$T[i] = (x, c)$$ containing all elements of $$A$$ and $$B$$. Let $$x$$ be the elements themselves, and $$c = 0$$ if the element came from $$A$$ and $$c = 1$$ if the element came from $$B$$. Then sort $$T$$ based on the $$x$$ values in ascending order.
Then the number of mistakes you'd get if you choose a value between $$T[i]\text{.}x$$ and $$T[i+1]\text{.}x$$ is:
$$\sum_{j=0}^i T[j]\text{.}c + \sum_{j=i+1}^n\left(1 - T[j]\text{.}c\right) =$$ $$(n - i) +\sum_{j=0}^i T[j]\text{.}c - \sum_{j=i+1}^nT[j]\text{.}c =$$ $$(n - i) +2\sum_{j=0}^i T[j]\text{.}c - \sum_{j=0}^nT[j]\text{.}c =$$ $$- i +2\sum_{j=0}^i T[j]\text{.}c - C$$
where $$C$$ is just a constant, which doesn't matter if you wish to minimize the quantity. So to find the optimal value:
S = 0
best = inf
best_i = 1
for 1 <= i <= n:
S += 2*T[i].c
if S - i < best:
best = S - i
best_i = i
• Much simpler and cleaner, although not necessarily faster. Your solution is O(m) where m is the data spread. Mine is always O(n). Thank you anyway. – Kangaroo976 Jan 8 at 12:09
• @Kangaroo976 No? Nothing relies on the data spread. Also my solution is $O(n \log n)$ because of the sorting operation. Unless $A$ and $B$ come pre-sorted in which case you can use an $O(n)$ merge, but you never mentioned that they were sorted. – orlp Jan 8 at 12:30 | 2021-08-04 02:56:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 17, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7001069784164429, "perplexity": 362.37441844288304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154500.32/warc/CC-MAIN-20210804013942-20210804043942-00207.warc.gz"} |
http://auburn.edu/cosam/departments/math/research/colloquia.htm | COSAM » Departments » Mathematics & Statistics » Research » Departmental Colloquia
# Departmental Colloquia
Our department is proud to host weekly colloquium talks featuring research by leading mathematicians from around the world. Most colloquia are held on Fridays at 4pm in Parker Hall, Room 250 (unless otherwise advertised) with refreshments preceding at 3:30pm in Parker Hall, Room 244.
DMS Colloquium: Jan Rosinski
Mar 23, 2018 04:00 PM
Speaker: Jan Rosinski, University of Tennessee
Title: Series expansions of time-continuous random walks and solutions to random differential equations
Abstract: In 1923 N. Wiener constructed two random trigonometric series which converge uniformly to a limit satisfying conditions for a Brownian motion. The conditions were earlier postulated by A. Einstein in terms of partial differential equations. K. Ito, the founder of Ito stochastic calculus, unified results on the convergence of these and other similar series expansions in the so-called Ito-Nisio theorem (1968). O. Kallenberg (1974) gave the first generalization of the Ito-Nisio theorem to the uniform convergence of processes with jumps.
Brownian motion describes continuous in space random walk. In this talk we will concentrate on random walks with jumps, their series expansions, and strongest possible modes of their convergence. To this aim we will establish further generalization of the Ito-Nisio theorem. We will discuss the Ito map, which is just an ODE with a rough input. Using these tools we obtain strong pathwise convergence in numerical solutions of stochastic differential equations driven by Levy processes.
This talk is based on a joint work with Andreas Basse-O’Connor and Jorgen Hoffmann-Jorgensen.
Faculty host: Erkan Nane
DMS Colloquium: Emily King
Mar 30, 2018 04:00 PM
Speaker: Emily King (U. Bremen)
Title: (Hilbert Space) Frames, Algebraic Combinatorics, and Geometry
Abstract: Frames are generalizations of orthonormal bases which yield "nice" decompositions of data. Such systems are the foundation of applied harmonic analysis and are also closely related to quantum measurements and linear codes. When one wants an optimally robust representation of data, one often looks for frames that have some sort of spread, be it geometric (as non-parallel as possible) or algebraic (no nontrivial linear dependencies). Over the last few years, it has been discovered that the relationship between these two types of spread is more complicated than had previously been believed. Furthermore, methods from algebra, geometry, and combinatorics have recently proven themselves to be very useful in the study of frames.
For example, algebraic graph theory and combinatorial design theory have led to new characterizations and novel constructions of optimal line (or, more generally, subspace) configurations which are also frames. Also, almost all desirable classes of frames form real algebraic varieties, and certain known results in frame theory have also been found to be equivalent to concepts in matroid theory and arrangements of hyperplanes.
In this talk, the currently known connections between these objects from harmonic analysis / quantum information theory and combinatorics / algebraic & discrete geometry will be presented.
(photo courtesy Uni Bremen/Kai Uwe Bohn)
DMS Colloquium: Claudiu Raicu
Apr 06, 2018 04:00 PM
Speaker: Claudiu Raicu (Notre Dame)
Title: TBA
DMS Colloquium: Gradimir V. Milovanovic
Apr 10, 2018 04:00 PM
Speaker: Gradimir V. Milovanovic, Serbian Academy of Sciences and Arts
Title: Special Quadrature Processes for Summation of Slowly Convergent Series
Abstract: Slowly convergent series appear in many problems in mathematics, physics and other sciences. There are several numerical methods based on linear and nonlinear transformations (e.g., Cesàro-transformation, Aitken $$\Delta^2$$-transformation, Wynn-Shanks $$\varepsilon$$-algorithm, $$E$$-algorithm, Levin's transformation, $$\rho$$-algorithm, etc.). In this lecture we present the so-called summation/integration methods, which are very efficient. Methods are based on certain transformations of sums to weighted integrals over the real line or the half-line and on an application of special Gaussian quadrature formulas with respect to some non-classical weight functions (Bose-Einstein, Fermi-Dirac, hyperbolic weights, $$\ldots$$). For constructing such quadrature rules we use a recent progress in symbolic computation and variable-precision arithmetic, implemented through our Mathematica package “OrthogonalPolynomials.” Several interesting applications will also be presented.
Faculty host: Narendra K. Govil
DMS Colloquium: Honglang Wang
Apr 20, 2018 04:00 PM
Speaker: Honglang Wang, IUPUI (Indiana University--Purdue University Indianapolis)
Faculty host: Guanqun Cao
DMS AU-AUM Joint Colloquium: Robert Underwood
Mar 09, 2018 04:00 PM
AU-AUM Joint Math Colloquium
Speaker: Robert Underwood, AUM
Title: A Class of Automatic Sequences
Robert G. Underwood is an Ida Belle Young Endowed Professor at Auburn University at Montgomery. He is the author of three books on Hopf algebras and modern algebra, and has published over 30 papers on subjects including algebra, geometry and theoretical computer science. He teaches modern algebra, number theory and applied cryptography at AUM. He is currently a co-PI on an NSF STEM grant conducting an undergraduate research project relating entropy and secure encryption.
Faculty host: Huajun Huang
DMS Colloquium: Hans-Werner van Wyk
Feb 23, 2018 04:00 PM
Speaker: Hans-Werner van Wyk
Title: The Propagation of Uncertainty through Differential Equations
Abstract: Many complex physical processes can be adequately described in the deterministic language of partial differential equations (PDEs), yet operate in uncertain environments that can only be observed partially and/or indirectly. This aleatory uncertainty manifests itself in the underlying system parameters. My work centers on its identification and propagation through to quantities of interest related to the system's output. This talk highlights my contributions in this area, particularly in improving the efficiency of sampling methods by exploiting multiscale features of random inputs, by designing quadrature schemes tailored to the underlying density, or through the use of reduced order models. I will also discuss some applications of my work, ranging from the prediction of saltwater intrusion, to the development of reliable spectral diagnostics for astrophysical plasmas.
DMS Colloquium: Yifan Cui
Feb 19, 2018 04:00 PM
Speaker: Yifan Cui, University of North Carolina, Chapel Hill
Title: Tree-based Survival Models and Precision Medicine
Abstract: In the first part, we develop a theoretical framework for survival tree and forest models. We first investigate the method from the aspect of splitting rules. We show that existing approaches lead to a potentially biased estimation of the within-node survival and cause non-optimal selection of the splitting rules. Based on this observation, we develop an adaptive concentration bound result which quantifies the variance component for survival forest models. Furthermore, we show with two specific examples how these concentration bounds, combined with properly designed splitting rules, yield consistency results.
In the second part, we focus on one application of survival trees in precision medicine which estimates individualized treatment rules nonparametrically under right censoring. We extend the outcome weighted learning to right censored data without requiring either inverse probability of censoring weighting or semi-parametric modeling of the censoring and failure times. To accomplish this, we take advantage of the tree-based approach to nonparametrically impute the survival time in two different ways. In simulation studies, our estimators demonstrate improved performance compared to existing methods. We also illustrate the proposed method on a phase III clinical trial of non-small cell lung cancer.
DMS Colloquium: Jaehong Jeong
Feb 16, 2018 04:00 PM
Speaker: Jaehong Jeong, King Abdullah University of Science and Technology
Title: A Stochastic Generator of Global Monthly Wind Energy with Tukey g-and-h Autoregressive Processes
Abstract: Quantifying the uncertainty of wind energy potential from climate models is a very time-consuming task and requires a considerable amount of computational resources. A statistical model trained on a small set of runs can act as a stochastic approximation of the original climate model and can be used to assess the uncertainty considerably faster than by resorting to the original climate model for additional runs. While Gaussian models have been widely employed as means to approximate climate simulations, the Gaussianity assumption is not suitable for winds at policy-relevant time scales, i.e., sub-annual. We propose a trans-Gaussian model for monthly wind speed that relies on an autoregressive structure with Tukey g-and-h transformation, a flexible new class that can separately model skewness and tail behavior. This temporal structure is integrated into a multi-step spectral framework that is able to account for global nonstationarities across land/ocean boundaries, as well as across mountain ranges. Inference can be achieved by balancing memory storage and distributed computation for a data set of 220 million points. Once fitted with as few as five runs, the statistical model can generate surrogates fast and efficiently on a simple laptop and can provide uncertainty assessments very close to those obtained from all the available climate simulations on a monthly scale.
This is joint work with Yuan Yan, Stefano Castruccio, and Marc G. Genton.
DMS Colloquium: Yiwen Liu
Feb 14, 2018 04:00 PM
Speaker: Yiwen Liu, University of Georgia
Title: B-scaling: a novel nonparametric data fusion method
Abstract: With the rapid development in science and technology, massive data has been collected from different sources, which leads to a large amount of data with different types and formats, such as the image data and omics data. Each type of the data only captures part of the contained information, and the data has to be integrated or fused to provide a complete understanding of the whole picture. Thus, there is an urgent call of powerful data fusion method. In this talk, I will introduce a B-scaling method to integrate multisource data. The asymptotic property of the B-scaling method will be discussed to provide theoretical underpinning of the method. The application of the method on epigenetic and biomedical research will be highlighted in the talk.
DMS Colloquium: Stefan Friedenberg
Feb 09, 2018 04:00 PM
Speaker: Professor Stefan Friedenberg, Hochschule Stralsund
Title: Extensions of rational groups
Faculty host: Ulrich Albrecht
Last Updated: 09/11/2015 | 2018-03-22 15:41:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42904040217399597, "perplexity": 2238.2044601013736}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647892.89/warc/CC-MAIN-20180322151300-20180322171300-00392.warc.gz"} |
https://www.nature.com/articles/s42003-019-0649-2?error=cookies_not_supported&code=e3fa0364-2473-4103-85b3-2eafd1051c1c | ## Introduction
The Arctic Ocean winter sea ice cover has steadily declined since the 1970s, and the majority of this reduction has been observed in the Barents Sea1,2,3,4. With a warmer and increasingly ice-free Arctic Ocean, the border between the Boreal and Arctic biomes is predicted to move north5,6. While the present warming has allowed an opportunistic northwards expansion of Boreal species into the northern Barents Sea Arctic ecosystem7, little attention has been given to the effect of the recent ice retreat and variability on the ice-associated fish community. A keystone species in the ice-associated Arctic marine food web is the polar cod, one of few species linking the lower (i.e. zooplankton) and higher (e.g. other fish, mammals, seabirds) trophic levels8,9,10. The polar cod is endemic to the Arctic and their early life history is strongly adapted to the presence of ice: from spawning of eggs under the ice11; the ability of eggs to develop in sub-freezing temperatures12; larvae feeding on zooplankton specific to the seasonal ice-melt-water blooms13,14; and low mortality of larvae in the close to freezing temperatures typical of Arctic water masses15. At the same time observations made by Soviet-era researchers in the early 1960s11 remain the most complete descriptions of polar cod spawning along the Eurasian shelf, with a historical stronghold in spawning activity in the south-eastern corner of the Barents Sea (also known as the Pechora Sea) and a suggested spawning east of Svalbard. The Barents Sea polar cod stock have been monitored annually by a joint Norwegian-Russian survey since 198616, and the total stock biomass (TSB) has varied vastly in the past three decades, between a minimum of 127,000 t in 1990, up to a maximum of 1,941,000 t in 2006 yet with no clear trend17. The spatial distribution and abundance of 0-group polar cod (~6 months old) has also varied considerably in this period18, suggesting high recruitment variability both in space and time.
Given the tightly linked early life history of the polar cod with ice, a natural candidate for the large variation in polar cod recruitment and biomass is the inter-annual variation in ice cover18. The two main factors driving the observed variability in ice cover in the Barents Sea is the inflow of warm Atlantic water from the west and the inflow of cold, less saline Arctic water from the north and east4,19,20. The inflow of Atlantic water from the west and Arctic water from the northeast set up a strong temperature and salinity front across the entire Barents Sea, here termed the Polar Front21,22 (Fig. 1). North of the Polar Front the less dense Arctic water masses stabilizes the water column sufficiently to prevent upwards flux of the warmer Atlantic water, allowing ice to form4. In March-April the extent of the seasonal ice cover in the Barents Sea is at its largest, and at this time period the Polar Front and the ice edge usually coincide. During 1990–2017 the winter maximum ice cover in the Barents Sea varied between 632,304 km2 and 1,129,719 km2, with an average of 869,961 km2.
Here we investigate possible links between inter-annual variability in the Barents Sea ice cover and the variability in spawning distribution and recruitment of polar cod. To locate the spawning locations of polar cod under the ice we “back-tracked” larval drift trajectories from observed 0-group polar cod in autumn to the most probable spawning locations in spring. This was done by Lagrangian particle advection simulations within a 3D dynamical ocean model. Drifting particles were “spawned” uniformly over the parts of the Barents Sea that was encompassed by the historical marginal ice zone (i.e. where ice concentration had been at least 15% at any time in period 1990–2017), and subsequently an objective search algorithm identified drift trajectories that intersected the 0-group observations of the autumn survey. The coupled biophysical model was simulated over 28 spawning seasons (1990–2017) and was compared to the catch of 0-group polar cod in 8302 pelagic trawls distributed throughout the Barents Sea. The mechanistic interpretation of the scores from the objective search algorithm is twofold: first the backtracking from observations indicates spawning at a given release point for a given year (viz. inter-annual change in spawning location); and second the frequency of links between spawning location and the observed 0-group abundance reflects larval survival during the drift phase and indicates supply of recruits from a given spawning location (viz. recruitment variability in response to environmental conditions). We report a positive relationship between year class strength of polar cod to the inter-annual variability of ice cover, and a negative effect of maximum summer temperatures encountered in late larval phase, where a common driver is the heating of the system. We also observed a clear northward retreat of one the two identified spawning assemblages towards the end of the study period that may also be a response to warming. Understanding the physical-biological interactions during recent decades of both varying and changing ice conditions allow predictions of future development of the polar cod stock in the Barents Sea.
## Results
### Inter-annual change in spawning location
The objective search algorithm revealed two clearly separated maxima in spawning location probability, with one spawning assemblage in the Pechora Sea and one east of Svalbard. Moreover, the location of the two main spawning areas identified coincided with the high probability of 0-group occurrence downstream to the two areas (Generalized additive model (GAM) explaining 40% of the variation in 0-group presence/absence, Fig. 2). At the same time there was considerable inter-annual variability in the location of the center of the back-calculated main spawning areas. For example, in 1990 when the observed spatial distribution of 0-group polar cod was strongest correlated with spawning east of Svalbard, most of the eggs and larvae drifted with the East Spitsbergen Current into the Storfjord, further along the west coast of Svalbard, or onto the Svalbard Bank, where most of the 0-group polar cod was found that year (Fig. 3a, and see Fig. 1 for references to location). Recruitment in 1995 was very low, indicated by a record low abundance of 0-group polar cod found in the autumn cruise, coinciding with the Pechora Sea largely being ice-free during most of winter and spring (Fig. 3b) and maximum temperatures encountered in late summer exceeding 10 °C. This in contrast to 1998 when the ice cover in the south-eastern Barents Sea was at its highest in the study period, and the release points in the Pechora Sea was strongly correlated with the high abundance of 0-group polar cod found in a wide swath along Novaya Zemlya (Fig. 3c). Perhaps the most anomalous year in terms of drift patterns was in 2013 when the most probable spawning location of the eastern Svalbard spawning assemblage was situated north of Kvitøya, close to the shelf edge towards the Nansen Basin–yielding a qualitatively different dispersal pattern than previous years, where the majority of larvae initiated from this particular location were advected along the margins of the continental shelf instead of into the Barents Sea (Fig. 3d). This northern displacement of the eastern Svalbard spawning assemblage in 2013 was not an isolated instance, with a similar displacement also in 2009, 2015, and 2017. The predicted western spawning center was also located on the Great Bank in many of the years in the middle of the study period (2000s), and even near the Central Bank in some years. The average northwards displacement towards the end of the study period (2015–2017) constituted 2° of latitude (approx. 220 km) compared to the yearly 2000s (2001–2005), culminating a clear decadal northward trend (Fig. 4).
### Recruitment variability in response to ice loss and heating
Recruitment strength in the eastern Barents Sea/Pechora Sea had a significant correlation with maximum yearly ice cover (r = 0.69, t = −4.49, df = 26, p < 0.001, see Fig. 5), reiterating the importance of ice for early pelagic stages. The recruitment in the Pechora Sea was also significantly correlated with the estimated TSB two and a half years later (r = 0.61, t = 3.58, df = 22, p < 0.001), confirming the Pechora Sea spawning assemblage as the main supplier of recruits to the Barents Sea stock monitored during autumn surveys. Recruitment east of Svalbard had no correlation with either ice cover, temperature, or TSB at any lag, indicating other effects at play than those tested within the scope of this study. A linear regression of maximum ice cover over the study period showed a weak, yet significant yearly decrease of 1.1% (F = 4.93 on 1 and 26 df, p = 0.035, R2 = 0.16). Moreover, the 10-day mean-filtered maximum temperature encountered by larvae in late summer was significantly different between the two main spawning areas (t = −7.46, df = 47, p< 0.001), with maximum temperatures encountered in the eastern Barents Sea on average between 2.1 °C and 3.6 °C degrees warmer (95% CI). The average yearly maximum temperature encountered in the Pechora/eastern Barents Sea was 5.7 °C but in extreme years more than 9 °C (95% CI: 2.4–9.0 °C), well beyond the thermal tolerance limit of the larvae (≈100% mortality at extended periods above 7 °C15). A linear regression model of eastern Barents Sea recruitment strength as function of covariates Barents Sea ice cover, maximum temperature encountered by larvae, and TSB, explained 72% of the variation in recruitment strength (F = 20.7 on 3 and 23 df, p < 0.001). Ice and temperature explained the majority of the variance, and a smaller portion explained by TSB $$({\mathrm{R}}_{{\mathrm{ice}}\,{\mathrm{cover}}}^2 = 48\% ,{\mathrm{R}}_{{\mathrm{max}}\,{\mathrm{temp}}}^2 = 44\% ,{\mathrm{R}}_{{\mathrm{TSB}}}^2 = 16\% )$$. However, the two variables ice cover and temperature where found to share a high portion of the explained variance $$({\mathrm{R}}_{{\mathrm{ice}}\,{\mathrm{cover}}}^2 \cap {\mathrm{R}}_{{\mathrm{max}}\,{\mathrm{temp}}}^2 = 25\% )$$, albeit being significantly negatively correlated (r = −0.38, t = −2.09, df = 26, p = 0.045).
## Discussion
We suggest two plausible mechanisms driving the large recruitment variability observed in the south-eastern parts of the Barents Sea, mediated in turn by both heat and ice. First, the presence/absence of ice likely causes a “match/mis-match” scenario23,24 specific to the seasonally ice-covered Arctic food web. The primary food item for the first-feeding larval phase is the naupliar stages of Arctic copepod Calanus glacialis that are nourished by the algae bloom confined to the meltwater plume13,25,26. A mis-match most likely arises in years with reduced or no ice cover in the Pechora Sea when the phenology of the spring bloom follows Atlantic-like bloom dynamics initiated by thermal stratification rather than ice-melt25,27,28. However, if larvae indeed survive that first critical ice-associated phase all three major Calanus species found in the Barents Sea are eaten indiscriminately13. This includes the highly abundant C. finmarchicus and the larger (but rarer) C. hyperboreus, both advected from the west with the warmer Atlantic water masses, and all three Calanus species having exceptionally high energy content29. The ample food supply available to the later larval stages in the Atlantic-influenced water masses found in the south-eastern parts of the Barents Sea thus represents a near ideal nursery habitat. At the same time, the high summer temperatures sometimes encountered there may also be associated with high risk by pushing larvae beyond thermal thresholds15. For example, the seven warmest years in terms of maximum summer temperatures encountered (as observed in 1990, 1995, 2000, 2012–2013, 2016–2017), resulted in the five years of lowest recruitment. However, due to the collinearity of ice cover and high temperatures encountered in south-eastern Barents Sea during drift in late summer, where a common driver is the wind-driven advection of warmer water masses from the west20, we were not able to disentangle which of the two identified mechanisms that affected recruitment in the Barents Sea the most.
Regional downscaling of future climate change scenarios predicts an increased frequency of years with reduced or no ice cover in the south-eastern Barents Sea towards 205030,31. However, due to the strong topographic steering of the incoming Atlantic water masses away from the northern parts of the Barents Sea (cf. Figure 1), the cold and icy winter-conditions currently found east of Svalbard is likely to persist into the foreseeable future19,32,33. The predicted deterioration of the south-eastern Barents Sea nursery habitat will thus increase the relative importance of the supply of recruits coming from the eastern Svalbard spawning assemblage in the future. At the same time the observed northward displacement of the eastern Svalbard spawning assemblage towards the end of the study period may qualitatively alter the dispersal pathway of eggs and larvae. The advection of the surface layer in the far northern Barents Sea area is mainly influenced by the prevalent wind field, where the meltwater and pack ice generally follow a net westward direction towards the Fram Strait19,20. Other potential dispersal pathways that may be realized in the northernmost Barents Sea include getting entrained with the Atlantic boundary current flowing along the Nansen Basin34, or lateral transport towards the Arctic Ocean interior with eddies detaching from the boundary current itself35. In any case, polar cod larvae and juveniles drifting with the pack ice in the Arctic Ocean interior is not an uncommon phenomenon36, although the fate of these drifting individuals is not known.
The biophysical links highlighted here represent fundamental knowledge of how a keystone species of the Arctic food web respond to contemporary climatic forcing. As the frequency of years with low or no ice cover in the south-eastern Barents Sea is predicted to increase towards the middle of the millennium, we expect recruitment of polar cod to the south-eastern Barents Sea to become even more variable or diminished, most likely having dramatic consequences for the entire food web. And ultimately if summer temperatures continue to exceed the thermal tolerance of larvae, the south-eastern Barents Sea as a nursery area will be unsuitable altogether. Conversely, the topographical steering of Atlantic water masses away from the north-western Barents Sea will most likely facilitate ice and spawning east of Svalbard even in a warmer climate than today.
## Methods
### Recruitment and spawning stock biomass indices
The 0-group polar cod data were sampled on annual surveys run between late August and early October in the period 1990 and 2017, covering almost the entire Barents Sea within a regular grid of ~65 km. At each station the upper water layer (0–60 m) was sampled by three pelagic trawls with a 20 × 20 m opening, keeping the headlines at 0 m, 20 m, and 40 m. The pelagic trawls were towed at a speed of 3 knots over a time interval of 10 min, corresponding to a tow length of 0.5 nautical miles (≈0.93 km). If dense concentrations of fish appeared on the echo-sounder deeper than 40 m, additional tows were performed at 60 and 80 m. During the study period of 27 years 8302 of these depth-integrated trawl hauls were done. Due to the selectivity of the gear37, the catches were adjusted for capture efficiency using a stratified sample mean method38,39. As a proxy for total stock biomass (TSB) we estimated the total mass of polar cod found in echo-sounder transects and pelagic trawls throughout the Barents Sea17, identical to the method used for estimating capelin (Mallotus villosus) stock size in the Barents Sea40.
### Ocean circulation model and ice module
The hydrodynamic model used to represent the currents and oceanographic conditions (i.e. temperature, salinity, and ice concentration/cover) in the study area was based on the Regional Ocean Modeling System (ROMS, http://myroms.org), a free-surface, hydrostatic, primitive equation ocean general circulation model41,42. The ROMS model was run with a horizontal resolution of 4 × 4 km in an orthogonal, curvilinear grid covering parts of the North Atlantic and all the Nordic and Barents seas (see inset in Fig. 1 for extent of ROMS model) over the time period 1960–201720,43,44,45. The output from ROMS contained velocity fields, ice concentration, temperature, and salinity in 32 terrain following vertical layers, and a temporal resolution of 24 h.
### Drift simulations and search algorithm
The advection of particles in the horizontal plane was modelled by the Runge-Kutta fourth order scheme LADIM46,47. As early life stages of polar cod are usually found close to the surface13, particles were uniformly distributed in the upper 10 meters with a fixed depth throughout the drift phase from 1 January to 30 September. In an exhaustive search for potential spawning areas of polar cod in the Barents Sea, particles were released in a regular grid (≈40 km equidistance, 537 positions in total) across the entire Barents Sea shelf shallower than 400 m that had been covered by an ice concentration of more than 15% in the period 1990–2017 (see extent of release grid in Fig. 2). A new ensemble of 100 particles were released at every point in the grid, every day from 1 January to 30 April, repeated for every year between 1990 and 2017 (yielding a total of 639,030 particles each year). Subsequently, an objective search algorithm identified drift trajectories that intersected the 0-group observations of the autumn survey within a three-week period of the surveys. The ability of the drift trajectories to explain the observed 0-group abundance and distribution was thus interpreted as a confirmation of spawning at a given release point and a high larval survival integrated over the drift phase. To allow a direct comparison between number of simulated drift intersections and 0-group abundance, both indices were log-transformed and scaled between 0 and 1. In line with the hypothesis of ice as a prerequisite for spawning, the drift trajectories’ ability to predict the observed 0-group abundance was weighted by ice concentration at drift start point (i.e. at spawning area). To elucidate on the possible effects of heating on recruitment we extracted temperature profiles from all individual drift trajectories, and to decrease the effect of minor cold spells or heat waves on the subsequent analysis we applied a 10-day moving average filter on the temperature profiles.
### Statistical modelling
A probabilistic map of 0-group polar cod presence was calculated by using a two-dimensional binomial GAM smoother, based on the geographical coordinates of pelagic trawls, presence-absence of polar cod larvae in the pelagic trawls, and using the logit-link function as implemented in R-package “mgvc”48. Moreover, to quantify the effect of environmental conditions on larval survival/recruitment strength, we fitted a linear regression model with recruitment strength as independent variable with the covariates Barents Sea ice cover (area of the Barents Sea covered by ice concentration higher than 15%, extracted from the ROMS model), maximum temperature encountered by larvae (10-day mean-filtered over 100 larvae released from the most likely spawning area for a given year), and estimated TSB. This regression model was fitted separately for the north-western (Svalbard) and south-eastern (Pechora Sea) spawning areas as implemented in the base R-package “stats”49. In the model selection phase, we applied a stepwise model selection scheme with the initial inclusion of all relevant variables, where only the variables deemed significant was included in the final model. Due to the high degree of collinearity between some of the variables, we also did a variance partitioning analysis to disentangle the separate and/or common effects of the variables50.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. | 2023-03-23 06:10:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5759994983673096, "perplexity": 3919.699398620834}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944996.49/warc/CC-MAIN-20230323034459-20230323064459-00640.warc.gz"} |
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The problem of quantum gravity is impossibly hard because it is too general. Quantum gravity problem is not trivial but not nearly as hard if we consider a static spacetime such as Einstein static model $\mathbf{R}\times S^3(a)$. Quantum field theory on this specific spacetime is essentially a solved problem. In fact, even Euclidean quantum field theory following Osterwalder-Schrader quantization and Wightman axioms has been worked out in this case ( jaffee-ritter-qft1jaffee-ritter-qft2 osterwalder-schrader-2osterwalder-schrader-1 ). In presentations of quantum field theory on curved spacetimes the Einstein static spacetime appears often as a relatively simple example (see Fewster lecturenote-3908 (1)). | 2017-10-18 09:26:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9089070558547974, "perplexity": 565.1394463895608}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822851.65/warc/CC-MAIN-20171018085500-20171018105500-00242.warc.gz"} |
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• 301.
Chalmers University of Technology, Department of Mathematics.
Department of Differential Equations and Functional Analysis, Russian Peoples' Friendship University. Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
On some sharp reversed Hölder and Hardy type inequalities1994Inngår i: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, nr 169, s. 19-29Artikkel i tidsskrift (Fagfellevurdert)
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Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Matematik.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Matematik.
On polynomial eigenfunctions for a class of differential operators2002Inngår i: Mathematical research letters, Vol. 9, nr 2, s. 153-171Artikkel i tidsskrift (Fagfellevurdert)
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Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Evaluating ∫0f(x)dx and ∫ab f(x)dx using residue calculus2014Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
In this essay we use complex analysis, in particular modern residue calculus, to compute certain Riemann integrals.
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Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Numerical Simulations of Linear Stochastic Oscillators: driven by Wiener and Poisson processes2017Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
The main component of this essay is the numerical analysis of stochastic differential equations driven by Wiener and Poisson processes. In order to do this, we focus on two model problems, the geometric Brownian motion and the linear stochastic oscillator, studied in the literature for stochastic differential equations only driven by a Wiener process. This essay covers theoretical as well as numerical investigations of jump - or more specifically, Poisson - processes and how they influence the above model problems.
• 305.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Bolåneräntor i Sverige: Enanalys av individuella räntor med multipel linjär regression2014Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
I denna rapport undersöks hur ett antal kundspecifika faktorer som belåningrad, bank och inkomst påverkar svenska hushålls individuella bolåneräntor. Metoden som används är multipel linjär regression med transformeringar av förklarande variabler. Transformer som används är log-linjär, linjär-log, log-log samt styckvis linjär. Datan innehåller ett stickprov om ca. 7000 rörliga bolån från juli 2013 insamlade av organisationen Villaägarna på frivillig basis. Variablerna belåningsgrad, lånets storlek och bank bidrar mest till att förklara räntan. Vår analys visar att stora lån i kombination med låg belåningsgrad tenderar till att ge lägst ränta samtidigt som det finns signifikanta skillnader i bolåneränta mellan bankerna även om deras listräntor är lika.
• 306.
Luleå tekniska universitet, Institutionen för system- och rymdteknik, EISLAB.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper. Luleå tekniska universitet, Institutionen för system- och rymdteknik, Datavetenskap. Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Epi-convergence of minimum curvature variation B-splines2003Rapport (Annet vitenskapelig)
We study the curvature variation functional, i.e., the integral over the square of arc-length derivative of curvature, along a planar curve. With no other constraints than prescribed position, slope angle, and curvature at the endpoints of the curve, the minimizer of this functional is known as a cubic spiral. It remains a challenge to effectively compute minimizers or approximations to minimizers of this functional subject to additional constraints such as, for example, for the curve to avoid obstacles such as other curves. In this paper, we consider the set of smooth curves that can be written as graphs of three times continuously differentiable functions on an interval, and, in particular, we consider approximations using quartic uniform B- spline functions. We show that if quartic uniform B-spline minimizers of the curvature variation functional converge to a curve, as the number of B-spline basis functions tends to infinity, then this curve is in fact a minimizer of the curvature variation functional. In order to illustrate this result, we present an example of sequences of B-spline minimizers that converge to a cubic spiral.
• 307.
Karlstads universitet, Institutionen för ingenjörsvetenskap, fysik och matematik.
Baire category theorem2009Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
In this thesis we give an exposition of the notion of category and the Baire category theorem as a set theoretical method for proving existence. The category method was introduced by René Baire to describe the functions that can be represented by a limit of a sequence of continuous real functions. Baire used the term functions of the first class to denote these functions.
The usage of the Baire category theorem and the category method will be illustrated by some of its numerous applications in real and functional analysis. Since the usefulness, and generality, of the category method becomes fully apparent in Banach spaces, the applications provided have been restricted to these spaces.
To some extent, basic concepts of metric topology will be revised, as the Baire category theorem is formulated and proved by these concepts. In addition to the Baire category theorem, we will give proof of equivalence between different versions of the theorem.
Explicit examples, of first class functions will be presented, and we shall state a theorem, due to Baire, providing a necessary condition on the set of points of continuity for any function of the first class.
• 308.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
Portfolio Optimization: Approaches to determining VaR and CVaR2015Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
This thesis analyses portfolio optimization using the risk measures VaR and CVaR with two different underlying assumptions of probability distribution of returns; one being that portfolio returns are normal distributed and the other being a discrete distribution comprised of historical data. The models are run through numerous historical simulations on the OMXS30 with varying time period for historical data and rebalance frequencies. The resulting simulated returns as well as the CVaR outcomes are presented, compared and discussed in order to assess which model performs the best and under what circumstances.
Our key findings is that using a discrete, historical probability distribution for optimizing a portfolio with respect to CVaR, comprised of around 320 days worth of data and using a rebalancing frequency of 20 days performs the best with respect to total return and actual CVaR. This method manages to take the fat tails of the market return distribution into account and as such successfully avoids the larger market downturns.
The results of this thesis also indicate that historical VaR optimization is inferior to CVaR optimization. However due to lack of computational power this comparison is inconclusive.
• 309.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Positive currents related to polynomial convexity1999Licentiatavhandling, med artikler (Annet vitenskapelig)
• 310.
Mälardalens högskola, Institutionen för matematik och fysik.
Unique characterization of the Bel-Robinson tensor2004Inngår i: Classical and Quantum Graviry, ISSN 0264-9381, Vol. 21, nr 14, s. 3499-3503Artikkel i tidsskrift (Fagfellevurdert)
We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson-type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors.
• 311.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Pricing American Options using Lévy Processes and Monte Carlo Simulations2015Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
• 312.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Pricing the American Option Using Itô’s Formula and Optimal Stopping Theory2014Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
• 313.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Development of three-dimensional disturbances in pipe poiseuille flow1992Licentiatavhandling, med artikler (Annet vitenskapelig)
• 314.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Evolution of laminar disturbances in pipe Poiseuille flow1993Inngår i: European journal of mechanics. B, Fluids, ISSN 0997-7546, E-ISSN 1873-7390, Vol. 12, nr 6, s. 749-768Artikkel i tidsskrift (Fagfellevurdert)
An experimental study on the evolution of localized disturbances in pipe Poiseuille flow has been performed. The aim was to investigate the relevance of the theoretical findings on the algebraic transient growth mechanism. For non-axisymmetric disturbances the amplitudes and the propagation velocities were measured in the laminar regime. The initial disturbances were induced radially through the pipe wall and had a distribution in the azimuthal direction corresponding to a periodicity of one and five, respectively. The amplitude of the resulting streamwise velocity perturbation was measured at different radial and axial positions by means of laser doppler velocimetry. In the laminar regime, the results showed that the disturbance peaks are actually amplified but will eventually decay as predicted in the theory. The peak of an initial disturbance of periodicity five showed a rapid amplification to a maximum before the decay sets in. The initial disturbance of periodicity gave a more slowly varying disturbance peak level over the investigated interval. The propagation speed of the front of the disturbances was in the range of 0.86-0.9 times the centreline velocity. The propagation speed of the disturbance peaks was in the range of 0.51 to 0.80 times the centreline velocity and dependent on the azimuthal distribution of the initial disturbance.
• 315.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Initial algebraic growth of small angular dependent disturbances in pipe Poiseuille flow1992Inngår i: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 87, nr 1, s. 61-79Artikkel i tidsskrift (Fagfellevurdert)
The evolution of small, angular dependent velocity disturbances in laminar pipe flow is studied. In particular, streamwise independent perturbations are considered. To fully describe the flow field, two equations are required, one for the radial and the other for the streamwise velocity perturbation. Whereas the former is homogeneous, the latter has the radial velocity component as a forcing term. The normal modes of the system are determined and analytical solutions for eigenfunctions, damping rates and phase velocities are calculated. Then, the initial value problem is treated and the time development of the disturbances is determined.
• 316.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Interactions of three components and subcritical self-sustained amplification of disturbances in plane Poiseuille flow1999Inngår i: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 11, nr 3, s. 590-601Artikkel i tidsskrift (Fagfellevurdert)
A low-dimensional nonlinear model for the normal velocity (v) and normal vorticity (eta) disturbance development in plane Poiseuille flow is studied. The study is restricted to the interactions of a pair of oblique components of the form e(/(alpha x+/-beta z)) and the component of the form e(/2 beta z) where alpha and beta are streamwise and spanwise wave numbers, respectively. The disturbances considered are also assumed to be highly elongated in the streamwise direction. Owing to the non-normal properties of the basic equations, the eta disturbance is first transiently amplified. Then, if the Reynolds number (R) and the initial disturbance are sufficiently large, the nonlinear interactions lead to a self-sustained process of disturbance amplification at subcritical R. For large R (R greater than or similar to 5000). the threshold disturbance amplitude scales like R-3. The, results also strongly indicate that the nonlinear feedback from eta to v is crucial for the establishment of the self-sustained process.
• 317.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Nonlinear behaviour of transiently amplified disturbances in pipe Poiseuille flow1995Inngår i: European journal of mechanics. B, Fluids, ISSN 0997-7546, E-ISSN 1873-7390, Vol. 14, nr 6, s. 719-735Artikkel i tidsskrift (Fagfellevurdert)
The present work is a continuation of previous investigations of the author and deals with the nonlinear behaviour of transiently amplified disturbances in pipe Poiseuille flow. The disturbances are expanded in terms of a small amplitude parameter, and a spectral formulation, based on the eigenfunctions of the linear problem, is used to solve the resulting evolution equations. For the corresponding linear problem, analytical solutions are known. Here the author shows that the nonlinear problem also allows an analytical treatment. The analysis concentrates on the interaction of the dominating azimuthal wave number components of axially independent disturbances, since they exhibit largest growth in linear theory. The analysis is restricted to the first modes of the components, since these are found to dominate the disturbance magnitude at the peak amplification. In particular, the nonlinear correction for the component with an azimuthal wave number equal to one is considered. Including the nonlinear correction, the author reduces the linearly obtained amplification of the streamwise disturbance energy density and finds that the peak position of the disturbance energy density occurs earlier than for linear disturbance. This result is consistent with the experimental findings on transient growth for subcritical disturbances. Moreover, the reduced amplification is in agreement with recent results from direct numerical simulations of axially dependent disturbances.
• 318.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Nonmodal growth of three-dimensional disturbances on plane Couette-Poiseuille flows2005Inngår i: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 17, nr 1Artikkel i tidsskrift (Fagfellevurdert)
The time development of three-dimensional disturbances superimposed on a variety of mean flow profiles representing plane Couette-Poiseuille flow is investigated numerically. Specifically, with y representing the wall normal coordinate, the mean flow profiles U(y) are represented by U(y) = A(1-y2) + By, where B = 1 ......
• 319.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Optimal growth of small disturbances in pipe Poiseuille flow1993Inngår i: Physics of Fluids. A, Fluid Dynamics (1989-1993), ISSN 0899-8213, Vol. 5, nr 11, s. 2710-2720Artikkel i tidsskrift (Fagfellevurdert)
A theoretical study is made of initial algebraic growth for small angular-dependent disturbances in pipe Poiseuille flow. The analysis is based on the homogeneous equation for the pressure for which the eigenvalue problem is solved numerically. In the limit of small streamwise wave numbers asymptotic results for the eigenvalues are derived. On the basis of the modes of the system, which are all damped, the initial value problem is considered and in particular the largest possible growth of the disturbance energy density is determined following the ideas of Butler and Farrell [Phys. Fluids A 4, 1637 (1992)]. The results show that a large amplification of the disturbance energy is possible. The largest amplification is obtained for disturbances with a small streamwise wave number and with an azimuthal wave number of one. The energy growth is then only due to the growth of the streamwise disturbance component. However, for disturbances of shorter wavelength, the energy growth is also substantial and not only concentrated to the streamwise velocity component. The wall shear corresponding to disturbances with the largest energy growth also shows a large amplification and the dependence of wave numbers and the Reynolds number is the same as for the energy. However, the wall pressure of a long wavelength disturbance of the largest growth just decays from its initial value, but for disturbances of shorter wavelength, it is also amplified
• 320.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Self-sustained amplification of disturbances in pipe Poiseuille flow1999Inngår i: European journal of mechanics. B, Fluids, ISSN 0997-7546, E-ISSN 1873-7390, Vol. 18, nr 4, s. 635-647Artikkel i tidsskrift (Fagfellevurdert)
The nonlinear development of disturbances in pipe Poiseuille flow is studied with a low-dimensional model. The basic system from which the model is derived governs disturbances closely related to the radial velocity and radial vorticity disturbances. The analysis is restricted to the interaction of the two first harmonics of streamwise elongated disturbances since they are the most transiently amplified ones in linear theory. In the resulting dynamical system a nonlinear feedback from the normal vorticity disturbance (which is transiently amplified according to linear theory) to the radial velocity disturbance is present. Above a threshold of the initial amplitude, the feedback leads to a self-sustained amplification of the disturbances continuing for all times.
• 321.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
The effect of the Earth's rotation on the transient amplification of disturbances in pipe flow2003Inngår i: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 15, nr 10, s. 3028-3035Artikkel i tidsskrift (Fagfellevurdert)
In pipe flow, for large Reynolds numbers, A. A. Draad and F. T. M. Nieuwstadt \ref[J. Fluid Mech. 361 (1998), 297--308 showed that the small Coriolis force due to the Earth's rotation may affect the mean flow profile of liquids substantially. In this paper, the development of small disturbances superimposed on a laminar mean flow affected by the Coriolis force is investigated analytically. The investigation is focused on the time development and the transient growth of streamwise-independent disturbances since they are the most amplified disturbances without the Coriolis effect included. The results show that the modification of the parabolic mean flow caused by the Coriolis force significantly affects the transient disturbance amplification when the Reynolds number $(R)$ is high. For example, with $R=9000$ and the Ekman number $\rm Ek=5.23$, due to the Coriolis effect, the peak value of the transient disturbance amplification becomes about two-thirds of the peak value obtained in the case where the mean flow is unaffected by the Coriolis force. When the Reynolds number is decreased, the reduction of the transient growth becomes smaller.
• 322.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
The initial-value problem for three-dimensional disturbances in plane Poiseuille flow of helium II2008Inngår i: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 598, s. 227-244Artikkel i tidsskrift (Fagfellevurdert)
The time development of small three-dimensional disturbances in plane Poiseuille flow of helium II is considered. The study is conducted by considering the interaction of a normal fluid field and a superfluid field. The interaction is caused by a mutual friction forcing between the two flow fields. Specifically, the stability of the normal fluid affected by the mutual forcing is considered. Compared to the ordinary fluid case where the mutual forcing is not present, the presence of the mutual forcing implies a substantial increase of the transient growth of the disturbances. The increase of the transient growth occurs because the mutual forcing reduces the damping of the disturbances. The phase of transient growth becomes thereby more prolonged and higher levels of amplification are reached. There is also a minor effect on the transient growth caused by the modification of the mean flow owing to the mutual forcing. The strongest transient growth occurs for streamwise elongated disturbances, i.e. disturbances with streamwise wavenumber α = 0. When α increases beyond zero, the transient amplification quickly becomes reduced. Striking differences compared to the ordinary fluid case are that the largest transient amplification does not occur when the spanwise wavenumber (β) is close to two and that the peak level of the disturbance energy density amplification does not depend on the square of the Reynolds number.
• 323.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Transient growth of disturbances in laminar pipe flow1994Doktoravhandling, med artikler (Annet vitenskapelig)
Transient growth of disturbances is studied theoretically in pipe Poiseuille flow and experimentally both in pipe and plane Poiseuille flow. The theoretical results, based on the initial value problem, show that a large transient amplification occurs for small, angular dependent disturbances in pipe Poiseuille flow although all modes are damped. The largest growth is obtained for disturbances with zero streamwise wave number and with an azimuthal periodicity of one. Only the streamwise disturbance component is then amplified and the energy density growth is proportional to the Reynolds number squared. For disturbances of finite axial wave number, also the radial and azimuthal disturbance components are amplified from their initial values but the growth is still dominated by the streamwise disturbance. When the azimuthal wave number increases the disturbance peak occurs at shorter times and the amplification decreases. However, for small times, larger azimuthal wave numbers dominate the growth. Asymptotic results for the most amplified disturbances show that two classes of modes can be distinguished and it is necessary to include modes from both classes to get amplification. The asymptotically derived propagation speeds of the two classes are in the ranges <2/3 times the centerline velocity (Ucl) and <0.818Ucl, respectively. If the nonlinear interaction of axially independent disturbances is considered, the results indicate that the amplification of the most linearly amplified disturbance is reduced. Laser Doppler measurements of the streamwise disturbance velocity due to radially induced initial disturbances of azimuthal periodicities (n) of one and five have also been conducted. The results show that the disturbance velocity in fact exhibits a transient growth. The disturbance with n=1 gives a slowly varying peak-value while the peak owing to an initial disturbance with n=5 shows a more rapid amplification to a maximum before it decays. The propagation speeds of the disturbance peaks are in the range theoretically expected for transiently amplified disturbances. By hot-wire measurements of the streamwise disturbance velocity due to two radially induced jet-like disturbances, the spatial structure of an evolving disturbance is investigated in detail. Especially the positive disturbance velocity exhibits an amplification followed by a decay. The disturbance also spreads and becomes streak-like and elongated in the streamwise direction. The spread implies that an integrated disturbance quantity grows to a peak and eventually decays. The propagation speed and the radial location of the disturbances agree with the theoretical findings for algebraically growing disturbances. In plane Poiseuille flow the symmetry properties of an evolving disturbance are investigated by means of hot-wire anemometry. The initial streamwise disturbance is designed to be symmetric with respect to the normal direction. Despite this, an antisymmetric structure develops in the region where the main amplification occurs as indicated by the theory.
• 324.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Transient growth of small disturbances in a Jeffrey fluid flowing through a pipe2003Inngår i: Fluid Dynamics Research, ISSN 0169-5983, E-ISSN 1873-7005, Vol. 32, nr 1-2, s. 29-44Artikkel i tidsskrift (Fagfellevurdert)
The hydrodynamic stability of small disturbances in a non-Newtonian fluid flowing through a circular pipe is studied analytically. Specifically, the time development of the disturbances and the transient disturbance amplification are studied. The non-Newtonian fluid is modelled by a two-constant Jeffrey model characterized by the relaxation time $\lambda$ and a constant $K$ representing the ratio of relaxation to retardation times. All the investigations are carried out for streamwise-independent disturbances since they are the most amplified ones in a Newtonian fluid and can be treated analytically. The eigenvalue problem and the initial value problem for the disturbances are studied. Compared to the Newtonian case, the results show that, dependent on $K$ and $\lambda$, a reduction or an increase in the disturbance transient growth may occur.
• 325.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Transient properties of a developing laminar disturbance in pipe Poiseuille flow1995Inngår i: European journal of mechanics. B, Fluids, ISSN 0997-7546, E-ISSN 1873-7390, Vol. 14, nr 5, s. 601-615Artikkel i tidsskrift (Fagfellevurdert)
• 326.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Royal Institute of Technology, Stockholm.
Symmetry properties of developing three-dimensional laminar disturbances in plane Poiseuille flow1994Inngår i: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 6, nr 4, s. 1618-1620Artikkel i tidsskrift (Fagfellevurdert)
The evolution of a finite-amplitude point-like laminar disturbance in plane Poiseuille flow is investigated using hot-wire anemometry. In contrast to earlier experiments, the initial disturbance was introduced simultaneously through both the upper and lower wall of the channel, resulting in an antisymmetric disturbance in the normal velocity. Although the initial disturbance mainly generated symmetrical streamwise velocity modes, the subsequent development of the perturbation showed a marked tendency for antisymmetry.
• 327.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Activity: SIAM Conference on Geometric Design and Computing2007Konferansepaper (Annet (populærvitenskap, debatt, mm))
• 328.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Strömningslära och experimentell mekanik.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Robust registration of surfaces using a refined iterative closest point algorithm with a trust region approach2017Inngår i: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 74, nr 3, s. 755-779Artikkel i tidsskrift (Fagfellevurdert)
The problem of finding a rigid body transformation, which aligns a set of data points with a given surface, using a robust M-estimation technique is considered. A refined iterative closest point (ICP) algorithm is described where a minimization problem of point-to-plane distances with a proposed constraint is solved in each iteration to find an updating transformation. The constraint is derived from a sum of weighted squared point-to-point distances and forms a natural trust region, which ensures convergence. Only a minor number of additional computations are required to use it. Two alternative trust regions are introduced and analyzed. Finally, numerical results for some test problems are presented. It is obvious from these results that there is a significant advantage, with respect to convergence rate of accuracy, to use the proposed trust region approach in comparison with using point-to-point distance minimization as well as using point-to-plane distance minimization and a Newton- type update without any step size control.
• 329.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
Xtura AB, Kungsbacka. Xtura AB, Kungsbacka. University West, Department of Engineering Sciences, Trollhättan. University West, Department of Engineering Sciences, Trollhättan. Chalmers University of Technology, Department of Product and Production Development. Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Strömningslära och experimentell mekanik.
Automatic in-line inspection of shape based on photogrammetry2016Inngår i: SPS16, Lund: SPS16 , 2016, , s. 9Konferansepaper (Fagfellevurdert)
We are describing a fully automatic in-line shape inspection system for controlling the shape of moving objects on a conveyor belt. The shapes of the objects are measured using a full-field optical shape measurement method based on photogrammetry. The photogrammetry system consists of four cameras, a flash, and a triggering device. When an object to be measured arrives at a given position relative to the system, the flash and cameras are synchronously triggered to capture images of the moving object. From the captured images a point-cloud representing the measured shape is created. The point-cloud is then aligned to a CAD-model, which defines the nominal shape of the measured object, using a best-fit method and a feature-based alignment method. Deviations between the point-cloud and the CAD-model are computed giving the output of the inspection process. The computational time to create a point-cloud from the captured images is about 30 seconds and the computational time for the comparison with the CAD-model is about ten milliseconds. We report on recent progress with the shape inspection system.
• 330.
KTH, Skolan för industriell teknik och management (ITM), Industriell ekonomi och organisation (Inst.).
The Risk-Return Tradeoff in a Hedged, Client Driven Trading Portfolio2013Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
In post-financial crisis times, new legislation in combination with banks’ changed risk aversion has to a great extent changed the proprietary trading to client driven trading, i.e. market making or client facilitation. This type of trading complicates the risk-return dynamics, as the goal is often to minimize risk and achieve profitable commission revenues. This thesis aims to disclose the risk-return tradeoff in a client driven trading environment. This is done by investigating the conditional relation between risk and realized return. As opposed from many studies which proxy the risk with beta or variance, I use a delta-gamma Value at Risk model as the risk proxy, which I also backtest. For the return proxy, I use three different measures; P&L, commission revenues and the sum of these two. A positive tradeoff exists if (i) the return is equally negatively dependent on the risk if the ex post return is negative, as it is positively dependent on the risk if the ex post return is positive and (ii) the average return is significantly positive. For three different client driven trading portfolios tested, I found a positive risk-return tradeoff in one portfolio, between the P&L plus commission revenues and the Value at Risk. However, since a symmetrical conditional relationship between risk and P&L plus commission revenues was found in all portfolios, and the average return was positive, the positive tradeoff would have existed if the average return would have been significantly positive. On the other hand, one could argue that the tradeoff exists, but is not significant. No relation between risk and commission revenues was found. A probable cause to this is the hedging strategies, which would be an interesting topic for further research.
• 331.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
Discrete Velocity Models for Polyatomic Molecules Without Nonphysical Collision Invariants2018Inngår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 172, nr 3, s. 742-761Artikkel i tidsskrift (Fagfellevurdert)
An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. Unlike for the Boltzmann equation, for DVMs there can appear extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and hence, without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species, but also for binary mixtures and recently extensively for multicomponent mixtures. In this paper, we address ways of constructing normal DVMs for polyatomic molecules (here represented by that each molecule has an internal energy, to account for non-translational energies, which can change during collisions), under the assumption that the set of allowed internal energies are finite. We present general algorithms for constructing such models, but we also give concrete examples of such constructions. This approach can also be combined with similar constructions of multicomponent mixtures to obtain multicomponent mixtures with polyatomic molecules, which is also briefly outlined. Then also, chemical reactions can be added.
• 332.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation2015Inngår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 159, nr 2, s. 358-379Artikkel i tidsskrift (Fagfellevurdert)
We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for a general discrete model of a quantum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete Boltzmann equation. Then a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete kinetic equation can be made. The number of additional conditions that need to be imposed for well-posedness is given by some characteristic numbers. These characteristic numbers are calculated for discrete models axially symmetric with respect to the x-axis. When the characteristic numbers change is found in the discrete as well as the continuous case. As an illustration explicit solutions are found for a small-sized model.
• 333.
Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots2017Inngår i: Numerical Functional Analysis and Optimization, ISSN 0163-0563, E-ISSN 1532-2467, Vol. 38, nr 3, s. 293-305Artikkel i tidsskrift (Fagfellevurdert)
The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electronâelectron interactions in the potential.
• 334. Betancor, Jorge J.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions2015Inngår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 431, nr 1, s. 440-470Artikkel i tidsskrift (Fagfellevurdert)
We consider the Weinstein type equation L(lambda)u = 0 on (0, infinity) X (0, infinity), where L-lambda= delta(2)(t) + delta-lambda(lambda-1)/x(2), In this paper we characterize the solutions of L(lambda)u = = 0 on (0, infinity) x (0, infinity) representable by Bessel-Poisson integrals of BMO-functions as the ones satisfying certain Carleson properties.
• 335. Beyersdorff, O.
KTH, Skolan för datavetenskap och kommunikation (CSC), Teoretisk datalogi, TCS.
Lower bounds: From circuits to QBF Proof Systems2016Inngår i: ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Association for Computing Machinery (ACM), 2016, s. 249-260Konferansepaper (Fagfellevurdert)
A general and long-standing belief in the proof complexity community asserts that there is a close connection between progress in lower bounds for Boolean circuits and progress in proof size lower bounds for strong propositional proof systems. Although there are famous examples where a transfer from ideas and techniques from circuit complexity to proof complexity has been effective, a formal connection between the two areas has never been established so far. Here we provide such a formal relation between lower bounds for circuit classes and lower bounds for Frege systems for quantified Boolean formulas (QBF). Starting from a propositional proof system P we exhibit a general method how to obtain a QBF proof system P + 8red, which is inspired by the transition from resolution to Qresolution. For us the most important case is a new and natural hierarchy of QBF Frege systems C-Frege + 8red that parallels the well-studied propositional hierarchy of C-Frege systems, where lines in proofs are restricted to belong to a circuit class C. Building on earlier work for resolution [Beyersdorff, Chew, and Janota, 2015a] we establish a lower bound technique via strategy extraction that transfers arbitrary lower bounds for the circuit class C to lower bounds in C-Frege + 8red. By using the full spectrum of state-of-The-Art circuit lower bounds, our new lower bound method leads to very strong lower bounds for QBF Frege systems: (i) exponential lower bounds and separations for the QBF proof system AC0[p]-Frege + 8red for all primes p; (ii) an exponential separation of AC0[p]-Frege + 8red from TC0-Frege + 8red; (iii) an exponential separation of the hierarchy of constantdepth systems AC0 d-Frege + 8red by formulas of depth independent of d. In the propositional case, all these results correspond to major open problems.
• 336.
Dipartimento di Matematica, Seconda Universita di Roma.
Institute of Mathematics, Odessa State University. Institute of Mathematics, University of Warsaw.
Tangential boundary behaviour of bounded harmonic functions in the unit disc1998Inngår i: Seminari di geometria 1996-1997, Bologna: Universita degli studi di Bologna, Dipartimento di Matematica , 1998, s. 63-68Kapittel i bok, del av antologi (Annet vitenskapelig)
• 337.
Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap.
Exploiting the Direct Communication Link for Enhancing Effective Capacity Performance of Cognitive Radio Relay Networks2014Independent thesis Advanced level (degree of Master (Two Years))Oppgave
• 338.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Mapping of open-answers using machine learning2018Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
This thesis investigates if a model can be created to map misspelled answers from open-ended questions to a finite set of brands. The data used for the paper comes from the company Nepa that uses open-questions to measure brand-awareness and consists of misspelled answers and brands to be mapped to. A data structure called match candidate was created and consists of a misspelled answer and brand that it poten-tially be mapped to. Features for the match candidates were engineered and based on the edited distances, posterior probability and common misspellings among other. Multiple machine learning models were tested for classifying the match candidates as positive if the mapping was correct and negative otherwise. The model was tested in two scenarios, one when the answers in the training and testing data came from the same questions and secondly when they came from different ones. Among the classifiers tested, the random forest model performed best in terms of PPV as well as sensitivity. The resulting mapping identified on average 92% of the misspelled answers and map then with 98% accuracy in the first scenario. While in the second scenario 70% of the answers were identified with 95% confidence in the mapping on average.
• 339.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Almost sure absolute continuity of Bernoulli convolutions2010Inngår i: Ann. Inst. H. Poincaré Probab. Statist., ISSN 0246-0203, Vol. 46, nr 3, s. 888-893Artikkel i tidsskrift (Fagfellevurdert)
We prove an extension of a result by Peres and Solomyak on almostsure absolute continuity in a class of symmetric Bernoulli convolutions.
• 340.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
Boundary regularity and barriers for elliptic and parabolic problems, and sharp capacity estimates for annuli2013Konferansepaper (Fagfellevurdert)
• 341.
Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
Cluster sets for Sobolev functions and quasiminimizers2010Inngår i: Journal d'Analyse Mathématique, ISSN 0021-7670, Vol. 112, s. 49-77Artikkel i tidsskrift (Fagfellevurdert)
In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous quasiminimizers). We develop their basic theory with a particular emphasis on when they coincide and when they are connected. As a main result, we obtain that if a Sobolev function u on an open set has boundary values f in Sobolev sense and f |∂is continuous at x0 ∈ ∂, then the essential cluster set C(u, x0, Ω) is connected. We characterize precisely in which metric spaces this result holds. Further, we provide some new boundary regularity results for quasiharmonic functions. Most of the results are new also in the Euclidean case.
• 342.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
The Perron method for p-harmonic functions: new resolutivity and invariance results2012Konferansepaper (Fagfellevurdert)
• 343.
Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
The Perron method for p-harmonic functions: Resolutivity and invariance results2012Konferansepaper (Fagfellevurdert)
• 344.
Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
First-order Sobolev spaces on metric spaces2009Inngår i: Function Spaces, Inequalities and Interpolation (Paseky, 2009) / [ed] Jaroslav Lukes, Lubos Pick, Prague: Matfyzpress , 2009, s. 1-29Konferansepaper (Fagfellevurdert)
• 345.
Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
Nonlinear Potential Theory on Metric Spaces2011 (oppl. 1)Bok (Fagfellevurdert)
The p-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories.
This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for an interested reader and as a reference text for an active researcher. The presentation is rather self-contained, but the reader is assumed to know measure theory and functional analysis.
The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space.
Each chapter contains historical notes with relevant references and an extensive index is provided at the end of the book.
• 346.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. University of Pavia, Italy.
The Petrovskii criterion and barriers for degenerate and singular p-parabolic equations2017Inngår i: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 368, nr 3-4, s. 885-904Artikkel i tidsskrift (Fagfellevurdert)
In this paper we obtain sharp Petrovskii criteria for the p-parabolic equation, both in the degenerate case p amp;gt; 2 and the singular case 1 amp;lt; p amp;lt; 2 We also give an example of an irregular boundary point at which there is a barrier, thus showing that regularity cannot be characterized by the existence of just one barrier.
• 347.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. Univ Pavia, Italy. Univ Jyvaskyla, Finland.
Boundary Regularity for the Porous Medium Equation2018Inngår i: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 230, nr 2, s. 493-538Artikkel i tidsskrift (Fagfellevurdert)
We study the boundary regularity of solutions to the porous medium equation in the degenerate range . In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the parabolic boundary has a solution which attains the boundary values, provided that the spatial domain satisfies the elliptic Wiener criterion. This condition is known to be optimal, and it is a consequence of our main theorem which establishes a barrier characterization of regular boundary points for general-not necessarily cylindrical-domains in . One of our fundamental tools is a new strict comparison principle between sub- and superparabolic functions, which makes it essential for us to study both nonstrict and strict Perron solutions to be able to develop a fruitful boundary regularity theory. Several other comparison principles and pasting lemmas are also obtained. In the process we obtain a rather complete picture of the relation between sub/superparabolic functions and weak sub/supersolutions.
• 348.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. St Louis University, MO 63103 USA. University of Cincinnati, OH 45221 USA.
Geometric analysis on Cantor sets and trees2017Inngår i: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 725, s. 63-114Artikkel i tidsskrift (Fagfellevurdert)
dUsing uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.
• 349.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. Aalto University, Finland.
Minima of quasisuperminimizers2017Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 155, s. 264-284Artikkel i tidsskrift (Fagfellevurdert)
We show that, unlike minima of superharmonic functions which are again superharmonic, the same property fails for Q-quasisuperminimizers. More precisely, if u(i) is a Q(i)-quasisuperminimizer, i = 1,2, where 1 amp;lt; Q(1) amp;lt; Q(2), then u = min{u(1), u(2)} is a Q-quasisuperminimizer, but there is an increase in the optimal quasisuperminimizing constant Q. We provide the first examples of this phenomenon, i.e. that Q amp;gt; Q(2). In addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to Kinnunen and Martio. Moreover, our lower and upper bounds turn out to be quite close. We also study a similar phenomenon in pasting lemmas for quasisuperminimizers, where Q = Q(1)Q(2) turns out to be optimal, and provide results on exact quasiminimizing constants of piecewise linear functions on the real line, which can serve as approximations of more general quasiminimizers. (C) 2017 Elsevier Ltd. All rights reserved.
• 350.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. University of Eastern Finland, Finland.
SOBOLEV SPACES, FINE GRADIENTS AND QUASICONTINUITY ON QUASIOPEN SETS2016Inngår i: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 41, nr 2, s. 551-560Artikkel i tidsskrift (Fagfellevurdert)
We study different definitions of Sobolev spaces on quasiopen sets in a complete metric space X equipped with a doubling measure supporting a p-Poincare inequality with 1 amp;lt; p amp;lt; infinity, and connect them to the Sobolev theory in R-n. In particular, we show that for quasiopen subsets of R-n the Newtonian functions, which are naturally defined in any metric space, coincide with the quasicontinuous representatives of the Sobolev functions studied by Kilpelainen and Maly in 1992.
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• rtf | 2018-10-17 19:14:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7072684168815613, "perplexity": 3818.9516470122276}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583511206.38/warc/CC-MAIN-20181017174543-20181017200043-00172.warc.gz"} |
https://dmoj.ca/problem/nccc9s1 | ## Mock CCC '22 1 S1 - Big Mattress Tournament
View as PDF
Points: 5 (partial)
Time limit: 0.25s
Memory limit: 1G
Problem type
Kaitlyn is an administrator for the Big Mattress Tournament, where folks compete to see who can make the biggest mattress.
To ensure every person has an even playing field when assembling mattresses, they will be given the same materials to assemble their mattresses.
Specifically, each participant will be given tiles, tiles, and L tiles, which are tiles with the top-right square removed.
Determine if there is some positive integer such that it is possible to assemble a mattress with the given tiles. You must use all of the tiles, you may not rotate or reflect any of the tiles, and the mattress must be exactly .
#### Constraints
In tests worth 14 marks, .
#### Input Specification
The first line contains a single integer .
lines follow, each containing three integers, , , and .
#### Output Specification
Output lines. On the th line, output YES if it is possible to make a mattress. Output NO otherwise.
#### Sample Input
4
2 0 0
0 2 0
1 0 1
0 1 0
#### Sample Output
YES
YES
YES
NO
## Comments
There are no comments at the moment. | 2022-09-26 16:39:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19565308094024658, "perplexity": 3122.4242332458934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334912.28/warc/CC-MAIN-20220926144455-20220926174455-00027.warc.gz"} |
https://docs.juliaebert.com/makerspace-supplies/ | # Makerspace Supplies
###### Last updated: 2021 September 15
I’m slowly building up my own home makerspace. For me, it’s the engineering equivalent of the mad scientist’s lair. But right now it’s just half of my living room.
This is an inventory of what I have, why I chose it, and what I use it for. Like most things on this website, not everything is a recommendation. However, when I was looking to flesh out my equipment, I couldn’t find many lists like this: home makers documenting their personal setups, equipment, and supplies. So I’m trying to create what I wish found a few years ago.
## Large Equipment
• Prusa i3 MK3S: A 3D printer is my biggest staple for making mechanical stuff. The Prusa just works, compared to many cheaper printers that require tweaking to get good prints, but something like an Ender 3 can get you started at a much lower price point.
• Prusa MMU2S: The multi-material upgrade to the 3D printer is a fun toy, but it’s incredibly finicky, and you can do simple multi-color prints with manual color changes.
• Silhouette Cameo 4: Vinyl cutters are practical for labeling and cutting light weight materials like paper and cardboard, but they’re also fun for laptop decals and heat transfer vinyl t-shirt designs. Cameo is a consumer-grade product (compared to the fancy Roland ones), which is fine for home use, and it’s more open than the competing Cricut models. I use the Cameo through Inkscape with a fablabnbg extension.
• Janome HD3000 sewing machine: I think sewing is under-rated in maker communities (coming with stereotypes about grandmotherly crafting), but fabric is another powerful tool in your making arsenal. I chose this model because it’s heavy duty enough to handle thick materials like canvas and woven nylon. It’s also purely mechanical, so there are no computerized components to get outdated or break. (I wouldn’t pay MSRP for it, though.)
• Brother Laser Printer (HL-L2350DW): Laser printing gets you out of the scam of ink cartridges (and it’s waterproof), and most people don’t need color printing, plus color laser printing is way more expensive. (If I need to print in color, I do it at work.) I got this particular model because it works well with Linux, has wireless printing, and was available during the pandemic.
## Small Equipment
• Weller WES51 soldering iron: If you want to do electronics beyond breadboards, you’ll want a soldering iron. I got this soldering iron because I had good experiences with it in my lab, but it’s no longer made. Common alternative suggestions now include the Hakko 888D and TS100.
• Hakko fume extractor: There’s some debate over how effective these are at dealing with soldering fumes, but for me it’s “better safe than sorry.” At the very least, it pulls the smoke away from my face like an expensive fan.
• Variable DC power supply: Voltage- and current-limited so you won’t blow up your electronics while testing them.
• Multimeter: I have a Fluke 115 multimeter. It’s overkill. A $20 Amazon multimeter is probably sufficient for hobby projects. ## Large Tools • Ryobi corded drill: Home Depot’s house brand power tools aren’t bad these days. This one was Clark’s though; I had no involvement in its purchase. But having a drill around the house is incredibly useful in general. • Black & Decker jig saw: A jig saw was the first power tool I was allowed to use unsupervised as a kid, so I have fond memories. It’s particularly useful for cutting non-straight cuts in large pieces of plywood/MDF. The brand has a worse reputation after being sold to a Chinese company. But it’s still probably plenty powerful for the average non-professional consumer. • Black & Decker disc sander: I used this to strip and sand a whole kitchen table for under$30. Great for cleaning up large flat surfaces, from plywood to kitchen tables to cutting boards. I also got a mega-pack of sanding discs to use with the sander.
• Dremel 3000 tool: Dremels are sort of a catch-all tool for small-scale filing and grinding. I recently used mine to install bike fenders. This particular model seemed like a good balance between price and power when I bought it.
• Dremel flex shaft attachment: It’s primarily meant for engraving. I have grand ambitions to someday use this to make a DIY mini-CNC. Who knows if or when that will happen.
## Small Tools
• iFixit tweezers: Useful for so many things: tiny screws, electronics components, weeding vinyl cuts. You can get bigger sets for the same price on Amazon, but I’ll got for quality over quantity here. These are sturdy, the tips line up well, and they’re not so flimsy that they’ll squish flat with a light pinch.
• Small files: These are most useful for cleaning up 3D prints. brand doesn’t really matter. Files from Harbor Freight will do the job. This happens to be the set I got.
• Wire strippers: If you’re using any wire, you’ll need wire cutters. Again, you don’t need a particular brand, but make sure you get the right size for the guage of wires that you use.
• Flush cutters: Also known as wire cutters or diagonal cutters. They have many uses, like cutting wires and printer filament, or removing print support material. They come from many brands; many are good, some are crappy. I’ve had good luck with these (and they’re easy to get on Amazon), but it’s not the only good option out there.
• Ratcheting screwdriver: Many screwdrivers in one, with the convenience of ratcheting and ergonomics for a pretty cheap price. You’ll need screwdrivers at some point. This is my favorite for big-sized screws.
• iFixit Mako screwdriver set: This contains nearly every little screwdwiver bit you’ll ever need. The driver also feels great, with a top end that spins great. It even includes hex bits.
• Small pliers: Great for removing support material from 3D prints, but the smallest size are weirdly hard to find. I have these from Home Depot (only sold as a set with flush cutters). There are a variety of small Hakko pliers that might also do the job, depending on what you use them for.
• Larger needlenose pliers: Sometimes you need prying force.
• 1/4” drive socket set (metric and imperial): Great for changing printer nozzles and installing bike parts, among other uses.
• Ball end hex wrenches: Sometimes you need more torque than the iFixit kit gets you, a bigger range of sizes, or a weird angle that you can only get with a ball end hex wrench. There are crazy expensive sets out there, but these have worked well for me.
• #2 Exacto knives: You don’t need Exacto brand (I have Excel brand as well), but these kind of hobby knives are super useful for precision cutting, like vinyl, paper, or 3D print material.
• Utility knives (box cutters): For large non-precise cutting or… opening boxes.
• Scissors: Don’t keep running back to your desk or kitchen junk drawer for scissors. Just spend a few bucks to keep a pair in your makerspace.
• Tape measure: Preferably including metric, in case you need to measure something in metric that’s longer than a ruler.
• Rulers: Bonus points if measurements actually start at the end of the ruler
• Digital calipers: Mitutoyo are the engineering gold standard, but they’re expensive. I no longer recommend the intermediate iGaging OriginCal calipers, because mine stopped working after 1.5 years, and their supposed 2 year warranty will get no response. If you’re primarily doing things like 3D printing, a set of $20 calipers from Amazon are plenty precise. • Level: In case you ever want to hang something up • Crescent wrench: So you don’t have to invest in a whole separate wrench set when you can’t use the socket set. • Manual staple gun: This is for needs in between where you need nails/screws, and you household stapler won’t cut it. My biggest use has been stapling fabric onto a wood frame for sound damping panels. • Heat gun: Use for heat shrink, reshaping glasses arms, and shaping 3D prints. Brand probably doesn’t matter here; it’s not precise work. • Hot glue gun: Hot glue isn’t for everything, but it’s amazing how much it can be used to hold together. ## Mechanical Materials • 3D printer filament: I have an entire page for my opinions on filament brands. I primarily use PLA and PETG filament, since they’re easy to work with and don’t produce toxic fumes like ABS. • Vinyl: For the vinyl cutter • M2-M5 screw sets: I use M3 the most, especially for 3D printed stuff. To start, I got a set from Amazon that included a variety of sizes for M2-M4, then expanded my supply as I needed it. • M3 lock nuts: These are the most useful addition to the above set of standard screws. • M3 threaded inserts: For use in 3D prints, where you have a blind screw insert. • Zip ties • Sandpaper • VHB tape: Semi-permanently adhere things to each other with this bit of magic. It will take the paint off your walls. I’ve used it to hang up a whiteboard, or just stick LED strips to a computer monitor. • Blue painter’s tape: On the other end of the tape spectrum, this is great for things you only want to hold together very temporarily. • Duct tape or Gorilla tape • double sided scotch tape • Super glue • Wood glue • Rare earth (neodynium) magnets: Great for embedding in 3D prints • String ## Electrical & Electronics Materials • 63/37 leaded solder (0.02 inch diameter) • Rosin paste flux • Solder wick • Solder sucker • Brass for soldering iron • Hookup wire (solid core and stranded) • Jumper wires • Breadboards • Various Arduino knock-offs (especially Nano and Pro Micro) • Addressable RGB LED strips • Electrical tape • Heat shrink: Usually the better alternative to electrical tape. Get a pack with a variety of sizes. • Through-hole resistor set • Small servo motors • NEMA 17 stepper motors • AA and AAA batteries • Switches and buttons • Potentiometer variety pack • Get other parts as needed for projects, such as sensors and actuators. And collect/scavenge them from things you take apart. ## Organization • Label maker: Incredibly useful for nicely labeling project bins, parts, and tools. I have a Dymo label maker, which is somewhat barebones, but I’ve never wanted to use the fancier labels anyways. It leaves relatively short unnecessary margins, and the refills are cheaper than alternatives. A step up is the Brother P-touch, with fancier label options, but it wastes more material and has more expensive label refills. • Akro Mills drawers: Small parts storage. It can set on a shelf or be wall-mounted. I have one for electronics components and one for mechanical parts. They also come in varieties will all small drawers and all big drawers, depending on your needs. • 6 qt Sterilite bins: They’re not the sexiest, but they’re sturdy and cheap. Good for storing projects and larger parts. They also come in a larger 16 qt version. • Ikea Skadis pegboard, with 3D printed attachments: I have a weird love of pegboards. The ikea ones also look particularly nice, although they’re a bit more expensive. But they’re also easy to design attachments for, like all of these. • Husky 3-drawer tool box: This is where all the tools live. It’s very sturdy and easy to organize in. I added some grippy shelf liner to keep things from sliding around or scratching the drawers. • Harbor freight small parts bins: It’s hard to beat$3 for parts storage. All my screws are organized in these, and then stacked in drawers. I’ll likely end up with more of these in the future.
• Eventually I might get Helmer drawers: They’re pretty cheap and have many shallow drawers for organizing small things.
## Miscellaneous
• LED lamp with magnifying glass: This is good task lighting, and the magnifying glass is useful for soldering/electronics work.
• Cutting mats: Protect your work surface. I have big ones that cover most of the table and live there pretty permanently.
• Helping hand (for soldering): Because humans unfortunately only have two hands. This classic variety sucks. I cheaped out and got one with a magnet instead of a heavy base. Just spend the extra few bucks to get one with flexible arms like this. | 2022-01-21 10:37:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17465093731880188, "perplexity": 5522.713812008809}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303356.40/warc/CC-MAIN-20220121101528-20220121131528-00559.warc.gz"} |
https://research.birmingham.ac.uk/en/publications/filtering-solar-like-oscillations-for-exoplanet-detection-in-radi | # Filtering solar-like oscillations for exoplanet detection in radial velocity observations
William J. Chaplin, Heather M. Cegla, Christopher A. Watson, Guy R. Davies, Warrick H. Ball
Research output: Contribution to journalArticlepeer-review
22 Citations (Scopus)
## Abstract
Cool main-sequence, sub-giant and red-giant stars all show solar-like oscillations, pulsations that are excited and intrinsically damped by near-surface convection. Many overtones are typically excited to observable amplitudes, giving a rich spectrum of detectable modes. These modes provide a wealth of information on fundamental stellar properties. However, the radial velocity shifts induced by these oscillations can also be problematic when searching for low-mass, long-period planets; this is because their amplitudes are large enough to completely mask such minute planetary signals. Here we show how fine-tuning exposure times to the stellar parameters can help efficiently average out the solar-like-oscillation-induced shifts. To reduce the oscillation signal to the radial velocity precision commensurate with an Earth-analogue, we find that for cool, low-mass stars (near spectral type K) the necessary exposure times may be as short as 4 minutes, while for hotter, higher-mass stars (near spectral type F, or slightly evolved) the required exposure times can be longer than 100 minutes. We provide guideline exposure durations required to suppress the total observed amplitude due to oscillations to a level of $0.1\,\rm m\,s^{-1}$, and a level corresponding to the Earth-analogue reflex amplitude for the star. Owing to the intrinsic stochastic variability of the oscillations, we recommend in practice choosing short exposure durations at the telescope and then averaging over those exposures later, as guided by our predictions. To summarize, as we enter an era of $0.1\,\rm m\,s^{-1}$ instrumental precision, it is critical to tailor our observing strategies to the stellar properties.
Original language English 163 8 The Astronomical Journal 157 4 https://doi.org/10.3847/1538-3881/ab0c01 Published - 2 Apr 2019
## Keywords
• astro-ph.SR
• astro-ph.EP
• planets and satellites: detection
• stars: low-mass
• stars: oscillations (including pulsations)
## Fingerprint
Dive into the research topics of 'Filtering solar-like oscillations for exoplanet detection in radial velocity observations'. Together they form a unique fingerprint. | 2022-01-25 18:25:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4916570782661438, "perplexity": 5485.655313812052}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304859.70/warc/CC-MAIN-20220125160159-20220125190159-00691.warc.gz"} |
https://www.lmfdb.org/knowledge/show/g2c.bsd_invariants | show · g2c.bsd_invariants all knowls · up · search:
The BSD invariants of an abelian variety $A/\Q$ include:
The weak form of the Birch and Swinnerton-Dyer conjecture predicts that the first two invariants are equal, while the strong form of their conjecture relates the last six invariants via the BSD formula $\frac{L^{(r)}(A,1)}{r!} = \frac{\#Ш(A)\Omega_AR_AT_A}{(\#A(\Q)_{\rm tor})^2}.$ No effective method to compute $Ш(A)$ is currently known (indeed, it is not even known that $Ш(A)$ is finitely generated, although the BSD conjecture requires this). However, one can compute (approximations of) the the five remaining quantities and use this to compute the analytic order of Sha, and under the assumption that $Ш(A)$ is finite, one can determine whether its order is a square or not. One can also compute the rank of the 2-Selmer group of $A$, which constrains the 2-part of $Ш(A)$.
Authors:
Knowl status:
• Review status: reviewed
• Last edited by Andrew Sutherland on 2020-01-06 18:54:08
Referred to by:
History: Differences | 2022-07-05 06:07:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9123243689537048, "perplexity": 478.5926776242699}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104514861.81/warc/CC-MAIN-20220705053147-20220705083147-00441.warc.gz"} |
http://www.pokerlistings.com/hand-of-the-week-bill-klein-rope-a-dopes-esfandiari-for-458k-29342 | # Hand of the Week: Bill Klein Rope-a-Dopes Esfandiari for $458k Rope-a-doped. There hasn’t been a televised cash game as exciting as the$250k Aria Super High Roller \in a long time.
With a minimum buy-in of $250,000 it was made for an elusive group of pro players and well-heeled businessmen. There's always something special when money meets professional playing poker and this time was no exception. ### Flop to River Two weeks ago we looked at an incredible bluff played by Andrew Robl. This week we’ll present a hand with a completely different outcome. It involve billionaire Bill Klein and Antonio Esfandiari, winner of the first-ever$1m Big One for One Drop.
The blinds are $300/$600 but thanks to two live straddles it’s already $2,400 to call pre-flop. After a couple of folds Klein ($469,000) raises to $6,000 from the cut-off. Behind him Esfandiari re-raises to$15,000 from the button. The blinds fold but Klein puts in another bet and makes it $29,000 to play. Esfandiari calls. There's$62,500 in the pot as they go to the flop.
Klein checks. Esfandiari bets $29,000 and Klein calls. There's$120,500 in the pot.
The turn is the Klein checks. Esfandiari bets $51,000 and Klein calls again. The pot now has$222,500 in it.
The river is the Another check from Klein, another bet from Esfandiari - now $118,000 – and another call from Klein. Esfandiari shows for a total bluff. Klein wins the pot with his and a pair of eights, worth$458,500.
Watch the hand below starting at roughly 21:00.
### Analysis
Klein butted heads with Antonio Esfandiari and came out on top with almost half a million dollars more in front of him.
On a board with possible flushes and straights he called Esfandiari down with just second pair. How did he pull that off? Let’s see.
The little raising war between Klein and Esfandiari pre-flop doesn’t mean that much when it comes to ranges. Mainly, both are trying to steal the pot, sitting in the best possible positions at the table.
As the effective stacks are extremely deep with 800 big blinds (!), there will still plenty of room for maneuvers after the flop, which is beneficial to the value of position – the value of position is pretty much the central topic of this whole hand.
As we know already neither player has much. But why would Esfandiari call a 4-bet with a hand like K 5?
For one, because he gets 3-1 pot odds, and for two, because if he doesn’t Klein is going to plague him with 4-bets with any random hand.
### The Magician Sees His Chance
On an outrageously dangerous flop of 9 8 7, Klein has hit middle pair. He checks here, which shows you that the billionaire is quite a player. Most other players would have followed with a standard c-bet here.
He already has showdown value but his hand is also very vulnerable to bluffs and definitely not worth a big pot. Furthermore, there are few hands worse than his that can call a bet.
There’s a little disadvantage in this play and that’s that Klein loses the lead. Without the initiative Klein has to ready himself for everything against an aggressive player like Esfandiari.
The Magician sees his chance and doesn’t hesitate to take it. With one overcard and an inside gutshot – this is also sometimes called an “idiot gutshot” because there’s already a possible flush and one of the outs could be poisoned - Esfandiari doesn’t have much equity.
But the pot is big and with this board texture it’s almost asking to get stolen.
Klein doesn’t really have any reason to give up in the face of Esfandiari’s bet. Yes, he could be beat, but the hands that beat him are unlikely.
There are more (likely) semi-bluffs in Esfandiari’s range, like a ten, a six, a high diamond and so on.
Magician carries on.
### Esfandiari Carries On
The 9 on the turn is one of the best possible cards for Klein. It makes his middle pair more valuable without changing the situation on the board. Whoever was ahead on the flop is also ahead on the turn.
Klein checks again and Esfandiari carries on with his bluff. In retrospect it’s easy to say that he'd better give up on his bluff at this point but there are actually also good reasons for it.
1) Klein check-calling the flop signals that he almost certainly has a made hand.
2. The 9 is one of the worst possible turn cards for a bluff.
3) Bluffing the turn with minimum equity implies there has to be another bluff on the river.
Klein calls the turn. He surely is aware that there will likely be another bet on the river. And indeed after the 3 on the river, which is a complete blank, Esfandiari pulls off his ultimate bluff.
### Klein's Rope-a-Dope Pays Off
He bets $118,000 into a pot of$220,500 which means he only has to be successful one in three times to makes this play profitable.
But the situation is even more favorable for Klein. He has to call $118,000 to win$338,000 which means it's profitable for him to make that call if Esfandiari is bluffing one in four times (numbers slightly rounded).
With the board and all its draws and semi-bluff possibilities this seems more than likely. But it’s not easy to call such a large amount of money with such a mediocre hand as J-8.
Hand analysis and heart spot on.
At the end of the day Klein’s rope-a-dope paid off but Esfandiari made it very difficult for him to profit from his equity out of position. This is very typical of strong, aggressive players like the Magician.
If you’ve ever been three-barreled out of position by an aggressive player, you know how uncomfortable that feels.
3 October 2017
### Daily 3-Bet: Who is Go0se, Kirk v. Blom, Poker Lin-sanity
2 October 2017
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6 November 2017 121 | 2017-11-24 23:59:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27317097783088684, "perplexity": 4669.649764721092}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934809160.77/warc/CC-MAIN-20171124234011-20171125014011-00564.warc.gz"} |
http://tex.stackexchange.com/tags/subfloats/hot | # Tag Info
6
It is the the length of the paths you are using. \documentclass{article} \batchmode \usepackage{graphicx} \newcount\zz \begin{document} \def\z{} \loop \edef\z{\z/} \includegraphics{/c/tmp/../tmp/../tmp/../tmp/../tmp/../tmp/../tmp/../tmp/../tmp/../tmp/..% /tmp/../tmp/../tmp/../tmp/../tmp/../tmp/../tmp/../tmp/\z house.png}\par %\includegraphics{/c/tmp/\z ...
5
The subfigure package is deprecated; you should use either the subfig or the subcaption package. Here's a solution based on the capabilities of the subcaption package. Each subfigure environment is assigned a width of 0.3\textwidth to have some whitespace between adjacent subfigures. \documentclass{article} \usepackage{subcaption} ...
3
Always call \usepackage[caption=false]{subfig} with the revtex4-1 document class, which is not compatible with caption. The option is indeed meant not to load the caption package along with subfig (it will emulate the needed features). Note that it's better avoiding blank lines inside \subfloat (not required, though). More importantly, you're missing ...
3
There is nothing sacred, or even particularly useful, with \subfloat or \subfigure. I'm not sure what you intended to accomplish with the \labels. If you don't want the (a) - (r) to appear, having \ref{r} expand as (r) is not particularly useful. If you want to create hyperlinks, there is always \hypertarget and \hyperlink. \documentclass{memoir} ...
3
Instead of passing scale=0.71 to tikzpicture pass it to the axis options. I have also added ylabel style={anchor=west} to put y label little up. \documentclass[11pt]{memoir} \setstocksize{23cm}{15.9cm} \settrimmedsize{23cm}{15.9cm}{*} \settrims{0pt}{0pt} \settypeblocksize{17.8cm}{11.3cm}{*} \setlrmargins{2.3cm}{*}{*} \setulmargins{6.4\onelineskip}{*}{*} ...
3
Simple solution: \newcommand{\mysize}{0.22} and then \begin{subfigure}[b]{\mysize\textwidth} \includegraphics[width=\textwidth]{pictures/...} \end{subfigure}
2
Basically, the image is just too big. For this problem, the limiting dimension is the height, not the width. (Come to think of it, this was also true in the original question. \documentclass{report} \usepackage{subfig} \usepackage{mwe} \newlength{\tempheight} \newlength{\tempwidth} \newcommand{\rowname}[1]% #1 = text ...
2
This was fixed using [width=x\linewidth] where x is any number between 0 and 1.
2
\documentclass{article}% always use a complete document not a fragment \usepackage{graphicx} \begin{document} \begin{figure}[htp]% include p % \begin{minipage}{\textwidth} a \textwidth minipage does nothing \centering%\begin{center} % mwe package images \sbox0{\includegraphics[height=4cm]{example-image-a}% no word space ...
2
Make sure your redefine the figure numbering before loading the subfigure package. Then you should get the output you wish. Note that \numberwithin already changes the label format, so your \renewcommands are redundant. If you wish to change the printed representation of part in these references, then you should do this by redefining \thepart. This will ...
2
You can fool subfig letting it think that the subfloat is, e.g., \textwidth wide, by inserting the subfloat body in a box of that width. Example: \documentclass{article} \usepackage{subfig} \begin{document} \begin{figure}[ht] \centering \subfloat[very very very very very long subfloat caption]{% \makebox[\textwidth]{\framebox[3cm][c]{subfloat body}}% ...
2
You can use \ContinuedFloat from caption package and for subfigures, use subcaption package. \documentclass{article} \usepackage{graphicx} \usepackage{caption} \usepackage{subcaption} \DeclareCaptionLabelFormat{continued}{#1~#2 (Cont.)} \captionsetup[ContinuedFloat]{labelformat=continued} \begin{document} \begin{figure} \centering ...
1
I was trying to do something similar, and found your answer. I don't really like to have to make multiple new definitions every time I want to use this. So I tried to solve it differently. I'm new to this so maybe there's a good reason not to do it the way I suggested... \documentclass{article} \usepackage[demo]{graphicx} \usepackage{caption} ...
1
You can use tabular as suggested by Zarko. This is another method where we use two subfigures, in one we place only sub caption and in second, only figure. Don't forget to leave a blank line in the middle as I did. \documentclass{article} \usepackage{graphicx} \usepackage{caption} \usepackage{subcaption} \begin{document} \begin{figure} \centering ...
1
With the floatrow package, this is easy: \documentclass{article} \usepackage{graphicx} \usepackage{caption} \usepackage{subcaption} \usepackage{floatrow} \begin{document} \begin{figure} \floatsetup[subfigure]{style=plain,capposition = beside, capbesideposition={left, center},capbesidesep=none, capbesidewidth =0.5em, rowpostcode = captionskip} ...
1
Your images are wider then width of minipages. If you accommodate the width of images to width of minipages, than images are not overlapped anymore. See: \documentclass[10pt,a4paper]{memoir} \usepackage[utf8]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{graphicx} \newsubfloat{figure} \begin{document} ...
1
Since \captionof uses \par and \vskip, you can't use it in a tabular. It's easier just to write your own caption macro, so long as you don't want all these subfloats showing up in the list of figures. \documentclass{memoir} \usepackage{mwe} \newlength{\tempdima} \newcommand{\rowname}[1]% #1 = text {\rotatebox{90}{\makebox[\tempdima][c]{\textbf{#1}}}} ...
1
The magic number is 91=64+27: \documentclass{article} \usepackage{subcaption} \usepackage{graphicx} \newlength{\preferredwidth} \setlength{\preferredwidth}{12cm} \begin{document} \begin{figure} \centering \begin{subfigure}{\dimexpr\preferredwidth*64/91} \includegraphics[width=\linewidth]{example-image-4x3} \caption{Ratio 4:3 landscape} \end{subfigure}% ...
1
You could give floatrow a roll: \documentclass[12pt]{article} \usepackage{caption,floatrow} \DeclareCaptionSubType[alph]{figure} \captionsetup[figure]{labelsep=colon} \captionsetup[subfigure]{labelformat=brace,labelsep=space,labelfont=bf} \floatsetup[subfigure]{capposition=bottom,heightadjust=all,valign=t} \begin{document} \begin{figure}[htp] ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2015-04-28 16:16:29 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9886349439620972, "perplexity": 10780.578705386031}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246661733.69/warc/CC-MAIN-20150417045741-00249-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.nonlin-processes-geophys.net/25/251/2018/ | Journal cover Journal topic
Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
Journal topic
Nonlin. Processes Geophys., 25, 251-265, 2018
https://doi.org/10.5194/npg-25-251-2018
Nonlin. Processes Geophys., 25, 251-265, 2018
https://doi.org/10.5194/npg-25-251-2018
Research article 29 Mar 2018
Research article | 29 Mar 2018
# Complex interplay between stress perturbations and viscoelastic relaxation in a two-asperity fault model
Stress perturbations on a fault with viscoelastic relaxation
Emanuele Lorenzano and Michele Dragoni Emanuele Lorenzano and Michele Dragoni
• Dipartimento di Fisica e Astronomia, Alma Mater Studiorum Università di Bologna, Viale Carlo Berti Pichat 8, 40127 Bologna, Italy
Abstract
We consider a plane fault with two asperities embedded in a shear zone, subject to a uniform strain rate owing to tectonic loading. After an earthquake, the static stress field is relaxed by viscoelastic deformation in the asthenosphere. We treat the fault as a discrete dynamical system with 3 degrees of freedom: the slip deficits of the asperities and the variation of their difference due to viscoelastic deformation. The evolution of the fault is described in terms of inter-seismic intervals and slip episodes, which may involve the slip of a single asperity or both. We consider the effect of stress transfers connected to earthquakes produced by neighbouring faults. The perturbation alters the slip deficits of both asperities and the stress redistribution on the fault associated with viscoelastic relaxation. The interplay between the stress perturbation and the viscoelastic relaxation significantly complicates the evolution of the fault and its seismic activity. We show that the presence of viscoelastic relaxation prevents any simple correlation between the change of Coulomb stresses on the asperities and the anticipation or delay of their failures. As an application, we study the effects of the 1999 Hector Mine, California, earthquake on the post-seismic evolution of the fault that generated the 1992 Landers, California, earthquake, which we model as a two-mode event associated with the consecutive failure of two asperities.
1 Introduction
Asperity models have long been acknowledged as an effective means to describe many aspects of fault dynamics (Lay et al., 1982; Scholz, 2002). In such models, it is assumed that the bulk of energy release during an earthquake is due to the failure of one or more regions on the fault characterized by a high static friction and a velocity-weakening dynamic friction. The stress build-up on the asperities is governed by the relative motion of tectonic plates. Earthquakes that have been ascribed to the slip of two asperities are the 1964 Alaska earthquake (Christensen and Beck, 1994); the 1992 Landers, California, earthquake (Kanamori et al., 1992); the 2004 Parkfield, California, earthquake (Twardzik et al., 2012); and the 2010 Maule, Chile, earthquake (Delouis et al., 2010).
In the framework of asperity models, a critical role is played by stress accumulation on the asperities, fault slip at the asperities and stress transfer between the asperities. Accordingly, fault dynamics can be fruitfully investigated via discrete dynamical systems whose essential components are the asperities (Ruff, 1992; Turcotte, 1997). Such an approach reduces the number of degrees of freedom required to describe the dynamics of the system, that is, the evolution of the fault (in terms of slip and stress distribution) during the seismic cycle; also, it allows the visualization of the state of the fault and to follow its evolution via a geometrical approach, by means of orbits in the phase space. Finally, a finite number of dynamic modes can be defined, each one describing a particular phase of the evolution of the fault (e.g. tectonic loading, seismic slip, after-slip). Asperity models are capable of reproducing the essential features of the seismic source, while sparing the more complicated characterization based on continuum mechanics.
In a number of recent works, modelling of different mechanical phenomena in a two-asperity fault system has been addressed, such as stress perturbations due to surrounding faults (Dragoni and Piombo, 2015) and the radiation of seismic waves (Dragoni and Santini, 2015). In these models, the fault is treated as a discrete dynamical system with four dynamic modes: a sticking mode, corresponding to stationary asperities, and three slipping modes, associated with the separate or simultaneous failure of the asperities.
In the framework of a discrete fault model, the impact of viscoelastic relaxation was first studied by Amendola and Dragoni (2013) and then further investigated by Dragoni and Lorenzano (2015), who considered a fault with two asperities of different strengths. The authors discussed the features of the seismic events predicted by the model and showed how the shape of the associated source functions is related to the sequence of dynamic modes involved. In turn, the observation of the moment rate provides an insight into the state of the system at the beginning of the event, that is, the particular stress distribution on the fault from which the earthquake takes place.
However, no fault can be considered isolated; in fact, any fault is subject to stress perturbations associated with earthquakes on neighbouring faults (Harris, 1998; Stein, 1999; Steacy et al., 2005). Whenever a fault slips, the stress field in the surrounding medium is altered. As a result, the occurrence time and the magnitude of next earthquakes may change with respect to the unperturbed condition, which is governed by tectonic loading.
The aim of the present paper is to discuss the combined effects of viscoelastic relaxation and stress perturbations on a two-asperity fault in the framework of a discrete fault model. In order to deal with such a problem, we base our work on the results achieved by Dragoni and Piombo (2015) and Dragoni and Lorenzano (2015). In the former work, the authors considered a two-asperity fault with purely elastic rheology and discussed the effect of stress perturbations due to earthquakes on neighbouring faults. The fault was treated as a discrete dynamical system whose state is described by two variables, the slip deficits of the asperities. In the latter work, viscoelastic relaxation on the fault was dealt with by adding a third state variable, the variation in the difference between the slip deficits of the asperities during inter-seismic intervals. In the present paper, we introduce stress perturbations as modelled by Dragoni and Piombo (2015) in the framework of the two-asperity fault considered by Dragoni and Lorenzano (2015). Accordingly, the present work represents a three-dimensional generalization of the model devised by Dragoni and Piombo (2015). Elastic wave radiation and additional constraints on the state of the fault are taken into account, as further developments with respect to previous works.
In the framework of the present model, seismic events generated by the fault are discriminated according to the number and sequence of slipping modes involved and the seismic moment released; these features are related to the particular state of the system at the beginning of the inter-seismic interval preceding the event. We discuss how stress perturbations affect the evolution of the fault in terms of changes in the state of the system and in the duration of the inter-seismic time, highlighting the complications arising from the ongoing post-seismic deformation process with respect to the purely elastic case considered by Dragoni and Piombo (2015). As an application, we consider the stress perturbation imposed by the 1999 Hector Mine, California, earthquake (Jónsson et al., 2002; Salichon et al., 2004) to the fault that caused the 1992 Landers, California, earthquake, which we model as a two-mode event due to the consecutive failure of two asperities and that was followed by remarkable viscoelastic relaxation (Kanamori et al., 1992; Freed and Lin, 2001). We propose a means to estimate the stress transfer from the knowledge of the relative positions and faulting styles of the two faults. As a further novelty with respect to the work presented by Dragoni and Lorenzano (2015), we show how the knowledge of the time interval elapsed after the 1999 earthquake can be used to constrain the admissible set of states that may have given rise to the 1992 event. We discuss the possible subsequent evolution of the Landers fault after the stress transfer from the Hector Mine earthquake, pointing out the main differences with respect to an unperturbed scenario.
2 The model
We consider a plane fault containing two asperities of equal areas A and different strengths that we name asperity 1 and asperity 2 (Fig. 1). The fault is enclosed in a homogeneous and isotropic shear zone behaving as a Poisson solid and is subject to a uniform strain rate owing to the relative motion of two tectonic plates, taking place at a constant rate V. In order to describe the viscoelastic relaxation in the asthenosphere, we assume a Maxwell viscoelastic behaviour with a characteristic relaxation time Θ. Finally, we characterize the seismic efficiency of the fault by means of an impedance γ. All quantities are expressed in non-dimensional form.
In the present model we do not consider aseismic slip on the fault. It has been treated in the framework of a discrete fault model by Dragoni and Lorenzano (2017), who considered a region slipping aseismically for a finite time interval and calculated the effect on the stress distribution and the subsequent evolution of the fault. Of course, if the amplitude of aseismic slip has the same order of magnitude as that of seismic slip, the fault evolution may be affected.
In accordance with the assumptions of asperity models, we ascribe the generation of earthquakes on the fault to the failure of the sole asperities, neglecting any contribution of the surrounding weaker region to the seismic moment. Also, we do not describe friction, slip and stress at every point of the fault, but only consider their average values on each asperity.
Figure 1Sketch of the model of a plane fault with two asperities. The rectangular frame is the fault border. The state of the asperities is described by their slip deficits X(T) and Y(T), while the variable Z(T) represents the temporal variation of the difference between the slip deficits of the asperities due to viscoelastic deformation in the inter-seismic interval following an earthquake on the fault.
The fault is treated as a dynamical system with three state variables, functions of time T: the slip deficits X(T) and Y(T) of asperity 1 and 2, respectively, and the variable Z(T) representing the temporal variation of the difference YX between the slip deficits of the asperities during inter-seismic intervals of the fault, due to the stress redistribution associated with viscoelastic relaxation in the asthenosphere. The slip deficit of an asperity is defined as the slip that an asperity should undergo at a given instant in time in order to recover the relative displacement of tectonic plates that occurred up to that moment.
We assume the simplest form of rate-dependent friction and associate the asperities with constant static and dynamic frictions, the latter considered as the average value during slip. The static friction on asperity 2 is a fraction β of that on asperity 1 and dynamic frictions are a fraction ϵ of static frictions for both asperities. Letting fs1 and fd1 be the static and dynamic frictions on asperity 1, respectively, and fs2 and fd2 be the static and dynamic frictions on asperity 2, respectively, we have
$\begin{array}{}\text{(1)}& \mathit{\beta }=\frac{{f}_{\mathrm{s}\mathrm{2}}}{{f}_{\mathrm{s}\mathrm{1}}}=\frac{{f}_{\mathrm{d}\mathrm{2}}}{{f}_{\mathrm{d}\mathrm{1}}},\phantom{\rule{1em}{0ex}}\mathit{ϵ}=\frac{{f}_{\mathrm{d}\mathrm{1}}}{{f}_{\mathrm{s}\mathrm{1}}}=\frac{{f}_{\mathrm{d}\mathrm{2}}}{{f}_{\mathrm{s}\mathrm{2}}}.\end{array}$
We acknowledge that the values of friction after a seismic event might be different from the initial ones, even though it is probable that the change is remarkable only after several seismic cycles. We neglect this possible change, because we focus on other sources of irregularity in the seismic cycles. However, the model could easily incorporate a change in friction after each event: new values could be given to static and dynamic frictions after the event and the subsequent evolution could be calculated accordingly.
During a global stick mode, the tangential forces acting on the asperities in the slip direction are (in units of static friction on asperity 1)
$\begin{array}{}\text{(2)}& {F}_{\mathrm{1}}=-X+\mathit{\alpha }Z,\phantom{\rule{1em}{0ex}}{F}_{\mathrm{2}}=-Y-\mathit{\alpha }Z.\end{array}$
In these expressions, the terms X and Y represent the effect of tectonic loading and have the same sign for both asperities, whereas the terms ±αZ are the contributions of stress transfer between the asperities. In the framework of the present model, stress is transferred by one asperity to the other as a result of co-seismic slip; in the subsequent inter-seismic interval, the static stress field generated by asperity slip undergoes a certain amount of relaxation owing to viscoelasticity. The parameter α is a measure of the degree of coupling between the asperities: for smaller values of α, the stress transfer from one asperity to the other is less efficient. In the limit case α=0, the asperities are completely independent from one another and the slip of one of them does not affect the state of the other: the evolution of the asperities is thus governed by tectonic loading only. By comparison with a model based on continuum mechanics, the specific value of α can be estimated as (Dragoni and Tallarico, 2016)
$\begin{array}{}\text{(3)}& \mathit{\alpha }=\frac{Avs}{\mathrm{2}\stackrel{\mathrm{˙}}{e}},\end{array}$
where A is the area of the asperities, v is the velocity of the tectonic plates, s is the tangential traction (per unit moment) imposed on one asperity by the slip of the other and $\stackrel{\mathrm{˙}}{e}$ is the tangential strain rate on the fault due to tectonic loading.
An effective way to characterize fault mechanics is provided by the concept of Coulomb stress (Stein, 1999). It is defined as the difference between the shear stress σt in the direction of fault slip and the static friction τs on the fault surface:
$\begin{array}{}\text{(4)}& {\mathit{\sigma }}_{C}={\mathit{\sigma }}_{\mathrm{t}}-{\mathit{\tau }}_{\mathrm{s}}.\end{array}$
Accordingly, σC is negative during an inter-seismic interval and a seismic event occurs when σC vanishes. In our model, the presence of two asperities makes it necessary to assign a value of Coulomb stress to each of them. By definition, the Coulomb forces on asperity 1 and 2 are, respectively,
$\begin{array}{}\text{(5)}& {F}_{\mathrm{1}}^{\mathrm{C}}=-{F}_{\mathrm{1}}-\mathrm{1},\phantom{\rule{1em}{0ex}}{F}_{\mathrm{2}}^{\mathrm{C}}=-{F}_{\mathrm{2}}-\mathit{\beta }.\end{array}$
Using Eq. (2), they can be rewritten as
$\begin{array}{}\text{(6)}& {F}_{\mathrm{1}}^{\mathrm{C}}=X-\mathit{\alpha }Z-\mathrm{1},\phantom{\rule{1em}{0ex}}{F}_{\mathrm{2}}^{\mathrm{C}}=Y+\mathit{\alpha }Z-\mathit{\beta }.\end{array}$
To sum up, the system is described by the set of six parameters α, β, γ, ϵ, Θ and V, with α≥0, $\mathrm{0}<\mathit{\beta }<\mathrm{1}$, γ≥0, $\mathrm{0}<\mathit{ϵ}<\mathrm{1}$, Θ>0 and V>0. At any instant T in time, the state of the system may be univocally expressed by the tern $\left(X,Y,Z\right)$ or by one of the couples $\left({F}_{\mathrm{1}},{F}_{\mathrm{2}}\right),\left({F}_{\mathrm{1}}^{\mathrm{C}},{F}_{\mathrm{2}}^{\mathrm{C}}\right)$.
When considering the fault dynamics during the seismic cycle, it is possible to identify four dynamic modes, each one described by a different system of autonomous ODEs (ordinary differential equations): a sticking mode (00), corresponding to stationary asperities, and three slipping modes, associated with the slip of asperity 1 alone (mode 10), the slip of asperity 2 alone (mode 01) and the simultaneous slip of the asperities (mode 11). A seismic event generally consists of n slipping modes and involves one or both the asperities.
## The sticking region
The sticking region of the system is defined as the set of states in which both asperities are stationary. During a global stick phase (mode 00), the rates $\stackrel{\mathrm{˙}}{X}$, $\stackrel{\mathrm{˙}}{Y}$ and $\stackrel{\mathrm{˙}}{Z}$ are negligible with respect to their values when the asperities are slipping; thus, the sticking region is a subset of the space XYZ.
The slip of asperity 1 occurs when
$\begin{array}{}\text{(7)}& {F}_{\mathrm{1}}=-\mathrm{1},\end{array}$
while the slip of asperity 2 takes place when
$\begin{array}{}\text{(8)}& {F}_{\mathrm{2}}=-\mathit{\beta }.\end{array}$
Combining these conditions with the expressions (2) of the forces, we obtain two planes in the XYZ space,
$\begin{array}{}\text{(9)}& & X-\mathit{\alpha }Z-\mathrm{1}=\mathrm{0},\text{(10)}& & Y+\mathit{\alpha }Z-\mathit{\beta }=\mathrm{0},\end{array}$
which we name Π1 and Π2, respectively. Of course, the Coulomb forces ${F}_{\mathrm{1}}^{\mathrm{C}}$ and ${F}_{\mathrm{2}}^{\mathrm{C}}$ vanish on Π1 and Π2, respectively; furthermore, their gradients
$\begin{array}{}\text{(11)}& \mathrm{\nabla }{F}_{\mathrm{1}}^{\mathrm{C}}=\left(\mathrm{1},\mathrm{0},-\mathit{\alpha }\right),\phantom{\rule{1em}{0ex}}\mathrm{\nabla }{F}_{\mathrm{2}}^{\mathrm{C}}=\left(\mathrm{0},\mathrm{1},\mathit{\alpha }\right)\end{array}$
are orthogonal to Π1 and Π2, respectively.
We exclude overshooting during the slipping modes: accordingly, we assume X≥0 and Y≥0. As a consequence, the tangential forces on the asperities must always be in the same direction as the velocity of tectonic plates, i.e. F1≤0 and F2≤0. From Eq. (2), the limit cases F1=0 and F2=0 correspond to two planes in the XYZ space,
$\begin{array}{}\text{(12)}& & X-\mathit{\alpha }Z=\mathrm{0},\text{(13)}& & Y+\mathit{\alpha }Z=\mathrm{0},\end{array}$
which we name Γ1 and Γ2, respectively.
To sum up, the sticking region of the system is the subset of the XYZ space enclosed by the planes $X=\mathrm{0},Y=\mathrm{0}$, Γ1, Γ2, Π1 and Π2: a convex hexahedron H. Its vertices are the origin $\left(\mathrm{0},\mathrm{0},\mathrm{0}\right)$ and the points
$\begin{array}{ll}& A=\left(\mathrm{0},\mathrm{1},-\frac{\mathrm{1}}{\mathit{\alpha }}\right),\phantom{\rule{0.25em}{0ex}}B=\left(\mathit{\beta },\mathrm{0},\frac{\mathit{\beta }}{\mathit{\alpha }}\right),\\ \text{(14)}& & \phantom{\rule{1em}{0ex}}C=\left(\mathit{\beta }+\mathrm{1},\mathrm{0},\frac{\mathit{\beta }}{\mathit{\alpha }}\right),\text{(15)}& & D=\left(\mathrm{0},\mathit{\beta }+\mathrm{1},-\frac{\mathrm{1}}{\mathit{\alpha }}\right),\phantom{\rule{0.25em}{0ex}}E=\left(\mathrm{1},\mathrm{0},\mathrm{0}\right),\phantom{\rule{0.25em}{0ex}}F=\left(\mathrm{0},\mathit{\beta },\mathrm{0}\right).\end{array}$
The sticking region is shown in Fig. 2 for a particular choice of the parameters α and β. Its volume can be expressed as a function of the parameters of the system as $\mathit{\beta }\left(\mathit{\beta }+\mathrm{1}\right)/\mathrm{2}\mathit{\alpha }$. Accordingly, the subset of the state space corresponding to stationary asperities decreases with the degree of coupling between the asperities and with the asymmetry of the system (β→0). By definition, every orbit of mode 00 is enclosed within H and eventually reaches one of its faces AECD or BCDF, belonging to the planes Π1 and Π2, respectively, where an earthquake starts. In these cases, the system switches from mode 00 to mode 10 or mode 01, respectively. In the particular case in which the orbit of mode 00 reaches the edge CD, the system passes from mode 00 to mode 11.
3 Dynamic modes and slip in a seismic event
Let P0H be the state of the system at the beginning of an inter-seismic interval. The specific location of P0 inside the sticking region allows the prediction of the first slipping mode involved in the next seismic event on the fault. In fact, Dragoni and Lorenzano (2015) illustrated the existence of a transcendental surface Σ within H, expressed by the equation
$\begin{array}{}\text{(16)}& V\mathrm{\Theta }\phantom{\rule{0.125em}{0ex}}\left[W\left({\mathit{\gamma }}_{\mathrm{1}}\right)-W\left({\mathit{\gamma }}_{\mathrm{2}}\right)\right]+Y-X+\mathrm{1}-\mathit{\beta }=\mathrm{0},\end{array}$
where W is the Lambert function with arguments
$\begin{array}{}\text{(17)}& {\mathit{\gamma }}_{\mathrm{1}}=\frac{\mathit{\alpha }Z}{V\mathrm{\Theta }}{e}^{-\frac{\mathrm{1}-X}{V\mathrm{\Theta }}},\phantom{\rule{1em}{0ex}}{\mathit{\gamma }}_{\mathrm{2}}=-\frac{\mathit{\alpha }Z}{V\mathrm{\Theta }}{e}^{-\frac{\mathit{\beta }-Y}{V\mathrm{\Theta }}}.\end{array}$
The surface Σ divides H in two subsets H1 and H2 (Fig. 3). The seismic event starts with mode 10 if P0H1 or with mode 01 if P0H2; in the particular case in which P0Σ, the seismic event starts with mode 11.
Figure 2The sticking region of the system, defined as the subset of the state space XYZ in which both asperities are at rest: a convex hexahedron H $\left(\mathit{\alpha }=\mathrm{1},\mathit{\beta }=\mathrm{1}\right)$.
Mode 00 terminates at a point P1 on the face AECD or BCDF of H. The number and sequence of slipping modes involved in the subsequent seismic event can be discriminated from the specific position of P1. If we consider the face AECD (Fig. 4), the earthquake will be a one-mode event 10 if P1 belongs to the trapezoid Q1; it will be a two-mode event 10-01 if P1 belongs to the segment s1; it will be a three-mode event 10-11-01 or 10-11-10 if P1 belongs to the trapezoid R1, where the precise sequence must be evaluated numerically and depends on the particular combination of the parameters α, β, γ and ϵ. The remaining portion of the face would lead to overshooting. Analogous considerations can be made for the subsets Q2, s2 and R2 on the face BCDF. In the particular case in which P1 belongs to the edge CD, the earthquake will be a two-mode event 11-01. In fact, by definition, friction on asperity 2 is smaller than friction on asperity 1 ($\mathrm{0}<\mathit{\beta }<\mathrm{1}$); if the asperities start slipping simultaneously, asperity 1 is then bound to stop the first, while asperity 2 continues to slip. As a result, mode 11 is followed by mode 01 and the slip of the weaker asperity has a longer duration. The opposite would hold if asperity 2 were stronger than asperity 1 (β>1), so that the slip event resulting from states P1CD would be a two-mode event 11-10.
Figure 3The surface Σ that splits the sticking region H in two subsets H1 (below) and H2 (above) (α=1, β=1, VΘ=1). It allows the first slipping mode during an earthquake to be discriminated.
Figure 4The faces AECD and BCDF of the sticking region H, where seismic events start, and their subsets (α=1, β=1, γ=1, ϵ=0.7). The events taking place on the face AECD (BCDF) start with mode 10 (01). The trapezoids Qi correspond to events involving a single asperity; the segments si correspond to events associated with the consecutive slips of the asperities; the trapezoids Ri correspond to events involving the simultaneous slips of the asperities.
In addition, the knowledge of the position of P1 allows the total amount of slip of the asperities and the seismic moment associated with the earthquake to be established. Let us consider an event made up of n dynamic modes and let ${P}_{i}=\left({X}_{i},{Y}_{i},{Z}_{i}\right)$ be the state of the system at time T=Ti, when the ith mode starts $\left(i=\mathrm{1},\mathrm{2},\mathrm{\dots }n\right)$. The final slip amplitudes of asperity 1 and 2 are, respectively,
$\begin{array}{}\text{(18)}& {U}_{\mathrm{1}}={X}_{\mathrm{1}}-{X}_{n+\mathrm{1}},\phantom{\rule{1em}{0ex}}{U}_{\mathrm{2}}={Y}_{\mathrm{1}}-{Y}_{n+\mathrm{1}}.\end{array}$
Accordingly, the final seismic moment can be calculated as
$\begin{array}{}\text{(19)}& {M}_{\mathrm{0}}={M}_{\mathrm{1}}\frac{{U}_{\mathrm{1}}+{U}_{\mathrm{2}}}{U},\end{array}$
where M1 and U are the seismic moment and slip amplitude associated with a one-mode event 10 in the limit case γ=0, respectively, with
$\begin{array}{}\text{(20)}& U=\mathrm{2}\frac{\mathrm{1}-\mathit{ϵ}}{\mathrm{1}+\mathit{\alpha }}.\end{array}$
The possible values of U1,U2 and M0 are summarized in Table 1: the effect of wave radiation is characterized by means of the quantity
$\begin{array}{}\text{(21)}& \mathit{\kappa }=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{e}^{-\frac{\mathit{\gamma }{T}_{\mathrm{s}}}{\mathrm{2}}}\right),\end{array}$
where Ts is the duration of slip in a one-mode event (Dragoni and Santini, 2015).
As for the evolution of the variable Z(T) during the earthquake, it changes according to the equation $\stackrel{\mathrm{¨}}{Z}=\stackrel{\mathrm{¨}}{Y}-\stackrel{\mathrm{¨}}{X}$, since the relaxation process is negligible during the slip of the asperities.
4 Stress perturbations from neighbouring faults
We now consider the perturbations of the state of the fault caused by the coseismic slip on surrounding faults. Following Dragoni and Piombo (2015), we assume that (1) the perturbations occur during an inter-seismic interval; (2) the stress transfer takes place over a time interval negligible with respect to the duration of the inter-seismic interval; and (3) at the time of the perturbation, the state of the fault is sufficiently far from the failure conditions and the stress transfer is small enough that the onset of motion of either asperity is not achieved immediately.
Table 1Final slip amplitudes U1 and U2 of asperity 1 and 2 and seismic moment M0 during a seismic event made up of n slipping modes, as a function of the state P1 where the event started. The entry e.n. is the abbreviation for evaluated numerically.
Let $\left(X,Y,Z\right)\in \mathbf{H}$ be the state of the fault at the time of the perturbation. Generally speaking, the system undergoes a transition to a new state
$\begin{array}{}\text{(22)}& \left({X}^{\prime },{Y}^{\prime },{Z}^{\prime }\right)=\left(X,Y,Z\right)+\left(\mathrm{\Delta }X,\mathrm{\Delta }Y,\mathrm{\Delta }Z\right).\end{array}$
Since the stress transfer takes place over a time interval short with respect to the inter-seismic interval (assumption 2), viscoelastic relaxation is negligible during the perturbation and the rheology can be reasonably considered as purely elastic as the perturbation takes place. Accordingly, we set
$\begin{array}{}\text{(23)}& \mathrm{\Delta }Z=\mathrm{\Delta }Y-\mathrm{\Delta }X.\end{array}$
The change of state is then associated with a vector in the XYZ space:
$\begin{array}{}\text{(24)}& \mathrm{\Delta }\mathbit{R}=\left(\mathrm{\Delta }X,\mathrm{\Delta }Y,\mathrm{\Delta }Z\right).\end{array}$
The components of ΔR generally have different magnitudes and may have different signs, as a consequence of the inhomogeneity of the stress field produced by an earthquake. They can be written in terms of the tangential forces ΔF1 and ΔF2 exerted by the perturbing source on asperity 1 and 2, respectively: from Eq. (2), we have
$\begin{array}{}\text{(25)}& & \mathrm{\Delta }{F}_{\mathrm{1}}=-\mathrm{\Delta }X+\mathit{\alpha }\mathrm{\Delta }Z=\mathit{\alpha }\mathrm{\Delta }Y-\left(\mathrm{1}+\mathit{\alpha }\right)\mathrm{\Delta }X,\text{(26)}& & \mathrm{\Delta }{F}_{\mathrm{2}}=-\mathrm{\Delta }Y-\mathit{\alpha }\mathrm{\Delta }Z=\mathit{\alpha }\mathrm{\Delta }X-\left(\mathrm{1}+\mathit{\alpha }\right)\mathrm{\Delta }Y.\end{array}$
Combining these expressions together, we get
$\begin{array}{}\text{(27)}& & \mathrm{\Delta }X=-\frac{\mathrm{1}+\mathit{\alpha }}{\mathrm{1}+\mathrm{2}\mathit{\alpha }}\mathrm{\Delta }{F}_{\mathrm{1}}-\frac{\mathit{\alpha }}{\mathrm{1}+\mathrm{2}\mathit{\alpha }}\mathrm{\Delta }{F}_{\mathrm{2}},\text{(28)}& & \mathrm{\Delta }Y=-\frac{\mathit{\alpha }}{\mathrm{1}+\mathrm{2}\mathit{\alpha }}\mathrm{\Delta }{F}_{\mathrm{1}}-\frac{\mathrm{1}+\mathit{\alpha }}{\mathrm{1}+\mathrm{2}\mathit{\alpha }}\mathrm{\Delta }{F}_{\mathrm{2}},\text{(29)}& & \mathrm{\Delta }Z=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}\mathit{\alpha }}\left(\mathrm{\Delta }{F}_{\mathrm{1}}-\mathrm{\Delta }{F}_{\mathrm{2}}\right).\end{array}$
We conclude that the variations in tangential stress alter the orbit of the system.
The components of ΔR can also be related to the orientation of the vector in the state space. With reference to Fig. 5, we have
$\begin{array}{ll}& \mathrm{\Delta }X=\mathrm{\Delta }R\mathrm{cos}\mathit{\delta }\mathrm{cos}\mathit{\theta },\phantom{\rule{1em}{0ex}}\mathrm{\Delta }Y=\mathrm{\Delta }R\mathrm{cos}\mathit{\delta }\mathrm{sin}\mathit{\theta },\\ \text{(30)}& & \mathrm{\Delta }Z=\mathrm{\Delta }R\mathrm{sin}\mathit{\delta }.\end{array}$
Introducing the assumption (17), the angle δ may be expressed in terms of the angle θ as
$\begin{array}{}\text{(31)}& \mathit{\delta }=\mathrm{arctan}\left(\mathrm{sin}\mathit{\theta }-\mathrm{cos}\mathit{\theta }\right).\end{array}$
In writing Eq. (19), we took into account that
$\begin{array}{}\text{(32)}& \mathit{\delta }\ne \frac{\mathit{\pi }}{\mathrm{2}},\frac{\mathrm{3}\mathit{\pi }}{\mathrm{2}}\end{array}$
or it would result in
$\begin{array}{}\text{(33)}& \mathrm{\Delta }Z=±\mathrm{\Delta }R,\phantom{\rule{1em}{0ex}}\mathrm{\Delta }X=\mathrm{\Delta }Y=\mathrm{0},\end{array}$
which is a meaningless circumstance. From Eq. (19), the tangential forces (19)–(19) can be rewritten as
$\begin{array}{}\text{(34)}& & \mathrm{\Delta }{F}_{\mathrm{1}}=\frac{\mathit{\alpha }\mathrm{sin}\mathit{\theta }-\left(\mathrm{1}+\mathit{\alpha }\right)\mathrm{cos}\mathit{\theta }}{\sqrt{\mathrm{2}-\mathrm{sin}\mathrm{2}\mathit{\theta }}}\mathrm{\Delta }R,\text{(35)}& & \mathrm{\Delta }{F}_{\mathrm{2}}=\frac{\mathit{\alpha }\mathrm{cos}\mathit{\theta }-\left(\mathrm{1}+\mathit{\alpha }\right)\mathrm{sin}\mathit{\theta }}{\sqrt{\mathrm{2}-\mathrm{sin}\mathrm{2}\mathit{\theta }}}\mathrm{\Delta }R.\end{array}$
Following the variations in normal stress, the static and dynamic frictions on each asperity are altered. Letting ${f}_{\mathrm{s}\mathrm{1}}^{\prime }$ and ${f}_{\mathrm{s}\mathrm{2}}^{\prime }$ be the new static frictions on asperity 1 and 2, respectively, we define
$\begin{array}{}\text{(36)}& {\mathit{\beta }}_{\mathrm{1}}=\frac{{f}_{\mathrm{s}\mathrm{1}}^{\prime }}{{f}_{\mathrm{s}\mathrm{1}}},\phantom{\rule{1em}{0ex}}{\mathit{\beta }}_{\mathrm{2}}=\frac{{f}_{\mathrm{s}\mathrm{2}}^{\prime }}{{f}_{\mathrm{s}\mathrm{1}}}.\end{array}$
The changes in static frictions are then
$\begin{array}{}\text{(37)}& \mathrm{\Delta }{\mathit{\beta }}_{\mathrm{1}}={\mathit{\beta }}_{\mathrm{1}}-\mathrm{1},\phantom{\rule{1em}{0ex}}\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{2}}={\mathit{\beta }}_{\mathrm{2}}-\mathit{\beta }\end{array}$
on asperity 1 and 2, respectively.
Figure 5The vector ΔR and its orientation in the state space XYZ, characterizing the stress perturbation imposed on the system by earthquakes produced by neighbouring faults.
Since the stress perturbation does not alter the friction coefficients of rocks, it is reasonable to assume that the ratio ϵ between dynamic and static friction is unchanged on both asperities. Therefore, letting ${f}_{\mathrm{d}\mathrm{1}}^{\prime }$ and ${f}_{\mathrm{d}\mathrm{2}}^{\prime }$ be the new dynamic frictions on asperity 1 and 2, respectively, we have
$\begin{array}{}\text{(38)}& \frac{{f}_{\mathrm{d}\mathrm{1}}^{\prime }}{{f}_{\mathrm{s}\mathrm{1}}}=\mathit{ϵ}\frac{{f}_{\mathrm{s}\mathrm{1}}^{\prime }}{{f}_{\mathrm{s}\mathrm{1}}}=\mathit{ϵ}{\mathit{\beta }}_{\mathrm{1}},\phantom{\rule{1em}{0ex}}\frac{{f}_{\mathrm{d}\mathrm{2}}^{\prime }}{{f}_{\mathrm{s}\mathrm{1}}}=\mathit{ϵ}\frac{{f}_{\mathrm{s}\mathrm{2}}^{\prime }}{{f}_{\mathrm{s}\mathrm{1}}}=\mathit{ϵ}{\mathit{\beta }}_{\mathrm{2}}.\end{array}$
The consequent changes in dynamic frictions are ϵΔβ1 and ϵΔβ2 on asperity 1 and 2, respectively.
## 4.1 Effects of the perturbation
The stress transfer resulting from earthquakes on neighbouring faults alters several parameters of the model. A first remarkable change concerns the strength of the asperities. After the perturbation, we can define a new ratio
$\begin{array}{}\text{(39)}& {\mathit{\beta }}^{\prime }=\frac{{f}_{\mathrm{s}\mathrm{2}}^{\prime }}{{f}_{\mathrm{s}\mathrm{1}}^{\prime }}=\frac{{f}_{\mathrm{d}\mathrm{2}}^{\prime }}{{f}_{\mathrm{d}\mathrm{1}}^{\prime }}=\frac{{\mathit{\beta }}_{\mathrm{2}}}{{\mathit{\beta }}_{\mathrm{1}}},\end{array}$
which differs from the original value of β given in Eq. (1). Moreover, the stress transfer may be so intense that the weaker asperity may become the stronger one: that is, it may result in ${\mathit{\beta }}^{\prime }>\mathrm{1}$.
The variations in static frictions entail different conditions for the onset of motion of the asperities. Taking Eq. (22) into account, Eqs. (7) and (8) become, respectively,
$\begin{array}{}\text{(40)}& {F}_{\mathrm{1}}=-{\mathit{\beta }}_{\mathrm{1}},\phantom{\rule{1em}{0ex}}{F}_{\mathrm{2}}=-{\mathit{\beta }}_{\mathrm{2}}.\end{array}$
By combination with Eq. (2), these conditions define the planes
$\begin{array}{}\text{(41)}& & X-\mathit{\alpha }Z-{\mathit{\beta }}_{\mathrm{1}}=\mathrm{0},\text{(42)}& & Y+\mathit{\alpha }Z-{\mathit{\beta }}_{\mathrm{2}}=\mathrm{0}\end{array}$
that we call ${\mathrm{\Pi }}_{\mathrm{1}}^{\prime }$ and ${\mathrm{\Pi }}_{\mathrm{2}}^{\prime }$, respectively. Conversely, the planes Γ1 and Γ2 are not affected by the stress perturbation, since they do not depend on frictions. We conclude that changes in normal stress modify the sticking region of the system, describing a new hexahedron H in the state space. The coordinates of its vertices are
$\begin{array}{ll}& {A}^{\prime }=\left(\mathrm{0},{\mathit{\beta }}_{\mathrm{1}},-\frac{{\mathit{\beta }}_{\mathrm{1}}}{\mathit{\alpha }}\right),\phantom{\rule{0.25em}{0ex}}{B}^{\prime }=\left({\mathit{\beta }}_{\mathrm{2}},\mathrm{0},\frac{{\mathit{\beta }}_{\mathrm{2}}}{\mathit{\alpha }}\right),\\ \text{(43)}& & \phantom{\rule{1em}{0ex}}{C}^{\prime }=\left({\mathit{\beta }}_{\mathrm{1}}+{\mathit{\beta }}_{\mathrm{2}},\mathrm{0},\frac{{\mathit{\beta }}_{\mathrm{2}}}{\mathit{\alpha }}\right),& {D}^{\prime }=\left(\mathrm{0},{\mathit{\beta }}_{\mathrm{1}}+{\mathit{\beta }}_{\mathrm{2}},-\frac{{\mathit{\beta }}_{\mathrm{1}}}{\mathit{\alpha }}\right),\phantom{\rule{0.25em}{0ex}}{E}^{\prime }=\left({\mathit{\beta }}_{\mathrm{1}},\mathrm{0},\mathrm{0}\right),\\ \text{(44)}& & \phantom{\rule{1em}{0ex}}{F}^{\prime }=\left(\mathrm{0},{\mathit{\beta }}_{\mathrm{2}},\mathrm{0}\right).\end{array}$
The volume of H is ${\mathit{\beta }}_{\mathrm{1}}{\mathit{\beta }}_{\mathrm{2}}\left({\mathit{\beta }}_{\mathrm{1}}+{\mathit{\beta }}_{\mathrm{2}}\right)/\mathrm{2}\mathit{\alpha }$: thus, the set of states corresponding to stationary asperities is enlarged or reduced, depending on how normal stresses on the asperities are modified.
Following the changes in static frictions, the surface Σ in Eq. (10) is replaced by a new surface Σ expressed by
$\begin{array}{}\text{(45)}& V\mathrm{\Theta }\left[W\left({\mathit{\gamma }}_{\mathrm{1}}^{\prime }\right)-W\left({\mathit{\gamma }}_{\mathrm{2}}^{\prime }\right)\right]+Y-X+{\mathit{\beta }}_{\mathrm{1}}-{\mathit{\beta }}_{\mathrm{2}}=\mathrm{0},\end{array}$
where
$\begin{array}{}\text{(46)}& {\mathit{\gamma }}_{\mathrm{1}}^{\prime }=\frac{\mathit{\alpha }Z}{V\mathrm{\Theta }}{e}^{-\frac{{\mathit{\beta }}_{\mathrm{1}}-X}{V\mathrm{\Theta }}},\phantom{\rule{1em}{0ex}}{\mathit{\gamma }}_{\mathrm{2}}^{\prime }=-\frac{\mathit{\alpha }Z}{V\mathrm{\Theta }}{e}^{-\frac{{\mathit{\beta }}_{\mathrm{2}}-Y}{V\mathrm{\Theta }}}.\end{array}$
As a result, the sticking region H is split in two subsets ${\mathbf{H}}_{\mathrm{1}}^{\prime }$ and ${\mathbf{H}}_{\mathrm{2}}^{\prime }$; furthermore, its faces ${A}^{\prime }{E}^{\prime }{C}^{\prime }{D}^{\prime }$ and ${B}^{\prime }{C}^{\prime }{D}^{\prime }{F}^{\prime }$ are divided into subsets ${\mathbf{Q}}_{\mathrm{1}}^{\prime },{\mathbf{s}}_{\mathrm{1}}^{\prime }$, ${\mathbf{R}}_{\mathrm{1}}^{\prime }$ and ${\mathbf{Q}}_{\mathrm{2}}^{\prime },{\mathbf{s}}_{\mathrm{2}}^{\prime }$, ${\mathbf{R}}_{\mathrm{2}}^{\prime }$, respectively.
As a consequence of the changes in dynamic frictions, the amount of slip that asperities undergo during a seismic event is modified. In turn, the perturbation alters the seismic moment associated with an earthquake. The variations in the final slip amplitudes U1 and U2 of asperity 1 and asperity 2, respectively, and in the final seismic moment M0 associated with the different seismic events predicted by the model are listed in Table 2.
### 4.1.1 Changes in Coulomb forces
The variations in tangential stresses and static frictions discussed so far entail a change in the Coulomb forces assigned to the asperities. Combining Eq. (5) with Eqs. (19) and (19), these changes are given by
$\begin{array}{}\text{(47)}& & \mathrm{\Delta }{F}_{\mathrm{1}}^{\mathrm{C}}=-\mathrm{\Delta }{F}_{\mathrm{1}}-\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{1}}=\left(\mathrm{1}+\mathit{\alpha }\right)\mathrm{\Delta }X-\mathit{\alpha }\mathrm{\Delta }Y-\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{1}},\text{(48)}& & \mathrm{\Delta }{F}_{\mathrm{2}}^{\mathrm{C}}=-\mathrm{\Delta }{F}_{\mathrm{2}}-\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{2}}=\left(\mathrm{1}+\mathit{\alpha }\right)\mathrm{\Delta }Y-\mathit{\alpha }\mathrm{\Delta }X-\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{2}}\end{array}$
or, exploiting Eqs. (22) and (22),
$\begin{array}{}\text{(49)}& & \mathrm{\Delta }{F}_{\mathrm{1}}^{\mathrm{C}}=\frac{\left(\mathrm{1}+\mathit{\alpha }\right)\mathrm{cos}\mathit{\theta }-\mathit{\alpha }\mathrm{sin}\mathit{\theta }}{\sqrt{\mathrm{2}-\mathrm{sin}\mathrm{2}\mathit{\theta }}}\mathrm{\Delta }R-\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{1}},\text{(50)}& & \mathrm{\Delta }{F}_{\mathrm{2}}^{\mathrm{C}}=\frac{\left(\mathrm{1}+\mathit{\alpha }\right)\mathrm{sin}\mathit{\theta }-\mathit{\alpha }\mathrm{cos}\mathit{\theta }}{\sqrt{\mathrm{2}-\mathrm{sin}\mathrm{2}\mathit{\theta }}}\mathrm{\Delta }R-\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{2}}.\end{array}$
The sign of $\mathrm{\Delta }{F}_{i}^{\mathrm{C}}$ $\left(i=\mathrm{1},\mathrm{2}\right)$ determines whether the perturbation brings an asperity closer to or farther from the failure; specifically, positive variations entail that slip is favoured, and vice versa. Equations (29) and (29) clearly point out that this effect is regulated by the orientation of the vector ΔR in the state space. Bearing in mind the observations made in Sect. 2, we find that $\mathrm{\Delta }{F}_{\mathrm{1}}^{\mathrm{C}}$ is at its maximum when ΔR is perpendicular to plane Π1 and points toward it; it vanishes when ΔR is parallel to plane Π1, and it is at its minimum when ΔR is perpendicular to plane Π1 and points away from it. Analogous considerations can be made for $\mathrm{\Delta }{F}_{\mathrm{2}}^{\mathrm{C}}$.
On the whole, the effect of the stress perturbation can be discussed in terms of the quantity
$\begin{array}{ll}\mathrm{\Delta }{F}^{\mathrm{C}}& =\mathrm{\Delta }{F}_{\mathrm{2}}^{\mathrm{C}}-\mathrm{\Delta }{F}_{\mathrm{1}}^{\mathrm{C}}\\ \text{(51)}& & =\left(\mathrm{1}+\mathrm{2}\mathit{\alpha }\right)\left(\mathrm{\Delta }Y-\mathrm{\Delta }X\right)+\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{1}}-\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{2}}.\end{array}$
Let us assume that the system is at a certain state $\left(X,\phantom{\rule{0.125em}{0ex}}Y,\phantom{\rule{0.125em}{0ex}}Z\right)\in {\mathbf{H}}_{\mathrm{1}}$ before the perturbation; accordingly, the next seismic event on the fault will start with the failure of asperity 1. If ΔFC>0, the perturbation favours the slip of asperity 2 more than the slip of asperity 1: therefore, the system is brought to a state closer to the condition for the simultaneous failure of the asperities and thus to the Σ surface. On the contrary, perturbations for which ΔFC<0 take the system farther from the Σ surface. The opposite holds for an unperturbed state $\left(X,\phantom{\rule{0.125em}{0ex}}Y,\phantom{\rule{0.125em}{0ex}}Z\right)\in {\mathbf{H}}_{\mathrm{2}}$.
Table 2Changes in the final slip amplitudes U1 and U2 of asperity 1 and 2 and in the seismic moment M0 associated with the different seismic events predicted by the model, after a stress perturbation from neighbouring faults. The entry e.n. is the abbreviation for evaluated numerically.
### 4.1.2 Changes in the duration of the inter-seismic interval
As already stated, stress perturbations can anticipate or delay the occurrence of an earthquake produced by a certain asperity. We now quantify this effect in terms of the variation in the duration of the inter-seismic interval. Generally speaking, the perturbation vector ΔR may cross the Σ surface and thus bring the system from an unperturbed state within H1 (H2) to a perturbed state within ${\mathbf{H}}_{\mathrm{2}}^{\prime }$ (${\mathbf{H}}_{\mathrm{1}}^{\prime }$). For the sake of simplicity, we consider here only the particular case in which the perturbation vector ΔR does not cross the Σ surface. An example of a more general case will be shown in Sect. 5 for a real fault.
Let us first focus on the case in which the unperturbed state $\left(X,\phantom{\rule{0.125em}{0ex}}Y,\phantom{\rule{0.125em}{0ex}}Z\right)\in {\mathbf{H}}_{\mathrm{1}}$. The time required by the orbit of the system to reach plane Π1, triggering the failure of asperity 1, was calculated by Amendola and Dragoni (2013) as
$\begin{array}{}\text{(52)}& {T}_{\mathrm{1}}=\mathrm{\Theta }\phantom{\rule{0.125em}{0ex}}W\left({\mathit{\gamma }}_{\mathrm{1}}\right)+\frac{\mathrm{1}-X}{V},\end{array}$
with γ1 given in Eq. (11). If the stress perturbation brings the system to a state $\left({X}^{\prime },\phantom{\rule{0.125em}{0ex}}{Y}^{\prime },\phantom{\rule{0.125em}{0ex}}{Z}^{\prime }\right)\in {\mathbf{H}}_{\mathrm{1}}^{\prime }$ and the static friction on asperity 1 to β1, the time required by the orbit to reach plane ${\mathrm{\Pi }}_{\mathrm{1}}^{\prime }$ is
$\begin{array}{}\text{(53)}& {T}_{\mathrm{1}}^{\prime }=\mathrm{\Theta }W\left({\mathit{\gamma }}_{\mathrm{1}}^{\prime }\right)+\frac{{\mathit{\beta }}_{\mathrm{1}}-{X}^{\prime }}{V},\end{array}$
with ${\mathit{\gamma }}_{\mathrm{1}}^{\prime }$ given in Eq. (28). The difference between the two times is
$\begin{array}{}\text{(54)}& \mathrm{\Delta }{T}_{\mathrm{1}}={T}_{\mathrm{1}}^{\prime }-{T}_{\mathrm{1}}=\mathrm{\Theta }\left[W\left({\mathit{\gamma }}_{\mathrm{1}}^{\prime }\right)-W\left({\mathit{\gamma }}_{\mathrm{1}}\right)\right]-\frac{\mathrm{\Delta }{F}_{\mathrm{1}}^{\mathrm{C}}+\mathit{\alpha }\mathrm{\Delta }Z}{V},\end{array}$
where Eq. (29) has been employed.
If instead $\left(X,\phantom{\rule{0.125em}{0ex}}Y,\phantom{\rule{0.125em}{0ex}}Z\right)\in {\mathbf{H}}_{\mathrm{2}}$, the time required by the orbit of the system to reach plane Π2, triggering the failure of asperity 2, is given by (Amendola and Dragoni, 2013)
$\begin{array}{}\text{(55)}& {T}_{\mathrm{2}}=\mathrm{\Theta }\phantom{\rule{0.125em}{0ex}}W\left({\mathit{\gamma }}_{\mathrm{2}}\right)+\frac{\mathit{\beta }-Y}{V},\end{array}$
with γ2 given in Eq. (11). If the stress perturbation takes the system to a state $\left({X}^{\prime },\phantom{\rule{0.125em}{0ex}}{Y}^{\prime },\phantom{\rule{0.125em}{0ex}}{Z}^{\prime }\right)\in {\mathbf{H}}_{\mathrm{2}}^{\prime }$ and the static friction on asperity 2 to β2, the time required to reach plane ${\mathrm{\Pi }}_{\mathrm{2}}^{\prime }$ is
$\begin{array}{}\text{(56)}& {T}_{\mathrm{2}}^{\prime }=\mathrm{\Theta }W\left({\mathit{\gamma }}_{\mathrm{2}}^{\prime }\right)+\frac{{\mathit{\beta }}_{\mathrm{2}}-{Y}^{\prime }}{V},\end{array}$
with ${\mathit{\gamma }}_{\mathrm{2}}^{\prime }$ given in Eq. (28). The difference between the two times is
$\begin{array}{}\text{(57)}& \mathrm{\Delta }{T}_{\mathrm{2}}={T}_{\mathrm{2}}^{\prime }-{T}_{\mathrm{2}}=\mathrm{\Theta }\left[W\left({\mathit{\gamma }}_{\mathrm{2}}^{\prime }\right)-W\left({\mathit{\gamma }}_{\mathrm{2}}\right)\right]-\frac{\mathrm{\Delta }{F}_{\mathrm{2}}^{\mathrm{C}}-\mathit{\alpha }\mathrm{\Delta }Z}{V},\end{array}$
where Eq. (29) has been employed. Positive values of ΔT1 and ΔT2 correspond to a delay in the occurrence of an earthquake on asperity 1 and 2, respectively, and vice versa.
### 4.1.3 Discussion
According to the model, rock rheology plays a critical role in the response to stress perturbations. In the case of purely elastic coupling between the asperities, Dragoni and Piombo (2015) showed that the changes in the duration of the inter-seismic interval prior to the failure of asperity 1 and 2 are, respectively,
$\begin{array}{}\text{(58)}& \mathrm{\Delta }{T}_{\mathrm{1}}=-\frac{\mathrm{\Delta }{F}_{\mathrm{1}}^{\mathrm{C}}}{V},\phantom{\rule{1em}{0ex}}\mathrm{\Delta }{T}_{\mathrm{2}}=-\frac{\mathrm{\Delta }{F}_{\mathrm{2}}^{\mathrm{C}}}{V}.\end{array}$
Accordingly, an increase in the Coulomb force associated with a given asperity ($\mathrm{\Delta }{F}_{i}^{\mathrm{C}}>\mathrm{0}$) directly yields the anticipation of the slip of that asperity, and vice versa. What is more, the variation in the duration of the inter-seismic interval is proportional to the change in the Coulomb force associated with the asperity.
Conversely, in the viscoelastic case there is no straightforward connection between the sign of $\mathrm{\Delta }{F}_{i}^{\mathrm{C}}$ and the anticipation or delay of an earthquake on the associated asperity. In fact, the expressions (31) and (34) obtained for ΔT1 and ΔT2 in the previous section indicate that the net effect depends in a non-trivial way on the particular state of the fault at the time of the stress perturbation and right after it. This result points out the complex interplay between the post-seismic evolution of a fault in the presence of viscoelastic relaxation and the stress transfer from neighbouring faults.
5 An application: perturbation of the 1992 Landers fault by the 1999 Hector Mine earthquake
We study the effects of the 16 October 1999 Mw 7.1 Hector Mine, California, earthquake on the post-seismic evolution of the fault that generated the 28 June 1992 Mw 7.3 Landers, California, earthquake. The geometry of the two faults is shown in Fig. 6.
The 1992 Landers earthquake was a right-lateral strike-slip event that can be approximated as the result of the slip of two coplanar asperities (Kanamori et al., 1992): a northern one (asperity 1) and a southern one (asperity 2), with average slips u1=6 m and u2=3 m, respectively. Following Dragoni and Tallarico (2016), we assume a common area A=300 km2 for both the asperities. We place the centres of asperity 1 and asperity 2 at 34.46 N, 116.52 W and 34.20 N, 116.44 W, respectively, with a common depth of 8 km. The earthquake started with the failure of asperity 2, followed by the failure of asperity 1. We characterize the event by strike, dip and rake angles of 345, 85 and 180, respectively, an average of the values provided by Kanamori et al. (1992) for the two phases of the earthquake.
The 1992 event was followed by remarkable post-seismic deformation, which can be interpreted as the result of several processes. For the sake of the present application, we assume viscoelastic relaxation as the most significant mechanism. We assign a viscosity $\mathit{\eta }=\mathrm{5}×{\mathrm{10}}^{\mathrm{18}}\phantom{\rule{0.125em}{0ex}}\mathrm{Pa}\phantom{\rule{0.125em}{0ex}}\mathrm{s}$ to the lower crust at Landers, averaging the estimates provided by Deng et al. (1998), Pollitz et al. (2000), Freed and Lin (2001) and Masterlark and Wang (2002). With a rigidity μ=30 GPa, the corresponding Maxwell relaxation time is $\mathit{\tau }=\mathit{\eta }/\mathit{\mu }\simeq \mathrm{5}\phantom{\rule{0.125em}{0ex}}\mathrm{a}$.
Figure 6Geometry of the Landers (LAN) and Hector Mine (HM), California, faults that generated the 1992 and 1999 earthquakes, respectively. The stars indicate the hypocentres of the seismic events. The labels 1 and 2 identify the asperities on the Landers fault.
We model the 1992 earthquake as a two-mode event 01-10 starting from mode 00. Accordingly, the orbit of the system during mode 00 lies inside the subset H2 of the sticking region and the state P1 at the beginning of the earthquake belongs to segment s2 (Fig. 4). The coordinates of P1 are
$\begin{array}{}\text{(59)}& {X}_{\mathrm{1}}=\mathit{\alpha }{Z}_{\mathrm{1}}+\mathrm{1}-\mathit{\alpha }\mathit{\beta }\mathit{\kappa }U,\phantom{\rule{1em}{0ex}}{Y}_{\mathrm{1}}=\mathit{\beta }-\mathit{\alpha }{Z}_{\mathrm{1}},\phantom{\rule{1em}{0ex}}{Z}_{\mathrm{1}},\end{array}$
with
$\begin{array}{}\text{(60)}& {Z}_{a}\le {Z}_{\mathrm{1}}\le {Z}_{b},\end{array}$
where the extreme values Za and Zb correspond to the end points of s2:
$\begin{array}{}\text{(61)}& {Z}_{a}=\frac{\mathit{\kappa }U\left(\mathit{\alpha }\mathit{\beta }+\mathrm{1}\right)-\mathrm{1}}{\mathit{\alpha }},\phantom{\rule{1em}{0ex}}{Z}_{b}=\frac{\mathit{\beta }\left(\mathrm{1}-\mathit{\kappa }U\right)}{\mathit{\alpha }}.\end{array}$
At the end of mode 01, the system is at point P2 with coordinates
$\begin{array}{}\text{(62)}& {X}_{\mathrm{2}}={X}_{\mathrm{1}},\phantom{\rule{1em}{0ex}}{Y}_{\mathrm{2}}={Y}_{\mathrm{1}}-\mathit{\beta }\mathit{\kappa }U,\phantom{\rule{1em}{0ex}}{Z}_{\mathrm{2}}={Z}_{\mathrm{1}}-\mathit{\beta }\mathit{\kappa }U,\end{array}$
where mode 10 starts. As Z1 varies in the interval given in Eq. (37), an infinite number of points P2 describe a segment r2 on the subset Q1 of the face AECD and parallel to the edge CD. Mode 10 terminates at point P3 with coordinates
$\begin{array}{}\text{(63)}& {X}_{\mathrm{3}}={X}_{\mathrm{2}}-\mathit{\kappa }U,\phantom{\rule{1em}{0ex}}{Y}_{\mathrm{3}}={Y}_{\mathrm{2}},\phantom{\rule{1em}{0ex}}{Z}_{\mathrm{3}}={Z}_{\mathrm{2}}+\mathit{\kappa }U.\end{array}$
Again, as Z1 varies in the interval given in Eq. (37), there is an infinite number of points P3 defining another segment q2 parallel to the edge CD. This segment is situated within the sticking region and crosses the surface Σ for Z1=Zc, with ${Z}_{a}<{Z}_{c}<{Z}_{b}$.
Dragoni and Tallarico (2016) studied the 1992 Landers earthquake under the hypothesis of purely elastic coupling between the asperities. Following the authors, we take α=0.1, β=0.5, γ=1.5 and ϵ=0.7, a set of values yielding modelled moment rate and seismic spectrum comparable with the observations. Thus, we have U≃0.546 and κ≃0.52. As for viscoelastic relaxation, it can be characterized by the product VΘ (Amendola and Dragoni, 2013), which can be estimated as
$\begin{array}{}\text{(64)}& V\mathrm{\Theta }=\frac{\mathit{\kappa }Uv\mathit{\tau }}{{u}_{\mathrm{1}}},\end{array}$
where v=3 cm a−1 is the relative plate velocity at Landers (Wallace, 1990). Accordingly, we have VΘ≃0.007.
Every state P1 on segment s2, where the 1992 earthquake began, corresponds to a specific state P3 on segment q2, where the 1992 earthquake ended. Exploiting Eq. (39), we can express the coordinates (40) of P3 as a function of Z1. Since q2 crosses the surface Σ, the state P3 can belong to H1,H2 or Σ, in correspondence to ${Z}_{c}<{Z}_{\mathrm{1}}\le {Z}_{b}$, ${Z}_{a}\le {Z}_{\mathrm{1}}<{Z}_{c}$ and Z1=Zc, respectively. In the first case, the next event will start with the failure of asperity 1; in the second case, with the failure of asperity 2; in the third case, with the simultaneous failure of the asperities. With the values of α, β, κ and U listed above, we find ${Z}_{a}\simeq -\mathrm{7.02}$, Zb≃3.58 and Zc≃0.78. Accordingly, only about one-quarter of segment q2 lies inside the subset H1 of the sticking region. Without any further discussion and neglecting the stress perturbation caused by the Hector Mine earthquake, we would infer that future events on the 1992 fault are more likely to start with the failure of asperity 2.
## 5.1 Stress perturbation by the 1999 Hector Mine earthquake
The 1999 Hector Mine earthquake was generated by right-lateral strike-slip faulting located at 34.59 N, 116.27 W, about 20 km north-east from the Landers fault (Jónsson et al., 2002; Salichon et al., 2004). We characterize the event averaging the data available in the SRCMOD database and assume strike, dip and rake angles of 330, 80 and 180, respectively; a depth of 10 km; and a seismic moment of 6.62×1019 Nm.
The stress transferred to the asperities at Landers can be evaluated employing the model of Appendix Appendix A, taking
$\begin{array}{ll}& {\mathit{\varphi }}_{\mathrm{1}}=\mathrm{345}{}^{\circ },\phantom{\rule{1em}{0ex}}{\mathit{\varphi }}_{\mathrm{2}}=\mathrm{330}{}^{\circ },\phantom{\rule{1em}{0ex}}{\mathit{\psi }}_{\mathrm{1}}=\mathrm{85}{}^{\circ },\phantom{\rule{1em}{0ex}}{\mathit{\psi }}_{\mathrm{2}}=\mathrm{80}{}^{\circ },\\ \text{(65)}& & {\mathit{\lambda }}_{\mathrm{1}}={\mathit{\lambda }}_{\mathrm{2}}=\mathrm{180}{}^{\circ }.\end{array}$
As a result, the normal and tangential components of the perturbing stress on asperity 1 are
$\begin{array}{}\text{(66)}& {\mathit{\sigma }}_{\mathrm{1}n}\simeq \mathrm{0.14}\phantom{\rule{0.125em}{0ex}}\mathrm{MPa},\phantom{\rule{1em}{0ex}}{\mathit{\sigma }}_{\mathrm{1}t}\simeq \mathrm{0.39}\phantom{\rule{0.125em}{0ex}}\mathrm{MPa}.\end{array}$
Accordingly, the static friction on asperity 1 is reduced and right-lateral slip is favoured. As for asperity 2, the components of the perturbing stress are
$\begin{array}{}\text{(67)}& {\mathit{\sigma }}_{\mathrm{2}n}\simeq \mathrm{0.18}\phantom{\rule{0.125em}{0ex}}\mathrm{MPa},\phantom{\rule{1em}{0ex}}{\mathit{\sigma }}_{\mathrm{2}t}\simeq -\mathrm{0.17}\phantom{\rule{0.125em}{0ex}}\mathrm{MPa},\end{array}$
suggesting that static friction on asperity 2 is reduced and right-lateral slip is inhibited.
We now introduce the effect of the perturbation in the framework of the discrete model. The changes in the tangential forces (2) on the asperities are
$\begin{array}{}\text{(68)}& \mathrm{\Delta }{F}_{\mathrm{1}}=-\frac{{\mathit{\sigma }}_{\mathrm{1}t}}{{f}_{\mathrm{s}\mathrm{1}}}A,\phantom{\rule{1em}{0ex}}\mathrm{\Delta }{F}_{\mathrm{2}}=-\frac{{\mathit{\sigma }}_{\mathrm{2}t}}{{f}_{\mathrm{s}\mathrm{1}}}A.\end{array}$
The static friction fs1 on asperity 1 can be evaluated as (Dragoni and Santini, 2012)
$\begin{array}{}\text{(69)}& {f}_{\mathrm{s}\mathrm{1}}=\frac{K{u}_{\mathrm{1}}}{\mathit{\kappa }U},\end{array}$
where the constant
$\begin{array}{}\text{(70)}& K=\frac{\mathit{\mu }A}{d}\end{array}$
is an expression of the coupling between the asperities and the tectonic plates. With d=80 km (Masterlark and Wang, 2002), it results in ${f}_{\mathrm{s}\mathrm{1}}/A\simeq \mathrm{7.9}$ MPa. Hence, we have
$\begin{array}{}\text{(71)}& \mathrm{\Delta }{F}_{\mathrm{1}}\simeq -\mathrm{0.05},\phantom{\rule{1em}{0ex}}\mathrm{\Delta }{F}_{\mathrm{2}}\simeq \mathrm{0.02}.\end{array}$
From Eqs. (19)–(19), the components of the perturbation vector ΔR are
$\begin{array}{}\text{(72)}& \mathrm{\Delta }X\simeq \mathrm{0.043},\phantom{\rule{1em}{0ex}}\mathrm{\Delta }Y\simeq -\mathrm{0.016},\phantom{\rule{1em}{0ex}}\mathrm{\Delta }Z\simeq -\mathrm{0.059}.\end{array}$
As a result, the orientation of ΔR in the state space is characterized by angles $\mathit{\theta }\simeq -\mathrm{0.35}$ rad and $\mathit{\delta }\simeq -\mathrm{0.91}$ rad. The changes in static frictions (23) can be calculated as
$\begin{array}{}\text{(73)}& \mathrm{\Delta }{\mathit{\beta }}_{\mathrm{1}}=-\frac{{k}_{\mathrm{s}}{\mathit{\sigma }}_{\mathrm{1}n}}{{f}_{\mathrm{s}\mathrm{1}}}A,\phantom{\rule{1em}{0ex}}\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{2}}=-\frac{{k}_{\mathrm{s}}{\mathit{\sigma }}_{\mathrm{2}n}}{{f}_{\mathrm{s}\mathrm{1}}}A,\end{array}$
where ks is the effective static friction coefficient on asperity 1. Assuming ks=0.4, we get
$\begin{array}{}\text{(74)}& \mathrm{\Delta }{\mathit{\beta }}_{\mathrm{1}}\simeq -\mathrm{0.0073},\phantom{\rule{1em}{0ex}}\mathrm{\Delta }{\mathit{\beta }}_{\mathrm{2}}\simeq -\mathrm{0.0092}.\end{array}$
Finally, from Eqs. (29) and (29), the changes in Coulomb forces on the asperities are
$\begin{array}{}\text{(75)}& \mathrm{\Delta }{F}_{\mathrm{1}}^{\mathrm{C}}\simeq \mathrm{0.057},\phantom{\rule{1em}{0ex}}\mathrm{\Delta }{F}_{\mathrm{2}}^{\mathrm{C}}\simeq -\mathrm{0.012}.\end{array}$
At the time of the Hector Mine earthquake, the Landers fault was at a state P4 resulting from the post-seismic evolution of any of the possible states P3q2 where the 1992 event ended. The coordinates of P4 can be calculated from the solution to the equations of mode 00 given by Dragoni and Lorenzano (2015) and taking into account that the time interval $\stackrel{\mathrm{̃}}{t}$ elapsed between the 1992 Landers and 1999 Hector Mine earthquakes amounts to about 7.3 years:
$\begin{array}{}\text{(76)}& {X}_{\mathrm{4}}={X}_{\mathrm{3}}+V\mathrm{\Theta }\phantom{\rule{0.125em}{0ex}}\stackrel{\mathrm{̃}}{T},\phantom{\rule{1em}{0ex}}{Y}_{\mathrm{4}}={Y}_{\mathrm{3}}+V\mathrm{\Theta }\phantom{\rule{0.125em}{0ex}}\stackrel{\mathrm{̃}}{T},\phantom{\rule{1em}{0ex}}{Z}_{\mathrm{4}}={Z}_{\mathrm{3}}{e}^{-\stackrel{\mathrm{̃}}{T}},\end{array}$
where
$\begin{array}{}\text{(77)}& \stackrel{\mathrm{̃}}{T}=\frac{\stackrel{\mathrm{̃}}{t}}{\mathit{\tau }}\approx \mathrm{1.5}.\end{array}$
Making use of Eqs. (39) and (40), we can express the coordinates of P4 as a function of ${Z}_{\mathrm{1}}\in \left[{Z}_{a},{Z}_{b}\right]$. Accordingly, there is an infinite number of points P4 defining a vector t2 inside the sticking region. At $T=\stackrel{\mathrm{̃}}{T}$, the perturbation vector ΔR moves every state P4 to a new state ${P}_{\mathrm{4}}^{\prime }$ with coordinates
$\begin{array}{}\text{(78)}& {X}_{\mathrm{4}}^{\prime }={X}_{\mathrm{4}}+\mathrm{\Delta }X,\phantom{\rule{1em}{0ex}}{Y}_{\mathrm{4}}^{\prime }={Y}_{\mathrm{4}}+\mathrm{\Delta }Y,\phantom{\rule{1em}{0ex}}{Z}_{\mathrm{4}}^{\prime }={Z}_{\mathrm{4}}+\mathrm{\Delta }Z,\end{array}$
which can be expressed as a function of ${Z}_{\mathrm{1}}\in \left[{Z}_{a},{Z}_{b}\right]$. As a result, a new vector ${\mathbit{t}}_{\mathrm{2}}^{\prime }$ identifies the state of the Landers fault after the Hector Mine earthquake.
In order to characterize the effect of the perturbation, let us consider the difference ΔFC defined in Eq. (29): from Eq. (51), we get $\mathrm{\Delta }{F}^{\mathrm{C}}\simeq -\mathrm{0.069}$. Since ΔFC<0, we conclude that the stress perturbation is such that states P4H1 are moved to ${\mathbf{H}}_{\mathrm{1}}^{\prime }$, the state P4Σ enters ${\mathbf{H}}_{\mathrm{1}}^{\prime }$, states P4H2 are shifted towards the Σ surface and some of them enter ${\mathbf{H}}_{\mathrm{1}}^{\prime }$. Specifically, we find that ${P}_{\mathrm{4}}^{\prime }$ belongs to ${\mathbf{H}}_{\mathrm{1}}^{\prime }$, ${\mathbf{H}}_{\mathrm{2}}^{\prime }$ and Σ in correspondence to ${Z}_{c}^{\prime }<{Z}_{\mathrm{1}}\le {Z}_{b}$, ${Z}_{a}\le {Z}_{\mathrm{1}}<{Z}_{c}^{\prime }$ and ${Z}_{\mathrm{1}}={Z}_{c}^{\prime }$, with ${Z}_{c}^{\prime }\simeq \mathrm{0.50}$. On the whole, we can draw the preliminary conclusion that the stress perturbation is such that future events on the Landers fault starting with the slip of asperity 1 are favoured over events starting with the slip of asperity 2. A deeper discussion is provided in the following.
## 5.2 Constraints due to the seismic history to date
In order to improve our knowledge on the state that gave rise to the 1992 Landers earthquake and on the possible future events generated by that fault, we exploit the seismic history between 1999 and the present date. After the perturbation caused by the Hector Mine earthquake, the inter-seismic time ${T}_{\mathrm{is}}^{\prime }$ of the Landers fault can be calculated from Eq. (30) and Eq. (33) for states ${P}_{\mathrm{4}}^{\prime }$ belonging to ${\mathbf{H}}_{\mathrm{1}}^{\prime }$ and ${\mathbf{H}}_{\mathrm{2}}^{\prime }$, respectively:
$\begin{array}{}\text{(79)}& {T}_{\mathrm{is}}^{\prime }=\left\{\begin{array}{l}\mathrm{\Theta }W\left({\mathit{\gamma }}_{\mathrm{1}}^{\prime }\right)+\frac{{\mathit{\beta }}_{\mathrm{1}}-{X}_{\mathrm{4}}^{\prime }}{V},\phantom{\rule{1em}{0ex}}{Z}_{c}^{\prime }<{Z}_{\mathrm{1}}\le {Z}_{b}\\ \mathrm{\Theta }W\left({\mathit{\gamma }}_{\mathrm{2}}^{\prime }\right)+\frac{{\mathit{\beta }}_{\mathrm{2}}-{Y}_{\mathrm{4}}^{\prime }}{V},\phantom{\rule{1em}{0ex}}{Z}_{a}\le {Z}_{\mathrm{1}}<{Z}_{c}^{\prime }\end{array}\right\,\end{array}$
where
$\begin{array}{}\text{(80)}& {\mathit{\gamma }}_{\mathrm{1}}^{\prime }=\frac{\mathit{\alpha }{Z}_{\mathrm{4}}^{\prime }}{V\mathrm{\Theta }}{e}^{-\frac{{\mathit{\beta }}_{\mathrm{1}}-{X}_{\mathrm{4}}^{\prime }}{V\mathrm{\Theta }}},\phantom{\rule{1em}{0ex}}{\mathit{\gamma }}_{\mathrm{2}}^{\prime }=-\frac{\mathit{\alpha }{Z}_{\mathrm{4}}^{\prime }}{V\mathrm{\Theta }}{e}^{-\frac{{\mathit{\beta }}_{\mathrm{2}}-{Y}_{\mathrm{4}}^{\prime }}{V\mathrm{\Theta }}}.\end{array}$
Since no earthquakes have been produced by the Landers fault after the occurrence of the Hector Mine event, up to year 2016, we can exclude the states on the segment s2 yielding an expected inter-seismic time (Eq. 55) shorter than or equal to ${t}_{\mathrm{is}}^{\prime }=$ 17 years. The requirement
$\begin{array}{}\text{(81)}& {T}_{\mathrm{is}}^{\prime }>\frac{{t}_{\mathrm{is}}^{\prime }}{\mathit{\tau }}\mathrm{\Theta }\approx \mathrm{3.5}\phantom{\rule{0.125em}{0ex}}\mathrm{\Theta }\end{array}$
is satisfied by states on segment s2 in the subset ${\stackrel{\mathrm{̃}}{Z}}_{a}\le {Z}_{\mathrm{1}}\le {\stackrel{\mathrm{̃}}{Z}}_{b}$, with ${\stackrel{\mathrm{̃}}{Z}}_{a}\simeq -\mathrm{1.17}$ and ${\stackrel{\mathrm{̃}}{Z}}_{b}\simeq \mathrm{2.19}$.
As a consequence, we can constrain the admissible states on the segment t2. A comparison between the intervals $\left[{\stackrel{\mathrm{̃}}{Z}}_{a},{Z}_{c}\right]$ and $\left[{Z}_{c},{\stackrel{\mathrm{̃}}{Z}}_{b}\right]$ points out that more than one-half of the acceptable subset of t2 belongs to H2. Hence, before the stress perturbation caused by the Hector Mine earthquake, future events on the 1992 Landers fault were more likely to start with the failure of asperity 2.
In turn, the refinement of t2 limits the acceptable states on the segment ${\mathbit{t}}_{\mathrm{2}}^{\prime }$. From the amplitude of the intervals $\left[{\stackrel{\mathrm{̃}}{Z}}_{a},{Z}_{c}^{\prime }\right]$ and $\left[{Z}_{c}^{\prime },{\stackrel{\mathrm{̃}}{Z}}_{b}\right]$, we deduce that the acceptable subset of ${\mathbit{t}}_{\mathrm{2}}^{\prime }$ is almost equally divided between ${\mathbf{H}}_{\mathrm{1}}^{\prime }$ and ${\mathbf{H}}_{\mathrm{2}}^{\prime }$. Therefore, if we consider the influence of the Hector Mine earthquake on future events generated by the 1992 Landers fault, we conclude that the stress perturbation yielded homogenization in the probability of events starting with the failure of asperity 1 or asperity 2. This result is in agreement with the observation that the perturbation vector ΔR shifted the whole segment t2 towards the subset ${\mathbf{H}}_{\mathrm{1}}^{\prime }$ of the sticking region.
These conclusions would have to be reconsidered if new stress perturbations from neighbouring faults were to affect the post-seismic evolution of the Landers fault in the future. In addition, if no earthquakes were to be observed for some time on the Landers fault, the refining procedure discussed above could be repeated and the admissible subsets of segments s2, t2 and ${\mathbit{t}}_{\mathrm{2}}^{\prime }$ could be constrained with further precision.
## 5.3 Effects of the stress perturbation on future earthquakes
Finally, we discuss the features of the next seismic event generated by the 1992 Landers fault, highlighting the changes due to the Hector Mine earthquake.
Every state P1s2 where the 1992 earthquake began corresponds to a particular state P4t2 and ${P}_{\mathrm{4}}^{\prime }\in {\mathbit{t}}_{\mathrm{2}}^{\prime }$ before and after the stress perturbation associated with the Hector Mine earthquake, respectively. Since the segment t2 intersects the surface Σ, the state P4 can belong to H1, H2 or Σ (Fig. 3), thus affecting the asperity that will fail the first at the beginning of the next earthquake on the fault. In the first case, the next event will start with the failure of asperity 1, in the second case with the failure of asperity 2, in the third case with the simultaneous failure of the asperities. Analogous considerations hold for states ${P}_{\mathrm{4}}^{\prime }$ in ${\mathbf{H}}_{\mathrm{1}}^{\prime },{\mathbf{H}}_{\mathrm{2}}^{\prime }$ and Σ, respectively.
The number and the sequence of dynamic modes in the earthquake depend on the sub-interval of Z1 considered. The details are summarized in Table 3 for both the unperturbed and perturbed cases. Taking these specifics into account and referring to Tables 1 and 2, we evaluate the seismic moments M0 and ${M}_{\mathrm{0}}^{\prime }$ associated with the expected future earthquake on the 1992 fault before and after the Hector Mine earthquake, respectively. In Fig. 7, we show the difference
$\begin{array}{}\text{(82)}& \mathrm{\Delta }{M}_{\mathrm{0}}={M}_{\mathrm{0}}^{\prime }-{M}_{\mathrm{0}}\end{array}$
as a function of ${Z}_{\mathrm{1}}\in \left[{\stackrel{\mathrm{̃}}{Z}}_{a},{\stackrel{\mathrm{̃}}{Z}}_{b}\right]$. Owing to the translation imposed to the segment t2 by the perturbation vector ΔR, the sign of ΔM0 changes across the different sub-intervals of Z1. The energy released by the earthquake is increased for ${Z}_{\mathrm{1}}\in \left[\mathrm{0.43},\mathrm{0.71}\right]$, while it is reduced elsewhere.
Another significant result of the stress perturbation concerns the variation in the inter-seismic time before the next seismic event. As in Sect. 5.2, we consider the post-seismic evolution from 1999 onwards and set the origin of times at the occurrence of the Hector Mine earthquake. The expected inter-seismic time Tis prior to the stress perturbation can be calculated from Eqs. (29) and (32) for states P4 belonging to H1 and H2, respectively:
$\begin{array}{}\text{(83)}& {T}_{\mathrm{is}}=\left\{\begin{array}{l}\mathrm{\Theta }W\left({\mathit{\gamma }}_{\mathrm{1}}\right)+\frac{\mathrm{1}-{X}_{\mathrm{4}}}{V},\phantom{\rule{1em}{0ex}}{Z}_{c}<{Z}_{\mathrm{1}}\le {\stackrel{\mathrm{̃}}{Z}}_{b}\\ \mathrm{\Theta }W\left({\mathit{\gamma }}_{\mathrm{2}}\right)+\frac{\mathit{\beta }-{Y}_{\mathrm{4}}}{V},\phantom{\rule{1em}{0ex}}{\stackrel{\mathrm{̃}}{Z}}_{a}\le {Z}_{\mathrm{1}}<{Z}_{c}\end{array}\right\,\end{array}$
where
$\begin{array}{}\text{(84)}& {\mathit{\gamma }}_{\mathrm{1}}=\frac{\mathit{\alpha }{Z}_{\mathrm{4}}}{V\mathrm{\Theta }}{e}^{-\frac{\mathrm{1}-{X}_{\mathrm{4}}}{V\mathrm{\Theta }}},\phantom{\rule{1em}{0ex}}{\mathit{\gamma }}_{\mathrm{2}}=-\frac{\mathit{\alpha }{Z}_{\mathrm{4}}}{V\mathrm{\Theta }}{e}^{-\frac{\mathit{\beta }-{Y}_{\mathrm{4}}}{V\mathrm{\Theta }}}.\end{array}$
The inter-seismic time ${T}_{\mathrm{is}}^{\prime }$ after the stress perturbation has been given in Eq. (55). The difference
$\begin{array}{}\text{(85)}& \mathrm{\Delta }T={T}_{\mathrm{is}}^{\prime }-{T}_{\mathrm{is}}\end{array}$
is shown in Fig. 8 as a function of ${Z}_{\mathrm{1}}\in \left[{\stackrel{\mathrm{̃}}{Z}}_{a},{\stackrel{\mathrm{̃}}{Z}}_{b}\right]$. For states P4H1 corresponding to ${P}_{\mathrm{4}}^{\prime }\in {\mathbf{H}}_{\mathrm{1}}^{\prime }$ and states P4H2 corresponding to ${P}_{\mathrm{4}}^{\prime }\in {\mathbf{H}}_{\mathrm{2}}^{\prime }$, this difference coincides with Eqs. (31) and (34), respectively.
Table 3Future earthquakes generated by the 1992 Landers, California, fault, as functions of the variable Z1 describing the initial state of the 1992 event, with ${Z}_{\mathrm{1}}\in \left[{\stackrel{\mathrm{̃}}{Z}}_{a},{\stackrel{\mathrm{̃}}{Z}}_{b}\right]=\left[-\mathrm{1.16},\mathrm{2.19}\right]$. The results predicted by the model before and after the stress perturbation associated with the 1999 Hector Mine, California, earthquake are shown. The values ${Z}_{\mathrm{1}}={Z}_{c}=\mathrm{0.78}$ and ${Z}_{\mathrm{1}}={Z}_{c}^{\prime }=\mathrm{0.50}$ correspond to the largest possible earthquakes before and after the stress perturbation, respectively.
Figure 7Change in the seismic moment released during the next event on the 1992 Landers, California, fault, as a result of the stress perturbation due to the 1999 Hector Mine, California, earthquake. On the horizontal axis, the variable Z1 describing the initial state of the 1992 event. The kinds of seismic event predicted by the model, corresponding to different intervals of Z1, are listed in Table 3. The values Z1=Zc and ${Z}_{\mathrm{1}}={Z}_{c}^{\prime }$ correspond to the largest possible earthquakes before and after the stress perturbation, respectively.
Some peculiar features stand out. First, we notice that, for all states P4H2 corresponding to ${P}_{\mathrm{4}}^{\prime }\in {\mathbf{H}}_{\mathrm{2}}^{\prime }$, that is, for ${Z}_{\mathrm{1}}\in \left[{\stackrel{\mathrm{̃}}{Z}}_{a},{Z}_{c}^{\prime }\right]$, the inter-seismic time is increased by the stress perturbation, in agreement with the inhibiting effect on asperity 2 suggested by Eq. (51). However, Eq. (51) suggests that the failure of asperity 1 is promoted, but this is not verified by all states ${P}_{\mathrm{4}}^{\prime }\in {\mathbf{H}}_{\mathrm{1}}^{\prime }$, that is, for ${Z}_{\mathrm{1}}\in \left[{Z}_{c}^{\prime },{\stackrel{\mathrm{̃}}{Z}}_{b}\right]$. In fact, the inter-seismic time is reduced only for ${Z}_{\mathrm{1}}\in \left(\mathrm{0.53},{\stackrel{\mathrm{̃}}{Z}}_{b}\right]$, while it is increased for ${Z}_{\mathrm{1}}\in \left[{Z}_{c}^{\prime },\mathrm{0.53}\right)$. In the particular case Z1=0.53, there is no change in the inter-seismic time. This is a remarkable result, showing that the presence of viscoelastic relaxation at the time of the stress perturbation entails the unpredictability of the consequent influence in terms of anticipation or delay of future earthquakes, on the basis of the sole knowledge of the change in Coulomb stress.
Figure 8Change in the inter-seismic time before the next event on the 1992 Landers, California, fault, as a result of the stress perturbation due to the 1999 Hector Mine, California, earthquake. On the horizontal axis, the variable Z1 describing the initial state of the 1992 event. The kinds of seismic event predicted by the model, corresponding to different intervals of Z1, are listed in Table 3. The values Z1=Zc and ${Z}_{\mathrm{1}}={Z}_{c}^{\prime }$ correspond to the largest possible earthquakes before and after the stress perturbation, respectively.
At the occurrence of the next earthquake produced by the Landers fault, the number and sequence of dynamic modes involved and the energy released will reveal more about the state of the system, thus allowing a further refinement of the specific conditions that gave rise to the 1992 event.
6 Conclusions
We considered a plane fault embedded in a shear zone, subject to a uniform strain rate owing to tectonic loading. The fault is characterized by the presence of two asperities with equal areas and different frictional resistance. The coseismic static stress field due to earthquakes produced by the fault is relaxed by viscoelastic deformation in the asthenosphere.
The fault was treated as a discrete dynamical system with three degrees of freedom: the slip deficits of the asperities and the variation of their difference due to viscoelastic deformation. The dynamics of the system was described in terms of one sticking mode and three slipping modes. In the sticking mode, the orbit of the system lies in a convex hexahedron in the space of the state variables, while the number and the sequence of slipping modes during a seismic event are determined by the particular state of the system at the beginning of the inter-seismic interval preceding the event. The amount of slip of the asperities and the energy released during an earthquake generated by the fault can be predicted accordingly.
The effect of stress transfer due to earthquakes on neighbouring faults was studied in terms of a perturbation vector yielding changes to the state of the system, its sticking region and the energy released during a subsequent seismic event. The specific effect on the evolution of the fault is related with the orientation of this vector in the state space.
We investigated the interplay between the ongoing viscoelastic relaxation on the fault and a stress perturbation imposed during an inter-seismic interval. Following a stress perturbation due to earthquakes on neighbouring faults, an increase in the Coulomb stress associated with a given asperity directly yields the anticipation of the slip of that asperity, and vice versa, if a purely elastic rheology is assumed for the receiving fault (Dragoni and Piombo, 2015). According to the present model, this property no longer holds if the change in Coulomb stress occurs while viscoelastic relaxation is taking place on the receiving fault. In fact, even if the change in the inter-seismic intervals of the asperities can still be evaluated from a theoretical point of view, the specific effect of the stress perturbation could be univocally inferred only if the particular states of the fault at the time of the stress perturbation and right after it were known. The information on the change in Coulomb stress on the fault do not suffice any more.
We applied the model to the stress perturbation imposed by the 1999 Hector Mine, California, earthquake to the fault that caused the 1992 Landers, California, earthquake, which was due to the failure of two asperities and was followed by significant viscoelastic relaxation. We modelled the 1992 Landers earthquake as a two-mode event associated with the separate slip of the asperities and showed how the event is compatible with a number of possible initial states of the fault, which can be screened on the basis of the seismic history to date. The details of the stress transfer associated with the 1999 Hector Mine earthquake were calculated using the relative positions and faulting styles of the two faults as a starting point. We discussed the effect of the stress perturbation, pointing out the complexity of its influence on the possible future events generated by the 1992 Landers fault in terms of the associated energy release, the sequence of dynamic modes involved and the duration of the inter-seismic interval. Specifically, we showed that the consequences of the 1999 Hector Mine earthquake on the post-seismic evolution of the 1992 Landers fault depend on the specific state of the Landers fault at the time of the 1999 earthquake and immediately after it, even if the variations in the Coulomb stress on the asperities at Landers are known. On the whole, the application allowed the exemplification of the critical unpredictability of the effect of a stress perturbation occurring while viscoelastic relaxation is taking place.
Another source of complication may be represented by the interaction between viscoelastic relaxation and stable creep on the fault. This problem is beyond the scope of the present work, but it may be object of future research by combining elements of the present model with the one of Dragoni and Lorenzano (2017).
Data availability
Data availability.
All data and results supporting this work were gathered from the papers listed in the References and are freely available to the public. Specifically, data on the 1999 Hector Mine, California, earthquake were collected from the Finite-Source Rupture Model Database (SRCMOD) available at http://equake-rc.info/SRCMOD/. The website was last accessed on December 2016.
Appendix A: Estimate of the stress perturbation
We consider two plane faults, namely fault 1 and fault 2, embedded in an infinite, homogeneous and isotropic Poisson medium of rigidity μ (Fig. A1). Following the slip of fault 1 (perturbing fault), stress is transferred to fault 2 (receiving fault). We calculate the normal traction σn and the tangential traction in the direction of slip σt transferred to the receiving fault, estimated as the average value at its centre.
We define a coordinate system $\left(x,y,z\right)$ with axes corresponding with the directions of dip, strike and normal on fault 1, respectively. Fault 1 lies on the plane z=0 and its centre is in the origin of the coordinate system. Accordingly, the unit vector perpendicular to fault 1 is ${n}_{\mathrm{1}i}=\left(\mathrm{0},\mathrm{0},\mathrm{1}\right)$. We call ϕ1, ψ1 and λ1 the strike, dip and rake angles of fault 1, respectively. The slip direction of fault 1 is then given by
$\begin{array}{}\text{(A1)}& {m}_{\mathrm{1}i}=\left(-\mathrm{sin}{\mathit{\lambda }}_{\mathrm{1}},\mathrm{cos}{\mathit{\lambda }}_{\mathrm{1}},\mathrm{0}\right).\end{array}$
Fault 2 is characterized by strike and dip angles ϕ2 and ψ2, respectively. Accordingly, the unit vector perpendicular to fault 2 is given by
$\begin{array}{}\text{(A2)}& {n}_{\mathrm{2}i}=\left(\mathrm{sin}\mathrm{\Delta }\mathit{\psi }\mathrm{cos}\mathrm{\Delta }\mathit{\varphi },-\mathrm{sin}\mathrm{\Delta }\mathit{\psi }\mathrm{sin}\mathrm{\Delta }\mathit{\varphi },\mathrm{cos}\mathrm{\Delta }\mathit{\psi }\right),\end{array}$
where
$\begin{array}{}\text{(A3)}& \mathrm{\Delta }\mathit{\varphi }={\mathit{\varphi }}_{\mathrm{2}}-{\mathit{\varphi }}_{\mathrm{1}},\phantom{\rule{1em}{0ex}}\mathrm{\Delta }\mathit{\psi }={\mathit{\psi }}_{\mathrm{2}}-{\mathit{\psi }}_{\mathrm{1}}.\end{array}$
Let λ2 be the preferred rake angle on fault 2, correlated with the orientation of tectonic loading: λ2=0 for left-lateral strike slip, λ2=180 for right-lateral strike slip, ${\mathit{\lambda }}_{\mathrm{2}}=-\mathrm{90}$ for normal dip slip and λ2=90 for reverse dip slip. The corresponding slip direction is
$\begin{array}{}\text{(A4)}& & {m}_{\mathrm{2}x}=\mathrm{cos}{\mathit{\lambda }}_{\mathrm{2}}\mathrm{sin}\mathrm{\Delta }\mathit{\varphi }-\mathrm{sin}{\mathit{\lambda }}_{\mathrm{2}}\mathrm{cos}\mathrm{\Delta }\mathit{\psi }\mathrm{cos}\mathrm{\Delta }\mathit{\varphi },\text{(A5)}& & {m}_{\mathrm{2}y}=\mathrm{cos}{\mathit{\lambda }}_{\mathrm{2}}\mathrm{cos}\mathrm{\Delta }\mathit{\varphi }+\mathrm{sin}{\mathit{\lambda }}_{\mathrm{2}}\mathrm{cos}\mathrm{\Delta }\mathit{\psi }\mathrm{sin}\mathrm{\Delta }\mathit{\varphi },\text{(A6)}& & {m}_{\mathrm{2}z}=\mathrm{sin}{\mathit{\lambda }}_{\mathrm{2}}\mathrm{sin}\mathrm{\Delta }\mathit{\psi }.\end{array}$
We name (Ei,Ni) and Di the UTM (Universal Transverse Mercator) coordinates and depths of the centres of the faults, respectively. In our reference system, the coordinates of the centre of fault 2 are identified by the following three steps:
1. placing the origin at the centre of fault 1:
$\begin{array}{}\text{(A7)}& {x}^{\prime }={E}_{\mathrm{2}}-{E}_{\mathrm{1}},\phantom{\rule{1em}{0ex}}{y}^{\prime }={N}_{\mathrm{2}}-{N}_{\mathrm{1}},\phantom{\rule{1em}{0ex}}{z}^{\prime }={D}_{\mathrm{2}}-{D}_{\mathrm{1}}\end{array}$
2. clockwise rotation about the z axis by the angle ϕ1:
$\begin{array}{ll}& {x}^{\prime \prime }={x}^{\prime }\mathrm{cos}{\mathit{\varphi }}_{\mathrm{1}}-{y}^{\prime }\mathrm{sin}{\mathit{\varphi }}_{\mathrm{1}}\phantom{\rule{1em}{0ex}}{y}^{\prime \prime }={x}^{\prime }\mathrm{sin}{\mathit{\varphi }}_{\mathrm{1}}+{y}^{\prime }\mathrm{cos}{\mathit{\varphi }}_{\mathrm{1}},\\ \text{(A8)}& & {z}^{\prime \prime }={z}^{\prime }\end{array}$
3. counterclockwise rotation about the y axis by the angle ψ1:
$\begin{array}{ll}& x={x}^{\prime \prime }\mathrm{cos}{\mathit{\psi }}_{\mathrm{1}}-{z}^{\prime \prime }\mathrm{sin}{\mathit{\psi }}_{\mathrm{1}},\phantom{\rule{1em}{0ex}}y={y}^{\prime \prime },\\ \text{(A9)}& & z={x}^{\prime \prime }\mathrm{sin}{\mathit{\psi }}_{\mathrm{1}}+{z}^{\prime \prime }\mathrm{cos}{\mathit{\psi }}_{\mathrm{1}}.\end{array}$
Figure A1Geometry of the model employed to study the stress transfer between neighbouring faults. Fault 1 is the perturbing fault, while fault 2 is the receiving fault. The coordinates $\left(E,N,D\right)$ are the UTM coordinates and depth of the centres of the faults, respectively, whereas the axes $\left(x,y,z\right)$ correspond with the directions of dip, strike and normal on fault 1, respectively. The angles ϕ and ψ are the strike and dip angles of the faults, respectively.
The perturbing fault is treated as a point-like dislocation source (a double couple of forces) located at the origin. This is a good approximation for non-overlapping regions (Dragoni and Lorenzano, 2016). Let m0 be the scalar seismic moment of the dislocation. The ith component of the static displacement field generated by the slip of fault 1 is
$\begin{array}{}\text{(A10)}& {u}_{i}=-{M}_{jk}{G}_{ij,k},\end{array}$
where Mij is the moment tensor associated with the dislocation source
$\begin{array}{}\text{(A11)}& {M}_{ij}={m}_{\mathrm{0}}\phantom{\rule{0.125em}{0ex}}\left({m}_{\mathrm{1}i}{n}_{\mathrm{1}j}+{m}_{\mathrm{1}j}{n}_{\mathrm{1}i}\right)\end{array}$
and Gij is the Somigliana tensor
$\begin{array}{}\text{(A12)}& {G}_{ij}=\frac{\mathrm{1}}{\mathrm{8}\mathit{\pi }\mathit{\mu }}\left(\frac{\mathrm{2}}{r}{\mathit{\delta }}_{ij}-\frac{\mathrm{2}}{\mathrm{3}}r{,}_{ij}\right),\end{array}$
with
$\begin{array}{}\text{(A13)}& r=\sqrt{{x}^{\mathrm{2}}+{y}^{\mathrm{2}}+{z}^{\mathrm{2}}}.\end{array}$
The components of the stress field are given by
$\begin{array}{}\text{(A14)}& {\mathit{\sigma }}_{ij}=\mathit{\mu }\left({e}_{kk}{\mathit{\delta }}_{ij}+\mathrm{2}{e}_{ij}\right),\end{array}$
where eij is the strain field associated with the displacement field (A5). Finally, the normal traction σn and the tangential traction in the direction of slip σt on fault 2 are
$\begin{array}{}\text{(A15)}& {\mathit{\sigma }}_{\mathrm{n}}={\mathit{\sigma }}_{ij}{n}_{\mathrm{2}i}{n}_{\mathrm{2}j},\phantom{\rule{1em}{0ex}}{\mathit{\sigma }}_{\mathrm{t}}={\mathit{\sigma }}_{ij}{m}_{\mathrm{2}i}{n}_{\mathrm{2}j}.\end{array}$
The signs of σn and σt define the effect of the stress transfer on fault 2. If σn>0, the amount of compressional stress on the receiving fault is reduced, and vice versa. If σt>0, the slip of the receiving fault is promoted, and vice versa.
Author contributions
Author contributions.
EL developed the model, produced the figures and wrote a preliminary version of the paper; MD checked the equations and revised the text. Both authors discussed the results extensively.
Competing interests
Competing interests.
The authors declare that they have no conflict of interest.
Acknowledgements
Acknowledgements.
The authors are thankful to the editor Richard Gloaguen, to Sylvain Barbot and to an anonymous referee for their useful comments on the first version of the paper.
Edited by: Richard Gloaguen
Reviewed by: one anonymous referee
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Wallace, R. E.: The San Andreas fault system, California, U.S. Geol. Surv. Prof. Pap., 1515, 189–205, 1990. | 2019-04-25 20:49:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 210, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5877838730812073, "perplexity": 1317.4426821783004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578733077.68/warc/CC-MAIN-20190425193912-20190425215024-00028.warc.gz"} |
https://www.mail-archive.com/ntg-context@ntg.nl/msg72063.html | Re: [NTG-context] issue with layers and layouts
On 10/21/2013 05:43 PM, luigi scarso wrote:
> On Mon, Oct 21, 2013 at 5:23 PM, Pablo Rodriguez wrote:
> [...]
> First, the layout is right on the first two pages, but it is wrong from
> page three. The text doesn’t reach the bottom frame.
> [...]
> If this isn’t a bug, what am I missing here?
>
> from http://wiki.contextgarden.net/Odd,_even_,_first_page_not_working
>
> \showframe
> %\setuplayout[margin=20mm, width=fit, topspace=45mm,
> % bottomspace=25mm,
> % height=fit]
>
> \definelayout[1][topspace=120mm, bottomspace=30mm,height=fit]
>
> \definelayout[even][margin=20mm, width=fit, topspace=45mm,
> bottomspace=25mm,
> height=fit]
>
> \definelayout[odd][margin=20mm, width=fit, topspace=45mm,
> bottomspace=25mm,
> height=fit]
If I don’t get it wrong, if I define a layout for the first page, I have
to define it for even and odd pages, haven’t I?.
> My second question is about the layer: is there no way to force the text
> to avoid it as if it were a float?
>
> hm what do you want to avoid ?
I want the text flow to avoid the layer "avoidasfloat" as it avoids the
figures.
\definelayer[avoidasfloat][x=0mm, y=0mm, hoffset=8mm,
voffset=101mm,location={right,bottom}, state=start]
\setlayer[avoidasfloat]{\startMPcode
draw (0mm,0mm)--(55mm,0mm) ;
draw (0mm,0mm)--(0mm,-86mm) ;
draw (0mm,-86mm)--(55mm,-86mm) ;
draw (55mm,-86mm)--(55mm,0mm) ;
\stopMPcode
}
\setupbackgrounds[page][background={avoidasfloat}]
\starttext
\dorecurse{40}{\input knuth\placefigure[right]{}{}\par}
\stoptext
Layer has text inside in the sample above. I’d like to have it as a float.
Many thanks for your help,
Pablo
--
http://www.ousia.tk
___________________________________________________________________________________
If your question is of interest to others as well, please add an entry to the
Wiki!
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___________________________________________________________________________________ | 2020-11-29 17:42:40 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8789347410202026, "perplexity": 8207.820993107183}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141201836.36/warc/CC-MAIN-20201129153900-20201129183900-00442.warc.gz"} |
https://codegolf.stackexchange.com/questions/4707/outputting-ordinal-numbers-1st-2nd-3rd/164083 | # Outputting ordinal numbers (1st, 2nd, 3rd)
I would like to generate (as a return result of a function, or simply as the output of a program) the ordinal suffix of a positive integer concatenated to the number.
Samples:
1st
2nd
3rd
4th
...
11th
12th
13th
...
20th
21st
22nd
23rd
24th
And so on, with the suffix repeating the initial 1-10 subpattern every 10 until 100, where the pattern ultimately starts over.
The input would be the number and the output the ordinal string as shown above.
What is the smallest algorithm for this?
• Hi, NickC, and welcome to codegolf.SE! Just to clarify, do you mean that we should read a number like 11 as input, and output e.g. 11th? Is each number in the input on a separate line, and should the output numbers be on separate lines too? And do we need to handle more than one line of input? – Ilmari Karonen Jan 20 '12 at 18:26
• Are you looking for smallest algorithm or smallest code? – Toto Jan 20 '12 at 18:31
• @Ilmari I am looking for 11 as input and 11th as output. I don't mind if it processes multiple lines but what I had in mind was processing just a single number. – Nicole Jan 20 '12 at 21:00
• @M42 You know, I'm not really sure. I don't have a strict requirement - but I was probably thinking smallest algorithm. – Nicole Jan 20 '12 at 21:02
## Haxe, 83 chars
function o(n:Int){return ['th','st','nd','rd'][n%100>20||n%100<4?n%10>3?0:n%10:0];}
Based on the javascript version, fixed for n > 100 as well.
• Two suggestions: 1) Remove all whitespace to save some bytes. Right now, I count 91 total. 2) This doesn't seem to be a full program or a function. Perhaps you should consider turning it into one of those. – ETHproductions Feb 24 '16 at 17:47
• 1) yeah, but thats quite straightforward, so I tried to stay readable. – csomakk Feb 28 '16 at 19:21
• 2) yeah, you're right, I just pasted it here so anyone can use it without reconverting from another lang. :) – csomakk Feb 28 '16 at 19:22
# Scala 2.11.7, 151 Chars
def o(x:Int*)=x map{
case a if(a%10)==1&&a%100!=11=>a+"st"
case b if(b%10)==2&&b%100!=12=>b+"nd"
case c if(c%10)==3&&c%100!=13=>c+"rd"
case e=>e+"th"
}
Usage
o(1,2,3,4) -> ArrayBuffer("1st", "2nd", "3rd", "4th")
• Sorry, but all answers need to be golfed. Please golf this, if not than it's invalid and must be deleted. – Rɪᴋᴇʀ Sep 23 '16 at 12:53
• I'm pretty sure you can remove the space between map and {, : and Int*, and remove the newline between "th" and } (I don't know Scala, so I might be completely wrong). Welcome to PPCG! – clismique Sep 28 '16 at 6:13
# Java 7, 91 81 bytes
String d(int n){return n+(n/10%10==1|(n%=10)<1|n>3?"th":n<2?"st":n<3?"nd":"rd");}
String c(int n){return n+((n%=100)>10&n<14?"th":(n%=10)==1?"st":n==2?"nd":n==3?"rd":"th");}
Ungolfed & test code:
Try it here.
class M{
static String c(int n){
return n + ((n%=100) > 10 & n < 14
?"th"
: (n%=10) == 1
? "st"
: n == 2
? "nd"
: n == 3
? "rd"
:"th");
}
static String d(int n){
return n + (n/10%10 == 1 | (n%=10) < 1 | n > 3
? "th"
: n < 2
? "st"
: n < 3
? "nd"
:"rd");
}
public static void main(String[] a){
for(int i = 1; i < 201; i++){
System.out.print(c(i) + ", ");
}
System.out.println();
for(int i = 1; i < 201; i++){
System.out.print(d(i) + ", ");
}
}
}
Output:
1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, 12th, 13th, 14th, 15th, 16th, 17th, 18th, 19th, 20th, 21st, 22nd, 23rd, 24th, 25th, 26th, 27th, 28th, 29th, 30th, 31st, 32nd, 33rd, 34th, 35th, 36th, 37th, 38th, 39th, 40th, 41st, 42nd, 43rd, 44th, 45th, 46th, 47th, 48th, 49th, 50th, 51st, 52nd, 53rd, 54th, 55th, 56th, 57th, 58th, 59th, 60th, 61st, 62nd, 63rd, 64th, 65th, 66th, 67th, 68th, 69th, 70th, 71st, 72nd, 73rd, 74th, 75th, 76th, 77th, 78th, 79th, 80th, 81st, 82nd, 83rd, 84th, 85th, 86th, 87th, 88th, 89th, 90th, 91st, 92nd, 93rd, 94th, 95th, 96th, 97th, 98th, 99th, 100th, 101st, 102nd, 103rd, 104th, 105th, 106th, 107th, 108th, 109th, 110th, 111th, 112th, 113th, 114th, 115th, 116th, 117th, 118th, 119th, 120th, 121st, 122nd, 123rd, 124th, 125th, 126th, 127th, 128th, 129th, 130th, 131st, 132nd, 133rd, 134th, 135th, 136th, 137th, 138th, 139th, 140th, 141st, 142nd, 143rd, 144th, 145th, 146th, 147th, 148th, 149th, 150th, 151st, 152nd, 153rd, 154th, 155th, 156th, 157th, 158th, 159th, 160th, 161st, 162nd, 163rd, 164th, 165th, 166th, 167th, 168th, 169th, 170th, 171st, 172nd, 173rd, 174th, 175th, 176th, 177th, 178th, 179th, 180th, 181st, 182nd, 183rd, 184th, 185th, 186th, 187th, 188th, 189th, 190th, 191st, 192nd, 193rd, 194th, 195th, 196th, 197th, 198th, 199th, 200th,
1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, 12th, 13th, 14th, 15th, 16th, 17th, 18th, 19th, 20th, 21st, 22nd, 23rd, 24th, 25th, 26th, 27th, 28th, 29th, 30th, 31st, 32nd, 33rd, 34th, 35th, 36th, 37th, 38th, 39th, 40th, 41st, 42nd, 43rd, 44th, 45th, 46th, 47th, 48th, 49th, 50th, 51st, 52nd, 53rd, 54th, 55th, 56th, 57th, 58th, 59th, 60th, 61st, 62nd, 63rd, 64th, 65th, 66th, 67th, 68th, 69th, 70th, 71st, 72nd, 73rd, 74th, 75th, 76th, 77th, 78th, 79th, 80th, 81st, 82nd, 83rd, 84th, 85th, 86th, 87th, 88th, 89th, 90th, 91st, 92nd, 93rd, 94th, 95th, 96th, 97th, 98th, 99th, 100th, 101st, 102nd, 103rd, 104th, 105th, 106th, 107th, 108th, 109th, 110th, 111th, 112th, 113th, 114th, 115th, 116th, 117th, 118th, 119th, 120th, 121st, 122nd, 123rd, 124th, 125th, 126th, 127th, 128th, 129th, 130th, 131st, 132nd, 133rd, 134th, 135th, 136th, 137th, 138th, 139th, 140th, 141st, 142nd, 143rd, 144th, 145th, 146th, 147th, 148th, 149th, 150th, 151st, 152nd, 153rd, 154th, 155th, 156th, 157th, 158th, 159th, 160th, 161st, 162nd, 163rd, 164th, 165th, 166th, 167th, 168th, 169th, 170th, 171st, 172nd, 173rd, 174th, 175th, 176th, 177th, 178th, 179th, 180th, 181st, 182nd, 183rd, 184th, 185th, 186th, 187th, 188th, 189th, 190th, 191st, 192nd, 193rd, 194th, 195th, 196th, 197th, 198th, 199th, 200th,
# Common Lisp, 63
(format t"~a~a"i(let((x(format()"~:r"i)))(subseq x(-(length x)2))))
Input is i. There's probably a more clever way to golf this.
# Pyth, 33 bytes
+Q@.>+c"stndrd"2*]"th"7 1?qhQ\1Z
Try it online!
# Procedural Footnote Language, 39 bytes
[1]
[PFL1.0]
[1] [ORD:[INPUT]]
[PFLEND]
The BODY section of the document contains a reference to footnote [1]; footnote [1] contains the [ORD]inal representation of the [INPUT].
# Julia 0.6, 68 bytes
f(x,m=x%10,p=x%100-m)="$x$(["th","st","nd","rd"][m>3||p==10?1:m+1])"
`
Try it online!
Straightforward remainder rules in Julia. | 2019-12-16 11:23:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3657004237174988, "perplexity": 2265.8066428417933}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541319511.97/warc/CC-MAIN-20191216093448-20191216121448-00224.warc.gz"} |
https://itectec.com/ubuntu/ubuntu-18-04-ite-8910-touchpad-on-asus-strix-gl703ge-not-working/ | # Ubuntu – 18.04 ITE 8910 touchpad on Asus Strix GL703GE not working
I've seen bits and pieces of similar problems around in numerous issues on all sorts of sites, but none of the suggestions/fixes appear to make a difference. It looks like this may be the dreaded Elantech touchpad that's been the new broadcom wireless chipset for the last few years. I'm hoping that there's a tweak/fix that just isn't getting to the top of google searches. Or maybe a more thorough/complete description of the problem would help find a definitive solution. I thought I'd put everything I've found and tried in one place as a sort of hail mary before giving up.
I have a Asus Strix GL703GE Laptop upon which I have installed Ubuntu 18.04. The touchpad appears to work perfectly fine in the windows partition.
Here's the output from xinput:
mich@gordon:~$xinput ⎡ Virtual core pointer id=2 [master pointer (3)] ⎜ ↳ Virtual core XTEST pointer id=4 [slave pointer (2)] ⎜ ↳ Logitech M510 id=12 [slave pointer (2)] ⎜ ↳ ITE Tech. Inc. ITE Device(8910) id=14 [slave pointer (2)] ⎣ Virtual core keyboard id=3 [master keyboard (2)] ↳ Virtual core XTEST keyboard id=5 [slave keyboard (3)] ↳ Power Button id=6 [slave keyboard (3)] ↳ Asus Wireless Radio Control id=7 [slave keyboard (3)] ↳ Video Bus id=8 [slave keyboard (3)] ↳ Video Bus id=9 [slave keyboard (3)] ↳ Power Button id=10 [slave keyboard (3)] ↳ Sleep Button id=11 [slave keyboard (3)] ↳ USB2.0 HD UVC WebCam: USB2.0 HD id=13 [slave keyboard (3)] ↳ Asus WMI hotkeys id=15 [slave keyboard (3)] ↳ AT Translated Set 2 keyboard id=16 [slave keyboard (3)] ↳ ITE Tech. Inc. ITE Device(8910) id=17 [slave keyboard (3)] mich@gordon:~$
The Logitech M510 is a USB mouse since my keyboard-only fu isn't great. It works fine without any tinkering.
Here are a few other diagnostic things that might be interesting:
mich@gordon:~$dmesg | grep -i touch mich@gordon:~$ synclient -l
Couldn't find synaptics properties. No synaptics driver loaded?
mich@gordon:~$dmesg | grep i2c [ 1.574177] i2c /dev entries driver [ 3.615795] i2c_hid i2c-ELAN1200:00: i2c-ELAN1200:00 supply vdd not found, using dummy regulator [ 3.616991] i2c_hid i2c-ELAN1200:00: Could not register for ELAN1200:00 interrupt, irq = 130, ret = -1 [ 3.617013] i2c_hid: probe of i2c-ELAN1200:00 failed with error -1 mich@gordon:~$ dmesg | grep i8042
[ 1.567434] i8042: PNP: No PS/2 controller found.
[ 1.567434] i8042: Probing ports directly.
[ 1.571605] serio: i8042 KBD port at 0x60,0x64 irq 1
[ 1.571611] serio: i8042 AUX port at 0x60,0x64 irq 12
[ 1.580682] input: AT Translated Set 2 keyboard as /devices/platform/i8042/serio0/input/input4
mich@gordon:~$ The synclient response seems weird to me as I have done an apt install of xserver-xorg-input-synaptics which didn't appear to change anything. Here's the dpkg output: mich@gordon:~$ dpkg -l *synaptics*
Desired=Unknown/Install/Remove/Purge/Hold
| Status=Not/Inst/Conf-files/Unpacked/halF-conf/Half-inst/trig-aWait/Trig-pend
|/ Err?=(none)/Reinst-required (Status,Err: uppercase=bad)
||/ Name Version Architecture Description
+++-=======================-================-================-===================================================
un xorg-driver-synaptics <none> <none> (no description available)
ii xserver-xorg-input-syna 1.9.0-1ubuntu1 amd64 Synaptics TouchPad driver for X.Org server
mich@gordon:~$ Trying to sudo apt install xorg-drivers-synaptics bounces to the driver that's already installed. I've tried all 8 variations of these three kernel params: i8042.reset i8042.kbdreset=1 i8042.nomux=1 I've checked BIOS for anything that might be disabling the touchpad. The only thing that I found was a setting for the "internal pointer", if I remember correctly. Whatever it was called, it's enabled. Here's the input device: mich@gordon:~$ cat /proc/bus/input/devices
....
I: Bus=0003 Vendor=0b05 Product=1869 Version=0110
N: Name="ITE Tech. Inc. ITE Device(8910)"
P: Phys=usb-0000:00:14.0-8/input0
S: Sysfs=/devices/pci0000:00/0000:00:14.0/usb1/1-8/1-8:1.0/0003:0B05:1869.0001/input/input9
U: Uniq=
H: Handlers=sysrq kbd event7 leds
B: PROP=0
B: EV=12001f
B: KEY=3007f 0 ffffffffffffffff ffffffffffffffff ffffffffffffffff ffffffffffffffff 130c130b17c007 ffbf7bfad941dfff febeffdfffefffff fffffffffffffffe
B: REL=40
B: ABS=ffffff0100000000
B: MSC=10
B: LED=1f
....
I've also tried enabling the device via xinput to no avail:
xinput set-prop 14 "Device Enabled" 1
I saw a suggestion to modify /usr/share/X11/xorg.conf.d/50-synaptics.conf, but I do not have that file. I do have a /usr/share/X11/xorg.conf.d/51-synaptics-quirks.conf that is filled with things that appear to be completely irrelevant as they all point to /dev/input/event*. I found a /usr/share/X11/xorg.conf.d/70-synaptics.conf. I modified that file for the touchpad catchall to look like this:
Section "InputClass"
Driver "synaptics"
# This option is recommend on all Linux systems using evdev, but cannot be
# enabled by default. See the following link for details:
# http://who-t.blogspot.com/2010/11/how-to-ignore-configuration-errors.html
Option "TapButton1" "1"
MatchDevicePath "/dev/input/event*"
EndSection
My changes were the TapButton1 Option line and adding two spaces to the MatchDevicePath to match the tabbing for the rest of the config.
In case it's of any help, here are the rest of the files in that directory:
mich@gordon:~$ls -l /usr/share/X11/xorg.conf.d/ total 36 -rw-r--r-- 1 root root 92 Mar 20 05:02 10-amdgpu.conf -rw-r--r-- 1 root root 206 Apr 18 10:01 10-nvidia.conf -rw-r--r-- 1 root root 1350 Apr 13 08:31 10-quirks.conf -rw-r--r-- 1 root root 92 Mar 20 05:17 10-radeon.conf -rw-r--r-- 1 root root 329 May 21 00:33 11-nvidia-prime.conf -rw-r--r-- 1 root root 945 Apr 11 00:50 40-libinput.conf -rw-r--r-- 1 root root 590 Mar 7 2017 51-synaptics-quirks.conf -rw-r--r-- 1 root root 1785 May 21 00:32 70-synaptics.conf -rw-r--r-- 1 root root 3025 Apr 3 00:39 70-wacom.conf mich@gordon:~$
I've also run this command which did not enable the touchpad:
sudo modprobe -r psmouse && sudo modprobe psmouse proto=imps
This is probably stating the obvious just a few weeks after release, but here's the kernel:
mich@gordon:~$uname -r 4.15.0-20-generic mich@gordon:~$
Did I do something wrong? Am I missing the magic switch? Is there anything else that I can try? Is there any more information that could help troubleshoot this to find a solution? I've tried to exhaust every reasonable solution that I could find. Thanks!
I've been taking blind stabs at more solutions, but haven't made any progress. I removed the synaptics apt package, but getting that out of the way hasn't made any difference. Here's a little more information that might be helpful:
mich@gordon:~$xinput --list-props "pointer:ITE Tech. Inc. ITE Device(8910)" Device 'ITE Tech. Inc. ITE Device(8910)': Device Enabled (174): 1 Coordinate Transformation Matrix (176): 1.000000, 0.000000, 0.000000, 0.000000, 1.000000, 0.000000, 0.000000, 0.000000, 1.000000 libinput Natural Scrolling Enabled (308): 0 libinput Natural Scrolling Enabled Default (309): 0 libinput Middle Emulation Enabled (310): 0 libinput Middle Emulation Enabled Default (311): 0 libinput Left Handed Enabled (312): 0 libinput Left Handed Enabled Default (313): 0 libinput Send Events Modes Available (293): 1, 0 libinput Send Events Mode Enabled (294): 0, 0 libinput Send Events Mode Enabled Default (295): 0, 0 Device Node (296): "/dev/input/event7" Device Product ID (297): 2821, 6249 libinput Drag Lock Buttons (314): <no items> libinput Horizontal Scroll Enabled (315): 1 mich@gordon:~$
Here's some xorg log. It's probably telling me something that I'm missing:
mich@gordon:~$cat /var/log/Xorg.0.log ... cropped for clarity ... [ 7.318] (II) config/udev: Adding input device ITE Tech. Inc. ITE Device(8910) (/dev/input/event7) [ 7.318] (**) ITE Tech. Inc. ITE Device(8910): Applying InputClass "libinput pointer catchall" [ 7.318] (**) ITE Tech. Inc. ITE Device(8910): Applying InputClass "libinput keyboard catchall" [ 7.318] (II) Using input driver 'libinput' for 'ITE Tech. Inc. ITE Device(8910)' [ 7.318] (II) systemd-logind: got fd for /dev/input/event7 13:71 fd 51 paused 0 [ 7.318] (**) ITE Tech. Inc. ITE Device(8910): always reports core events [ 7.318] (**) Option "Device" "/dev/input/event7" [ 7.318] (**) Option "_source" "server/udev" [ 7.319] (II) event7 - ITE Tech. Inc. ITE Device(8910): is tagged by udev as: Keyboard Mouse Joystick [ 7.319] (II) event7 - ITE Tech. Inc. ITE Device(8910): device is a pointer [ 7.319] (II) event7 - ITE Tech. Inc. ITE Device(8910): device is a keyboard [ 7.319] (II) event7 - ITE Tech. Inc. ITE Device(8910): device removed [ 7.319] (II) libinput: ITE Tech. Inc. ITE Device(8910): needs a virtual subdevice [ 7.319] (**) Option "config_info" "udev:/sys/devices/pci0000:00/0000:00:14.0/usb1/1-8/1-8:1.0/0003:0B05:1869.0004/input/input9/event7" [ 7.319] (II) XINPUT: Adding extended input device "ITE Tech. Inc. ITE Device(8910)" (type: MOUSE, id 14) [ 7.319] (**) Option "AccelerationScheme" "none" [ 7.319] (**) ITE Tech. Inc. ITE Device(8910): (accel) selected scheme none/0 [ 7.319] (**) ITE Tech. Inc. ITE Device(8910): (accel) acceleration factor: 2.000 [ 7.319] (**) ITE Tech. Inc. ITE Device(8910): (accel) acceleration threshold: 4 [ 7.319] (II) event7 - ITE Tech. Inc. ITE Device(8910): is tagged by udev as: Keyboard Mouse Joystick [ 7.319] (II) event7 - ITE Tech. Inc. ITE Device(8910): device is a pointer [ 7.319] (II) event7 - ITE Tech. Inc. ITE Device(8910): device is a keyboard ... later ... [ 7.327] (**) ITE Tech. Inc. ITE Device(8910): Applying InputClass "libinput pointer catchall" [ 7.327] (**) ITE Tech. Inc. ITE Device(8910): Applying InputClass "libinput keyboard catchall" [ 7.327] (II) Using input driver 'libinput' for 'ITE Tech. Inc. ITE Device(8910)' [ 7.327] (II) systemd-logind: returning pre-existing fd for /dev/input/event7 13:71 [ 7.327] (**) ITE Tech. Inc. ITE Device(8910): always reports core events [ 7.327] (**) Option "Device" "/dev/input/event7" [ 7.327] (**) Option "_source" "_driver/libinput" [ 7.327] (II) libinput: ITE Tech. Inc. ITE Device(8910): is a virtual subdevice [ 7.327] (**) Option "config_info" "udev:/sys/devices/pci0000:00/0000:00:14.0/usb1/1-8/1-8:1.0/0003:0B05:1869.0004/input/input9/event7" ... end of relevant section ... After removing the synaptics package, I just had the default /usr/share/X11/xorg.conf.d/40-libinput.conf file. This didn't work, so I added a slightly modified /usr/share/X11/xorg.conf.d/50-elantech-touchpad.conf that i found here. Here it is with my modifications: mich@gordon:~$ cat /usr/share/X11/xorg.conf.d/50-elantech-touchpad.conf
Section "InputClass"
MatchProduct "8910"
MatchDevicePath "/dev/input/event*"
Driver "libinput"
Option "Tapping" "on"
Option "TappingButtonMap" "lmr"
Option "DisableWhileTyping" "on"
Option "DisableWhileTyping" "on"
Option "NaturalScrolling" "on"
Option "NaturalScrolling" "twofinger"
Option "TappingDrag" "on"
Option "TappingDragLock" "on"
Option "AccelSpeed" "0.1"
EndSection
mich@gordon:~$ I changed the MatchProduct to match the name in xinput and added the MatchDevicePath just to make sure it didn't go to the wrong place. Edit2: Added: Installing kernel v4.17rc6 without making other changes didn't fix the touchpad. #### Best Answer • This has been an ongoing problem and I have been working on it with the amazing Ubuntu community. You will need to update to at least kernel 4.17.2 from kernel.org. My first bug report: https://bugs.launchpad.net/ubuntu/+source/linux/+bug/1777679 My second bug report: https://bugs.launchpad.net/ubuntu/+source/linux/+bug/1778087 Some information which will help you get it to work: https://bugs.launchpad.net/ubuntu/+source/linux/+bug/1777679/comments/28 You might need an extra script for when the touchpad disconnects: #!/bin/bash if [ -z$1 ]
then
echo 'rsmod unloads and reloads kernel modules with modprobe'
echo 'usage: rsmod <kernelmodulename>'
echo 'Requires root privileges'
exit 1
fi
pkexec bash -c "modprobe -r $1; modprobe$1"
Save that as /usr/local/bin/rsmod and call it with hid-multitouch when the touchpad disconnects. (Unfortunately there is no work-around for this, if you want to use it on linux it will disconnect occasionally until the drivers get updated)
Edit: You have an ELAN1200 touchpad not Elantech, and it is currently not detected at all by your kernel. If it was, in xinput --list you would see ELAN1200 as well as ITE8910.
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Online Help for CXC CSEC Mathematics, Past Papers, Worksheets, Tutorials and Solutions Robert A. vectors in mathematics pdf download | 2020-01-26 13:02:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48540544509887695, "perplexity": 1407.5960421829914}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251688806.91/warc/CC-MAIN-20200126104828-20200126134828-00301.warc.gz"} |
https://electronics.stackexchange.com/questions/406768/what-are-the-initial-conditions-used-to-find-the-coefficients-in-the-current-equ | # What are the initial conditions used to find the coefficients in the current equations?
Suppose a RLC series circuit. R, L and C are in series with a battery and a switch. The switch is open. L and C are discharged.
At t=0, the switch is closed and the battery (V1) feeds the circuit.
I apply KVL to the circuit and find three equations:
CRITICALLY DAMPED
$$\ V_C(t) = (At + B) \thinspace e^{-\alpha t} \$$
OVERDAMPED
$$\ V_C(t) = Ae^{m_1t} + Be^{m_2t} \$$
UNDERDAMPED
$$\ V_C(t) = e^{- \alpha t}[K_1 \thinspace Cos(\omega_d t) + K_2 \thinspace Sin(\omega_d t) ] \$$
To find the coefficients of those equations I apply the two initial conditions:
1. I solve the equations for t=0 and for the initial voltage across the capacitor.
2. I take the derivative of the equation and solve for t=0.
Now lets talk about the current equations.
I never understood why but apparently the current equations are the same, or
CRITICALLY DAMPED
$$\ i(t) = (At + B) \thinspace e^{-\alpha t} \$$
OVERDAMPED
$$\ i(t) = Ae^{m_1t} + Be^{m_2t} \$$
UNDERDAMPED
$$\ i(t) = e^{- \alpha t}[K_1 \thinspace Cos(\omega_d t) + K_2 \thinspace Sin(\omega_d t) ] \$$
What are the two conditions I must use for the current equations to find the coefficients?
In the voltage equations I used the initial voltage across the capacitor and the derivative of voltage (current).
Now I have the current equations.
One condition must be to solve the equations for t=0, but what about the second condition? How do I find the coefficients of the current equations?
• @SpaceDog I've made a minor edit to my answer. You're correct that the current in the circuit is still zero at $t = 0^{+}$. However, current will start to flow, and eventually the inductor will look just like a wire, right? And correct, when the capacitor is fully charged, the current stops. But this time, the voltage across the capacitor is different than it was at $t = 0$. – Shamtam Nov 14 '18 at 17:24
The current through the inductor cannot change abruptly. So, iL(0-) = iL(0+). You can use inductor current at time instant 0 i.e. iL(0) as an initial condition.
For the second initial condition you can use the following formula diL(0)/dt = vL(0)/L. To find vL(0), you can apply KVL at your circuit for the time instant 0. | 2020-09-20 05:38:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 6, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7957531809806824, "perplexity": 377.00458361480486}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400193391.9/warc/CC-MAIN-20200920031425-20200920061425-00630.warc.gz"} |
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Calculate the gravitational potential at point P, include an appropriate unit for your answer. Description. It is measured in Nkg⁻¹ or ms⁻² which are equivalent. 5.4k SHARES. Gravitational potential = workmassHence, SI unit of gravitational potential = unit of workunit of mass = Jkg = J-kg-1 Book Store. The potential energy caused by the gravitational force. SI Unit: Joule (J) The potential energy which you will deal with the most in physics is gravitational potential energy. The unit ‘kgf’ or ‘kilogram force’ is a gravitational unit of force. NEET Physics : Units and Measurement. Gravitational potential may be expressed as; ψ g (mass) per unit mass = Eg/M = MgZ/M = gZ This is because the gravitational force is an extremely weak force as compared to other fundamental forces. It is denoted by V g. Gravitational potential, V g = W / m = – GM / r. Its SI unit is J / kg and it is a scalar quantity. 4. For the gravitational force the formula is P.E. Overview of SI Unit Of Gravitational Potential Energy Assume an object of mass ( m ) is lifted to a height ( h ) against the gravitational force. Simply, it is the gravitational potential energy possessed by a unit test mass ⇒ V = U/m ⇒ V = -GM/r ⇒ Important Points: The gravitational potential at a point is always negative, V is maximum at infinity. Which of the following is the SI unit of measurement for gravitational potential energy? Electric Potential Energy. It is defined as Gravitational potential energy per unit mass. Gravitational potential energy is the energy possessed or acquired by an object due to a change in its position when it is present in a gravitational field. 6 7 –1 [1] b. Type of energy associated with the motion of an object. 7. unit of gravitational potential is 4.3k LIKES. Gravitational Field Unit: SI unit is N/m. In the SI unit system of measurement, the potential energy is measured in Joules. The diagram shows the path of an asteroid as it moves past the planet. Physics. Its dimensional formula is … The SI units for energy and work JOULES 2. The dimensional formula is M 0 L 2 T-2. SI unit for power that is equivalent to Joules/second. Gravitational potential at any point in gravitational field is equal the work done per unit mass in bringing a very light body from infinity to that point. Assuming a point at a height Z above a reference level, the gravitational potential energy, Eg is: Eg = MgZ = ρ w gvZ . Sometimes given the symbol U, V or Φ. Subject. 6. Download books and chapters from book store. Jkg⁻¹ (small 'k' for kilo) is the SI unit for gravitational *potential* (energy per kilogram). Watts. 1:32 28.7k LIKES. Explain what is meant by gravitational potential energy and give examples of objects that possess it. (b) Approximately how much gravitational potential energy, in terms ofm, g, N, and y, is stored in the entire pile? Use the gravitational potential energy equation to solve problems. ‘1 kgf’ is the force exerted by the earth for a body with mass 1 kg. In SI units, the 2010 CODATA-recommended value of the gravitational constant (with standard uncertainty in parentheses) is: Define energy and name 5 forms of energy. Type of force that does not change the total mechanical energy of a system when it does work on it. is always positive. The S.I. KINETIC 4. is independent of the point of reference. where as Gravitational potential is a scalar. The SI unit of gravitational potential is J/Kg. 8. Gravitational Potential. Both gravitational and electric field strengths can be described by similar equations written in the form 5 € (a) €€€€Complete the following table by writing down the names of the corresponding quantities, together with their SI units, for the two types of field. Currently only available for. Type of energy associated with the motion of an object. Compare the result to the SI base units … GRAVITATIONAL POTENTIAL 5. WATTS 3. To calcualte Potential at any point a test mass m is assumed at that point. The newton (symbol: N) is the unit of force (any kind of force, not only the gravitational one) derived in the SI system; it is equal to the amount of force required to accelerate a mass of one kilogram at a rate of one metre per second per second. This can be derived using the mass of the object, the gravitational field strength, and height above a reference point (usually the vertical height from the ground). In the second column, write an SI unit for each quantity. 9.0k VIEWS. g = Acceleration due to gravity, cm s-2 . = mgh, where m is the mass in kilograms, g is the acceleration due to gravity (9.8 m / s 2 at the surface of the earth) and h is the height in meters. The GPE formula GPE = mgh shows that it depends on the mass of the object, the acceleration due to gravity and the height of the object. State the law of conversation of energy. SI unit for power that is equivalent to Joules/second. The gravitational constant is perhaps the most difficult physical constant to measure to high accuracy. Kinetic Energy POTENTIAL ENERGY: 5. Type of stored energy associated with the position of an object in a gravitational field. ... How is the SI unit energy defined. 5. The gravitational potential at a point in a gravitational field is the work done per unit mass that would have to be done by some externally applied force to bring a massive object to that point from some defined position of zero potential, usually infinity.It is the gravitational potential difference between the chosen point and the position of zero potential. Gravitational intensity definition is - a vector quantity related to the condition at any point under gravitational influence the measure of which is the gravitational force exerted upon a unit mass placed at the point in question. In the third column indicate whether the quantity is a scalar or a vector. Define power and state its SI unit. CBSE. Notice that gravitational potential energy has the same units as kinetic energy, kg m 2 / s 2. View All. a. Text Solution. Estimate the speed of the asteroid at point P … 3. Let’s consider the Gravitational field around a spherical object of mass M. The gravitational force experienced by the mass, m is given by F = G Mm/r^2. Gravitational Field Dimensional Formula: Dimensional Formula is … (a)€€€€ Complete the table of quantities related to fields. So the field strength at this point in space, g = Gravitational force per unit mass =F/m So, g = G M/r^2.This is the formula of Gravitational field strength.. Write down the formula of Gravitational field strength The constant of proportionality, G, is the gravitational constant. What is the SI unit of energy?-Joule, the unit of work or energy in the International System of Units (SI) 3. What are the two forms of mechanical energy? Simply, it is the gravitational potential energy possessed by a unit test mass ⇒ V = U/m ⇒ V = -GM/r. A. Watt B. Jouile C. Ampere D. Newton - 6894734 ρ w = Density of water, g cm-3 . QUESTION 24 The SI units of angular momentum, L, are kg.m/s2 kg.m/s3 kg.m2/s2 kg.m2/s QUESTION 25 The gravitational potential energy: is always negative. where, M = Mass of water, g . What is the gravitational potential energy of a two-particle system of a 5.3 kg particle and a 2.3 kg particle with a gravitational attraction of magnitude {eq}2.4\times10^{-12} N {/eq}? The amount of work done in moving a unit test mass from infinity into the gravitational influence of source mass is known as gravitational potential. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, without directly … can be positive or negative. Reference level. 4. Gravitational potential energy (GPE) is an important physical concept that describes the energy something possesses due to its position in a gravitational field. Mechanical energy is the energy due to the position of something or the movement of something. In simple terms, it can be said that gravitational potential energy is an energy which is related to gravitational force or to gravity. The object is lifted in vertical direction by an external force, so the force to lift the box and the force due to gravity, F g F_g F g are parallel. Answer : B Related Video. v = Volume of water, cm 3. 5.4k VIEWS. In SI units, the 2018 CODATA-recommended value of the gravitational constant (with standard uncertainty in parentheses) is: As the clock runs, the mass is lowered. gravitational potential energy = 360 J Question Calculate the energy transferred to the gravity store when a rocket of mass 4000 kg reaches 10 km up in the atmosphere. 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https://www.jiskha.com/archives/2014/03/22 | # Questions Asked onMarch 22, 2014
1. ## Chemistry
Complete and balance the equation for this reaction in basic solution? Redox reaction Cr is oxidized to CrO42– and Fe3 is reduced to Fe2...? For a particular redox reaction Cr is oxidized to CrO42– and Fe3 is reduced to Fe2 . Complete and balance the
2. ## Math
Suppose that Ramos contributes $6000/year into a traditional IRA earning interest at the rate of 2%/year compounded annually, every year after 35 until his retirement age at age 65. At the same time, his wife Vanessa deposits$4900/year into a Roth IRA
3. ## math
solve the equation 5coshx + 3sinhx = 4
4. ## Physics
Consider two diffraction gratings. One grating has 3000 lines per cm, and the other one has 6000 lines per cm. Both gratings are illuminated with a beam of the same monochromatic light. Which grating produces the greater dispersion? Both gratings produce
5. ## Urgent Chemistry
The Heisenberg Uncertainty Principle: A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of mm = 0.100fg\rm fg (where a femtogram, fg\rm fg, is 10−15g\rm 10^{-15}\; g) and is swimming at a velocity of vv =
6. ## Chemistry
an extremely soluable salt is added to water, and all of it dissolves. is it at equilibrium? explin.
7. ## chemistry
calculate the freezing point of an aqueous solution of 15.5 grams of glucose dissolved in 150 grams of water
8. ## physics
A 2400 kg car moving at 11 m/s crashes into a tree and stops in 0.26 s. calculate the force the seat belt exerts on passenger in the car to bring him to a halt the mass of the passenger is 72 kg.
9. ## college physics 2
A double-slit diffraction pattern is formed on a distant screen. If the separation between the slits decreases, what happens to the distance between interference fringes? Assume the angles involved remain small. A double-slit diffraction pattern is formed
10. ## physics
In a double-slit experiment, two beams of coherent light traveling different paths arrive on a screen some distance away. What is the path difference between the two waves corresponding to the third bright band out from the central bright band? The path
11. ## Calculus
A rancher wants to build a rectangular fence next to a river, using 100 yd of fencing. What dimensions of the rectangle will maximize the area? What is the maximum area? (Note that the rancher should not fence the side next to the river.)
12. ## BIOLOGY
Which of the following statements is NOT true about gamete formation in mammals? a)In the female, the separation of cytoplasm is unequal during meiosis. b)Only one of the four cells resulting from meiosis goes on to become an oocyte. c)Only one of the four
45. ## Science
a layer of ice has an area of 35m2 and is 2.5cm thick and is -7.5'c. how much water (in kilograms) is needed to melt the ice?
46. ## Accounting 205
help with paper on basic transaction processing fr Richard Simmons
47. ## science
an object being pulled to earth by gravity will have the force of air pushing back opposite directions
48. ## Biology
(This question ties into the previous question I posted about mRNA). Use the Universal triplet code to determine the sequence of amino acids that would be generated for each of the mRNA sequences that you generated in question 2. Remember that the reading
49. ## Biology
What 3' to 5' DNA sequence would correspond to a 5'-AUG-3' start codon?
50. ## Statisics
4. The purpose of the investigation was to examine the effects of the Tweed edgewise technique on mandibular growth and rotation. It employed 36 Class II Tweed edgewise patients, all of whom had 4 bicuspids extracted during the course of treatment.
51. ## Physic II
Why would you expect a plate to hold more excess charge on each of its conducting surfaces when the voltage difference between the two pieces of conductor increases?
52. ## focus high school
What percentage of all infertile American women are infertile because of tubal damage caused by untreated STD?
53. ## science
what is the substance that melts at 10°C?
54. ## Math
Solve the system using elimination. y = -x + 3 y = x^2 + 1
55. ## Math
Doublets? Change one word to another word one letter at a time. Create linking words. DINNER serve COFFEE. Has anybody got a clue how to do this? You have to end with the word coffee? Any help would me much appreciated. Thank You
56. ## Math
Thanks Steve, but can anyone else help? Start with the word dinner change one letter at a time to form new until you have the word coffee Thank soooooooo much to everybody.
find y' and y” by implicit differentiation. 2x^3 + 3y^3 = 8
58. ## english
What are some websites that are against animal abuse & the actions they have taken to stop it. Thank you!
59. ## Math
-7x + 9y = -29 8x + 5y = -28
60. ## Math
What is 66 2/3 of 39? | 2020-02-29 10:38:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31077131628990173, "perplexity": 3141.1468874528305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875148850.96/warc/CC-MAIN-20200229083813-20200229113813-00311.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcds.2008.21.353 | # American Institute of Mathematical Sciences
January 2008, 21(1): 353-366. doi: 10.3934/dcds.2008.21.353
## On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets
1 Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
Received December 2006 Revised August 2007 Published February 2008
Let $M^{N\times n}$ be the space of real $N\times n$ matrices. We construct non-negative quasiconvex functions $F:M^{N\times n}\to R_+$ of quadratic growth whose zero sets are the graphs $\Gamma_f$ of certain Lipschitz mappings $f:K\subset E\to$ $E^$⊥, where $E\subset M^{N\times n}$ is a linear subspace without rank-one matrices, $K$ a compact subset of $E$ with $E^$⊥ its orthogonal complement. We show that the gradients $DF:M^{N\times n}\to M^{N\times n}$ are strictly quasimonotone mappings and satisfy certain growth and coercivity conditions so that the variational integrals $u\to \int_{\Omega}F(Du(x))dx$ satisfy the Palais-Smale compactness condition in $W^{1,2}$. If $K$ is a smooth compact manifold of $E$ without boundary and the Lipschtiz mapping $f$ is of class $C^2$, then the closed $\epsilon$-neighbourhoods $(\Gamma_f)_\epsilon$ for small $\epsilon>0$ are quasiconvex sets.
Citation: Kewei Zhang. On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 353-366. doi: 10.3934/dcds.2008.21.353
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2019 Impact Factor: 1.338 | 2020-07-08 08:45:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6796029806137085, "perplexity": 5849.143415577905}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655896905.46/warc/CC-MAIN-20200708062424-20200708092424-00177.warc.gz"} |
http://crypto.stackexchange.com/questions/8845/finding-a-keylength-in-a-repeating-key-xor-cipher?answertab=oldest | # Finding a keylength in a repeating key XOR cipher
In an old cryptography FAQ, I found the following step described for determining a the length of the key a cipher was repeatedly XORed against:
1. Discover the length of the key by counting coincidences. (See Gaines [GAI44], Sinkov [SIN66].) Trying each displacement of the ciphertext against itself, count those bytes which are equal. If the two ciphertext portions have used the same key, something over 6% of the bytes will be equal. If they have used different keys, then less than 0.4% will be equal (assuming random 8-bit bytes of key covering normal ASCII text). The smallest displacement which indicates an equal key is the length of the repeated key.
The wording of the bolded text is the part I'm confused with. Given that my plaintext equals the string "This is a secret message", would I be comparing the first n number of bytes with a second set of n number of bytes that is offset from the first (e.g. the xor'd version of "Th" with "is", if I was comparing 2-byte chunks of the cipher against itself)?
If not, what's the correct way to interpret this passage? Thank you.
-
of related interest: reverseengineering.stackexchange.com/a/2063/1854 – mikeazo Jun 25 '13 at 0:35
This attack only works if your ciphertext is relatively long compared to the key length, and if the plaintext is some human-language text (with words occuring multiple times). – Paŭlo Ebermann Jun 25 '13 at 15:19
@Paŭlo: Actually, you don't need human-language text; the attack works as long as the plaintext character distribution is sufficiently uneven (and the message sufficiently long) that you can statistically distinguish different key characters by their corresponding ciphertext character distributions. – Ilmari Karonen Jun 25 '13 at 19:17
This looks like a sliding window approach to calculating the index of coincidence. So you would have something like:
ABCDE FGHIJ KLMNO
OACBD EFGHI JKLMN
Given enough cyphertext, you'll discover a length at which the IC is high; this is a candidate keylength for the cyphertext, because you've shifted the two texts by one keylength. Multiples of this size will also return high ICs.
Another way to brute force the key length of a Vigenère cypher by iterating through keysizes of different lengths, computing the Hamming distance of adjacent cyphertext blocks of that length, normalizing the distance to the keylength, and taking the smallest value as your keylength. In this case, given cyphertext:
CYPHERCYPHERCYPHERCYPHER
for a guess of keysize 4 you'd split the text into
CYPH
ERCY
PHER
CYPH
ERCY
PHER
and get the Hamming distances that way. You'd keep going for a few more keysizes and should eventually 'settle' on 6 (in this highly contrived example, at least).
-
Thanks for the explanation. Your last paragraph nails what I'm trying to figure out with this post: to break this type of cipher, I'd need to compare (via hamming distance) adjacent N-length blocks of the ciphertext? So the hypothetical example in my post shows the right idea with how to compare the cipher against itself, and I've understood what the passage meant to convey? – hlh Jun 25 '13 at 13:05
well, you wouldn't be comparing plaintext blocks, but the cyphertext ones; also, as far as the particular chunk of text you posted, my note on the index of coincidence, and @D.W.'s answer, are the appropriate interpretations. I've provided a different way of breaking the cypher for educational purposes =) – xorbyte Jun 25 '13 at 18:51
I corrected that! I upvoted both D.W.'s answer and your answer because they explained the text I was looking at, but I'm marking yours as 'correct' because your bit about comparing adjacent blocks of the cyphertext is what helped me solve the problem I was working on. Thanks to all! – hlh Jun 26 '13 at 13:26
You'd be trying each possible displacement (offset).
Suppose the ciphertext is CXEKCWCOZKUCAYZEKW. Here's displacement 1:
CXEKCWCOZKUCAYZEKW
CXEKCWCOZKUCAYZEKW
At displacement 1, there are no matches (nothing where the a letter in the top line is equal to the letter immediately below it).
Here's displacement 2:
CXEKCWCOZKUCAYZEKW
CXEKCWCOZKUCAYZEKW
^
You can see that, at displacement 2, there is one match.
Here's displacement 3:
CXEKCWCOZKUCAYZEKW
CXEKCWCOZKUCAYZEKW
At displacement 3, there are no matches.
In this way, you can count the number of matches at each displacement.
The idea is that, if you line up the ciphertext with itself displaced by $k$, where $k$ is the period of the keystream (i.e., the length of the key), then you get a match in the ciphertext (offset by $k$ places) if and only if there is a match in the plaintext (offset by $k$ places). Now it's a property of English language that the frequency distribution of English letters is not uniform: some are more likely than others. If you pick two positions at random from an English text, there's about a 6% chance (say) that those two positions have the same letter. Consequently, when you've guessed $k$ correctly, there's about a 6% chance that any particular ciphertext letter matches the one $k$ positions later.
In contrast, when you line up the ciphertext with itself displaced by something displacement that does not match the key length, the chances of a match at any particular position are much smaller (1/256, if ciphertexts are bytes; 1/26, if ciphertexts are letters).
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https://bioriental.net/%EF%BB%BFsupplementary-materialssupplementary-information-dmm-11-031146-s1/ | # Supplementary MaterialsSupplementary information dmm-11-031146-s1
Supplementary MaterialsSupplementary information dmm-11-031146-s1. invasion and epithelial-to-mesenchymal changeover (EMT) in the imaginal disc. Further evidence comes from cell culture studies, in which we expressed Hipk in human breast cancer cells and showed that it enhances proliferation and migration. Past studies show that Hipk can promote the actions of conserved pathways implicated in EMT and tumor, such as for example Wnt/Wingless, Hippo, JNK and Notch. We present that Hipk phenotypes aren’t likely to occur from activation VX-809 (Lumacaftor) of an individual focus on, but through a cumulative VX-809 (Lumacaftor) influence on numerous focus on pathways rather. Most tumor versions involve mutations in multiple genes, like the well-known RasV12 model, where EMT and invasiveness take place after the extra lack of the tumor suppressor gene Our research reveals that raised degrees of Hipk independently can promote both hyperproliferation and intrusive cell behavior, recommending that Hipk family could possibly be potent motorists and oncogenes of EMT. a fantastic program for the analysis of metastasis VX-809 (Lumacaftor) and tumorigenesis. Many signaling pathways have already been implicated in the introduction of tissues overgrowth and/or metastatic behavior in the journey. Nearly all these studies have got described tumor versions that want the mix of multiple hereditary aberrations to be able to express hyperproliferation in conjunction with intrusive behaviors. The initial metastasis model included activated Ras coupled with lack of the tumor suppressor (Pagliarini and Xu, 2003). Notch pathway activation in conjunction with modifications in histone epigenetic marks also resulted in a tumor model (Ferres-Marco et al., 2006). Following studies have determined further factors involved with both Ras- and VX-809 (Lumacaftor) Notch-driven tumorigenesis (Doggett et al., 2015). Various other tumor research involve Epidermal development aspect receptor (Egfr) signaling (Herranz et al., VX-809 (Lumacaftor) 2012) as well as the Sin3A histone deacetylase (HDAC) (Das et al., 2013). The Hippo pathway is certainly a powerful tumor suppressor pathway that’s needed is to avoid hematopoietic disorders (Milton et al., 2014). Activated JAK/STAT signaling causes leukemia-like hematopoiesis flaws in (Harrison et al., 1995; Luo et al., 1997). Homeodomain-interacting proteins kinases (Hipk) are evolutionarily conserved, and vertebrates have Hipk1-Hipk4, whereas and also have only 1 Hipk each. Hipk family are portrayed in powerful spatial and temporal patterns, highlighting their essential roles during advancement (evaluated by Blaquiere and Verheyen, 2017). Hipk proteins levels are extremely governed by post-translational adjustment and proteasomal degradation (Saul and Schmitz, 2013). Hipk family are reported to possess specific and contradictory results on cell proliferation and tissues development. Overexpressing Hipk causes tissue overgrowths in the wing, vision and legs in a dose-dependent manner (Chen and Verheyen, 2012; Lee et al., 2009a; Poon et al., 2012). In reduces the number of proliferating cells and size of the mitotic region (Berber et al., 2013). and vertebrate Hipks can modulate Wnt signaling in many ways (Hikasa and Sokol, 2011; Hikasa et al., 2010; Kuwahara et al., 2014; Lee et al., 2009b; Louie et al., 2009; Shimizu et al., 2014; Swarup and Verheyen, 2011; Wu et al., 2012). Hipk proteins modulate the Hippo pathway in loss of function can suppress the effects of constitutively active Yki (YkiS168A). Hipks have also been shown to regulate Jun N-terminal kinase (JNK) signaling in numerous contexts (Hofmann et al., 2003, 2005; Huang et al., 2011; Lan et al., 2007, 2012; Rochat-Steiner et al., 2000; Song and Lee, 2003; Chen and Verheyen, 2012). Hipk is required for the full effect of JAK/STAT signaling, because loss of through somatic clonal analysis causes loss of Stat92E-GFP reporter and, furthermore, loss of can suppress lethality and tumor frequency in the constitutively Rabbit Polyclonal to CKLF3 active allele (Blaquiere et al., 2016 preprint). Hipk2 is the best-characterized vertebrate Hipk family member. Studies in cell culture and cancer samples reveal conflicting results (Blaquiere and Verheyen, 2017). For example, Hipk2 acts as a tumor suppressor in the context of p53-mediated cell death after lethal DNA damage (Hofmann et al., 2013), and reduced expression of Hipk proteins is seen in several malignancy types (Lavra et al., 2011; Pierantoni et al., 2002; Ricci et al., 2013; Tan et al., 2014). By contrast, Hipk2 is usually elevated. | 2022-08-10 19:22:32 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8728806376457214, "perplexity": 12307.807430906963}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571210.98/warc/CC-MAIN-20220810191850-20220810221850-00529.warc.gz"} |
http://technomediaenterprise.com/luahu/parts-and-function-of-differential-system-46e817 | Four wheels drive layout – differential is located at the front as well as rear. To differentiate the speed of two rear wheels when the vehicle makes a turn, i.e. Given the following system of differential equations (assuming ##y \neq 0##) \begin{equation*} A gear is a tooth element,These tooths are mounted on a wheel which rotates and transmits motion. 7. A limited-slip differential increases the safety of a vehicle because it increases the control a driver has over the vehicle. The pinion gears, side gears, and other parts are similar to a standard differential. The relationship between these functions is described by equations that contain the functions themselves and their derivatives. Spider gears: The spider gear lies at the heart of the differential and special mention should be made about its rotation. If What you looking for is not here, you may send a request to modify articles, Notes, and Content modification, Feedback, Suggestions here (Team will reply as early as Possible ): Contact Form or Using Whatsapp, © 2021 Copyright Learn Mechanical Engineering, When a vehicle travels in a straight line, the two rear wheels turn on the road exactly at the same speed and there is no relative movement. The torque transmitted to each rear wheel is equal in this case, although their speed is different. We carry an extensive inventory of new differential parts and components for all makes and models, from light-, medium- and heavy-duty vehicles to farm, construction and material movement vehicles, and more. The outward pressure of the side gears presses the friction discs and steel plates together between side gears and case. Components of Differential: The components of Differential are as follows: Ring Gear or Crown Wheel; Planet Pinion; Drive Shaft; Bevel Pinion; Half Shafts; Sun Gears – 2 No’s; Ring Cage; Cross Pin; Let me give you the overview of these in detail. The inner wheel makes a larger angle than. A standard differential, also referred to as an “Open” differential, is the OEM standard for many vehicles. Introduction to Seals : Sachin is a B-TECH graduate in Mechanical Engineering from a reputed Engineering college. When the vehicle takes a turn, the inner wheel experiences resistance and tends to rotate in the opposite direction. To differentiate the speed of road wheels while taking a turn. Thank you For Visiting. Differential gear is representative of the automotive components that incorporate such bevel gears. 3 side gears. If What you looking for is not here, you may send a request to modify articles, Notes, and Content modification, Feedback, Suggestions here (Team will reply as early as Possible ). The derivatives re… Sachin is a B-TECH graduate in Mechanical Engineering from a reputed Engineering college. The function of differential: 1. 4) Both driving wheels can rotate in the opposite direction at a different speed. Make a variational formulation of the system u 1 = u 2, u 2 = fwith u 1 = 0 and u 2 = 0 on the boundary. The Jeep Wrangler’s off-roading model, the Rubicon, features Tru-Lok front and rear electronic locking differentials that can be initiated with the flip of a switch. Problem 2. 1. Then, the pinion shaft meshes with the two differential side gears connected to the inner ends of the axle shafts. A differential is a device, usually but notnecessarily employing gears, capable oftransmitting torque and rotation throughthree shafts, almost always used in one oftwo ways. The system is called homogeneous if all fj = 0, otherwise it is called non-homogeneous. The two most common types of limited-slip differential are as follows. Axle – The shaft and connecting parts that transmits torque from … Besides these, other parts of a manual transmission include gaskets, snap rings, bushing kits … The rear wheels and the differential case tum at the same speed. The function of differential gears: 1. The pinion gears, side gears, and other parts are similar to a standard differential. Function: Transmission to the differential. Differential serves this purpose. Let u and v be functions of the variable x. This is the situation when the vehicle is travelling along a straight line. Matrix Notation for Systems. As part of the front and/or rear axle assembly, the differential plays an integral role in how your car makes turns. We have stock replacement and performance products, including ring-and-pinion gear sets, bearing and overhaul kits, spider gear kits and more. To differentiate the speed of road wheels while taking a turn. Functions and Roles. The clutch friction discs are made of steel covered with friction material. The assembly consists of differential, rear axles, wheels, and bearings. Your email address will not be published. y is the response of the measurement system component, x is the input to the measurement system component, n is the order of the system. This function provides proportional RPMs between the left and right wheels. BRAKE SYSTEM. Differential gear Box – Diagram, parts, Types, Working, Advantages, Multi Plate Clutch : Parts , types , Working, Advantages, Cone Clutch -Parts , Working, Advantages , Disadvantages, Applications, Epicyclic Gearbox – Diagram , Working , Advantages of Epicyclic gearbox, Automobile Clutch | Function and Major Types Of Clutches, Single Plate Clutch – Parts, Diagram, Working, Advantages, Diaphragm clutch | Diagram , Working , Advantages, Automatic Differential Unit Locking System Report Download. The differential has three jobs: To aim the engine power at the wheels To act as the final gear reduction in the vehicle, slowing the rotational speed of the transmission one final time before it hits the wheels Both clutch-plate and cone differentials require a special limited-slip gear oil. You can buy used Differential which is a part of the front and rear axle assembly. Whenever the discs and plates are pressed together, the splined and dogged connections ensure the side gear: and differential cases are locked together. It allows the differential to operate similarly to a standard differential when making turns. Introduction to Differential Equation Solving with DSolve The Mathematica function DSolve finds symbolic solutions to differential equations. Also, the ends of planet shafts are left loose in notches provided on the differential cage. In this case, we speak of systems of differential equations. Differential – The device, usually in the axle housing, that allows the two wheels on an axle to rotate at different speeds. Given are the functions aij(t) and fj(t) on some interval a < t < b. Differential is a mechanical system which consists of three shafts that rotates on the speed with average to that of the other. While it might seem to be a somewhat cumbersome method at times, it is a very powerful tool that enables us to readily deal with linear differential equations with discontinuous forcing functions. The rear wheels and the differential case tum at the same speed. Front engine rear-wheel-drive layout – it is located at the rear in between two half shafts. So, it makes two clutch members move away from each other. The standard differential sends almost all engine power to the slipping wheel. The differential system allows the two wheels of the rear axle (or any powered axle) to rotate with respect to each other while transferring the power from the propeller shaft to both of them. In Front-engine front-wheel-drive layout – differential is located at the front next to gearbox. 1. Renal system, in humans, organ system that includes the kidneys, where urine is produced, and the ureters, bladder, and urethra for the passage, storage, and voiding of urine. The derivatives re… Limited Slip Differentials are also used on handy on icy or dirt roads. Introduction to Slotting Machine : The power produced inside the engine cylinder is ultimately aimed to turn the wheels so that the vehicle can move on the road. Since, the pinion and side gears are beveled gears, their teeth try to come out of engagement when the differential is transmitting engine torque. P in = P out1 + P out2, or P in = (T 1 x w 1) + (T 2 x w 2), where P in is the power input from the transmission to the differential, and P out is the power output from the differential to the wheels. The ring gear is bolted to a flange on the differential case. Matrix Notation for Systems. In Front-engine front-wheel-drive layout – differential is located at the front next to gearbox. The machine operates... LearnMech.Com is a Mechanical Project-oriented platform run by Sachin Thorat who is a B-Tech Graduate in Mechanical Engineering. T is … Figure. I. Clutch-plate differential 2. :) Note: Make sure to read this carefully! 4 ring gear. Therefore the outer road wheel runs faster than the inner road wheel and covers a more distance. Hello, There is a function that can solve SYMBOLICALLY a differential equation and a system of differential equations automatically in Mathcad? We will restrict ourselves to systems of two linear differential equations for the purposes of the discussion but many of the techniques will extend to larger systems of linear differential equations. The system is called homogeneous if all fj = 0, otherwise it is called non-homogeneous. So, the drive pinion is rotated when the drive shaft turns. there is relative movement between two rear wheels. Fortunately, Inland has the differential specialists and experience to handle any differential repair need you may have. shows the basic parts of the type of differential used in rear-wheel-drive cars. The type of part-time system typically found on four-wheel-drive pickups and older SUVs works like this: The vehicle is usually rear-wheel drive. Brake – The device, usually in the axle housing, that stops the motion of the tractor. Select the option that gives the particular integral. Therefore, the side gear sends power to the wheel with the most traction. But when the vehicle takes a turn the outer wheel travels a longer radius than the inner wheel i.e. An epicyclic differential can use epicyclic gearing to split and apportion torque asymmetrically between the front and rear axles. 2. Differential (mechanical device) 3. Now, if we start with $$n = 1$$then the system reduces to a fairly simple linear (or separable ) first order differential equation. 2. The operation is similar to the clutch-plate differential. The purpose of the compressor in the refrigeration cycle is to accept the low-pressure dry gas from the evaporator and raise its pressure to that of the condenser. Small Parts, Big Impact Manual transmissions are essentially made up of gears and shafts. The differential has three jobs: To aim the engine power at the wheels To act as the final gear reduction in the vehicle, slowing the rotational speed of the transmission one final time before it hits the wheels 5. The differential of the sum (difference) of two functions is equal to the sum (difference) of their differentials: d(u±v)=du±dv. Currently, he is working in the sheet metal industry as a designer. When the vehicle is moving straight ahead, the clutch-plate differential operates similarly to a standard differential. Required fields are marked *. Pinion shaft or Cross pin (or) spider. Differential & Axle Parts Specialists We have your differential parts in stock ready to ship today. Actual Practical Differential Gearbox Working Video : 1) Both driving wheels can rotate in the same direction at the same speed. Enjoy! To transmit the power from the propeller shaft at the right angle to the axle shafts for moving the wheel. The differential case is mounted with two-wheel axles and differential side gears. At t… Summary:: We want to find explicit functions ##g(y,t)## and ##f(y,t)## satisfying the following system of differential equations. If What you looking for is not here, you may send a request to modify articles, Notes, and Content modification, Feedback, Suggestions here (Team will reply as early as Possible ). y is the response of the measurement system component, x is the input to the measurement system component, n is the order of the system. A differential is a device, usually but not necessarily employing gears, capable of transmitting torque and rotation through three shafts, almost always used in one of two ways: in one way, it receives one input and provides two outputs--this is found in most automobiles--and in the other way, it combines two inputs to create an output that is the sum, difference, or average, of the inputs. Your email address will not be published. (The Mathe- matica function NDSolve, on the other hand, is a general numerical differential equation solver.) However, on very slippery surfaces such as icy or muddy roads, a lack of driving force, called traction force, can cause rear wheels to slip because the standard differential will drive the wheel with the least traction. thus the vehicle negotiates the turn safely. The outer wheel turns faster and covers a larger distance than the inner wheel. Since oil is a liquid it has the tendency to 'leak' through every gas/slot it finds during movement. Oil under pressure is moving in every hydraulic circuit. SUSPENSION SYSTEM When the vehicle is making a tum, a high torque caused by the outer wheel rotating faster than the case and it causes the clutch pack to slip. Solving ODEs by using the Complementary Function and Particular Integral An ordinary differential equation (ODE)1 is an equation that relates a summation of a function () and its derivatives. The main function of the transmission is similar to the chain on a bicycle: it keeps the engine turning in time with the wheels, regardless of what gear the vehicle is in. 2. The wheel lying on soft ground spins or rotates around its own axis due to differential action, while the wheel on the solid ground is not driven and remains stationary. Preload spring and side gear pressures force the cone into a dished depression in the differential case. Many measurement system components can be modeled by either first or second order differential equations. 4. Engine – Crankshaft – flywheel – clutch – transmission box – differential – final drives – axle – drive wheels. The differential case has bearings that rotate two axle shafts. West Coast Differentials stocks a complete line of light duty axle parts for Chevrolet, Chrysler, Dana, Ford, GM, Jeep and Toyota and more! The differential gear is a part of the power transmission device. In one way, it receives one inputand provides two outputs; this is … 2. This leakage of... Slotter Machine - Types, Parts, Operations, Diagram, Specification. The function of differential gears: 1. Read more about this portal or Sachin Thorat click on below button! Now, when it encounters a turn, the differential comes into action. Changes the direction of axis of rotation of the power by 90o i.e. Finding the Differential of a Function There are many different types of functions in various formats, therefore we need to have some general tools to differentiate a function based on what it is. If the resistance at both wheels is equal, the planet gear (green) does not rotate, and … The discs and plates slide against each other discs turn with side gears, plates turning with case allowing different rotating speeds between the case and side gears. 3. We carry an extensive inventory of new differential parts and components for all makes and models, from light-, medium- and heavy-duty vehicles to farm, construction and material movement vehicles, and more. Read more about this portal or Sachin Thorat click on below button! Order before 4PM and most parts ship out the SAME DAY! But when vehicle takes a turn the outer wheel travels on a longer radius than the inner wheel. It consists one input, drive shaft and 2 outputs (drive wheels), but the rotation of the drive wheels is linked to the road. In general, wetted parts (such as diaphragms) that are in contact with the fluid provide movement or force that is related to the differential pressure across the transmitter pressure ports. Axle – The shaft and connecting parts that transmits torque from the differential or final gear reduction, to the wheels. It allows the differential to operate similarly to a standard differential when making turns. The unknowns are the functions x1(t), ..., xn(t). Differential (mechanical device) 1. P in = P out1 + P out2, or P in = (T 1 x w 1) + (T 2 x w 2), where P in is the power input from the transmission to the differential, and P out is the power output from the differential to the wheels. Small Parts, Big Impact Manual transmissions are essentially made up of gears and shafts. On the inner ends of each axle a smaller bevel gear called differential side gear is mounted. 1. Differential (mechanical device) A cutaway view of an automotive final drive unit which contains the differential Input torque is applied to the ring gear (blue), which turns the entire carrier (blue), providing torque to both side gears (red and yellow), which in turn may drive the left and right wheels. 3. The differential of a constant is zero: d(C)=0. Whenever the discs and plates are pressed together, the splined and dogged connections ensure the side gear: and differential cases are locked together. The outward pressure of the side gears presses the friction discs and steel plates together between side gears and case. Therefore, rear wheels rotate at different speeds. The unknowns are the functions x1(t), ..., xn(t). It helps the outer drive wheel to rotate faster than the inner wheels. Differential serves this purpose. Two bevel gears are put together to mesh both driving and driven shafts at an angle of 90°. 3. The clutch-plate differential uses several friction discs which are similar to small manual clutch discs. Locked Differential: The locked or locking differential is a variant found on some vehicles, primarily … The clutch packs are applied but they are not needed. The quality of the differential pressure flow transmitter signal can be described by its performance specifica… The outer, wheel turns faster and covers a larger distance than the inner wheel. The main function of the brake system is to decelerate or decrease the speed of a vehicle. 2. West Coast Differentials stocks a complete line of light duty axle parts for Chevrolet, Chrysler, Dana, Ford, GM, Jeep and Toyota and more! Systems of differential equations Handout Peyam Tabrizian Friday, November 18th, 2011 This handout is meant to give you a couple more example of all the techniques discussed in chapter 9, to counterbalance all the dry theory and complicated ap-plications in the differential equations book! Front engine rear-wheel-drive layout – it is located at the rear in between two half shafts. In this document we consider a method for solving second order ordinary differential equations of the form 2 … Non-slip or self-locking type differential overcomes this drawback. The function of the differential is to permit the relative movement between inner and outer wheels when vehicle negotiates (takes) a turn. What is the function of differential gears? Differential case assembly: Hold the gear and drive the axel. Or cones to slip and vibrate during turns and rear axle Impact Manual transmissions are essentially up. Measurement system components can be modeled by either first or second order equations... Depression in the differential plays an integral role in how your car makes turns initial conditions don t... X1 ( t ) carried to the axle shafts an important component of the axle shafts moving. \Neq 0 # # y \neq 0 # # ) \begin { equation * system... Slip differentials are also used on handy on icy or dirt roads by additional gear include,. Differential are as follows driver has over the vehicle in motion, stop the vehicle can move on the ground. The renal system in this browser for the next time I comment pushing action the... Whole torque is then applied to the road wheels through-axle half shafts ),..., xn t..., radius than the inner road wheel and covers a larger distance than the wheel. Discs which are similar to a standard differential when making turns opposite direction to an. Friction discs are made of steel covered with friction material, wheel faster! Brake – the device parts and function of differential system which will divide the input torque of the power transmission device rear-wheel.., side gears and case drive pinion is assembled with the drive pinion rotated... Is similar to small Manual clutch discs differential parts in stock ready ship... Most traction every gas/slot it finds during movement differentiate the speed of road wheels while taking a.... B-Tech graduate in Mechanical Engineering it will cause the discs and plates are applied by the Mechanical pressure the! Lock with and turn differential to operate similarly to a shaper: oil pressure... And drive the axel e-lockers send an electrical current that activates the system... A Mechanical system which consists of three shafts that rotates on the side and. And sand, while the other is on soft mud are having resistance. Makes turns Thorat click on below button he is working in the metal..., radius than the inner wheel parts and function of differential system a larger distance than the inner wheel i.e by sending the DAY... Carrier, is the next version of the power by 90o i.e direction! Order to simplify things, let us assume that the vehicle makes a turn a B-TECH graduate in Engineering. 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In rear engine rear-wheel-drive layout – it is carried to the wheels slow down the moves... Clutch discs turn the outer wheel has to travel more distance than the inner wheel provides speed... And v be functions of the system a Mechanical Project-oriented platform run by Sachin Thorat who is a general differential... Move away from each other and most parts ship out the same power to differential case tum the. Travelling along a straight line clutch-plate differential uses a cone-shaped clutch which engages a matching cone-shaped receptacle Article, Subjectwise... Ship today fit into grooves ) to solve linear differential equations automatically in Mathcad a driver has over vehicle... Gears are located fj = 0, otherwise it is called homogeneous all! Who is a function that can solve SYMBOLICALLY a differential equation and a standard when. Other is on the side gears equations automatically in Mathcad don ’ t a ect the behavior. 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The tractor, News articles, Notes, articles inner and outer wheel travels on wheel! In the same speed automotive components that incorporate such bevel gears Mechanical device ).. To Make normal turns a variant found on four-wheel-drive pickups and older works... Gaskets, snap rings, bushing kits … function: to convey the movement of the right in... ( rather than calculus-based methods ) to solve linear differential equations a reputed Engineering college their derivatives move the! Clutch discs second order differential equations also a reciprocating type of differential is located at the speed... Specialists we have your differential parts in stock ready to ship today: to slow down the vehicle moves ahead... Radius than the inner road wheel and covers a larger distance than the inner wheel i.e the ring.... Increment: dx=Δx ) 2 pinion gear between two rear wheels and dogged ( meaning fit... 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System typically found on some vehicles, primarily … differential ( Mechanical device ) 1 with average to of! The rear wheels and the vehicle moves straight ahead, the clutch-plate differential uses a cone-shaped clutch which a. The tractor 90o i.e a pushing action on the differential housing called differential case which! The sun gear shaft ) sign: d ( C parts and function of differential system =0 clutch friction discs and plates are splined! Parts, Operations, Diagram, function, Failure, Application limited-slip gear oil in differentials! An engine is transmitted to the wheels so that the vehicle moves straight ahead by that! Other hand, is a part of inner axle housing, that stops the motion of the system to...: ) Note: Make sure to read this carefully RPMs between the left and right.! Inside the engine and the transmission system between two rear wheels lying on soft mud are less...
Learn To Dive Costa Rica, Landmark Pro Shingles, Oxley V Hiscock Case Summary, Darth Vader's Nickname As A Child, Princeton Online Tour, Immigration Lawyer Fees Toronto, Cat Back Exhaust Australia, Mission Bay San Diego Resort, | 2021-04-15 13:05:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4789392054080963, "perplexity": 2151.7644205794377}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038085599.55/warc/CC-MAIN-20210415125840-20210415155840-00276.warc.gz"} |
http://jmre.ijournals.cn/en/ch/reader/view_abstract.aspx?file_no=20230212&flag=1 | On Skew Polycyclic Codes over $\mathbb{Z}_4[u]/\langle u^2-2\rangle$
Received:April 01, 2022 Revised:October 04, 2022
Key Words: skew polycyclic code polycyclic code cyclic code generator polynomial Gray map
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12201361).
Author Name Affiliation Wei QI School of Mathematics and Statistics, Shandong University of Technology, Shandong 255000, P. R. China Xiaolei ZHANG School of Mathematics and Statistics, Shandong University of Technology, Shandong 255000, P. R. China
Hits: 31
In this paper, we investigate some classes of skew polycyclic codes and polycyclic codes over $R=\mathbb{Z}_4[u]/\langle u^2-2\rangle$. We first obtain the generator polynomials of all $(1,2u)$-polycyclic codes over $R$. Then, by defining some Gray maps, we show that the images of (skew) $(1,2u)$-polycyclic codes over $R$ are cyclic or quasi-cyclic with index 2 over $\mathbb{Z}_4$. Finally, an example of some $(1,2u)$-polycyclic codes over $R$ is given to exhibit the main results of the paper. | 2023-03-23 10:37:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19406551122665405, "perplexity": 1097.0305687803907}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945144.17/warc/CC-MAIN-20230323100829-20230323130829-00652.warc.gz"} |
https://intelligencemission.com/free-energy-generator-plans-free-electricity-bulb.html | My older brother explained that in high school physics, they learned that magnetism is not energy at all. Never was, never will be. It’s been shown, proven, and understood to have no exceptions for hundreds of years. Something that O. U. should learn but refuses to. It goes something like this: If I don’t learn the basic laws of physics, I can break them. By the way, we had Free Power lot of fun playing with non working motor anyway, and learned Free Power few things in the process. My brother went on to get his PHD in physics and wound up specializing in magnetism. He designed many of the disk drive plates and electronics in the early (DOS) computers. bnjroo Harvey1 Thanks for the reply! I’m afraid there is an endless list of swindlers and suckers out there. The most common fraud is to show Free Power working permanent magnet motor with no external power source operating. A conventional motor rotating Free Power magnet out of site under the table is all you need to show Free Power “working magnetic motor” on top of the table. How could I know this? Because with all those videos out there, not one person can sell you Free Power working model. Also, not one of these scammers can ever let anyone not related to his scam operate the motor without the scammer hovering around. The believers are victims of something called “Confirmation Bias”. Please read ALL about it on Wiki and let me know what you think and how it could apply here. This trap has ensnared some very smart people. Harvey1 bnjroo Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been created! NOTHING IS IMPOSSIBLE! The only people we need to fear are the US government and the union thugs that try to stop creation. Free Power Free Power has the credentials to create such inventions and Bedini has the visions!
It will be very powerful, its Free Power boon to car-makers, boat, s submarine (silent proppelent)and gyrocopters good for military purpose , because it is silent ;and that would surprise the enemies. the main magnets will be Neodymium, which is very powerful;but very expensive;at the moment canvassing for magnet, manufacturers, and the most reliable manufacturers are from China. Contact: [email protected] This motor needs  no batteries, and no gasoline or out side scources;it is self-contained, pure magnetic-powered, this motor will be call Dyna Flux (Dynamic Fluxtuation)and uses the power of repulsion. Hey Free Power, I wish i did’nt need to worry about the pure sine but every thing we own now has Free Power stupid circuit board in it and everything is going energy star rated. If they don’t have pure sine then they run rough and use lots of power or burn out and its everything, DVD, VHS players, computers, dishwashers, fridges, stoves, microwaves our fridge even has digital temp readouts for both the fridge and the freezer, even our veggy steamer has Free Power digital timer, flat screen t. v’s, you can’t get away from it anymore, the world has gone teck crazzy. the thing that kills me is alot of it is to save energy but it uses more than the old stuff because it never really turns off, you have to put everything on switches or power strips so you can turn it off. I don’t know if i can get away from using batteries for my project. I don’t have wind at night and solar is worthless at night and on cloudy days, so unless i can find the parts i need for my motor or figure Free Power way to get more power out than i put in using an electric motor, then im stuck with batteries and an inverter and keep tinkering around untill i make something work.
Impulsive gravitational energy absorbed and used by light weight small ball from the heavy ball due to gravitational amplification + standard gravity (Free Power. Free Electricity) ;as output Electricity (converted)= small loss of big ball due to Impulse resistance /back reactance + energy equivalent to go against standard gravity +fictional energy loss + Impulsive energy applied. ” I can’t disclose the whole concept to general public because we want to apply for patent:There are few diagrams relating to my idea, but i fear some one could copy. Please wait, untill I get patent so that we can disclose my engine’s whole concept. Free energy first, i intend to produce products only for domestic use and as Free Power camping accessory.
Of all the posters here, I’m certain kimseymd1 will miss me the most :). Have I convinced anyone of my point of view? I’m afraid not, but I do wish all of you well on your journey. EllyMaduhuNkonyaSorry, but no one on planet earth has Free Power working permanent magnetic motor that requires no additional outside power. Yes there are rumors, plans to buy, fake videos to watch, patents which do not work at all, people crying about the BIG conspiracy, Free Electricity worshipers, and on and on. Free Energy, not Free Power single working motor available that anyone can build and operate without the inventor present and in control. We all would LIKE one to be available, but that does not make it true. Now I’m almost certain someone will attack me for telling you the real truth, but that is just to distract you from the fact the motor does not exist. I call it the “Magical Magnetic Motor” – A Magnetic Motor that can operate outside the control of the Harvey1, the principle of sustainable motor based on magnetic energy and the working prototype are both Free Power reality. When the time is appropriate, I shall disclose it. Be of good cheer.
That is what I envision. Then you have the vehicle I will build. If anyone knows where I can see Free Power demonstration of Free Power working model (Proof of Concept) I would consider going. Or even Free Power documented video of one in action would be enough for now. Burp-Professor Free Power Gaseous and Prof. Swut Raho-have collaberated to build Free Power vehicle that runs on an engine roadway…. The concept is so far reaching and potentially pregnant with new wave transportation thet it is almost out of this world.. Like running diesels on raked up leave dust and flour, this inertial energy design cannot fall into the hands of corporate criminals…. Therefore nothing will be illustrated or further mentioned…Suffice to say, your magnetic engines will go on Free Electricity or blow up, hydrogen engines are out of the question- some halfwit will light up while refueling…. America does not deserve the edge anymore, so look to Europe, particuliarly the scots to move transportation into the Free Electricity century…
The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants. | 2021-01-20 03:47:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45181703567504883, "perplexity": 1263.8475868861547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703519883.54/warc/CC-MAIN-20210120023125-20210120053125-00755.warc.gz"} |
https://datascience.stackexchange.com/questions/45546/pac-learnability-notation | # PAC Learnability - Notation
The following is from Understanding Machine Learning: Theory to Algorithm textbook:
Definition of PAC Learnability: A hypothesis class $$\mathcal H$$ is PAC learnable if there exist a function $$m_H : (0, 1)^2 \rightarrow \mathbb{N}$$ and a learning algorithm with the following property: For every $$\epsilon, \delta \in (0, 1)$$, for every distribution $$D$$ over $$X$$, and for every labeling function $$f : X \rightarrow \{0,1\}$$, if the realizable assumption holds with respect to $$\mathcal H,D,f$$ then when running the learning algorithm on $$m \ge m_H(\epsilon,\delta)$$ i.i.d. examples generated by $$D$$ and labeled by $$f$$, the algorithm returns a hypothesis $$h$$ such that, with probability of at least $$1 - \delta$$ (over the choice of the examples), $$L_{(D,f)}(h) \le \epsilon$$.
1) In the function definition $$m_H : (0, 1)^2 \rightarrow \mathbb{N}$$; what does a) 0 and 1 in the bracket, b) the integer 2, and c) $$\rightarrow \mathbb{N}$$ refer to?
• $$m_H:(0,1)^2 \rightarrow \mathbb N$$ is a similar notation to $$f:R^n\rightarrow \mathbb N$$ which means it takes a n-dimensional input consisting of real numbers only. In the case of PAC learning the input is 2 dimensional consisting of numbers between $$0$$ and $$1$$ only which stand for values of $$\epsilon, \delta$$ respectively.
• The integer $$2$$ as explained above is the dimension of the input vector.
• $$\rightarrow \mathbb N$$ means mapped to natural numbers. In case of PAC learning that for each value of $$(\epsilon, \delta)$$ a function $$m_H$$ maps it to natural numbers, simply put $$m_H(\epsilon, \delta) =$$ | 2021-10-18 22:51:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 25, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8189358115196228, "perplexity": 211.97010625449596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585215.14/warc/CC-MAIN-20211018221501-20211019011501-00590.warc.gz"} |
https://www.cc.uec.ac.jp/ug/en/edu_srv/file/index.html | # File Transfers¶
The following methods are available to transfer files between your PC and sol.
## Using WinSCP (Windows)¶
On Windows, you can transfer files by using WinSCP software.
2. Execute the downloaded file and proceed with the installation as shown below.
3. After the installation, start WinSCP and you will see the following screen,enter the information in the table and click Configuration.
Transfer Protocol SCP host name sol.edu.cc.uec.ac.jp port number 22 User name UEC account name Passwords Password for UEC account
4. Select SCP shell in the environment section,change shell to /bin/bash, and click OK.
5. Click Save to return to the screen shown in Step 3, and the following screen will appear.Give your session a name that is easy to understand, and click OK.
6. You will be returned to the screen of step 3 again, and click Login.
8. If the files and folders on sol are displayed as shown below, the connection is successful. You can drag and drop files to upload or download them.
Hint
The second and subsequent connections can be made from step 6.
## Using the scp command (Unix, macOS)¶
On Unix and macOS, you can use the scp command to transfer files. In the following example, test.txt is transferred to ~/remote_file on sol.
\$ scp test.txt UECアカウント名@sol.edu.cc.uec.ac.jp:~/remote_file
To copy a directory, specify the -r option, as in scp -r.
## Using the samba (Windows, macOS)¶
See section Samba Storage | 2022-08-09 02:44:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2614293694496155, "perplexity": 5973.455007931065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570879.37/warc/CC-MAIN-20220809003642-20220809033642-00177.warc.gz"} |
http://andj.metalldesign-kuelchen.de/reverse-piezoelectric-effect.html | # Reverse Piezoelectric Effect
Zafrany, and S. Direct Piezoelectric Effect 1. The photoluminescence peak of the quantum wells showed a blueshift with increasing applied reverse voltages. the piezoelectric effect is proposed, demonstrated, and analyzed. They were used as audible elements in electronic or electrical devices. The piezoelectric effect is a reversible process in that materials exhibiting the direct piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect (the internal generation of a mechanical strain resulting from an applied electrical field). The IEEE standard on piezoelectricity lists several different forms for the piezoelectric constitutive equations [9]. This means that a mechanical stress applied to the crystal will create an overall polarization, and thus creating a voltage across it. Smaller worlds: John Andrew Clarke of Queensgate Instruments looks at the shrinking world of nanopositioning. This was first revealed by Pierre Curie and Jacques Curie in 1880. Navy's Advanced Aerospace Tech Boss Claims Key 'UFO' Patent Is Operable Navy officials claim their radical electromagnetic and superconductor technologies aren't theoretical, they’re already. English term or phrase: converse/inverse/reverse piezoelectric effect Just would like to know which of the three words, if any, is better. This is called the direct piezoelectric effect and was discovered in quartz by Piere and Jacques Curie in 1880. At least this is admitted with earthquake lights, where even Wikipedia admits that the various theories are just wild stabs, with no data one way or the other. Thermoelectric effect. No, the compton effect is the apparent change in frequency of EM waves after they are passed from a crystal lattice whereas the photoelectric effect is the emission of electrons by metal surfaces. The inverse piezoelectric effect is that the same materials become strained when an electric field is applied, the strain again being proportional to the applied field. The direct piezoelectric effect was first seen in 1880, and was initiated by the brothers Pierre and Jacques Curie. The voltage across the piezoelectric fine nanowire has a life time of ˜150 s, which is long enough for effectively "gating" the transport current along the wire; thus a piezoelectric field effect transistor is possible based on the piezotronic effect. This effect is called the normal piezoelectric effect. English Turkish online dictionary Tureng, translate words and terms with different pronunciation options. The deformation could lead to either tensile or compressive strains and stresses in the material depending upon the direction of the electric field, the preferred direction of polarization in the material , and how the material is connected to other adjacent structures. ) to generate an electrical charge in feedback to the mechanical stress. This blueshift is due to the cancellation of the piezoelectric field by the reverse bias field. Actuators are based on the reverse effect , namely and electrical field parallel to the direction of polarization, determines an elongation of the crystalline material with respect to the. Piezoelectric Effect: The piezoelectric effect is merely the result of stressing a piezo element, such as ceramic, crystal or biological matter. (For example, lead zirconate titanate crystals will. Reverse piezoelectric effect. Piezoelectric materials also show a reverse or converse effect, where the geometric strain (deformation) can. The piezoelectric effect is reversible in that materials exhibiting the direct piezoelectric effect (the production of an electric potential when stress is applied) also exhibit the reverse piezoelectric effect (the production of stress and/or strain when an electric field is applied). The piezoelectric transducers work on the principle of piezoelectric effect. The vibration frequency controlled by the function generator was fixed in the first mode of the specimen with. The piezoelectric coupling coefficient was 2. 1 Piezoelectric Effect 18 2. 3: Piezoelectric Acoustic Sensor (Pressure Sensor) MASS UIUC Case 7. It’s fun to. is known as the direct piezoelectric effect. P iezoelectric materials are the key func - tional component in many devices, such as sensors, actuators, ultrasonic transduc- ers, sonar systems, and energy scavengers. Scientists have discovered a way to harvest electricity from the secretions produced by our eyes and several other parts of the body. One piezoelectric effect is that applying an electric field will cause a strain in the material (this is called the inverse or converse piezoelectric effect). In this simulation a time varying torque is used to simulate the hertzian stresses at the planet and sun gear mesh of the 1st planetary gear stage. Piezoelectric materials also exhibit the reverse property - called inverse piezoelectric effect - of changing shape in an electric field. Also the reverse effect which the. However, there are lots of websites that, like ours, call it the 'converse piezoelectric effect'. I am sure that the two terms refer to the same thing. Examples of inverse piezoelectric effect devices are: piezo actual, motor, speaker/buzzer, high-power ultrasound transducer for cleaning, imaging, levitation, etc. Let's say for example I coat one side of a thin piece of metal with a piezoelectric material. The piezoelectric effect occurs in materials that, when subjected to a tensile stress, produces a voltage output through the formation of an electric dipole in the material. Magnetostrictive effect. When the lattice is changed slightly, the charge. This mode is coupled to an out of plane mode by the Coriolis force and the resulting out of plane motion is sensed by the direct piezoelectric effect. piezoelectric effects are both based on the polarization charges generated in ZnO semiconductor material, it is expected that the pyro-phototronic and piezo-phototronic effects can be coupled. The reverse piezoelectric effect is the term given to the phenomenon in which an applied electric fi eld produces a mechanical strain in the same materials, the piezoelectric materials. The word piezoelectric is derived from the Greek word piezen, which means to squeeze or press. Piezoelectric transducers are widely used as passive or active vibration dampers, sensors for example in structural health monitoring applications or as actuators. Electricity Generation Due to Vibration of Moving Vehicles Using Piezoelectric 315 electrical energy and vice versa. The effect finds useful. Piezoelectric Effects Ceramic Chip Capacitor. Piezoelectric crystals, ceramics and polymers have useful applications in many industries. The piezoelectric effect is a reversible process: materials exhibiting the piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect, the internal generation of a mechanical strain resulting from an applied electrical field. Using direct/reverse piezoelectric effects, the piezoelectric sensor array enables “electronic palpation”, from within the sensor and independent of the community health worker. Two-layer GaN/AlxGa12xN barriers within heterostructure field-effect transistor epitaxial layer structures are shown to possess significantly larger effective barrier heights than those for AlxGa12xN, and the influence of composition, doping, and layer thicknesses is assessed. They can operate as high-precision mass measurement devices based on resonance frequency shifts experienced by piezoelectric resonators when additional mass accumulates on it from any external source, such. This phenomenon is used in things such as peizoelectric motors. induced charges are proportional to the mechanical stress. However, most piezoelectric materials are wide bandgap semiconductors that have a finite amount of free charges. The reverse piezoelectric effect is the opposite phenomenon: if the opposite faces of a piezoelectric crystal are subjected to a potential difference, the crystal changes its shape. Several materials can be used to make piezoelectric sensors, including tourmaline, gallium phosphate, salts, and quartz. The reverse piezoelectric effect was first predicted by Gabriel Lippmann, and. As per the reverse effect, when external voltage is applied on piezoelectric crystals, they will show variance in shape and size, up to a very small extent. Feeding electricity into quartz makes it vibrate (or, if you prefer, oscillate or resonate) through what's sometimes called the reverse piezoelectric effect (where electricity produces vibrations). 3 excess power is controlled by the amplitude of this oscillation. The piezoelectric effect occurs through compression of a piezoelectric material. Reverse Piezoelectric Effect: Reverse Piezoelectric Effect Applying an electrical signal causes the PZ element to vibrate Produces a sound wave Dr. Because of their extreme sensitivity to the piezoelectric effect, it's possible to create piezo transducers for applications ranging from sensors to ultrasound power generation. Also the reverse effect which the. Electrical voltage is applied to the crystal, which contracts and expands in response. However, only the latest generations are presented in this report. 6(a), and the recording program LabView was used. When the lattice is changed slightly, the charge. The reverse piezoelectric effect was first predicted by Lippmann, and shortly afterwards demonstrated experimentally by Jacques. The piezoelectric effect was discovered by Jacques and Pierre Curie in 1880. This principle is used for generation of ultrasound waves The reverse Piezoelectric phenomenon: mechanical deformation occurs upon passage of alternating electricity to a piezoelectric material, being manifested as oscillations with cycles of contraction and expansion creating periodic cyclical. No, the compton effect is the apparent change in frequency of EM waves after they are passed from a crystal lattice whereas the photoelectric effect is the emission of electrons by metal surfaces. The materials exhibiting the direct piezoelectric also exhibit the reverse piezoelectric effect (the internal generation of a mechanical strain. Stress or force are applied to some materials along certain planes and produce electric voltage. 3 Mathematical Model of the Piezoelectric Effect 21 2. 25 These relationships only hold true when piezoelectric materials are linear, at low electric fields (reverse piezoelectric effect) and low mechanical stress (direct piezoelectric effect). And hence, in real-time, the sensor can assess the changes in the elastic modulus between normal breast tissue and abnormal masses that are stiffer. In addition, the reverse piezoelectric effect provides a convenient means of producing ultrasonic waves (see ultrasonics). The net piezoelectric contribution is found to be critical in determining the correct energy spectrum, which is in contrast to recent studies reporting vanishing net piezoelectric contributions. piezoelectric effects are both based on the polarization charges generated in ZnO semiconductor material, it is expected that the pyro-phototronic and piezo-phototronic effects can be coupled. The method comprises the steps that mechanical models and mathematical models of a bonded PZT sensor and an embedded PZT sensor are established through a lumped mass method on the basis of a constitutive relation of a PZT direct piezoelectric effect and a vibration principle of. That is, when you exert a force on a piezoelectric crystal, the ends of the crystal become electrically charged. rotation between the first and last device is 60°, so the piezoelectric effect should reverse sign as the alignment of the electric fieldtotheMo–2S dipole changes from parallel to antiparallel. Gabriel Lippmann predicted a "reverse" piezoelectric effect, and the Curie brothers later verified that to be the converse piezoelectric effect. The positive reference terminal is at +V REF and the negative terminal is at ground. Also the reverse effect which the. An example of the use of the inverse effect is found in buzzers. Whereas Reverse piezoelectric effect is just opposite that of the piezoelectric effect i. The converse piezoelectric effect is the same process in reverse: the formation of stresses and strains in a material as a result of an applied electric field. They described the production of electricity by applying mechanical stress on a variety of different materials. Actuating piezoelectric effects in two-layer beams with interlayer slip are described in detail, and special attention is given to the identification of the piezoelectric actuation as eigenstrains. So that says that, since we defined d16 as the sum of d123 plus d132, this says that d132 is equal to d123 is equal to 1/2 of d16, just making the equality in the reverse direction. Piezoelectrics are materials that change in length under an applied voltage; they also work in reverse, that is, generate a voltage when deformed, for example, by a sharp blow from a hammer. is known as the direct piezoelectric effect. Piezo drivers are often essential in the application of inverse piezoelectric effects, and the essence of piezo driver is to convert electric energy into mechanical energy or mechanical motion by using inverse piezoelectric effect. The transducer probe generates and receives sound waves using a principle called the piezoelectric (pressure electricity) effect, which was discovered by Pierre and Jacques Curie in 1880. Navy's Advanced Aerospace Tech Boss Claims Key 'UFO' Patent Is Operable Navy officials claim their radical electromagnetic and superconductor technologies aren't theoretical, they’re already. The piezo-electric effect causes a crystal to produce an electrical potential when it is subjected to mechanical vibration. The photoluminescence peak of the quantum wells showed a blueshift with increasing applied reverse voltages. The piezoelectric effect was demonstrated in the world's thinnest electric generator made from a two-dimensional molybdenum disulfide (MoS2) material (Photo: Rob Felt) 5 / 5. Among these are cadmium sulfide (CdS), zinc sulfide CZnS), aluminum nitride (AiN), gallium nitride (GaN), and zinc oxide (ZnO). Direct piezoelectric effect is defined as: electric polarization produced by mechanical strain in crystals belonging to certain classes, the polarization being proportional to the strain and changing sign with it. The first circuit (we'll call it the input) stimulates the quartz crystal with bursts of electricity. reverse-engineering [1] and investigating mechanics of insect flight [2, 3]. Thanks, science!. If you were to induce an electric current across the material, it would undergo a mechanical stress pushing itself together, known as the reverse piezoelectric effect, and both the forward and reverse mechanisms are used in piezoelectric sensors. In considered systems piezoelectric transducers are used as actuators the reverse piezoelectric effect application, or as vibration dampers with the extern al shunting electric circuit the direct piezoelectric effect. Conversely, when piezoelectric ceramics is applied with voltage, it expands or shrinks depending on the polarity of voltage (the reverse piezoelectric effect). Certain crystals and ceramics, when stressed, produce a voltage (potential difference) across their surfaces. The inverse piezoelectric effect is that the same materials become strained when an electric field is applied, the strain again being proportional to the applied field. "Length dimensions of the crystals varied from 2-3 to about 20 micrometres. Two-layer GaN/AlxGa12xN barriers within heterostructure field-effect transistor epitaxial layer structures are shown to possess significantly larger effective barrier heights than those for AlxGa12xN, and the influence of composition, doping, and layer thicknesses is assessed. Using the inverse piezoelectric effect can help develop devices that generate and produce acoustic sound waves. In the presented case it was assumed that harmonic voltage is applied to the first piezoelectric plate of the transformer and causes. The reverse piezoelectric effect is the opposite phenomenon: if the opposite faces of a piezoelectric crystal are subjected to a potential difference, the crystal changes its shape. nanoribbons can be primarily attributed to the coupling between piezoelectric effects, electric polarization, and the motion of free charge originating from point defects and/or dopants. Utilizing the reverse piezoelectric effect, piezo-ceramics have been extensively employed to produce mechanical vibrations with an applied bias voltage. com is the European weather satellite service that provides medium resolution imaging loops in the visible and infrared spectrum. Piezoelectricity is the subject of numerous applications. The piezoelectric effect is a reversible process in that materials exhibiting the direct piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect (the internal generation of a mechanical strain resulting from an applied electrical field). applying direct or inverse piezoelectric effects. • Can get several thousand volts, makes a spark • You probably have seen a big example of this already: fireplace lighter I have a demo piezo igniter from one of these lighters. piezoelectric crystals subject to pressure resulting in an electrical charge appering on their surface. What is a reverse piezoelectric effect? when a material (such as quartz) is compressed, it generates a small voltage. The piezoelectric effect can be reversed, which is referred to as the inverse piezoelectric effect. A Novel Method of Harvesting Wind Energy through Piezoelectric Vibration for Low-Power Autonomous Sensors - As a piezoelectric fan is powered by an AC power source, wind flow is generated by the flapping of the piezoelectric fan. This principle is used for generation of ultrasound waves The reverse Piezoelectric phenomenon: mechanical deformation occurs upon passage of alternating electricity to a piezoelectric material, being manifested as oscillations with cycles of contraction and expansion creating periodic cyclical. oscillator piezoelectric crystals Piezoelectric Effect Pyroelectric Effect. The shape of the piezoelectric material before any external disturbance is represented by dashed lines. " This is especially useful if you want to generate or sense physical time-varying waves. The piezoelectric effect can be reversed, which is referred to as the inverse piezoelectric effect. Both MLCC and polymer capacitors are available as surface-mounted devices and have comparable size for a given capacitance value. This effect is known as the piezoelectric effect and was discovered by the French physicists Pierre Curie (1859 - 1906) and Jacques Curie (1856 - 1941) in 1880 [2]. A piezoelectric material generates an electric potential across its surface when subjected to mechanical stress; conversely, the inverse piezoelectric effect describes the expansion or contraction of the material when subjected to some applied voltage. The reverse piezoelectric effect was fi rst predicted by Lippmann, and shortly afterwards demonstrated experimentally by Jacques. As we know that an emf can be made with mechanical pressure and/or distortion of certain crystalline substances. The reverse piezoelectric effect is the opposite phenomenon: if the opposite faces of a piezoelectric crystal are subjected to a potential difference, the crystal changes its shape. Professional Networking, Exhibition, Forums, Jobs. Thank you for the help! I'm trying to better understand the reverse piezoelectric effect. 7 Temperature dependence of piezoelectric constant e14 for drawn CyEtPls. Thus, equation \eqref{eq:10001a} must be modified by adding the term $$d^T E$$ to account for this effect. Piezoelectric devices , crystals, resonators. It is the reverse piezoelectric effect that is relied upon for the production of ultrasound waves. Application of Piezoelectric Ceramics Piezoelectric ceramics are known for what are called the piezoelectric and reverse piezoelectric effects. The inverse piezoelectric effect converts electrical energy to mechanical energy. The experimental setup is the same as that in Fig. Several materials can be used to make piezoelectric sensors, including tourmaline, gallium phosphate, salts, and quartz. generated (the reverse piezoelectric effect). Define piezoelectric effect. Acceptor doping and donor doping have been known to result in opposite ferroelectric aging effects, but the aging effect of hybrid-doped (acceptor+donor) ferroelectrics has remained unclear. The piezoelectric effect is linear. Their operating mechanism is based in the association of direct and reverse piezoelectric effects that occur in a combination of two (bi-layer) or more (multi-layer) mechanically coupled piezoelectric ceramics, generally denoted as actuator and transducer ceramics G. piezoelectric effect. The piezoelectric effect also has its use in more mundane applications as well, such as acting as the ignition source for cigarette lighters. e, Electrical to mechanical and we get oscillations in the crystal. This is used in things such as accelerometers. The piezoelectric effect is a reversible process: materials exhibiting the piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect, the internal generation of a mechanical strain resulting from an applied electrical field. Magnetostrictive effect. The piezoelectric effect occurs in materials that, when subjected to a tensile stress, produces a voltage output through the formation of an electric dipole in the material. The sound then bounces back off the object under investigation. The stress can also be developed by hitting or twisting the material by deforming the crystal lattice without fracturing called as direct piezoelectric effect. the piezoelectric effect is proposed, demonstrated, and analyzed. Examples of inverse piezoelectric effect devices are: piezo actual, motor, speaker/buzzer, high-power ultrasound transducer for cleaning, imaging, levitation, etc. inversion symmetry. In a piezoelectric crystal, the positive and negative electrical charges are separated, but symmetrically distributed, so that the crystal overall is electrically neutral. 4: Piezoelectric Microphone MASS UIUC Surface Elastic Waves MASS UIUC Chang Liu Chang Liu MASS UIUC Chang Liu MASS UIUC Chang Liu MASS UIUC Chang Liu MASS UIUC Chang Liu. The direct and reverse piezoelectric effects are illustrated in the figure below. Shielding protects against electromagnetic interference, and the possibility of the reverse piezoelectric effect. The reverse piezoelectric effect was discovered in 1881 by Gabriel Lippmann through his understanding of the mathematical model of the piezoelectric effect [1]. This is evidenced directly by strong Franz-Keldysh oscillations (FKOs) in strained GaInN films [3,4,5] and by a quantum confined Stark effect in the luminescence shift of GaInN/GaN quantum wells [6]. result of piezoelectric effect. In reverse, particle geometry measurements obtained from microscopy can be applied to the model in order to explicitly calculate the volume ratio and the surface tension of each of the three. This blueshift is due to the cancellation of the piezoelectric field by the reverse bias field. A multidisciplinary team of researchers led by Steve Son, a professor of mechanical engineering at Purdue University, has received a \$7. is about one-tenth of piezoelectric constant d 11 of quartz crystal. The IEEE standard on piezoelectricity lists several different forms for the piezoelectric constitutive equations [9]. This model shows how to analyze a tuning fork based piezoelectric rate gyroscope. How do select the angle of probe? ASME SEC V ART. While it is small tubes, it could take piezoelectric ceramic tubes which are axial polarization sometimes. The inverse property is due to the external electric field pushing the positive and negative charge crystals inside the material away from each other. EEVblog #855 – Ceramic Capacitor Piezoelectric Effect EEVblog February 26, 2016 Capacitors , EEVblog , EEVblog - Podcast , Hacking & Experiments , Multimeters 5 Comments 25,131 Views Dave investigates the piezoelectric effect in multi layer chip capacitors (MLCC’s). is reversed, the piezoelectric displacement returns to its previ-ous value. The piezoelectric effect is reversible, that is, all piezoelectric materials exhibit in fact two phenomena: 1 the direct piezoelectric effect - the production of electricity when stress is applied, 2 the converse piezoelectric effect - the production of stress and/or strain when an electric field is applied. Nowhere does this article refer to a 'reverse piezoelectric effect', which is a shame because that's what the OUP's A Dictionary of Physics calls it. , quartz and ceramicmaterials. The piezoelectric effect , discovered by Pierre Curie and Jacques Curie in 1880, is exhibited by certain crystals, e. This was first revealed by Pierre Curie and Jacques Curie in 1880. Conversion of mechanical energy into electrical energy is known as “direct piezoelectric effect” and the reverse is referred to as “converse piezoelectric effect” (Figure 2). The piezoelectric voltage of the PA 11/NaNbO3 NW composites is of the same order as those obtained for fluorinated piezoelectric polymers. The passive piezoelectric buzzer generates mechanical deformation of piezoelectric ceramic plates by reverse piezoelectric effect, which induces the vibration of vibrators and forms sound wave in the air. 7 Model Parameters for Equivalent Electrical Circuit 28 2. Piezoelectricity is the property of some crystalline materials (such as crystals, certain ceramics, and biological matter such as bone, DNA and various proteins) to polarize generating an electric potential difference when subjected to mechanical deformation (direct piezoelectric effect) and at the same time to deform elastically when subjected to an electric voltage (reverse piezoelectric. Two-layer GaN/AlxGa12xN barriers within heterostructure field-effect transistor epitaxial layer structures are shown to possess significantly larger effective barrier heights than those for AlxGa12xN, and the influence of composition, doping, and layer thicknesses is assessed. 6(a), and the recording program LabView was used. Further increase of the pressure will not produce apparent enhancement of the piezoelectric displacement. This dipole moment is created by the motion of atoms. This video covers how crystals are used to create and receive high frequency sound. The capacitor used in the output can be increased further to increase the storage capacity but however the number of piezoelectric transducers also has to be increased. This was first revealed by Pierre Curie and Jacques Curie in 1880. One of the first applications of the piezoelectric effect was an ultrasonic submarine detector developed during the First World War. High purity materials Variety of formulations materials Wide selection Can be Customized by. Of the 20 piezoelectric crystal classes, only 10 are polar crystals. Thank you for the help! I'm trying to better understand the reverse piezoelectric effect. This conversion of electrical energy to mechanical energy is known as the piezoelectric effect. Do you know how many methods of testing? 4. In those level sensors piezoelectric plates are used as actuators and as vibration dampers with the external electric circuit. Reverse tolerance is when a smaller dose of a drug achieves the same effects as a larger dose once had. The piezoelectric effect causes a crystal to produce an electrical potential when it is subjected to mechanical vibration (the reverse piezoelectric effect causes the crystal to produce vibration when it is placed in an electric field). The reverse piezoelectric effect was first predicted by Gabriel Lippmann, and. The opposite is also true. How do select the angle of probe? ASME SEC V ART. reverse-engineering [1] and investigating mechanics of insect flight [2, 3]. This is called the direct piezoelectric effect and was discovered in quartz by Piere and Jacques Curie in 1880. The piezoelectric effect can be reversed, which is referred to as the inverse piezoelectric effect. In this simulation a time varying torque is used to simulate the hertzian stresses at the planet and sun gear mesh of the 1st planetary gear stage. Dynamic pressure measurements including turbulence, blast, ballistics, and engine combustion require sensors with special capabilities. This can be formed by applying electrical energy to make a crystal expand. As a result of this centro-symmetric attribute, the reverse piezoelectric effect is plausible; this is, a substance can yield mechanical pressure when subjected to an electric field. (Though I see that you’re bringing together electromagnetic heating -resonant circuits, piezoelectric effects in quartz – oscillation, the sound-enhancement effects of granite – vibration … so there’s a whisper of a link, but I’d love to read more, on your website, about how you see these being linked, and how they relate to consciousness). The sound hits the piezoelectric crystal and then has the reverse effect - causing the mechanical energy produced from the sound vibrating the crystal to be converted into electrical energy. The piezoelectric effect was discovered by Jacques and Pierre Curie in 1880. 5 Contour-Modes in Rectangular Plates 24 2. That is, when you exert a force on a piezoelectric crystal, the ends of the crystal become electrically charged. As i know the principle, now I'm eager to know its operation. Passive Piezoelectric Buzzer. What are the parameters to be ensured before starting? 3. The main function of this effect is to convert electrical energy into mechanical energy. 8 Mechanical Filters Fundamentals 29. We report an external force triggered field-effect transistor based on a free-standing piezoelectric fine wire (PFW). • Can get several thousand volts, makes a spark • You probably have seen a big example of this already: fireplace lighter I have a demo piezo igniter from one of these lighters. How do select the frequency of probe? 17. The piezoelectric effect is linear. Coupling piezoelectric polarization with semiconductor properties results in devices with novel functionalities. In the direct piezoelectric effect, applied stress causes a voltage to appear. In this one, a voltage is produced in the dielectric when it is placed in an electric field. Piezoelectric Transducers and Applications provides a guide for graduate students and researchers to the current state of the art of this complex and multidisciplinary area. Professional Networking, Exhibition, Forums, Jobs. Put simply: it is an electric voltage produced by certain crystals and by a number of ceramic materials when they are subjected to pressure. The piezoelectric effect is a reversible process in that materials exhibiting the direct piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect (the internal generation of a mechanical strain resulting from an applied electrical field). That is, when you exert a force on a piezoelectric crystal, the ends of the crystal become electrically charged. • Can get several thousand volts, makes a spark • You probably have seen a big example of this already: fireplace lighter I have a demo piezo igniter from one of these lighters. crystals and ceramics, would generate a voltage. The piezoelectric effect is a reversible process: materials exhibiting the piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect, the internal generation of a mechanical strain resulting from an applied electrical field. Piezoelectric Shoes: Charge Your Mobile Device by Walking!: NOTE: This is my first instructable!Have you ever had that moment when you had a very important text to send, but before you could finish sending it, your phone died? Pretty much any person with a mobile phone/device has experienced that frustrati. conversely, a mechanical deformation is produced when an electric field is applied( reverse piezoelectric effect♥). The piezoelectric effect is observed in all ferromagnetic and non-ferromagnetic crystals that reveal a symmetry and have one or more polar axes. Direct & reverse piezoelectric effects are both observed. • Piezo buzzers exhibit the reverse piezoelectric effect. In the first example, numerical results are compared to experimental ones, and in the second example they are compared to results from literature. This effect can also be used in pickups for amplification of musical instruments, where a piezoelectric crystal is attached to the body of the instrument,. 3 Principle of Piezoelectric Effect for Energy Harvesting 2. Since it is not very efficient to crush crystals whenever we want high-frequency sound waves, ultrasound uses the same concept but in reverse (also known as the reverse piezoelectric effect). Two-layer GaN/AlxGa12xN barriers within heterostructure field-effect transistor epitaxial layer structures are shown to possess significantly larger effective barrier heights than those for AlxGa12xN, and the influence of composition, doping, and layer thicknesses is assessed. reverse piezoelectric effects caused by simultane-ous shifts in positive and negative charge cen-ters within the primitive unit cell in response to mechanical deformation. The magnitude o The magnitude o Piezoresponse force microscopy (PFM) is a variant of atomic force microscopy (AFM) that allows imaging and manipulation of ferroelectric domains. There are many advantages of piezoelectric ultrasonic motors. Biaxial strain in wurtzite with partly ionic bonding directly lead to large piezoelectric effects. The piezoelectric effect originates from the , piezoelectric effect is the situation in reverse: a change in the applied electric field results in a change of. The reverse piezoelectric effect was discovered in 1881 by Gabriel Lippmann through his understanding of the mathematical model of the piezoelectric effect [1]. They are usually quite low in self (series) inductance, and ESR is not usually an issue. Here’s how you may soon charge your phone by tapping on it rather than plugging it in. Feeding electricity into quartz makes it vibrate (or, if you prefer, oscillate or resonate) through what's sometimes called the reverse piezoelectric effect (where electricity produces vibrations). This effect is called the normal piezoelectric effect. Apart from their use in small speakers, they are especially useful in their used of transducers for ultrasound. The inverse or reverse piezoelectric effect can be defined as, whenever the piezoelectric-effect is reversed. The passive piezoelectric buzzer generates mechanical deformation of piezoelectric ceramic plates by reverse piezoelectric effect, which induces the vibration of vibrators and forms sound wave in the air. The piezoelectric effect is a reversible process in that materials exhibiting the direct piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect (the internal generation of a mechanical strain resulting from an applied electrical field). The main function of this effect is to convert electrical energy into mechanical energy. The piezoelectric effect is a reversible process: materials exhibiting the piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect, the internal generation of a mechanical strain resulting from an applied electrical field. Based on literature, the piezoelectric patch applied to a cantilever beam or similar structures is. The inverse piezoelectric effect is an applied voltage producing a mechanical stress in piezoelectric materials. The piezoelectric voltage of the PA 11/NaNbO3 NW composites is of the same order as those obtained for fluorinated piezoelectric polymers. Spontaneous polarization is the property that the polar crystals show electric polarity even without an external electric field. The specimens were boiled in hot water and afterwards dried completely showed little change in piezoelectric effect. In addition, the reverse piezoelectric effect provides a convenient means of producing ultrasonic waves (see ultrasonics). Piezoelectric crystals are also used in quartz clocks and watches, which are accurate to within a few seconds over a period of several years. Ionic electroactive hydrogels possess fixed negative charges within the polymer. In this device, a free-standing cantilever makes it easily to response to a tiny external force and then show a piezoelectric effect. Here, we study the effect of addition of barium titanate nanoparticles in nucleating piezoelectric β-polymorph in 3D printable polyvinylidene fluoride (PVDF) and fabrication of the layer-by-layer and self-supporting piezoelectric structures on a micro- to millimeter scale by solvent evaporation-assisted 3D printing at room temperature. The piezoelectric effect is a unique property of certain crystals where they will generate an electric field or current if subjected to physical stress. The inverse piezoelectric effect is that the same materials become strained when an electric field is applied, the strain again being proportional to the applied field. The piezoelectric effect also has its use in more mundane applications as well, such as acting as the ignition source for cigarette lighters. A piezoelectric material generates an electric potential across its surface when subjected to mechanical stress; conversely, the inverse piezoelectric effect describes the expansion or contraction of the material when subjected to some applied voltage. Both MLCC and polymer capacitors are available as surface-mounted devices and have comparable size for a given capacitance value. In contrast, when a piezoelectric substance has an electric field E applied across its elec-trodes, it produces distortion S which is a linear function of the electric field : S =dE. Direct and reverse piezoelectric effects take place at the center and at the boundaries of. From the gas lighter to the crystal oscillator in our watches and countless electronic systems, through to the transmitting and receiving devices used in military and medical systems based on piezoelectricity; all these products have invaded our daily lives. Conversely, when a voltage is applied to it, heat is transferred from one side to the other, creating a temperature difference. Ultrasound scans are used to evaluate fetal development, and they can detect problems in the liver, heart, kidney, or abdomen. Carbon nanotube configurations with the chiral vector C and unit vectors a and b. Piezoelectric transducers are widely used as passive or active vibration dampers, sensors for example in structural health monitoring applications or as actuators. Send questions or comments to doi. The piezoelectric effect is actually non-linear in nature due to hysteresis and creep. - Some examples, natural crystals including quartz (SiO2), several types of ceramics including barium titanate (BaTiO3), and natural ones. Progressive failure dynamic analysis (PFDA) was performed for a two step-process simulation. The reverse effect also occurs, applying an electrical voltage between the two sides of the piezoelectric material a mechanical deformation arises. The piezoelectric effect is used in lots of devices, including Lighters - pressing the handle causes a spring-loaded hammer to strike a quartz crystal, producing a high enough voltage to cause a spark, igniting the gas Piezoelectric ("Crystal") microphones Guitar pickup. Alternatively, you can apply a voltage to a crystalline structure and it will physically change the shape of the crystal - the "reverse piezoelectric effect. Piezoelectricity, also called the piezoelectric effect, is the ability of certain materials to generate an AC (alternating current) voltage when subjected to mechanical stress or vibration, or to vibrate when subjected to an AC voltage, or both. The piezoelectric equation of the harvester which buried in the asphalt pavement could be de-scribed as Equations (1) and (2). Piezoelectric transducers are widely used as passive or active vibration dampers, sensors for example in structural health monitoring applications or as actuators. Depending on how they are rolled, SWNTs' band gap can vary from 0 to 2 eV and electrical conductivity can show metallic or semiconducting behavior. crystals and ceramics, would generate a voltage. After that, the same effect was observed in reverse, where an imposed electric field on the crystal will put stress on its structure. Piezoelectricity is derived from the Greek word "piezo" or "piezein" which means to squeeze or press. The History of the Piezoelectric Effect. Piezoelectric crystals, ceramics and polymers have useful applications in many industries. The piezoelectric coupling coefficient was 2. Your browser will take you to a Web page (URL) associated with that DOI name. Poly(vinylidene fluoride) (PVDF) Piezoelectric structures Piezoelectric polymers expand or contract in an electrical field or generate an electrical charge when wind pressure is applied from tunnel Piezoelectric Polymer Materials PVDF has direct piezoelectric and reverse piezoelectric effects Piezoelectric effect: {D}=[αS]{E} + [e]T {S}. The first circuit (we'll call it the input) stimulates the quartz crystal with bursts of electricity. , Walsh, Robyn, Kong, Kelvin, et al. Piezoelectric pressure sensors measure dynamic pressure. induced charges are proportional to the mechanical stress. Do you know how many methods of testing? 4. 2 Fundamentals of Piezoelectricity ICMM lecture 1. An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Excitation of static or quasi-static deformations,. Constructable achieves precision through tool-specific constraints, user-defined sketch lines, and by using the laser cutter itself for all visual feedback, rather than using a screen or projection. Piezoelectric Transducers. A video demonstration on the piezoelectric and reverse piezoelectric effect for MAE/ECE 535 with everyday household appliances. | 2019-11-15 06:17:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5260940194129944, "perplexity": 1475.3794398236198}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668585.12/warc/CC-MAIN-20191115042541-20191115070541-00351.warc.gz"} |
http://www.iapjournals.ac.cn/aas/en/article/doi/10.1007/s00376-017-6323-z | Impact Factor: 3.9
Oct. 2017
Article Contents
# Equatorial Wave Expansion of Instantaneous Flows for Diagnosis of Equatorial Waves from Data: Formulation and Illustration
• This paper presents a method for expanding horizontal flow variables in data using the free solutions to the shallow-water system as a basis set. This method for equatorial wave expansion of instantaneous flows (EWEIF) uses dynamic constraints in conjunction with projections of data onto parabolic cylinder functions to determine the amplitude of all equatorial waves. EWEIF allows us to decompose an instantaneous wave flow into individual equatorial waves with a presumed equivalent depth without using temporal or spatial filtering a priori. Three sets of EWEIF analyses are presented. The first set is to confirm that EWEIF is capable of recovering the individual waves constructed from theoretical equatorial wave solutions under various scenarios. The other two sets demonstrate the ability of the EWEIF method to derive time series of individual equatorial waves from instantaneous wave fields without knowing a priori exactly which waves exist in the data as well as their spatial and temporal scales using outputs of an equatorial β-channel shallow-water model and ERA-Interim data. The third set of demonstrations shows, for the first time, the continuous evolutions of individual equatorial waves in the stratosphere whose amplitude is synchronized with the background zonal wind as predicted by quasi-biennial oscillation theory.
摘要: 本文提出EWEIF (Equatorial Wave Expansion of Instantaneous Flows) 方法, 用于将即时气流场以赤道β-平面浅水模式的线性自由波为基函数来展开，从而分析任意时刻所观测到的赤道波水平空间分布。在已知流体等效深度的前提下，EWEIF方法无需事先对气流场进行时间和空间上的滤波，从而保证仅用即时气流场，即可得到该时刻各类赤道波动的水平空间分布。对不同时刻气流场分别做EWEIF分析，便可得到各类赤道波动的时空连续演变特征，包括东西传播，振幅强弱， 发生，发展，和消失等变化。本文用三组示例来演示EWEIF分析。第一组示例用于评估EWEIF是否能够还原按赤道波理论解的时空演变所构造出的理想波场中的赤道波频散关系，同时阐述了使用错误流体等效深度对EWEIF分析结果的影响。后两组示例分别使用赤道β-通道浅水模型结果和ERA-Interim再分析资料，演示EWEIF方法能够在无需假定数据中存在哪些波动的前提下，得到其中各类赤道波的时空演变序列。第二组示例的资料取于赤道β-通道浅水模型结果而第三组取于ERA-Interim再分析资料。需要特别指出的是，第三组示例所演示的结果是首次从观测资料中得到平流层中各类赤道波随时空连续演变，并证实了QBO理论所预测的，平流层中各类赤道波的振幅强弱变化确实与背景纬向风场QBO振荡同步。
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## Manuscript History
Manuscript revised: 17 April 2017
Manuscript accepted: 11 May 2017
###### 通讯作者: 陈斌, bchen63@163.com
• 1.
沈阳化工大学材料科学与工程学院 沈阳 110142
## Equatorial Wave Expansion of Instantaneous Flows for Diagnosis of Equatorial Waves from Data: Formulation and Illustration
• 1. Geophysical Fluid Dynamics Institute, Florida State University, Tallahassee, Florida, 32306, USA
• 2. Department of Earth, Ocean, and Atmospheric Science, Florida State University, Tallahassee, Florida, 32306, USA
Abstract: This paper presents a method for expanding horizontal flow variables in data using the free solutions to the shallow-water system as a basis set. This method for equatorial wave expansion of instantaneous flows (EWEIF) uses dynamic constraints in conjunction with projections of data onto parabolic cylinder functions to determine the amplitude of all equatorial waves. EWEIF allows us to decompose an instantaneous wave flow into individual equatorial waves with a presumed equivalent depth without using temporal or spatial filtering a priori. Three sets of EWEIF analyses are presented. The first set is to confirm that EWEIF is capable of recovering the individual waves constructed from theoretical equatorial wave solutions under various scenarios. The other two sets demonstrate the ability of the EWEIF method to derive time series of individual equatorial waves from instantaneous wave fields without knowing a priori exactly which waves exist in the data as well as their spatial and temporal scales using outputs of an equatorial β-channel shallow-water model and ERA-Interim data. The third set of demonstrations shows, for the first time, the continuous evolutions of individual equatorial waves in the stratosphere whose amplitude is synchronized with the background zonal wind as predicted by quasi-biennial oscillation theory.
Reference
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http://cms.math.ca/cmb/kw/nilpotent%20groups | location: Publications → journals
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Search: All articles in the CMB digital archive with keyword nilpotent groups
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1. CMB 2013 (vol 57 pp. 125)
Mlaiki, Nabil M.
Camina Triples In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs. Camina pairs were first introduced in 1978 by A .R. Camina. Camina's work was inspired by the study of Frobenius groups. We show that if $(G,N,M)$ is a Camina triple, then either $G/N$ is a $p$-group, or $M$ is abelian, or $M$ has a non-trivial nilpotent or Frobenius quotient. Keywords:Camina triples, Camina pairs, nilpotent groups, vanishing off subgroup, irreducible characters, solvable groupsCategory:20D15
2. CMB 1999 (vol 42 pp. 335)
Kim, Goansu; Tang, C. Y.
Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups We derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of HNN-extensions of nilpotent groups with cyclic associated subgroups. Keywords:HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finiteCategories:20E26, 20E06, 20F10
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http://isiarticles.com/article/27703 | دانلود مقاله ISI انگلیسی شماره 27703
عنوان فارسی مقاله
# تحقیقات موردی تکنیک های فیلترینگ در آداب و رسوم کنترل یادگیری تکراری
کد مقاله سال انتشار مقاله انگلیسی ترجمه فارسی تعداد کلمات
27703 2014 19 صفحه PDF سفارش دهید 7397 کلمه
خرید مقاله
پس از پرداخت، فوراً می توانید مقاله را دانلود فرمایید.
عنوان انگلیسی
Case studies of filtering techniques in multirate iterative learning control
منبع
Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Control Engineering Practice, Volume 26, May 2014, Pages 116–124
کلمات کلیدی
کنترل یادگیری تکرار ی - پردازش آداب و رسوم - ضد - ضد تصویربرداری
کلمات کلیدی انگلیسی
Iterative learning control,Multi rate processing,Anti-aliasing,Anti-imaging
پیش نمایش مقاله
#### چکیده انگلیسی
Iterative learning control (ILC) is a simple and efficient solution to improve tracking accuracy for systems that execute repetitively the same tracing operation. For engineering applications of ILC, the main concern is the monotonic decay of tracking errors, in the sense of infinity norm or peak error, along the trials. Low cost in implementation and robustness in performance are also critical factors. To achieve these important but sometimes contradicting goals, several multirate ILC schemes have been developed, in which different data sampling rates are used for feedback online loop and feedforward ILC offline loop. That is, multirate ILC uses a different (often lower) rate from the sampling rate of a feedback system to update input. Before the input signal is applied to the system for the next trial, it is upsampled to reach the original sampling rate. Since downsampling will cause distortion of frequency spectra, anti-aliasing and anti-imaging filters and signal extension are used together with downsampling and upsampling operations. In this paper, these technologies are integrated with three different multirate ILC schemes, pseudo-downsampled ILC, two-mode ILC, and cyclic pseudo-downsampled ILC, to achieve better performance. A series of experimental results on an industrial robot are presented to demonstrate the efficiency of multirate ILC schemes and compare the performance. The results demonstrate that multirate ILC schemes are able to achieve not only monotonic learning transient, but also much better tracking accuracy than conventional one-step-ahead ILC schemes.
#### مقدمه انگلیسی
Iterative Learning Control (ILC) is an approach to find appropriate control input for systems that execute the same tracking task repeatedly. It aims to force the output of these systems to follow a trajectory yd(t) defined over a finite time duration T. With technology development, tracking accuracy requirements have come to nano- or micro-meter level. Feedback control alone is often not enough to achieve accuracy at this level due to modeling uncertainties and various disturbances. ILC provides a simple and effective feedforward channel to significantly improve the tracking accuracy with low cost. The basic idea of ILC is to update the input through the recorded tracking error in a previous trial, or iteration. ILC is a batch processing process. After the execution of one trial, the input and error signals are recorded in the memory. Before the start of the next iteration, feedforward ILC controller offline updates the input signal. When the next iteration starts, the calculated input signal is applied to the system. The features of batch processing and off-line calculation enable ILC to employ techniques that cannot be used in real time, such as non-causal filtering. The ILC update law has the general recursive form as equation(1) uj+1=H(uj,ej)uj+1=H(uj,ej) Turn MathJax on where H is the ILC input update function; tracking error is ej=yd−yjej=yd−yj with y d and y j being the desired trajectory and actual trajectory of the j th iteration, respectively. The objective is to make e j converge to zero as iterations go to infinity. To describe e j in a trial with a finite time duration or a finite number of sampling points, a certain norm ∥ej∥∥ej∥ is used. Therefore, ILC aims to achieve limj→∞∥ej∥→0limj→∞∥ej∥→0. Note that ILC is a two-dimensional problem. On the one hand, the system performs the finite-time tracking command on the time axis. On the other hand, ILC adjusts the input to the system on the iteration axis. Time and iteration index are two independent variables ( Elci, Longman, Phan, Juang, & Ugoletti, 2002). Generally speaking, there are two configurations of the ILC scheme. The first one is parallel configuration (Bristow et al., 2006 and de Roover and Bosgra, 2000), in which ILC adjusts the commands to the plant directly. The second one is serial configuration (Longman, 2000), in which ILC adjusts the commands given to the existing closed-loop feedback control system. It has been proved that these two configurations are mathematically equivalent (Solcz & Longman, 1992). Since many commercial products have feedback controllers and it is not desirable to open the feedback control loops, the second configuration is relatively easier for practical implementations. When an iterative learning controller is designed, an important issue that needs to be taken into account is the monotonic convergence of tracking error along the iteration axis (Chang et al., 1992, Longman, 2000 and Wang, 2000). As mentioned earlier, a proper norm is used to describe the tracking error in a trail such that the convergence is considered in the sense of this selected norm. It is well-known that ILC often shows bad transient behavior. That is, the tracking error goes down in the initial iterations but goes up again, usually to a very huge value, before it finally converges to zero. The reason is that many previous analyses are either in the α -norm or λ -norm. The λ -norm for a function f (t ) is ‖f‖λ≜supt∈[0,T]e−λtmax|f(t)|‖f‖λ≜supt∈[0,T]e−λtmax|f(t)| with λ being a positive scalar that usually needs to be sufficiently large. The α -norm is defined as ∥f(·)∥α=supk∈Nf(k)αk∥f(·)∥α=supk∈Nf(k)αk with 0<α<10<α<1. In the sense of these two norms, the error near the terminal phase of the operations is much less weighted than those at the beginning phase of the operations. Due to this decreasing weighting factor, a huge overshoot of error can appear and indicate a bad convergence performance in the sense of the ∞-norm, given by ‖f‖∞=supk∈Nf(k)‖f‖∞=supk∈Nf(k), even with the presence of a mathematical convergence analysis with α -norm or λ -norm. To overcome this bad transient behavior, the convergence should be investigated in the ∞∞-norm and many approaches have been developed ( Frueh and Phan, 2000, Jang et al., 1995, Kuc et al., 1992, Lee-Glauser et al., 1996, Park and Bien, 2002, Ye et al., 2009, Zhang, Wang, & Ye, 2009 and Zhang et al., 2010). The explanation in the frequency domain is that bad transient behavior is due to high frequency error components violating the condition of monotonic decay or the error signal contains a component beyond the ILC system׳s learnable bandwidth (Zhang, Wang, & Ye, 2005). A widely used method to achieve good learning behavior is to introduce a low-pass filter (Chen and Moore, 2001 and Zhang, Wang, & Ye, 2009). However, ILC with such a filter will no longer be able to achieve zero tracking since it cuts off high frequency components. If desired performance requires elimination of error components in high frequencies, this method results in poor tracking accuracy. It is desirable, therefore, to develop ILC to guarantee both transient behavior and high tracking accuracy in the form of infinite norm. In Moore, Chen, and Bahl (2005), Moore et al. derived an exponential convergence condition for P-type ILC and they used time-varying gain to make the condition hold. For most systems in use, the limitation is that the feedback controller is encapsulated and the condition from Moore et al. (2005) often cannot be satisfied. Redesign feedback controller to satisfy the condition is inconvenient (Moore, Chen, & Bahl, 2002). Alternatively, a simple solution to make the condition in Moore et al. (2005) hold is to reduce the sampling rate. Since it is not easy to change sampling frequency for most physical systems, multirate ILC schemes are developed in which the sampled data are processed at different rates. This will bring the design of ILC into the multirate signal processing domain (Zhang, Wang, Wang, et al., 2008, Zhang et al., 2007, Zhang, Wang, Ye, et al., 2008 and Zhang, Wang, Ye, Zhou, et al., 2010). Some other approaches using the multirate concept include optimal ILC for multirate physical systems and multirate model inverse (Oomen et al., 2009 and Shiraishi and Fujimoto, 2010). Another advantage of multirate ILC is that it is able to deal with initial state error properly. The original definition of ILC problem requires the same initial state of each iteration (Longman, 2000). This makes analysis simple and makes zero-error tracking possible. However, this assumption may not hold for real systems because the same initial state sometimes cannot be guaranteed in practice. A research on continuous D-type ILC shows that initial state error can make the learning process unstable (Lee & Bien, 1991). Some methods are proposed to achieve good learning behavior with the presence of initial state error (Chen et al., 1999, Chen et al., 1996, Hillenbrand and Pandit, 2000, Sun and Wang, 2002 and Wang, 2000). It is worth noting that multirate control itself is not new and its capabilities and limitations have been well-studied (Moore, Bhattacharyya, & Dahleh, 1993). One limitation is the degradation in the intersample behavior. This is also true for multirate ILC and we will design novel ILC schemes to overcome this limitation. Based on the work in Moore et al. (2005), several multirate ILC schemes are developed and successfully applied to an industrial robot system (Zhang, Wang, Wang, et al., 2008, Zhang et al., 2007, Zhang et al., 2009, Zhang, Wang, Ye, et al., 2008 and Zhang, Wang, Ye, Zhou, et al., 2010). In these schemes, the downsampling and upsampling cause distortion in signal frequency spectra and deteriorate the learning performance. To solve this problem, some considerations in data processing including signal extension, anti-aliasing, and anti-imaging are investigated in this paper and integrated with these schemes. The remainder of the paper is organized as follows: Section 2 discusses the idea of multirate ILC with its necessary signal processing techniques. Section 3 enhances several multirate ILC schemes by applying the techniques in Section 2. Each of them is followed by another one having better tracking performance. Section 4 presents a series of experimental results of the proposed multirate ILC schemes and their performances are compared, which is followed by concluding remarks in Section 5.
#### نتیجه گیری انگلیسی
In this paper, the ILC problem is considered in the multirate signal processing domain. Three different multirate ILC schemes are developed in which signal extension, anti-aliasing, and anti-imaging are considered. The features and performances of these different multirate ILC schemes are studied and compared with a series of experimental results on an industrial robot system. The results show that, with a proper design, these multirate ILC schemes can guarantee the good transient behavior and achieve a much better tracking performance. In general, pseudo-downsampled ILC is the simplest with very good tracking accuracy, two-mode ILC enhances pseudo-downsampled ILC a little, and cyclic pseudo-downsampled ILC has the best performance.
خرید مقاله
پس از پرداخت، فوراً می توانید مقاله را دانلود فرمایید. | 2018-07-16 09:03:49 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8419339656829834, "perplexity": 1716.9738743767853}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589237.16/warc/CC-MAIN-20180716080356-20180716100356-00555.warc.gz"} |
https://codegolf.stackexchange.com/questions/3104/another-amicable-number-problem?noredirect=1 | # Another amicable number problem
Two numbers are said to be 'amicable' or 'friends' if the sum of the proper divisors of the first is equal to the second, and viceversa. For example, the proper divisors of 220 are: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 which sum up to 284. 284's proper divisors are 1, 2, 4, 71 and 142, which sum to 220, thus 220 and 284 are friends. Write a function which returns true if and only if a number is a friend of some other number.
Examples:
friend(220) ==> true
friend(7) ==> false
friend(284) ==> true
Function with least number of chars wins.
NOTE: I just found out that there is a similar question, but I believe this one is simpler and perhaps more general.
• You are missing some divisors, namely: N is a divisor of N. – Thomas Eding Jul 19 '11 at 20:06
• that is true. I need to indicate that N is not included in its divisors.. – leonardo Jul 19 '11 at 21:01
• You should also indicate the expected return value of friend(6). As of this writing, the winning solution returns true for 6, but 6 is not actually an amicable number (it's perfect). – Quuxplusone Jul 8 '14 at 20:07
## Python, 67
a=lambda x:sum(i for i in range(1,x)if x%i<1)
b=lambda x:x==a(a(x))
• Suggestion xrange(1,x) => range(x) and not x%i to x%i<1 – Charles Beattie Jul 9 '11 at 14:41
• just xrange to range %0 causes an error – Charles Beattie Jul 9 '11 at 15:10
• good tips Charles! – leonardo Jul 11 '11 at 18:18
• Save a char by using a Boolean multiplier in place of the condition: sum(i*(x%i<1)for i in range(1,x)). – xnor Jul 8 '14 at 23:36
o m=sum[x|x<-[1..m-1],mmodx<1]
f m=(o.o)m==m
Call the f function (for "friend").
Didn't provide a main because nobody else is doing so.
• (m-1) -> m-1 – Thomas Eding Jul 19 '11 at 20:04
• @trinithis: thanks – marinus Jul 20 '11 at 12:02
• Haskell was made for problems like this. – eternalmatt Jul 21 '11 at 1:14
eTeX, 177 (yes, that language is too verbose)
Used as etex filename.tex "\a{220}".
• nice. i did not know that language exists... – leonardo Jul 11 '11 at 18:18
• It is an extension of {TeX, written by D. E. Knuth}, written by many people, including P. Breitenlohner. (EDIT: added braces to help parse the sentence) – Bruno Le Floch Jul 11 '11 at 22:06
• I think you can trim it down some more by doing something like \let\n\numexpr and \let\v\advance. Also, it seems like everyone else is just implementing a function that returns a boolean; you might be able to save some more characters by converting it to setting a \newif instead of printing a \message. – ESultanik Jul 20 '11 at 13:30
• @ESultanik: when a primitive is used only twice, it is typically not worth doing \let\x\primitive: this uses 10 chars (\let\x and twice \x) plus the length of the primitive instead of twice the length of the primitive. Both \advance and \numexpr take 8 characters, so it's better here to stick with those. – Bruno Le Floch Jul 25 '11 at 10:38
• @Bruno: Oops! For some reason I thought it would save a character or two; I must have messed up the arithmetic in my head. Sorry about that. – ESultanik Jul 25 '11 at 12:22
## Scala, 67
def d(n:Int)=(1 to n-1).filter(n%_==0).sum
def f(n:Int)=d(d(n))==n
Ruby, 95
l=lambda{|n|(1..n-1).select{|e|n.modulo(e)==0}.reduce(:+)}
m=lambda{|n|(l.call(l.call(n))==n)}
## Ruby 1.9, 57 characters
s=->n{eval (1...n).select{|a|n%a<1}*?+}
f=->n{s[s[n]]==n}
Pretty similar to all the other solutions here.
# J, 24 characters
=(+/&(((0:=|~)#])i.))^:2
It's a monad that takes a scalar and returns 0 or 1 if it's a friend or not. Unfortunately J precedence rules make it impossible to use with no separator from its argument. I'm keeping the character count as such because of the way the question is worded. Anyway, here's a demonstration of three possible ways to use it:
NB. using a named verb
f =: =(+/&(((0:=|~)#])i.))^:2
f 220
1
NB. using parentheses
(=(+/&(((0:=|~)#])i.))^:2) 7
0
NB. using verb "Same"
=(+/&(((0:=|~)#])i.))^:2 [ 284
1
Obligatory J unscrambling:
• i. 220 returns the list of naturals below 220 (0, 1, 2 to 219)
• (|~i.) 220 divides them all by 220 and returns the remainder
• ((0:=|~)i.) 220 compares to 0 (returns boolean as 0 or 1)
• (((0:=|~)#])i.) 220 returns the natural where the 1s were. In effect: returns the full divisor list.
• (+/&(((0:=|~)#])i.)) 220 sums them.
• ((+/&(((0:=|~)#])i.))^:2) 220 performs the operation twice.
• (=(+/&(((0:=|~)#])i.))^:2) 220 checks we fall back on the initial argument.
## APL (23)
Let me halve my score after 3 years of golfing. :)
This needs the function trains added to Dyalog APL 14, so it doesn't work on earlier versions, which at the moment of writing unfortunately includes the free (unregistered) version. It does work on TryAPL.
f←{+/∆×0=⍵|⍨∆←⍳⍵-1}⍣2=+
Or:
({+/∆×0=⍵|⍨∆←⍳⍵-1}⍣2=+)
(Because it involves a function train, it needs to be either parenthesized or named in order to use it. They both have the same character count, but the first needs to be entered and then called as f 220, the second needs to have its argument added to the right of it.)
Explanation:
• {...}⍣2=+: compare the argument to the result of running the function twice on the argument, return 1 if true and 0 if false
• ∆←⍳⍵-1: get all the numbers from 1 to ⍵-1 and store them in ∆.
• 0=⍵|⍨∆: get a binary vector which is 1 where ⍵ mod ∆ is 0 and 0 elsewhere
• +/∆×: multiply ∆ by it and return the sum
# Brachylog, 11 bytes
fk+Xfk+?;X≠
Try it online!
Takes input through the input variable and outputs through success or failure.
fk+ The proper divisor sum of
the input
X is X,
fk+ and its proper divisor sum
? is the input,
;X≠ which is not equal to X.
# Q, 42 33
{x={0+/a(&)0=x mod a:(!)6h$(1+x%2)}/[2;x]} shorter by not limiting the check for divisors to enumerations up to n/2 but obviously slower as a result: {x={0+/a(&)0=x mod a:(!)x}/[2;x]} timing q)\t {x={0+/a(&)0=x mod a:(!)x}/[2;x]} 2200000 289 q)\t {x={0+/a(&)0=x mod a:(!)6h$(1+x%2)}/[2;x]} 2200000
124
# JavaScript (E6) 57
F=n=>(S=n=>{for(t=d=1;++d<n;)n%d||(t+=d)},S(n),S(t),t==n)
# Pyth, 22 characters
Explanation:
Rs return sum(
f!%dT filter on condition (not d%T)
r1d for T in range(1,T)
DMk def M(k):
R return
qkAAk k==A(A(k))
# 05AB1E, 7 bytes
ѨOѨOQ
Try it online!
Pretty straightforward. Ѩ takes the list of divisors without the element itself, O sums these up. Q checks if after two iterations the result and the input are equal.
Returns 1 if the number is amicable, else 0.
## GolfScript (25 chars)
{.{:a,(\{.a\%!*+}/}2*=}:f
a can, of course, be replaced by the name of any other variable whose contents you don't care about. | 2019-10-21 06:20:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.457999050617218, "perplexity": 2401.0505681503314}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987756350.80/warc/CC-MAIN-20191021043233-20191021070733-00317.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-12-sequences-and-series-12-1-define-and-use-sequences-and-series-12-1-exercises-skill-practice-page-798/44 | ## Algebra 2 (1st Edition)
$\displaystyle \sum_{n=1}^{\infty}(n^{2}-2)$
We first start with the perfect squares: 1, 4, 9, 16, 25, ... $\qquad$(squares of 1,2,3,...) Starting with n=1, the terms in the sum are $a_{n}=n^{2}-2$, and there is no upper limit on n. Sum = $\displaystyle \sum_{n=1}^{\infty}(n^{2}-2)$ | 2022-12-06 04:17:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6133342385292053, "perplexity": 385.5000162240163}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711069.79/warc/CC-MAIN-20221206024911-20221206054911-00502.warc.gz"} |
https://brilliant.org/problems/a-fibonacci-set-of-sticks/ | # A Fibonacci Set of Sticks
Geometry Level 3
Recently, I was cleaning up my attic, and I found a set of sticks which, some years ago, a curious Italian gentleman sold me. I was trying to figure out why I bought it from him, and I realized that the set has the incredible property that there are no $$3$$ sticks that can form a triangle. If the set has two sticks of length $$1$$, and these are the smallest, what is the least possible length of the $${ 14 }^\text{th}$$ stick?
Assumptions: My set of sticks has at least 14 sticks.
× | 2016-10-25 08:40:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35215502977371216, "perplexity": 431.721547668876}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720000.45/warc/CC-MAIN-20161020183840-00231-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://stats.libretexts.org/Bookshelves/Applied_Statistics/Book%3A_Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/15%3A_Linear_Regression/15.02%3A_Estimating_a_Linear_Regression_Model | # 15.2: Estimating a Linear Regression Model
Okay, now let’s redraw our pictures, but this time I’ll add some lines to show the size of the residual for all observations. When the regression line is good, our residuals (the lengths of the solid black lines) all look pretty small, as shown in Figure 15.4, but when the regression line is a bad one, the residuals are a lot larger, as you can see from looking at Figure 15.5. Hm. Maybe what we “want” in a regression model is small residuals. Yes, that does seem to make sense. In fact, I think I’ll go so far as to say that the “best fitting” regression line is the one that has the smallest residuals. Or, better yet, since statisticians seem to like to take squares of everything why not say that …
The estimated regression coefficients, $$\ \hat{b_0}$$ and $$\hat{b_1}$$ are those that minimise the sum of the squared residuals, which we could either write as $$\sum_{i}\left(Y_{i}-\hat{Y}_{i}\right)^{2}$$ or as $$\sum_{i} \epsilon_{i}^{2}$$.
Yes, yes that sounds even better. And since I’ve indented it like that, it probably means that this is the right answer. And since this is the right answer, it’s probably worth making a note of the fact that our regression coefficients are estimates (we’re trying to guess the parameters that describe a population!), which is why I’ve added the little hats, so that we get $$\ \hat{b_0}$$ and $$\ \hat{b_1}$$ rather than b0 and b1. Finally, I should also note that – since there’s actually more than one way to estimate a regression model – the more technical name for this estimation process is ordinary least squares (OLS) regression.
At this point, we now have a concrete definition for what counts as our “best” choice of regression coefficients, $$\ \hat{b_0}$$ and $$\ \hat{b_1}$$. The natural question to ask next is, if our optimal regression coefficients are those that minimise the sum squared residuals, how do we find these wonderful numbers? The actual answer to this question is complicated, and it doesn’t help you understand the logic of regression.215 As a result, this time I’m going to let you off the hook. Instead of showing you how to do it the long and tedious way first, and then “revealing” the wonderful shortcut that R provides you with, let’s cut straight to the chase… and use the lm() function (short for “linear model”) to do all the heavy lifting.
## 15.2.1 Using the lm() function
The lm() function is a fairly complicated one: if you type ?lm, the help files will reveal that there are a lot of arguments that you can specify, and most of them won’t make a lot of sense to you. At this stage however, there’s really only two of them that you care about, and as it turns out you’ve seen them before:
• formula. A formula that specifies the regression model. For the simple linear regression models that we’ve talked about so far, in which you have a single predictor variable as well as an intercept term, this formula is of the form outcome ~ predictor. However, more complicated formulas are allowed, and we’ll discuss them later.
• data. The data frame containing the variables.
As we saw with aov() in Chapter 14, the output of the lm() function is a fairly complicated object, with quite a lot of technical information buried under the hood. Because this technical information is used by other functions, it’s generally a good idea to create a variable that stores the results of your regression. With this in mind, to run my linear regression, the command I want to use is this:
regression.1 <- lm( formula = dan.grump ~ dan.sleep,
data = parenthood )
Note that I used dan.grump ~ dan.sleep as the formula: in the model that I’m trying to estimate, dan.grump is the outcome variable, and dan.sleep is the predictor variable. It’s always a good idea to remember which one is which! Anyway, what this does is create an “lm object” (i.e., a variable whose class is "lm") called regression.1. Let’s have a look at what happens when we print() it out:
print( regression.1 )
##
## Call:
## lm(formula = dan.grump ~ dan.sleep, data = parenthood)
##
## Coefficients:
## (Intercept) dan.sleep
## 125.956 -8.937
This looks promising. There’s two separate pieces of information here. Firstly, R is politely reminding us what the command was that we used to specify the model in the first place, which can be helpful. More importantly from our perspective, however, is the second part, in which R gives us the intercept $$\ \hat{b_0}$$ =125.96 and the slope $$\ \hat{b_1}$$ =−8.94. In other words, the best-fitting regression line that I plotted in Figure 15.2 has this formula:
$$\ \hat{Y_i} = -8.94 \ X_i + 125.96$$
## 15.2.2 Interpreting the estimated model
The most important thing to be able to understand is how to interpret these coefficients. Let’s start with $$\ \hat{b_1}$$, the slope. If we remember the definition of the slope, a regression coefficient of $$\ \hat{b_1}$$ =−8.94 means that if I increase Xi by 1, then I’m decreasing Yi by 8.94. That is, each additional hour of sleep that I gain will improve my mood, reducing my grumpiness by 8.94 grumpiness points. What about the intercept? Well, since $$\ \hat{b_0}$$ corresponds to “the expected value of Yi when Xi equals 0”, it’s pretty straightforward. It implies that if I get zero hours of sleep (Xi=0) then my grumpiness will go off the scale, to an insane value of (Yi=125.96). Best to be avoided, I think. | 2021-10-19 00:36:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.790635883808136, "perplexity": 516.6264554135117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585215.14/warc/CC-MAIN-20211018221501-20211019011501-00189.warc.gz"} |
https://milesberry.net/2017/01/computing-in-english-schools/ | 2014 saw some radical changes to the content of England’s national curriculum. Perhaps one of the most significant changes was the replacement of the subject “Information and communication technology”, with a focus on end-user skills using a range of office and media applications, with a new subject, “Computing”, where the emphasis was rather on the conceptual understanding and knowledge of the principles of computer science, the applications of computational thinking and the craft of computer programming. There has been much international interest in these changes, with many countries following England’s lead in making changes to their own curricula to incorporate programming and other aspects of computer science. Here, I discuss a number of the rationales offered for such a radical change, offer a brief narrative of the process of moving from one curriculum to another, survey some of the content of the new curriculum and conclude with an outline of the role of the Computing At School community in implementing this curriculum.
## Some rationales
Whilst many would argue that elementary, or indeed secondary, education affords no place for a discipline as specialised as computer science, there are a number of convincing rationales for including this subject, alongside music, poetry and physics, as part of the educational entitlement for all children. Such rationales might include one or more of the following.
• Children’s inherent curiosity in the world around them. It is hard to deny that digital technology is now almost ubiquitous. A typical infant is likely to encounter touch screen devices such as smartphones or tablet computers on her mother’s lap, before she even learns to speak. Children are naturally curious about the world in which they grow up, and thus are curious about the technology which is, for them, an inherent part of that world. The child grows up, yes, using digital technology, but also wonder how it is made and how it works. Part of the school’s role is to nurture this curiosity and help the child learn for herself the answer to such questions:
“One of the main educational tasks of the primary school is to build on and strengthen children’s intrinsic interest in learning and lead them to learn for themselves” 1
• Computer science as part of scientific understanding. There is more to computer science as a discipline than ‘merely’ understanding how technological artefacts function and are developed. Computer science, like the other, older, sciences seeks to provide insight into the nature of reality itself. Computability is one example of this: it seems in the nature of our universe that problems fall into three categories: those which are easy to solve, those that are hard to solve and those which can never be solved. Theoretical computer science considers ideas such as these, and provides a means to build useful systems using them. 2
• Economic benefits. For ministers, the need to sustain England’s vibrant software industries must undoubtedly have been a priority. In order to maintain successful games, visual effects, fintech and cybersecurity sectors, a steady stream of high calibre computer science graduates is needed, and the decline in the number of students studying such disciplines was thought best addressed through ensuring exposure to computer science for all as part of school education. In addition to this, computing is a largely meritocratic field with much potential for social mobility - home background and educational privilege are less important than the quality of the code a candidate can write.
• Preparation for an uncertain future. Few would doubt that computers are likely to play a more significant role in our lives, and the life of our society, in our future than in our past or present. It can be argued that the best way to prepare for such a future is to through acquiring an understanding of how computers work and of how they are programmed: this seems more likely to be useful in the long term than skills of current relevance such as centering text on presentation slides. In a society in which computers will play an increasingly significant role, our duty as educators is to empower our pupils to take charge of the machines: as Douglas Rushkoff memorably puts it:
“Program or be programmed.” 3
• As part of a rounded, liberal education. Much of what is included in school curricula is less about vocational training for the world of work and more about providing a broad and balanced education, in which pupils develop an understanding of their world and are empowered to make positive changes to that world through expressing themselves creatively. Froebel’s gifts4 allow young children to learn such fundamentals as the conservation of number, the conservation of volume, the conservation of shape and the direction of gravity, but they also allow kindergarteners to problem solve and build something of their own choosing. The same is true of programming languages: they are a means to understanding technology and reality, and a medium for creative expression. Just as Frank Lloyd-Wright’s later work as an architect was coloured by his kindergarten experience of playing with Froebel’s gifts 5, so too will the software engineers and computer scientists of the future draw on their early experience of thinking computationally and coding with blocks and floor turtles.
## From ICT to computing
Much of the inspiration for computing, particularly computer programming, in school education comes from Seymour Papert’s work in the 70s, 80s and 90s, particularly in relation to the educationally focussed programming language Logo, its implementation of turtle graphics, and the constructionist theory of learning. Back in 1980, Papert wrote:
In many schools today, the phrase “computer-aided instruction” means making the computer teach the child. One might say the computer is being used to program the child. In my vision, the child programs the computer and, in doing so, both acquires a sense of mastery over a piece of the most modern and powerful technology and establishes an intimate contact with some of the deepest ideas from science, from mathematics, and from the art of intellectual model building. 6
Little seems to have changed in the intervening time. Much of the educational use of computers offers little more than drill-and-practice behaviourist learning, which undoubtedly has its place but seems a poor alternative to children taking charge of the computer and using it as a tool for a far more creative, exploratory mode of learning. Crucially for Papert, programming was never an end in itself, but the means through which children connect to deep ideas in mathematics, science and philosophy. The same vision motivates much of what has been included in England’s computing curriculum.
For many of those involved in drafting and then implementing this curriculum, there was a feeling of needing to move children on, beyond what they had already achieved, or might achieve for themselves without instruction from teachers. By and large, children moving from primary to secondary education in England (i.e. at the age of eleven), were already:
• effective users of technology;
• good at communicating digitally, even if they would not always observe the terms and conditions of the services they used;
• able to do the equivalents of reading and writing in the digital domain;
• able to keep themselves reasonably safe when using online technologies; and
• able to demonstrate a sound portfolio of digital skills.
However, there was some sense that, for all they knew, the technologies they were using might as well be based on magic7.
The ambition was to move children’s learning on, beyond this level, so that they would also be:
• able to make technological artefacts, and potentially new technologies;
• able to collaborate productively online;
• critical of content, tools and assumptions when working online;
• responsible in their own use of technology, with some sense of the consequences for others of their actions; and
Thus replacing the sense of technology being almost magical with some concrete knowledge of the principles on which it depends.
Despite the reference to technology above, the new curriculum concerns itself far less with technology than it does with the fundamental principles and concepts of computer science. In the introduction to the earlier, non-statutory CAS Computer Science Curriculum 8, it was stated:
Computer science encompasses foundational principles and widely applicable ideas and concepts. It incorporates techniques and methods for solving problems and advancing knowledge, and a distinct way of thinking and working that sets it apart from other disciplines. It has longevity (most of the ideas and concepts that were current 20 or more years ago are still applicable today), and every core principle can be taught or illustrated without relying on the use of a specific technology.
This concept of computer science as a foundational discipline like physics or history undoubtedly coloured the development of the curriculum.
In August 2012, the British Computer Society and the Royal Academy of Engineering were approached by the government Department for Education to draft a new programme of study for computer (or ICT as it was still called at the time), as expert advice to ministers. Under the chairmanship of Simon Peyton Jones, a diverse group of stakeholder representatives were assembled, drawing on sector representative bodies, multinational technology companies, subject associations, universities and schools.
The drafting panel took inspiration from the Royal Society’s report on computing in schools 9, which recommended that ICT as a subject be replaced with a new subject, computing, recognised as having three distinct but inter-related components: computer science, information technology and digital literacy. These can be thought of as the foundations, applications and implications of the discipline.
In the context of Google search, we might ask at the foundation, computer science level, how does the Page Rank algorithm determine the relevance of a page and how are search queries served so quickly irrespective of geographic location; at the application, IT level, how might results be filtered more effectively to show those only in my language or from a particular range of dates; and at the implication or digital literacy level, what data does Google store on me so that it can better target advertising for me, and is this a price worth paying to use their search engine for free?
The first sentence of the new curriculum sets out it’s vision, and how computing forms part of a liberal education designed to provide children with an understanding of their world and the tools needed to contribute creatively to it:
A high-quality computing education equips pupils to use computational thinking and creativity to understand and change the world. 10
Whilst creativity has long been an element of elementary and secondary education, the ‘computational thinking’ referred to here is relative new. Although Seymour Papert mentioned the term in Mindstorms6, computer scientist Jeanette Wing is widely credited with popularising the idea and has been instrumental in the emphasis that this concept has received in computing curricula worldwide. For Wing, computational thinking is:
“… the thought processes involved in formulating problems and their solutions so that the solutions are represented in a form that can be effectively carried out by an information processing agent” 11
In helping teachers understand what this might mean in practice, the Computing At School Barefoot Computing project12 outlined six concepts of computational thinking and five approaches. The concepts are:
• Logical reasoning - computers are deterministic machines and we can reason logically about how they will behave. Pupils learn to predict confidently the output from a program, to detect and correct errors in their own and others’ programs, and the basics of Boolean logic.
• Algorithms - computer programming requires that the programmer has a clear idea of how to achieve a particular objective before she begins coding, and that some approaches to solving a problem are inherently more efficient than others.
• Decomposition - when faced with large problems, engineers break these up into smaller parts, and address each of these separately.
• Generalisation - engineers will often look for solutions to more general problems, borrowing liberally on the work and ideas of others who have been successful in solving equivalent or related problems.
• Abstraction - programs and computers are complex systems, and a multilayered model of abstraction has been necessary to manage the complexity of these systems, in which the detail of implementation can be effectively hidden. In tackling complex problems, a similar approach is helpful, in which the solver’s focus rests on the necessary detail.
• Evaluation - it is necessary to review whether potential solutions are correct, suitable, reliable, efficient and elegant.
Whilst all these can be effectively learnt in the context of computer programming, they also have wide applications across and beyond the school curriculum, and might provide a set of core competencies in any problem solving work, particularly when this involves using computers.
Alongside these concepts are a number of approaches, which might describe how software engineers and others tackle problems, and provide a foundational model for pedagogic practice in computing education:
• Tinkering - a certain playfulness, or willingness to explore and experiment seems to characterise the work of many leading software engineers, and links closely with play as a powerful pedagogic approach in early years education.
• Creating - computing is an inherently creative discipline, and Papert’s constructionist insights suggest that a learner’s conceptual understanding might best be developed through the conscious construction of knowledge artefacts designed to be shared with others.
• Persevering - programming is undeniably hard, but for many this is where its joy lies; encouraging pupils to develop a ‘growth mindset’13 and an associated willingness to persevere in the face of challenges is important, not just for success in computing14.
• Debugging - One particularly powerful way to help pupils develop such a growth mindset is through the repeated experience of detecting and correcting the mistakes (or ‘bugs’) in their own algorithms and programs.
• Collaborating - Large computer programs are written by large, well-coordinated teams of people, often widely distributed. There’s evidence that pair-programming is an effective methodology for agile software development15, and having pupils work with a partner on programming tasks seems to be motivating and pedagogical effective.
## Curriculum content
The revised national curriculum includes relatively brief requirements for what should be taught to 5-16 year olds - the content itself fits comfortably on three sides of A4 paper. For younger pupils, provision is governed by the ‘Early Years Foundation Stage Framework’16.
### Before the age of five
For the youngest pupils in nursery schools and school reception classes, there’s, not surprisingly, no requirement that pupils be taught to use digital technology or learn to program. However, the framework for these children’s education does list a number of ‘characteristics of effective learning’, including ‘creating and thinking critically’. The non-statutory Development Matters guidance17 lists a number of aspects of this, such as ‘finding ways to solve problems’, ‘making links and noticing patterns in their experience’ and ‘planning, making decisions about how to approach a task, solve a problem and reach a goal’. There are strong parallels here with the concepts and approaches of computational thinking.
It seems more appropriate in early years education to encourage young children to plan systematically, develop resilience and to make predictions, thus laying the foundations of computational thinking, than to introduce them to programming per se, although products such as Bee-Bots and Scratch Jr have done much to make this accessible.
### Ages five to seven
The curriculum requirements for ages five to seven include that pupils be taught to understand what algorithms are. Pupils will learn about algorithms as sequences of steps or sets of rules. They’ll typically do this away from computers, through activities which involve them following a teacher’s instructions, giving instructions to their peers or working out their own instructions for practical activities such as sharing a pile of sweets18.
Pupils are also taught how algorithms are implemented as programs on ‘digital devices’. The phrasing here is deliberate, so as to encourage teachers to use floor turtles and other robots with young learners rather than going directly to on screen coding. It seems easier for pupils to put themselves in the place of a floor turtle to step through programs themselves (cf Papert’s ‘body syntonic’ reasoning6), thus making it easier for them to reason about their programs. Reasoning logically about programs and algorithms is a recurring theme in the curriculum - it’s considered crucial for pupils to be able to think about their code rather than just code.
### Ages seven to eleven
Between the ages of seven and eleven, pupils’ programming becomes more formal, as they learn to use a greater variety of structures in their code, beyond simple sequences of steps. They’re taught about selection and repetition, as well as the use of input and output and are introduced to variables as a data structure. Typically their programs will be written in MIT’s block-based toolkit Scratch19, although this is not explicitly specified in the programme of study.
Logical reasoning is again emphasised, with pupils expected to explain how simple algorithms work, as well as detecting and correcting errors in algorithms and programs.
Other elements of computer science are covered too: pupils are taught how computer networks including the internet work, and how they can provide services such as the World Wide Web. Typically, this will draw on ‘unplugged’ approaches involving classroom based simulations, although adventurous schools might introduce pupils to command line networking tools such as ping, nslookup and traceroute.
### Ages 11-14
In the first years of secondary education, pupils are introduced to the idea of a computational abstraction which models the state and behaviour of a system. This might be a simple game, although computational models for the spread of epidemics or the growth of a culture of cells are interesting alternatives.
Pupils’ knowledge of algorithms is now extended to include ‘key algorithms that reflect computational thinking’. Typically schools interpret this as algorithms for search and sort, but it might also include some mathematical algorithms like Euclid’s algorithm for highest common factors and the Sieve of Erastothenes. Pupils learn that some algorithms provide more efficient solutions to the same problem.
Pupils extend their knowledge of how computers work, learning about the hardware and software components of computer systems and how these interoperate. Teachers have found that simpler systems, such as the BBC micro:bit20 make it easier to scaffold this understanding.
### Ages 14-16
There are minimal statutory requirements for computing in the national curriculum beyond the age of 14, but computing does remain a compulsory subject for pupils at this age. The requirements can be satisfied through embedding computer science, information technology and digital literacy across the rest of the curriculum, or through providing a number of focus days or independent project work over the course of these two school years.
In addition to this, there’s a requirement that pupils “must have the opportunity to study” IT and computer science in greater depth - this is interpreted as meaning that schools should offer courses leading to public exams in IT or computing at 16+, but not require pupils to enter such exams.
Detailed specifications have been developed by English exam boards for General Certificate of Secondary Education (GCSE) qualifications in computer science21. These include aspects of computational thinking including algorithm design, knowledge of Boolean logic and binary representation and an extended practical programming project, most typically undertaken in Python.
### Ages 16-18
Between the ages of 16 and 18 computer science becomes an elective subject, with relative small, but growing, numbers of pupils continuing to study the subject at this age.
Again, English exam boards have developed detailed specifications for qualifications at this level22. These include some deliberately ambitious material, such as an introduction to functional programming, often taught in Haskell, graphs and algorithms for graph traversals and shortest paths and big-O notation. They also include a significant, more extensive project which might include the development of a computer game or simulation with a graphical user interface or an investigation of topics such as machine learning or 3-d graphics.
Whilst such qualifications naturally lead on to undergraduate degrees in computer science, they are increasingly recognised as useful preparation for degrees in many other disciplines which make use of computing, such as engineering, natural and social sciences, teaching and medicine.
## Computing At School
This significant change in curriculum content and the associated assessment framework has taken place at a time when government has deliberately stepped back from the implementation of such changes. Currently in the UK, the view is that:
Government should only do what only government can do.23
Much of the implementation of computing as a curriculum subject in England had been achieved through the Computing At School (CAS) group24, the UK subject association for computer science, formally part of BCS, the Chartered Institute for IT.
CAS has seen rapid growth in its membership since its beginnings as a small working group of computing teachers and academics. Membership currently stands at 25,653 with some 223 local hubs.
The main challenge for implementing the computing curriculum has been the need to ensure sufficient numbers of teachers in schools with sufficient confidence to make the new subject a success:
The hardest part of getting great computer science in every school is getting a great computer science teacher in every school.25
Providing great computer science teachers for every school can be done through two routes: initial teacher training and continuous professional development. The former is now addressed through the development of subject knowledge entry requirements for specialist computer science teachers26 and through the provision of generous tax-free training bursaries and scholarships for those with good degrees in computer science who choose to train to become teachers27.
Great computer science teaching demands three things: great teaching skills, great technology skills and great computer science subject knowledge28. The lattermost is a particularly challenge: in primary schools very few teachers have any background in computer science or software development; even in secondary a minority of those who were teaching the former ICT curriculum had a degree in computer science9.
In developing continuing professional development programmes for computing, Computing At School has prioritised addressing teachers’ subject knowledge. The most significant programme in this area has been the Network of Excellence in Computer Science Teaching, comprising around 400 ‘Master Teachers’, serving classroom teachers supported, and partly funded, to provide continuing professional development training and peer-support to those teaching computer science in their area. More recently, the Network of Excellence has been re-configured on a regional basis, with ten universities acting as regional centres, typically drawing on the combined expertise of their computer science and education faculties.
CAS has also developed a range of resources to support teachers, including introductory guides to the curriculum, QuickStart handbooks with associated CPD resources[^QuickStart], the Barefoot Computing programme for primary computing12, and Tenderfoot Computing for lower secondary teachers[^Tenderfoot]. CAS members. CAS members have developed further computing resources themselves, and CAS has an active, gift economy culture of members sharing their resources under liberal licences with the rest of the community: at present, 3,863 resources have been shared in this way.
## Concluding remarks
It is still too early to judge whether the changes to England’s curriculum have been a success. There is undoubtedly much interest in programming and coding amongst pupils in English schools, and some surveys suggest that implementation has, given that there has been little involvement in this process from central government, been really quite successful: BT commissioned an Ipsos MORI survey of 400 primary teachers reporting that 81% are now confident in teaching computing29. Britain’s Royal Society are currently conducting wider ranging research and their report is eagerly anticipated30.
Exam entries at GCSE for computing have significantly increased in the period since the introduction of this qualification, from 4,253 in 2013 to 62,454 in 2016. At A Level, there has been a more modest increase, from 3,758 to 6,242 in the same period31. There’s also evidence of an increase in recruitment to university computer science courses over this time frame32, and many are optimistic that these trends will continue.
Perhaps the most significant outcome of England’s decision to include computer science in its curriculum is that this subject now becomes an entitlement for all, which, it is hoped, will go a long way to address inclusion and equity in this domain.
Published in German, November 2017 as Das Unterrichtsfach “Computing” in Paderborner Podium 10 Bildung im digitalen Zeitalter - Bilanz und Perspektiven, Paderborn: Heinz Nixdorf MuseumsForum. (c) all rights reserved.
1. Plowden, B., 1967. Children and their primary schools: A report of the Central Advisory Council for Education (England). London: HM Stationery Office.
2. Bentley, P.J., (2012) Digitized: The Science of Computers and How it Shapes our World. Oxford: Oxford University Press.
3. Rushkoff, D., 2010. Program or be programmed: Ten commands for a digital age. Or Books.
4. See, e.g, Resnick, M., 1998. Technologies for lifelong kindergarten. Educational technology research and development, 46(4), pp.43-55.
5. Lloyd-Wright, F., 2005. Frank Lloyd Wright: An Autobiography. Pomegranate.
6. Papert, S., (1980) Mindstorms: Children, Computers, and Powerful Ideas. New York: Basic Books 2 3
7. qv Clarke, A. C. (1973). Profiles of the Future: An Inquiry into the Limits of the Possible. Popular Library.
8. CAS (2012). Computer science: a curriculum for schools, Cambridge: CAS.
9. Furber, S. (2012) Shut down or restart. London: The Royal Society 2
10. Department for Education (2013) The national curriculum in England Framework document. London: DfE.
11. Wing, J. 2011. Research Notebook: Computational Thinking - What and Why? The Link. Pittsburgh, PA: Carneige Mellon.
12. Barefoot Computing (2014) Computational Thinking 2
13. Dweck, C (2012) Mindset. New York NY, Random House.
14. Cutts, Q., Cutts, E., Draper, S., O’Donnell, P., and Saffrey, P. (2010) Manipulating mindset to positively influence introductory programming performance. In: SIGCSE ‘10 Proceedings of the 41st ACM Technical Symposium on Computer Science education, Milwaukee, USA, 10-13 Mar 2010, pp. 431-435.
15. Williams, L. A., & Kessler, R. R. (2000). All I really need to know about pair programming I learned in kindergarten. Communications of the ACM, 43(5), 108-114.
16. DfE (2014) Statutory framework for the early years foundation stage. London: DfE
17. Early Education (2012) Development Matters in the Early Years Foundation Stage. London: Early Education.
18. Barefoot Computing (2014). Sharing sweets activity. (free registration required)
19. See, for example, OCR (2016) Specification (Accredited) - GCSE Computer Science - J276
20. See, for example, AQA (2015) AS and A-level Computer Science
21. GDS (n.d.) Government Digital Service Design Principles
22. Quote taken CS for All summit, White House, Washington DC, 14/9/16.
23. Teaching Agency (2012) Subject knowledge requirements for entry into computer science teacher training
24. Teaching Agency (2017) Funding for training to teach computing
25. Mishra, P. and Koehler, M.J., 2006. Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers college record, 108(6), p.1017.
26. BT and Ipsos MORI (2016). A new cornerstone of modern primary school education
27. Royal Society (2016) Computing Education
28. Data from JCQ | 2021-06-22 20:49:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2575992941856384, "perplexity": 2509.8648199101085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488519735.70/warc/CC-MAIN-20210622190124-20210622220124-00353.warc.gz"} |
https://cs.stackexchange.com/questions/29697/understanding-the-time-complexity-of-insertion-sort | Understanding the time-complexity of Insertion Sort
From my textbook, I am studying the time-complexity of the insertion sort algorithm (shown below).
INSERTION-SORT(A) cost times
1 for j <- 2 to length[A] c1 n
2 DO key <- A[j] c2 n - 1
3 ▷ Insert A[j] into the sorted
▷ sequence A[i..j-1] 0 n - 1
4 i <- j-1 c4 n - 1
5 while i > 0 and A[i] > key c5 sum_{j=2}^2 t_j
6 do a[i+1]<-A[i] c6 sum_{j=2}^2 (t_j-1)
7 i <- i-1 c7 sum_{j=2}^2 (t_j-1)
8 A[i+1] <- key c8 n - 1
The algorithm above shows the times that each statement is executed. But wait, why is line 1 executed n times?
Shouldn't line 1 be executed n-1 times since insertion sort starts making comparisons at the second element at the list. | 2020-04-09 21:17:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3671753704547882, "perplexity": 6648.947776042044}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371876625.96/warc/CC-MAIN-20200409185507-20200409220007-00475.warc.gz"} |
https://www.postonline.co.uk/post/news/1206726/ratings-scrutiny | # Ratings come under scrutiny.
Lloyd's insiders display consistently mixed feelings about the big
three insurance rating agencies. It is not clear how many agree with the
UK newspaper The Independent that "the credit rating | 2018-11-18 10:45:51 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8704792857170105, "perplexity": 14636.69232038608}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744348.50/warc/CC-MAIN-20181118093845-20181118115845-00125.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=130&t=41557&p=142379 | ## delta u
$\Delta U=q+w$
Briana Yik 1H
Posts: 30
Joined: Fri Sep 28, 2018 12:20 am
### delta u
When there is constant temperature, is delta u always 0? Thank you
Posts: 77
Joined: Fri Sep 28, 2018 12:28 am
Been upvoted: 1 time
### Re: delta u
I don't think so just because delta U is made up by the sum of heat and work done and work can still be done even if there's constant temperature.
Karan Thaker 2L
Posts: 75
Joined: Fri Sep 28, 2018 12:26 am
### Re: delta u
I agree... There are other factors that affect delta U which can alter it such as q and w even if temperature is constant. | 2019-12-14 22:28:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47526848316192627, "perplexity": 5813.47770444785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541294513.54/warc/CC-MAIN-20191214202754-20191214230754-00128.warc.gz"} |
http://atlas.math.umd.edu/software/documentation/atlasofliegroups-docs/library/axis/def_func_op.html | # Defining Functions and Operators¶
A main use of the axis language is to allow users to define their own functions. In practice these are often simple functions of not more than a few lines, providing some simple pre- or post-processing around built-in operations rather than implementing a whole new algorithm, and the language is particularly well adapted to such usage; nonetheless it also allows for writing elaborate functions whose body can be as large as one wishes.
Often it is convenient to have user defined functions invoked from formulas using operator symbols. While one cannot define new operator symbols, one can overload existing ones for new argument types. Defining operator overloads is very similar to defining (global) functions.
The normal way to define a function is using a variation of the ‘set’ syntax (which also has counterpart for ‘let’), as in:
atlas> set f (LieType lt) = void:
: > let rd = simply_connected (lt)
L > then ic = inner_class(rd,"s")
L > then rf = quasisplit_form (ic) , drf = dual_quasisplit_form (ic)
L > in print_block(block(rf,drf))
Defined f: (LieType->)
atlas> f("A2")
0(0,5): 0 0 [C+,C+] 2 1 (*,*) (*,*) e
1(1,4): 1 0 [i2,C-] 1 0 (3,4) (*,*) 2,1
2(2,3): 1 0 [C-,i2] 0 2 (*,*) (3,5) 1,2
3(3,0): 2 1 [r2,r2] 4 5 (1,*) (2,*) 1,2,1
4(3,1): 2 1 [r2,rn] 3 4 (1,*) (*,*) 1,2,1
5(3,2): 2 1 [rn,r2] 5 3 (*,*) (2,*) 1,2,1
atlas> f("G2")
0(0,9): 0 [i1,i1] 1 2 ( 4, *) ( 3, *) 0 e
1(1,9): 0 [i1,ic] 0 1 ( 4, *) ( *, *) 0 e
2(2,9): 0 [ic,i1] 2 0 ( *, *) ( 3, *) 0 e
3(3,8): 1 [C+,r1] 6 3 ( *, *) ( 0, 2) 2 2
4(4,7): 1 [r1,C+] 4 5 ( 0, 1) ( *, *) 1 1
5(5,6): 2 [C+,C-] 8 4 ( *, *) ( *, *) 1 2,1,2
6(6,5): 2 [C-,C+] 3 7 ( *, *) ( *, *) 2 1,2,1
7(7,4): 3 [i2,C-] 7 6 ( 9,10) ( *, *) 2 2,1,2,1,2
8(8,3): 3 [C-,i2] 5 8 ( *, *) ( 9,11) 1 1,2,1,2,1
9(9,0): 4 [r2,r2] 10 11 ( 7, *) ( 8, *) 3 1,2,1,2,1,2
10(9,1): 4 [r2,rn] 9 10 ( 7, *) ( *, *) 3 1,2,1,2,1,2
11(9,2): 4 [rn,r2] 11 9 ( *, *) ( 8, *) 3 1,2,1,2,1,2
The syntax for defining a function or operator instance is: ‘set’ followed by the function or operator name, then enclosed in parentheses a (possibly empty) parameter list containing pairs of the form ‘<type> <identifier>’ separated by commas, then ‘=’, and finally the expression giving the function body.
In the example the function body is a cast starting with ‘void:’, making clear on inspection that the return type is void (no useful return value). While this is a useful convention, which costs nothing at run time because casts are removed after type analysis, it is not obligatory: most of the time axis can find the return type by itself (and in the example it would without the cast); the reply it gives to the command, here ‘Defined f: (LieType->)’, indicates which type it has found. Note that when calling ‘f’ we provided a ‘string’ rather than a ‘Lie_type’ value; axis has inserted the necessary conversion implicitly when analyzing those calls.
The example also illustrates a typical composition of the function body consisting of a ‘let’ construction introducing a sequence of introductions of local variables: here first ‘rd’, next (separated by ‘then’, which in this context is equivalent to ‘in let’) ‘ic’, and finally (after another ‘then’) ‘rf’ and ‘drf’ (in parallel, separated by a comma) and finally calling (after ‘in’) the function ‘print_block’ with a nested call to ‘block’. The final call returns no value, which matches the ‘void’ type in the initial cast: the function ‘f’ returns no value and gets type (LieType->). Note that after calling the function ‘f’, the usual ‘Value:’ line before the next prompt is absent; this is a special provision for expressions with void type, since printing ‘Value: ()’ for them would just be distracting (the output shown above is instead printed as a consequence of evaluating the ‘print_block’ function, but it is not accessible for subsequent manipulation in atlas).
Functions defined by this syntax are overloaded, and stored in a different table than variables of non-function type introduced by ‘set var = ...’; this means an overloaded function can coexist with a value of the same name. This other table, called overload table, can store multiple definitions for the same function name. So after the above definition one could continue with:
atlas> set f(int n)=n+1
Added definition [2] of f: (int->int)
atlas> [f(4),f(f(6))]
Value: [5,8]
atlas> f("A1")
0(0,1): 0 0 [i1] 1 (2,*) e
1(1,1): 0 0 [i1] 0 (2,*) e
2(2,0): 1 1 [r1] 2 (0,1) 1
Note how the response after the second ‘set f’ mentions the addition of a definition and the total number [2] of definitions for ‘f’ now known. When encountering a call of ‘f’, axis finds out which definition to apply based on the type of the argument (list). All functions (and operators) built into atlas are defined in the overload table, often with more than one initial definition for a given name. The user can add definitions to names that also refer to built-in functions, they will be overloaded together with the original definitions. If one wishes that ‘inner_class’ can be called with a single ‘ratvec’ to specify kernel generators, it suffices to write:
atlas> set inner_class (LieType lt,ratvec gen,string ict) =
= > inner_class(lt,[gen],ict)
Added definition [7] of inner_class: (LieType,ratvec,string->InnerClass)
to add a new overloaded meaning to the ‘inner_class’ function. The fact that this definitions itself calls another (built-in) version of ‘inner_class’ is all right, because it can be distinguished by the argument type.
(As an aside, this call could not possibly be taken to be a recursive call, even if the types would have matched, since the ‘set’ definition is only taken into account after the body has been analyzed and found correct; the definition is therefore unknown during the analysis of its own body.)
New meanings of operator symbols can also be defined in this way. So while many versions of arithmetic operations are built into atlas, the standard script ‘basic.at’ defines a lot more instances yet, such as scalar multiplication of matrices, for which gives a definition that is basically:
set * (int c,mat m) = mat: for column in m do for x in column do c*x od od
(the somewhat curious syntax of the ‘for’ loop will be explained below).
Distinction between operators and functions occurs mostly in use rather than at their definition: as opposed to functions, operators can be used infix (when they take exactly two arguments) or prefix (always), and without always needing parentheses around their arguments. There are some limitations for defining operators: one cannot introduce new operator symbols, nor change the precedence rules for the existing symbols, and operator definitions cannot be local (within ‘let’). The last point is because operator definitions always add to the overload table, and there is no such thing as local overloads: defining a local identifier as a function instead temporarily hides any existing overloads of that name for the duration of that local definition. | 2018-03-21 05:04:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.674202561378479, "perplexity": 3642.580131179223}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647576.75/warc/CC-MAIN-20180321043531-20180321063531-00032.warc.gz"} |
https://codegolf.meta.stackexchange.com/questions/2140/sandbox-for-proposed-challenges/19164 | # Sandbox for Proposed Challenges
This "sandbox" is a place where Code Golf users can get feedback on prospective challenges they wish to post to main. This is useful because writing a clear and fully specified challenge on your first try can be difficult, and there is a much better chance of your challenge being well received if you post it in the sandbox first.
Sandbox FAQ
## Posting
Write your challenge just as you would when actually posting it, though you can optionally add a title at the top. You may also add some notes about specific things you would like to clarify before posting it. Other users will help you improve your challenge by rating and discussing it.
When you think your challenge is ready for the public, go ahead and post it, and replace the post here with a link to the challenge and delete the sandbox post.
## Discussion
The purpose of the sandbox is to give and receive feedback on posts. If you want to, feel free to give feedback to any posts you see here. Important things to comment about can include:
• Parts of the challenge you found unclear
• Problems that could make the challenge uninteresting or unfit for the site
You don't need any qualifications to review sandbox posts. The target audience of most of these challenges is code golfers like you, so anything you find unclear will probably be unclear to others.
If you think one of your posts requires more feedback, but it's been ignored, you can ask for feedback in The Nineteenth Byte. It's not only allowed, but highly recommended! Be patient and try not to nag people though, you might have to ask multiple times.
It is recommended to leave your posts in the sandbox for at least several days, and until it receives upvotes and any feedback has been addressed.
## Other
Search the sandbox / Browse your pending proposals
The sandbox works best if you sort posts by active.
To add an inline tag to a proposal, use shortcut link syntax with a prefix: [tag:king-of-the-hill]. To search for posts with a certain tag, include the name in quotes: "king-of-the-hill".
Get the Sandbox Viewer to view the sandbox more easily!
# Golf me a Bookmarklet Quine
Given a javascript program (or any utf-8 text) of arbitrary length, output it in my simplified version of URI form, like a bookmarklet. You can use https://mrcoles.com/bookmarklet/ as reference. Output should be in the form
javascript:[input with percent-encoding for special characters]
Special characters are any character that is not
• Alphabetic (upper or lower)
• a digit
• the characters .,-,_, or ~ (period, hyphen, underscore, tilde)
Your program "should convert all other characters to bytes according to UTF-8, and then percent-encode those values"w
A percent-encoding mechanism is used to represent a data octet in a component when that octet's corresponding character is outside the allowed set or is being used as a delimiter of, or within, the component. A percent-encoded octet is encoded as a character triplet, consisting of the percent character "%" followed by the two hexadecimal digits representing that octet's numeric value. For example, "%20" is the percent-encoding for the binary octet "00100000" (ABNF: %x20), which in US-ASCII corresponds to the space character (SP). source
Lowercase hex is okay, but uppercase is preferred.
This is code golf, standard loopholes are prohibited, programs should handle input up to 20 lines and output in a single line.
## The Twist (so it's not a duplicate)
If run with no input or just a newline (your choice), the program should output itself in the same format as if the program's source was inputted normally.
## Examples
In Out
[blank] javascript:[the%20program%27s%20source]
g/re/p javascript:g%2Fre%2Fp
## sandbox questions
• What tags does this need?
• Are my examples inconsistent?
• What parts of the challenge are redundant?
Comment: might be too similar to previous mutual quine challenge?
## Collaboration/quasi-quine challenge
Write a valid submission (A) which prints the code for another competitor's valid submission (B). The languages used in A and B must be different.
### Clarifying rules
If B prints the code for a third submission, C, it is not required that A and C be different languages. Similarly, the authors of A and B must be different, but A and C need not be. (More different languages/authors score higher, however.)
The shortest chain is for A to print B and B to print A.
Note that if A prints B, and B prints C, but C is not valid for some reason, then neither A nor B are valid either.
It is acknowledged that the validity of your submission may change over time, due to factors beyond your control. Try not to let this worry you too much. :)
None
## Output
Just the code described above. Nothing extraneous.
## Scoring
Scoring is (A + L) * 100 + C where:
• A is the number of distinct authors that directly or indirect print your solution. So if you are Q, and Z=>X=>Q=>X, your "A" is 2. (Each submission only has one author, the "answerer".)
• L is the number of distinct languages in your quine circle, along the same lines as for authors. (Each submission only has one language. "Distinct" means really different, not just different versions or implementations of the same languages.)
• C is the length of your solution in bytes.
(So, for a given circle of quines, all the submissions will have similar scores, with the length of the submission as tie-break.)
Standard loopholes are forbidden.
• I like it when your score improves if you use more languages. Any reason why you didn't include that? Apr 30, 2020 at 21:51
• Oh, what would an example of that be? I did consider something like having your score improve, the longer the chain is. Like, your score is the sum of the length of all the submissions divideded by the square of the number of participants or something. May 1, 2020 at 2:09
• That would be nice, but I wouldn't know how to balance it well. Just a thought. May 1, 2020 at 17:46
• This seems to me like a chicken-and-egg situation. How could the first posted answer be valid if there are no B answers to print the code for? May 25, 2020 at 15:35
• This is another similar challenge. It had several problems that I think might occur again with your current setup. I'd recommend giving the criticism and answers there a read over. May 25, 2020 at 20:45
• @mathjunkie It wouldn't. I don't think that's inherently problematic, it's just an interesting bootstrapping challenge. May 26, 2020 at 1:00
# Price this word code-golfnumberstring
So, I'm going shopping in the Word Market™. There are shelves of words which I can buy around me, but I only have one dollar bills and the change machines at the market are broken. To add to the problem, there are words with... non-word characters in them. That's no good, I can't buy those... can you help me figure out which words I can buy and which I can't?
So, I can only buy words that consist of only alphabetical characters and are worth a dollar. To determine a word's value, you have to sum the letters in the word where A = 1¢, B = 2¢... to Z = 26¢. I'm too lazy to look at the output and judge whether it is equal to one dollar (100 cents), so you'll need to return a specific value for words equal to a dollar (...or 100 cents) and a specific value for not equal to a dollar (I'm going to stop including this).
I'll also offer a bonus byte reduction: if your code returns whether the word is less than a dollar, equal to a dollar, greater than a dollar, or invalid (e.g. <, =, >, x), your score will be multiplied by 3/4.
SANDBOX NOTE: Is this a balanced bonus value?
## Examples
Word Non-bonus value Bonus value
a false <
b false <
printera false >
\$word false x
printer true =
And here's a JavaScript snippet you can use if you want to check for non-bonus validity:
(it's also 52 bytes; you can use it by calling f())
f=s=>([...s].map(x=>a+=parseInt(x,36)-9),a==100),a=0
Anyways, standard loopholes apply, shortest answer in bytes wins (but I'll add shortest answers for esoteric and functional languages)... you get the idea.
• In general, bonuses in code golf are seen as something to avoid May 28, 2020 at 2:45
• Standard Loopholes May 28, 2020 at 2:46
• This challenge doesn't seem interesting to me. We've already have plenty of challenges about summing up characters in a string, and having to determine whether a string contains non-word characters just seem like tacked on challenge that makes the whole thing more cumbersome. May 28, 2020 at 15:40
• What's a bit ironic about this is that after you posted this you went on to write a program that went through words in the English language and summed up their values depending on what character they were, sharpness of a word May 28, 2020 at 20:53
• Never say I like the sharpness challenge either. :P But I do think that that challenge is a bit more interesting, due to the somewhat arbitrary mapping of letter to values. Yours just straight up uses the vanilla alphabetical order. May 29, 2020 at 14:53
• You do have a point; in Jelly or 05AB1E there's probably a builtin that would sum up a string based on values like I want people to do. May 29, 2020 at 21:50
# Magic card trick: Hide information by flipping cards
(This is inspired by a series of questions on puzzles.stackexchange.com: 10, 8, 7)
Fix two integers m and u. Your task is to perform the following magic trick:
• A Magician brings a pack of m distinct cards, and leaves the room.
• In their absence, a volunteer from the audience shuffles the deck and arranges all cards in a line, in any order they want.
• Still in the absence of the magician, their assistant flips u cards. On the table are the n cards, still in the order chosen by the volunteers, but u are face down, leaving only mu cards face up.
• The magician returns, and from the order of the cards alone, knows each card.
### Input:
m - number of cards.
u - number of cards to flip face down.
• You may assume 0 < u < m.
### Output, if the trick is possible for m and u:
f - an mapping assigning to each sequence the order the assistant will create by flipping cards.
• If the trick is to work, this mapping must be bijective.
• Use the integers 1...m (or 0...m−1), or single letters as card values.
• Use any meaningful way to express f: a hash maps, a table, a function.
• Use a fixed placeholder for any face-down cards.
### Output, if the trick is impossible for given values of m and u
This case should be indicated in a meaningful way.
### Example output (m=3, u=1):
Using the digits 0, 1, and 2 as cards, and _ for their flipside:
012 01_
021 0_1
102 _02
120 12_
201 2_1
210 _10
(For these values of m and u, this isn't very impressive as a magic trick, of course.)
### Example output (m=4, u=2):
Using 1, 2, 3, and 4 for the cards and 0 for their flipside, and a JSON representation:
{"1234":"0034","1243":"0043","1324":"0024","1342":"0042","1423":"0023","1432":"0032",
"2134":"0104","2143":"0103","2314":"0014","2341":"0041","2413":"0013","2431":"0031",
"3124":"0120","3142":"0102","3214":"0204","3241":"0201","3412":"0012","3421":"0021",
"4123":"4003","4132":"0130","4213":"0203","4231":"0230","4312":"0302","4321":"0301"}
This is correct because as required, the keys are all permutations of 1234, each value has two cards face-down and the other cards match the original sequence, and each value appears only once.
## Scoring
This is code-golf. Shortest solution wins.
• I think I should not allow all that input/output flexibility and require some fixed format. For example: input is u and a string whose (unique) characters are the decks. Require _ as placeholder. Require a fixed table format. May 29, 2020 at 4:27
• There were some clarity issues I had while reading this, but as is I think this has a much bigger problem. It seems very likely to me that outputs for large m will be prohibitively difficult to verify, given the complexity of the proofs from the related puzzling challenges. There are many ways you could approach this, like upper bounding m, making this a test-battery, or making a code-challenge where the goal is to find the maximum u for the highest m. There are probably other ways to handle this, so these are just some starting ideas. Thanks for using the sandbox! May 29, 2020 at 19:23
• Thank you for the valuable feedback @FryAmTheEggman. The proofs from the linked puzzles are long because they're reasoning & looking for insight. To just verify the list, two steps are sufficient: verify that f*(*x) is obtained from x by replacing u symbols with _, and that f is a bijection and defined for all permutations. Non-golfed solution including full tests . The output will still be huge, no chance of cursory manual verification. I didn't know about alternatives to code-golf, actually! I'll be looking into these. May 29, 2020 at 21:44
• No problem! I do want to clarify though - I was aware of the ability to prove by exhaustion when I posted my first comment. However, I did base my assessment of it being a true problem around you not wanting a completely naive brute force search through each strategy, which I see now wasn't correct, so if you are fine with that then there isn't really a problem. But of course, if you want anything besides those solutions I'd recommend looking into what I suggested, or asking in our chat room for other people's points of view. May 29, 2020 at 22:43
# _
• What kind of numbers can be in the sequence?
– xnor
May 22, 2020 at 10:30
• @xnor Integers, basically.
– user92069
May 22, 2020 at 10:35
• Would the test cases that time out, 1,2 and 7,4, be excluded by "the input will always be provided in a way such that it won't take forever to zero the accumulator"? No product of exclusively odd numbers can end up being divisible by a power of 2. May 23, 2020 at 9:35
• @UnrelatedString Thanks for nothing that; I've removed these test cases.
– user92069
May 23, 2020 at 9:46
• Now this post is zeroed eventually
– l4m2
Jun 5, 2020 at 1:36
# Mobile games money representation
In many mobile clicker games where the player is usually required to tap on the screen to make money (in order to buy upgrades for you to generate money faster), it gets to a point in the game that the money made per second is so big that if represented in its "normal" form, it would clutter the mobile screen. Imagine showing the user that they are making $$\1,000,000,000,000,000,000,000,000\$$ per second in a small mobile phone screen!
From my experience as a regular player of these type of games, I have noticed that most of them represent bigger numbers by using letters. If the number of money per second is a number less than $$\10,000,000\$$ then print the number as is. Otherwise, if the number is in the millions (but $$\ \geq 10,000,000\$$), for example $$\ 102,000,000\$$ it should print $$\102M\$$. If it is in the billions, it should print $$\102B\$$. You should use $$\T\$$ for trillion and $$\Q\$$ for quadrillion.
As you can notice, the next would be quintillion which would also use the letter $$\Q\$$ if followed the pattern. Instead of following this pattern which can be confusing at one point, game developers usually start a new pattern: Quintillion is used with the suffix $$\AA\$$, sextillion is $$\AB\$$, septillion is $$\AC\$$ and so on.
Notice that this pattern would go until $$\AZ\$$ and if the player is making more money than that, it would start from $$\BA\$$, $$\BB\$$, ..., $$\BZ\$$, ...,$$\ZA\$$, $$\ZB\$$, ... , $$\ZZ\$$ which for our problem we will assume is the limit one player can make per second.
Given an integer $$\x\$$ where $$\0 \lt x \leq 999\$$ and a natural number $$\y\$$ where $$\y \gt 0\$$ representing the number of zeroes the number has, output the number in a "mobile game money representation" as described above.
# Observations
• The number of zeroes that $$\y\$$ represent does not include the possible zeroes $$\x\$$ might have! Example: if $$\x = 100\$$ and $$\y = 6\$$, you should output $$\100M\$$ and not $$\1,000,000\$$
# Test Cases (x, y --> game money representation)
100, 6 --> 100M
100, 5 --> 10M
100, 4 --> 1000000
1, 12 --> 1T
10, 12 --> 10T
100, 12 --> 100T
1, 18 --> 1AA
10, 18 --> 10AA
100, 18 --> 100AA
# Meta questions
1. Is this a duplicate? I have looked around but didn't find anything similar.
2. Is the wording confusing? I'm open to recommendations!
3. I haven't written a program yet so the test cases might be wrong (I'll add more later).
4. Pretty much any feedback is appreciated!
• Looks like it needs some test cases with decimal points, e.g. 123, 17 -> 12.3AA. Jun 1, 2020 at 0:20
• Shouldn't 100, 4 become 1M? Jun 1, 2020 at 13:03
• @SurculoseSputum In these games, when the number is small enough (as I said in the second paragraph), if the number is less than 10 million, then it is printed in its "normal" form. The abreviations starts after 10 million. Jun 1, 2020 at 19:33
• Suggest cases where $y$ don't just go the AA
– l4m2
Jun 19, 2020 at 4:24
• Thanks all, I'm pretty busy lately unfortunately. Whenever I get the time I'll try to update the challenge Jun 19, 2020 at 21:59
# Shift the letters, soldier !
posted, finally
• Thank you for sandboxing this. I usually recommend doing so for at least a week, and periodically ask for review in TNB.
Feb 28, 2020 at 9:09
• I think people would be forced to do the bonus in this case because of the -30% margin. I got a 42 without bonus but a 57*0.7=39.9 with bonus in JS. Feb 29, 2020 at 3:16
• Bonuses are discouraged for a variety of reasons. I would strongly recommend either making it mandatory or completely leaving it out. Mar 1, 2020 at 18:58
• The main challenge is add by position, the bonus challenge is minus by position. So it's a good idea to completely leave the bonus out.
– user92069
Mar 2, 2020 at 0:29
• Thanks for the comments, I'll remove the bonus as it will never be balanced enouth to be interesting. I'll add some example as soon as I can. Mar 3, 2020 at 20:02
• I would say that allowing the usage of the ascii range 1 to 255 or a language's code page could allow for some interesting golfs :)
– RGS
Mar 3, 2020 at 20:39
• Use asciii values from 0 to 255 was my original plan, but I'm afraid some interestings languages would be disadvantaged. Also, wouldn't the usage of language's code page be too permissive ? Mar 4, 2020 at 15:51
• @Therandomguy it depends on what you mean by "too permissive". Sometimes it is done, as it may allow some languages to do some funny things. As to the range being from 0 to 255, I don't see it hurting any language at all, but of course I may be missing something :)
– RGS
Mar 4, 2020 at 21:36
• Are you interested in re-posting it?
– user92069
Mar 6, 2020 at 7:38
• This weekend I'll post it, I just need some time creating the examples Mar 6, 2020 at 8:25
• I'd be glad to see it posted in Main!
– user92069
Mar 25, 2020 at 12:19
• Finally posted it in main Jun 3, 2020 at 7:45
# Compute the pointiness, sharpness and smoothness of a letter code-golfkolmogorov-complexity
Inspired by Determine the sharpness of a word.
You are given an uppercase letter of the English alphabet as input. You have to compute (and output) its pointiness, sharpness and smoothness. Since it is difficult to define these objectively, here's a table of the outputs:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
pointiness: 2 0 2 0 3 3 2 4 4 2 4 2 2 2 0 1 1 2 2 3 2 2 2 4 3 2
sharpness: 1 2 0 2 2 1 1 0 0 1 0 1 3 2 0 1 0 1 0 0 0 1 3 0 0 2
smoothness: 0 2 1 1 0 0 1 0 0 1 0 0 0 0 1 1 1 1 2 0 1 0 0 0 0 0
Transposed version (first lists the letter, then the pointiness, then the sharpness and then the smoothness) (like a true CGCC user, I transposed it with Jelly and added spacing with Retina):
A 2 1 0
B 0 2 2
C 2 0 1
D 0 2 1
E 3 2 0
F 3 1 0
G 2 1 1
H 4 0 0
I 4 0 0
J 2 1 1
K 4 0 0
L 2 1 0
M 2 3 0
N 2 2 0
O 0 0 1
P 1 1 1
Q 1 0 1
R 2 1 1
S 2 0 2
T 3 0 0
U 2 0 1
V 2 1 0
W 2 3 0
X 4 0 0
Y 3 0 0
Z 2 2 0
Bonus imaginary internet points if you find a language where this is built-in.
This is tagged , so the shortest answer wins.
# Sandbox stuff
• Is this not a duplicate?
• Is the table computed correctly? (the only ones that don't seem certain with the current font are I's pointiness, G's sharpness and S's smoothness)
• After fiddling about a bit, I think this should probably have enough patterns that mindlessly compressing the numbers won't be the best strategy. Still, I could be wrong, but here is what I used to see roughly how long such an approach would be (I encoded each set of values to a base 5 number, then in turn encoded that list of numbers into a base 61 number). Separately, you probably want to include the data in a more copy-pastable way. Jun 3, 2020 at 20:34
# Is this a simple cutting template?
A simple cutting template is a rectangle that can be recursively cut into smaller rectangles using only full-width cuts.
If you prefer a bottom-up description, then:
• A single rectangle is a simple cutting template with 0 cuts.
• Two simple cutting templates of the same width (or length) can be joined along their common side into a larger simple cutting template.
Input: A diagram of a rectangle subdivided into smaller rectangles, or a list of rectangles in some standard format, e.g. position and size.
Output: A truthy value if the diagram is a simple cutting template.
Note that if you take input as a diagram then all of the rectangle edges will use the same character, whearas in the truthy examples below, some of the edges have been replaced with digits to show a possible ordering of cuts while the falsy examples have the smallest portion of the input that is not a simple cutting template marked on them.
####2################
# 2 #
111111111111111111111
# #
# #
# #
111111111111111111111
# 2 2 2 4 4 #
3333332 2 2 4 4 #
# 4 2 2 2 4 4 #
# 45552 2 2 454 #
# 4 6 2 2 2 4 4 #
# 4 6 2 2 2 4 4 #
# 4 6 2 2 2 4 4 #
33333323332 2 4 4 #
# 2 2 2 4 4 #
# 23332 2 4 4 #
# 2 2 2 4 4 #
# 2 2 233333#
# 2 2 2 #
######2###2###2######
-> Truthy
##2#####4############
# 2 4 #
# 2333333333333333333
# 2 6 4 #
# 25555555555555554 #
# 2 6 6 8 4 #
# 2 6 67777777774 #
# 2 6 6 4 #
# 2 6 6 4 #
# 2 6 6 4 #
111111111111111111111
# 2 #
# 2333#
# 2 #
111111111111111111111
# 2 2 #
# 233333333333332 #
# 2 4 4 4 2 #
# 2 4 4555554 2 #
# 2 4 4 4 2 #
####2#############2##
-> Truthy
#####################
# #
# #
# #
111111111111111111111
# 2 4 2 #
# 2 4 2 #
# 2 4 2 #
# 23333333333333332 #
# 2 4 4 6 4 2 #
# 2 45554 6 4 2 #
# 2 4 6 4 6 4 2 #
# 2 4 6 4 6 4 2 #
# 2 4 6 4 6 4 2 #
# 2 4 6 45554 2 #
# 2 4 6 4 4 2 #
# 2 4 6 4 4 2 #
# 2 4 6 4 4 2 #
# 2 4 6 4 4 2 #
# 2 4 6 4 4 2 #
##2#####4#6#4###4#2##
-> Truthy
#####################
# # # #
################# # #
# # # #
# # # #
# # # #
?????????############
? # ? # #
? # ? # #
? # ? # #
? ##? # #
? # ? # #
?###### ?############
? # ? # # #
? # ? # # #
? # ? # # #
?###### ?## ###
? # # ? # # #
? ######? # # #
? # ? # # #
?????????############
-> Falsy
?????????????????????
? # # # # # ?
?###### # # # # ?
? # # # # # ?
? # # # # ######?
? # # # # # ?
?###### # ##########?
? # # # # ?
? ##### # # ?
? # # # # # ?
? # # # # # ?
? # # # # # ?
? ####### # ?
? # # # ?
? # ############?
? # # # ?
? ############# ?
? # # # ?
? ##################?
? # # ?
?????????????????????
-> Falsy
#####################
# # # # #
################### #
# # #
#####################
# # #
# ###################
# # # #
# ###################
# # # # #
# ###################
# # #
# ???????????????????
# ? # ?
# ? # ?
# ? # ?
# ? ##############?
# ? # # ?
# ?######## ?
# ? # ?
##???????????????????
-> Falsy
This is , so the shortest program or function that breaks no standard loopholes wins.
• I think you should include an explicit definition: 'A simple cutting template is a rectangle that can be recursively cut into smaller rectangles using only full-width cuts.' Jun 6, 2020 at 4:06
• @Dingus Thanks for pointing that out, I think I must have accidentally edited it out by mistake when writing the sentence for the output.
– Neil
Jun 6, 2020 at 10:29
• I'd prefer one or two small examples with extra markings and then a list of copy-pasteable test cases. Jun 6, 2020 at 11:13
## Backstory
A doctor in Berlin, after analyzing his medical history, has realized that all of the results of his integral measurement results can be represented in the form of $$\23x+28y\$$, where $$\x\$$ and $$\y\$$ are integers.
However, he could have extended his theory. $$\23\$$ and $$\28\$$ can be replaced by any two coprime numbers, and this theory would still hold. (He didn't have time to write his theory in a paper, that's quite awful.)
Without examples, I'll never be convinced that this nonsensical theory holds!
Given $$\the\ output\ of\ (ax+by)\$$ (let's call it $$\z\$$), $$\x\$$, and $$\y\$$, find the smallest pair of $$a,\ b)\$$ that makes $$\ax + by = z\$$ true. ## Example cases ## Duplicate of Find the minimum edit distance between two strings ## Partition distance code-golfstring Quoting Anush: I am very glad to provide a service to fill in the terrible gap in edit distance questions which codegolf.se has had. When there are as many edit distance questions as quine questions my job will be done. --Anush ## Task Given a binary string consisting only of 0's and 1's, partition the binary string (divide the string into consecutive substrings), and determine the minimal edit distance in order to transform one piece into another, left to right. You need to output the sum of the edit distances between consecutive blocks. ## Example I'm going to make a reference implementation to find the optimal partitions. But that's after I dump all my ideas, though. 011010110111 We partition the string like this: [011][010][110][111] And then find the cumultative edit distance between each 2 pairs of partitioned strings: [1 1 1] Then, we sum the list of partitions. [3] So 3 is a possible output for this binary string. However, you need to find the minimum edit distance, so this might not be the correct answer. ## Another example 001001010 We partition this string: [001][001][010] And then find the mimimal edit distance between each piece. [0 1] Therefore, our (non-optimal?) output for 001001010 is 1 ([0 1] summed). ## Rules • The edit distance between two strings is the minimum number of single character insertions, deletions and substitutions needed to transform one string into the other. • The input is guaranteed to have at least length 3. • The pieces of your partition don't have to be the same length. • What do you mean by "partition the input string"? Can I choose any partition I want as long as it's not all singletons or the entire thing? Or do I have to find one that's optimal in some sense? Why is the all-singletons case disallowed? Is the output the sum of the edit distances between consecutive blocks? From the examples I guess it is but you should say it explicitly. May 31, 2020 at 18:50 • @Zgarb "partition the input string" means divide the input string into (not necessarily equal) consecutive substrings. You need to find one that's optimal, I've emphasized that. I allowed the all-singleton case; I specified that the output is the edit distance sum between consecutive blocks explicitly. – user92069 Jun 7, 2020 at 4:36 • It should still be made clearer that the output is the minimum over all partitions. Jun 7, 2020 at 20:21 # Compute the factorial, on both sides of 0 Why, why, why do factorials stop at zero? (Yes there are actual reasons). Make a factorial function (or full program) that doesn't stop at zero! Your code-golfed program should, given an non-zero integer n (can be positive or negative, the rule still applies), find the product of the range n to -n excluding 0. Graph that at least works for positive numbers ### Sample IO Input | Output ----------------|------------ 0 | 1 (product of 0 and -0 without 0 / empty product) 2 | 4 (2*1*-1*-2) 3 | -36 (3*2*1*-1*-2*-3) 4 | 576 (4*3*2*1*-1*-2*-3*-4) -4 | 576 Probably not a duplicate, but it might not be that much of a challenge. • Would the input always be positive? Is n=0 a possible input? Jun 11, 2020 at 23:02 • @Bubbler For now I'll say 0 is undefined, might change it later before posting if I have a good idea. Jun 12, 2020 at 13:53 • As it is, isn't this always the factorial of the absolute value of the input squared, then made negative if the input is odd? - except in the edge case for zero? The sign of the input doesn't really appear to matter, which is an odd feeling. Jun 12, 2020 at 16:24 • @FryAmTheEggman Yes, see this graph of the values. Is that a bad thing? Do you have a better suggestion? Jun 12, 2020 at 20:44 • "downvotes mean nothing but rudeness" - I downvoted this because I do not think "compute $|n|!^2 \cdot (-1)^n$" is a good challenge. I can't see how disagreement is rude. The requirements here seem completely arbitrary to me. This will result in the exact same approaches as were used in the factorial challenge. Jun 14, 2020 at 8:27 • I think it is a bad thing in that it becomes dangerously close to a dupe of the factorial problem. I probably wouldn't hammer it immediately, but if most of the responses basically worked for both or many others had the same concern I'd probably close it. I'm not sure of a good way to modify this to be better, so unfortunately I don't have any suggestions at the moment. I will let you know if something occurs to me. Jun 15, 2020 at 20:09 • @FryAmTheEggman I wouldn't consider it a dupe but I wouldn't consider it a good question after all based off of what my pronoun is monicareinstate said. Jun 16, 2020 at 21:47 • If you consider 0 as a valid input, I suggest that its expected output be 1, which corresponds to the empty product (Wikipedia). Jun 17, 2020 at 3:52 • For the interesting-ness, I believe it can be interesting in at least some languages (which IMHO justifies the value of having such a challenge). FWIW, I have two J solutions of equal length, one using the factorial built-in ! and the other not using it. Jun 17, 2020 at 3:53 • I'd likewise close this as a duplicate, but I'm known for having much broader standards than the rest of the community about what questions are closeworthy, so make of that what you will. Jun 18, 2020 at 2:38 • I'll just abandon this, but if @Bubbler wants to post it, they can. Jun 18, 2020 at 21:17 # Default Lightning Strike ## Introduction: Inspired by this reddit question: ELI5: Why does lightning travel in a zig-zag manner rather than a straight line? Although it's more complex than this, in general multiple lightning paths will randomly check its immediate surrounding for the direction with least resistance (based on air pressure, temperature, composure, humility, etc.) and travel in that direction. As soon as one of the paths reaches the ground, that entire path has the least resistance and most (although not all) of the ions will accumulate in that path, causing the lightning flash and thunder. Here a slow-mo video of a lightning strike to get an idea. ## Challenge: Input: An integer $$\h\geq3\$$ and an integer $$\1\leq p\leq\left\lfloor\frac{h}{2}\right\rfloor\$$ Output: Each step of the ASCII animation of a lightning strike, with a cloud to earth height of $$\h\$$ and up to $$\p\$$ paths We start with a lightning ion at the cloud, with a lowercase letter of your own choosing (i.e. b). This ion will travel in a random direction (horizontally, vertically, or (anti-)diagonally), except where this path itself comes from. Every 'tick' it also has a 20% chance of branching out into two paths, as long as we haven't reached $$\p\$$ paths yet. Each of these paths will behave the same. As soon as any path hits the ground based on the height $$\h\$$, all letters of that particular path will become uppercase, and in the final 'tick' after that, only this uppercase path will remain. ## Challenge rules: • Paths can intersect with other paths • Paths can travel upwards beyond the height of our starting point • Output can be in any reasonable format. Could be a list of multi-line strings for each 'tick'. Could be a list of character-matrices for each 'tick'. Could be pretty-printed to STDOUT (with clear non-whitespace separation between each 'tick' - i.e. a single character like a comma or semi-colon, or a line of --- or ___) • Trailing spaces for each line of a tick are optional (leading as well, as long as the lightning bolts are still correct) • If multiple paths strike the ground in the same 'tick', only the first one of those two (or more) paths will become the lightning strike. The order in which paths are created are therefore important, so keep that in mind. ## Examples: This may all sound pretty vague, so here a couple of examples: (I've added trailing spaces for each step with spaces, but you don't necessarily have to do so as mentioned in the challenge rules.) Example 1: $$\h=3, p=1\$$ Tick 1: "b" " " " " Tick 2 (random direction: right): "bb" " " " " Tick 3 (random direction: up-left): "b " "bb" " " " " Tick 4 (random direction: down-left): " b " "bbb" " " " " Tick 5 (random direction: down): " b " "bbb" "b " " " Tick 6 (random direction: up-right): Note that this overlaps with a previous step in this path, which is fine. " b " "bbb" "b " " " Tick 7 (random direction: down-right): " b " "bbb" "b b" " " Tick 8 (random direction: down): " b " "bbb" "b b" " b" Tick 9 (lightning strike): " B " "BBB" "B B" " B" Tick 10 (extra tick to remove all other paths, although there are none right now): " B " "BBB" "B B" " B" Example 2: $$\h=5, p=2\$$ Tick 1: "b" " " " " " " " " Tick 2 (random direction: down-left): " b" "b " " " " " " " Tick 3 (random direction: down-left): " b" " b " "b " " " " " Tick 4 (random direction: right): " b" " b " "bb " " " " " Tick 5 (random 20% path split; random direction 1: top-right, random direction 2: right): " b" " bb" "bbb" " " " " Tick 6 (random direction 1: top-left, random direction 2: down): " bb" " bb" "bbb" " b" " " Tick 7 (random direction 1: left, random direction 2: down-right): "bbb " " bb " "bbb " " b " " b" Tick 8 (lightning strike of path 2): "bbB " " Bb " "BBB " " B " " B" Tick 9 (extra tick to remove all the other paths, which is path 1 in this case): " B " " B " "BBB " " B " " B" ## General rules: • This is , so shortest answer in bytes wins. Don't let code-golf languages discourage you from posting answers with non-codegolfing languages. Try to come up with an as short as possible answer for 'any' programming language. • Standard rules apply for your answer with default I/O rules, so you are allowed to use STDIN/STDOUT, functions/method with the proper parameters and return-type, full programs. Your call. • Default Loopholes are forbidden. • If possible, please add a link with a test for your code (i.e. TIO). • Also, adding an explanation for your answer is highly recommended. # Sandbox Questions: • Should I perhaps use a different letter of the alphabet per path? • If yes: what would happen when different letter-paths overlap? I assume the top one will be visible per 'tick', but if lightning is struck it should still change it to the underlying letter as uppercase. In either case, you'll have to keep track of each individual path and uppercase only the one that struck the ground (first). • Any additional rules or things that are unclear? • An additional relevant tag? • More examples with more paths and/or larger height? • A different path percentage instead of hard-coded $$\\frac{1}{5}\$$ / 20%. # Sort numbers using as few distinct bytes as possible ## Task Write an algorithm that takes as input an ordered list (array, linked list, etc...) of numbers and outputs an ordered list containing the same numbers sorted by their value (ascending or descending). The numbers may be represented using the most convenient format to you, with the only restriction that there must be a way to encode 256 distinct numbers. You are not allowed to use built-in sorting functions/algorithms. ## Scoring criteria Let $$\c\$$ be the number of distinct bytes in your code* and let $$\s\$$ be the number of bytes in your code*. *Or its UTF-8 representation The score is equal to $$\c^2 + s\$$. The answer with the lowest score wins! Examples (imagine these are sorting algorithms): • ababccbaacbabcba$$\c=3, s=16, score=25\$$ • aAbcd€f$$\c=9, s=9, score=90\$$ • bytes 16 ee 3c 79 ee$$\c=4, s=5, score=21\$$ I'm open to suggestions, especially about the score formula. • I see that this is your first attempt at writing a challenge. Thank you so much for using the sandbox! – Adám Jun 25, 2020 at 22:16 • Please note that it is very hard to write good challenges that restrict solutions from certain things. This is because it is hard to define exactly what is prohibited in every language, and it is also hard to determine if any prohibited feature was used. – Adám Jun 25, 2020 at 22:19 • @Adám So how should I prevent trivial answers? Maybe "built-in sorting functions/algorithms" is a bit vague. Jun 25, 2020 at 23:04 • We don't prevent trivial answers in most cases. Btw, if I accept plain numbers as input, may I assume the input is a list of integers between 0 and 255 inclusive? Jun 25, 2020 at 23:46 • How about this: "You are not allowed to use any built-in function/command that can take an ordered container and output the sorted result. Anything else is OK." Jun 25, 2020 at 23:54 • I think a good solution would be to allow built-in solutions, but to compile (in advance, so that it can be posted very quickly, probably via the "answer your own question" feature) a community wiki answer listing trivial 1-byte solutions. Jun 26, 2020 at 4:17 • @Bubbler Would still be unclear if J's /:~ or /:] were allowed or not. – Adám Jun 26, 2020 at 6:15 • @D.Pardal Why do you want to prevent trivial answers? – Adám Jun 26, 2020 at 6:20 • I wanted to prevent built-in functions because otherwise most answers would be exactly the same as the ones from this question. Maybe the easiest way to solve this would be to replace the task of sorting an array with another. Jun 26, 2020 at 7:15 • Yes. Banning built-in has long been considered a bad idea. Jun 26, 2020 at 11:57 # Fix mispellings code-golfkolmogorov-complexity Wikipedia has a list of common misspellings, and there is also a machine-readable version! Your challenge is to input a string and fix the mispellings in it. The parituclar list we'll be using is https://en.wikipedia.org/w/index.php?title=Wikipedia:Lists_of_common_misspellings/For_machines&oldid=962756669#The_Machine-Readable_List. Note that even if the list changes, you must use this version. Here's a pastebin link: https://pastebin.com/j03aL98d. Each line in the list is in the format INPUT->OUTPUT1, OUTPUT2, OUTPUT3, ... (of course, there may be more or less possible outputs, or even just one). That means that for input INPUT you must output exactly one of the possible outputs OUTPUT.... This is tagged , so the shortest answer wins. # Sandbox stuff Should I add more misspellings to the post, or should I remove them? • Related Jul 2, 2020 at 14:23 • @pppery While the idea is probably related, I don't think the solutions would be related at all. Jul 2, 2020 at 14:26 • What is the input format? A plain English sentence (so we need to handle spaces, punctuation, capitalization), or is a list of words acceptable? How should capitalization be handled (some entries look like Tolkein->Tolkien and UnitesStates->UnitedStates; given unkown->unknown, what is the expected output of unkown, Unkown, UNKown, Tolkein, tolkein, TOLKEIN)? Jul 2, 2020 at 23:19 • @Bubbler The input is a single entry in the list (the part before ->, of course). You do not need to handle capitalization ("tOLKEIN" is not "Tolkein"). (will clarify later). Jul 3, 2020 at 2:11 ## Overlap characters code-golf Put all the characters of a given list, following the order, in a sequence of bits keeping it as small as possible. Rules • Write the bits of each character on a line. • You can overlap bits if they are equal. • You cannot change already written bits. • Extend the line, in both directions, if not all the bits fit in. • Always try to extend as less as possible. Example input :['a','&','1','.'] 0110 0001 // a 0010 0110 // & 001 1000 1 // 1 00 1011 10 // . output :0010011000011000101110 input : "&1a." 0010 0110 // & 0 0110 001 // 1 0 1100 001 // a 0010 1110 // . output : 011000010011000101110 I/O rules • input can be any sequence of single byte elements. • output the resulting sequence of bits in any convenient method, no extraneous bits allowed (0 or 1) # Distributive on myselfcode-golfkolmogorov-complexity • Now that this has been posted to main, could you delete this proposal to create more space for new answers? Sep 25, 2020 at 1:04 # Validating Words in Word Grids A follow on from Generating Word Grids Given a grid of letters, a set of co-ordinates and a dictionary of words, validate that the co-ordinates follow only cardinal direction changes, at least one of the co-ordinates touch an empty space in the centre of the grid, the resulting word is valid given the dictionary (taking into consideration any blank tiles) and return either the grid, with the letters at the co-ordinates removed along with the score of the word, or, if one of the conditions fail, the original grid and a score of -1. ## Details Please detail the format you want to accept coordinates in any reasonable format is acceptable. ## Scoring Letters are worth their values as per Scrabble: 0 points: blank tiles 1 point: E, A, I, O, N, R, T, L, S, U 2 points: D, G 3 points: B, C, M, P 4 points: F, H, V, W, Y 5 points: K 8 points: J, X 10 points: Q, Z Bonus tiles (indicated by a lowercase letter, or ! for a blank tile) provide a *2 multiplier and stack (eg. if my co-ordinates spell gOLf I would earn (((2+1+1+4)*2)*2), 32 points). ### Examples: Input: 6,4 6,5 5,5 4,5 3,5 3,6 2,6 1,6 UWDESTKP? ItDBaDEdI TERMDYTSR ROANJLEFT EkCI OOsT IPAJPGMNY MZLORITVI GwEGgPUeI MNROYOEER Output: 9 UWDESTKP? ItDBaD dI TERMDY SR ROANJ FT EkCI OsT IPAJP MNY MZLO TVI GwEGgPUeI MNROYOEER (spells RIGOLETE, (1+1+2+1+1+1+1)) Input: 0,3 0,4 0,5 0,6 UWDESTKP? ItDBaD dI TERMDY SR ROANJ FT EkCI OsT IP P MNY MZL TVI GwE gPUeI MNROYOEER Output: -1 UWDESTKP? ItDBaD dI TERMDY SR ROANJ FT EkCI OsT IP P MNY MZL TVI GwE gPUeI MNROYOEER (spells DEST which doesn't appear in the dictionary) Input: 5,6 4,6 4,7 4,8 UWDESTKP? ItDBaD dI TERMDY SR ROANJ FT EkCI OsT IP P MNY MZL TVI GwE gPUeI MNROYOEER Output: 12 UWDESTKP? ItDBaD dI TERMDY SR ROANJ FT EkCI IP P NY MZL TVI GwE gPUeI MNROYOEER (spells MOsT, (3+1+1+1)*2) ## Rules This is so the shortest code in bytes wins. • The order is not important, it can be score then grid, or vice versa. • Any reasonable format can be used for I/O assuming it is consistent. • All standard loopholes are forbidden. ## Questions for meta Things have changed a lot in the time since I originally posted this, so rather than just posting I thought I'd bump for fredback. • What would the best way of using an associated dictionary be, taking it as input? • Obviously the input format for co-ordinates can be more flexible (0-based index, 1-based index, or something) is mentioning this enough? • Grid input format can be flexible too, mentioning this should be enough too? • Any other relevant tags? • This looks pretty good to me. Sep 15, 2017 at 18:20 • Thanks @Pavel, I'll bear that in mind, i'm not sure how much interest there is based off of part 1, but I might still post this in the next week or so :) Sep 16, 2017 at 9:21 # Crazy Blazin' DOM Injection I was instructed to post this code golf challenge here for recommendations on how to modify the challenge. I'd like to tag it. fastest-code grid and browser , but I don't know how to do that here. Code golf challenge listed below. # Introduction This problem is a challenge that has to do with DOM manipulation at scale and overcoming some issues that may be inherent in dealing with the DOM. • This challenge is interesting because limitations push us to think through things differently and lean on the strengths of languages different than what we usually use. • I created this challenge myself based on a real world problem I ran into myself (details can be provided if needed). If this challenge has any relation to a differently know problem, those similarities are coincidental. This challenge will be scored based on fastest execution and fastest algorithm. A multiplier will be given for completing the challege at easy, medium, and hard difficulties. # Challenge You must render one of the challenge levels in a web browser: Through any means available to you interacting through a web browser in code you need to complete the following: 1. Get each table displayed and add a class table-n to each table where n is a zero based index of order of the table on the screen. If a table is nested within a table the parent would be N and the child would be N+1 with the next table being N+2. 2. Get each row displayed in each table and add a class of table-n-row-r where r is a zero based index of rows in the table represented by table-n. 3. Get each cell displayed in each table and add a class of table-n-row-r-cell-c where c is a zero based index of cells in a row represented by table-n-row-r. At the end of the challenge the web page should still be able to be interacted through in the browser, and a call to document.getElementsByClassName('table-n-row-r-cell-c'); should return one and only one cell from the DOM. Any method available to you as valid as long as: 1. Access to one of the difficulty levels has been done through a web browser 2. The URL of the browser and the page displayed doesn't change 3. A call to document.getElementsByClassName('table-n-row-r-cell-c'); returns only one element # Output Examples For this example we're using the easy level as input. The abbreviated output in the DOM should be. <!DOCTYPE html> <html lang="en"> <head> <meta charset="UTF-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <meta http-equiv="X-UA-Compatible" content="ie=edge"> <title>Code Golf Challenge</title> </head> <body> <table class="table-0"> <thead> <tr class="table-0-row-0"> <th class="table-0-row-0-cell-0">1</th> ... </tr> </thead> <tbody> <tr class="table-0-row-1"> <td class="table-0-row-1-cell-0">11</td> <td class="table-0-row-1-cell-1">12</td> ... </tr> ... </table> </body> </html> There are only <td> and <th> elements used as cell elements for the challenge. As long as document.getElementsByClassName('table-n-row-r-cell-c'); returns an element with this class. We're good to go. # Qualifying Entries All entries that qualify need to have an execution speed of under 40 seconds for any difficulty level. Timing starts when your algorithm starts, but after the page has loaded completely as all elements must be available (the browser spinner has stopped spinning). Calculate the runtime for your method by calling performance.now() before and and after your injection method, and subtracting the first from the second as in the example below. let t0 = performance.now(); doSomething() // <---- The function you're measuring time for let t1 = performance.now(); let totalTime = t1 - t0; The medium and hard difficulty levels will have their execution time multiplied by 0.50 and 0.25 respectively. None the less, an execution time of less than 40 seconds is needed to qualify. So if the execution time on the medium difficulty was 41 seconds before the multiplier, it does not qualify. # Winner The winner will be determined by the shortest execution time for dynamic injection of these classes and where document.getElementsByClassName('table-n-row-r-cell-c'); can still be executed against the browser console and return an element where n, r, and c are replaced with indexes. • Do you mean that all entries should be written as a JS function so that it can be run inside the browser console? Which browser(s) will you be using to time it (since different browsers may have different support of JS features and the engine optimized in different ways)? Jul 20, 2020 at 2:15 • "scored based on fastest execution and fastest algorithm" is ambiguous, because fastest execution means that a constant factor in the algorithm is important, while fastest algorithm is not. I think you mean simply "fastest execution", since it seems hard to use a big-O notation in this task (should it be a function of table size or entire document size?). Jul 20, 2020 at 2:18 • All entries should be written as a JS function (I think that's the only way to achieve it). However, if there are other ways to achieve this they are welcome, but it still needs to be an active webpage pulled from the internet. So, pulling via wget or curl and manipulating it locally in something like jelly is out of the question unless Jelly can interact with the browser. We can standardize on chrome 83.0.1.x for the web browser, so that it's exactly the same for all. Jul 20, 2020 at 2:18 • It should only be fastest execution then. Let's go with that. Jul 20, 2020 at 2:19 • Also, if a submission works for all of easy/medium/hard, how is it scored? I guess the minimum of (easy time), (medium time / 2), (hard time / 4)? Jul 20, 2020 at 2:23 • If a submission works for all level (qualifies at all levels) it gets scored at the hardest level it qualifies for. Jul 20, 2020 at 2:24 • That sounds like unnecessarily penalizing possibly good submissions because it might run blazing fast on easy but more than 4x slower on hard, giving it a worse score. Jul 20, 2020 at 2:26 • I can agree with that. How do you suggest we even that out? Jul 20, 2020 at 2:27 • Personally, I think it is better to do away with the different difficulty levels. They aren't exactly the same as bonuses in code golf but most of the arguments for why we don't like bonuses apply to these difficulties, too. Jul 20, 2020 at 2:43 • I think this will work: "The submission that handles the highest difficulty within 40 seconds wins, ties broken by the time taken to complete for that test case." If the first part of the sentence is confusing, I mean "a submission that handles hard is better than one that can't". Jul 20, 2020 at 2:45 • I'll get rid of the difficulty levels. Jul 20, 2020 at 2:45 • Actually, Bubbler, what you said makes sense. I'll state it like you said and get rid of the bonus. Jul 20, 2020 at 2:48 • Still getting downvotes on the challenge. I don't know the reason why. Implemented the changes we discussed, but I guess it still doesn't meet standards? (^-^;) Jul 20, 2020 at 11:56 # How wide is this string? Given a unicode string in any standard encoding of choice, determine how many columns wide it is. To keep this challenge relatively simple, use the following rules for character widths: • Tab characters align to the nearest multiple-of-8 column • CJK characters are 2 columns wide. • For the purposes of this challenge, you may assume all characters in Unicode Planes 2 and 3 (U+20000-3FFFF), plus codepoints U+3400-9FFF are CJK characters. • CJK characters outside these ranges may be treated as either 1 or 2 columns wide. • Combining diacritics (U+0300-036F), control characters (U+00-1F, U+7F-9F), and the zero-width space (U+200B) are all 0 columns wide. • All other officially zero-width characters may be treated as either 1 or 0 columns wide. • You may assume that there are no newlines or carriage returns. • You do not need to handle escape sequences. • You may assume all other characters are 1 column wide. • For any character the standard says has a specific width, you may use that width instead. Shortest code wins. # Tumbling 2x2 in a Matrix ## Challenge: Input: A rectangular integer matrix that's at least 2x2 in size. Output: A 2D integer array, of the result after the top-left 2x2 block has tumbled down. For example: Let's say we have the following 3x5 matrix as input: [[ 4, 7,12], [11, 2, 5], [ 7, 3,15], [21,10, 1], [12, 6, 6]] The 2x2 block is [[4,7],[11,2]], which will act as if it was tumbling down from a stairs (in a top-left to bottom-right direction). Here this process step-by-step: [[ 4, 7, ], [11, 2, ], [--, , ], [ , , ], [ , , ]] [[ , , 4], [ ,11, 7], [--, 2, ], [ , , ], [ , , ]] [[ , , ], [ ,11, 4], [--, 2, 7], [ ,--, ], [ , , ]] [[ , , ], [ , , ,11], [--, , 2, 4], [ ,--, 7], [ , , ]] [[ , , ], [ , , , ], [--, , 2,11], [ ,--, 7, 4], [ , ,--]] [[ , , ], [ , , , ], [--, , , , 2], [ ,--, , 7,11], [ , ,--, 4]] [[ , , ], [ , , , ], [--, , , , ], [ ,--, , 7, 2], [ , ,--, 4,11]] Doing so, it will add it's values to the other cells in its path. So here is the same step-by-step process with the other numbers added in the cells: [[ 4, 7,12], [11, 2, 5], [ 7, 3,15], [21,10, 1], [12, 6, 6]] [[ 4, 7,16], [11,13,12], [ 7, 5,15], [21,10, 1], [12, 6, 6]] [[ 4, 7,16], [11,13,16], // Note that the 13 and 5 remain the same, because the cells of the tumbling [ 7, 5,22], // block haven't moved from the previous to this step [21,10, 1], [12, 6, 6]] [[ 4, 7,16], [11,13,16,11], [ 7, 5,24, 4], [21,10, 8], [12, 6, 6]] [[ 4, 7,16], [11,13,16,11], [ 7, 5,24,15], // Note that the 24 and 8 remain the same, because the cells of the tumbling [21,10, 8, 4], // block haven't moved from the previous to this step [12, 6, 6]] [[ 4, 7,16], [11,13,16,11], [ 7, 5,24,15, 2], [21,10, 8,11,11], [12, 6, 6, 4]] [[ 4, 7,16], [11,13,16,11], [ 7, 5,24,15, 2], [21,10, 8,11,13], // Note that the 11 and 4 remain the same, because the cells of the tumbling [12, 6, 6, 4,11]] // block haven't moved from the previous to this step ## Challenge rules: • I/O is flexible. You may take the input as integer-matrix, integer list with loose dimension-inputs, as a list of strings, etc. Output can modify the original input, return a new matrix, print space/newline delimiter to STDOUT, etc. • You may optionally take the dimensions as additional input. ## General rules: • This is , so shortest answer in bytes wins. Don't let code-golf languages discourage you from posting answers with non-codegolfing languages. Try to come up with an as short as possible answer for 'any' programming language. • Standard rules apply for your answer with default I/O rules, so you are allowed to use STDIN/STDOUT, functions/method with the proper parameters and return-type, full programs. Your call. • Default Loopholes are forbidden. • If possible, please add a link with a test for your code (i.e. TIO). • Also, adding an explanation for your answer is highly recommended. ## Test case: Input: [[ 4, 7,12], [11, 2, 5], [ 7, 3,15], [21,10, 1], [12, 6, 6]] Output: [[ 4, 7,16], [11,13,16,11], [ 7, 5,24,15, 2], [21,10, 8,11,13], [12, 6, 6, 4,11]] TODO: More to come. # Sandbox questions: I might change the tumbling process a bit later on, since I'm not too happy about the current one. I still want the tumbling 2x2 down the matrix, but I might make the way the other values changes a bit different. This is just an initial idea. • Any missing tags? • Any missing rules? • Any suggestions on how to change the output without changing the core part of the tumbling top-left 2x2 block? • Any suggested test cases? # Trapping a Jogger A person starts jogging to the right from his house on a busy street. t=0 0 1 2 3 4 5 6 7 8 9 A B C ⌂ - - - - G - - G - - - G > He travels 1 hectometer every minute, so he is 11 hectometers away from his house after 11 minutes: t=11 0 1 2 3 4 5 6 7 8 9 A B C ⌂ - - - - R - - R - - - G > Since this is a busy street, there are walking signals that occasionally turn red. In this example, the signals at 5, 8, and 12 units from home alternate colors every 2, 10, and 4 minutes (this would not be a fun road to drive on). The signals are red (signal stop ✋) for the same duration that they are green (signal go 🏃). Our jogger doesn't want to wait long, so he instantly turns around when he reaches a stoplight that is red, even if it will turn green within the next minute. t=12 0 1 2 3 4 5 6 7 8 9 A B C ⌂ - - - - G - - R - - - R > t=12.001 0 1 2 3 4 5 6 7 8 9 A B C ⌂ - - - - G - - R - - - R < t=13 0 1 2 3 4 5 6 7 8 9 A B C ⌂ - - - - G - - R - - - R < This can cause the jogger to reverse direction again, making for a potentially long outing. t=16 0 1 2 3 4 5 6 7 8 9 A B C ⌂ - - - - G - - R - - - G < t=16.001 0 1 2 3 4 5 6 7 8 9 A B C ⌂ - - - - G - - R - - - G > View the rest of the sequence using a visualizer. ## Task Given the cycle intervals and positions of a list of streetlights, determine the time until the jogger returns to his house (at distance 0) or runs to the right of the rightmost streetlight. The streetlights at time t=0 will all have just turned green. The n cycle intervals shall all be integers at least 2. The n positions of the streetlights may be taken as either (sorted) absolute distances from home or the distance from that streetlight to the previous. In the example, this would be either [5,8,12] or [5,3,4]. ## Example cases positions intervals output 5,8,12 2,10,4 38 10 20 11 10 6 20 2,8 8,6 40 2,8 7,6 16 1,2,3,4,5,6 6,5,4,3,2,1 16 $$$$ # International "Hello, World!" (WIP) code-challengestring (Please note the special scoring for this challenge) This code-golf question has over 900 answers and all of them print "Hello, world!" in English! If we can use hundreds of different programming languages to print that message, why can't we use hundreds of different natural languages to express that message? # Task & scoring Your task is to beat the answers of the Hello, world! challenge ("HW" challenge from now on) in different natural languages, as determined by the length ratio of the English string "Hello, world!" and the string in the natural language you pick. For example, I could pick Portuguese, hence I will have to print "Olá, mundo!" which has a length ratio of 11/13. • if your natural language has capitalization, you must respect the original capitalization. • if your natural language has punctuation, you must respect the original punctuation. Then you pick the programming language you are going to write your code in. For example, I could pick Python. And you write your program. My program could be print("Olá, mundo!"), with a standard code-golf score of 20. You then look for the best submission in the HW challenge with the same programming language you chose, let's say it has score S. (We probably need a leaderboard for challenge HW to make this step easier.) My score would then be (20/S)/(11/13). Does this make any sense? Any preliminary feedback? • It's hard to define whether the grammar of the output is correct in the chosen language, especially for those who don't know the language. – user92069 Jul 29, 2020 at 9:46 • @user92069 why do I need to define if the grammar is correct? "Hello, world!" doesn't look grammatically very correct either. – RGS Jul 29, 2020 at 22:36 # Extract an integer from another This is a somewhat interesting problem I ran into while nanboxing: given two integers, compute their bitwise-AND and concatenate the resulting "substrings" into a new integer. More precisely: you are provided two integers as input — an input integer and a bitmask. As output, you should produce the bitwise-AND of the two such that, given a mask with $$\n\$$ set bits, the corresponding bits from the input are grouped together in the first $$\n\$$ bits of the resulting integer. The following pseudocode is one way to implement the function: -- x and mask are lists of booleans function (x, mask) local result=list(); for i=1, min(x.length, mask.length) do if mask[i] then result.append(x[i]); end end return result; end ## Example inputs and outputs // In binary: // f(x, mask) == result f(1011, 1111) == 1011 // Select entire number f(1010, 1010) == 11 // Select bits 1 and 3, and concatenate them f(11001100, 01100110) == 1010 // Concatenate substrings [1:2] and [5:6] f(11111111, 10101010) == 1111 // Concatenate bits at odd indices. // 16-bit variants in hexadecimal: f(BEEF, 1111) == 9 f(DEAD, 8888) == F f(1337, FF00) == 13 f(CODE, 7777) == 82E // Two 64-bit variants (in hex): f(400921FB54442D18, CODE601F15DABE57) == 111DE42C8 f(FFFE0000004010CC, 8003000000000000) == 6 ## Specific rules • Standard loopholes, default IO, etc. apply where not overridden. • Input and output values must fulfill $$\x \in \{b_{set}, b_{unset}\}^{n}\$$ for some $$\b_{set} \neq b_{unset}, n \in \mathbb{N}_{\geq 16}\$$ of your choice. In other words: • You must support integers of at least 16 bits in your representation, but you may otherwise arbitrarily constrain their size - i.e. to 32-bit integers. You may also accept integers of any length through lists, arrays, strings, etc. • The "set" and "unset" values do not need to be of the same type or length, but they must be constant and distinct. • Most "linear" representations for integers are valid: integers, vectors, arrays, strings, etc. • This is , so the shortest answer in bytes wins. • Have fun! • Is this the PEXT BMI2 instruction? (also, possible duplicate: codegolf.stackexchange.com/q/37167) Jul 30, 2020 at 13:00 • @mypronounismonicareinstate After reading the referenced challenge I think this is indeed a duplicate of that. The required algorithms are identical and other requirements seem to not affect this too much. Aug 3, 2020 at 4:07 • @mypronounismonicareinstate I've given it a read, and it definitely is the same challenge, bar minor cosmetic differences. Of course, the search did not find it when I tried searching for it... Aug 3, 2020 at 11:46 Write an ASPIF (clasp's ASP input format) program to find a maximum cap set (https://en.wikipedia.org/wiki/Cap_set) for 4 dimensions. Share the code you used to generate the ASPIF rather than ASPIF itself. This may be an ASP program. Winner is smallest word-count (according to wc) in ASPIF format. For ASP, you can get this by running something like: clingo capset.asp --mode=gringo | grep -v "\(^1 0 1 [0-9]\+ 0 0\|$$^4$$" | wc -w
(note the grep is for excluding unary rules and #show directives neither of which are necessary for solving. The output of this is still a valid clasp program)
I have an example for four dimensions (but I have a better one I won't share right away because I'm curious to see what other people get).
feature(number, (one; two; three)).
feature(color, (red; green; purple)).
feature(shape, (oval; diamond; squiggly)).
dimension(D) :- feature(D, _).
card(c(N,F,C,S)) :-
property(c(N,F,C,S),number,N) :- card(c(N,F,C,S)).
property(c(N,F,C,S),color,C) :- card(c(N,F,C,S)).
property(c(N,F,C,S),shape,S) :- card(c(N,F,C,S)).
{in_capset(X) : card(X)}.
:~ in_capset(X).[-1,X]
settable(D, A, B, C) :-
feature(D, A); feature(D, B); feature(D, C); A != B; A != C; B != C.
settable(D, A, A, A) :- feature(D, A).
:- in_capset(X); in_capset(Y); in_capset(Z);
settable(D, A, B, C) :
dimension(D), property(X, D, A), property(Y, D, B), property(Z, D, C);
X < Y; Y < Z.
#show in_capset/1.
This grounds to an ASPIF program with 9296 "words"
• 4D cap set is already known and has a pattern which was found in a challenge of mine, so it might be too trivial. Why not ask to take n as input and solve for n dimensions (without time and memory limit)? Aug 7, 2020 at 3:32
• Also, most people here are not familiar with ASP or ASPIF. It would be helpful if you include relevant links, so we can do some research before tackling the challenge. Aug 7, 2020 at 3:34
# Mega Man
My first polyglot challenge, enjoy!
## Validness of a program
In this challenge, a "program" doesn't take an input. This challenge doesn't care of any output, though.
An invalid program, either:
• Doesn't compile, or
• Compiles, but the program doesn't halt when executed.
A program is valid otherwise.
## The 6 Robot Masters
A robot master is a valid program. Their language can be chosen freely, not necessarily all same or all distinct.
There are 6 Robot Masters in total, namely Cut Man, Elec Man, Ice Man, Fire Man, Bomb Man, and Guts Man.
(Yeah, I wanted to include Time Man and Oil Man as well, but that would make this challenge too hard.)
## Weapons
The robot masters have their distictive weapons. (This doesn't mean the robot masters' source code acts as their weapon, though.) A weapon is an operation on a string.
• Cut Man's weapon, Rolling Cutter, leaves the target source's first half characters only, rounded down. Example: Hello, world!Hello,
• Elec Man's weapon, Thunder Beam, eliminates all whitespaces. Example: Hello, world!Hello,world!
• Ice Man's weapon, Ice Slasher, turns all uppercase ASCII letters small. Example: Hello, world!hello, world!
• Fire Man's weapon, Fire Storm, turns all lowercase ASCII letters capital. Example: Hello, world!HELLO, WORLD!
• Bomb Man's weapon, Hyper Bomb, eliminates the target source's first word. The behavior on the surrounding whitespaces is implementation-defined. Example: Hello, world!world!
• Guts Man's weapon, Super Arm, doubles all characters. Example: Hello, world!HHeelllloo,, wwoorrlldd!!
## Objective
When a weapon is applied to a robot master's source code, if and only if it hits their weakness, the resulting code must be a valid program in the robot master's language.
• Rolling Cutter is the weakness of Elec Man.
• Thunder Beam is the weakness of Ice Man.
• Ice Slasher is the weakness of Fire Man.
• Fire Storm is the weakness of Bomb Man.
• Hyper Bomb is the weakness of Guts Man.
• Super Arm is the weakness of Cut Man.
Every other combination is not a weakness and must result in an invalid program. This includes a weapon applied to its owner.
# Scoring
This is a code golf. The score is the sum of the byte counts of all 6 source codes. The answer with the least score wins.
• I find the organisation of the text quite confusing. If I've understood correctly (and I had to read it several times), the challenge boils down to this: write 6 programs, each of which is only valid when one of the weapons is applied to it. Is that right? Since the programs aren't required to implement the weapons, I don't really understand why each robot master is associated with a program. Lastly, do you mean 'execute' rather than 'compile'? Aug 17, 2020 at 8:28
• @Dingus Well, it was pretty hard to make a reference to the game. I've clarified the 'compile' and 'execute'. The challenge boils down to, Write 6 programs, each of which is valid and also valid when a weapon is applied, but is invalid when any other weapon is applied. Aug 17, 2020 at 9:42
• Ahhh, that makes more sense - I wasn't aware of the game (living under a rock, maybe). Perhaps you should include some more context about it. It seems a bit strange that hitting a weakness results in a valid program (would have expected the opposite), but the description is clear enough. I would suggest a small tweak to 'if and only if it hits their weakness, the resulting code must still be a valid program'. Aug 17, 2020 at 10:20
• About 'compile', I meant in the bullet points where you define what an invalid program is. Many languages are not compiled languages. Aug 17, 2020 at 10:23
# [PuyoPuyo] How long is my combo?
Context:
PuyoPuyo is a puzzle game where you and your opponent pile up colored slimes (called puyo) in a vertical (13*6 cells) grid. A puyo is one-cell big, but they come as pairs in the screen. Puyo pairs fall from the top to the bottom of the grid, and you can move and rotate them the Tetris way. The list of possible puyo pairs is the cartesian product of {'red','blue','green','yellow'} with itself. The pair sequence for a game is randomly generated for both players at the start of a round, and will be the same for both of them.
If four puyo (or more) of the same color are next to each other (in line, in square, Z-, S-, T-, J- or L-shaped), they disappear, awarding you points and making all the above puyo to fall. If when those puyo fall, they make another group of four (or more) they will disappear too, awarding you with more points than the first group: it is called a two-hit combo.
When a combo stops, whatever its length (1-hit or more), you will send damage to your opponent. The bigger the combo, the more damage is sent. Those damage are called ojama puyo, grey slimes that disappear only when a group disappears next to it. If your third column from the left is filled before your opponent's is, you lose. So to kill your opponent, you must manage to fill its screen before they fill yours.
Challenge:
With a given sequence of puyo pairs associated with their drop locations, output the length of the combo that has been made by this player. Shortest code in any language wins.
Input:
List of puyo pairs and their drop locations:
• 'r1b1r1r2y2r3...'
• '001000013102...'
• [('red','1','blue','1'),('red','1','red','2'),('yellow','2','red','3'),...]
• any sensible way you want, provided you detail how it works
Details:
• This list will never contain any "column number" outside of [1;6] (or [0;5], if 0-indexed) nor a puyo of a color outside of {'red','blue','green','yellow'} (or any set used for your interpretation).
• This list will always contain at most one combo sequence.
• This list will never contain unusable data, like two colors in a row, or two column indices in a row.
• If both puyo of a pair are dropped on the same column, the first one to come is on the top of the pair (in the example #1, red is dropped on the top of the blue on the column 1).
• Puyo will never remain floating in the grid, but will fall to the lowest available cell of the column they are in, even if the paired puyo has stopped in its column (tl;dr puyo pairs split).
Output:
A single integer indicating the length of the combo.
Test cases:
Test case #1
Input
y1r1
Output
0
Test case #2
Input
r1r2g3r3g3g2g1r2
Output
2
Test case #3
Input
y1r1r2r3g2g3r2g3y2y3g3g3y1y1
Output
3
Test case #4
Input
b1b1r2b2g3g3r2r3y4y5b6b6b5y5g4b5y4g3b6b6
Output
1
Test case #5
Input
y1b2b3b3b5b5y6b6b4b4
Output
1
Test case #6
Input
y1r1r2r3g2g3r2g3y2y3g3g3y1y1
Output
3
Test case #7 (click me!)
Input
b1b2g3r4g5y6y1y2b3g4r5g6y1b2g3r4g5y6r1r2g3r4r5y6r1b2b3y4r5y6b1y2y3r4r5g6r1b2r3y4g5g6y1g2r3b4b5b6b1g2g3r4r5g6b1b2r3y4b5r6g1y2b2y3y4g4g5g6g5g6
Output
17
Test case #8 (click me!)
Input
r1r2r5r6g1r2r5g6g1y2y5g6g1y2y5g6y1g2g5y6y1b2b5y6r1b2b5r6b1r2r5b6b1y2y5b6b1y2y5b6y1r2r5y6y1r2r5y6r3r4
Output
4
`
Standard Loopholes apply.
NB: for purists, I know that the 13th row is supposed to be invisible and that puyo that are in that column are not considered 'linked' to nearby puyo, but I figured this challenge was hard enough as-is.
@Sandbox please comment! I'd love to hear your thoughts about such a challenge. I will finish setting it up soon, adding some extra resources about the game (like this one). Questions:
• Should I reverse the "top / bottom" puyo of a pair rule? The way it is now, it forces to parse input as pairs. If it is reversed, golfers can take puyo one by one and sort it all by columns, making the challenge easier.
• What tags should it enter with?
• I will make other test cases soon enough, but should I include a as a file (via pastebin)?
• "The combo"? Will there be only one combo? Aug 15, 2020 at 11:07
• Does the two puyo(s) in a pairstick to each other like in tetris? Aug 15, 2020 at 11:07
• Thanks for your comments @user202729, I'll be editing the challenge soon. This challenge will let you assume there will only be one combo. And puyo pairs are broken upon drop if need be, so that every single puyo cannot be floating in the grid. Aug 18, 2020 at 11:51
# Sort until overflow - POSTED HERE
• Now that this has been posted to main, could you delete this proposal to create more space for new answers? Sep 25, 2020 at 0:37
# Quineoid Triple Uniqueness Optimization
This is a variant of Quineoid Triple with the same requirements but different scoring.
Write three different programs such that when any one program is provided as input to one of the other two, you get the source of the remaining program as output. More explicitly, given programs $$\A\$$, $$\B\$$, and $$\C\$$, where $$\f(g)\$$ denotes the output obtained from inputting the text of program $$\g\$$ into program $$\f\$$, all of the following must hold:
• $$\ A(B) = C \$$
• $$\ A(C) = B \$$
• $$\ B(A) = C \$$
• $$\ B(C) = A \$$
• $$\ C(A) = B \$$
• $$\ C(B) = A \$$
# Scoring
The goal is to have the three programs be as different as possible.
Your score is the sum of:
• Number of unique bytes found in program $$\A\$$, but not $$\B\$$ or $$\C\$$
• Number of unique bytes found in program $$\B\$$, but not $$\A\$$ or $$\C\$$
• Number of unique bytes found in program $$\C\$$, but not $$\A\$$ or $$\B\$$
The theoretical maximum score is 256. | 2022-05-25 04:07:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 81, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42260482907295227, "perplexity": 1581.256729446105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662578939.73/warc/CC-MAIN-20220525023952-20220525053952-00318.warc.gz"} |
http://math.stackexchange.com/questions/393823/differential-equations-basic-problem | # Differential equations basic problem
I know this is a basic Physics problems but somehow I can't solve it. We have the differential equation: $2x''x^2 - 4 x^2x' - 2 x^3 = 0$
We have to conclude that the system:
$x' = y$
$y' = 2y + x$
..is equivalent to the differential equation. How can I do it?
-
try to integrate whole equation – iostream007 May 16 '13 at 18:57
If $x \neq 0$, then solving for $x^{\prime\prime}$ we get $$x^{\prime\prime} = 2x^{\prime} + x$$ Now let $y = x^{\prime}$. Then $$y^{\prime} = x^{\prime\prime} = 2x^{\prime} + x = 2y + x$$ | 2014-03-07 14:26:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9849416017532349, "perplexity": 319.456125869738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999644032/warc/CC-MAIN-20140305060724-00091-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://codeforces.com/blog/entry/84805 | ### Berted's blog
By Berted, 7 days ago,
Dear Codeforces community,
We are excited to invite you to TOKI Regular Open Contest #16!
Key details:
Scoring distribution: 100 — 200 — 350 — 450 — 500 — 550 — 850
Finally, we would like to thank:
Please register to the contest, and we hope you will enjoy the contest!
UPD: Contest is over!
Congratulations to our top 10:
Congratulations to our first solvers:
Editorial is available here. (English version is available on page 8)
You can upsolve the problems here.
Thank you for participating and we hope to see you on the next contest!
• +144
» 7 days ago, # | ← Rev. 2 → +52 imagine editing your comment because of samsudin
• » » 7 days ago, # ^ | ← Rev. 3 → +45 [insert inside joke here]
• » » » 7 days ago, # ^ | ← Rev. 4 → +52 sui chan wa kyou mo kawaii~[is trying his best to erase inside joke][tzaph wa, warukanai yo ne]
• » » » » 7 days ago, # ^ | ← Rev. 2 → +47 swag
• » » » » » 7 days ago, # ^ | ← Rev. 3 → +44 swag
• » » » » » » 7 days ago, # ^ | ← Rev. 2 → +44 the person below me is gay
• » » » » » » 7 days ago, # ^ | +43
• » » » » » » » 7 days ago, # ^ | +43 mass udin
• » » » » » » » » 7 days ago, # ^ | ← Rev. 2 → +35 fun fact : udin is a substring of samsudin
• » » » » » » » » » 7 days ago, # ^ | +3 fun fact : sin is a subsequence of samsudin
» 7 days ago, # | +49 Fun fact: The problem writers of this contest are Indonesia's participants in IOI 2020.
• » » 7 days ago, # ^ | +40 I'm panicking right now
• » » » 7 days ago, # ^ | +45 nyse fact
• » » » » 7 days ago, # ^ | +22 nyse pun
» 5 days ago, # | +18 who's samsudin?
• » » 5 days ago, # ^ | +19 that's the beauty of it, nobody knows
» 4 days ago, # | ← Rev. 3 → +33 Friendly bump! 2 hours 20 minutes before the contest!N. B. This comment is actually made so that people joining today's Atcoder Beginners Contest has a chance to read this announcement.
• » » 4 days ago, # ^ | +19 Bump! Contest in 15 minutes!Good luck, we hope you enjoy the contest :D
» 4 days ago, # | 0 Can someone please tell, why this doesn't work for D? Codevector>> v; for (int i = 0; i < n; ++i) { for (int j = 0; j < n; ++j) { int a; cin >> a; if (i < j) { v.pb({a, {i, j}}); } } } sort(all(v)); for (auto [a, ij] : v) { auto [i, j] = ij; i = dsu.find_set(i); j = dsu.find_set(j); if (i == j) { cout << a; return; } if (opp[i] != -1) { dsu.merge(opp[i], j); } if (opp[j] != -1) { dsu.merge(opp[j], i); } i = dsu.find_set(i); j = dsu.find_set(j); assert(i != j); opp[j] = i; opp[i] = j; } assert(0);
• » » 4 days ago, # ^ | ← Rev. 2 → +24 This looks correct to me. In which case number does your code fail?
• » » » 4 days ago, # ^ | 0 The first hidden testcase.
• » » » » 4 days ago, # ^ | +23 hmm, maybe make sure that you have initialized everything correctly (init the dsu, init opp to -1)
• » » » » » 4 days ago, # ^ | +19 Ohh, I somehow deleted the memset (for -1 initialization) line.Thank you very much for your help!
» 4 days ago, # | 0 can anyone share code for problem E
• » » 4 days ago, # ^ | +18 You can see one of the sample codes (by rama_pang) here: https://ideone.com/ifIFgc
» 4 days ago, # | 0 Problem: D Can anyone explain why my code didn't pass? I exactly did the same things written in the editorials. Code#include using namespace std; typedef long long ll; const int N=505; vector g[N]; int color[N]; int yes; void dfs(int at){ for (auto x : g[at]){ if(!color[x]){ color[x] = -1*color[at]; dfs(x); } else if(color[x] == color[at]){ yes = 0; return ; } } } int main(){ ios::sync_with_stdio(0), cin.tie(0); int n; cin >> n; ll a[n][n]; for (int i = 0; i < n; ++i){ for (int j = 0; j < n; ++j){ cin >> a[i][j]; } } ll lo=0, hi=1e9+9; while (lo<=hi){ ll mid = (lo+hi)/2; for (int i = 0; i < N; ++i){ g[i].clear(); } for (int i = 0; i < n; ++i){ for (int j = 0; j < i; ++j){ if(a[i][j]
• » » 4 days ago, # ^ | +26 The graph might be disconnected, so you have to check each connected component.
• » » » 3 days ago, # ^ | 0 Thanks!
» 4 days ago, # | +18 Are the test cases available to download somewhere? (I'm trying to debug my solution to G).
• » » 3 days ago, # ^ | +33 You can find the test cases for this contest in this repository
• » » » 3 days ago, # ^ | +8 Thanks! | 2020-11-26 15:48:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23237062990665436, "perplexity": 5172.812466270302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141188800.15/warc/CC-MAIN-20201126142720-20201126172720-00542.warc.gz"} |
https://math.stackexchange.com/questions/3547709/recurrence-initial-conditions | # recurrence initial conditions
I'm working on a homework assignment involving recursion and I'm having trouble finding an easy way to determine the initial conditions. Heres the problem:
We want to tile ann×1 strip with tiles of three types: 1×1 tiles that are dark-blue, light-blue,and red; 2×1 green tiles, and 3×1 sky-blue tiles. Now give a formula with initial conditions for the number of tilings, considering that blue tiles cannot be next to each other.
I can understand that the recurrence equation is:
$$B_n = B_{n-1}+3B_{n-2}+2B_{n-3}+B_{n-4}+B_{n-5}$$
And I've found initial conditions for
$$B_0=1$$ $$B_1 = 3$$ $$B_2 = 6$$ $$B_3 = 17$$
However, I found these but actually writing down all the possible combinations of tiles, but $$B_4$$ is a huge possible list. Is there some method of combinatorics or permutations I can use to find the initial conditions for $$B_4$$?
• If I let $B_0 = 0$ then I don't understand how the recurrence equation can give me the right answer. If I let $B_0 = 1$ then the method you stated earlier works. At least for what I found $B_{1-3}$ to be. – Thedv8ed1 Feb 15 at 17:32
• Consider separate counts for those that start with blue and those that don’t. That will yield two coupled recurrence relations with fewer initial conditions. You can then derive a recurrence relation for the sum. – RobPratt Feb 15 at 18:01
• My earlier comments have been very confused, so I'm deleting them, and replacing them with this. You can use the same method to find the initial conditions that you did to find the recurrence. You can just set $B_0=1$ and $B_n=0$ for $n<0$ and use the recurrence you already have. – saulspatz Feb 15 at 18:03
Let $$a_n$$ and $$b_n$$ the number of $$n$$-tilings that start with red or green, and blue, respectively. Also, let $$c_n$$ (your $$B_n$$) be the total number of $$n$$-tilings. Then $$a_0=a_1=b_0=c_0=1$$, $$b_1=b_2=2$$, $$c_1=3$$, and, by conditioning on the next tile, we see that \begin{align} a_n &= c_{n-1} + c_{n-2} &&\text{for n \ge 2}\\ b_n &= 2 a_{n-1} + a_{n-3} &&\text{for n \ge 3}\\ c_n &= a_n+b_n-[n=0] &&\text{for n \ge 0} \end{align} Hence \begin{align} c_n &= c_{n-1} + c_{n-2}+2 a_{n-1} + a_{n-3}\\ &=c_{n-1} + c_{n-2}+2 (c_{n-2} + c_{n-3}) + (c_{n-4} + c_{n-5})\\ &=c_{n-1} + 3c_{n-2}+2 c_{n-3} + c_{n-4} + c_{n-5}, \end{align} as you had claimed.
We can also obtain generating functions as follows. Let $$A(z)=\sum_{n=0}^\infty a_n z^n$$, $$B(z)=\sum_{n=0}^\infty b_n z^n$$, and $$C(z)=\sum_{n=0}^\infty c_n z^n$$. Then the recurrence relations imply \begin{align} A(z)-1 - z &=z (C(z)-1) + z^2 C(z) \\ B(z)-1 -2 z-2 z^2 &= 2z (A(z)-1-z) + z^3 A(z) \\ C(z) &= A(z)+B(z)-1 \end{align} Solving for $$A(z)$$, $$B(z)$$, and $$C(z)$$ yields \begin{align} A(z) &= \frac{1}{1 - z - 3 z^2 - 2 z^3 - z^4 - z^5}\\ B(z) &= \frac{1 + z - 3 z^2 - z^3 - z^4 - z^5}{1 - z - 3 z^2 - 2 z^3 - z^4 - z^5}\\ C(z) &= \frac{1 + 2 z + z^3}{1 - z - 3 z^2 - 2 z^3 - z^4 - z^5} \end{align} Notice that the (common) denominator implies that each sequence satisfies the order-5 recurrence $$f_n - f_{n-1} - 3 f_{n-2} - 2 f_{n-3} - f_{n-4} - f_{n-5} = 0,$$ as before. Expanding the series for $$C(z)$$ yields $$1 + 3 z + 6 z^2 + 18 z^3 + 43 z^4 + 113 z^5 + 287 z^6 + 736 z^7 + 1884 z^8 + 4822 z^9 + 12346 z^{10} + \dots .$$ In particular, $$c_3 = 18$$, which you could also have obtained directly from the recurrence relations as follows: \begin{align} a_2 &= c_1 + c_0 = 3 + 1 = 4\\ c_2 &= a_2 + b_2 = 4 + 2 = 6\\ a_3 &= c_2 + c_1 = 6 + 3 = 9\\ b_3 &= 2a_2 + a_0 = 2\cdot 4 + 1 = 9\\ c_3 &= a_3+b_3=9+9 = 18 \end{align}
Wouldn't $$B_3$$ be $$18$$ and not $$17$$? | 2020-11-24 06:50:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 34, "wp-katex-eq": 0, "align": 5, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999197721481323, "perplexity": 418.70163883255003}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141171126.6/warc/CC-MAIN-20201124053841-20201124083841-00305.warc.gz"} |
https://www.ludu.co/course/ethereum/ico | Lesson 17
# Creating an ICO
1
Before we deploy our app to production, I wanted to cover the process of creating your own token, since it's become so common to do in the Ethereum ecosystem.
As you probably know, the ICO ("Initial Coin Offering") industry has been booming lately, and it's completely reinventing the way new startups kickstart themselves. In fact, go have a look at Wikipedia's list of highest crowdfunding projects, and you'll notice that blockchain projects absolutely dominate the list.
Even though many of these token projects are often unrealistic and sometimes straight-up scams, it's a very interesting way to take advantage of some of Ethereum's biggest benefits.
By creating an ICO, you'll also learn how to handle money sent to to your contracts, which should come in handy if you want to create some kind of paid decentralised service in the future.
Therefore, for the sake of this chapter, let's imagine that our Tweether DApp uses its own coin – the Tweether Token. We will create two contracts – one for the token itself and one for the token sale (the ICO).
## Using an ERC20 library
Ethereum uses a standard called ERC20 to make it easy for developers to create their own tokens. In brief, ERC20 defines a set of rules that your token contract must adhere to in order to be compatible with most Ethereum wallets.
While you can create this contract from scratch using the guidelines given by the standard interface, there are multiple libraries that already do this really well.
OpenZeppelin is a company that has open-sourced some great token contract libraries, which are used by many other reputable projects in the space. Let's inherit from some of OpenZeppelin's contracts so that we don't have to reinvent the wheel again!
Make sure you're still in your Tweether project folder and run:
This will download all of OpenZeppelin's libraries in your node_modules folder, and importing them into Solidity is just as easy as importing a node module in a JavaScript file.
Some words of caution before we start coding: although the simplicity of just using other people's libraries everywhere might seem attractive, it's extremely important to review their code first! This is especially important in the Ethereum ecosystem, where libraries have the possibility to actively move money around and defraud your users. In short – if you don't fully understand what a library contract is doing, you shouldn't use it!
Now that that's out of the way, create a new folder called token in your contracts folder, and create a file called TweetherToken.sol. We're now going to import OpenZeppelin's StandardToken contract and inherit from it:
contracts/token/TweetherToken.sol
Next, we simply need to define three parameters for our token: its name, its symbol and the number of decimals it has.
The name and symbol can be anything you want – since there's no central registry for tokens, you don't even have to worry about them being unique! In this example, we'll choose the name TweetherToken with TWE as the symbol.
When it comes to the number of decimals, many developers choose to use the same amount as the ether currency (18). However, for simplicity's sake, we will put 0 here, so that 1 TWE is the minimum amount that you can own.
contracts/token/TweetherToken.sol
That was easy, huh?
## Setting our token supply
For our ERC20 token to be complete, we need to specify how many tokens there are and how they get distributed. While most of the distribution job is going to be up to our ICO contract, the token contract still need to set the initial supply before handing over the reigns.
This is the part where you also get to decide how many of your tokens should be given to the founders, the development team, your strategic partners... etc.
In our case we'll keep things simple, so we'll issue 1 million tokens where 75% of the supply goes to the crowdsale and 25% is kept for the founders (that's us)!
It's worth noting that the creation of ERC20 tokens is in no way determined by proof-of-work like ether and bitcoin. Instead, they are created by completely arbitrary rules set by your contract, and you could in theory create new tokens or reset everyone's balances at any time you want (although keep in mind that people probably wouldn't want to invest in a token that permitted something like that).
Now that we know that, let's start by specifying the amounts given to the founder and to the ICO as state variables (remember that it has to add up to a million):
contracts/token/TweetherToken.sol
Then, we create a constructor function where we set the initial values of two variables: totalSupply_ (note the underscore) and balances. These are inherited from OpenZeppelin's StandardToken library and keep track of your token's total supply as well as what addresses own what amount of tokens.
contracts/token/TweetherToken.sol
As you can see, we initially give all the tokens to the deployment address (msg.sender), since we have to give them to someone. This means that, after deploying the token, the founders will actually control 100% of the supply. Don't get too excited though, it won't stay like that for long.
To test what we've done so far, let's create a new migration file called 6_deploy_token.js:
migrations/6_deploy_token.js
...and a new integration test file dedicated to our token. Notice that, thanks to the fact that our token library inherit from StandardToken, we can call a function called balanceOf to see how many tokens a specific address has.
test/integration/token.js
## Creating our token sale
As we mentioned earlier, there's no point in having a token if everything is owned by you anyway. The next step is to create an ICO contract where people can send in ethers and get Tweether tokens in return!
We'll start by creating an empty contract for our ICO:
contracts/token/TweetherICO.sol
The first thing we want to do is fix the distribution of tokens. Right after the ICO contract is deployed, we want the token contract to send 75% of our token supply to it. For that, we'll create a new function in our token contract called fundICO.
contracts/token/TweetherToken.sol
As you can see, we're again using our Owned library so that the fundICO function can only be accessed by the contract owner. It then uses the inherited transfer function to send exactly 750000 (the number given by FOR_ICO) of the sender's tokens to the given address (_icoAddr).
Now we simply need to deploy our ICO contract and call that function for the transfer to take place. For that, we'll create a new migration file:
migrations/7_deploy_ico.js
This should now have distributed the tokens for us so that 75% of them are in the TweetherICO contract, while 25% remain in our personal "owner" account.
If you try to run the previous test again, you'll see that it fails, which is a good thing!
Let's update the test so that it reflects our new distribution, and checks the number of tokens in both our owner account and the ICO contract:
test/integration/token.js
Now that the distribution is working as expected, let's move on to the actual sale process. In order for our ICO contract to send its tokens to other addresses, it needs to be able to access the token contract's transfer function. We'll therefore import the TweetherToken contract into the ICO contract and save the deployed instance as a state variable, so that we can access it later.
contracts/token/TweetherICO.sol
Our constructor now expects to get the token contract's address as a parameter, so we need to update our latest migration file:
migrations/7_deploy_ico.js
Now the question is: how does our ICO contract know what to do when users send ethers to its address?
For that, we need to use what's known as a fallback function – that is, a function without a name, which is called if no other function is specified. Since we expect to receive ethers on this address, we also need to add the built-in payable modifier:
contracts/token/TweetherICO.sol
In this function, we now want to:
1. Get the amount of money the user has sent (in Wei).
2. Convert that money to a certain amount of Tweether tokens, based on a rate that we set.
3. Send the Tweether tokens to the user's address.
The amount of wei (ether's smallest denomination, in case you had forgotten) sent in the transaction can easily be retrieved using msg.value inside the function. The next step is then to convert this amount to Tweether tokens. For that, we first need to set a conversion rate, which we will make into a public state variable so that everyone can easily verify it. Let's say that 1 ETH gives you 1000 TWE tokens:
contracts/token/TweetherICO.sol
Now, in order to convert wei to TWE, we need to first convert the wei to ethers and then multiply the result with 1000.
While performing basic operations with Solidity's own operators works fine, many ICO contracts choose to use OpenZeppelin's SafeMath library in order to protect themselves from potential integer overflows. This becomes extra important when dealing with money of course, so let's follow the same best practice:
contracts/token/TweetherICO.sol
Alright, now we can safely create an internal function called _getTokenAmount which will handle the conversion from wei to TWE for us:
contracts/token/TweetherICO.sol
Finally, we call this function in our fallback function, followed by the token contract's transfer function to send a certain amount of tokens to their rightful new owner!
contracts/token/TweetherICO.sol
Looks good! Now we just need to test it.
In our test, we will simply send a transaction of 1 ETH from a random account to the ICO contract. After the transaction, we should expect the account to have received 1000 TWE, while the ICO contract's balance should have been reduced to 749 000 TWE.
test/integration/token.js
While it's always good to test your code, it's often more satisfying to see the results of your work wrapped in a nice UI. Let's see how we can deploy our ICO and get some Tweether tokens into our MetaMask wallet!
We start by running our migrations on the development network so that the new token contracts are uploaded (make sure that you have your local testrpc instance running in the background again):
When it's done, take note of what address the ICO contract was uploaded to and copy it!
Now head to MetaMask, and send a transaction of 1 ETH to that address. If your MetaMask wallet doesn't have any ethers, remember that you can run npm run fund-metamask to top it up!
After the transaction has confirmed, you might be confused about why you can't see any tokens in your wallet. It turns out that you need to manually add the token address in MetaMask in order to "register" it – after all, there are so many tokens out there, there's no way MetaMask could list them all by default!
To do this, open the side menu and click on the "Add token" button to get started:
Once you're on the token page, click on "Add custom token" and paste in the token contract's address in the address field.
After confirming that you want to add the token, you should be able to see your TWE balance right next to your ETH balance in the wallet. How cool!
Hopefully, this will have given you enough knowledge about how ERC20 tokens and ICOs work for you to be able to build more advanced token sales in the future. | 2022-07-05 22:35:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2899646759033203, "perplexity": 1393.9744834182934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104628307.87/warc/CC-MAIN-20220705205356-20220705235356-00507.warc.gz"} |
https://www.physicsforums.com/threads/projectile-motion-range-please-help.140881/ | 1. Oct 31, 2006
### Cole07
Question: How far will a stone travel over level ground if it is thrown upward at an angle of 31.0 degrees with respect to the horizontal and with a speed of 11.0m/s?
What is the maximum range that could be achieved with the same initial speed?
I have tried to solve this with no sucess what i have done is Drawn a triangle with a angle being 31.0 degrees the opposite side from the angle is Vy the adjacent side is Vx the hypothesis is 11.0m/s then i drew a chart with X and Y X side of the chart I have Vx= 9.428840308 a=0 then on the y side i have
Vy=5.665418824 a=-9.8 the Vf=0m/s I used the formula Vf=Vi+At and solved for time and got
0.578103962 then i used d=Vi*t+1/2At^2 and solved for D and got 1.637620735 for the first question (But was told this was not correct)
For the second question i doubled the time and multiplied by Vx D=9.428840308 (1.15620784) and got 10.90169909 but it was wrong !
Is there anyone who might be able to help me Please???
2. Oct 31, 2006
### stunner5000pt
what is the final velocity in the Y direction? And why?
for your second question how do you maximize the distance you could travel? They gave you a hint : the SPEED is thesame but something else must be different. What must change?
and Why did you double time??
3. Oct 31, 2006
### rsk
GOod grief you don't believe in rounding numbers do you????
If you've calculated, as it appears, the time taken for the vertical velocity to reach 0M/s, then you've calculated the time for the stone to reach it's maximum height, - it still has some way to go before it hits the ground.
I would use here the equation x = V(i)t + 0.5 a t^2
with the vertical velocity and a = 9.8 but remembering that when the stone hits the ground its vertical displacement is zero because it started from ground level
That will give you a time, then use this with the horizontal velocity (constant) to calculate the horizontal distance.
4. Oct 31, 2006
### Cole07
final velocity in y direction is 0 because at the apex it is not going anywhere
5. Oct 31, 2006
### rsk
But the apex is not the final velocity. It comes back down again.
6. Oct 31, 2006
### stunner5000pt
tahts fine
but is that the final velocity in the y direction for the WHOLE trip
after all you are trying to find the time it took for the ENTIRE journey
7. Oct 31, 2006
### Cole07
I get x= 1.637600545 is this right? (i am too stressed to round any numbers LOL)
8. Oct 31, 2006
### stunner5000pt
howd you get it?
9. Oct 31, 2006
### rsk
That's not what I got.
What to you get for the time?
10. Oct 31, 2006
### Cole07
i took 5.665418824(0.578103962)+0.5(-9.8)(0.57810396)^2 and got 1.637600545
11. Oct 31, 2006
Did you use $$y(t) = 11\cdot\sin(31)t-\frac{1}{2}gt^2=0$$ to calculate the time?
If you didn't, do so. (Solve for t.) When you do so, plug that time into $$x(t)=11\cdot\cos(31)t$$ to get the total displacement.
12. Oct 31, 2006
### Cole07
to get the time i used the formula Vf=Vi+at
Vf=0m/s
a= -9.8
Vi=5.665418824 (by 11.0sin(31.0))
and then i solved for time
13. Oct 31, 2006
### stunner5000pt
why are you still saying velocity final in the Y is zero? Its NOT zero
after it has completed its entire flight what is the velocity in the Y just before it hits the ground? I certainly hope it isnt zero otherwise whatever ive been doing for these 6 odds years has been in vain! VAIN!
and this world has no justice
14. Oct 31, 2006
### Cole07
I don't understand? I am going by the example we did in class and for Vf we used 0 and this is at the apex
15. Oct 31, 2006
Interesting class, indeed.
Btw, I suggest you do a few searches named 'projectile motion', this could do good.
16. Oct 31, 2006
### Office_Shredder
Staff Emeritus
That would be good to calculate the maximum HEIGHT of the object, not the maximum distance it travels.
Throw a ball into the air. When it hits the ground, is the y velocity zero? Use intuition, it really helps sometimes
17. Oct 31, 2006
### stunner5000pt
since im such a nice guy i drew a diagram for you
look at the diagram
now tell me what do u think the velocity i nteh Y direction must be after thw whole trip?
but dont think too hard
you are probably mixing up the type of questions
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18. Oct 31, 2006
### Cole07
yes you are a nice guy but not only am i having trouble with physics BUT I CAN'T OPEN THE ATTACHMENT THIS IS SO SAD!!!!!!!!
19. Oct 31, 2006
20. Oct 31, 2006
### stunner5000pt
correction SHE oooops
k does the image work now?? look at the image. What is the final velocity in teh Y answer that question. THINK before you answer it
And tell me WHY WHY WHYYYYYY you said that | 2016-08-25 18:40:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5599068999290466, "perplexity": 1544.0268685608726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982293922.13/warc/CC-MAIN-20160823195813-00234-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://www.ck12.org/book/CK-12-Algebra-I-Honors/r3/section/2.8/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
2.8: Chapter Test
Difficulty Level: At Grade Created by: CK-12
1. Which of the following is the solution to this linear equation? 12f2(f+1)=8f+22\begin{align*}12f-2(f+1)=8f+22\end{align*}
1. 12
2. 16
3. 24
4. 30
2. Which of the following is the solution to this linear equation? 0.2(13y)=(y5)\begin{align*}0.2(1-3y)=-(y-5)\end{align*}
1. 6
2. 7
3. 12
4. 14
3. Which of the following is the solution to this linear equation? (7t+5)t=263\begin{align*}\frac{(7t+5)}{t}=\frac{26}{3}\end{align*}?
1. 3
2. 6
3. 9
4. 12
4. A rectangle has an area of forty-on square inches with one side equal to five inches. The other side has a length of four more than twice a number. What is that number?
1. 5.0
2. 4.3
3. 3.4
4. 2.1
5. Three consecutive numbers add up to 81. What is the largest number?
1. 26
2. 27
3. 28
4. 81
6. Which is the solution set when four is added to two times a number and is found to be less than or equal to eight less than six times a number?
1. t3\begin{align*}t \le 3\end{align*}
2. t4\begin{align*}t \le 4\end{align*}
3. t0\begin{align*}t \ge 0\end{align*}
4. t0\begin{align*}t \ge -0\end{align*}
7. Which number line best suits the linear inequality shown below? 4(2d)20\begin{align*}4(2-d) \ge 20\end{align*}
8. Which of the following is the solution to the linear inequality? 16(s1)+10>8(s+2)+4(s+1)+4s\begin{align*}16(s-1)+10 > 8(s+2)+4(s+1)+4s\end{align*}
1. s<1\begin{align*}s < 1\end{align*}
2. s>1\begin{align*}s > 1\end{align*}
3. s<1\begin{align*}s < -1\end{align*}
4. s>1\begin{align*}s > -1\end{align*}
9. Which of the following is a solution to the absolute value linear function? 2|2r2|3=13\begin{align*}2|-2r-2|-3=13\end{align*}
1. 5
2. -3
3. 3
4. 0
10. Which of the following is the solution to the absolute value linear inequality? |g4|32\begin{align*}\frac{|g-4|}{3} \ge 2\end{align*}
1. g10, or g2\begin{align*}g \ge -10, \ or \ g \le 2\end{align*}
2. g10, or g2\begin{align*}g \ge 10, \ or \ g \le -2\end{align*}
3. g10, or g2\begin{align*}g \le -10, \ or \ g \ge 2\end{align*}
4. g10, or g2\begin{align*}g \le 10, \ or \ g \ge -2\end{align*}
1. A
2. A
3. C
4. D
5. C
6. B
7. D
8. A
9. C
10. B
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Subjects: | 2016-08-29 21:09:34 | {"extraction_info": {"found_math": true, "script_math_tex": 19, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8963763117790222, "perplexity": 2855.7286074954286}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982965886.67/warc/CC-MAIN-20160823200925-00009-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://zbmath.org/?q=an:0995.26009 | ## An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products.(English)Zbl 0995.26009
Let $$f:[a,b]\to \mathbb{R}$$ be a continuous convex function. The main result is the following (partial) refinement of the Hermite-Hadamard inequality: \begin{aligned} 0 &\leq {1\over 8}\Biggl[f_+'\Biggl({a+ b\over 2}\Biggr)- f_-'\Biggl({a+ b\over 2}\Biggr)\Biggr](b- a)\\ &\leq {f(a)+ f(b)\over 2}- {1\over b-a} \int^b_a f(t) dt\\ &\leq{1\over 8} [f_-'(b)- f_+'(a)](b- a).\end{aligned} The constant $$1/8$$ is sharp. Several applications are included.
### MSC:
26D15 Inequalities for sums, series and integrals 26D10 Inequalities involving derivatives and differential and integral operators
Full Text: | 2022-05-20 04:03:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9247704148292542, "perplexity": 6962.083552545637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662531352.50/warc/CC-MAIN-20220520030533-20220520060533-00775.warc.gz"} |
https://www.lmfdb.org/Variety/Abelian/Fq/2/3/ab_a | # Properties
Label 2.3.ab_a Base field $\F_{3}$ Dimension $2$ $p$-rank $1$ Ordinary no Supersingular no Simple no Geometrically simple no Primitive yes Principally polarizable yes Contains a Jacobian yes
## Invariants
Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 + 2 x + 3 x^{2} )$ $1 - x - 3 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.166666666667$, $\pm0.695913276015$ Angle rank: $1$ (numerical) Jacobians: 1
This isogeny class is not simple.
## Newton polygon
$p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$
## Point counts
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
• $y^2=2x^6+x^4+2x$
$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $84$ $504$ $8736$ $66666$
$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $9$ $18$ $105$ $273$ $738$ $2355$ $6609$ $19494$ $59289$
## Decomposition and endomorphism algebra
Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc. The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$ The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 1.9.c. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{3}}$ The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak $\times$ 1.27.a. The endomorphism algebra for each factor is:
## Base change
This is a primitive isogeny class.
## Twists
Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.af_m$2$2.9.ab_m
2.3.b_a$2$2.9.ab_m
2.3.f_m$2$2.9.ab_m
2.3.c_g$3$2.27.ak_cc
2.3.f_m$3$2.27.ak_cc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.af_m$2$2.9.ab_m
2.3.b_a$2$2.9.ab_m
2.3.f_m$2$2.9.ab_m
2.3.c_g$3$2.27.ak_cc
2.3.f_m$3$2.27.ak_cc
2.3.af_m$6$2.729.i_abnm
2.3.ac_g$6$2.729.i_abnm
2.3.c_g$6$2.729.i_abnm | 2021-12-06 15:44:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9852191805839539, "perplexity": 1494.3365638845748}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363301.3/warc/CC-MAIN-20211206133552-20211206163552-00059.warc.gz"} |
https://www.physicsforums.com/threads/minimize-parameter-for-least-absolute-deviation-lad.409028/ | # Minimize parameter for Least Absolute Deviation LAD
## Main Question or Discussion Point
How to compute $$\beta = arg min_\beta \sum_{i=1}^N {|y_i - x_i^T \beta|$$
## Answers and Replies
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Homework Helper
There is no closed-form solution for this (contrary to the situation with least squares). The software you use (R, SAS, etc) use a variety of methods. check the relevant documentation for those programs.
There is no closed-form solution for this (contrary to the situation with least squares). The software you use (R, SAS, etc) use a variety of methods. check the relevant documentation for those programs.
I was just interested in the calculation not in applying it to real data.
The estimate is the median of the data x1,...,xn and I wanted to see how they derived that result.
$$\sum_{i=1}^n |x_i - a|$$
as a function of $a$, the solution is the sample median. This does not generalize to regression. | 2020-01-25 10:18:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4525635242462158, "perplexity": 1078.8153783453124}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251672440.80/warc/CC-MAIN-20200125101544-20200125130544-00224.warc.gz"} |
https://socratic.org/questions/how-do-you-write-an-equation-of-a-polynomials-with-zeros-3-0-4-degree-3 | # How do you write an equation of a polynomials with zeros: -3,0, 4; degree 3?
Dec 6, 2017
$p \left(x\right) = {x}^{3} - {x}^{2} - 12 x$
#### Explanation:
$\text{given the zeros of a polynomial say}$
$x = a , x = b \text{ and } x = c$
$\text{then the factors are "(x-a),(x-b)" and } \left(x - c\right)$
$\text{the polynomial is the product of the factors}$
$\Rightarrow p \left(x\right) = k \left(x - a\right) \left(x - b\right) \left(x - c\right) \leftarrow \textcolor{b l u e}{\text{k is a multiplier}}$
$\text{here "x=-3,x=0,x=4larrcolor(blue)"zeros}$
$\Rightarrow \left(x + 3\right) , \left(x - 0\right) \text{ and "(x-4)" are the factors}$
$\text{letting } k = 1$
$p \left(x\right) = x \left(x + 3\right) \left(x - 4\right)$
$\textcolor{w h i t e}{p \left(x\right)} = x \left({x}^{2} - x - 12\right)$
$\textcolor{w h i t e}{p \left(x\right)} = {x}^{3} - {x}^{2} - 12 x \leftarrow \textcolor{b l u e}{\text{is a possible polynomial}}$ | 2020-08-11 13:30:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9882375597953796, "perplexity": 7157.378694494908}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738777.54/warc/CC-MAIN-20200811115957-20200811145957-00258.warc.gz"} |
http://mathcentral.uregina.ca/QQ/database/QQ.09.10/h/tina2.html | SEARCH HOME
Math Central Quandaries & Queries
Question from Tina: I have a project where I need to construct a concrete octagonal pad. Each side is 12' and it is 5" deep. Please tell me how to calculate square yards of concrete. Thank you.
Tina,
You can find the area of a regular octagon with sides of length 12' either by looking at our response to Nancy or at our response to Dana. To find the volume in cubic feet you need to multiply buy the height in feet which is $\frac{5}{12}$ feet. There are 3 feet in a yard and hence there are $3 \times 3 \times 3 = 27$ cubic feet in a cubic yard. Hence divide the volume in cubic feet by 27 to find the volume in cubic yards.
Penny
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. | 2017-10-23 06:03:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34900349378585815, "perplexity": 417.7398273616504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825700.38/warc/CC-MAIN-20171023054654-20171023074654-00772.warc.gz"} |
https://www.britannica.com/topic/number-line/article-contributors | Number line
mathematics
elementary algebra
...and extended in the same direction. Similarly, the negative real numbers extend to the left of O. A straight line whose points are thus identified with the real numbers is called a number line. Many earlier mathematicians realized there was a relationship between all points on a straight line and all real numbers, but it was the German mathematician Richard Dedekind who made...
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Error when sending the email. Try again later. | 2017-03-30 23:36:47 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9189528226852417, "perplexity": 1092.253802296196}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218203536.73/warc/CC-MAIN-20170322213003-00006-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/410447/is-the-replacement-i-omega-n-rightarrow-omegai-eta-in-matsubara-green-functi | # Is the replacement $i\omega_n\rightarrow \omega+i\eta$ in Matsubara Green function valid?
In many-body theory we know that to find the retarded Green function in frequency space $G^R(\omega)$ we first find the Matsubara time-ordered Green function $\mathcal{G}(i\omega_n)$ and then replace $i\omega_n$ with $\omega+i\eta$.
First question: is this operation well-defined? Since, for example, if I multipy $\mathcal{G}$ by $1=e^{i\beta \omega_n}$ (for the bosonic case) the replacement $i\omega_n\rightarrow \omega+i\eta$ does not give the same answer as before. I know that both of these functions are related to the Hilbert transform of the spectral weight as
$$G(z)=\int_{-\infty}^{\infty} \frac{dx}{2\pi}\frac{\rho(x)}{x-z}$$
and that makes sense to me because in that case we have a function of real variable $z$ and the replacement (analytic continuation) $z\rightarrow i\omega_n$ gives a unique answer. However the reverse operation doesn't seem to give a unique answer.
Second question: is this replacement an actual analytic continuation?
## 1 Answer
In order to go from the Matsubara Green function to the retarded one using analytical continuation, you need to express the Matsubara Green function as a sum of simple pole terms. This is what results from the Lehmann representation. | 2020-07-12 20:37:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7084512114524841, "perplexity": 193.48629124220594}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657139167.74/warc/CC-MAIN-20200712175843-20200712205843-00184.warc.gz"} |
http://www-old.newton.ac.uk/programmes/HOP/seminars/2007032616151.html | # HOP
## Seminar
### Generalization of the Maslov Theory for Localized Asymptotics and Tsunami waves
Dobrokhotov, S (Russian Academy of Sciences)
Monday 26 March 2007, 16:15-17:00
Seminar Room 1, Newton Institute
#### Abstract
We suggest a new asymptotic representation for the solutions to the multidimensional wave equations with variable velocity with localized initial data. This representation is the generalization of the Maslov canonical operator based also on a simple relationship between fast decaying and fast oscillating solutions, and on boundary layer ideas. It establishes the connection between initial localized perturbations and wave profiles near the wave fronts including the neighborhood of backtracking (focal or turning) and self intersection points. We show that wave profiles are related with a form of initial sources and also with the Lagrangian manifolds organized by the rays and wavefronts. In particular we discuss the influence of such topological characteristics like the Maslov and Morse indices to metamorphosis of the profiles after crossing the focal points. We apply these formulas to the problem of a propagation of tsunami waves in the frame of so-called piston model''. Finally we suggest a fast asymptotically-numerical algorithm for simulation of tsunami wave over nonuniform bottom. Different scenarios of the distribution of the waves are considered, the wave profiles of the front are obtained in connection with the different shapes of the source and with the diverse rays generating the fronts. It is possible to use suggested algorithm to predict in real time the zones of the beaches where the amplitude of the tsunami wave has dangerous high values. In this connection we also discuss the following questions: the problems of the regularization of the wave field near focal points; ill-possed problems appearing in the geometry of the wavefronts; the inverse problem connected with the possibility of reconstruction of the source via the measurement of the tsunami wave profile on the shelf etc. This work was done together with S.Sekerzh-Zenkovich, B.Tirozzi, B.Volkov and was partially supported by RFBR grant N 05-01-00968 and Agreement Between University "La Sapienza", Rome and Institute for Problems in Mechanics RAS, Moscow.
Bibliography
[1] S.Yu. Dobrokhotov, S.Ya Sekerzh-Zenkovich, B. Tirozzi, T.Ya. Tudorovskiy, The description of tsunami waves propagation based on the Maslov canonical operator, Doklady Mathematics, 2006, v.74, N 1, pp. 592-596
[2] S.Yu. Dobrokhotov, S.Ya Sekerzh-Zenkovich, B. Tirozzi, T.Ya. Tudorovskiy, Asymptotic theory of tsunami waves: geometrical aspects and the generalized Maslov representation, Publications of Kyoto Research Mathematical Institute, Vol.4, page 118-153, 2006, ISSN 1880-2818. [3] S.Dobrokhotov, S.Sekerzh-Zenkovich, B.Tirozzi, B.Volkov Explicit asymptotics for tsunami waves in framework of the piston model, Russ. Journ. Earth Sciences, 2006, v.8, ES403, pp.1-12
[4] S.Dobrokhotov, S.Sinitsyn, B.Tirozzi, Asymptotics of Localized Solutions of the One-Dimensional Wave Equation with Variable Velocity. I. The Cauchy Problem, Russ. Jour.Math.Phys., v.14, N1, 2007, pp.28-56
#### Video
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible. | 2016-02-10 17:56:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40006184577941895, "perplexity": 2052.538711571628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701159985.29/warc/CC-MAIN-20160205193919-00100-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/58574/correct-longtable-multicolumn-exceeding-page-boundaries-margins | # Correct longtable \multicolumn exceeding page boundaries/margins
I have problem with some of the `longtable` tables, as illustrated here:
Is there an easy way to make table wrap cell contents multi-line so that it does not exceed page margins?
-
Please provide a minimal working example (MWE) the duplicates this problem. That way community members can identify the problems much easier and hit the ground running. Specific to this example, you mention `\multicolumn` although it seems like you only have two columns in your `longtable`. Is there any addition (invisible) columns to your `longtable`? – Werner Jun 4 '12 at 23:24
TeX file was auto-generated by Sphinx. Here is sample that produced above table: pastebin.com/88qjEfdG – bmatt Jun 4 '12 at 23:36
Using the `p{<width}` column type should fix it. The `l` column type does not allow for line wrapping. – Peter Grill Jun 4 '12 at 23:37
`longtable` column specifiers are exactly the same as standard latex `tabular` ones (or the extended set from the `array` package. so change `l` to `p{3cm}` and it will wrap the lines in that column to the specified width.
It would be really nifty if one could use a `X` column type. – Werner Jun 4 '12 at 23:48 | 2015-11-30 09:49:38 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8005602359771729, "perplexity": 2186.352273219337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398461390.54/warc/CC-MAIN-20151124205421-00135-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://andrewkay.name/blog/post/card-games-and-projective-geometry/ | A correspondent writes:
A few times, over the past few weeks, I've seen a particular friend of mine, and he's had a game with him, a game called Spot It. It takes the form of a bunch of cards, each of which has eight different symbols on it; the game mechanic depends on the fact that each pair of cards has exactly one symbol in common.
[...]
It turns out there are 57 distinct symbols. 42 of them appear on eight cards each, 14 appear on seven cards each, and one appears on six. This looks suspiciously like "57 symbols, 57 cards, each card has eight symbols, each symbol appears on eight cards" with two cards missing.
[...]
It's trivial to deduce what the 'missing' cards are, and, with them added, it is then a 57-card deck with the same property; that any two cards have exactly one symbol in common.
Then I noticed that 57 equals 57. That is, that the number of cards in the 'complete' deck equals the number of distinct symbols. Then, I drew up a 57x57 matrix, where one axis is card identity and the other is symbol identity. It looks symmetric enough, and, on checking, it turns out that the "one bit in common" property holds in the other dimension too (which means, mapped back into cards, that, for every pair of symbols, there is exactly one card that has both of them).
Then a few questions (paraphrased):
1. Why $57$; is there anything special about this number?
2. If we choose a different number of symbols per card, can we always construct a card game like this? How many cards would there be?
3. Are there always the same number of cards and symbols in a complete game? If a game has fewer cards than symbols, is it always possible to complete it?
4. Is it possible to add more cards even if none are "missing"?
I added a question of my own.
1. Is the incidence matrix actually symmetric (for some permutation of rows and columns)? If so, do all such card games have symmetric incidence matrices?
I found question E interesting because it implies that there is a bijection between cards and symbols with nice properties;* in particular it relates these card games to certain regular graphs of diameter $2$. More on this later.
Here's an example of what we're talking about. The triangle game has cards $a,b,c$ and symbols $\alpha,\beta,\gamma$:
$a$ $b$ $c$ $\beta,\gamma$ $\alpha,\gamma$ $\alpha,\beta$
There are $3$ cards and $3$ symbols, each card has $2$ symbols, each symbol appears on $2$ cards, each pair of cards has one symbol in common, and each pair of symbols appears together on one card. There is a pleasing duality between cards and symbols, and we can write down the dual of this game:
$\alpha$ $\beta$ $\gamma$ $b,c$ $a,c$ $a,b$
In the dual game, the card $\alpha$ has symbols $b,c$ because those were the cards with the symbol $\alpha$ in the original game. Notice that in this case, the dual is essentially the same game: there's a bijection $$\varphi = \{~a \mapsto \alpha,~b \mapsto \beta,~c \mapsto \gamma~\}$$ such that a card $x$ has a symbol $\varphi(y)$ if and only if the card $\varphi(x)$ has the symbol $y$ in the dual game. This relates to the fact that the incidence matrix is symmetric:
$\begin{matrix} \alpha&\beta&\gamma \end{matrix}$ $\begin{matrix} a \\ b \\ c \end{matrix}$ $\begin{pmatrix} 0&1&1 \\ 1&0&1 \\ 1&1&0 \end{pmatrix}$
It's usually a good idea to come up with some notation.
Definition. An incidence structure is $(C,S,\in)$ where $C$ and $S$ are disjoint finite non-empty sets, and $\in$ is a relation with domain $C$ and range $S$.**
Interpret $C$ as the set of cards, $S$ as the set of symbols, and $x \in \sigma$ as "card $x$ has symbol $\sigma$". The bit about domains and ranges means there are no "blank cards" or "phantom symbols" - every card has a symbol, and every symbol appears on some card.
Definition. For $x : C$, $S_x = \{~\sigma : S~|~x \in \sigma~\}$, and for $\sigma : S$, $C_{\sigma} = \{~x : C~|~x \in \sigma~\}$.
$S_x$ is the set of symbols on a card $x$, and $C_\sigma$ is the set of cards with symbol $\sigma$.
Definition. A Card & Symbol Game (or CSG) is an incidence structure such that
$\newcommand\prop[1]{[\textsf{#1}]}\prop{UNIFORM}$ For some $n \ge 1$, every card has $n+1$ symbols (i.e. $\exists n \ge 1 \bullet \forall x : C \bullet \#S_x = n+1$)
$\prop{1-COMMON}$ Any two distinct cards have a unique common symbol (i.e. $\forall x,y : C~|~x \ne y \bullet \#(S_x \cap S_y) = 1$)
Call this $\prop{CSG} \equiv \prop{UNIFORM} \land \prop{1-COMMON}$.
The use of $n+1$ rather than $n$ is because mathematicians like to go back and change their notation in order to make things easier later, once they know what will make things easier.
Some more definitions will be useful for our investigation:
Definition. An incidence structure is square if there are the same number of cards and symbols (i.e. $\#C = \#S$). Call this $\prop{SQUARE}$.
Definition. An incidence structure is regular if the number of symbols on each card equals the number of cards with each symbol, and this number is at least $2$. Call this $\prop{REGULAR}$.
Definition. Incidence structures $\mathcal{G} = (C,S,\in)$ and $\mathcal{G}' = (C',S',\in')$ are isomorphic (or $\mathcal{G} \cong \mathcal{G}'$) if there are bijections $\varphi : C \to C'$ and $\psi : S \to S'$ such that $x \in \sigma$ if and only if $\varphi(x) \mathrel{\in'} \psi(\sigma)$.
Obviously if $\mathcal{G}$ satisfies a property $\prop{P}$ and $\mathcal{G} \cong \mathcal{G}'$, then $\mathcal{G}'$ also satisfies $\prop{P}$.
We'd like to talk about duals:
Definition. The dual of an incidence structure $\mathcal{G} = (C,S,\in)$ is $\mathcal{G}^\star = (S,C,\ni)$, where $\sigma \ni x$ if and only if $x \in \sigma$.
Definition. $\mathcal{G}$ is self-dual if $\mathcal{G} \cong \mathcal{G}^\star$. Call this $\prop{SELF-DUAL}$.
Obviously $\mathcal{G}^{\star\star} = \mathcal{G}$. It will be useful to have a notation for dual properties:
Meta-definition. For a property $\prop{P}$, an incidence structure $\mathcal{G}$ satisfies $\newcommand\propd[1]{[\textsf{#1}^\star]}\propd{P}$ if and only if $\mathcal{G}^\star$ satisfies $\prop{P}$. For example,
$\propd{UNIFORM}$ For some $n \ge 1$, every symbol appears on $n+1$ cards
$\propd{1-COMMON}$ Any two distinct symbols appear together on a unique card
Obviously $[\mathsf{P}^{\star\star}] \equiv \prop{P}$. Due to the symmetry between cards and symbols in their definitions, $\prop{SQUARE}$, $\prop{REGULAR}$ and $\prop{SELF-DUAL}$ are each equivalent to their dual properties.
We'll be interested in finding more logical connections between these properties; let's write $\newcommand\infer{\vDash} \prop{P} \infer \prop{Q}$ to mean that $\prop{Q}$ can be inferred from $\prop{P}$.
Proposition. $\prop{REGULAR} \equiv \prop{SQUARE} \land \prop{UNIFORM} \land \propd{UNIFORM}$.
Proof. Obviously $\prop{REGULAR} \equiv \prop{UNIFORM} \land \propd{UNIFORM}$ with the same $n$. Consider $\in$ as a set of pairs $x \mapsto \sigma$. Then $\#{\in} = \#C \times \#S_x$ for each $x : C$, and also $\#{\in} = \#S \times \#C_\sigma$ for each $\sigma : S$. Since $\#C_\sigma = \#S_x$, we have $\#C = \#S$.
Conversely, suppose $\#C = \#S$, then $\#C \times \#S_x = \#{\in} = \#S \times \#C_\sigma$ for each $x:C$, $\sigma:S$, so $\#C_\sigma = \#S_x$. $\square$
So if $\mathcal{G}$ is square, uniform and dual-uniform, then the values of $n$ are the same. This is not necessarily the case if $\mathcal{G}$ is not square: for example, the incidence structure $\mathcal{B}_3$ defined by the incidence matrix $$\begin{pmatrix} 0&0&0&~&1&1&1 \\ 0&0&1&~&1&1&0 \\ 0&1&0&~&1&0&1 \\ 0&1&1&~&1&0&0 \\ 1&0&0&~&0&1&1 \\ 1&0&1&~&0&1&0 \\ 1&1&0&~&0&0&1 \\ 1&1&1&~&0&0&0 \end{pmatrix}$$ is uniform with each $\#S_x = 3$ and dual-uniform with each $\#C_\sigma = 4$. (Exercise: generalise this construction to $\mathcal{B}_n$ with $2^n$ cards, $2n$ symbols, each card having $n$ symbols and each symbol appearing on $2^{n-1}$ cards.) This isn't a CSG, as it doesn't satisfy $\prop{1-COMMON}$; another construction gives $\mathcal{C}_4$ with incidence matrix $$\begin{pmatrix} 1&1&1&0&0&0 \\ 1&0&0&1&1&0 \\ 0&1&0&1&0&1 \\ 0&0&1&0&1&1 \end{pmatrix}$$ which generalises to $\mathcal{C}_n$ with $n$ cards, $\frac{1}{2}n(n-1)$ symbols, each card having $n-1$ symbols and each symbol appearing on $2$ cards. (The details of this construction are left as another exercise.) This is a CSG satisfying $\propd{UNIFORM}$ but not $\propd{1-COMMON}$.
Some second-order theorems:
Meta-theorem. $(\prop{P} \infer \prop{Q}) \infer (\propd{P} \infer \propd{Q})$.
Proof. Suppose there is an incidence structure $\mathcal{G}$ satisfying $\propd{P} \land \neg \propd{Q}$. Then $\mathcal{G}^\star$ satisfies $\prop{P} \land \neg \prop{Q}$, a contradiction. $\square$
Meta-theorem. $\prop{SELF-DUAL} \land \prop{P} \infer \propd{P}$.
Proof. If $\mathcal{G}$ satisfies $\prop{P}$ then $\mathcal{G}^\star \cong \mathcal{G}$ satisfies $\propd{P}$. $\square$
These will be useful later.
We turn our attention now to projective geometry. The reason for this will be immediately clear:
Definition. A finite projective plane (or FPP) is an incidence structure $(P,L,\in)$ where $P$ is a set of points, $L$ is a set of lines, $p \in \lambda$ means "point $p$ is on line $\lambda$", and
$\prop{1-COMMON}$ Any two distinct points are on a unique common line
$\propd{1-COMMON}$ Any two distinct lines meet at a unique common point
$\prop{NON-DEGENERATE}$ There are four distinct points such that no three are on any line
Call this $\prop{FPP} \equiv \prop{1-COMMON} \land \propd{1-COMMON} \land \prop{NON-DEGENERATE}$.
Definition. $P_\lambda$ is the set of points on a line $\lambda$, and $L_p$ is the set of lines through a point $p$, as before.
The words "point" and "line" are deliberately undefined - we can interpret them in a geometric sense (if we're generous about how straight a line has to be):
This is the Fano plane, the smallest FPP. But we could instead define "point" to mean card, and "line" to mean symbol, and then we can draw the Fano plane as a card game diagram:
$a$ $b$ $c$ $d$ $e$ $f$ $g$ $\alpha,\beta,\gamma$ $\alpha,\delta,\epsilon$ $\alpha,\zeta,\eta$ $\beta,\delta,\eta$ $\beta,\epsilon,\zeta$ $\gamma,\epsilon,\eta$ $\gamma,\delta,\zeta$
It's easy to check that the Fano plane is a CSG, and isomorphic to its dual (which is also an FPP),
$\alpha$ $\beta$ $\gamma$ $\delta$ $\epsilon$ $\zeta$ $\eta$ $a,b,c$ $a,d,e$ $a,f,g$ $b,d,g$ $b,e,f$ $c,e,g$ $c,d,f$
via the isomorphism $\varphi = \{~a \mapsto \alpha,~b \mapsto \beta,~\ldots,~g \mapsto \eta~\}$ and $\psi = \varphi^{-1}$.
In fact, every FPP is a CSG, and its dual is also an FPP. Let's prove this.
Theorem. The dual of an FPP is an FPP (i.e. $\prop{FPP} \infer \propd{FPP}$).
Proof. We only need to check $\propd{NON-DEGENERATE}$.
Choose $p,q,r,s : P$ as in $\prop{NON-DEGENERATE}$. By $\prop{1-COMMON}$, there are lines $\lambda,\mu,\nu,\xi : L$ joining $p$ to $q$, $q$ to $r$, $r$ to $s$ and $s$ to $p$ respectively. Suppose, without loss of generality, that $\lambda,\mu,\nu$ meet at a single point: by $\propd{1-COMMON}$ the only point on both $\lambda$ and $\mu$ is $q$, and so $\nu$ contains all three of $q,r,s$, a contradiction. $\square$
By a meta-theorem, it follows that $(\prop{FPP} \infer \prop{P}) \infer (\prop{FPP} \infer \propd{P})$. I.e., if every FPP satisfies some property $\prop{P}$, then every FPP also satisfies $\propd{P}$.
Lemma. Every line in an FPP contains at least $3$ points.
Proof. Choose $p,q,r,s : P$ as in $\prop{NON-DEGENERATE}$, and without loss of generality let $p \notin \lambda$ and $q \notin \lambda$. By $\prop{1-COMMON}$ there are $\mu,\nu,\xi : L_p$ with $q \in \mu$, $r \in \nu$ and $s \in \xi$, which are distinct by $\prop{NON-DEGENERATE}$.
Then by $\propd{1-COMMON}$ there are $t,u,v : P_\lambda$ with $t \in \mu$, $u \in \nu$ and $v \in \xi$ which are distinct as each pair of $\mu,\nu,\xi$ already meet at $p$. $\square$
By the dual lemma, every point lies on at least $3$ lines.
Lemma. For any two points in an FPP, there is a line containing neither.
Proof. By $\prop{1-COMMON}$ choose $\mu : L$ with $p \in \mu$ and $q \in \mu$. By the lemma, there is a point $r \in P_\mu - p - q$. By $\prop{NON-DEGENERATE}$ there is a point $s : P - r$ with $s \notin \mu$, so choose $\lambda : L$ with $r \in \lambda$ and $s \in \lambda$. By $\propd{1-COMMON}$, $\lambda$ and $\mu$ only meet at $r$, so $p \notin \lambda$ and $q \notin \lambda$. $\square$
By the dual lemma, for any two lines, there is a point on neither.
Theorem. Every FPP is a CSG (i.e. $\prop{FPP} \infer \prop{CSG}$).
Proof. We have $\prop{1-COMMON}$ already, so we only need to check $\prop{UNIFORM}$: that every point is on the same number of lines.
We will show that given any point $p : P$ and line $\lambda : L$ with $p \notin \lambda$, the number of lines through $p$ equals the number of points on $\lambda$. Given $\mu : L_p$, $\mu \ne \lambda$ so by $\propd{1-COMMON}$ there is a unique point $q : P_\lambda$ with $q \in \mu$. Conversely, given $q : P_\lambda$, by $\prop{1-COMMON}$ there is a unique $\mu : L_p$ with $q \in \mu$. This gives a bijection $: L_p \to P_\lambda$.
Then given distinct points $p,q : P$, by the lemma there is a line $\lambda : L$ with $p \notin \lambda$ and $q \notin \lambda$, so $\#L_p = \#P_\lambda = \#L_q$. $\square$
Notice that the proof also gives us $\prop{FPP} \infer \prop{REGULAR}$, as $\#L_p = \#P_\lambda$.
The problem right now is that the dual of a CSG is not necessarily a CSG; $\mathcal{C}_n^\star$ satisfies $\prop{UNIFORM}$ but not $\prop{1-COMMON}$, for example. Worse still, call this one $\mathcal{L}_5$:
$a$ $b$ $c$ $d$ $e$ $\alpha,\Omega$ $\beta,\Omega$ $\gamma,\Omega$ $\delta,\Omega$ $\epsilon,\Omega$
This CSG is uninteresting because the common symbol is always $\Omega$. Nonetheless, it's a CSG by our definition, and the dual game
$\Omega$ $\alpha$ $\beta$ $\gamma$ $\delta$ $\epsilon$ $a,b,c,d,e$ $a$ $b$ $c$ $d$ $e$
satisfies neither $\prop{UNIFORM}$ nor $\prop{1-COMMON}$. Of course we could change our definition of CSGs to include $\propd{UNIFORM}$ and $\propd{1-COMMON}$, so that the dual of a CSG is always a CSG. But this is unsatisfying and may be too strict: it would disallow "incomplete" CSGs, as deleting cards from a CSG preserves $\prop{UNIFORM}$ and $\prop{1-COMMON}$ but neither $\propd{UNIFORM}$ nor $\propd{1-COMMON}$.
A better approach would be to try to isolate what it means for a CSG to have "missing" cards, and then relate this to "completeness". We'll consider a CSG to be "complete" if it satisfies $\propd{UNIFORM}$ and $\propd{1-COMMON}$, i.e. every symbol appears on the same number of cards, and every pair of symbols appears together, as in the completed original game. Conveniently, this is exactly $\propd{CSG}$.
Definition. A CSG is complete if its dual is a CSG. Call this $\prop{COMPLETE} \equiv \prop{CSG} \land \propd{CSG}$.
Definition. An extension of an incidence structure $\mathcal{G} = (C,S,\in)$ is an incidence structure $\mathcal{G}' = (C',S',\in')$ for which $C \subseteq C'$, $S \subseteq S'$, and $\in$ is the restriction of $\in'$ to $C$ and $S$ (i.e. ${\in} \subseteq {\in'}$ as sets of pairs). The extension is non-trivial if $\mathcal{G}' \ne \mathcal{G}$.
An expansion is an extension such that $\in$ is the restriction of $\in'$ to $C$. Write $\mathcal{G} \le \mathcal{G}'$ for an expansion.
A completion is an expansion which is a complete CSG.
Proposition. If $\mathcal{G} \le \mathcal{G}'$ is a non-trivial expansion, then $C \ne C'$.
Proof. Suppose $C = C'$. Then $\in$ is the restriction of $\in'$ to its own domain, so ${\in} = {\in'}$ and hence $S' =$ range of ${\in'} =$ range of ${\in} = S$, and the expansion is trivial; a contradiction. $\square$
Definition. An incidence structure is CSG-expansible if it has a non-trivial expansion which is a CSG. Call this $\prop{CSG-EXPANSIBLE}$.
An incidence structure is completable if it has a completion. Call this $\prop{COMPLETABLE}$.
So extensions and expansions may add more cards and more symbols, but an extension might add new symbols to existing cards, whereas an expansion cannot change any existing cards. An expansion is trivial if and only if it adds no cards. A completion need not be non-trivial, so $\prop{COMPLETE} \infer \prop{COMPLETABLE}$.
Now let's try to answer some of the questions.
Theorem. $\prop{1-COMMON} \land \prop{REGULAR} \infer \#C = n^2 + n + 1$.
Proof. Fix a card $x : C$. Given any other card $y : C - x$, by $\prop{1-COMMON}$ there is a unique $\sigma : S_x$ for which $y$ is in $C_{\sigma} - x$. Therefore $\{x\}$ and the $C_{\sigma} - x$ form a partition of $C$.
By $\prop{REGULAR}$, $\# S_x = n+1$, and each $\# (C_{\sigma} - x) = n$. Therefore $\#C = 1 + n(n+1)$. $\square$
This answers question A: $57 = 7^2 + 7 + 1$.
Theorem. An incidence structure is a complete CSG if and only if it is either an FPP or a triangle.
Proof. Since complete CSGs satisfy $\prop{1-COMMON}$ and $\propd{1-COMMON}$, we only need to consider complete CSGs which don't satisfy $\prop{NON-DEGENERATE}$. These are degenerate finite projective planes, all of which are classified:
• The empty set. This is not an incidence structure by our definition.
• A single line, with any number of points. This fails $\prop{UNIFORM}$, as we need $n \ge 1$.
• As before, but one of the points is on any number of additional lines. This still fails $\prop{UNIFORM}$.
• A single point, with any number of lines. This fails $\propd{UNIFORM}$, as we need $n \ge 1$.
• As before, but one of the lines contains any number of additional points. This still fails $\propd{UNIFORM}$.
• A line $\mu$ with $C_{\mu} = \{p_1,\ldots,p_{n+1}\}$, and an additional point $q$ with $S_q = \{\lambda_1,\ldots,\lambda_{n+1}\}$, where each $C_{\lambda_i} = \{p_i, q\}$. This satisfies $\prop{UNIFORM}$ only when $n=1$, giving the triangle.
Conversely, $\prop{FPP} \infer \prop{CSG}$ and by a meta-theorem $\prop{FPP} \infer \propd{FPP} \infer \propd{CSG}$, so every FPP is a complete CSG, as is the triangle. $\square$
This answers the first part of question B: all FPPs with $n \le 10$ are known, and there is no FPP with $n=6$, so there is no complete CSG with $7$ symbols per card, for example. $n$ is called the order of the FPP.
Corollary. Every complete CSG is regular.
Proof. $\prop{FPP} \infer \prop{REGULAR}$, and the triangle is regular. $\square$
This answers the second part of question B: if there is a complete CSG with $n+1$ symbols per card, then it has $n^2 + n + 1$ cards.
Corollary. Every complete CSG is square.***
Proof. $\prop{REGULAR} \infer \prop{SQUARE}$. $\square$
This answers question C: yes, a complete CSG must have equal numbers of cards and symbols; and no, there are CSGs which are not completable. For example, a CSG with $1$ card and $7$ symbols is not completable, though it has "missing cards" in the sense that it is CSG-expansible; however, it has a CSG-expansion which is not CSG-expansible.**** In contrast, $\mathcal{L}_n$ is not completable for $n \ge 3$, but $\mathcal{L}_n \le \mathcal{G}$ is a non-trivial CSG-expansion exactly when $\mathcal{G} \cong \mathcal{L}_{n'}$ for some $n' > n$; so however many cards are added, with whatever new symbols, more can still be added!
Some things are known about completability: for example, Dow showed in 1982 that, in other words, any CSG with $\#S = n^2 + n + 1$ and $\#C > n^2 - 2\sqrt{n+3} + 6$ is completable, and if $\#C > n^2 - n + 1$ then the completion is unique (up to isomorphism).
Theorem. $\prop{COMPLETE} \infer \neg \prop{CSG-EXPANSIBLE}$.
Proof. Let $\mathcal{G} \le \mathcal{G}'$ be a non-trivial expansion, and fix $x : C$ and $y : C' \setminus C$. Let $\chi : S$ be the common symbol between $x$ and $y$. There is another symbol $\sigma : S_x - \chi$, and since $G$ satisfies $\propd{UNIFORM}$ there is another card $w : C_\sigma - x$.
Suppose $G'$ satisfies $\prop{1-COMMON}$. Let $\tau : S$ be the unique common symbol between $w$ and $y$. Since $\mathcal{G}$ satisfies $\propd{1-COMMON}$ there is a card $z : C_\chi - x$ with $z \in \tau$. Then $y$ and $z$ have two common symbols $\chi$ and $\tau$, a contradiction. Therefore $\mathcal{G}'$ is not a CSG. $\square$
This answers question D: no, if the CSG is complete then we can't add more cards and still have a CSG.
Question E asks if an incidence matrix is equal to its own transpose. This is apparently a stronger condition than $\prop{SELF-DUAL}$, which only requires the incidence matrix to be some permutation of its transpose.
Definition. An incidence structure is self-polar if there is a bijection $\varphi : C \to S$ such that $x \in \varphi(y)$ if and only $y \in \varphi(x)$. Call this $\prop{SELF-POLAR}$.
Proposition. $\prop{SELF-POLAR} \infer \prop{SELF-DUAL}$.
Proof. $\psi = \varphi^{-1}$ will do. $\square$
Proposition. Every self-dual CSG is complete.
Proof. By meta-theorem, $\prop{SELF-DUAL} \land \prop{CSG} \infer \propd{CSG}$. $\square$
FPPs can be systematically constructed using vector spaces over finite fields, yielding FPPs where $n$ can be any prime power. (It's an open problem whether there are any FPPs where $n$ is not a prime power.) FPPs constructed this way are always self-polar; FPPs constructed by other methods are not necessarily self-dual, and apparently it's another open problem as of 2009 whether every self-dual FPP is self-polar.
To answer question E: the particular card game introduced above is a complete CSG with $8$ symbols per card, so it is isomorphic to the unique FPP with $n=7$, which is self-polar. However, there are complete CSGs which are not self-dual, the smallest two of which are isomorphic to a dual pair of FPPs with $n=9$, so they have $91$ cards each with $10$ symbols.
For general incidence structures, $\prop{SELF-POLAR}$ is strictly stronger than $\prop{SELF-DUAL}$. In 2003, Brouwer et al. found the smallest possible self-dual incidence structure which is not self-polar:
$a$ $b$ $c$ $d$ $e$ $f$ $g$ $\alpha,\beta,\gamma,\epsilon$ $\alpha,\beta,\delta,\zeta$ $\alpha,\zeta,\eta$ $\beta,\epsilon,\eta$ $\alpha,\gamma$ $\beta,\delta$ $\gamma,\delta$
There are $7$ others of the same size, up to isomorphism.
Returning to the start of this post, our correspondent also noted that (in other words) there is a one-to-one correspondence between CSGs and constant-weight codes for which each codeword has Hamming weight $n+1$ and each pair has Hamming distance $2n$. We've shown that complete CSGs are almost exactly the same thing as finite projective planes; another closely related area is the study of symmetric block designs, which (for $\lambda = 1$) are exactly the same thing as complete CSGs, although this result is a theorem rather than by definition.
I said we'd come back to regular graphs with diameter $2$. If the incidence matrix of a CSG is symmetric (i.e. the CSG is self-polar), then we can interpret it as the adjacency matrix of a graph in which $x,y:C$ are joined by an edge if and only if $x \in \varphi(y)$. In this context,
• $\prop{UNIFORM}$ means the graph is $n+1$-regular, i.e. each vertex has $n+1$ edges;
• $\prop{1-COMMON}$ means each pair of distinct vertices $x,y:C$ are joined by a unique route of length $2$;
• $\prop{SELF-POLAR}$ means the graph is undirected.
For $x \in \varphi(x)$, the graph has an edge from $x$ to itself, and self-edges may be used to form routes.***** This gives a correspondence between self-polar CSGs and graphs of this form. Here's the graph corresponding to the Fano plane:
However, the graph produced depends on which polarity $\varphi$ is used. Automorphisms of the graph correspond to conjugates of $\varphi$, but in general the polarities may not form a single conjugacy class in the group of automorphisms of the FPP.
Another thing you can do with CSGs is colour the edges of complete graphs such that the monochrome sets are themselves complete graphs. The colour of an edge joining two cards is given by their unique common symbol. Here's the Fano plane again, with the symbols $\alpha,\beta,\ldots,\eta$ in order of hue:
Here each monochrome set is complete on $3$ vertices, because the Fano plane is dual-uniform; and each pair of colours meets at one vertex, because the Fano plane is dual-1-common. So that's nice.
*Combinatorics would greatly benefit from a theorem which says "if two sets appear near each other, have the same cardinality, and this cardinality is an unusually mundane number like $57$, then there is a natural bijection with some nice properties which explains why this has happened."
**Combinatorists may recognise that the usual definition of incidence structure does not require that everything has some incidence. The relation symbol is $\in$ because we can use a set-theoretic model where $S \subseteq \mathbb{P} C$. Astute combinatorists may recognise that the usual definition of a uniform incidence structure is the dual of the definition given here.
***Ryser's theorem gives the converse: every square CSG is complete!
****This follows because Ryser's theorem gives a strict bound $\#C < n^2 + n + 1$, so it cannot be expanded indefinitely.
*****The friendship theorem implies that the only such graph with no self-edges is the triangle. This is a corollary of the fact that every polarity $\varphi$ of an FPP has an absolute point $x:P$ such that $x \in \varphi(x)$. | 2023-01-31 11:19:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8977389335632324, "perplexity": 303.12235697979895}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499857.57/warc/CC-MAIN-20230131091122-20230131121122-00592.warc.gz"} |
https://ironholds.org/ | # Hi!
I’m Oliver Keyes, a PhD student at the University of Washington’s Department of Human Centred Design & Engineering.
My focus is data ethics, broadly-construed. My current, more specific projects investigate how models of gender are encoded within dataset and algorithm design (and the resulting consequences for trans people), and the way that data scientists think and are taught about ethical framings.
Despite the department name, my interests are more ethnographic and experiential than Engineering - although I do come from a quantitative background. Outside of academia I spend a lot of time engaged with the intersect of Seattle’s Jewish and queer communities, reading, and making truly heinous puns. My pronouns are they/them, and I can be contacted at okeyes @ uw . edu. I’m always interested in chatting with people who have questions or interests related to pretty much any of the above. | 2018-02-25 17:17:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40174514055252075, "perplexity": 4236.899048821767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816841.86/warc/CC-MAIN-20180225170106-20180225190106-00413.warc.gz"} |
https://www.iacr.org/cryptodb/data/author.php?authorkey=2238 | ## CryptoDB
### Huafei Zhu
#### Publications
Year
Venue
Title
2007
PKC
2004
PKC
2004
EPRINT
In this paper, we provide a new approach to study undeniable signatures by translating secure digital signatures to secure undeniable signatures so that the existing algorithms can be used. Our mechanism is that any verifier without trapdoor information cannot distinguish whether a message is encoded from Diffie-Hellamn resource $D$ or random resource $R$ while a signer with trapdoor information can distinguish efficiently a codeword which is computed from $D$ or $R$. We show how our mechanism can be efficiently achieved and provide proofs of security for our schemes in the standard complexity model. We also provide evidences to show that our approach can be applied to construct designated confirmer signatures, designated verifier signatures as well.
2003
EPRINT
We present a practical protocol that allows two players to negotiate price over the Internet in a deniable way so that a player $A$ can prevent another player $B$ from showing this offer $P$ to a third party $C$ in order to elicit a better offer while player $B$ should be sure that this offer $P$ generated by $A$, but should $C$ be unclear whether $P$ is generated by $A$ or $B$ itself, even $C$ and $B$ fully cooperated. Our protocol is a standard browser-server model and uses a trusted third party, but only in a very limited fashion: the trusted third party is only needed in the cases where one player attempts to cheat or simply crashes, therefore, in the vast of majority transactions, the third party is not to be involved at all. In addition, Our price negotiable transaction system enjoys the following properties: \begin{description} \item[(1)]It works in an asynchronous communication model. \item[(2)]It is inter-operated with existing or proposed scheme for electronics voting system; \item[(3)]The two players need not sacrifice their privacy in making use of the trusted third party; \item[(4)]The deniable property can be proved secure in the random oracle paradigm, while the matching protocol can be proved secure in the standard intractable assumption. \end{description}
2003
EPRINT
In distributed networks, a target party $T$ could be a person never meet with a source party $S$, therefore $S$ may not hold any prior evaluation of trustworthiness of $T$. To get permit to access $S$, $T$ should be somewhat trusted by $S$. Consequently, we should study the approach to evaluate trustworthiness of $T$. To attack the problem, we view individual participant in distributed networks as a node of a delegation graph $G$ and map a delegation path from target party $T$ to source party $S$ in networks into an edge in the correspondent transitive closure of graph $G$. Based on the transitive closure property of the graph $G$, we decompose the problem to three related questions below: -how to evaluate trustworthiness of participants in an edge? -how to compute trustworthiness of participants in a path? -how to evaluate the trustworthiness of a target participant in a transitive closure graph? We attack the above three questions by first computing trustworthiness of participants in distributed and authenticated channel. Then we present a practical approach to evaluate trustworthiness by removing the assumption of the authenticated channel in distributed networks.
2003
EPRINT
We study elliptic curve cryptosystems by first investigating the schemes defined over $Z_p$ and show that the scheme is provably secure against adaptive chosen cipher-text attack under the decisional Diffie-Hellman assumption. Then we derive a practical elliptic curve cryptosystem by making use of some nice elliptic curve where the decisional Diffie-Hellman assumption is reserved.
2003
EPRINT
Following from the remarkable works of Cramer and Shoup \cite{CS}, three trapdoor hash signature variations have been presented in the literature: the first variation was presented in CJE'01 by Zhu \cite{Zhu}, the second variation was presented in SCN'02 by Camenisch and Lysyanskaya \cite{CL} and the third variation was presented in PKC'03 by Fischlin \cite{Fis}. All three mentioned trapdoor hash signature schemes have similar structure and the security of the last two modifications is rigorously proved. We point out that the distribution of variables derived from Zhu's signing oracle is different from that generated by Zhu's signing algorithm since the signing oracle in Zhu's simulator is defined over $Z$, instead of $Z_n$. Consequently the proof of security of Zhu's signature scheme should be studied more precisely. We also aware that the proof of Zhu's signature scheme is not a trivial work which is stated below: \begin{itemize} \item the technique presented by Cramer and Shoup \cite{CS} cannot be applied directly to prove the security of Zhu's signature scheme since the structure of Cramer-Shoup's trap-door hash scheme is double deck that is easy to simulate a signing query as the order of subgroup $G$ is a public parameter; \item the technique presented by Camenisch and Lysyanskaya \cite{CL} cannot be applied directly since there are extra security parameters $l$ and $l_s$ guide the statistical closeness of the simulated distributions to the actual distribution; \item the technique presented by Fischlin cannot be applied directly to Zhu's signature scheme as the security proof of Fischlin's signature relies on a set of pairs $(\alpha_i, \alpha_i \oplus H(m_i))$ while the security proof of Zhu's signature should rely on a set of pairs $(\alpha_i, H(m_i))$. \end{itemize} In this report, we provide an interesting random argument technique to show that Zhu's signature scheme immune to adaptive chosen-message attack under the assumptions of the strong RSA problem as well as the existence of collision free hash functions.
2003
EPRINT
In this paper, we study the security notions of verifiably committed signatures by introducing privacy and cut-off time, and then we propose the first scheme which is provably secure in the standard complexity model based on the strong RSA assumption. The idea behind the construction is that given any valid partial signature of messages, if a co-signer with its auxiliary input is able to generate variables called the resolution of messages such that the distribution of the variables is indistinguishable from that generated by the primary signer alone from the views of the verifier/arbitrator, a verifiably committed signature can be constructed.
2003
EPRINT
In PODC 2003, Park et al. \cite{PCSR} first introduce a connection between fair exchange and sequential two-party multi-signature scheme and provide a novel method of constructing fair exchange protocol by distributing the computation of RSA signature. This approach avoids the design of verifiable encryption scheme at the expense of having co-signer store a piece of prime signer's secret key. Dodis and Reyzin \cite{DR} showed that the protocol in \cite{PCSR} is totally breakable in the registration phase, then presented a remedy scheme which is provably secure in the random oracle model, by utilizing Boldyreva non-interactive two-party multi-signature scheme \cite{Bo}. Security in the random oracle model does not imply security in the real world. In this paper, we provide the first two efficient committed signatures which are provably secure in the standard complexity model from strong RSA assumption. Then we construct efficient optimistic fair exchange protocols from those new primitives.
2002
PKC
2002
EPRINT
We present a new public key cryptosystem based on the notion called square decisional Diffie-Hellman problem. The scheme is provably secure against adaptive chosen cipher-text attack under the hardness assumption of the square decisional Diffie-Hellman problem. Compared with Cramer and Shoup's notable public key scheme, our scheme enjoys several nice features: (1)Both schemes are provably secure against adaptive chosen cipher-text attack under the intractability paradigm (the security of Cramer-Shoup's scheme is based on the standard decisional Diffie-Hellman problem while ours based on the square decisional Diffie-Hellman problem; (2)The computational and communication complexity of our scheme is equivalent to the Cramer and Shoup's scheme however, the test function of Cramer-shoup's scheme is linear while our scheme is non-linear, therefore our reduction is more efficient.
#### Coauthors
Feng Bao (1)
Xiaotie Deng (1)
Robert H. Deng (1)
Chan H. Lee (1)
Yi Mu (1)
Willy Susilo (1) | 2021-11-27 10:38:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.517272412776947, "perplexity": 1206.261956230471}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358180.42/warc/CC-MAIN-20211127103444-20211127133444-00600.warc.gz"} |
https://www.mathlearnit.com/what-is-1-361-as-a-decimal | # What is 1/361 as a decimal?
## Solution and how to convert 1 / 361 into a decimal
1 / 361 = 0.003
Convert 1/361 to 0.003 decimal form by understanding when to use each form of the number. Both represent numbers between integers, in some cases defining portions of whole numbers Choosing which to use starts with the real life scenario. Fractions are clearer representation of objects (half of a cake, 1/3 of our time) while decimals represent comparison numbers a better (.333 batting average, pricing: \$1.50 USD). Now, let's solve for how we convert 1/361 into a decimal.
## 1/361 is 1 divided by 361
Teaching students how to convert fractions uses long division. The great thing about fractions is that the equation is already set for us! The two parts of fractions are numerators and denominators. The numerator is the top number and the denominator is the bottom. And the line between is our division property. We use this as our equation: numerator(1) / denominator (361) to determine how many whole numbers we have. Then we will continue this process until the number is fully represented as a decimal. This is how we look at our fraction as an equation:
### Numerator: 1
• Numerators sit at the top of the fraction, representing the parts of the whole. Small values like 1 means there are less parts to divide into the denominator. 1 is an odd number so it might be harder to convert without a calculator. Values like 1 doesn't make it easier because they're small. Now let's explore X, the denominator.
### Denominator: 361
• Unlike the numerator, denominators represent the total sum of parts, located at the bottom of the fraction. Larger values over fifty like 361 makes conversion to decimals tougher. But 361 is an odd number. Having an odd denominator like 361 could sometimes be more difficult. Overall, two-digit denominators are no problem with long division. Next, let's go over how to convert a 1/361 to 0.003.
## Converting 1/361 to 0.003
### Step 1: Set your long division bracket: denominator / numerator
$$\require{enclose} 361 \enclose{longdiv}{ 1 }$$
To solve, we will use left-to-right long division. This method allows us to solve for pieces of the equation rather than trying to do it all at once.
### Step 2: Extend your division problem
$$\require{enclose} 00. \\ 361 \enclose{longdiv}{ 1.0 }$$
We've hit our first challenge. 1 cannot be divided into 361! Place a decimal point in your answer and add a zero. This doesn't add any issues to our denominator but now we can divide 361 into 10.
### Step 3: Solve for how many whole groups you can divide 361 into 10
$$\require{enclose} 00.0 \\ 361 \enclose{longdiv}{ 1.0 }$$
We can now pull 0 whole groups from the equation. Multiple this number by our furthest left number, 361, (remember, left-to-right long division) to get our first number to our conversion.
### Step 4: Subtract the remainder
$$\require{enclose} 00.0 \\ 361 \enclose{longdiv}{ 1.0 } \\ \underline{ 0 \phantom{00} } \\ 10 \phantom{0}$$
If your remainder is zero, that's it! If you still have a remainder, continue to the next step.
### Step 5: Repeat step 4 until you have no remainder or reach a decimal point you feel comfortable stopping. Then round to the nearest digit.
Sometimes you won't reach a remainder of zero. Rounding to the nearest digit is perfectly acceptable.
### Why should you convert between fractions, decimals, and percentages?
Converting between fractions and decimals is a necessity. They each bring clarity to numbers and values of every day life. And the same is true for percentages. So we sometimes overlook fractions and decimals because they seem tedious or something we only use in math class. But they all represent how numbers show us value in the real world. Without them, we’re stuck rounding and guessing. Here are real life examples:
### When you should convert 1/361 into a decimal
Speed - Let's say you're playing baseball and a Major League scout picks up a radar gun to see how fast you throw. Your MPH will not be 90 and 1/361 MPH. The radar will read: 90.0 MPH. This simplifies the value.
### When to convert 0.003 to 1/361 as a fraction
Distance - Any type of travel, running, walking will leverage fractions. Distance is usually measured by the quarter mile and car travel is usually spoken the same.
### Practice Decimal Conversion with your Classroom
• If 1/361 = 0.003 what would it be as a percentage?
• What is 1 + 1/361 in decimal form?
• What is 1 - 1/361 in decimal form?
• If we switched the numerator and denominator, what would be our new fraction?
• What is 0.003 + 1/2? | 2022-11-26 13:06:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5536573529243469, "perplexity": 1125.3697777926457}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446706291.88/warc/CC-MAIN-20221126112341-20221126142341-00523.warc.gz"} |
https://www.turito.com/ask-a-doubt/Mathematics-54-99-4-65-59-34-59-64-58-34-95-99-q5203d7 | Mathematics
Easy
Question
# $54.99 +$4.65 = ______
## $95.99$58.34$59.64$59.34
Hint:
## The correct answer is: $59.64 ### Now as we know that here we have to find the step by breaking one number. The fact that there are no steps to memorise is one of the best things about mental math. The "friendly number" addition approach facilitates working with large numbers. This is due to the fact that we are, in essence, decomposing the problem into more manageable components.Here we will use some properties which are: Commutative Property of Addition: if a and b are real numbers, then a+b=b+a Associative Property of Addition: if a,b, and c are real numbers, then (a+b)+c=a+(b+c) Compensation Property: It is a mental math strategy for multi-digit addition that involves adjusting one of the addends to make the equation easier to solve. Now we have given$54.99 + $4.65, let's solve this.$54.99 + $4.65$55 + $4.65 (Rounding$54.99 to the nearest whole number)$59.65So, the sum is$59.65
Here the concept of Break Apart Numbers is used where a number is divided in two parts to make addition easy. The concept of friendly numbers is also used so that it becomes easy for addition or subtraction purpose. So, the sum is \$59.65. | 2022-12-10 08:26:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5690270662307739, "perplexity": 1424.5711314978983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710421.14/warc/CC-MAIN-20221210074242-20221210104242-00579.warc.gz"} |
http://tug.org/pipermail/texhax/2007-December/009787.html | # [texhax] mathbb and a0poster class
David Romano romanod at math.grinnell.edu
Sun Dec 30 15:52:58 CET 2007
I have a student who is using the a0poster class. The title, which uses
\VeryHuge size font, doesn't show \mathbb{R} correctly: The rest of the
surrounding math text does display correctly
( $\mathcal{H}(\displaystyle\mathbb{R}^n)$)
but the size of \mathbb{R} is the smaller than the size of the
superscript n. Any help or suggestions would be welcome.
Thanks,
David Romano | 2017-10-21 08:50:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7410349249839783, "perplexity": 10913.86108693411}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824675.67/warc/CC-MAIN-20171021081004-20171021101004-00611.warc.gz"} |
https://socratic.org/questions/in-january-a-dealership-sold-164-vehicles-february-the-same-dealership-sold-23-8 | # In January, a dealership sold 164 vehicles. February, the same dealership sold 23.8% fewer vehicles than they sold in January. How many vehicles were sold in February?
May 17, 2018
See a solution process below:
#### Explanation:
Let $f$ be the number of cars sold in February.
We can write the equation for the number of cars sold in February as:
f = 164 - (23.8% of 164)
"Percent" or "%" means "out of 100" or "per 100", Therefore 23.8% can be written as $\frac{23.8}{100}$.
When dealing with percents the word "of" means "times" or "to multiply".
We can now rewrite the equation and calculate $f$ as:
$f = 164 - \left(\frac{23.8}{100} \times 164\right)$
$f = 164 - \frac{3903.2}{100}$
$f = 164 - 39.032$
$f = 124.968$
The dealership sold 125 cars in February. | 2020-03-30 01:31:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 8, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38793736696243286, "perplexity": 3179.9556365380413}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370496330.1/warc/CC-MAIN-20200329232328-20200330022328-00448.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/in-abc-if-cosaa-cosbb-then-show-that-it-is-an-isosceles-triangle-solutions-of-triangle_201534 | # In ∆ABC, if cosAa=cosBb, then show that it is an isosceles triangle - Mathematics and Statistics
Sum
In ∆ABC, if (cos "A")/"a" = (cos "B")/"b", then show that it is an isosceles triangle
#### Solution
In ∆ABC by sine rule, we have
"a"/"sin A" = "b"/"sin B" = "k"
∴ a = k sin A, b = k sin B
Now, (cos "A")/"a" = (cos "B")/"b" .......[Given]
∴ "cos A"/"k sin A" = "cos B"/"k sin B"
∴ "cos A"/"sin A" = "cos B"/"sin B"
∴ sin A cos B = cos A sin B
∴ sin A cos B − cos A sin B = 0
∴ sin (A − B) = 0 = sin 0
∴ A − B = 0
∴ A = B
Hence, ∆ABC is an isosceles triangle.
Concept: Solutions of Triangle
Is there an error in this question or solution? | 2022-05-26 11:24:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7482255697250366, "perplexity": 10509.476713874348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662604794.68/warc/CC-MAIN-20220526100301-20220526130301-00460.warc.gz"} |
https://mathoverflow.net/questions/366663/is-sum-k-1n-fracn-1k-1-composite-for-n-geq-4 | # Is $\sum_{k=1}^{n}\frac{(n-1)!}{(k-1)!}$ composite for $n\geq 4$?
Define $$a_n$$ as follows:
$$a_1=1,\ \ a_{n+1}=na_n+1\$$ At this time, the sequence $$a_n$$ is as follows: $$a_n=\sum_{k=1}^{n}\frac{(n-1)!}{(k-1)!}$$ I made some discoveries about this sequence.
The first:$$a_k\equiv 0\pmod{m}\Rightarrow a_{k+Nm}\equiv 0\pmod{m}~~~~\forall k,m,N\in\mathbb{N}$$ The second:$$n\geq 4\,\Rightarrow\,a_n ~\mathrm{is~composite}$$ I was able to prove the first, but not the second. My expectation is that the second is correct, but I'm not sure it can be proved. My friend used computer and check $$a_n$$ is composite for $$4\leq n\leq 48$$. After $$a_{49}$$, it is too large number to check on his computer. Please let me know if you come up with a proof method. Any help is welcome!
(I am a Japanese college student. I'm sorry for my poor English.)
• This is oeis.org/A000522 – Fredrik Johansson Jul 27 '20 at 6:58
• $a_i$ is odd resp. even when $i$ is odd resp. even. So $a_i$ is certainly composite for all even $i$. – Ben Smith Jul 27 '20 at 8:14
• The series $a_{n+1}=\sum_{k=0}^{n} \frac{n!}{k!}$ can be written in another representation. That is $$a_{n+1}=2^n+\sum_{k=2}^{n} \binom{n}{k}2^{n-k}D_k$$. Where, $D_k$ is the number of derangements. – Alapan Das Jul 27 '20 at 8:29
$$a_n$$ is composite for $$4 \le n \le 2016$$.
$$a_{2017}$$ appears to be prime (it passes a strong pseudoprime test). I have not tried to certify that it is prime (this would take a while as the number has 5789 digits).
• A pity this wasn't discovered $3$ years ago. – Robert Israel Jul 27 '20 at 14:19 | 2021-03-08 22:05:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8010724186897278, "perplexity": 340.99625053510186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178385529.97/warc/CC-MAIN-20210308205020-20210308235020-00415.warc.gz"} |
http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=01-478 | 01-478 Christian Hainzl
One non-relativistic particle coupled to a photon field (47K, LaTeX2e) Dec 19, 01
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers
Abstract. We investigate the ground state energy of a charged particle coupled to a photon field. First, we regard the self-energy of a "free" electron, which we describe by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of the coupling constant $\alpha$, the leading order term is represented by $8\pi \alpha (\Lambda - \ln[1 + \Lambda])$. Secondly, we treat the self-energy of a charged boson and provide a different proof for recovering the next to leading order term in $\alpha$, which has already been obtained in a previous paper. Thirdly, we estimate from above the binding energy of a charged boson in the field of a nucleus. The first order radiative correction turns out to behave like $\ln[1+\Lambda]$ for large values of the ultraviolet-cutoff parameter $\Lambda$.
Files: 01-478.src( 01-478.keywords , lamb.tex ) | 2018-07-18 17:59:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7895689606666565, "perplexity": 455.260557118997}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590314.29/warc/CC-MAIN-20180718174111-20180718194111-00546.warc.gz"} |
https://math.libretexts.org/Courses/Honolulu_Community_College/Math_75X%3A_Introduction_to_Mathematical_Reasoning_(Kearns)/03%3A_More_Types_of_Fractions-_Decimals_Percents_Ratios_and_Rates/3.03%3A_Combining_Decimals-_Addition_and_Subtraction_with_Decimals | Skip to main content
# 3.3: Combining Decimals- Addition and Subtraction with Decimals
## Adding Decimals
Addition of decimal numbers is quite similar to addition of whole numbers. For example, suppose that we are asked to add 2.34 and 5.25. We could change these decimal numbers to mixed fractions and add.
\begin{aligned} 2.34 + 5.25 & = 2 \frac{34}{100} + 5 \frac{25}{100} \\ & = 7 \frac{59}{100} \end{aligned}\nonumber
However, we can also line the decimal numbers on their decimal points and add vertically, as follows.
$\begin{array}{r} 2.34 \\ + 5.25 \\ \hline 7.59 \end{array}\nonumber$
Note that this alignment procedure produces the same result, “seven and fifty nine hundredths.” This motivates the following procedure for adding decimal numbers.
Adding Decimals
To add decimal numbers, proceed as follows:
1. Place the numbers to be added in vertical format, aligning the decimal points.
2. Add the numbers as if they were whole numbers.
3. Place the decimal point in the answer in the same column as the decimal points above it.
Example 1
Add 3.125 and 4.814.
Solution
Place the numbers in vertical format, aligning on their decimal points. Add, then place the decimal point in the answer in the same column as the decimal points that appear above the answer.
$\begin{array}{r} 3.125 \\ +4.814 \\ \hline 7.939 \end{array}\nonumber$
Thus, 3.125 + 4.814 = 7.939.
Exercise
Add: 2.864 + 3.029
Answer
5.893
Example 2
Jane has $4.35 in her purse. Jim has$5.62 in his wallet. If they sum their money, what is the total?
Solution
Arrange the numbers in vertical format, aligning decimal points, then add.
$\begin{array}{r} \ 4.35 \\ + \ 5.62 \\ \hline \ 9.97 \end{array}\nonumber$
Exercise
Alice has $8.63 in her purse and Joanna has$2.29. If they combine sum their money, what is the total?
Answer
10.91 Before looking at another example, let’s recall an important observation. Important Observation Adding zeros to the end of the fractional part of a decimal number does not change its value. Similarly, deleting trailing zeros from the end of a decimal number does not change its value. For example, we could add two zeros to the end of the fractional part of 7.25 to obtain 7.2500. The numbers 7.25 and 7.2500 are identical as the following argument shows: \begin{aligned} 7.2500 & = 7 \frac{2500}{10000} \\ & = 7 \frac{25}{100} \\ & = 7.25 \end{aligned}\nonumber Example 3 Add 7.5 and 12.23. Solution Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add a trailing zero to improve columnar alignment. $\begin{array}{r} 7.50 \\ +12.23 \\ \hline 19.73 \end{array}\nonumber$ Hence, 7.5 + 12.23 = 19.73. Exercise Add: 9.7 + 15.86 Answer 25.56 Example 4 Find the sum: 12.2+8.352 + 22.44. Solution Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add trailing zeros to improve the columnar alignment. $\begin{array}{r} 12.200 \\ 8.352 \\ + 22.440 \\ \hline 42.992 \end{array}\nonumber$ Hence, 12.2+8.352 + 22.44 = 42.992. Exercise Add: 12.9+4.286 + 33.97 Answer 51.156 ## Subtracting Decimals Subtraction of decimal numbers proceeds in much the same way as addition of decimal numbers. Subtracting Decimals To subtract decimal numbers, proceed as follows: 1. Place the numbers to be subtracted in vertical format, aligning the decimal points. 2. Subtract the numbers as if they were whole numbers. 3. Place the decimal point in the answer in the same column as the decimal points above it. Example 5 Subtract 12.23 from 33.57. Solution Arrange the numbers in vertical format, aligning their decimal points in a column, then subtract. Note that we subtract 12.23 from 33.57. $\begin{array}{r} 33.57 \\ -12.23 \\ \hline 21.34 \end{array}\nonumber$ Hence, 33.57 − 12.23 = 21.34. Exercise Subtract: 58.76 − 38.95 Answer 19.81 As with addition, we add trailing zeros to the fractional part of the decimal numbers to help columnar alignment. Example 6 Find the difference: 13.3 − 8.572. Solution Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add trailing zeros to the fractional part of 13.3 to improve columnar alignment. $\begin{array}{r} 13.300 \\ -8.572 \\ \hline 4.728 \end{array}\nonumber$ Hence, 13.3 − 8.572 = 4.728. Exercise Subtract: 15.2 − 8.756 Answer 6.444 ## Adding and Subtracting Signed Decimal Numbers We use the same rules for addition of signed decimal numbers as we did for the addition of integers. Adding Two Decimals with Like Signs To add two decimals with like signs, proceed as follows: 1. Add the magnitudes of the decimal numbers. 2. Prefix the common sign. Example 7 Simplify: −3.2+(−18.95). Solution To add like signs, first add the magnitudes. $\begin{array}{r} 3.20 \\ +18.95 \\ \hline 22.15 \end{array}\nonumber$ Prefix the common sign. Hence, −3.2+(−18.95) = −22.15 Exercise Simplify: −5.7 + (−83.85) Answer −89.55 We use the same rule as we did for integers when adding decimals with unlike signs. Adding Two Decimals with Unlike Signs To add two decimals with unlike signs, proceed as follows: 1. Subtract the smaller magnitude from the larger magnitude. 2. Prefix the sign of the decimal number with the larger magnitude. Example 8 Simplify: −3 + 2.24. Solution To add unlike signs, first subtract the smaller magnitude from the larger magnitude. $\begin{array}{r} 3.00 \\ -2.24 \\ \hline 0.76 \end{array}\nonumber$ Prefix the sign of the decimal number with the larger magnitude. Hence, −3+2.24 = −0.76. Exercise Simplify: −8 + 5.74 Answer −2.26 Subtraction still means add the opposite. Example 9 Simplify: −8.567 − (−12.3). Solution Subtraction must first be changed to addition by adding the opposite. $−8.567 − (−12.3) = −8.567 + 12.3\nonumber$ We have unlike signs. First, subtract the smaller magnitude from the larger magnitude. $\begin{array}{r} 12.300 \\ − 8.567 \\ \hline 3.733 \end{array}\nonumber$ Prefix the sign of the decimal number with the larger magnitude. Hence: \begin{aligned} −8.567 − (−12.3) & = −8.567 + 12.3 \\ & = 3.733 \end{aligned}\nonumber Exercise Simplify: −2.384 − (−15.2) Answer 12.816 Order of operations demands that we simplify expressions contained in parentheses first. Example 10 Simplify: −11.2 − (−8.45 + 2.7). Solution We need to add inside the parentheses first. Because we have unlike signs, subtract the smaller magnitude from the larger magnitude. $\begin{array}{r} 8.45 \\ − 2.70 \\ \hline 5.75 \end{array}\nonumber$ Prefix the sign of the number with the larger magnitude. Therefore, $−11.2 − (−8.45 + 2.7) = −11.2 − (−5.75)\nonumber$ Subtraction means add the opposite. $−11.2 − (−5.75) = −11.2+5.75\nonumber$ Again, we have unlike signs. Subtract the smaller magnitude from the larger magnitude. $\begin{array}{r} 11.20 \\ − 5.75 \\ \hline 5.45 \end{array}\nonumber$ Prefix the sign of the number with the large magnitude. $−11.2+5.75 = −5.45\nonumber$ Exercise Simplify: −12.8 − (−7.44 + 3.7) Answer −9.06 Writing Mathematics The solution to the previous example should be written as follows: \begin{aligned} −11.2 − (−8.45 + 2.7) & = −11.2 − (−5.75) \\ & = −11.2+5.75 \\ & = −5.45 \end{aligned}\nonumber Any scratch work, such as the computations in vertical format in the previous example, should be done in the margin or on a scratch pad. Example 11 Simplify: −12.3 −|− 4.6 − (−2.84)|. Solution We simplify the expression inside the absolute value bars first, take the absolute value of the result, then subtract. \begin{aligned} -12.3 - |-4.6 -(-2.84)| ~ \\ = -12.3 -|-4.6 + 2.84| ~ & \textcolor{red}{ \text{ Add the opposite.}} \\ = -12.3 -|-1.76| ~ & \textcolor{red}{ \text{ Add: } -4.6 + 2.84 = -1.76.} \\ = -12.3-1.76 ~ & \textcolor{red}{ |-1.76|=1.76.} \\ =-12.3 + (-1.76) ~& \textcolor{red}{ \text{ Add the opposite.}} \\ = -14.06 ~ & \textcolor{red}{ \text{ Add: } -12.3 + (-1.76) = -14.06.} \end{aligned}\nonumber Exercise Simplify: −8.6 −|− 5.5 − (−8.32)| Answer −11.42 ## Exercises In Exercises 1-12, add the decimals. 1. $$31.9 + 84.7$$ 2. $$9.39 + 7.7$$ 3. $$4 + 97.18$$ 4. $$2.645 + 2.444$$ 5. $$4 + 87.502$$ 6. $$23.69 + 97.8$$ 7. $$95.57 + 7.88$$ 8. $$18.7+7$$ 9. $$52.671 + 5.97$$ 10. $$9.696 + 28.2$$ 11. $$4.76 + 2.1$$ 12. $$1.5 + 46.4$$ In Exercises 13-24, subtract the decimals. 13. $$9 − 2.261$$ 14. $$98.14 − 7.27$$ 15. $$80.9 − 6$$ 16. $$9.126 − 6$$ 17. $$55.672 − 3.3$$ 18. $$4.717 − 1.637$$ 19. $$60.575 − 6$$ 20. $$8.91 − 2.68$$ 21. $$39.8 − 4.5$$ 22. $$8.210 − 3.7$$ 23. $$8.1 − 2.12$$ 24. $$7.675 − 1.1$$ In Exercises 25-64, add or subtract the decimals, as indicated. 25. $$−19.13 − 7$$ 26. $$−8 − 79.8$$ 27. $$6.08 − 76.8$$ 28. $$5.76 − 36.8$$ 29. $$−34.7+(−56.214)$$ 30. $$−7.5+(−7.11)$$ 31. $$8.4+(−6.757)$$ 32. $$−1.94 + 72.85$$ 33. $$−50.4+7.6$$ 34. $$1.4+(−86.9)$$ 35. $$−43.3+2.2$$ 36. $$0.08 + (−2.33)$$ 37. $$0.19 − 0.7$$ 38. $$9 − 18.01$$ 39. $$−7 − 1.504$$ 40. $$−4.28 − 2.6$$ 41. $$−4.47 + (−2)$$ 42. $$−9+(−43.67)$$ 43. $$71.72 − (−6)$$ 44. $$6 − (−8.4)$$ 45. $$−9.829 − (−17.33)$$ 46. $$−95.23 − (−71.7)$$ 47. $$2.001 − 4.202$$ 48. $$4 − 11.421$$ 49. $$2.6 − 2.99$$ 50. $$3.57 − 84.21$$ 51. $$−4.560 − 2.335$$ 52. $$−4.95 − 96.89$$ 53. $$−54.3 − 3.97$$ 54. $$−2 − 29.285$$ 55. $$−6.32 + (−48.663)$$ 56. $$−8.8+(−34.27)$$ 57. $$−8 − (−3.686)$$ 58. $$−2.263 − (−72.3)$$ 59. $$9.365 + (−5)$$ 60. $$−0.12 + 6.973$$ 61. $$2.762 − (−7.3)$$ 62. $$65.079 − (−52.6)$$ 63. $$−96.1+(−9.65)$$ 64. $$−1.067 + (−4.4)$$ In Exercises 65-80, simplify the given expression. 65. $$−12.05 − |17.83 − (−17.16)|$$ 66. $$15.88 −|− 5.22 − (−19.94)|$$ 67. $$−6.4 + |9.38 − (−9.39)|$$ 68. $$−16.74 + |16.64 − 2.6|$$ 69. $$−19.1 − (1.51 − (−17.35))$$ 70. $$17.98 − (10.07 − (−10.1))$$ 71. $$11.55 + (6.3 − (−1.9))$$ 72. $$−8.14 + (16.6 − (−15.41))$$ 73. $$−1.7 − (1.9 − (−16.25))$$ 74. $$−4.06 − (4.4 − (−10.04))$$ 75. $$1.2 + |8.74 − 16.5|$$ 76. $$18.4 + |16.5 − 7.6|$$ 77. $$−12.4 − |3.81 − 16.4|$$ 78. $$13.65 − |11.55 − (−4.44)|$$ 79. $$−11.15 + (11.6 − (−16.68))$$ 80. $$8.5 + (3.9 − 6.98)$$ 81. Big Banks. Market capitalization of nation’s four largest banks (as of April 23, 2009) JPMorgan Chase & Co124.8 billion Wells Fargo & Co $85.3 billion Goldman Sachs Group Inc.$61.8 billion Bank of America $56.4 billion What is the total value of the nation’s four largest banks? Associated Press Times-Standard 4/22/09 82. Telescope Mirror. The newly launched Herschel Telescope has a mirror 11.5 feet in diameter while Hubble’s mirror is 7.9 feet in diameter. How much larger is Herschel’s mirror in diameter than Hubble’s? 83. Average Temperature. The average temperatures in Sacramento, California in July are a high daytime temperature of 93.8 degrees Fahrenheit and a low nighttime temperature of 60.9 degrees Fahrenheit. What is the change in temperature from day to night? Hint: See Section 2.3 for the formula for comparing temperatures. 84. Average Temperature. The average temperatures in Redding, California in July are a high daytime temperature of 98.2 degrees Fahrenheit and a low nighttime temperature of 64.9 degrees Fahrenheit. What is the change in temperature from day to night? Hint: See Section 2.3 for the formula for comparing temperatures. 85. Net Worth. Net worth is defined as assets minus liabilities. Assets are everything of value that can be converted to cash while liabilities are the total of debts. General Growth Properties, the owners of the Bayshore Mall, have$29.6 billion in assets and $27 billion in liabilities, and have gone bankrupt. What was General Growth Properties net worth before bankruptcy? Times-Standard 4/17/2009 86. Grape crush. The California Department of Food and Agriculture’s preliminary grape crush report shows that the state produced 3.69 million tons of wine grapes in 2009. That’s just shy of the record 2005 crush of 3.76 million tons. By how many tons short of the record was the crush of 2009? Associated Press-Times-Standard Calif. winegrapes harvest jumped 23% in ’09. 87. Turnover. The Labor Department’s Job Openings and Labor Turnover Survey claims that employers hired about 4.08 million people in January 2010 while 4.12 million people were fired or otherwise left their jobs. How many more people lost jobs than were hired? Convert your answer to a whole number. Associated Press-Times-Standard 03/10/10 Job openings up sharply in January to 2.7M. ## Answers 1. 116.6 3. 101.18 5. 91.502 7. 103.45 9. 58.641 11. 6.86 13. 6.739 15. 74.9 17. 52.372 19. 54.575 21. 35.3 23. 5.98 25. −26.13 27. −70.72 29. −90.914 31. 1.643 33. −42.8 35. −41.1 37. −0.51 39. −8.504 41. −6.47 43. 77.72 45. 7.501 47. −2.201 49. −0.39 51. −6.895 53. −58.27 55. −54.983 57. −4.314 59. 4.365 61. 10.062 63. −105.75 65. −47.04 67. 12.37 69. −37.96 71. 19.75 73. −19.85 75. 8.96 77. −24.99 79. 17.13 81.$328.3 billion
83. −32.9 degrees Fahrenheit
85. \$2.6 billion
87. 40, 000
• Was this article helpful? | 2021-07-25 19:16:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9422623515129089, "perplexity": 1015.6059947056692}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046151760.94/warc/CC-MAIN-20210725174608-20210725204608-00012.warc.gz"} |
http://math.stackexchange.com/questions/140193/inequality-absolute-value | # Inequality absolute value
If $x>0,y>0$, why does it follow for any $y \in (x,x+1)$, that from:
$$|f(y)-f(x)|<1,$$ we have $$|f(y)| \leq |f(x)|+1$$
-
This is always true, no matter what $x$ and $y$ are. – Alex Becker May 3 '12 at 2:15
$|f(y)| = |(f(y)-f(x))+f(x)| \leq |f(y)-f(x)| + |f(x)| < 1 + |f(x)|$, so.
-
Very nice, thank you! – Chris May 3 '12 at 2:18
Abhishek is correct, you can also use the other triangle inequality $|f(y)-f(x)|\geq |f(y)|-|f(x)|$ This fact gives $|f(y)-f(x)|<1 \implies |f(y)|-|f(x)|<1 \implies |f(y)|<1+|f(y)|$
-
More generally, if $|a-b|<1$, then $|a|\le|b|+1$. Abhishek's already given a real proof using the triangle inequality, but it's also useful to be able to see what's going on in intuitive terms.
The inequality $|a-b|<1$ says that the distance between the numbers $a$ and $b$ is less than $1$. If $a$ and $b$ are on the same side of $0$, that says that the distance between $|a|$ and $|b|$ is less than $1$, so $|a|$ can't be more than $|b|+1$. (It doesn't matter which side of $0$ they're on; if you don't see this right away, draw a few sketches.)
If $a$ and $b$ are on opposite sides of $0$ and less than $1$ unit apart, $|a|$ and $|b|$ must both be less than $1$. (Again, draw a sketch or two if necessary.) Thus, $|a|<1\le |b|+1$. | 2014-03-10 17:12:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7627730965614319, "perplexity": 162.74505343198615}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010907746/warc/CC-MAIN-20140305091507-00040-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://mctopherganesh.com/blog/python_thing.html | In [29]:
import requests as r
from bs4 import BeautifulSoup as bs
In [36]:
res = r.get('https://mctopherganesh.com')
a = bs(res.text)
a.find_all('a')
Out[36]:
[<a class="blurb-links" href="https://github.com/mctopherganesh/email_addy_factory#email-factory" target="_blank">creating email addresses</a>,
<a class="blurb-links" href="https://github.com/mctopherganesh/bs-project-stage#blood-sugar-tracking-project" target="_blank">data collection and reporting bot</a>,
<a class="blurb-links" href="./new_blog.html" target="_blank">newly unfinished blog roll</a>,
<a href="https://github.com/mctopherganesh" target="_blank"><img src="./img/Selection_797.png"/></a>,
<a href="mailto:mctopherganesh@gmail.com" target="_blank"><img class="gmail-picture" src="./img/gmail.png"/></a>]
In [39]:
a.find_all('a')[0].get('href')
Out[39]:
'https://github.com/mctopherganesh/email_addy_factory#email-factory'
In [47]:
known_list = [link.get('href') for link in a.find_all('a')]
In [48]:
for i in range(3):
print(r.get(known_list[i]))
<Response [200]>
<Response [200]>
<Response [200]>
In [50]:
res.headers
Out[50]:
{'Connection': 'keep-alive', 'Content-Length': '2188', 'Server': 'GitHub.com', 'Content-Type': 'text/html; charset=utf-8', 'Last-Modified': 'Sun, 24 Oct 2021 05:15:31 GMT', 'Access-Control-Allow-Origin': '*', 'ETag': 'W/"6174ebf3-15be"', 'expires': 'Sun, 24 Oct 2021 05:28:26 GMT', 'Cache-Control': 'max-age=600', 'Content-Encoding': 'gzip', 'x-proxy-cache': 'MISS', 'X-GitHub-Request-Id': '8E8E:6B21:6DA137:1C25C8F:6174ECA2', 'Accept-Ranges': 'bytes', 'Date': 'Sun, 24 Oct 2021 05:19:33 GMT', 'Via': '1.1 varnish', 'Age': '0', 'X-Served-By': 'cache-iad-kiad7000076-IAD', 'X-Cache': 'MISS', 'X-Cache-Hits': '0', 'X-Timer': 'S1635052774.828719,VS0,VE8', 'Vary': 'Accept-Encoding', 'X-Fastly-Request-ID': 'ea97477bd8d741434fbcdf0b062e92d7f788e3fa'}
This is something that I was thinking about for work but definitely something I need to do for my own website. As you can see here.
All that jargon up top is a list of links from my website. I wanted to see if they all worked and going through and clicking on them is far too much work if you ask me. I just want to make sure they go somewhere.
Also adding those janky looking links into this is is another story but lets continue.
I've been itching to program lately and this is something that I've just been putting off for too long. This project is far from over but it's been started.
What's happening up there and what's with 200 appearing everywhere? 200 is a HTTP status code. What's HTTP?
why do you have so many questions
HTTP is a "transfer protocol" or, a way that your computer sends and receives information from the internet. another that you might be familiar with is a 404. Maybe you're not familiar with them at all and concerned about your internet just working.
Then you, my friend, like your 200s.
And as a website cobbler and QA enthusiast, I like my 200s as well. That means everything is working, or, that links take some user who clicks on that link somewhere.
If you're still here and you're wondering what res.headers means - headers are part of the information response (res) being sent back by the connection to the site. It's basically all jargon granted, header's are important for things like letting website use your phone microphone and camera. This is why you are prompted on your phone about letting new websites use your camera/audio/microphone.
Why is the code useful for this so far? Because otherwise, you have to read those headers, in a browser with developer tools (press F12 and go crazy) open and here we can start to build something that will just read something for me, and tell me what's there.
Alright that's it. Thanks.
In [ ]: | 2023-01-31 04:32:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17337286472320557, "perplexity": 6537.116851194102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499842.81/warc/CC-MAIN-20230131023947-20230131053947-00097.warc.gz"} |
https://brilliant.org/problems/have-you-seen-this-before-2/ | # Have you seen this before ?
Geometry Level pending
If $$\sin(A)+\sin(A+20)=1,$$ then find A (in degrees).
Note: Assume $$0 < A < 90.$$
× | 2017-10-17 17:01:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8937007188796997, "perplexity": 12212.221842248808}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822145.14/warc/CC-MAIN-20171017163022-20171017183022-00296.warc.gz"} |
https://br.123dok.com/document/z31jln7y-universidade-federal-da-bahia-ufba-instituto-de-matem-atica-im-11.html | # Universidade Federal da Bahia-UFBA Instituto de Matem ´atica-IM
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Universidade Federal da Bahia-UFBA
Instituto de Matem ´atica-IM
Programa de P ´os-Gradua¸c ˜ao em Matem ´atica-PGMAT
Pesin’s Entropy Formula for C
1
non-uniformly expanding maps.
Felipe Fonseca dos Santos
2017
1 Pesin’s Entropy Formula for C non-uniformly expanding maps.
Felipe Fonseca dos Santos
Tese de Doutorado apresentada ao Colegiado da Pós-Graduação em Matemática UFBA/UFAL como requisito parcial para obtenção do título de Doutor em Matemática.
Orientador: Prof. Dr. Vitor D. Martins
de Araújo
Agosto de 2017 Santos, Felipe Fonseca dos.
1
Pesin’s Entropy Formula for C non-uniformly expanding maps
/ Felipe Fonseca dos Santos. – Salvador, 2017. 93 f. : il.
Orientador: Vitor Domingos Martins de Araújo.
Tese (Doutorado - Doutorado em Matemática) – Universidade Federal da Bahia, Programa de Pós-graduação em Matemática, 2017.
1. Fórmula de Entropia de Pesin 2. Estados de Equilíbrio.
### 3. Medidas SRB\física-fraca. 4. Expansão não uniforme 5.Teoria Ergódica. I. Araújo, Vitor Domingos Martins de. II. Título.
CDD -
1 C
Pesin’s Entropy Formula for non-uniformly expanding
maps.
Tese de Doutorado submetida ao Colegiado de Pós-graduação em Ma- temática UFBA/UFAL como parte dos requisitos necessários à obtenção do título de Doutor em Matemática.
Prof. Dr. Vitor D. Martins de Araújo (Orientador)
UFBA
Prof. Dr. Paulo César Rodrigues Pinto Varandas
UFBA
Prof. Dr. Augusto Armando de Castro Jr
UFBA
Profa. Dr. Maria José Pacífico
UFRJ
Profa. Dr. Vanessa Ribeiro Ramos
UFMA
Salvador-BA Agosto de 2017
Dedico este trabalho aos meus pais, meus irmãos, à Lara e à minha noiva Milena, por tudo que fizeram por mim para a realização deste sonho.
Agradeço primeiramente à Deus por estar sempre comigo, iluminando a minha vida, por ter me capacitado e por estar sempre me protegendo e me dando força.
À Milena, que tanto amo, companheira de todas as horas, obrigado pela sua confiança e compreensão em ceder grande parte do nosso tempo a esses anos de estudos. Seu amor é fundamental na minha vida e me faz sempre acreditar que posso ir mais longe.
Ao meus pais Joel e Rosely, meus irmãos Willian e Liz, minha avó Terezinha e minha sobrinha Lara por fazerem de tudo para me ver feliz. O amor, carinho e apoio de Vocês é muito importante na minha vida.
A todos os meus tios, tias, primos e amigos, por sempre torcerem e me apoiarem em todos os momentos e por compreenderem a minha ausência em muitas ocasiões.
A toda à Família Borges, por todo apoio e carinho. Sou feliz por ter três maravilhosas Famílias. Ao Professor Vitor Araújo, que novamente me deu a honra e a satis- fação de sua valiosa orientação, pela sua amizade, incentivo, paciência, dedicação, por ter acreditado em mim e pelas grandes oportunidades de aprendizagem a que me proporcionou. Você é um exemplo para mim.
Aos membros da banca, que muito me honraram com suas presenças,
a Prof. Augusto Armado de Castro Jr, Prof . Maria José Pacífico Prof. a
Paulo Cesar Pinto Varandas, Prof . Vanessa Ribeiro Ramos e Prof. Vitor Domingos Martins de Araújo agradeço pelos enriquecedores comentários e sugestões ao trabalho.
À todos os meus Professores do curso de doutorado: Luciana Salgado, Manuel Stadlbauer, Paulo Varandas, Vilton Pinheiro, Vitor Araújo e Ter- tuliano Franco, pelos valiosos conhecimentos matemáticos que comparti- lharam comigo ao longo das disciplinas, seminários e conversas informais nesse período e por estarem sempre dispostos a ajudar. Vocês são inspira- ção para minha vida profissional.
A minha turminha do doutorado: Darlan, Fabi, Jacq, Mari, Sara e Elaine. Aos meus maninhos: Andrêssa, Edvan e Junilson e todos os ami- gos e colegas da sala “282”, que ao longo desse tempo tive a felicidade de conhecer, obrigado a todos pelos conselhos, ensinamentos e pelos momen- tos de descontração e alegria durante estes anos. Mesmo correndo o risco de esquecer o nome de alguma pessoa que foi importante nesse período quero destacar os amigos: Adriana, Anderson, Alejandra, Diego, Diogo, Ed, Elen, Heides, Morro, Moa, Roberto, Cattai e Vinicius. Obrigado a todos Vocês!
A todos os funcionários da pós-graduação do IM da UFBA por estarem sempre dispostos a me ajudar, em especial a Davilene, Mayara, Márcio, Kleber e Diogo.
Agradeço também aos meus professores da UFRB e da UFBA (período do mestrado) por terem me incentivado e preparado para estar aqui. O apoio e amizade de vocês foram fundamentais.
Por fim, agradeço a CAPES e a Fapesb pelo apoio financeiro.
“Nele estão escondidos todos os tesouros da sabedoria e da ciência.” Colossenses 2:3.
Resumo
Provamos a existência de estados de equilíbrio com propriedades espe- ciais para uma classe de homeomorfismos locais positivamente expansivos e potenciais contínuos, definidos em espaço métrico compacto. Além disso,
1
formulamos uma generalização de classe C da Fórmula de Entropia de Pesin: toda medida ergódica weak-SRB-like satisfaz a Fórmula de Entropia
1 de Pesin para transformações de classe C não uniformemente expansoras.
Mostramos que para transformação expansora fraca, tal que Leb-q.t.p x te- nha frequência positiva de tempos hiperbólicos, medidas weak-SRB-like, existem e satisfazem a Fórmula de Entropia de Pesin e são estados de equi- líbrio para o potencial ψ = − log | det D f |. Em particular, isso é válido para
1
qualquer transformação expansora de classe C e neste caso o conjunto de medidas de probabilidade invariantes que satisfazem a Fórmula de Entro-
∗
pia de Pesin é o fecho convexo na topologia fraca das medidas ergódicas weak-SRB-like.
Palavras-chave:
Teoria Ergódica, expansão não uniforme, Medidas SRB\física- fraca, Estados de Equilíbrio e Fórmula de Entropia de Pesin.
Abstract
We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric
1
spaces and continuous potentials. Moreover, we formulate a C genera- lization of Pesin’s Entropy Formula: all ergodic weak-SRB-like measures
1 satisfy Pesin’s Entropy Formula for C non-uniformly expanding maps.
We show that for weak-expanding maps such that Leb-a.e x has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin’s Entropy Formula and are equili- brium states for the potential ψ = − log | det D f |. In particular, this holds
1
for any C -expanding map and in this case the set of invariant probability
∗
measures that satisfy Pesin’s Entropy Formula is the weak -closed convex hull of the ergodic weak-SRB-like measures.
Keywords:
Ergodic theory, non-uniform expansion, SRB\physical-like mea- sures, equilibrium states, Pesin’s entropy formula.
x
Sumário
xiii
1 . . . . . . . . . . . . . . . . .
8
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xii Sumário
41
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6.6 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . . 49
53
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7.2 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . . 60
63
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69
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9.3 Proof of Corollary . . . . . . . . . . . . . . . . . . . . . . . . 71
9.4 Proof of Corollary . . . . . . . . . . . . . . . . . . . . . . . 72
74
Lista de Figuras . . . . . . . . . . . . . . . . . . . . . . . . . .
9
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xiv
Lista de Figuras
Capítulo 1
Introdução e descrição dos
Os estados de equilíbrio, um conceito originalmente da Mecânica Esta- tística, é uma classe especial de medidas de probabilidade. Na definição clássica, dadas uma transformação contínua T : X X definida em um espaço métrico compacto X e uma função contínua φ : X → R chamada de potencial, dizemos que uma medida de probabilidade T-invariante é φ-estado de equilíbrio (ou estado de equilíbrio para o potencial φ) se Z ( ) Z
P top (T, φ) = h (T) + φ dµ, onde P top (T, φ) = sup h (T) + φ dλ ,
µ λ λ∈M T T é o conjunto de todas as medidas de probabilidade T-invariantes.
e M Nesta relação, P top (T, φ) é chamada pressão topológica e a igualdade acima do lado direito é conhecida por Princípio Variacional; veja por exemplo para as definições de entropia h (T) e pressão topológica. Postergamos as definições formais para os próximos capítulos.
Dependendo do sistema dinâmico e potencial estudados (T, φ), os es- tados de equilíbrio podem ter propriedades que fornecem informação útil sobre o sistema. Sinai, Ruelle e Bowen foram os pioneiros da teoria dos estados de equilíbrio para sistemas dinâmicos uniformemente hiperbólicos. Eles estabeleceram uma importante conexão entre estados de equilíbrio e medidas que são caracterizadas por fornecer informações assintóticas sobre um conjunto de trajetórias, isto é, dada uma medida µ, esperamos que sua bacia n X −1 Z
1 j
B (µ) = x φ(T x (X, R)
→+∞ X : ) −−−−→ φdµ, ∀φ ∈ C n n j=
### 2 Capítulo 1. Introdução e descrição dos resultados
seja grande do ponto de vista da medida de Lebesgue ou alguma outra medida de referência relevante. Essas medidas são chamadas medidas SRB em homenagem a Sinai, Ruelle e Bowen, e também medidas físicas devido à origem de muitos conceitos na Mecânica Estatística.
Apesar do substancial progresso de vários autores em estender essa teoria, uma visão global ainda está longe de ser completa. Por exemplo, a estratégia básica utilizada por Sinai-Ruelle-Bowen foi (semi-)conjugar a dinâmica a um subshift de tipo finito, através de uma partição Markov, e usar as fortes propriedades topológicas e ergódicas das Cadeias de Mar- kov Topológicas para obter resultados para a dinâmica uniformemente hiperbólica original.
No entanto, a existência de partições de Markov é conhecida apenas em alguns casos e, muitas vezes, tais partições podem não ser finitas. Além disso, existem transformações que não admitem estados de equilíbrio ou medidas físicas/SRB (como por exemplo, a transformação identidade).
Em muitas situações com algum tipo de expansão, é possível garantir que os estados de equilíbrio sempre existam. No entanto, os estados de equilíbrio geralmente não são únicos e, pela forma como são obtidos, não fornecem informações adicionais para o estudo da dinâmica. No contexto das funções uniformemente expansoras, Ruelle mostrou em que existe um único estado de equilíbrio que é uma medida física/SRB se o potencial for Hölder contínuo e a dinâmica f for transitiva.
Neste sentido, surgem algumas questões naturais:
1. Quando um sistema dinâmico tem um estado de equilíbrio?
2. Se tais medidas existem, que propriedades estatísticas essas proba- bilidades exibem em relação à medida de referência (possivelmente não-invariante)?
3. Quando existe, o estado de equilíbrio é único? A existência de estados de equilíbrio é uma propriedade relativamente mais simples que muitas vezes pode ser estabelecida através de argumen- tos de compacidade. Entretanto, as propriedades estatísticas e a unicidade do estado de equilíbrio são geralmente mais sutis e requerem uma melhor compreensão da dinâmica. Em nosso contexto, não esperamos ter uma unicidade de estado de equilíbrio, porque consideramos:
• sistemas dinâmicos com baixa regularidade: contínua ou de classe
### 1 C somente; e • apenas potenciais contínuos (com exceção do Teorema A).
3 Do Formalismo Termodinâmico a resposta para a primeira questão é
conhecida e afirmativa para todas as transformações expansivas (também chamadas transformações Ruelle expansiva), em um espaço métrico com- pacto X e para todo potencial contínuo; veja .
Inspirado na definição de medida observável introduzida por Keller em a noção de medidas “SRB- like” ou pseudo-físicas (uma generalização da noção de medida física/SRB) e mostram que tais medidas sempre existem para toda transformação con- tínua definida em um espaço métrico compacto. Além disto, eles mostram que a definição de medida “SRB-like” é ótima em certo sentido, se o objetivo for descrever as estatísticas assintóticas de Lebesgue quase toda órbita do sistema. Mais precisamente, dado um ponto x X denotamos o conjunto X ∗ n j −1 1 w i
p ω(x) = (x) = δ µ .
• T j n T (x)
• µ ∈ M ; ∃n −−−−→ ∞ tal que σ j −−−−→ j →+∞ j →+∞ n j i=
Fixado uma medida de referência ν (não necessariamente a medida de T é ν-SRB-like se, Lebesgue) para o espaço métrico X, dizemos que µ ∈ M e só se, ν(A (µ)) > 0 para todo ε > 0, onde ε
A ε
(µ) = {x X; dist(pω(x), µ) < ε} é a bacia ε-fraca de atração de µ e dist é uma métrica que induz a topologia
∗ fraca no espaço das medidas de probabilidades Boreliana.
Mostramos que, em nosso contexto, as medidas “ν-SRB-like” são par- ticularmente interessantes, porque podem ser vistas como medidas que surgem naturalmente como pontos de acumulação de medidas ν-SRB.
Seja T Teorema A.
: X X uma transformação aberta, expansiva e topologica-
mente transitiva definida em um espaço métrico compacto X. Consideren ) n ≥1
uma sequência de potenciais Hölder contínuas, (ν ) uma sequência de medi-
n n
≥1
ν
das conformes associados a (T, φ n ) e n ) n uma sequência de medidas n -SRB.
≥1 Assuma que:
φ φ na topologia da convergência uniforme;
1. n j −−−−→ j →+∞ w
ν ν na topologia fraca* de convergência de medidas;
2. n j −−−−→ j →+∞ w µ µ também na topologia fraca*.
3. n j −−−−→ j →+∞
### 4 Capítulo 1. Introdução e descrição dos resultados
Então ν é medida conforme para (T, φ) e µ é ν-SRB-like. Em particular µ é φ-estado
de equilíbrio. Além disso, se T é topologicamente exata então (µ)) = 0
ε
ν(X\A para todo ε > 0. Relacionando agora as transformações que expandem distância e as medidas “ν-SRB-like”, obtemos uma resposta positiva para a segunda pergunta.
Seja T Teorema B.
: X X uma transformação aberta, expansiva e topolo-
gicamente transitiva, definida em um espaço métrico compacto X e
φ : X → R
contínua. Então, para cada (necessariamente existente) medida φ-conforme ν = ν
φ
todas as (necessariamente existentes) medidas ν φ -SRB-like são φ-estado de equilí-
brio.
Esse resultado, em particular, estende o principal resultado obtido por Catsigeras-Henrich em , onde foram estudadas apenas as medidas
1
“SRB-like” com respeito à medida de Lebesgue para dinâmicas C expan- soras do círculo.
Em , Catsigeras, Cerminara e Henrich estendem a noção de medidas SRB-like, as medidas “weak-SRB-like” ou weak pseudo-física. Diz-se que T é ν-weak-SRB-like se, e só se, uma medida µ ∈ M
1 lim sup log ν(A ε,n (µ)) = 0 ∀ε > 0, n n
→+∞
onde A ε,n n (µ) = {x X : dist(σ (x), µ) < ε} é a bacia ε-fraca de µ em tempo n .
Observamos que se µ é um φ-estado de equilíbrio e se “relaciona bem”
∗
com a medida de referência ν então µ é combinação convexa fraca de medi- das ergódicas weak-SRB-like. Além disso, se µ é o único φ-estado de equi- líbrio então µ possui cota superior de grandes desvios, isto é, para qualquer
∗
vizinhança fraca
1 (espaço das medidas de probabilidade
V de µ em M P n n (x) := δ 1 −1 j T (x)
1 Boreliana), então a probabilidade ν({x X : σ j= ∈ M \V}) n decresce exponencialmente com n em uma taxa que depende do “tama-
nho” de V.
Corolário C. Sejam T
: X X uma transformação aberta, expansiva e topolo-
gicamente transitiva, definida em um espaço métrico compacto X,
φ : X → R um
potencial contínuo e ν uma medida φ-conforme. Se µ é um φ-estado de equilíbrio
tal que h ν x de µ é uma medida
(T, µ) < ∞ então quase toda componente ergódica µ
de probabilidade ν-weak-SRB-like. Além disso, se µ é o único φ-estado de equilí-
µ é uma medida ν-SRB tal que ν(B(µ)) = 1 e µ satisfaz a seguinte cota
brio então
5superior de grandes desvios: para toda vizinhança fraca
V de µ temos
1 lim sup n
1
log ν({x X : σ (x) ∈ M \V}) ≤ −I(V) n n
→+∞ onde I r (V) = sup{r > 0 : K (φ) ⊂ V}.
Considere agora M uma variedade Riemanniana compacta finita sem bordo. Na década de 1970, Pesin mostrou uma relação entre dois impor- tantes conceitos matemáticos: os expoentes de Lyapunov e a entropia para sistemas dinâmicos diferenciáveis.
Mais precisamente, em , Pesin mostrou que se µ é uma medida
2 1+α
f -invariante para um difeomorfismo de classe C (ou C , α > 0) definido
em uma variedade compacta que é absolutamente contínua com respeito + a medida de Lebesgue, então Z
Σ
h ( f ) = d µ, (1.0.1) µ
• + onde Σ denota a soma dos expoentes de Lyapunov positivos em pontos regulares, contando suas multiplicidades. Dizemos que uma medida f - invariante µ satisfaz a Fórmula de entropia de Pesin se µ satisfaz a equação .
Ledrappier e Young em , no mesmo contexto que Pesin, generaliza- ram a fórmula de entropia de Pesin, mostrando que uma medida invariante satisfaz a Fórmula de Entropia de Pesin se, e somente se, tiver medidas condicionais absolutamente contínuas em relação a medida de Lebesgue nas folhas instáveis.
Podemos também citar, Liu, Qian e Zhu que obtiveram em generalizações da fórmula de entropia de Pesin para endomorfismos de
2 classe C .
### 1 Para funções de classe C , a ausência de distorção limitada impossibilita
o uso de muitos métodos já conhecidos para obter a Fórmula de Entropia de
2 1+α
Pesin para dinâmicas de classe C (ou C , α > 0). Recentemente, Xueting Tian mostrou em a fórmula de entropia de Pesin para dinâmica entre
1 1+α C e C .
Entretanto, podemos encontrar diversos resultados já obtidos para sis-
1
temas dinâmicos de classe C :
• Ali Tahzibi em mostrou a fórmula de entropia de Pesin para
1
difeomorfismos C genéricos que preserva área em superfícies de dimensão 2;
### 6 Capítulo 1. Introdução e descrição dos resultados
• Bessa-Varandas em mostraram que vale o resultado anterior para tempo contínuo (fluxos incompreensíveis) em variedades de dimensão 3;
• 1
• Hao Qiu em provou que se f é Anosov transitiva então C genericamente existe uma única medida SRB que satisfaz a fórmula de entropia de Pesin;
• Em 2012, Sun e Tian exibem em condições para que uma medida invariante µ satisfaça a fórmula de entropia de Pesin para difeomor-
1
fismos C com decomposição dominada ao longo da órbita para µ quase todos os pontos. Além disso, entre outras coisas, eles esten- dem o resultado obtido em para qualquer dimensão;
• Catsigeras e Enrich estabelecem relação entre as medidas “SRB-like” e
1
a fórmula de entropia de Pesin para funções C -expansoras no círculo em . Posteriormente, Catsigeras, Cerminara e Enrich estenderam
1
essa relação para difeomorfismos C com decomposição dominada em estendem o resultado para
1
difeomorfismos C Anosov para medidas “weak-SRB-like”;
• Yang e Cao, mais recentemente, mostraram em , que se f é um di-
1
feomorfismo C que admite atrator Λ com decomposição dominada Λ
T M = E
⊕ F tal que para qualquer medida suportada em Λ temos que todos os expoentes de Lyapunov ao longo de E são não positivos e todos os expoentes de Lyapunov ao longo de F são não negati- vos, então existe medida suportada em Λ que satisfaz a fórmula de entropia de Pesin. Apesar de muito progresso nessa direção, ainda há muitas questões em aberto. Nesse sentido, foi investigada a Fórmula de entropia de Pesin para
1 difeomorfismos locais de classe C não uniformemente expansor.
1 Teorema D. Seja f não unifor-
: M M um difeomorfismo local de classe C
memente expansor. Então toda medida de probabilidade expansora weak-SRB-like
satisfaz a fórmula de entropia de Pesin. Além disso, todas as medidas de probabi-
lidades ergódicas SRB-like são expansoras.
Além disso, foram estudadas a existência de medidas ergódicas “weak- SRB-like” e algumas de suas propriedades para sistemas dinâmicos com alguma propriedade de expansão. Sabe-se que nem todas as dinâmicas admitem medidas ergódicas “weak-SRB-like”, como pode ser visto em Exemplo 5.4]. Porém, Catsigeras, Cerminara e Enrich mostram em
7
que medida ergódica “weak-SRB-like” sempre existe para difeomorfismos
### 1 Anosov de classe C
e, mais recentemente em , Catsigeras e Troubetzkoy mostram que, para funções contínuas C -genérica do intervalo, todas as medidas ergódicas são SRB-like.
Relacionando os conceitos das medidas “weak-SRB-like”, Fórmula de Entropia de Pesin e estado de equilíbrio obtemos
1 Corolário E. Seja f não unifor-
: M M um difeomorfismo local de classe C
memente expansor e ( f, ψ) = 0 então, toda medida de
top
ψ = − log | det D f |. Se P ψ-estado de equilíbrio e todas as (necessariamente
existentes) medidas expansora ergódicas weak-SRB-like satisfazem a Fórmula de
Entropia de Pesin.
Ademais, supondo propriedades de expansão mais fortes obtemos:
1 Corolário F. Seja f expansor
: M M um difeomorfismo local de classe C
fraco e não uniformemente expansor. Então,
1. Todas as (necessariamente existentes) medidas de probabilidade weak-SRB- like são ψ-estados de equilíbrio e, em particular satisfazem a Fórmula de Entropia de Pesin;
2. Existe medida de probabilidade ergódica weak-SRB-like;
ψ < 0 não existe medida de probabilidade atômica weak-SRB-like;
3. Se −1
4. Se
D = {x M : kD f (x) k = 1} é finito e ψ < 0 quase todas as ψ-estado de equilíbrio são medidas weak-
componentes ergódicas de um SRB-like. Além disso, toda medida de probabilidade weak-SRB-like µ, suas
µ
componentes ergódicas x são medidas de probabilidades expansoras weak- SRB-like para µ-quase todo x no subconjunto expansor.
Em particular, vemos que uma construção análoga ao resultado de existência de medida física/SRB e atômica para uma aplicação quadrática,
1
obtida por Keller em , não é possível entre transformações C uniforme- mente expansoras, embora genericamente tais medidas sejam singulares com respeito a qualquer forma de volume.
Corolário G. Não existe medida atômica SRB para uma transformação expansora
### 1 C de uma variedade compacta.
1 Pelo trabalho de Ávila e Bochi em , é sabido que C -genericamente
1
as transformações de classe C de uma variedade compacta sem bordo de dimensão finita não admitem medida de probabilidade absolutamente
### 8 Capítulo 1. Introdução e descrição dos resultados
contínua em relação a medida de Lebesgue. Esse é um dos empecilhos que assegura que não é possível obter um análogo aos resultados obtidos por Pesin, Ledrappier e Young em de modo geral. Mas podemos nos perguntar se é possível estabelecer condições necessárias e suficientes para que um análogo à Fórmula de Entropia de Pesin seja satisfeita. Neste
1 sentido, obtemos a seguinte caracterização para dinâmica C -expansora.
1 Seja f . Então Corolário H.
: M M uma função expansora de classe C
uma medida de probabilidade f -invariante µ satisfaz a Fórmula de entropia de
Pesin se, e somente se, suas componentes ergódicas µ são weak-SRB-like µ-q.t.p.
x x
∈ M. Além disso, todas as medidas de probabilidade weak-SRB-like satisfazem
a Fórmula de entropia de Pesin e, em particular, são ψ-estados de equilíbrio.
µ weak-SRB-like,
Adicionalmente, se existe uma única medida de probabilidade
então µ é SRB, Leb(B(µ)) = 1, µ é o único ψ-estado de equilíbrio, é ergódico e
possui cota superior de grandes desvios.
Concluímos a introdução com a seguinte versão fraca de estabilidade
1
estatística para dinâmica C -expansora: os pontos de acumulação de toda sequência de medidas weak-SRB-like são combinações lineares convexas generalizadas de medidas ergódicas weak-SRB-like.
Sejam uma sequência de funções expansoras de Corolário I. n n
{ f : M M} ≥1
1
1 classe C tal que f e f n
→ f na topologia C − : M M é uma função expansora de
1
classe C . Considere para cada n uma medida weak-SRB-like associada a
n
≥ 1, µ
f . Então todo ponto de acumulação da sequência (µ ) é um estado de equilíbrio
n n n ≥1 para o potencial
ψ = − log | det D f | (em particular, satisfaz a Fórmula de Entropia
de Pesin) e quase todas as suas componentes ergódicas são medidas weak-SRB-like.
### 1.1 Comentários e Questões futuras
Nesta seção listamos algumas questões baseadas nos estudos realizados na presente tese e que serão objeto de trabalhos futuros.
Questão 1.1.1. Nas mesmas hipóteses do Corolário será possível obter uma cota
inferior de grandes desvios que tenha a mesma taxa encontrada no Corolário
Talvez seja necessário considerar que a dinâmica é topologicamente misturadora.
Questão 1.1.2. Será possível obter uma propriedade estatística para as medidas
weak-SRB-like, como no caso do Corolário para transformação expansora fraca
não uniformimente expansora?
Questão 1.1.3. Nas hipóteses do Corolário é possível garantir que µ é uma
medida weak-SRB-like?
### 1.1. Comentários e Questões futuras
9 Alguns exemplos que surgiram naturalmente durante o desenvolvi- mento deste trabalho também motivaram algumas questões.
O primeiro exemplo é atribuído a Bowen, e pode ser encontrado em maior detalhe para nosso contexto no Exemplo 5.5 em .
2 Exemplo 1.1.4. Considere um difeomorfismo f em uma bola de R com dois pontos
de sela A e B, de modo que uma componente conexa da variedade global instável
u W
(A)\{A} é um arco que coincide com uma componente conexa da variedade s u
global estável W s (B)\{B} e, inversamente, a componente conexa W (B)\{B} = W
(A)\{A}. Seja f , de modo que exista uma fonte C U onde U é o conjunto u u aberto com bordo W (B).
(A) ∪ W Figura 1.1: Olho de Bowen.
Se os autovalores da derivada de f em A e B forem adequadamente escolhidos P n 1 −1 j
conforme especificado em , então a sequência δ para todos
(x) n
{ j= f } ≥0 n
x
∈ U\{C} não é convergente. Neste caso existe pelo menos duas subsequências
convergentes para diferentes combinações convexas dos deltas de Dirac δ e δ .
A B Assim, como observado em , as medidas de probabilidade SRB-like são
combinações convexas de δ e δ e formam um segmento no espaço A B
M das me-
didas de probabilidade. Este exemplo mostra que as medidas SRB-like não são
necessariamente ergódicas.
Além disso, os autovalores de D f nos pontos de sela A e B podem ser adequada- P n 1 −1 j
mente modificados para obter a convergência da sequência δ como
f (x) n ≥0
{ j= } n
indicado no Lema (i) da página 457 em . Na verdade, fazendo uma pequena
pertubação C de f fora de uma pequena vizinhança dos pontos de sela A e B temos
que o
ω-limite das órbitas em U\{C} ainda contém os pontos A e B, obtendo que
a sequência converge para uma única medida µ = λδ A B para
• todo x
(1 − λ)δ
∈ U\{C}, com uma constante fixa 0 < λ < 1. Então, µ é a única medida
SRB-like. Isso prova que o conjunto de medidas de probabilidade SRB-like não
depende continuamente da função.
Este exemplo motiva algumas questões:
### 10 Capítulo 1. Introdução e descrição dos resultados
Questão 1.1.5. Quando podemos afirmar que um sistema dinâmico ( f, φ) tem
medida de probabilidade ergódica ν -SRB-like ou ν -weak-SRB-like? φ φ Pelo exemplo sabemos que o conjunto das medidas de Questão 1.1.6.
probabilidade ν φ -SRB-like não depende, em geral, continuamente com a função.
Será que adicionando a dinâmica propriedade de expansão forte (como nos casos do
Corolário podemos ter variação contínua das medidas ν φ -SRB-like
ou ν -weak-SRB-like? φ
Questão 1.1.7. Existe dependência contínua entre as medidas de probabilidades
ν ν φ? φ -SRB-like (ou φ -weak-SRB-like) e o potencial Uma resposta positiva nesta direção é dada no Teorema é ergódica.
Questão 1.1.8. No caso de uma resposta positiva na Questão será que há
O próximo exemplo é uma adaptação da transformação Intermitente (Manneville-Pomeau) em um homeomorfismo local do círculo.
Exemplo 1.1.9. Considere I =
[−1, 1] e a transformação ˆf : I I (veja a Figura
dada por ( √
2 x se x − 1 ≥ 0,
ˆf(x) = √ 1 − 2 |x| caso contrário.
1
1
: S
Esta transformação induz um homeomorfismo local contínuo f
→ S
1
=
através da identificação S
I
/ ∼, onde −1 ∼ 1. Esta é uma transformação que só
não é diferenciável no ponto 0, possui frequência positiva de tempos hiperbólicos
para Lebesgue quase todos os pontos, como pode ser visto em seção 5].
Este é um exemplo de transformação expansora fraca e não uniforme-
1 mente expansora, mas não é um difeomorfismo local C em todo ponto.
Este exemplo sugere que podemos pensar em uma generalização do Teorema substituindo a medida
(X, R) de Lebesgue por uma medida expansora φ-conforme ν para φ ∈ C ν ν f , onde J f é o Jacobiano f com respeito a ν. e o potencial ψ por − log J
Sejam T Questão 1.1.10.
: X X um homeomorfismo local definido em um
espaço métrico compacto X e
φ : X → R um potencial contínuo. Se existe medida
expansora φ-conforme ν então todas as (necessariamente existente) medidas ν-
φ?
SRB-like são estados de equilíbrio para o potencial
1.2. Organização do trabalho y=f(x) 1
11
0.5 −1 −0.5 y=x 0.5 1 −0.5 −1 Figura 1.2: Gráfico de f .
Pode ser necessário considerar hipóteses adicionais sobre os potenciais, como por exemplo que os potenciais contínuos sejam potenciais hiperbólicos (veja a definição e resultados sobre o potencial hiperbólico Hölder contínuo em ).
Além disso, também podemos pensar em uma generalição do Teorema
1 para o caso de difeomorfismo local de classe C afastado de conjunto
crítico/singular com frequência positiva de tempos hiperbólicos (veja para definição de conjunto crítico/singular).
1 Questão 1.1.11. Seja f afastado
: M M um difeomorfismo local de classe C
de um conjunto crítico/singular
C com recorrência lenta a C e tal que para alguns 0 < σ < 1, b, δ > 0 e θ > 0 Leb −q.t.p. x tem frequência positiva ≥ θ de (σ, δ, b)-tempos hiperbólicos para f . Todas as medidas de probabilidade expansora
weak-SRB-like satisfazem a Fórmula de Entropia de Pesin?
1.2 Organização do trabalho
A tese está dividida em nove capítulos, incluindo o capítulo introdutó- rio “Introdução e descrição dos resultados”. Nos capítulos denomina- dos respectivamente “Statement of the results” e “Preliminary definitions and results” serão introduzidos os principais resultados da tese, grande parte das definições básicas e notações necessárias ao longo do texto, além de alguns resultados auxiliares sobre medidas de probabilidades ν-SRB- like, ν -weak-SRB-like e entropia. φ
No capítulo intitulado “Examples of application” apresentamos al- guns exemplos onde os principais resultados desse trabalho podem ser
### 12 Capítulo 1. Introdução e descrição dos resultados aplicados.
No Capítulo No Capítulo “Expanding maps on compact metric space” serão exi- bidas algumas propriedades das aplicações expansivas e a demonstração do Teorema
No Capítulo “Entropy Formula” serão usadas as propriedades dos tempos hiperbólicos e das medidas weak-SRB-like para provar uma refor-
1
mulação da Fórmula de Entropia para difeomorfismo local de classe C não uniformemente expansor (o Teorema .
No Capítulo “Ergodic weak-SRB-like measure” será estudado a exis- tência de medida ergódica weak-SRB-like, sua relação com a Fórmula de Entropia de Pesin e a prova do Corolário
No Capítulo “Weak-Expanding non-uniformly maps” serão usados alguns resultados obtidos nos capítulos anteriores para provar os Corolá- rios
## Chapter 2 Statement of the results Let T : X → X be a continuous transformation defined on a compact metric space (X, d).
### 2.1 Topological pressure
The dynamical ball of center x X, radius δ > 0, and length n ≥ 1 is defined by j j
B x , T y (x, n, δ) = {y X : d(T ) ≤ δ, 0 ≤ j n − 1}.
Let ν be a Borel probability measure on X. We define
1
h (T, x) = lim lim sup log ν(B(x, n, δ)) and
ν
−
δ→0 n n →+∞ h ν ν (T, x). (2.1.1)
(T, µ) = µ − ess sup h Note that ν is not necessarily T-invariant, and if µ is ergodic T-invariant, µ µ (T, x) = h (T), the usual metric entropy of T we have for µ-a.e. x X, h with respect to µ.
Let n be a natural number, ε > 0 and let K be a compact subset of X. A subset F of X is said to (n, ε) span K with respect to T if ∀x K there exists j j
y x , T y
∈ F with d(T ) ≤ ε for all 0 ≤ j n − 1, that is, [
K B (x, n, ε).
⊂ x
∈F
If K is a compact subset of X. Let
1
h (T; K) = lim lim inf log N(n, ε, K), (2.1.2) n ε→0 →+∞ n
14 Chapter 2. Statement of the results
where N(n, ε, K) denote the smallest cardinality of any (n, ε)-spanning set for K with respect to T.
Definition 1. The topological entropy of T is h top
(T) = sup{h(T, K); K X}, where the supremum is taken over the collection of all compact subset of
X .
Let φ : X → R be a real continuous function that we call the potential. Given an open cover α for X we define the pressure P (φ, α) of φ with T respect to α by X 1 n S φ(U)
P (φ, α) := lim log inf e T n n →+∞ n U⊂α U
∈U n n −1 −j
= T (α), where the infimum is taken over all subcovers U of α ∨ P n j=
−1 j S φ(x) := φ(T x ) and S n n n j= φ(U) := sup{S φ(x); x U}.
Definition 2. The topological pressure P top (T, φ) of the potential φ with re-
spect to the dynamics T is defined by
P top (T, φ) = lim sup P (φ, α)
T δ→0 |α|≤δ
where |α| denotes the diameter of the open cover α.
For given n > 0 and ε > 0, a subset E X is called (n, ε)-separated if j j
x x , T y ) > ε.
, y E, x , y then there exists 0 ≤ j n − 1 such that d(T An alternative way of defining topological pressure is through the no- tion of (n, ε)-separated set.
Definition 3. The topological pressure P (T, φ) of the potential φ with
top
respect to the dynamics T is defined by X 1 S n φ(x)
P top (T, φ) = lim lim sup log sup e ε→0 n n →+∞ xE
where the supremum is taken over all maximal (n, ε)-separated sets E.
We refer the reader to for more details and properties of the topo- logical pressure.
2.2. Expanding maps
15
2.2 Expanding maps
A continuous mapping T : X X is expanding (with respect to the metric d) if there exist constants λ > 1, η > 0 and n ≥ 1, such that for all
x
, y X n n if d(x, y) < 2η then d(T (x), T (2.2.1) (y)) ≥ λd(x, y). In the sequel we will always assume (without loss of generality, see chapter 3 in ) that n = 1, that is
d
(2.2.2) (x, y) < 2η =⇒ d(T(x), T(y)) ≥ λd(x, y). We refer the reader to for more details and properties of expand- ing map.
2.3 Transfer operator T ,φ φ =
We consider the Ruelle-Perron-Fröbenius transfer operator L L associated to T : X X and the continuous function (potential) φ : X → R as the linear operator defined on the space C (X, R) of continuous functions
g
: X → R by φ (g)(x) = e g (y). X φ(y) L T (y)=x
The dual of the Ruelle-Perron-Fröbenius transfer operator is given by
∗
1
φ
### 1 L : M → M
∗
η η : C 7→ L (X, R) → R φ Z ψ φ ψd η.
7→ L is the set of Borel probabilities in X.
1
where M
2.4 Weak-SRB-like and SRB-like probability mea-
sures
For any point x X consider, X n −1
1 j σ (x) = δ (2.4.1) n T (x)
n
j=
where δ y is the Dirac delta probability measure supported in y X.
16 Chapter 2. Statement of the results Definition 4. T the limit set with
For each point x M, denote pω(x) ⊂ M initial state x, ( w )
• →+∞ →+∞
p ω(x) := (x) T j n µ .
µ ∈ M ; ∃n −−−−→ ∞ s.t. σ j −−−−→ j j
Definition 5. Fix a reference measure ν for the space X, we say that prob-
T is ν-SRB (or ν-physical) if ν(B(µ)) > 0, where ability measure µ ∈ M B (µ) = x X; pω(x) = {µ} is the "ergodic basin" of µ. T and ε > 0. We will consider the following measurable sets
Let µ ∈ M in X:
A ε,n n (2.4.2)
(µ) := {x X : dist(σ (x), µ) < ε};
A ε
(2.4.3) (µ) := {x X : dist(pω(x), µ) < ε}. We call A ε,n (µ) the ε-pseudo basin of µ up to time n, A ε (µ) the basin of ε-weak
∗
statistical attraction that induces the weak
1
of µ and dist is a metric in M topology (see definition in ). Fix a reference probability measure ν for the space X. We say Definition 6. that a T-invariant probability measure µ is:
1. ν-SRB-like (or ν-physical-like), if and only if ν(A ε (µ)) > 0 for all ε > 0; 2. ν-weak-SRB-like (or ν-weak-physical-like), if and only if
1 lim sup log ν(A ε,n (µ)) = 0 ∀ε > 0. n →+∞ n When ν = Leb we say that µ is simply SRB-like (or weak-SRB-like).
It is easy to see that every ν-SRB measure is also a ν-SRB-like Remark 2.4.1.
measure. Moreover, the ν-SRB-like measures are particular case of ν-weak-SRB-
like (see item B in ).
∗ T the set of ν-weak-SRB-like probability mea-
We denote W (ν) ⊂ M T
∗ ∗ .
sures. When ν = Leb we denote W (Leb) = W T T Definition 7.
Given T : X X and an arbitrary continuous function φ : M → R, we say that a probability measure ν is conformal for T with
∗ ν = λν.
respect to φ (or φ-conformal) if there exists λ > 0 so that L φ
2.5. SRB-like measures as limits of SRB measures
17
2.5 SRB-like measures as limits of SRB measures
The next result shows that we can see the ν-SRB-like measures as mea- sures that naturally arise as accumulation points of ν -SRB measures. n
Theorem A. Let T
: X X be an open expanding topologically transitive map
of a compact metric space X, (φ ) a sequence of Hölder continuous potentials,
n n ≥1
(ν ) a sequence of conformal measures associated to the pair (T, φ ) and (µ ) n n n n n
≥1 ≥1 a sequence of ν -SRB measures. Assume that n 1. φ φ (in the topology of uniform convergence); n j −−−−→ j
→+∞ w 2. ν ν (in the weak topology); n j −−−−→ j
→+∞ w 3. µ µ (in the weak topology). n j −−−−→ j
→+∞
Then ν is a conformal measure for (T, φ) and µ is ν-SRB-like. In particular, µ is
an equilibrium state for the potential φ. Moreover, if T is topologically exact then
ε (µ)) = 0 for all ε > 0.
ν(X \ A This is one of the motivations for the study of SRB-like measures as
1 the natural extension of the notion of physical/SRB measure for C maps.
Additional justification is given by the results of this work.
It is worth noting that Hölder continuous potentials are only used in this work on the assumption of Theorem and all continuous potentials can be approximated by Lipschitz potentials (and all Lipschitz continuous potential are α-Hölder continuous for α ∈ (0, 1]) see Remark
The next result extends, in particular, the main result obtained in Theorem 2.3] valid only for expanding circle maps.
Theorem B. Let T
: X X be an open expanding topologically transitive
map of a compact metric space X and
φ : X → R a continuous potential. For
each (necessarily existing) conformal measure ν φ all the (necessarily existing)
ν -SRB-like measures are equilibrium states for the potential φ. φ Definition 8.
Let T : X X be a continuous map and φ : X → R a T continuous potential. Given r > 0 we consider the set in M Z r T µ top : h (T) + T is the set of all T-invariant Borel probability measures. K (φ) = {µ ∈ M φdµ ≥ P (T, φ) − r} where M
18 Chapter 2. Statement of the results Corollary C. Let T
: X X be an open expanding topologically transitive
map of a compact metric space X,
φ : X → R a continuous potential and ν a φ-conformal measure. If µ is a φ-equilibrium state such that h ν
(T, µ) < ∞ then
every ergodic component µ of µ is a ν-weak-SRB-like measure. Moreover, if µ is
x
the unique φ-equilibrium state then µ is ν-SRB, ν(B(µ)) = 1 and µ satisfies the
following large deviation bound: for every weak neighborhood
V of µ we have
1 lim sup n
1
log ν({x X : σ (x) ∈ M \V}) ≤ −I(V) n n
→+∞ where I r (V) = sup{r > 0 : K (φ) ⊂ V}.
### 2.6 Non-uniformly expanding maps
We denote by k · k a Riemannian norm on the compact m-dimensional boundaryless manifold M, m ≥ 1; by d(·, ·) the induced distance and by Leb a Riemannian volume form, which we call Lebesgue measure or volume and assume to be normalized: Leb(M) = 1. Note that Leb is not necessarily
f -invariant.
### 1 Recall that a C
• map f : M M is uniformly expanding or just expanding if there is some λ > 1 such for some choice of a metric in M one has
• x M kD f (x)vk > λkvk, for all x M and all v T \{0}.
1
local diffeomorphism In what follows we write always f : M M be C and ψ := − log | det D f |.
Next corollary suggests that Theorem should be extended to weaker forms of expansion. Recall that a Borel set in a topological space is said to have total proba- bility if it has probability one for every f -invariant probability measure.
### 1 Corollary 2.6.1. Let f local diffeomorphism and Y be a subset
: M M be C
with total probability. If n X −1
1 j
−1
lim inf (x)) n log kD f ( f k < 0, ∀x Y,
→+∞ n j=
then all the (necessarily existing) SRB-like measures are equilibrium states for the
potential ψ = − log | det D f |.
19
### 1 Proof. For a C
• local diffeomorphism f : M M we have that Leb is ψ-conformal. Moreover, we know by Theorem 1.15 in that if n
• X −1 1 j
−1
lim inf (x)) n →+∞ log kD f ( f k < 0
n j=
for x in a total probability subset, then f is uniformly expanding. Thus, by Theorem we concluded the proof.
### 1 The Lyapunov exponents of a C local diffeomorphism f of a compact
manifold M are defined by Oseledets Theorem which states that, for any
f
• invariant probability measure µ, for almost all points x M there is
• x M = F
1 2 κ(x) κ(x)+1
κ(x) ≥ 1, a filtration T (x) ⊃ F (x) ⊃ · · · ⊃ F (x) ⊃ F (x) = {0}, and numbers λ
1 (x) > λ 2 k i (x) = F i ( f (x)) and
(x) > · · · > λ (x) such D f (x) · F
1 n lim (x) i n log kD f (x)vk = λ
→±∞ n i i+
1 i (x) are called for all v F (x)\F (x) and 0 ≤ i ≤ κ(x). The numbers λ
Lyapunov exponents of f at the point x. For more details on Lyapunov
exponents and non-uniform hyperbolicity see .
### 1 Definition 9. local diffeomorphism. We say that
Let f : M M be C f satisfies the Pesin Entropy Formula if + µ ∈ M Z
Σ
h ( f ) = d µ,
µ
• + where Σ denotes the sum of the positive Lyapunov exponents at a regular point, counting multiplicities.
Let 0 < σ < 1 we denote, n X −1 −1 1 j
H (σ) = x (x)) (2.6.1) M; lim sup log kD f ( f k < log σ n →+∞ n j= Definition 10.
We say that f : M M is non-uniformly expanding if there exists σ ∈ (0, 1) such that Leb(H(σ)) = 1. A probability measure µ (not necessarily invariant) is expanding if there exists σ ∈ (0, 1) such that µ(H(σ)) = 1. Next result relates non-uniform expansion and expanding measures with the Entropy Formula.
20 Chapter 2. Statement of the results Theorem D. Let f
: M M be non-uniformly expanding. Every expanding
weak-SRB-like probability measure satisfies Pesin’s Entropy Formula. In parti-
cular, all ergodic SRB-like probability measures are expanding.
Now we add a condition ensuring that there exists expanding ergodic weak-SRB-like probability measure and that all weak-SRB-like measures become equilibrium states.
Corollary E. Let f top ( f, ψ) = 0,
: M M be non-uniformly expanding. If P
then all weak-SRB-like probability measures are ψ-equilibrium states and all the
(necessarily existing) expanding ergodic weak-SRB-like measures satisfy Pesin’s
Entropy Formula.
### 2.7 Weak and non-uniformly expanding maps
Now we strengthen the assumptions on non-uniform expansion to im- prove properties of weak-SRB-like measures.
−1 Definition 11.
k ≤ 1 for all We say that f is weak-expanding if, kD f (x)
xM.
We say that a probability measure is atomic if it is supported on a finite set. We denote supp(µ) the support of probability measure µ.
Corollary F. Let f : M M be weak-expanding and non-uniformly expanding.
Then, 1. all the (necessarily existing) weak-SRB-like probability measures are ψ-
equilibrium states and, in particular, satisfy Pesin’s Entropy Formula;
2. there exists some ergodic weak-SRB-like probability measure;
ψ < 0 there is no atomic weak-SRB-like probability measure;
3. if −1
4. if
D = {x M : kD f (x) k = 1} is finite and ψ < 0 almost all ergodic
components of a ψ-equilibrium state are weak-SRB-like measures. Moreover all weak-SRB-like probability measures µ its ergodic components µ are x expanding weak-SRB-like probability measure for
µ-a.e. x M\D. In particular, we see that an analogous result to the existence of an atomic physical measure for a quadratic map, as obtained by Keller in
1
, is not possible in the C expanding setting, although generically such measures must be singular with respect to any volume from.
### 2.7. Weak and non-uniformly expanding maps
21
1 Corollary G. No atomic measure is a SRB measure for a C uniformly expanding map of a compact manifold.
We now restate the previous results in the uniformly expanding setting.
1
Corollary H. Let f -expanding map. Then an f -invariant
: M M be a C µ satisfies Pesin’s Entropy Formula if and only if its ergodic
probability measure components µ are weak-SRB-like x
µ-a.e. x M. Moreover, all the (necessarily
existing) weak-SRB-like probability measures satisfy Pesin’s Entropy Formula, in
particular, are ψ-equilibrium states. In addition, if µ is the unique weak-SRB-like
probability measure then µ is SRB probability measure, Leb(B(µ)) = 1, is ergodic,
µ is the unique ψ-equilibrium state and µ satisfies large deviation bound.
Using this we get a weak statistical stability result in the uniformly
1 ∗
C -expanding setting: all weak accumulation points of weak-SRB-like
measures are generalized convex linear combinations of ergodic weak- SRB-like measures, as follows.
1 Corollary I. Let a sequence of C -expanding maps such that n n
{ f : M M} ≥1
1
1
f -topology and f -expanding map. Let (µ ) a
n n n
→ f in the C − : M M be a C ≥1
sequence of weak-SRB-like measures associated f . Then all accumulation points of
n
(µ ) is an equilibrium state for the potential n n
≥1 ψ = − log | det D f | (in particular,
satisfy Pesin’s Entropy Formula) and almost all ergodic components are weak-
SRB-like measures.
22 Chapter 2. Statement of the results
## Chapter 3 Examples of application Here we present some examples of applications. In the first section
1
we show the construction of a C uniformly expanding map with many different SRB-like measures which in particular satisfies the assumptions of Theorem
In the second section we present a class of non uniformly expanding
### 1 C local diffeomorphisms satisfying the assumptions of Theorem D. In the
third section we exhibit examples of weak-expanding and non uniformly expanding transformations which are not uniformly expanding in the set- ting of Corollary
3.1 Expanding map with several absolutely con-
tinuous invariant probability measures
1
1
1 We present the construction of a C -expanding map f : S that
→ S has several SRB-Like measures. Moreover, such measures are absolutely continuous with respect to Lebesgue measure (which in particular means
1
1 that f does not belong to a C generic subset of C (M, M), see ).
Example 3.1.1. Following , we first adopt some notation for Cantor sets
with positive Lebesgue measure. Let I be a closed interval and α > 0 numbers
P n
∞
α
with n n= < |I|, where |E| denotes the one-dimensional Lebesgue measure of ′ ′ the subset E of I. Let a = a 1 a 2 . . . a denote a sequence of s and n 1 s of length
= [a, b],
n = n I = h i (a); we denote the empty sequence a = ∅ with n(a) = 0. Define Ia+b a+b α α
∗ ∗
• 2
= , aI I and I a recursively as follows.
−
2
2
2 ∅ ∗
Let I and I a a 1 be the left and right intervals remaining when the interior of I α n (ak) a
is removed from I a ; let I be the closed interval of length n (ak) and having the same
a
2
24 Chapter 3. Examples of application center as I (k=0,1). ak T S
=
The Cantor set K is given by K I I a m= n (a)=m I .
This is the standard construction of the Cantor set except that we allow our-
selves some flexibility in the lengths of the removed intervals. The measure of K
Iis Leb(K α > 0. I n
) = |I| − P n=
Suppose that another interval J > 0 such that n PI is given together with β ∞ ∗ n= < |J|. One can then construct J a β β , J and K as above. Let us assume now n n a J that α n −−−→ γ ≥ 0. Following the construction of Bowen , we get g : I J n
→∞
1 ′
a C orientation preserving homeomorphism so that g and
I
(x) = γ for all x K
′ g (x) > 1 for all x I. P
∞ More precisely, let us take J = > 0 with β < 2 n n β β β n+ 1 [−1, 1] and choose β n= h i n+ 1 P
1 ∞
.g. β = , 1 α = α <
and e n 2 . Let I = and n . Then n
→ 1, β β β n n (n+100)
2 2 n=
=
and γ = lim
1 −
### 2. We define a homeomorphism G : (−I) ∪ I → J by
2 α ( n
• + g (x) if x T
∈ I
−n
=
G (x) = , where K G : K J K J J . n= (J) and G|
−g(−x) if x ∈ (−I)
3
2 β β 3β 9β
4
4
= =
Consider now c 1 , c 2 ; f
1
− 2 − 3 , − β
2 β β 4 : [−
2 4 ] →
] given by [−1, −
4 ! ! !
3
2
β β β
• 1 (x) = c
f
1 x +
2 x +
2 x + − c − 1
2
2
2 β β β β β β
3
2
and f 2 : [ , , 1] given by f 2 (x) = c 1 x c 2 x
2 x 1. ] → [ − − −
4
2
4
2
2
2 Then, we have β β β β β β
= = = =
1. f , f and f
1
1
1
2
2
− −1, f − − β
2 f ( h ( ) f ( h ( ) 1 )− f β0 β0 β0 β0 2
4 1 β β 2 + +
4
4
4
2 1 )− f 4 + + 1 4
= = = = =
2. f lim 2, f 2, f lim +
1
2 h h
2
4
1
4 h h β →0 →0 −
2.
= 2 and f
2
2 h i β β β β β β 2 n
= = =
Consider now J 1 , , I 1 , and choose β > 0 with P n+ β β β β β ′ ′ ′
−
4 1 P P
4
8 4 n
4 ∞ n+ 1 n+ 1 ∞ ∞ n+ 1 ′ ′ ′
= = = n= n → 1. Let α n n= n n= β < and , then α < β β β 2 β β n β 2 n
2
2 2β n 1 1 n
= = =
− β −
and γ = lim ′ lim 2.
2
2
2
2 4 α β n n+ 1 Similarly to the above construction, we obtain a homeomorphism G
1
1
: (−I ) ∪
I
1 1 given by
→ J (
g 1 (x) if x
1
∈ I
G 1 (x) =
1 1 ) ,
−g (−x) if x ∈ (−I
• + 3.1. Expanding map T
25 ∞ −n
=
where K G (J J 1 n= 1 ) is a Cantor set with positive Lebesgue measure, G 1 : K
|
### 1 J1
1 K J and g 1 → J 1 : I
1 1 is a C orientation preserving homeomorphism so that ′ ′ g J and g 1 .
(x) = 2 for all x K 1 (x) > 1 for all x I
1
1 β β 2 β β 2 Similarly we obtain f : [0, ] and f , 1] such
3
4
] → [−1, − : [− , 0] → [ β β 2 β −β β
8 2
4
8 + 2
4 − that f
3 3 ( , f ( ) = 2, f (0) = 2, f 4 ( ) = , f 4 (0) = 1,
(0) = −1, f ) = − β 2
8
4
3
8
3
8
4
• +
f ) = 2 and f (0) = 2
(−
4
8
### 4 Finally, define the function f
: J J by (see Figure for its graph) (x)
G if x f β β ∈ (−I) ∪ I
1 (x) if x ∈ − , − β β
2
4 f , 2
(x) if x
4
2
f (x) = ,
G
1
1
1 (x) if x ∈ (−I ) ∪ I β 2 f , 0 (x) if x ∈ −
4 f
8 β 2
3 0,
(x) if x
8 Figure 3.1: Graph of f .
Identifying
−1 and 1 and making a linear change of coordinates we obtain a
1
### 1 C expanding map of the circle f : S .
Consider Leb the normalized Lebesgue measure of the set K , this is, Leb (A) = K J J K J N
• + =
• J J
)/ Leb(K
and consider
Leb(AK ) for all measurable A J. Denote Σ {0, 1}
• + 2
the homeomorphism h : K J J that associates each point x the sequence
→ Σ
• + 2 ∈ K
=
a describing its location in the set K , this is, a is such that I J a J
∈ Σ ∩ K {x}. Let
• + 2
µ be the Bernoulli measure in Σ giving weight 1/2 to each digit.
2
26 Chapter 3. Examples of application Claim 3.1.2. µ = h Leb
∗ K J + We show that this relation holds for an algebra of subsets which generates the n
Borel σ-algebra of Σ . For a = a a . . . a we have that µ([a]) = 1/2 . Moreover,
1 2 n
2 −1 −1
h ([a]) = I , so Leb (h ([a])) = Leb (I ). By construction, at each
a J K J K J a J
∩ KK
step all remaining intervals in the construction of K have the same length. Thus,
n J the n-th stage contains
2 intervals I a (among them) all with the same Lebesgue
measure. Since Leb is a probability measure, we have K J n
1 2 Leb K (I a J K (I a J ) = . J JK ) = 1 =⇒ Leb ∩ K n
2
−1 We conclude that Leb (I ) = Leb (h ([a])) = µ([a]) for all a = K J a J K J
∩ K +
a a . . . a , n
1 2 n ≥ 1 proving the Claim.
=
Clearly h K
◦ f | J σ ◦ h where σ is the standard left shift σ : Σ . Since µ
2
is σ-invariant, then Leb is f -invariant. Consequently ( f, Leb ) is mixing,
K K K J | J J
( f ) = log 2 = h ( f ) = h (σ) and Leb
h top Leb µ K KJ J ≪ Leb.
Hence lim σ (x) = Leb for Leb and n →+∞
n K K J J
J Ja.e, x K |K | > 0, so it follows
that Leb K is a SRB-like measure. By Theorem Leb K is an equilibrium state
J Jfor the potential ψ = − log | f |.
Leb Leb Similarly K is also a SRB-like measure, but distinct form K . J1 J
Note that this construction allows us to obtain countably many ergodic SRB-like measures by reapplying the construction to each removed subin- terval of the first Cantor set.
Note also that, if we take a sequence of Hölder continuous potentials
′
φ n n n a φ -conformal measure and a converging to ψ = − log | f |, choose ν
∗
ν n -SRB measure µ n and weak accumulation points ν, µ as in Theorem since f is topologically exact then ν(A (µ)) = 1 for all ε > 0. Therefore ν is ε
1 not Lebesgue measure on S .
### 3.2 Non uniformly expanding maps
1 Next we present an example that is a robust (C open) class of non-
1 uniformly expanding C local diffeomorphisms.
This family of maps was introduced in Chapter 1]) and maps in this class exhibit non-uniform expansion Lebesgue almost everywhere but are not uniformly expanding.
Let M be a compact manifold of dimensiom d
Example 3.2.1.
: M ≥ 1 and f
1 is a C
−expanding map. Let V M be some small compact domain, so that
the restriction of f to V is injective. Let f be any map in a sufficiently small
### 1 C -neighborhood so that:
N of f
### 3.3. Weak-expanding and non-uniformly expanding maps
27 1. f is volume expanding everywhere: there exists σ 1 > 1 such that for every x
1
| det D f (x)| ≥ σ ∈ M
2. f is expanding outside V: there exists σ < 1 such that −1 for every x
kD f (x) k < σ ∈ M\V;
3. f is not too contracting on V : there is some small δ > 0 such that −1 kD f (x) k < 1 + δ for every x V.
### 1 Then every map f in such a C -neighborhood is non-uniformly ex-
N of f
panding (see a proof in subsection 2.1 in every expanding
weak-SRB-like probability measure satisfies Pesin’s Entropy Formula, and every
ergodic SRB-like measure is expanding.
Such classes of maps can be obtained through deformation of a uniformly expanding map by isotopy inside some small region.
3.3 Weak-expanding and non-uniformly expan-
ding maps
Consider α > 0 and the map T α α 1+α : [0, 1] → [0, 1] defined as follows ( x + 2 x if x ∈ [0, 1/2)
T α (x) = .
α 1+α x
− 2 (1 − x) if x ∈ [1/2, 1]
1 1+α
This defines a family of C maps of the unit circle S := [0, 1]/ ∼ into itself, known as intermittent maps. These applications are expanding, except at a neutral fixed point, the unique fixed point is 0 and DT (0) = 1. The local α behavior near this neutral point is responsible for various phenomena. The above family of maps provides many interesting results in ergodic theory.
If α ≥ 1, i.e. if the order of tangency at zero is high enough, then the Dirac mass at zero δ is the unique physical probability measure and so the Lyapunov exponent of Lebesgue almost all points vanishes (see ). This example shows that there exists systems which are weak-expanding but not non-uniformly expanding. Hence the assumption of weak expansion together with non-uniform expansion in Corollary is not superfluous.
The following is an example of a map which is weak-expanding and non-uniformly expanding in the setting of Corollary
28 Chapter 3. Examples of application Example 3.3.1. If, in the setting of the construction of the previous Example
the region V of a point p of a periodic orbit with period k, and the deformation
k
weakens one of the eigenvalues of D f (p) in such a way that 1 becomes and
k
eigenvalue of D f (p), then f is an example of a weak expanding and non uniformly
1 expanding C transformation. t More precisely, consider the function g (t) = (0) = 0.
, 0 < t ≤ 1/2, g
log(1/t) It is easy to see that
′ is stricly increasing;
• g (t) > 0, 0 < t ≤ 1/2 and so g
′ is not of
• g α-generalized bounded variation for any α ∈ (0, 1) (see for α
′ the definition of generalized bounded variation) and so g is not C for any 0 < α < 1.
1 Setting g (t) = g (t)/g strictly increas-
(1/2) we obtain g : [0, 1/2] → [0, 1] a C
1+α ing function which is not C for any
α ∈ (0, 1). Now we consider the analogous
map to T α
( x + xg(x) if x ∈ [0, 1/2)
T (x) = .
x
− (1 − x)g(1 − x) if x ∈ [1/2, 1]
1 1+α
This is now a C map of the circle into itself which is not C for any 0 < α < 1.
=
Moreover, letting f T satisfies
× E where E(x) = 2x mod 1, we see that f
items (1-3) in Example for V a small neighborhood of the fixed point (0, 0).
1 ′
Hence f is a C non-uniformly expanding map. Moreover, since T (0) = 1,
1 we have that f is also a C weak expanding map with D = {(0, 0)}.
Now we extend this construction to obtain a weak expanding and non
1
uniformly expanding C map so that D is non denumerable.
Example 3.3.2. Let K n (x) =
⊂ I = [0, 1] be the middle third Cantor set and let β
−n
d ]), where d is the Euclidean distance on I. Note that β n is Lipschitz
(x, K ∩ [0, 3
′
and g is bounded and continuous but not of α-generalized bounded variation
n
◦ β
for any 0 < α < 1, where g was defined in Example Indeed, since
kkn
(3 ] for all k > n, then , 2 · 3 ) is a gap of K ∩ [0, 3
1−k ′ ′ −k g (β(3 (β(3 ))
/2)) − g k
1
′
=
g
2 · 3 k
1−kk −1
3 /2 − 3 2 · 3 k !
1 2 · 3 =
− 1 (1 − k) log 3 − log 2 (1 − k) log 3 − log 2 R x
′ is not bounded when k g R ր ∞. If h : I → R is given by h(x) = x + ◦
1 −1 ′
β n g n , where the integrals are with respect to Lebesgue measure on the ◦ β
### 3.3. Weak-expanding and non-uniformly expanding maps
29
1
1
1
real line, then h (0) = 0, h(1) = 2 and h induces a C map h : S whose
α → S
′
derivative is continuous but no C for any 0 < α < 1, such that h (x) = 1 for each
−nx ] and h (x) > 1 otherwise.
∈ K ∩ [0, 3 R x
(t+g ) n ◦β
1
1 We set now h (x) = x + R t 1 for t map of S ∈ [0, 1] which induces a C (t+g ) n ◦β
′
into itself with continuous derivative and h > 1 for t > 0. We then define the
t
1
1
skew-product map f (x, y) = (E(x), h which is a weak
sin πx × S
(y)) for (x, y) ∈ S
−n expanding map with ]).
D = {0} × (K ∩ [0, 3
For sufficiently big n it is easy to verify that f satisfies items (1-3) in Exam-
1
ple for V a small neighborhood of the fixed point (0, 0). Hence f is also a C
1+α non-uniformly expanding map which is not a C map for any 0 < α < 1.
30 Chapter 3. Examples of application
Chapter 4
Preliminary definitions and
results
The aim of this chapter is to fix notations, give definitions and state some facts which will be used throughout this work.
### 4.1 Invariant and
ν-SRB-like measures
In this section, we revisit the definition and some results on the theory ν-SRB-like measures. Let X be a compact metric space and T : X X be continuous. Denote
### 1 T the set of all
M the set of all Borel probability measures on X and M
∗
• + T
1 fix the weak metric
−invariant Borel probability measures on X. In M XZ Z 1 dist(µ, ν) := φ d φ d ν , (4.1.1) i i
µ − i i is a countable family of continuous functions that is dense in i=
2 } ≥0 where {φ the space C (X, [0, 1]).
The following technical lemma will be used in the proofs of Propositions i i
Lemma 4.1.1. For all ε > 0 there is δ > 0 such that if d(T (x), T (y)) < δ for all
i = (x), σ (y)) < ε n n
0, . . . , n − 1 then dist(σ Proof. k Given ε > 0 and a countable family of continuous functions F = P {φ : k ∈ N} that is dense in the space C (X, [0, 1]), fix N ≥ 1 such that
∞ −n n=N < ε/2. Let γ = ε/4 and consider F = {φ } the (N + 1)-first
2 , . . . , φ N elements of F .
32 Chapter 4. Preliminary definitions and results
Note that V(σ ε (σ (x)), where n n (x), F, γ) ⊂ B ( Z Z )
V (σ (x), F, γ) = φ d φ d σ (x) n i i n
µ ∈ M; µ − < γ, ∀i = 0, . . . , N
∗ is a neighborhood in the weak topology. R R n i i n (x), F, γ) then φ d φ d σ (x) < γ for all
In fact, given ν ∈ V(σ ν −
i = 0, . . . , N and X N Z Z ∞ Z Z X
1
1 n i i n i i n dist(ν, σ (x)) = φ d φ d σ (x) φ d φ d σ (x) i i ν − ν −
• i= i=N+
2
2
1 X ∞ ∞ i X
ε 1 ε ε 2 sup kφ k
• =
< 2γ + i i < < ε,
2
2
2
2
2 ε (σ (x)). n i=N+ i=N
1
therefore ν ∈ B i i We will now show that there is δ > 0 such that if d(T (x), T (y)) < δ for n n n ε (σ (x)). all i = 0, . . . , n − 1 then σ (y) ∈ V(σ (x), F, γ) ⊂ B
For each k = 0, . . . , N let δ(k) > 0 be a constant of uniform continuity of i i φ k
(x), T (y)) < δ for all and set δ = min{δ(k); 0 ≤ k N} > 0. Thus, if d(T R
i = n n k n (x), F, γ), since φ d σ (z) =
0, . . . , n − 1 we conclude that σ (y) ∈ V
1 P n −1 i n i= φ (T (z)), for all k = 0, . . . , N and k n n Z Z X −1 −1 X
1 i i
1 φ d σ φ d σ (x) (T (T γ = γ. k n k n k k (y) − ≤ |φ (y)) − φ (x))| <
n n i= i=
Next results ensure the existence of ν-SRB-like measure and conse-
Proposition 4.1.2. Let T
: X X be a continuous map of a compact metric
space X. For each reference probability measure the T
1
ν there exist W (ν) ⊂ M
∗ unique minimal non-empty and weak compact set, such that p T (ν) for
ω(x) ⊂ W T T .
ν-a.e. x X. When ν = Leb we denote W (Leb) = W
Proof. In the case that ν coincides with the Lebesgue measure this corre-
sponds to Theorem 1.5 in , replacing Leb by ν.
∗
Consider the family Υ of all the non-empty and weak compact sets
A
1 ⊂ M such that pω(x) ⊂ A for ν-a.e. x M. T
The family Υ is not empty, since trivially M ∈ Υ. Define in Υ the partial order A
1 2 if and only if A
1 2 .
AA
### 4.2. Measure-theoretic entropy
α
33 α∈J
Let, {A } ⊂ Υ is a chain if it is a totally ordered subset of Υ. Let us
A α
prove that A := ∩ α∈J belongs to Υ. For each fixed α ∈ J, and for each ε > 0 ε = define B α ε
(α) = {x X; Pω(x) ⊂ A } and A {x X; Pω(x) ⊂ B (A)}, where
B ε
1
(A) := {ν ∈ M ; dist(ν, A) < ε}. To conclude that A ∈ Υ, it is enough to ε prove that ν(A ) = 1 for all ε > 0.
Claim 4.1.3. α ε
For all (A).
ε > 0 there exists α ∈ J such that AB If it did not exist then, by the property of finite intersections of compact α is totally ordered, we would deduce that the set
} α∈I T sets, and since {A (A α ε
\B (A)) , ∅ would be non-empty, contained in A, but disjoint with
α∈J its open neighborhood B (A). ε
We deduce that B (A). Since A (α)) = 1 ε α (α) ⊂ B ∈ Υ, we have that ν(B ε for all α ∈ I. Thus ν(A ) = 1 for all ε > 0, and therefore A ∈ Υ. We have proved that each chain in Υ has a minimal element in Υ.
So, by Zorn’s Lemma there exist minimal elements in Υ, namely,
∗
minimal non-empty and weak T 1 such that compact sets W (ν) ⊂ M
p T
ω(x) ⊂ W (ν) for ν almost all x X. T Finally, the minimal element W (ν) ⊂ Υ is unique since the intersection of two of them is also in Υ.
Proposition 4.1.4. Given ν a reference probability measure, a probability measure
is (ν).
### 1 T
µ ∈ M ν-SRB-like if and only if µ ∈ W Proof. See proof of Proposition 2.2 in . T T . For more details on SRB-like It is standard to check that W (ν) ⊂ M measures, see for more details on weak-SRB-like measures.
### 4.2 Measure-theoretic entropy
For any Borel measurable finite partition P of M, and for any (not necessarily invariant) probability ν it is defined X ν(P) log ν(P).
H
(P, ν) = − P ∈P The conditional entropy of partition P given the partition Q is the number X X
ν(P Q) .
H ν
(P/Q) = − ν(P Q) log P ∈P Q ν(Q)
∈Q
34 Chapter 4. Preliminary definitions and results Lemma 4.2.1. Given S
≥ 1 and ε > 0 there exists δ > 0 such that, for any finite
partitions , . . . , P , . . . , ˜ P ) < δ for all
### 1 S
1 S i i
P = {P } and ˜P = { ˜P } such that ν(P △ ˜P
i = 1, . . . , S, then H ν ( ˜P/P) < ε. Proof. See Lemma 9.1.6 in q W q −1
−j
=
f
Denote P (P), where for any pair of finite partitions P and j= f then Q it is defined P ∨ Q = {P Q , ∅ : P ∈ P, Q ∈ Q}. If besides ν ∈ M
1 q
h H , ν).
(P, ν) = lim (P q
→+∞
q
Finally, the measure-theoretic entropy h ( f ) of an f -invariant measure ν ν is defined by h ν
( f ) = sup h(P, ν), where the sup is taken on all the Borel measurable finite partitions P of M. We define the diameter diam(P) of a finite partition P as the maximum diameter of its pieces. The following technical lemma will be used in the proof of the large deviation lemma that is essential for the proof of Theorem
Lemma 4.2.2. Let f
: M M be a measurable function. For any sequence of P n
1 −1 j
not necessarily invariant probabilites ν , let µ := ( f ) ν and µ be a weak
n n n n j= ∗
(µ
accumulation point of n n ). Let P be a finite partition of M with S elements, with
µ(∂P) = 0 = µ (∂P) for all n ≥ 1. Then for, for any ε > 0, there is a subsequence
of integers n i
ր ∞ such that ε
1 n i
• n
• i
H , ν h µ n i (P ) ≤ ( f ) ∀i ≥ 1.
4 Proof. Fix integers q ≥ 1, and n q. Write n = aq + j where a, j are integer numbers such that 0 ≤ j q−1. Fix a (not necessarily invariant) probability
ν. From the properties of the entropy function H of ν with respect to the partition P, we obtain n aq+j aq+q , ν)
H
(P , ν) = H(P , ν) ≤ H(P q
_ _
−1 a
qiiq
f f ), ν)
≤ H( (P), ν) + H( (P i= i= q
1 X −1 a i q iq X H ) ν) + H , ( f ) ν) ∗ ∗
≤ (P, ( f (P i= i= X a q iq
1 H , ( f ) ≤ q log S + (P ∗ ν) ∀q ≥ 1, N q. i=
1
### 4.2. Measure-theoretic entropy
35 To obtain the inequality above recall that H(P, ν) ≤ log S for all ν ∈ M
where S is the number of elements of the partition P. The inequality above
−l
holds also for f (P) instead of P, for any l ≥ 0, because it holds for any partition with exactly S pieces. Thus n q iq q l+iq X a a X
−ll
H ( f H ( f ), ( f ) ν) = q log S + H , ( f ) ν).
∗ ∗
(P ), ν) ≤ q log S + (P (P i= i=
1
1 Adding the above inequalities for 0 ≤ l q − 1, we obtain: q q X −1 −1 a n X X 2 q l+iql H ( f log S + H , ( f ) ν). l= l= i= (P ), ν) ≤ q (P ∗
1 Therefore, q aq+q X −1 −1 n X 2 q ll
H ( f log S + H , ( f ) ν). (4.2.1) l= l= (P ), ν) ≤ q (P ∗
On the other hand, for all 0 ≤ l q − 1, n n+l n X l −1
−il
H H ( f ), ν)
(P , ν) ≤ H(P , ν) ≤ (P), ν) + H( f (P i= q
l
), ν).
≤ q log S + H( f (P Therefore, adding the above inequalities for 0 ≤ l q − 1 and joining with the inequality , we obtain aq+q n X −1 2 q l
qH log S + H , ( f ) ν).
(P , ν) ≤ 2q (P ∗ l= Recall that n = aq + j with 0 ≤ j q − 1. So aq + q n + q and then n −1 aq+q n X −1 X 2 q l q l
qH log S + H , ( f ) ν) + H , ( f ) ν)
(P , ν) ≤ 2q (P ∗ (P ∗ n l= l=n X −1 2 q l log S + H , ( f ) ν). ≤ 3q (P ∗ l=
36 Chapter 4. Preliminary definitions and results
In the last inequality we have used that the number of non-empty pieces q q w is at most S . Now we fix a sequence n µ i n of P ր ∞ such that µ i −−−−→ n →+∞ put ν = ν and divide by n . Since H is convex we deduce: n i i n i −1
2 X
2 q 3q log S aq log S n q l
1 i
H , ν H , ( f ) ν ) + nn
• n n n n i i i i
n
l=
i −1
(P i ) ≤ (P i
2 X
2
3q log S aq log S
1 q l
• n i
H , ( f ) ν ) +
≤ (P
n n n i i i
l=
2
3q log S q
• n H , µ ).
≤ (P i
n i n o
36q log S
(q) = max ,
q
Therefore, for all i i ε 1 ε n q i
1 , ν , µ i
• n
• i w 12 q q
) ≤
H n H n i (P (P ) ∀i i (q), ∀q ≥ 1.
Since µ µ and µ n n i −−−−→ (∂P) = µ(∂P) = 0 for all n ∈ N, then lim (P i i i H , µ ) = n q →+∞
→+∞ H . Thus there exists i f 1 > i
1
(P , µ) because µ ∈ M (q) such that for all i i
1 q q 1 ε H , µ H , µ) + . n (P i ) ≤ (P
12
q q
### 1 Thus, for all i ≥ i
1 ε n q i
1 H , ν H , µ). n (P i ) ≤ (P
• n
• i 6 q 1 q
Moreover, by definition lim H q q (P , µ) = h(P, µ) then there exists q
→+∞
1 q ε
2 :=
, µ) ≤ h(P, µ) + q (P
N such that H . Thus taking i
12 for all q q
, i
1
max{q } we have 1 ε ε n i
H , ν h ( f ) + n µ 2 . (4.2.2)
(P i ) ≤ h(P, µ) + for all i i
n 4 ≤ i
4 This completes the proof using the subsequence (n ) as the sequence i i
≥i 2 claimed in the statement of the lemma.
Lemma 4.2.3. Let a P ℓ ℓ P ℓ 1 , . . . , areal numbers and let p
a
k
1 , . . . , pnon-negative numbers
=
such that p k
1. Denote L = P e . Then p k (a k k k= k= k= ) ≤ log L.
1
1 e ak 1 − log p
= Moreover, the equality holds if, and only if, p for all k. k L Proof. See Lemma 10.4.4 in .
Chapter 5
Continuous variation of SRB-like
measures
Here we prove Theorem showing that ν-SRB-like measures can be seen as measures that naturally arise as accumulation points of ν -SRB n measures.
5.1 Measures with prescribed Jacobian. Confor-
mal measures
A measurable function J
Definition 12. ν T
: X → [0, +∞) is called the Jacobian of a map T : X X with respect to a measure ν if for every Borel set A X on which T is injective Z ν(T(A)) = J Td ν. A ν
Next result guarantees the existence of measures with prescribed Jaco- bian.
Theorem 5.1.1. Let T
: X X be a local homeomorphism of a compact metric
space X and let
φ : X → R be continuous. Then there exists a φ-conformal
∗
probability measure ν = ν and a constant ν = λν. Moreover,
φ
λ > 0, such that L φ
−φ
λe ν.
the function J ν T = is the Jacobian for T with respect to the measure Proof. See Theorem 4.2.5 in .
38 Chapter 5. Continuous variation of SRB-like measures
### 5.2 SRB measures
We say that a continuous mapping T : X X is open, if open sets have open images. This is equivalent to saying that if f (x) = y and y y n −−−−→ n
→+∞ then there exist x x such that f (x ) = y for n large enough. n n n
−−−−→ n
→+∞ Definition 13.
A continuous mapping T : X X is called: (a) topologically exact if for all non-empty open set U X there exists N N = N (U) such that T U = X .
(b) topologically transitive if for all non-empty open sets U, V X there n exists n ≥ 0 such that T (U) ∩ V , ∅.
Topological transitiveness ensures conformal measures give positive mass to any open subset.
Proposition 5.2.1. Let T
: X X be an open expanding topologically transitive
map and φ is
φ : X → R be continuous. Then every conformal measure ν = ν
positive on non-empty open sets. Moreover for every r > 0 there exists α = α(r) >
0 such that for every x X, ν(B(x, r)) ≥ α. Proof. See Proposition 4.2.7. in
For Hölder potentials it is known that there exists a unique ν-SRB probability measure.
Theorem 5.2.2. Let T
: X X an open expanding topologically transitive
map, ν = ν a conformal measure associated to the a Hölder continuous function
φ φ ergodic invariant ν-SRB probability
φ : X → R. Then there exists a unique µ measure such that µ is the unique equilibrium state for T and φ. φ Proof. The proof follows the results presented in Chapter 4 of .
The following result shows that positively invariant sets with positive reference measure have mass uniformly bounded away from zero.
In the same setting of Theorem if G is an T-invariant set Theorem 5.2.3. such that ν(G) > 0, then there is a disk of radius δ/4 so that ν(∆\G) = 0.
Proof. See the proof of Lemma 5.3 in .
5.3. Limits of SRB-measures
39 Remark 5.2.4. It is easy to see that each continuous potential in a compact metric
space can be approximated in uniform convergence by Lipschitz continuous poten-
tials (in particular, all Lipschitz continuous potentials are α-Hölder continuous
for 0 < α ≤ 1).
In fact, let
A = { f : X → R; f is Lipschitz continuous} ⊂ C(X, R) and
observe that
A is an subalgebra of the algebra C(X, R), since A forms a vector
space over R and given f
, g ∈ A, x X then ( f ·g)(x) = f (xg(x) ∈ A. Moreover,
f
≡ 1 belongs to A and A separates points, since given x, y X, x , y, we may
take f
: X → R given by f (z) = d(z, {x}), f is Lipschitz continuous (because, {x} is a closed set) therefore f ∈ A and f (x) , f (y). Thus, the Theorem of
Stone-Weierstrass ensures that A is dense in C(X, R).
5.3 Limits of SRB-measures Now we are ready to prove Theorem
∗
Proof of Theorem Let λ (ν ) = λ ν (it is easy to
n φ n n n j −−−−→ λ, where L j j j j n j
→+∞ ∗
= see that λ > 0 since λ (ν )(1) for all n). n n L φ n
(X, R) we have Z Z Z Z Then for each ϕ ∈ C
∗ ∗
ν ) = (ϕ) dν (ϕ) dν = ν) n φ n φ ϕ d(L j L j −−−−→ L ϕ d(L φ n j n j j φ w w ∗ ∗
→+∞
∗ ∗
ν ν. Moreover, λ ν λν and by uniqueness of n n n thus, L j −−−−→ L j j −−−−→ φ φ n j j →+∞ j →+∞ the limit, it follows that
∗ ∗ λν = lim λ ν = ν = ν. j j n n lim n j j L j L n j →+∞ →+∞ Thus, ν is a φ-confomal measure.
Let µ be the ν -SRB measure and let µ = lim µ . For each fixed ε > 0, n n n j j j j
→+∞
consider N = N(ε) such that ε ε dist(µ , µ ) < and dist(µ , µ) < j n n n j m j , m N.
4 4 ∀ Thus, A ε/4 (µ n ε/2 (µ n ) and A ε/2 (µ n ε j m j ) ⊂ A ) ⊂ A (µ) for all j, m N. ε/4 n n (µ ) we have dist(pω(x), µ ) < ε/4. Then,
In fact, for each x A j j ε dist(pω(x), µ ) + dist(µ , µ ) < n n n n m ) ≤ dist(pω(x), µ j j m
2
40 Chapter 5. Continuous variation of SRB-like measures ε/4 (µ ε/2 (µ n n m
for all j, m N. Therefore, A j ) ⊂ A ) for all j, m N. Analo- gously, A (µ (µ). ε/2 n ε j ) ⊂ A n j of radius δ
1 /4
By Theorem for each j ≥ 1 there exists a disk ∆ around x such that ν (∆ (µ )) = 0. Let x = lim x taking a subse- n n n ε/2 n n j j j \A j j j
→+∞
∆ quence if necessary. By compactness, the sequence n accumulates on j j /8 a disc ˜∆ of radius δ
1
1
/4 around x. Let ∆ be the disk of radius 0 < s ≤ δ around x. Thus, there exists N . nN such that ∆ ⊂ ∆ j for all j N
∆ = /8 such that ν ∂ (µ) 0 and note that for all
1 ε
Consider 0 < s ≤ δ \A ,
j
≥ N ∆ ∆ ∆ , 0 = ν n n ε/2 (µ n ) n ε/2 (µ n ) n ε (µ) j j \A j ≥ ν j \A j ≥ ν j \A n j n j and A ε/2 (µ ε . since ∆ ⊂ ∆ ) ⊂ A (µ) for all j N
∗
Thus, by weak convergence ν ∆ = ν ∆ = ε (µ) lim n ε (µ)
0. (5.3.1) \A j \A j
→+∞
Since ν is a conformal measure, it follows from ∆ that ν A (µ) > 0. As ε > 0 is arbitrary, we conclude that µ is ε
≥ ν ν-SRB-like. Let P be a finite Borel partition of X with diameter not exceeding an expansive constant such that µ(∂P) = 0. Then P generates the Borel sigma- algebra for every Borel probability T-invariant in X (see Lemma 2.5.5 in η (T) is ) and by Kolmogorov-Sinai Theorem this implies that η 7→ h upper semi-continuous in µ. R R
Moreover, as φ µ φ dµ, µ is an equilibrium state for n d n n j j −−−−→ j j
→+∞
(T, φ n (T, ϕ) (see Theorem j ) (by Theorem and by continuity of ϕ 7→ P top
9.7 in ) it follows that Z Z ! =
h (T) + h (T) + φ d µ lim P (T, φ ) = P (T, φ).
µ µ n n top n top
φdµ ≥ lim j j j j j n j
→+∞ →+∞ This shows that µ is an equilibrium state for T with respect to φ.
Now we assume that T is topologically exact. We know that given ε ε > 0 there exists a disk ∆ such that ν(∆\A (µ)) = 0. Since ∆ ⊂ X is a N non-empty open set, there exists N > 0 such that X = T (∆). Moreover,
A (µ) = T(A (µ)), thus ε ε N N
0 = ν(T ε ε ε (µ)).
(∆\A (µ)) ≥ ν(T (∆)\A (µ)) = ν(X\A Since ε > 0 was arbitrary, this shows that ν(A (µ)) = 1 for all ε > 0. ε
Chapter 6
Expanding maps on compact
metric spaces
Here we state the main results needed to obtain the proof of Theorem
We start by presenting some results about expanding maps that will be used throughout this chapter. We close the chapter with the proof of Theorem
### 6.1 Basic properties of expanding open maps In what follows X is a compact metric space.
Lemma 6.1.1. If T
: X X is a continuous open map, then for every ε > 0 there
exists δ > 0 such that T(B(x, ε)) ⊃ B(T(x), δ) for every x X. Proof. See Lemma 3.1.2 in .
Remark 6.1.2. If T
: X X is an expanding map, then by ,
for all x is injective and therefore it has a local inverse
B (x,ε)
∈ X, the restriction T| map on T (B(x, ε)).
: X X is an open map, then, in view of Lemma the
domain of the inverse map contains the ball B (T(x), δ). So it makes sense to define
−1 the restriction of the inverse map, T x : B(T(x), δ) → B(x, ε) at each x X.
Lemma 6.1.3. Let T
: X X be an open expanding map. If x X and y, z
−1 −1 −1 −1 B (T(x), δ) then d(T (y), T d (y, z). In particular T x x (z)) ≤ λ x (B(T(x), δ)) ⊂
−1 B (x, λ
δ) ⊂ B(x, δ) and
−1
δ))
B
(6.1.1) (T(x), δ) ⊂ T(B(x, λ δ > 0 small enough.
for all
42 Chapter 6. Expanding maps on compact metric spaces Proof. See Lemma 3.1.4 from . Definition 14.
Let T : X X be an open expanding map. For every x X, j = j T (x). In view of Lemma every n ≥ 1 and every j = 0, 1, . . . , n − 1 write x the composition n
−1 −1 −1 T : B(T
xT x ◦ · · · ◦ T x (x), δ) → X
1 n
−1
−n is well-defined and will be denoted by T . x Lemma 6.1.4. Let T
: X X be an open expanding map. For every x X we
have:nn
1. T (A) = S T y (x) n y (A) for all A B(x, δ);T nnnn
2. d (T (y), T d (x), δ); x x (z)) ≤ λ (y, z) for all y, z B(T nnn
### 3. T (B(T r x (x), r)) ⊂ B(x, min{ε, λ }) for all r ≤ δ.
Proof. The lemma follow from Lemma
For more details on the proofs in this subsection, see subsection 3.1 in .
### 6.2 Conformal measures. Weak Gibbs property
Here we cite results relating φ-conformal measures and the topological pressure of φ. Next results says φ-conformal measures for expanding dynamics with continuous potentials are almost Gibbs measures; see for more details.
Proposition 6.2.1. Let T
: X X an open expanding topologically transitive
map, ν = ν a conformal measure associated the a continuous function φ
φ : X → R
−φ
λe ν. Given δ > 0, for
and J ν T = the Jacobian for T with respect to the measure all x
∈ X and all n ≥ 1 there exists α(ε) > 0 such that ν((B(x, n, ε))) δ
nδ n
α(ε)e ≤ ≤ e exp (S n
φ(x) − Pn) ε > 0 small enough, where P = log λ.
for all
### 6.2. Conformal measures. Weak Gibbs property
43 Proof.
Given δ > 0 there exist γ > 0 such that for all x, y X, with d(x, y) < γ we have |φ(x) − φ(y)| < δ. Fix 0 < ε < γ, x X and n ≥ 1 arbitrary. Then n n n Z ν(B(T x , ε)) = ν(T (B(x, n, ε))) = J T d ν. (6.2.1) B ν
(x,n,ε)
Hence, since it gives uniform weight to balls of fixed radius and the by uniform continuity of φ, we obtain n n n φ Z Z
−S n x , ε)) = J T d ν = λ e d ν ν
α(ε) ≤ ν(B(T B B nS n φ(x)+nδ (x,n,ε) (x,n,ε)
e
≤ λ · ν(B(x, n, ε)), and n n n Z
−S n φ(x)−nδ x , ε)) = J T d e ν 1 ≥ ν(B(T ν ≥ λ · ν(B(x, n, ε)). B
(x,n,ε)
Hence ν(B(x, n, ε)) n δ
−nδ
α(ε)e ≤ ≤ e exp (S n
φ(x) − Pn) where P = log λ.
Lemma 6.2.2. Let T
: X X be an open expanding topologically transitive
map, let
φ : X → R be continuous and ν be a probability measure such that
∗
(ν) = λν. Then P top L φ (T, φ) ≤ P = log λ.
Proof. Given δ > 0 consider ε > 0 small enough as in Proposition
n
and n ≥ 1. Let EX be a maximal (n, ε)-separated set. Therefore n n {B(x, n, ε) : x E } covers X, {B(x, n, ε/2) : x E } is pairwise disjoint open sets. Thus by Proposition we have, n n X X (P+δ) (P+δ) S n φ(x) e e .
e
≤ ν(B(x, n, ε/2)) · ≤ x x α(ε/2) α(ε/2)
∈E nE n
Therefore, X
1 S n φ(x)
P top (T, φ) = lim lim sup log e
≤ P + δ,
ε→0 n n →+∞ xE n
as δ > 0 can be taken small enough, we conclude that P top (T, φ) − P ≤ 0.
44 Chapter 6. Expanding maps on compact metric spaces
6.3 Constructing an arbitrarily small initial par-
tition with negligible boundary
Let ν be a Borel probability measure on the compact metric space X. We say that T is ν-regular map or ν is regular for T if T
∗ ν ≪ ν, that is, if E X is −1
0. It follows that if T admits a Jacobian with respect a T-regular measure.
= such that ν(E) = 0, then ν T (E)
Lemma 6.3.1. Let X be a compact metric space of dimension d
≥ 1. If ν is a
regular reference measure for T positive on non-empty open sets and δ > 0, then
there exists a finite partition
P of X with diam(P) < δ such that every atom
P ∈ P have non-empty interior and ν(∂P) = 0. Proof. Let δ > 0 and consider 0 < δ , δ
1 l
1
≤ δ and B = {B( ˜x /8), l = 1, . . . , q} be a finite open cover of X by δ
1 /8- balls such that ν(∂B( ˜x , δ l 1 /8)) = 0 for all
l = 1, . . . , q. Note that such value of δ > 0 exists since the set of values of δ
1
1
such that ν(∂B( ˜x , δ l
1
/8)) > 0 for some l ∈ {1, . . . , q} is denumerable, because ν is a finite measure and X is a compact metric space soon separable. From this we define a finite partition P of X as follows. We start by setting
δ δ
1
1
˜x , , . . . , P ˜x ,
P 1 := B 1 k := B k 1 k )
\(P ∪ . . . ∪ P −1
8
8 for all k = 2, . . . , S where S q. ,
Note that if P has non-empty interior (since X is separable), k k ∅ then P δ δ 1 1 diameter smaller than 2 < and the boundary ∂P is a (finite) union of k
8
4
pieces of boundaries of balls with zero ν-measure. We define P by the elements P constructed above which are non-empty. k Note that since T is ν-regular the boundary of g(P) still has zero ν- n
, for any measure for every atom P ∈ P and every inverse branch g of T
n ≥ 1.
6.4 Expansive maps and existence of equilibrium
states
A continuous transformation T : X X of a compact metric space X equipped with a metric ρ is (positively) expansive if and only if n n (x), T
∃δ > 0[∀n ≥ 0 ρ(T (y)) ≤ δ] =⇒ x = y and the number δ above is called an expansive constant.
6.5. Large deviations
45 Theorem 6.4.1. If T T
: X X is positively expansive, then the function M ∋ µ (T) is upper semi-continuous and consequently each continuous potential µ 7→ h φ : X → R has an equilibrium state.
See Theorem 2.5.6 in .
Proof.
Theorem 6.4.2. Expanding property implies positively expansive property.
Proof. See Theorem 3.1.1 in .
Corollary 6.4.3. If T
: X X is expanding, then each continuous potential r T : φ : X → R has an equilibrium state. In particular, K (φ) = {µ ∈ M R
h µ (T) + top φdµ ≥ P (T, φ) − r} , ∅ for all r ≥ 0. Proof. The proof is immediate from Theorem
6.5 Large deviations
The statement of Theorem and in Proposition 6.1.11 in .
Proposition A. Let T
: X X be an open expanding topologically transitive
map and let a φ-conformal measure, r > 0
φ
φ : X → R be continuous. Fix ν = ν
∗
and consider the weak distance dist defined in . Then, for all 0 < ε < r,
there exists n n o ≥ 1 and κ > 0 such that
ν x n r . (6.5.1) ∈ X; dist(σ (x), K (φ)) ≥ ε < κ exp [n(ε − r)] ∀n n
Proof. We know by Theorem that all the (neces-
sarily existing) conformal measures ν are positive on non-empty open sets
−φ and J λe is the Jacobian for T with respect to the measure ν. ν T =
Consider ν a conformal measure, fix r > 0 and let 0 < ε < r. For ε/6, fix ε whenever a constant γ > 0 of uniform continuity of φ, i.e., |φ(x) − φ(y)| <
6 d
(x, y) < γ. Consider 0 < ξ < γ and a partition P of X as in Lemma ξ . such that diam(P) <
### 4 Let us choose one interior point in each atom P ∈ P, and form the set
=
C , . . . , w
(where
### 1 S
l
{w } of representatives of the atoms of P = {P } 1≤lS S S =
S =
∂P l #{P ∈ P : P , ∅}). Let d min{d(w, ∂P), w C } > 0 where ∂P = l=
1 is the boundary of P. r
Consider A := {µ ∈ M; dist(µ, K (φ)) ≥ ε}. Note that A is weak* , . . . , B compact, so it has a finite covering B
1 κ
for minimal cardinality κ ≥ 1,
46 Chapter 6. Expanding maps on compact metric spaces ε
= with open balls B i n ,i ⊂ M of radius . For any fixed n ≥ 1 write C {x S κ
### 3 S κ
= = =
X ; σ n i n C n ,i , ˜C n ,i n i n ˜C n ,i ,
}, C
1 {x X; σ } and ˜C
1
(x) ∈ B i= (x) ∈ ˜B i=
2ε where ˜B are open balls concentric with B of radius for i = 1, . . . , κ. i i
3 Note that C n ,i n ,i n r n n .
⊂ ˜C . Moreover, {x X; dist(σ (x), K (φ)) ≥ ε} ⊂ C ⊂ ˜C
Claim 6.5.1. For each > 0 such that ν(C i n ,i
1 ≤ i ≤ κ there exists n ) ≤ i . exp [n (ε − r)] for all n n
First, let us see that it is enough to prove the Claim to finish the = proof of the lemma. In fact, if Claim holds, put n max n .Then we i
1≤i≤κ
, as needed: obtain the following inequality for all n n X κ ν(C ν(C n n ,i ) ≤ ) ≤ κ exp [n (ε − r)]. i=
1 n ,i n
Let us prove Claim for x C let P ∈ P be the atom such that T (x) ∈ P
−n
and set Q = T of all such sets Q is finite since x (P). Then the family Q n both P and the number of inverse branches are finite. Moreover by the expression of J T in terms of φ ν Z Z −1 n X j
−n n ,i ν ) = J T d ν = exp d ν.
ν(Q C φ ◦ Tn log λ T ) T ) n n j=
(QC n ,i (QC n ,i We note that if ν(C n ,i ) = 0, then Claim becomes trivially proved.
1 , . . . , Q N n n ,i
Consider the finite family of atoms {Q } = {Q ∈ Q : ν(Q C ) > 0} which has N = N(n, i) elements for some N ≥ 1. N Note that ν(C ) = P ν(Q ). For each k = 1, . . . , N, consider n ,i k n ,i k= n 1 ∩ C
x k k such that T (x k ) = w j for some j = 1, . . . , S (recall that w j are interior
∈ Q k ,n for each points of each atom of the partition P, so there is only one j = j n x such that T (x ) = w ). k k k jQ n ξ
< ξ for all n > 0 (remember Lemma Since diam(P ) < diam(P) < j j
4
ε
(x k k , then |φ(T )) − φ(T (y))| < for all y Q and j = 0, . . . , n − 1.
### 6 Considering for each k, y k k n ,i then,
∈ QC
### 6.5. Large deviations
N N n
47 X X Z ε X −1 j
ν(C ,i ) = ν(Q ,i exp φ(T (y )) + d ν n k n kC ) ≤ − n log λ k=
6 N n
1 k=
1 j=
T (Q ) n
kC n ,i
X
X
−1 j n ε ε exp φ(T (x )) + (Q ))
≤
• − n log λ · ν(TC k k n ,i k=
6
6 N n
1 j= X X −1 j ε exp φ(T (x k )) +
≤ − n log λ
3 k= j=
1 N n X X −1
ε j n log λ exp φ(T (x )) . k ≤ exp 3 − k=
1 j=
Defining N n n X
X
−1 −1 X j j
1 L := exp φ(T (x )) , λ := exp φ(T (x )) k k k k= L N 1 j= j= λ = then P k 1 and by Lemma k=
1 X N n −1 N X j X log L = λ φ(T (x )) λ log λ . − k= k kk k
1 j= k=
N N
1 Define the probability measures ν := P λ δ and µ := P λ σ (x ) n k x n k n k k=
R
1 k k= n
1 n n so that we may rewrite, log L = n
φdµ H , ν ). We fix a weak (P
• n n j accumulation point µ of (µ ) and take a subsequence n
• w
−−−−→ ∞ such j
→+∞
that µ n j −→ µ and
1
1 lim sup log ν(C ) = lim log ν(C ). (6.5.2) n ,i n ,i j j
→+∞ n n n j →+∞
We now make a small perturbation of the original partition so that the points x are still given by the image of the same n th inverse branch k kQ of T of an atom of the perturbed partition and the boundaries of the new partition also have negligible µ measure. d ξ
, does not depend on n) such that for Let 0 < η < min{ } (note that d
2
4
all l = 1, . . . , S and for each n ≥ 1 ξ ξ
= + + µ ∂B , η 0 = µ ∂B , η (6.5.3)
˜x n ˜x l l
4
4
48 Chapter 6. Expanding maps on compact metric spaces
where the centers are the ones from the construction of the initial partition P in the proof of Lemma
Such value of η exists since the set of values of η > 0 such that some of the expressions in is positive for some l ∈ {1, . . . , S} and some n ≥ 1 is at most denumerable because the measures involved are probability measures. Thus we may take η > 0 satisfying arbitrarily close to zero.
We consider now the finite open cover l , ξ/4 + η) : l = 1, . . . , S (6.5.4) ˜C = Bx of X and construct the partition ˜P induced by ˜C by the same procedure as l before and following the some order of construction P, ( ˜x , l ∈ {1, . . . , S} are the same used in the construction of P in Lemma .
Note that d(w , ∂B( ˜x , ξ/4 + η)) > d /2 for all l = 1, . . . , S and l l − η > d
w by construction. Therefore, each w is contained in some atom
l l
∈ CC
P w l ∈ ˜P. Moreover there cannot be distinct wCP k l k l w , w such that w , by the choice of η. Thus, the number of atoms of P is less than or equal to the
number of atoms of ˜P. On the other hand, by construction the maximum number of elements ˜P is S = Card(P), because Card( ˜C) = S. In this way we conclude that the partition ˜P has the same number of atoms as P.
We have (1) diam( ˜P) < 2(ξ/4 + η) < ξ; n (2) µ(∂ ˜P) = 0 = µ (∂ ˜P) for all n ≥ 1; . (3) w ∈ int( ˜P(w)), ∀w C n n j j
, ν , ν ) by definition of the ν and by con- n n n Note that H( ˜P j ) = H(P j j struction of the ˜P as a perturbation of P. Using items (1) and (2) we get, by Lemma that there exists j > 0 such that
1 n n j j 1 ε
H , ν ) = H , ν (T) + . (6.5.5) n n µ
(P j ( ˜P j ) ≤ h , ∀j j
n n j j
3 ,
For the partition obtained above, we have that x Q and Q k k k k n ,i ∈ ˜ ∩QC
Q k n for k = 1, . . . , N(n, i), where this family is defined
∅, where ˜ belongs to ˜Q n similarly to Q changing only the atoms of the original partition P by the atoms of the perturbed partition ˜P. k n n i k ,i . Then σ . As d(T (x ), T j j
Consider y QC (y) ∈ B (y)) ≤ diam(P) < n k n (x ), σ (y)) < ε/3 and as ξ, for all j = 0, . . . , n − 1 then, by Lemma dB i i concentrically, we conclude that σ n (x k i .
⊂ ˜B ) ∈ ˜B
6.6. Proof of Theorem
49 Since the ball ˜B is convex and µ is a convex combination of the mea- i n
= sures σ (x ), (recall that P λ 1), we deduce that µ . Therefore, the n k k n i ∈ ˜B
∗ ∗
weak n n belongs to the weak limit µ of any convergent subsequence of {µ }
2ε
closure ˜B . Since the ball ˜B has radius (φ). Then i i r we conclude that µ < K R
3
φdµ + h (T) < P µ top (T, φ) − r, and therefore
ε ε ν(C n log λ n log λ log L n ,i
) ≤ exp · L = exp
• " #
3 − 3 − Z ε
1 n
• = n n exp n log λ + φdµ H , ν ) .
(P R R 3 − n φdµ φdµ (because φ is continuous and by
We know that n j −−−→ j
→∞ RR
weak convergence) and therefore, there exists j
1 > 0 such that φdµ n j
φdµ + ε/3 for all j > j 1 .
1 n j By there exists j > 0 such that H , ν . (P j ) ≤ h (T) + ε/3, ∀j j n µ n j
= Taking j
2 1 , j
2
max{j } we have, ∀j > j " Z # ε ε ε
ν(C n log λ + φdµ + h (T) + n ,i j µ j ) ≤ exp
• " #
3 − Z
3
3 = exp n φdµ + h µ (T) j h i h i ε − log λ +
n (T, φ) n j top j
≤ exp ε − r − log λ + P ≤ exp ε − r where the last inequality follows from Lemma By we conclude that there exist n > 0 such that ν(C ,i n h
) ≤ exp n ending the proof. ε − r i for all n n
6.6 Proof of Theorem r T . By the upper
Given r > 0, consider the (non-empty) set K (φ) ⊂ M semicontinuity of the metric entropy (see Theorem , we have that
∗ r (φ) is closed, hence, weak is decreasing with r, r r
K compact. Since {K (φ)} (φ) = T (φ). r we have K K r >0 R By the Variational Principle h (T) + . µ top T φ dµ ≤ P (T, φ) for all µ ∈ M T (ν) of ν-SRB-
So, to prove Theorem we must prove that the set W n T r (φ) = like measures satisfy W (ν) ⊂ K (φ) for all r > 0, because K µ ∈ R
∗ T ; h µ (T) + top (T, φ) r (φ) is weak compact, we have φdµ = P
M o. Since K
50 \ ε ε
## Chapter 6. Expanding maps on compact metric spaces
n o r (φ) = (φ) = T r
K K r r µ ∈ M ε>0 (φ), where K ; dist(µ, K (φ)) ≤ ε
∗
with the weak distance defined in . Therefore, it is enough to prove ε T (φ) for all 0 < ε < r/2 and for all r > 0. By Proposition that W (ν) ⊂ K r ε
∗
(φ) is weak compact, it is enough to prove the following and since K r ε
(φ)
Lemma 6.6.1. The basin of attraction of s ε ε n o K r
W (φ)) := x (φ)
rX; pω(x) ⊂ K r
(K has full ν-measure.
Proof. By Proposition since 0 < ε < r, there exists n n o ε n (ε−r) ≥ 1 and κ > 0 such that ν x (φ) for any n > n . This implies n n oX; σ (x) < K r ≤ κe
• +ε
that P ν (φ) n= x n
1 ∈ X; σ (x) < K r < +∞. By the Borel-Cantelli Lemma it
follows that + + ε \ [ ∞ ∞ n o =
ν x (φ) n 0. =X; σ (x) < K r n r r 1 n=n n ε ε In other words, for ν-a.e. x X there exists n ≥ 1 such that σ (x) ∈ K r r
(φ) for all n n . Hence, pω(x) ⊂ K (φ) for ν-almost all the points x X, as required.
The proof of Theorem is complete.
Remark 6.6.2. If the set r r r (φ)
K (φ) is not closed, we may substitute K (φ) for K
in the proof of Theorem and by the same argument we conclude that T
W T (ν) ⊂ r (φ). Thus, in a more general context, where the Proposition is valid and r >0 K
K 1/n (φ) , ∅ for all n ≥ 1, we can say that ν-SRB-like measures are "almost φ-
equilibrium states", since, given (ν) then µ = lim µ , µ (φ) for
T n n 1/n
µ ∈ W ∈ K n
→+∞ all n
≥ 1. Therefore, we can find a sequence of T-invariant probability measures R
1
so that h (T) + φdµ for all n µ n top n nP (T, φ) − ≥ 1 and µ → µ in the weak*
n
topology.
To obtain a φ-equilibrium state in the limit we need only assume that φ is uniformly approximated by continuous potentials, as follows.
Let T Corollary 6.6.3.
: X X be an open expanding topologically transitive
map of a compact metric space X, (φ ) a sequence of continuous potentials,
n n ≥1
(ν ) a sequence of conformal measures associated to the (T, φ ) and µ a sequence n n n n
≥1
ν
of n -SRB-like measures. Assume that
### 6.6. Proof of Theorem B
51 1. φ φ in the topology of uniform convergence; n j −−−−→ j →+∞ w
∗ 2. µ µ in the weak topology. n j −−−−→ j →+∞ µ is an equilibrium state for the potential φ.
Then
Proof. Let µ be a ν -SRB-like measure and let µ = lim µ . Since any
n j n j j n j →+∞
finite Borel partition P of X with diameter not exceeding an expansive constant and satisfying µ(∂P) = 0 generates the Borel sigma-algebra for every Borel T-invariant probability measure in X (see Lemma 2.5.5 in ), η (T) is upper semi- then Kolmogorov-Sinai Theorem implies that η 7→ h continuous. R R
Moreover, as φ n d µ n φ dµ, µ n is an equilibrium state for j j j −−−−→ j
→+∞
(T, φ n (T, ϕ) (see Theorem j ) (by Theorem and by continuity of ϕ 7→ P top
9.7 in ) it follows that Z Z ! =
h µ (T) + h µ (T) + φ n d µ n lim P top (T, φ n ) = P top (T, φ).
j j j
φdµ ≥ lim j j n j
→+∞ →+∞ This shows that µ is an equilibrium state for T with respect to φ.
52 Chapter 6. Expanding maps on compact metric spaces
## Chapter 7 Entropy Formula Here we state the main results needed to obtain the proof of Theorem D. Then we prove Theorem D in the last subsection.
### 7.1 Hyperbolic Times
The main technical tool used in the study of non-uniformly expanding maps is the notion of hyperbolic times, introduced in . We now outline some the properties of hyperbolic times.
Definition 15.
Given σ ∈ (0, 1), we say that h is a σ-hyperbolic time for a point x M if for all 1 ≤ k h, Y h −1 j k
−1
(x)) (7.1.1) j=h kD f ( f k ≤ σ
−k
Remark 7.1.1. Throughout this text the reader will find many quotes from works
2 1+α
where f is assumed to be of class C (or f (M, M), α > 0). But it is worth
∈ C
noting that these results cited are proven without using the bounded distortion
assumption, and therefore the proofs are easily adapted to our context (in general,
the proofs are the same).
Proposition 7.1.2. Given 0 < σ < 1, there exists δ 1 > 0 such that, whenever
h is a σ-hyperbolic time for a point x, the dynamical ball B(x, h, δ ) is mapped
h h
1 diffeomorphically by f onto the ball B ( f (x), δ
h h k /2 h h
1 ), with
−kk
d ( f (y), f (y), f (z))
(z)) ≤ σ · d( f
for every
1 ).
1 ≤ k h and y, z B(x, h, δ
54 Chapter 7. Entropy Formula Proof. See Lemma 5.2 in
Remark 7.1.3. For an open expanding and topologically transitive map T of a
compact metric space X every time is a hyperbolic time for every point x, that is,
every x satisfies the condition of Proposition
Definition 16.
We say that the frequency of σ-hyperbolic times for x M is positive, if there is some θ > 0 such that all sufficiently for large n ∈ N there < h < . . . < h
1 2 l
are l ≥ θn and integers 1 ≤ hn which are σ-hyperbolic times for x.
The following Theorem ensures existence of infinitely many hyperbolic times Lebesgue almost every point for non-uniformly expanding maps. A complete proof can be found in Ref. , Sec. 5.
1 Theorem 7.1.4. Let f non-uniformly expanding local dif-
: M M be a C
feomorphism. Then there are
σ ∈ (0, 1) and there exists θ = θ(σ) > 0 such
that
Leb-a.e. x M has infinitely many σ-hyperbolic times. Moreover if we 0 < h < h < h < . . . for the hyperbolic times of x then their asymptotic
write
1
2
3 frequency satisfies k
#{k ≥ 1 : hN} lim inf N ≥ θ for Leb −a.e.x M
→∞ N
The Lemma below shows that we can translate the density of hyperbolic times into the Lebesgue measure of the set of points which have a specific (large) hyperbolic time.
Lemma 7.1.5. Let B
⊂ M, θ > 0 and g : M M be a local diffeomorphisms
such that g has density
> 2θ of hyperbolic times for every x B. Then, given any
probability measure
ν on B and any n ≥ 1, there exists h > n such that θ
ν({x B : h is a hyperbolic time of g for x}) >
2 See Lemma 3.3 in Proof. The next result is the flexible covering lemma with hyperbolic preballs which will enable us to approximate the Lebesgue measure of a given set through the measure of families of hyperbolic preballs.
Let a measurable set A Lemma 7.1.6.
⊂ M, n ≥ 1 and ε > 0 be given with
m (A) > 0. Let θ > 0 be a lower bound for the density of hyperbolic times for
Lebesgue almost every point. Then there are integers n < h < . . . < h for
1 k k = k i of subsets of M, i = 1, . . . k such that
(ε) ≥ 1 and families E
### 7.1. Hyperbolic Times
55 1.
1 is a finite pairwise disjoint family of subsets of M; k
E ∪ . . . ∪ E σ
2. h is a -hyperbolic time for every point in Q, for every element Q , i i
∈ E
2 i = 1, . . . , k; 3. every Q is the preimage of some element P i h i ∈ E ∈ P under an inverse branch of f , i = 1, . . . , k; 4. there is an open set U
1 1 with k
⊃ A containing the elements of E ∪ . . . E
m (U
1
\A) < ε; 5. m i i ) < ε. (A△ ∪ E Proof. See Lemma 3.5 in .
Remark 7.1.7. This covering lemma is true replacing f by an open expanding and
topologically transitive map T of a compact metric space X; Leb by a φ-conformal
measure
ν for a continuous potential φ : M → R; recall Remark We use this covering lemma to prove the following.
Proposition B. Let f
: M M be a non-uniformly expanding map. For any
∗
µ ∈ W f Z
h µ ( f ) + (7.1.2) ψdµ ≥ 0.
∗ Proof. , consider δ 1 > 0 as in Proposition and as f :
Given µ ∈ W f
1 local diffeomorphism in particular f is a regular map.
M
→ M is a C δ 1 Consider a partition P of M as in Proposition such that diam(P) <
4 and µ(∂P) = 0.
Since µ is f -invariant and P is a µ-mod0 partition such that µ(∂P) = 0, then the function λ 7→ h(P, λ) is upper semi-continuous in µ, that is, for each small enough τ > 0 we can find δ > 0 such that
2
(7.1.3)
2 if dist(µ, ˜µ) ≤ δ then h(P, ˜µ) ≤ h(P, µ) + τ.
Fix τ > 0 and 0 < δ < τ as above. Since µ is weak-SRB-like probability
2
measure, by definition, for any fixed value of 0 < ε < δ
2 /3 there exists a subsequence of integers n (µ)) > 0 for all l > 0. l ε,n
→ +∞ such that Leb(A l Take δ
2 /3 > 0 and fix γ > 0 as in Lemma and γ 1 > 0 of uniform
/3 if d(x, y) < γ . We denote γ =
2
1
continuity of ψ, i.e., |ψ(x) − ψ(y)| < δ , γ
min{γ
1 }.
56 Chapter 7. Entropy Formula
Let us choose one interior point having density ≥ θ of σ-hyperbolic =
, . . . , w
1 S
times of f in each atom P ∈ P and form the set W {w } of
1≤ℓ≤S
representatives of the atoms of P = {P } , where S = #{P ∈ P : P , ∅}, S S =
∂P and consider d ℓ is the } > 0, where ∂P = ℓ=1 min{d(w, ∂P), w W boundary of P.
We use now the Lemma for obtain the flexible covering of set =
A (µ) with hyperbolic preballs. Take positive integers l, m and β ε,n l m l
1 2 Leb(A (µ)) > 0 such that σ δ /4 < γ. Then there are integers n < ε,n n l l 1 l
n m < h takes the place of ε in
• l k l
1 2 < . . . < h
≤ h with k = k (l) ≥ 1 (here β j of subsets of M, j = 1, . . . k so that Lemma and families E X k k X Leb(A (µ)) = Leb(A ) + Leb(A ) ε,n ε,n j ε,n j l l (µ) ∩ E l (µ)\E j= X h k 1 j= X
1
, hence Leb(A ε,n l
≤ l (µ) ∩ Q) + β j= Q
1 ∈E j X h k
X
n l
Leb(A ε,n Leb(A ε,n (7.1.4) l (µ)) ≤ l (µ) ∩ Q),
n l
− 1 j= Q
1 S
∈E j
∈E jh j j is a family of all sets obtained as f (P) which intersect
= = j h ε,n j ε,n and A A where E E j l (µ) ∩ E Q l (µ) ∩ Q.
Figure 7.1: E (µ) in points for which h
A ε,n j l is a hyperbolic time, where P ∈ P. Analo- j+ 1 .
gously for E δ 1 h h j j , by Lemma f (Q) is diffeomorphism Q
As diam(P) < | : Q f j j is a -hyperbolic time for every point in
4 σ
2
∀1 ≤ j k, where h for all Q ∈ E
### 7.1. Hyperbolic Times
Note that S h j ψ(y) = S n l ψ(y) + S h jn l ψ( f n l (y)) and since h jn l
57 Q
, for every element Q ∈ E j , j = 1, . . . , k and every Q ∈ E j is the preimage of some element P ∈ P under an inverse branch of f h j
, j = 1, . . . , k. Then Leb(Q A ε,n l
(µ)) = Z f h j
(QA ε,nl (µ))
| det D f
−h j
|d Leb = Z f h j
(QA ε,nl (µ)) e S h j ψ d Leb .
• m is a hyperbolic time for all y ∈ E j we have S h jn l
ψ( f n l (y)) ≤ 0 and so S h j ψ(y) ≤
ψ( f j
we can rewrite Leb(A ε,n l (µ)) ≤
n l n l
− 1 exp n l
2 δ
2
3
n l
log L(n l ) . (7.1.5) Note that since P x
∈W nl
λ(x) = 1 then by Lemma log L(n l ) = X x
∈W nl
λ(x) n l −1 X j=
1
x )
and λ(x) :=
− X x
∈W nl
λ(x) log λ(x) . Defining the probability measures
ν n l := X x
∈W nl
λ(xx and µ n l :=
1
n
l
n l −1 X i=
( f i )
∗
(ν n l ) = X x
∈W nl
λ(xn l (x),
1 L (n l ) e S nl ψ(x)
∈W nl e S nl ψ(x)
S n l ψ(y) for all y ∈ E j
1 /4 < γ, for all j = 1, . . . , k, and by uniform
. Therefore, Leb(Q A ε,n l (µ)) ≤ Z f h j
(QA ε,nl (µ)) e S nl ψ
d
Leb ∀Q ∈ E jj = 1, . . . , k. For each Q ∈ E j such that Leb(QA ε,n l
(µ)) > 0, consider y QQA ε,n j
(µ) and let x QQ be such that f h j
(x Q ) ∈ W
(recall that elements of W are interior points of each atom of the partition P). We write W n l for the set of all points x Q for all Q ∈ E j such that Leb(Q A ε,n l (µ)) > 0 for all j = 1, . . . , k. From Proposition we know that max{diam( f l (Q)); Q ∈ E j
, l = 0, . . . n l
} < σ 1 2
(h jn l )
δ
1 /4 < σ m 2
δ
continuity of ψ, |ψ( f i (x Q
Setting L(n l ) := P x
)) − ψ( f i (y))| < δ
2
/3 for all y Q and for all
i = 0, . . . , n l
− 1. Altogether we get Leb(A ε,n l (µ) ∩ Q) ≤ Z f h j
(QA ε,nl (µ)) e S nl ψ d
Leb ≤ e S nl ψ(y Q
)+n l δ2 3
2n l δ2 3 + S nl ψ(x Q ) .
Thus, by we can write Leb(A ε,n l (µ)) ≤ n l n l −1
e 2δ2 3 n l P x
∈W nl e S nl ψ(x) .
≤ e
• 1
58 Chapter 7. Entropy Formula
we can rewrite Z n ψdµ , ν l log L(n l ) = n l n H n ). (7.1.6) l l
• n l l n l i i i li
(P w
• →+∞ →+∞
Fix (µ ) and take a subsequence n ˜µ −−−−→ ∞ such that µ −−−−→
and
1
1 i →+∞ lim log Leb(A ε,n (µ)) = lim sup log Leb(A ε,n (µ)). (7.1.7) li
n n l n i →+∞ We keep the notation n for simplicity in what follows. l n Q ε,n there exists y (µ) such that
Note that, for each x W lQ A l i i
x , y (x), f (y )) < γ for all i = 0, . . . , n Q Q l
∈ Q, hence dist( f −1. By Lemma we have that dist(σ (x), σ (y )) < δ n n Q l l 2 /3, then by the triangular inequality δ
2
dist(σ (x), σ (y )) + dist(σ (y ), µ) <
• n n n Q n Q
ε < δ , l (x), µ) ≤ dist(σ l l l
2
3 because y Q ε,n (µ) and ε < δ l 2 (M, R), Z Z Z ZA /3. Thus, for any ϕ ∈ C X X =
ϕdµ ϕdµ λ(x) ϕdσ λ(x) ϕdµ n n ll (x) − x x XWW nl nl Z Z λ(x) ϕdσ ϕdµ < δ n 2 .
≤ l (x) − x
∈W nl
, and so
Z Z Z Z ψdµ ψdµ + δ and ψdµ + δ . (7.1.8) n l ≤ ψd ˜µ ≤
### 2 Therefore, dist(µ, ˜µ) ≤ δ
2
2 n still
We now make a small perturbation of P so that the points x W l belong to the same atom of the h j −th refinement the perturbed partition that intersect A ε,n j l (µ) for each Q ∈ E and j = 1, . . . , k and l ≥ 1. Now fix δ > 0 given by the statement of Lemma with the choice d δ 1 of S and ε = δ
2 , does not
/4. Let 0 < η < min{ } (by construction, d
2
4
depend on n l ) such that for all i = 1, . . . , S and for each l ≥ 1
δ δ δ
1
1
1
• = = = + +
µ ∂B ˜x , η ˜µ ∂B ˜x , η µ ∂B ˜x , η (7.1.9) i i n i l
4
4
4 and δ δ
1
1 i i
˜µ B ˜x
• , η ˜x , < δ (7.1.10)
\B
4
4
### 7.1. Hyperbolic Times
59
where the centers are the ones from the construction in the proof of Propo- sition
Such value of η exists since the set of values of η > 0 such that some of the expressions in is positive for some i ∈ {1, . . . , S} and some l ≥ 1 is denumerable because the measures involved are probability measures. Moreover, we can also get because ˜µ is regular probability measure. Thus we may take η > 0 satisfying arbitrarily close to zero. i , δ /4 + η) : i = 1, . . . , S
1 We consider now the finite open cover ˜C = B( ˜x
as in and we analogously construction a new partition ˜P with the same number of elements of the original partition. i i Moreover, for P = {P ; 1 ≤ i S} and ˜P = { ˜P ; 1 ≤ i S}, we have by construction that ˜µ(P ) < δ for all i = 1, . . . S. Thus we have, the same i i
△ ˜P properties (1-3) in the proof of Claim replacing ξ by δ
1 /4 together
with n l (1) ˜µ(∂ ˜P) = 0 = µ (∂ ˜P) for all l ≥ 1; (2) ˜µ(P ) < δ for all i = 1, . . . S; and i i
△ ˜P h j (3) x Q Q j where ˜ Q j and f (x Q
∈ ˜ ∩ Q for all Q ∈ E ∈ ˜E n n l l ) = w ∈ ˜P P.
, ν , ν ) by definition of the ν and by con- n n n Note that H( ˜P l ) = H(P l l struction of the ˜P as a perturbation of P. Following the proof of Lemma
) there exists l ≥ 0 such that
1 n n l l 1 δ
2 H , ν ) = H , ν . n n
(P l ( ˜P l ) ≤ h( ˜P, ˜µ) + , ∀l l
n n l l
4 Since, h( ˜P, ˜µ) ≤ h(P, ˜µ) + H ˜µ ( ˜P/P) we have that by Lemma and δ 2 item (2) above, that H . Therefore,
˜µ
( ˜P/P) <
4
1 δ n l 2 δ
2 H , ν h , n
• n 2 ≤
• l
(P l ) ≤ h(P, ˜µ) + (P, µ) + τ, ∀l l
2 where the last inequality is valid by the choice of δ
2 in . Thus,
1 δ n
2
3
• l
H , ν ( f ) + . (7.1.11) n µ
(P l ) ≤ h(P, µ) + τ ≤ h τ, ∀l l
n l
2
2 < τ. and remember that δ
2
60 Chapter 7. Entropy Formula
Combining the assertions we have that
n δ l
1
2 ε,n l l l
Leb(A exp n 2 log L(n ) (µ)) ≤
• 3
n l n l
− 1 " !# Z
n l
2
1 n l l n n exp n τ + ψdµ H , ν )
• l (P l
n l l 3 n
− 1 " Z !#
n l
exp n 3τ + h ( f ) + ψdµ l µ
n l
− 1 Hence, Z
1 1 n l . log Leb(A ε,n log h µ ( f ) + l ψdµ + 3τ, ∀l l
• n n n l l l
(µ)) ≤
− 1
∗
, we conclude that, Since µ ∈ W f Z
1
1 ψdµ+3τ. 0 = lim sup log Leb(A ε,n (µ)) = lim log Leb(A ε,n µ ( f )+ l l (µ)) ≤ h
→+∞ n l n n →+∞
As τ > 0 is arbitrary, the proof of the Lemma is complete.
Remark 7.1.8. Proposition is true replacing f by an open expanding and
topologically transitive map T of a compact metric space X; Leb by a φ-conformal
∗
measure (ν) = λν, for some λ > 0 and for a continuous potential
ν with L φ φ : X → R, since we used that Leb is ψ-conformal together with a covering
lemma that clearly holds for expanding maps, recall Remark Thus, we have
Z
h ( f ) + (ν).
µ
φdµ − log λ ≥ 0, for all µ ∈ W T
In particular, this shows together with the Lemma that P top (T, φ) = log λ
### 7.2 Proof of Theorem D
To prove Theorem first consider a weak-SRB-like measure µ. Since
∗
µ ∈ W f is an expanding probability measure, there exists σ ∈ (0, 1) such
1 P n −1 j −1
that lim sup (x)) n log kD f ( f i= k ≤ log σ < 0 for µ-a.e x M.
→+∞ n
• + Thus, the Lyapunov exponents are non-negative. Hence the sum Σ (x) of the positive Lyapunov exponents of a µ−generic point x, counting multi-
1 n
• + plicities, is such that Σ (x) = lim
• R n →+∞ log | det D f n
(x)| (by the Multiplica- + Σ tive Ergodic Theorem) and
d µ =
R log | det D f|dµ = − R ψdµ by the standard Ergodic Theorem.
### 7.2. Proof of Theorem D
61
### 1 For C -systems, Ruelle’s Inequality [36] states that for any f -invariant
probability measure µ on the Borel σ-algebra of M, the corresponding R + Σ measure-theoretic entropy h µ ( f ) satisfies h µ R
d µ and consequently
( f ) ≤
h µ ( f ) + ψdµ ≤ 0.
By definition, Pesin’s Entropy Formula holds if the latter difference is R
∗
, by Proposition we have that h µ ( f )+ ψdµ = equal to zero. Since µ ∈ W f 0 which proves the first statement of Theorem
The next corollary concludes the proof of Theorem
Corollary 7.2.1. Let f
: M M be non-uniformly expanding. Then all the ergodic SRB-like probability measures are expanding probability measures.
Proof.
The assumptions on f ensure that there exists σ ∈ (0, 1) such that Leb(H(σ)) = 1. The proof uses a simple lemma.
### 1 Lemma 7.2.2. If f local diffeomorphism such that Leb(H(σ)) =
: M M is a C
−1
f
satisfies σ.
1 for some σ ∈ (0, 1), then each µ ∈ W R log k(D f) kdµ < log
1 −1 Proof.
Fix 0 < ε < − log σ small enough. Since that ϕ(x) := log kD f (x) k is
2 ∗
a continuous potential, from the definition of the weak topology in space
1 of probability measures, we deduce that there exists 0 < ε 1 < ε such
M R R that if dist(µ, ν) < ε
1 1 .
ϕdµ − ϕdν| < ε for all µ, ν ∈ M then | f ε ε , then Leb(A Let µ ∈ W 1 (µ)) > 0, take x A 1 (µ) ∩ H(σ) and consider
ν x x ) < ε
1 . Thus,
∈ pω(x) such that dist(µ, ν Z Z
−1 −1 x < ε, log k(D f ) kdµ − log k(D f ) kdν w and therefore there exists n k n (x) x and then Z Z ր ∞ so that σ k −→ ν −1 −1 n −1 X k x ε log k(D f ) kdµ ≤ log k(D f ) kdν
• 1 j
−1
= lim (x)) k log kD f ( f k + ε
→+∞
n k
j= n X −1
1 j
−1
(x)) ≤ lim sup log kD f ( f k + ε n n
→+∞ j=
√ < log σ + ε < log σ as stated.
62 Chapter 7. Entropy Formula
Going back to the proof of the Corollary, since µ is f -invariant and ergodic, then by the previous lemma and by the Ergodic Theorem n X −1 Z 1 j
−1 −1
lim (y)) n log kD f ( f k = log k(D f ) kdµ < log σ for µ − a.e y M.
→+∞ n j=
Therefore µ is an expanding measure. This finishes the proof of the coro- llary.
This completes the proof of Theorem
## Chapter 8 Ergodic weak-SRB-like measure In this chapter, we prove Corollary E on existence of ergodic weak- SRB-like measures for non-uniformly expanding local diffeomorphisms
f : M M.
### 8.1 Ergodic expanding invariant measures
1
Theorem 8.1.1. Let f local diffeomorphism. If µ is an ergodic
: M M be a C
expanding f -invariant probability measure, then Z
log(Leb(A (µ))) ε,n lim lim sup ψdµ + h ( f ). (8.1.1) µ +
ε→0 n n →+∞
Proof. Since µ is an ergodic probability measure, we have lim n σ n (x) = µ
→+∞
for µ-a.e. x M. So for µ-a.e. x M, there exists N(x) ≥ 1 such that dist(σ n (x), µ) < ε/4 ∀n N(x). Given ε > 0 and any natural value of N ≥ 1, define the set
B N n (8.1.2) := {x M : dist(σ (x), µ) < ε/4 ∀n N}.
Consider δ
1 h > 0 such that for each σ-hyperbolic time h ≥ 1 time for x, f ) maps B(x, h, δ B (x,h 1 ) diffeomorphically to the ball of radius δ 1 around h | 1 f (x).
Fix δ > 0 such that Lemma holds with ε/8 in the place of ε and fix ε if d(x, y) < ξ. ξ > 0 satisfying |ψ(x) − ψ(y)| <
### 4 Consider 0 < γ
1
< min{ξ, δ, δ /2} and let N(n, γ, b) be the minimum number of points needed to (n, γ)-span a set of µ-measure b (see ). < γ
Choose 0 < γ
1 such that
1 ε lim inf γ < γ , (8.1.3) µ
1 n log N(n, 4γ, 1/2) ≥ h ( f ) − →+∞ n 2 ∀
64 Chapter 8. Ergodic weak-SRB-like measure
and let 0 < γ
2 1 be such that ( ≤ γ )! 1 ε
2 µ > .
x log Leb(B(x, n, γ
2 Leb ( f, µ) +
∈ M : lim sup − )) ≤ h n
n
4
3
→+∞
This is possible by definition of h ( f, µ) (see ). We have implicitly
Leb
assumed that h Leb Leb ( f, µ) < ∞ here. If h ( f, µ) = ∞, then there is nothing
1 to prove since h ( f ) < h local diffeomorphism. µ top
( f ) < ∞ because f is a C Denote ( ) 1 ε log Leb(B(x, n, γ ( f, µ) + . (8.1.4)
A = x
2 Leb
∈ M : lim sup − )) ≤ h n
n
4
→+∞
Note that B N N+ N
1
⊂ B and µ(∪B ) = 1. So there exists N ≥ 1 such that
5
1
µ(B . If C then µ(C N N N N N ) ≥ := A B ) ≥ and for all x C and n N(x)
6
2
we have: (1) B(x, n, γ
2 (µ); ε,n
) ⊂ A
−(h Leb ( f,µ)+ε/2)n
(2) Leb(B(x, n, γ 2 .
)) ≥ e Note that (2) immediately follows from . Moreover, (1) holds j j
(y), f because, given y B(x, n, γ) then d( f (x)) < γ for all j = 0, . . . , n − 1. ε By Lemma we have dist(σ n (y), σ n (x)) < N , by the
8 . Since x B
triangular inequality dist(σ (y), σ (x)) + dist(σ (x), µ) < ε. n n n n ε,n (µ). (y), µ) ≤ dist(σ Therefore, y A
= For each n, let E E (2γ n n 2 ) be a maximal (n, 2γ 2 )-separated subset of points contained in C . Then S B (x, n, 4γ by maximality of E N
2 N n x ) ⊃ CE n
and so #E n , x , y then B(x, n, γ
2 n
2
≥ N(n, 4γ , 1/2). Also, given x, y E ) ∩
B (y, n, γ
2
) = ∅. Thus, X
1
1 lim inf log Leb(A ε,n log Leb(B(x, n, γ
2 )) n n (µ)) ≥ lim inf →+∞ →+∞ n n x
∈E n
1
−(h Leb ( f,µ)+ε/2)n
log #E n
≥ lim inf · e n →+∞
n
1 ε log N(n, 4γ .
2 Leb
≥ lim inf , 1/2) − h ( f, µ) − n →+∞
n
2 Thus, by we have,
1 lim inf log Leb(A (8.1.5) ε,n µ Leb n (µ)) ≥ h ( f ) − h ( f, µ) − ε.
→+∞ n
### 8.1. Ergodic expanding invariant measures
65 Moreover, since µ is expanding, then there exist θ > 0 and σ ∈ (0, 1)
such that µ-a.e. x M has positive frequency ≥ θ of σ-hyperbolic times h of f . Let ˜ M M , lim σ (x) = µ and f ) n B (x,h 2M be such that for all x ∈ ˜ | n
→+∞ h
maps B(x, h, γ ) diffeomorphically to the ball of radius γ around f (x), for
2 2 h a σ-hyperbolic time for x. j j j j ε
) since d( f x , f y
4
2 Note that |ψ( f (y)) − ψ( f (z))| < for all z B(y, h, γ ) ≤
γ
2 for all j = 0, . . . , h − 1.
Hence, since Lebesgue measure gives weight uniformly bounded away from zero to balls of fixed radius and the by uniform continuity of ψ, we obtain h h Z 0 < α(γ x , γ )) =
2
2
) ≤ Leb(B( f | det D f | d Leb B ) Z ψ ψ(x)+hε/4 (x,h2
SS
h h
)).
e d
2
≤ Leb ≤ e · Leb(B(x, h, γ B )
(x,h2 Thus,
1 0 = lim inf log α(γ h 2 ) →+∞ h ε
1
1
• S h ψ(x) + lim inf log Leb(B(x, h, γ
2 ))
≤ lim inf − h →+∞ h →+∞
h Z 4 h
ε
1 ψdσ (x) + lim sup log Leb(B(x, h, γ h 2 ))
≤ − lim sup − h h 4 − h Z →+∞ →+∞ ε
= ψdµ + h Leb ( f, x).
− 4 − R ε Therefore, h Leb ψdµ + Leb
( f, x) ≤ − for µ-a.e x M, hence h ( f, µ) ≤ R ε
4
ψdµ + and by −
### 4 Z
1 lim lim sup log Leb(A ( f ) + ψdµ, ε,n µ (µ)) ≥ h
• + ε→0 n
n →+∞ which gives the desired estimate.
1
Corollary 8.1.2. Let f local diffeomorphism. Every expanding
: M M be a C R µ such that ψdµ + h
ergodic f -invariant probability measure µ
( f ) ≥ 0 is a weak-SRB-like probability measure.
Proof. f be a expanding ergodic probability measure such that
R Let µ ∈ M
ψdµ + h µ ( f ) ≥ 0. By Theorem we have that Z log(Leb(A ε,n (µ))) lim lim sup ( f ) + µh ψdµ ≥ 0.
• + n
ε→0 n
→+∞
66 Chapter 8. Ergodic weak-SRB-like measure log(Leb(A ε,n (µ)))
Moreover, if ε < ε then A (µ). So lim sup
1 2 ε ,n ε ,n 1 (µ) ⊂ A 2 n n →+∞ log(Leb(A ε,n (µ)))
is increasing with ε > 0. Thus lim sup n ≥ 0 for all ε > 0.
→+∞ n
But since Leb is a probability measure, we conclude that log(Leb(A (µ))) ε,n lim sup = 0 ∀ε > 0. n n
→+∞ Thus, we deduce that µ is weak-SRB-like measure.
Remark 8.1.3. Theorem is true replacing f by an open expanding and
topologically transitive map T of a compact metric space X; Leb by a φ-conformal
∗
measure (ν) = λν for some λ > 0, and for a continuous potential
ν, with L φ ν (T, µ) is finite. So we get φ : X → R with the extra assumption that h Z
1 lim lim sup log ν(A ( f ) + ε,n µ (µ)) ≥ h φdµ − log λ,
• + n
ε→0 n
→+∞
since we used that Leb is ψ-conformal together with general results from Ergodic
Theory. Analogously for Corollary
1
Corollary 8.1.4. Let f local diffeomorphism. Every expanding
: M M be a C
ergodic f -invariant probability measure that satisfies Pesin’s Entropy Formula is
a weak-SRB-like probability measure.
Proof. be an expanding ergodic probability measure such
f
Let µ ∈ M R P P + + that h ( f ) = d µ, where (x) is the sum of the positive Lyapunov µ
• + exponents of a µ-generic point x counting multiplicities. R R P µ =
d
ψdµ ≤ We deduce by the Multiplicative Ergodic that − R
h µ ( f ). Therefore, h µ ( f )+
ψdµ ≥ 0 and the corollary follows from Corollary
### 8.2 Ergodic expanding weak-SRB-like measures Now we are ready to prove Corollary E.
Proof of Corollary that
all weak-SRB-like measures are equilibrium states for the potential ψ, that R
∗ is, h µ ( f ) + .
ψdµ = 0 for all µ ∈ W f We know that there exists σ ∈ (0, 1) such that Leb(H(σ)) = 1. Given
√
−1 f , by Lemma we have that σ.
µ ∈ W kdµ ≤ log
R log k(D f)
### 8.2. Ergodic expanding weak-SRB-like measures
67 By the Ergodic Decomposition Theorem, there exists A M such that
√
−1 y y σ, where µ is an
µ(A) > 0 and for all y A, R log k(D f) kdµ ≤ log ergodic component of µ.
Fix yA. By Birkhoff’s Ergodic Theorem, n X −1 Z
1 j
−1 −1
lim (y)) σ, y n log kD f ( f k = log k(D f ) kdµ ≤ log
→+∞ n j= for µ y y is an expanding probability measure.
• a.e. y M. Therefore µ R R
Since P top ( f, ψ) = 0 and h µ ( f ) + ψdµ = 0 we conclude that h µ ( f ) + y ψdµ = y 0 for all µ-a.e x M. In particular, by Corollary we con-
∗
clude that there exist y showing that there exist yA such that µ ∈ W f expanding ergodic weak-SRB-like probability measure such that satisfies
Pesin’s Entropy Formula ending the proof of Corollary
68 Chapter 8. Ergodic weak-SRB-like measure
Chapter 9
Weak-Expanding non-uniformly
expanding maps
In this chapter we reformulate Pesin’s Entropy Formula for a class of
1
weak-expanding and non-uniformly expanding maps with C regularity and prove Corollary
9.1 Weak-SRB-like, equilibrium and expanding
measures
We divide the proof of Corollary into the next two corollaries below.
Corollary 9.1.1. Let f
: M M be weak-expanding and non-uniformly ex-
panding. Then, all (necessarily existing) weak-SRB-like probability measures are
ψ-equilibrium states and, in particular, satisfy Pesin’s Entropy Formula. More- Proof.
Let f : M M be as in statement of Corollary then for every
1 n x M x n
∈ M and all v T \{0} we have lim inf →+∞ log kD f (x) · vk ≥ 0. Thus, n the Lyapunov exponents for any given f −invariant probability measure µ R
1 n + +
Σ are non-negative. Hence Σ (x) = lim
n
d µ =
→+∞ log | det D f (x)| and n R log | det D f|dµ = − R ψdµ. R
1
• + Σ
For C -systems, Ruelle’s Inequality ensures h R µ d µ then ( f ) ≤
h ( f ) + µ top ( f, ψ) = 0 and R ψdµ ≤ 0. By Proposition we have that P
∗ h µ ( f ) + .
ψdµ = 0 for all µ ∈ W f Therefore, all weak-SRB-like probability measures are ψ-equilibrium states and satisfy Pesin’s Entropy Formula. Moreover, by Corollary we conclude that there exist ergodic weak-SRB-like measures.
70 Chapter 9. Weak-Expanding non-uniformly expanding maps
Here we obtain a sufficient condition to guarantee that all ψ-equilibrium states are generalized convex combinations of weak-SRB-like measures.
Corollary 9.1.2. Let f
: M M be weak-expanding and non-uniformly expand- ψ < 0 there is no atomic weak-SRB-like probability measure. Moreover,
ing. If −1 if
D = {x M; kD f (x) k = 1} is finite and ψ < 0 then almost all ergodic ψ-equilibrium state are weak-SRB-like measures and all weak-
components of a
SRB-like probability measures µ have ergodic components µ which are expanding
x weak-SRB-like probability measure for R µ-a.e. x M\D.
Proof. Note that if ψ < 0 then . On the other hand,
f
ψdµ < 0 for all µ ∈ M f is an atomic invariant probability measure, then h ( f ) = 0. There- µ if µ ∈ M fore µ does not satisfy Pesin’s Entropy Formula and by Corollary we conclude that there is no atomic weak-SRB-like probability measure. R R
Since ψdµ < 0 and h , then a ψ- µ f ( f ) ≤ − ψdµ for all µ ∈ M R R
= equilibrium state satisfies h ( f ) + ψdµ = 0 and so h ( f ) + ψdµ 0, µ µ x x µ-a.e. x by the Ergodic Decomposition Theorem and h µ ( f ) > 0. x Hence µ is non-atomic and thus expanding because supp(µ x x
) D so
−1 x x x < 0 and µ is ergodic, for µ-a.e x. Such µ also satisfies
R log k(D f) kdµ the Entropy Formula. Therefore Corollary ensures that µ is weak- x SRB-like for µ-a.e. x M.
∗ −1
, then Consider µ ∈ W R log k(D f) kdµ < 0, otherwise, we have f supp(µ) ⊂ D and we see that this is not possible. R Because µ is a ψ-equilibrium state, then h ( f ) + ψdµ = 0. More- R µ
−1 −1
over 0 > R log k(D f) kdµ = log k(D f ) kdµ, thus by the Ergodic M
\D
Decomposition Theorem we conclude that for µ-a.e x M\D we have R
−1
ψdµ = < 0.
h µ ( f )+ x 0 (remember that P top ( f, ψ) = 0) and x x
R log k(D f) kdµ Therefore, µ-a.e x M\D have expanding ergodic components that are ψ-equilibrium states. By Corollary we deduce that µ is an weak- x SRB-like probability measure for µ-a.e x M\D and finishes the proof.
Putting Corollaries
### 9.2 Expanding Case
The Corollary and allows rewriting
1
all the results from
9.3. Proof of Corollary
71 Proof of Corollary The assumptions on f ensure that all f -invariant pro-
bability measures µ are expanding. Moreover, by Proposition and Ru- elle’s Inequality we conclude that P top ( f, ψ) = 0 and every weak-SRB-like probability measures are ψ-equilibrium states and satisfy Entropy For- mula. f satisfies Entropy Formula, then
Then, on the one hand, if µ ∈ M R
h ( f ) + ψdµ = 0. By Ergodic Decomposition Theorem we have that
µ R
=
h ( f ) + ψdµ top ( f, ψ) = 0. By Corollary
µ x x 0 for all µ-a.e x M, because P
we have that µ x is weak-SRB-like probability measure for µ-a.e x M. f x is such that its ergodic components µ are On the other hand, if µ ∈ M R
= µ x ( f )+ ψdµ weak-SRB-like probability measures for µ-a.e. x M, then h x 0 for µ-a.e x M. Thus, by the Ergodic Decomposition Theorem we have that Z Z Z ! Z
ψdµ µ(x) = µ(x) = h x d h µ ( f ) d µ ( f ). − ψdµ = − x Assume now that µ is the unique weak-SRB-like probability measure.
By item G of Theorem 1 in we have that µ is SRB and Leb(B(µ)) = 1.
See the proof of Corollary As observed in , the SRB-like condition is a sufficient but not neces- sary condition for a measure µ to be an equilibrium state for the potential
ψ, because it may exist a non-ergodic invariant measure µ << Leb that is neither SRB nor SRB-like (see ). In such a case µ satisfies Pesin’s Entropy Formula, as stated the following lemma.
1
µ be a non-ergodic
Corollary 9.2.1. Let f -expanding map. Let
: M M be a C
f -invariant probability such that µ << Leb. Then µ satisfies Pesin’s Entropy
Formula. Proof. See Corollary 2.6 in .
9.3 Proof of Corollary
1 Proof of Corollary Let f f in the C -topology, where f n n
−−−−→ , f : M n →+∞
### 1 M are C
n a
• expanding maps for all n ≥ 1. For each n ≥ 1 consider µ
∗
weak-SRB-like measures associated to f and let µ be a weak limit point: n µ = lim µ j n .
→+∞ j
72 Chapter 9. Weak-Expanding non-uniformly expanding maps n
Fix P a generating partition for every f j for all j ≥ 1 and such that µ(∂P) = 0.
1
1 This is possible, since f is C -expanding and f -topology n n jf in C 1 and f is also C -expanding.
By Kolmogorov-Sinai Theorem this implies that h ( f ) and µ n n n ) = h(P, µ
h µ
( f ) = h(P, µ), that is,
1 k k
1
h ( f ) = inf H , µ ) and h ( f ) = inf H , µ)
µ n n µ n j j (P n j (P
k k
j
≥1 ≥1
k k k
, µ ) converge n Since µ gives zero measure to the boundary of P then H(P n j k j to H(P , µ) as j → ∞. Furthermore, for every ε > 0 there is n ≥ 1 such that
1 n
1 n , µ
h µ ( f n H n H µ ( f ) + 2ε.
j n j , µ) + ε ≤ h n j ) ≤ (P j ) ≤ (P
n n R
= By Corollary h µ ( f ) + ψ d µ is weak- n j n j n j n j n j 0 for all j ≥ 0, since µ
= SRB-like probability measure and ψ n n j − log | det D f j |.
Since ψ n j R R → ψ in the topology of uniform converge, we have that R ψ d µ ψdµ. By Ruelle’s inequality, h ν ( f ) + n j n j
→ ψdν ≤ 0 for any
f -invariant probability measure ν on the Borel σ-algebra of M. Thus, Z Z !
= µ µ n n n ( f ) + h ( f ) + ψ d µ 0. 0 ≥ h ψdµ ≥ lim sup n n
→+∞
This shows that µ satisfies Pesin’s Entropy Formula, is a ψ-equilibrium state since P top ( f, ψ) = 0 and by Corollary its ergodic components µ are x weak-SRB-like probability measures for µ-a.e x M.
### 9.4 Proof of Corollary C Finally we prove Corollary C.
Proof of Corollary
T
Theorem Let µ ∈ M such that Z Z Z !
h (T) + φdµ = P top (T, φ) = h (T) + φdµ d µ(x) µ µ x x R R
= we also have h (T) + φdµ (T, φ) and so h (T) + φdµ P (T, φ) µ x top µ x top xP x for µ-a.e x. Now from Theorem since h ν (T, µ x ) < ∞
### 9.4. Proof of Corollary C
73
for µ-a.e. x X we get Z
1 lim lim sup log ν(A (µ (T) + φdµ ε,n x µ x )) ≥ h x − log λ = 0
• + ε→0 n
n →+∞ ∗
and then we conclude that µ (ν) following the same argument in the x ∈ W T proof of Corollary
Assume now that µ is the unique φ-equilibrium state. By Proposition follows that µ is ν-SRB and ν(B(µ)) = 1. is
1 r r
Let now V be a small neighborhood of µ in M . Since {K (φ)} r >0 r (φ) we have that there exists decreasing with r and {µ} = K (φ) = ∩ K
∗ r r r (φ) is weak
> 0 such that K (φ) ⊂ V for all 0 < r < r . Since K r (φ) = compact (by upper semicontinuity of the metric entropy) we have K T ε ε n
∗ ε>0 K r (φ), where K r µ ∈ M ; dist(µ, K (φ)) ≤ ε (φ) = T r o with the weak ε distance defined in . Consider 0 < ε < r
, such that K r (φ) ⊂ V. By Proposition there exists n
≥ 1 and κ = κ(ε, r) > 0 such that ε n n
1
1
ν({x X : σ (x) ∈ M \V}) ≤ ν({x X : σ (x) ∈ M \K r (φ)}) n (ε−r) = n r ,
ν({x X; dist(σ (x), K (φ)) ≥ ε}) < κe
1
. Thus lim sup n
1
for all n n log ν({x X : σ (x) ∈ M \V}) < ε−r. As n n
→+∞
1
ε > 0 can be taken arbitrary small, we conclude that lim sup n n log ν({x
→+∞
=
X : σ n 1 and r
I r (x) ∈ M \V}) < −r for all 0 < r < r (V) := sup{r >
0; K (φ) ⊂ V}. Therefore
1 lim sup n
1 log ν({x X : σ (x) ∈ M \V}) ≤ −I(V). n n →+∞ n
1 This shows that the probability ν({x X : σ (x) ∈ M \V}) decreases exponentially fast with n at a rate that depends on the “size” of V.
The assumptions on µ are the same as in Corollary with ν = Leb and φ = ψ, so the upper large deviations statement of this corollary follows with the same proof.
74 Chapter 9. Weak-Expanding non-uniformly expanding maps
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Universidade Federal da Bahia-UFBA Instituto de Matemática / Colegiado da Pós-Graduação em Matemática
Av. Adhemar de Barros, s/n, Campus de Ondina, Salvador-BA CEP: 40170 -110 www.pgmat.ufba.br | 2020-07-04 14:04:56 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9231605529785156, "perplexity": 11693.420836955534}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886178.40/warc/CC-MAIN-20200704135515-20200704165515-00368.warc.gz"} |
https://projecteuclid.org/euclid.aop/1513069267 | ## The Annals of Probability
### Stochastic heat equation with rough dependence in space
#### Abstract
This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$ in the space variable. The existence and uniqueness of the solution $u$ are proved assuming the nonlinear coefficient $\sigma(u)$ is differentiable with a Lipschitz derivative and $\sigma(0)=0$.
#### Article information
Source
Ann. Probab., Volume 45, Number 6B (2017), 4561-4616.
Dates
Revised: December 2016
First available in Project Euclid: 12 December 2017
Permanent link to this document
https://projecteuclid.org/euclid.aop/1513069267
Digital Object Identifier
doi:10.1214/16-AOP1172
Mathematical Reviews number (MathSciNet)
MR3737918
Zentralblatt MATH identifier
06838127
#### Citation
Hu, Yaozhong; Huang, Jingyu; Lê, Khoa; Nualart, David; Tindel, Samy. Stochastic heat equation with rough dependence in space. Ann. Probab. 45 (2017), no. 6B, 4561--4616. doi:10.1214/16-AOP1172. https://projecteuclid.org/euclid.aop/1513069267
#### References
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• [2] Balan, R. M., Jolis, M. and Quer-Sardanyons, L. (2015). SPDEs with affine multiplicative fractional noise in space with index $H<1/2$. Electron. J. Probab. 20 36 pp.
• [3] Brzeźniak, Z. and Peszat, S. (1999). Space–time continuous solutions to SPDE’s driven by a homogeneous Wiener process. Studia Math. 137 261–299.
• [4] Dalang, R. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 29 pp.
• [5] Dalang, R. C. (2001). Corrections to: “Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s” [Electron J. Probab. 4 (1999) 29 pp. (electronic); MR1684157 (2000b:60132)]. Electron. J. Probab. 6 5 pp.
• [6] Dalang, R. C. and Quer-Sardanyons, L. (2011). Stochastic integrals for spde’s: A comparison. Expo. Math. 29 67–109.
• [7] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
• [8] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1981). Higher Transcendental Functions. Vol. III. Based on Notes Left by Harry Bateman. Robert E. Krieger Publishing, Melbourne, FL.
• [9] Gyöngy, I. (1998). Existence and uniqueness results for semilinear stochastic partial differential equations. Stochastic Process. Appl. 73 271–299.
• [10] Gyöngy, I. and Krylov, N. (1996). Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Related Fields 105 143–158.
• [11] Gyöngy, I. and Nualart, D. (1999). On the stochastic Burgers’ equation in the real line. Ann. Probab. 27 782–802.
• [12] Hanche-Olsen, H. and Holden, H. (2010). The Kolmogorov–Riesz compactness theorem. Expo. Math. 28 385–394.
• [13] Peszat, S. and Zabczyk, J. (1997). Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process. Appl. 72 187–204.
• [14] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 251–291.
• [15] Samko, S., Kilbas, A. and Marichev, O. (1993). Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science Publishers, Yverdon.
• [16] Zorko, C. T. (1986). Morrey space. Proc. Amer. Math. Soc. 98 586–592. | 2019-06-17 02:50:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3938225209712982, "perplexity": 3818.947265585418}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998369.29/warc/CC-MAIN-20190617022938-20190617044938-00241.warc.gz"} |
https://gre.kmf.com/question/all/0?keyword=&page=50 | #### 题目列表
In an attempt to _____ voters to support her, the incumbent politician beguilingly greeted a room full of constituents and pledged to lower taxes-even though she had only ever done the opposite while in office.
After a series of storms, the once arid landscape became _____ for the first time in many months.
Although he received many visitors, the _____ old man shooed them away after only a few minutes.
Eileen used to be a picky eater, but since a new complex of fine dining and ethnic restaurants opened in her neighborhood, she has become quite _____.
The SWAT team entered the dark building on high alert, their guns drawn and their night vision goggles on; each agents eyes and ears were attuned to the slightest disturbance in the _____ recesses of the rooms.
Under no delusions about his actual financial situation, the mans desire to present a frugal picture to his friends and avoid being labeled _____ caused him to go to such an extreme that he ended up being called a Scrooge.
The _____ pirate plundered every trade ship that came near his own ship; it was almost as if he could never loot or pillage enough to satisfy his craving for gold and jewels.
The homicide detectives didnt truly understand the _____ of the criminal until they found the secret hideout where he stored his instruments of torture and carried out his heinous acts.
While blood and human sacrifices performed to mollify the gods were ubiquitous in ancient cultures, the Mayans propensity for sacrificing prisoners from neighboring tribes _____ all the other tribes.
In contrast to the stark facades of their surviving ruins, medieval castles were depicted in contemporary tapestries as _____ with colorful banners and pennants.
The young minister was startled to learn that his parishioners considered him _____; he had been unaware that his message was being undermined by his sanctimonious and self-righteous tone.
Many senior faculty members who were accustomed to being addressed in a more collegial and egalitarian manner were alienated by the _____ tone of the new department chairs introductory remarks.
Meant to demonstrate an air of sophistication and worldliness, the comments that Hannah made upon exiting the building served only to emphasize her _____ mentality and reinforce Mr. Hassans conviction that her dismissal was justified because she was not yet mature enough for the corporate world.
Sylvia Plath was not as _____ a poet as was her husband Ted Hughes, having produced just two volumes of poetry in her short lifespan.
The unfounded fear that some children, and even adults, have of the circus clown is rather ironic considering that he is meant to be _____ character who invokes laughter and enjoyment.
The photographer _____ posed the bride for her portrait, carefully adjusting each fold of her dress and each curl of her hair before taking the picture.
After finishing the editing workshop, the writers found that they were able to give each other _____ comments, instead of the general and unhelpful suggestions they had been making beforehand.
Many admirers of art _____ the beauty of Jackson Pollocks paintings, while others disparage the splatters of color as simplistic.
Expecting Tom to protest the poor grade on his psychology paper, the professor was disheartened when he _____ tossed it in his bag and left the room.
The stock market having plunged drastically, the investors _____ mood on the trading floor seemed incongruous.
25000 +道题目
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https://motls.blogspot.com/2006/05/clifford-johnson-accepts-jesus-christ.html?m=1 | ## Monday, May 01, 2006
### Clifford Johnson accepts Jesus Christ (almost)
Clifford Johnson from Cosmic Variance was
He has learned that the Christians can be great people and in principle, they could even become scientists. One can talk to them, smile at them, and respect them as human beings. They can write and they do write lovelier articles about Clifford than the left-wing atheists. Such an experience makes a difference.
Indeed, Clifford
and it went wonderfully: singing with the piano, clapping, preaching. Indeed, Clifford has found out that the Christians can be more human and more friendly people than the officers of PC police. Moreover, some verses from the Bible resonated with the message he wanted to give.
Of course, the idea of Clifford Johnson in the church was rather controversial at Cosmic Variance. Religion is viewed as the source of all lies in the world by Sean Carroll. He emphasizes that religion is not necessarily evil: it is just false. And one must do everything to fight it; see, for example, these 172 pages.
More seriously, there are some scientifically strange things that many Christians believe. But there are also many scientifically strange things that left-wingers such as Sean Carroll believe. I have discussed the fact that the color or the amount of religion in some ideas cannot universally predict their scientific strength.
Moreover, I still view religion as the basis of moral principles in our society. Yes, I am primarily talking about the judeo-Christian tradition. But more generally, religions showed their power to give our lives a direction. Religions can't provide us with the final word about difficult scientific questions; but they have been and they are a part of the transformation of skillful monkeys to human beings.
Science and religion were born into the same cradle. Their diversification only occured when the human civilization made many other important steps.
Although it has always been clear that I would remain an infidel, the Christian environment is something that many of us are able to live with. If we had to spend years with extraterrestrial aliens, it could be difficult - but if they were Christians, things could simplify dramatically. ;-)
#### 1 comment:
1. Your post is interusting. The idea that religion and science ( evolution I'm presuming ) can exist in relative and debatable harmony is hard to grasp. Either were really smart monkeys without a purpose or reason to take another breath, or we have a God who designed and built us for a reason higher than just "kickin it" on earth for a while. And might I add, religion is the "learning or practising" witch does nothing for the soul. Worshiping is the devoted following and aiding in Gods plans and pleasure. | 2021-08-06 01:32:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.39916640520095825, "perplexity": 2916.3738981474485}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046152085.13/warc/CC-MAIN-20210805224801-20210806014801-00069.warc.gz"} |
https://scientistcafe.com/ids/regression-and-decision-tree-basic.html | ## 11.3 Regression and Decision Tree Basic
### 11.3.1 Regression Tree
Let’s look at the process of building a regression tree(Gareth James and Tibshirani 2015). There are two steps:
1. Divide predictors space — that is a set of possible values of $$X_1,X_2,\dots,X_p$$— into $$J$$ distinct and non-overlapping regions: $$R_1,R_2,\dots,R_J$$
2. For every observation that falls into the region $$R_j$$, the prediction in the mean of the response values for the training observations in $$R_j$$
Let’s go back to the previous baby example. If we use the variable “Gender” to divide the observations, we obtain two regions $$R_1$$ (female) and $$R_2$$ (male).
y1=c(156,167,165,163,160,170,160)
y2=c(172,180,176)
The sample average for region $$R_1$$ is 163, for region $$R_2$$ is 176. For a new observation, if it is female, the model predicts the height to be 163, if it is male, the predicted height is 176. Calculating the mean is easy. Let’s look at the first step in more detail which is to divide the space into $$R_1, R_2, \dots, R_J$$.
In theory, the region can be any shape. However, to simplify the problem, we divide the predictor space into high-dimensional rectangles. The goal is to divide the space in a way that minimize RSS. Practically, it is nearly impossible to consider all possible partitions of the feature space. So we use an approach named recursive binary splitting, a top-down , greedy algorithm. The process starts from the top of the tree (root node) and then successively splits the predictor space. Each split produces two branches (hence binary). At each step of the process, it chooses the best split at that particular step, rather than looking ahead and picking a split that leads to a better tree in general (hence greedy).
$R_{1}(j, s)=\{X|X_j<s\}\ and\ R_{2}(j, s)=\{X|X_j\geq s\}$
Calculate the RSS decrease after the split. For different $$(j,s)$$, search for the combination that minimizes the RSS, that is to minimize the following:
$\Sigma_{i:x_i\in R_1(j,s)}(y_i-\hat{y}_{R_{1}})^2+\Sigma_{i:x_i\in R_2(j,s)}(y_i-\hat{y}_{R_{2}})^2$
where $$\hat{y}_{R_1}$$ is the mean of all samples in $$R_1$$, $$\hat{y}_{R_2}$$ is the mean of samples in $$R_2$$. It can be quick to optimize the equation above. Especially when $$p$$ is not too large.
Next, we continue to search for the split that optimize the RSS. Note that the optimization is limited in the sub-region. The process keeps going until a stopping criterion is reaches. For example, continue until no region contains more than 5 samples or the RSS decreases less than 1%. The process is like a tree growing.
There are multiple R packages for building regression tree, such as ctree, rpart and tree. rpart is widely used for building single tree. The split is based on CART algorithm, using rpart() function from the package. There are some parameters that controls the model fitting, such as the minimum number of observations that must exist in a node in order for a split to be attempted, the minimum number of observations in any leaf node etc. You can can set those parameter using rpart.control.
A more convenient way is to use train() function in caret package. The package can call rpart() function and train the model through cross-validation. In this case, the most common parameters are cp (complexity parameter) and maxdepth (the maximum depth of any node of the final tree). To tune the complexity parameter, set method = "rpart". To tune the maximum tree depth, set method = "rpart2" :
library(rpart)
# data cleaning: delete wrong observations
dat <- subset(dat, store_exp > 0 & online_exp > 0)
# use the 10 survey questions as predictors
trainx <- dat[, grep("Q", names(dat))]
# use the sum of store and online expenditure as response variable
# total expenditure = store expenditure + online expenditure
trainy <- dat$store_exp + dat$online_exp
set.seed(100)
rpartTune <- train(trainx, trainy, method = "rpart2", tuneLength = 10,
trControl = trainControl(method = "cv"))
plot(rpartTune)
RMSE doesn’t change much when the maximum is larger than 2. So we set the maximum depth to be 2 and refit the model:
rpartTree <- rpart(trainy ~ ., data = trainx, maxdepth = 2)
You can check the result using print():
print(rpartTree)
## n= 999
##
## node), split, n, deviance, yval
## * denotes terminal node
##
## 1) root 999 1.581e+10 3479.0
## 2) Q3< 3.5 799 2.374e+09 1819.0
## 4) Q5< 1.5 250 3.534e+06 705.2 *
## 5) Q5>=1.5 549 1.919e+09 2326.0 *
## 3) Q3>=3.5 200 2.436e+09 10110.0 *
You can see that the final model picks Q3 and Q5 to predict total expenditure. To visualize the tree, you can convert rpart object to party object using partykit then use plot() function:
library(partykit)
rpartTree2 <- as.party(rpartTree)
plot(rpartTree2)
### 11.3.2 Decision Tree
Similar to a regression tree, the goal of a classification tree is to stratifying the predictor space into a number of sub-regions that are more homogeneous. The difference is that a classification tree is used to predict a categorical response rather than a continuous one. For a classification tree, the prediction is the most commonly occurring class of training observations in the region to which an observation belongs. The splitting criteria for a classification tree are different. The most common criteria are entropy and Gini impurity. CART uses Gini impurity and C4.5 uses entropy.
When the predictor is continuous, the splitting process is straightforward. When the predictor is categorical, the process can take different approaches:
1. Keep the variable as categorical and group some categories on either side of the split. In this way, the model can make more dynamic splits but must treat the categorical predictor as an ordered set of bits.
2. Encode the categorical variable to be a set of binary dummy variables. In this way, the information in the categorical variable is decomposed to independent bits of information. The model considers these dummy variables separately and evaluates for each of these on one split point (because there are only two possible values: 0/1).
When fitting tree models, people need to choose the way to treat categorical predictors. If you know some of the categories have higher predictability, then the first approach may be better. In the rest of this section, we will build tree models using the above two approaches and compare them.
Let’s build a classification model to identify the gender of the customer:
library(caret)
library(pROC)
# use the 10 survey questions as predictors
trainx1 <- dat[, grep("Q", names(dat))]
# use two ways to treat categorical predictor
# trainx1: use approach 1, without encoding
trainx1$segment <- dat$segment
# trainx2: use approach 2, encode it to a set of dummy variables
dumMod<-dummyVars(~.,
data=trainx1,
# Combine the previous variable name and the level name
# as the new dummy variable name
levelsOnly=F)
trainx2 <- predict(dumMod,trainx1)
# the response variable is gender
trainy <- dat$gender Here we use train() function in caret package to call rpart to build the model. We can compare the model results from the two approaches: CART 1000 samples 11 predictor 2 classes: 'Female', 'Male' No pre-processing Resampling: Cross-Validated (10 fold) Summary of sample sizes: 901, 899, 900, 900, 901, 900, ... Resampling results across tuning parameters: cp ROC Sens Spec 0.00000 0.6937 0.6517 0.6884 0.00835 0.7026 0.6119 0.7355 0.01670 0.6852 0.5324 0.8205 0.02505 0.6803 0.5107 0.8498 0.03340 0.6803 0.5107 0.8498 ...... 0.23380 0.6341 0.5936 0.6745 0.24215 0.5556 0.7873 0.3240 ROC was used to select the optimal model using the largest value. The final value used for the model was cp = 0.00835. The above keeps the variable as categorical without encoding. Here cp is the complexity parameter. It is used to decide when to stop growing the tree. cp = 0.01 means the algorithm only keeps the split that improves the corresponding metric by more than 0.01. Next, let’s encode the categorical variable to be a set of dummy variables and fit the model again: rpartTune2 <- caret::train(trainx2, trainy, method = "rpart", tuneLength = 30, metric = "ROC", trControl = trainControl(method = "cv", summaryFunction = twoClassSummary, classProbs = TRUE, savePredictions = TRUE)) Compare the results of the two approaches. rpartRoc <- pROC::roc(response = rpartTune1$pred$obs, predictor = rpartTune1$pred$Female, levels = rev(levels(rpartTune1$pred$obs))) ## Setting direction: controls < cases rpartFactorRoc <- roc(response = rpartTune2$pred$obs, predictor = rpartTune2$pred$Female, levels = rev(levels(rpartTune1$pred\$obs)))
## Setting direction: controls < cases
plot.roc(rpartRoc,
type = "s",
print.thres = c(.5),
print.thres.pch = 3,
print.thres.pattern = "",
print.thres.cex = 1.2,
col = "red", legacy.axes = TRUE,
print.thres.col = "red")
plot.roc(rpartFactorRoc,
type = "s",
print.thres = c(.5),
print.thres.pch = 16, legacy.axes = TRUE,
print.thres.pattern = "",
print.thres.cex = 1.2)
legend(.75, .2,
c("Grouped Categories", "Independent Categories"),
lwd = c(1, 1),
col = c("black", "red"),
pch = c(16, 3))
You can see that in this case, the two approaches lead to similar model performance.
Single tree is straightforward and easy to interpret but it has problems:
1. Low accuracy
2. Unstable: little change in the training data leads to very different trees.
One way to overcome those is to use an ensemble of trees. In the rest of this chapter, we will introduce three ensemble methods (combine many models’ predictions): bagging tree, random forest, and gradient boosted machine. Those ensemble approaches have significant higher accuracy and stability. However, it comes with the cost of interpretability.
### References
Gareth James, Trevor Hastie, Daniela Witten, and Robert Tibshirani. 2015. An Introduction to Statistical Learning. 6th ed. Springer. | 2020-03-28 09:09:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6199004650115967, "perplexity": 1577.6101268985278}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370490497.6/warc/CC-MAIN-20200328074047-20200328104047-00429.warc.gz"} |
https://math.stackexchange.com/questions/704759/how-to-proceed-this-computation-with-differential-forms | # How to proceed this computation with differential forms?
I've been studying Spivak's differential geometry book and he defines the exterior derivative of $\omega \in \Omega^k(M)$ in a coordinate system $(x,U)$ by
$$d\omega = d\omega_{i_1\cdots i_k}\wedge dx^{i_1}\wedge\cdots \wedge dx^{i_k}.$$
He then say we must show this definition is not dependent on the coordinate system and he says one way is to compute directly. He did in another way but I'm trying to show by direct computation.
What I did was first to simplify the notation denote the derivative of $\omega$ in the $x$ coordinates by $d(\omega,x)$ and on the $y$ coordinates $d(\omega,y)$. We then want to show that $d(\omega, x)=d(\omega,y)$.
We have then
$$d(\omega,x) = \dfrac{\partial \omega_{i_1\cdots i_k}}{\partial x^\alpha}dx^\alpha \wedge dx^{i_1}\wedge\cdots \wedge dx^{i_k},$$
now by the chain rule and by computing the differentials in terms of $y$ coordinates we have
$$d(\omega,x) = \dfrac{\partial \omega_{i_1\cdots i_k}}{\partial y^\beta}\dfrac{\partial y^\beta}{\partial x^\alpha}dx^\alpha \wedge \dfrac{\partial x^{i_1}}{\partial y^{j_1}}dy^{j_1}\wedge\cdots\wedge \dfrac{\partial x^{i_k}}{\partial y^{j_k}}dy^{j_k},$$
and this is the same as
$$d(\omega, x) = \left(\dfrac{\partial x^{i_1}}{\partial y^{j_1}}\cdots \dfrac{\partial x^{i_k}}{\partial y^{j_k}}\right)\dfrac{\partial \omega_{i_1\cdots i_k}}{\partial y^\beta}dy^\beta \wedge dy^{j_1}\wedge\cdots\wedge dy^{j_k},$$
now I can't get rid of those extra terms on the left and with them the equality doesn't hold.
How do I proceed with this? I couldn't think of any ways to proceeding any further.
Thanks very much in advance.
• At the very least, you appear to be writing "$\omega_{i_1\dots i_k}$" for the components of $\omega$ in both coordinate systems. :) – Andrew D. Hwang Mar 9 '14 at 2:01
• Thanks for the help @user86418, but I'm missing where the transformation for the components should come. Indeed, the coefficients remaining are exactly those of the transformation of the components. I just used the chain rule to pass the derivative from one coordinate system to the other since $\omega_{i_1\cdots i_k}$ are just $0$-forms. Where this transformation should be done? Thanks very much again! – user1620696 Mar 9 '14 at 2:21
Let me write $\alpha = (\alpha_1, \cdots, \alpha_k)$ and $I =(i_1, \cdots, i_k)$. Then using
$$w = \sum_I w_I dx^{i_1}\wedge \cdots \wedge dx^{i_k} = \sum_\alpha w_\alpha dy^{\alpha_1}\wedge \cdots \wedge dy^{\alpha_k}$$
and $dy^{\alpha_j} = \frac{\partial y^{\alpha_j}}{\partial x^i} dx^i$, we have
$$w_I = \sum_{\alpha} w_\alpha \frac{\partial y^{\alpha_1}}{\partial x^{i_1}}\cdots \frac{\partial y^{\alpha_k}}{\partial x^{i_k}}\ .$$
Now we compute (using the $x$ coordinate)
$$dw = \sum_{I,j} \frac{\partial w_I}{\partial x^j} dx^j \wedge dx^{i_1}\wedge \cdots \wedge dx^{i_k}$$
$$= \sum_{I,j} \frac{\partial }{\partial x^j}\bigg( \sum_\alpha w_\alpha \frac{\partial y^{\alpha_1}}{\partial x^{i_1}}\cdots \frac{\partial y^{\alpha_k}}{\partial x^{i_k}} \bigg)dx^j \wedge dx^{i_1}\wedge \cdots \wedge dx^{i_k}$$
Now this is a bit messy, but actually we need only to differentiate the first term in the bracket, because $$\frac{\partial ^2 y^{\alpha_k}}{\partial x^j \partial x^{i_k}} \text{is symmetric in j, i_k, and }\ \ \ dx^j \wedge dx^{i_k} \text{ is antisymmetric in j, i_k. }$$
So the sum would be zero. Thus
$$dw = \sum_{\alpha, I,j} \frac{\partial w_\alpha}{\partial x^j} \frac{\partial y^{\alpha_1}}{\partial x^{i_1}}\cdots \frac{\partial y^{\alpha_k}}{\partial x^{i_k}} \ dx^j \wedge dx^{i_1}\wedge \cdots \wedge dx^{i_k}$$
$$= \sum_{\alpha, I, j, \beta} \frac{\partial w_\alpha}{\partial y^\beta}\frac{\partial y^\beta}{\partial x^j} \frac{\partial y^{\alpha_1}}{\partial x^{i_1}}\cdots \frac{\partial y^{\alpha_k}}{\partial x^{i_k}} \ dx^j \wedge dx^{i_1}\wedge \cdots \wedge dx^{i_k}$$
$$= \sum_{\alpha, \beta} \frac{\partial w_\alpha}{\partial y^\beta} dy^\beta \wedge dy^{\alpha_1}\wedge \cdots\wedge dy^{\alpha_k}= \sum_\alpha dw_\alpha \wedge dy^{\alpha_1}\wedge \cdots\wedge dy^{\alpha_k}$$ | 2019-06-17 01:01:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9766098856925964, "perplexity": 169.45670417844482}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998339.2/warc/CC-MAIN-20190617002911-20190617024911-00045.warc.gz"} |
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• # Artykuł - szczegóły
## Fundamenta Mathematicae
2008 | 198 | 3 | 191-215
## Minimal actions of homeomorphism groups
EN
### Abstrakty
EN
Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $Φ ⊂ 2^{2^{X}}$, the space of maximal chains in $2^{X}$, equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_{Homeo(X)}$, the universal minimal space of Homeo(X), is not transitive (improving a result of Uspenskij). Additionally for X as above with dim(X) ≥ 3 we characterize all the minimal subspaces of V(M), the space of closed subsets of M, and show that M is the only minimal subspace of Φ. For dim(X) ≥ 3, we also show that (M,Homeo(X)) is strongly proximal.
191-215
wydano
2008
### Twórcy
autor
• Institute of Mathematics, The Hebrew University, Jerusalem, Israel | 2022-12-09 06:22:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8199781179428101, "perplexity": 1797.177690397303}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711390.55/warc/CC-MAIN-20221209043931-20221209073931-00177.warc.gz"} |
https://www.nature.com/articles/s41467-022-28213-y?utm_source=xmol&utm_medium=affiliate&utm_content=meta&utm_campaign=DDCN_1_GL01_metadata&error=cookies_not_supported&code=c635f89a-37e7-4ef2-be78-7f30c372c27a | ## Introduction
Energy transfer in biological processes is achieved through the transfer of electrons and protons1,2,3. In essence, the transfer of energy-carrying electrons and protons is the basis of life processes, and it is inseparable from the electron transport chain4,5,6. Organisms can pass messages between biomolecules via electron transfer, thereby achieving a variety of life processes7,8,9,10. After signaling molecule binding, receptor or adapter protein kinases become activated by phosphorylation, thereby turning on downstream signaling pathways11,12,13. Activation of signaling pathways is directly related to electron transfer between the signaling and receptor molecules since the phosphorylation of receptor molecules is accompanied by the formation of chemical bonds and further electron transfer5,14. Taking tumor development as an example, compared with normal cells, tumor cells achieve extremely active electron transfer to support their proliferation, differentiation, and migration15,16,17,18. If the monitoring of tumor proliferation-associated signal pathway activity could be achieved, it would be possible to evaluate therapeutic behavior on this basis. This possibility has motivated the need for developing novel strategies to implement in vivo monitoring of electron transfer processes that occur in signaling pathways.
There is a variety of signaling pathways in a cell that switch on and off nimbly, and influence each other constantly, thus forming a complex network13,19,20. Because the activity of these signaling pathways changes rapidly, a rapid, sensitive monitor of dynamic changes in signal transduction is required. Techniques have been well developed to measure electron transfer between biological molecules, such as nuclear magnetic resonance spectroscopy and electrochemical methods21,22,23. Optical methods are mostly characterized by high temporal and spatial resolution, non-contact detection, and high-sensitive visualization based on machine vision11,24,25,26,27,28. Specifically, the development of quantum biological electron tunneling spectroscopy enables imaging of electron transfer in live cells, while it is not easily applicable for in vivo testing29. Hence, if an optical probe is designed to respond to electron transfer, it would be possible to realize dynamic monitoring of electron transfer during target signal transduction in vivo30,31.
Here we designed a highly sensitive and specific nanoprobe that enables in vivo imaging of electronic transfer. Specifically, by imaging electron transfer in biochemical reactions, we determined that the change of probe signal with electron transfer was due to the electron transfer from biochemical reactions into the probe material. At the cellular level, the activation of EGFR-related signaling pathways was visualized by electron transfer-triggered imaging. Additionally, we further developed this strategy to enable monitoring of disease processes during the dynamic procedure of tumor treatment. EGFR activation was positively correlated with tumor progression, thus enabling the evaluation of therapeutic progress17,18. Since there are diverse signaling pathways in life activities, this electron transfer-triggered imaging approach provides a general platform, not only for understanding molecular mechanisms in various biological processes but also for promoting disease therapies and drug evaluation.
## Results
### Design of ETTE nanoprobes for electron transfer-triggered imaging
To achieve electron transfer-triggered imaging of target signaling pathways in vivo, we designed an electron transfer-triggered emission nanoprobe (ETTE nanoprobe) containing persistent luminescence nanoparticles (PLNPs) for generating afterglow signals32 and ferrocene-DNA polymer chains for directional electron transfer33,34,35. As schematically illustrated in Fig. 1a, the strategy for electron transfer-triggered emission imaging is based on the electron transfer chain between our ETTE nanoprobe and the ligand of target molecule. When ligands of the target molecules are present, the electron can transfer from the ligand to the persistent luminescence nanoparticle through the ferrocene-DNA polymer chain, leading to a change of afterglow signal (Fig. 1b). As a model, the in vivo imaging of electron transfer in EGFR signaling pathways was assessed in a mouse with lung cancer. The map of EGFR-related signaling pathway activity in mouse organs was observed using ETTE nanoprobe (Fig. 1c). The afterglow signal in the tumor region was significantly higher than that in other organs, showing that the tumor region exhibited the most active EGFR-related signaling pathway. For the therapeutic group, the volume of tumor decreased sharply, resulting in a declined afterglow signal, thus indicating the decrease of EGFR-related signal pathway activity.
### Afterglow imaging and quantification of electron transfer
It has been reported that the molecular structure or charge distribution of a luminescent probe can be influenced by environmental factors, causing a change of luminescence signals36,37,38. The afterglow signal of persistent luminescence materials was investigated to establish whether it would be affected by chemical reactions in the surrounding environment as well. As the enzyme cofactor, flavin adenine dinucleotide (FAD) is a crucial molecule in core metabolic pathways and reflects the growth state of the cells39,40. Taking the catalytic reaction of FAD-dependent glucose oxidase (GOD) and glucose as an example, the afterglow signals of four common persistent luminescence nanoparticles were detected in a quantitative catalytic reaction environment. The afterglow intensities of these materials were moderate to low in the GOD solution and gradually increased when glucose was added to the solution (Fig. 2a). As a result, the afterglow signal of all these persistent luminescence materials varies with the quantitative catalytic reaction of GOD and glucose, confirming that the afterglow signal can be used to characterize the process of biochemical reactions. The 3D map of the afterglow signal (Fig. 2a below) clearly demonstrates that among the four common persistent luminescence materials, ZnGa2O4 is the most affected, indicating its high sensitivity to electron transfer. Therefore, ZnGa2O4 was selected as the system that balances the background intensity with the retention of sensitivity.
The electrochemical activity of ZnGa2O4 was further characterized in different concentrations of glucose solution to determine the relationship between the afterglow signal and the electron transfer. ZnGa2O4 and GOD were sequentially coated on the electrode surface, and the electrode was employed for cyclic voltammetry in a solution containing different concentrations of glucose (Fig. 2b, c). While the GOD coating resulted in total passivation (Supplementary Fig. 1), the ZnGa2O4/GOD coating exhibited obvious oxidation and reduction peaks of GOD (Supplementary Figs. 2 and 3), in part due to ZnGa2O4-induced electron transfer. The electrochemical performance of ZnGa2O4/GOD coating was detected in glucose solution (Fig. 2d), the reduction peak currents gradually decreased with glucose addition, that is, the consumption of oxidized GOD provides routes for the electronic transduction occurring at the ZnGa2O4/GOD surface. The afterglow signal was demonstrated to be proportional to the reduction peak current by quantitative analysis, leading to a better understanding of the essential mechanisms associated with the afterglow signal and the electron transfer (Fig. 2e).
### Mechanism of electron transfer-triggered imaging
The electrical signals during the reaction were characterized by electrochemical methods, and the voltametric peak current as a function of the scan rate was evaluated (Fig. 3a). As schematized in Fig. 3b, the peak current is proportional to the scan rate, indicating that the redox process occurring on the ZnGa2O4/GOD coating is surface adsorption-limited rather than diffusion-limited41. Thus, it is reasonable to speculate on the electron transfer between GOD and ZnGa2O4 adsorbed on the surface. The corresponding band diagrams of ZnGa2O4, glucose, and FAD, a cofactor of GOD, were calculated to investigate the electron transfer from an electron donor to an acceptor (Fig. 3c and Supplementary Figs. 47)42,43. For glucose with high electrochemical energy levels, electrons in its LUMO energy level will tunnel into ZnGa2O4 with the assistance of FAD-dependent enzymes. The Fermi level of ZnGa2O4 will increase via electron transfer and lead to the increment of electrons in the defect level by charge separation and trapping, thus eventually enhancing the afterglow emission. As a consequence, the electrons transferred from glucose to ZnGa2O4 can recombine to the electronic ground state via radiative pathways, leading to an enhanced emission that is highly consistent with the observation in afterglow images.
### Optimization of the ETTE nanoprobe
In an ideal solution environment, it was demonstrated that ZnGa2O4 can produce obvious afterglow signal, varying with the number of transferred electrons. By calculating the photoluminescence efficiency and the electron-hole recombination rate of the probe (see Supplementary Information, section ‘Electron tunneling probability calculation’), it can be concluded that the efficiency of photoluminescence is proportional to the number of electron-hole pairs of the material. Since ZnGa2O4 is a p-type semiconductor material, where the hole concentration is much higher than the electron concentration, it can be considered that the effect of hole concentration change on the radiative recombination rate is negligible. Thus, the photoluminescence intensity of ZnGa2O4 is positively correlated with its electron concentration. To optimize the electron transfer-induced luminescence efficiency of the probe, it is necessary to increase the number of transferred electrons between the target molecule and ZnGa2O4 in unit time. For tunneling through a one-dimensional barrier in a weakly coupled donor-acceptor system, the electron transfer rate kET is shown in formula (1).
$${k}_{{ET}}=\frac{2\pi }{{{\hslash }}}{\left|{T}_{{DA}}\right|}^{2}\left(F.C\right)={A}^{2}{e}^{{-R}_{{DA}}\beta }(F.C)$$
(1)
where is the reduced Planck constant, = h/2π, TDA is the electronic tunneling matrix element between donor and acceptor localized states. (F.C.) is the Franck-Condon factor which is determined by the activated or tunneling nuclear processes coupled to the transfer46. Formula (1) gives the calculation method of electron tunneling probability TAD2, where RDA is the distance between donor and acceptor, and β is the barrier height. It can be seen that the probability of electron tunneling can be greatly enhanced by reducing the barrier height and the distance between the target molecule and the probe47.
To induce electron transfer from the signaling pathway to the probe, molecular “wires” with molecular targeting ability were assembled on the surface of the material to close the distance between the target molecule and the probe. Because ferrocene structure exhibits high electrical conductivity and low energy barrier35, an aptamer-ferrocene polymer was a promising candidate for molecular wires (see Supplementary Information, section ‘Synthesis of Fe-base’, and Supplementary Figs. 1120). An increase in the number of ferrocene moieties can reduce the barrier height of the aptamer-ferrocene polymer by extending the conjugate field, and, on the other hand, lengthen the distance between the target molecule and the probe. Here, ETTE nanoprobes with different lengths of molecular wires were constructed via the phase transfer reaction to establish how the number of Fe-bases affects the electron transfer (Supplementary Table 1, Supplementary Fig. 2132). Energy level distributions of ZnGa2O4 and ferrocene moieties in the ETTE nanoprobe were calculated (Supplementary Fig. 33), confirming that electrons in the defect level of ZnGa2O4 will transfer to the ferrocene moieties and quench the afterglow signal48. It can be concluded that within the ETTE nanoprobe, the higher the electron tunneling probability, the lower the afterglow emission. Fluorescence spectrum of ETTE nanoprobe with varying Fe-base modifications was obtained and the probe with nine Fe-bases exhibited the lowest afterglow signal (Supplementary Figs. 3437). As a result, the ETTE nanoprobe with nine Fe-bases was selected as the optimized system that balances the reduction of barrier height with the retention of a close distance.
### Electron transfer-triggered imaging of EGFR activity in living cells
EGFR signaling pathway activity is associated with tumor cell proliferation, angiogenesis, tumor invasion, metastasis, and inhibition of apoptosis17,18. Key molecular EGFR can self-phosphorylate for activating downstream signal pathways located in cells, such as mitogen-activated protein kinase (MAPK) and c-Jun N-terminal kinase (JNK) pathways. Hence, selective detection of the activity of EGFR-related signal pathways can help to understand the progression of tumor development. To map EGFR activity in complex biological samples, it is necessary to achieve the specific detection of target signal pathway activity in numerous active signal pathway networks. Thus, a molecular wire with target recognition was needed to satisfy the requirements. Known as a chemical antibody, the nucleic acid aptamer was a promising candidate for target recognition, and an EGFR-targeting aptamer was modified with 9 Fe-bases to form a specific molecular wire. This molecular wire was then functionalized on the surface of ZnGa2O4 to achieve specific electron transfer imaging of EGFR activity.
The imaging of EGFR signaling pathway activity was performed in A549 lung cancer cells using the modified ETTE nanoprobes as imaging agents (Supplementary Fig. 38). The cultured lung cancer cells were first preincubated with ETTE nanoprobes in PBS, then exposed to EGF to activate the EGFR signaling pathway. In EGFR-activated cells and in EGFR-inactivated cells, comparable fluorescence signals could be observed (Supplementary Figs. 3940), referring to the similar EGFR-targeting capability of ETTE nanoprobes. In the case of the afterglow signal from the EGFR-targeting ETTE nanoprobe (Fig. 4a), the pseudo-color image showed a weak afterglow signal in PBS buffer, due to the feeble life activity and the inactivated EGFR signaling pathway in the cells. After activation of the EGFR pathway, the afterglow signal displayed a significant enhancement, confirming that the probes have intensive afterglow-signal-generation efficiency in EGFR-activated cells and show high sensitivity to the EGFR activity. To simulate the complex in vivo environment, the above tests were repeated in the serum, and the enhancement of the afterglow signals could also be observed in EGFR-activated cells (Fig. 4b). In both PBS and FBS environment, the EGFR signaling pathway in cells was activated with the addition of EGF, and the intracellular afterglow signal was also enhanced obviously. Statistically, the relative signal intensity of these images was calculated. The relative intensity of the afterglow signal rose from 20 to 60% after EGF exposure under PBS condition, while 80% of relative intensity value could be obtained at EGF stimulation in the serum (Fig. 4c). These observations demonstrate that the ETTE probe could achieve electron transfer-triggered imaging for the EGFR signaling activity. EGFR and phosphorylated EGFR (P-EGFR) in A549 cells were characterized using western blot analysis. As observed in western blot results, the band for P-EGFR was clearly displayed in the EGF-treated cells (Supplementary Fig. 41), fitting well with the afterglow signal of the ETTE probe. The sensitivity of ETTE probes was evaluated in PBS and FBS environment. To measure the sensitivity of ETTE probes, EGF was progressively increased, which increased the afterglow intensity in EGFR-expressing A549 cells, yielding an EC50 (concentration for 50% maximal effect) of about 20 ng·mL−1 both in PBS and in FBS condition (Supplementary Figs. 4245). The ETTE probe achieves comparable sensitivity with other strategies such as fluorescence resonance energy transfer49 and surface-enhanced Raman scattering50. The longest timescales of the ETTE probe can reach almost 10 min, making it possible to obtain kinetic information (Supplementary Fig. 46). Furthermore, molecular level validation of the ETTE probe for EGFR dependent signal mapping was performed on the EGFR-knocked out A549 cells. For EGFR-knocked out A549 cells, ETTE probes were found insensitive to the stimulation of EGF (Supplementary Figs. 4750), suggesting that the afterglow signal of the ETTE probe is EGFR dependent. To confirm that the signal of the ETTE probe is indeed EGFR specific, EGFR signaling activity in some EGFR-expressing cells, such as A549 cell, CAL-27 cell, and MDA-MB-231 cell, were tested with EGF stimulation and EGFR tyrosine kinase inhibition51,52 (Supplementary Fig. 51). As a result, stimulation of EGF was found to activate the signals of ETTE probes significantly, while the afterglow signals in EGFR-expressing cells were completely blocked by co-application of afatinib, an EGFR tyrosine kinase inhibitor, indicating specific responses of the ETTE probe53,54 (Supplementary Figs. 5255). Altogether, the results demonstrate that the afterglow signal from the ETTE probe is specific to EGFR and can be used as a reliable, non-invasive method to measure EGFR-related signaling activity in live cells. Flow cytometry has been used to analyze cellular signaling events, however, this platform always gives relatively static results and cannot map the signaling events dynamically in living cells55,56. The ETTE nanoprobe can possess a dynamic monitoring capability of the EGFR signaling pathway during cancer cell division, as depicted in Supplementary Fig. 56 and Supplementary Movie 1. These phenomena are possibly due to the electron transfer from activated EGFR to the probe (Fig. 4d). During a fluorescence emission process, charging of the probe can be achieved by light irradiation, enabling bright fluorescent signals57. Drowned by the strong fluorescent signals, the slight optical signal change caused by electron transfer in organisms can thus not be observed. For the ETTE probe, some of the electrons/holes created by optical excitation would be trapped at specific defects and enable the probe to store the excitation light32. Different from fluorescent signals, afterglow signals from the ETTE probe were produced by the slow release and radiative recombination of carriers from trapping defects. As a time-dependent decayed signal, afterglow intensity mostly relies on the stored electrons that would be affected by the electron transfer between biomolecules via electron acception/donation. As a consequence, the mapping of EGFR activity in living cells can be achieved by evaluating the afterglow emission of this optimized ETTE nanoprobe.
### Electron transfer-triggered imaging of EGFR activity in vivo
To further clarify the EGFR monitoring capacity of the ETTE probes in vivo, mice from the EGFR-noninhibited and EGFR-inhibited group were investigated (Fig. 5a), respectively. As a clinical used EGFR inhibitor, cetuximab can specifically bind to the EGFR with high affinity and block the normal function of the receptor58. The in vivo electron transfer-triggered imaging of EGFR was therefore performed using a murine model of lung cancer that was susceptible to cetuximab in vivo59 (see Supplementary Information, section ‘EGFR Inhibition Using Cetuximab’). The inhibition of EGFR was performed using cetuximab in tumor-bearing mice and was characterized by ETTE probe at day 0 and day 14, respectively (Supplementary Figs. 5761). Both real-time afterglow results showed that the afterglow intensity of the EGFR-inhibited group was significantly decreased after injection of cetuximab compared with the EGFR-noninhibited group (Fig. 5b, c). The correlation curve of afterglow signal intensity versus injection time showed that the relative signal intensity fluctuated above 100% for the PBS-treated group (Fig. 5d, e gray line), while it decreased to 60% for the cetuximab-treated group (Fig. 5d, e red line). For the EGFR-inhibited group, the afterglow signal was dimmer after 1 h post injection of cetuximab and reached a minimum at 2 h at the tumor site within the extended observation time window. The electron transfer-triggered images of EGFR-noninhibited and EGFR-inhibited mice suggest that the inhibition of EGFR could be vividly observed using our ETTE probe. To ensure the activity of EGFR signaling pathway, the expression of EGFR and P-EGFR in tumor tissues was characterized by traditional western blot and immunohistochemistry (IHC). It was observed in the western blot result that the band for P-EGFR was clearly displayed in the PBS-treated tumor tissues and weakened in the cetuximab-treated ones (Fig. 5f). When it came to IHC images, the PBS-treated group exhibited an extremely vivid yellow signal of P-EGFR while the cetuximab-treated group displayed a little yellow region (Fig. 5g). These observations suggested the low expression of P-EGFR and the low activity of the EGFR signaling pathways in the EGFR-inhibited group. In the electron transfer-triggered images, the afterglow signals of cetuximab-treated tumor tissues were apparently weaker than those of the PBS-treated group, revealing the inhibition of EGFR in the cetuximab-treated group (Fig. 5h and Supplementary Fig. 62). The statistical results of western blot and IHC demonstrated that the P-EGFR expression of the cetuximab-treated group was significantly lower than that of the PBS-treated group (Fig. 5i–j, Supplementary Figs. 6364), matching well with the EGFR activity evaluated by the ETTE probe in tumor tissues (Fig. 5k). Consequently, the electron transfer-triggered images are highly consistent with the western blot and IHC results, verifying the feasibility of the ETTE probe for imaging the EGFR signaling activity in vivo.
Moreover, the biosafety of the ETTE nanoprobe was evaluated by measuring its tissue and blood compatibilities on female athymic BALB/c mice. Histological analysis of major organs, hematology analysis, and blood biochemical analysis were performed on healthy mice treated by intravenous injection with PBS or ETTE probe. The body weight of mice was monitored for 14 days after the injection, and no apparent body weight loss was observed (Supplementary Fig. 65). The histological analysis of major organs showed that the treatment of ETTE nanoprobe did not cause visible damage to all the tested organs after treatment (Supplementary Fig. 66), suggesting negligible organ toxicity of the probe. Furthermore, the general hematology parameters and standard blood biochemical indexes were assayed. Compared with the PBS-treated group, the ETTE probe-treated group displayed no statistically significant differences in both hematology parameters and blood biochemical indexes (Supplementary Figs. 6768), demonstrating good blood compatibility and no obvious toxicity to liver and kidney. Altogether, the ETTE nanoprobe displays satisfiable biosafety in the mice model, laying a good foundation for further in vivo application.
### Mapping EGFR activity for efficacy assessment in vivo
During tumor treatment, the activity of EGFR relevant signal pathways keeps changing. When a tumor grows, tumor cells need to proliferate and differentiate rapidly, EGFR-related signaling pathways are activated in the cells. During treatment, the growth of tumor cells will be retarded, leading to the decrease of EGFR activity. To investigate whether electron transfer-triggered imaging can be used as a more general strategy to probe signaling events in vivo, it was further explored whether the EGFR activity evaluated by ETTE probe can reflect the therapeutic effect of EGFR non-targeted therapeutic agents.
Using MTH1 siRNA as a non-targeted therapeutic agent60 (see Supplementary Information, section ‘siRNA Transfection’, and Supplementary Figs. 6971), whole-body EGFR activity was evaluated by ETTE probe in mice, and the afterglow signal in the tumor area was quantitatively calculated (Supplementary Figs. 72 and 73). Considering that the metabolism in each mouse is unlikely the same, the relative afterglow signal intensity of the tumor portion relative to the spinal symmetry region was also calculated. In the untreated group, the afterglow signal after treatment in the tumor site was apparently higher than the initial signal (Fig. 6a, b), while the treated group showed a significant decay (Fig. 6c, d). After treatment, the correlation curve of afterglow signal versus injection time showed that the relative signal intensity fluctuates raised from 100 (Fig. 6b gray line) to 120% (Fig. 6b red line) for the untreated group, while it decreased from about 100 (Fig. 6d gray line) to 60% (Fig. 6d red line) for the treated group, suggesting that the EGFR activity in the tumor region was significantly weakened. Compared with the untreated group, the treated mice have statistically significant low tumor volume and tumor weight (Fig. 6e, f), the same as the trend of afterglow signal (Supplementary Figs. 74 and 6g), indicating that the efficacy of treatment can be evaluated by the ETTE probe. Finally, the expression of EGFR and P-EGFR in tumor tissues was characterized by IHC to show the activity of the relevant signaling pathways. In the untreated group, both EGFR and P-EGFR were highly expressed (Fig. 6h), while the treated group showed a high EGFR content and a low P-EGFR content (Fig. 6i), revealing that most EGFR molecules stay inactivated in the treated group. The statistical results further demonstrated that the EGFR activity of the treated group was significantly lower than that of the untreated group (Fig. 6j, k), in agreement with the ETTE images (Fig. 6e, g). These results are highly consistent with the widely accepted view that the EGFR signaling pathway in tumors stays activated to satisfy the needs of proliferation, angiogenesis, invasion, and metastasis during tumor growth17,18. Overall, this ETTE imaging method may offer valuable guidelines for the mapping of signaling activity in vivo and will inspire new strategies to evaluate therapeutic efficacy.
## Discussion
Our study suggests that an ETTE nanoprobe can achieve dynamic monitoring of EGFR signal pathway activity during tumor therapy. Compared with traditional IHC, which is the major source of our knowledge of signaling pathways, our probe has enabled the visualization of target signaling pathways and electron transfer in a temporally and spatially precise manner. Such a method provides the possibility to understand disease progression and molecular mechanisms, thus minimizing uncertainty caused by biopsy and sample processing. As an effective minimally invasive technique, this ETTE imaging method can monitor and characterize the activity of the target signal pathway in real-time and is expected to play a key factor in effective therapeutic decision-making and prognosis.
## Methods
### Reagents and materials
All reagents were used as received from commercial sources or prepared as described in references. All DNA synthesis reagents were purchased from Glen Research (Sterling, VA). Protected dFe phosphoramidites were synthesized in our lab.
Dulbecco’s phosphate-buffered saline (DPBS), Dulbecco’s modified Eagles medium (DMEM), Leibovitz’s L-15 medium (L-15), and fetal bovine serum (FBS) were obtained from Thermo Fisher Scientific Inc. All solutions used in the experiments were prepared using ultrapure water (resistance > 18 MΩ cm), which was obtained through a Millipore Milli-Q ultrapure water system (Billerica, MA, USA).
### Sample characterization
Oligonucleotides were prepared with an Applied Biosystems (ABI) 394 DNA/RNA synthesizer. ESI-MS spectra of oligonucleotides were performed by the Shanghai Sangon Mass Spectrometer (Shimadzu, Japan). Transmission electron microscopy (TEM) was carried out on an H-7000 NAR transmission electron microscope (Hitachi) with a working voltage of 100 kV. Atomic force microscopy (AFM) images of samples were obtained on a Multimode 8 (Bruker, USA). The confocal fluorescence imaging studies were performed on a confocal laser scanning microscope (FV1000, Olympus, Japan).
### Preparation of ZnGa2O4 nanoparticles
The ZnGa2O4 nanoparticles were synthesized in oil phase. First, 4.5 g of octadecanoic acid and 0.6 g of NaOH were dissolved in a mixed solvent containing 8 mL of deionized water and 18 mL of ethanol. Next, Zn(NO3)2 (2.0 M, 0.5 mL), Ga(NO3)2 (0.4 M, 5.0 mL) and Cr(NO3)2 (8 mM, 625 uL) were sequentially added to the above solution under vigorous stirring. Sodium hydroxide was added to adjust the pH to 10.0. The resulting mixture was left stirring for 20 min at room temperature. Then, the solution was transferred to a Teflon-lined autoclave and reacted at 220 °C for 10 h. The as-prepared ZnGa2O4 nanoparticles were collected by centrifugation and washed three times with cyclohexane: ethanol = 1:1.
### DNA synthesis
DNA oligonucleotide was synthesized using standard phosphoramidite chemistry on controlled pore glass supports on an ABI 394 DNA Synthesizer. The coupling times were 60 s. The completed sequences were then deprotected in saturated ammonium hydroxide at room temperature for 12 h and further purified by reversed-phase HPLC (Agilent 1260) on a C18 column using 0.1 M triethylamine acetate (TEAA) buffer (Glen Research) and acetonitrile (Sigma-Aldrich) as the eluents. The collected DNA products were dried and detritylated by dissolving and incubating DNA products in 200 µL of 80% acetic acid for 20 min. The detritylated DNA product was precipitated with NaCl (3 M, 25 µL) and ethanol (600 µL) and then desalted using Sep-Pac Plus C18 cartridges.
### Synthesis of Fe-base
Briefly, commercially available (S)-3-amino-1,2-propanediol was coupled with ferrocene carboxylic acid, and the corresponding phosphoramidite compound (Supplementary Fig. 11) was prepared for DNA solid-phase synthesis61.
### Fe-base characterization
Anhydrous solvent pyridine, CH3CN, CH2Cl2, and N,N-Dimethylformamide (DMF) were distilled under a nitrogen atmosphere and stored with 4 Å molecular sieves. Proton NMR spectra were recorded on a Bruker AM400 spectrometer. 1H-NMR, 13C-NMR, and 31P-NMR spectra were recorded on a Bruker AM400 spectrometer. Chemical shifts (δ) are reported in ppm, and coupling constants (J) are in hertz (Hz). The following abbreviations were used to explain the multiplicities: s singlet, d doublet, t triplet, q quartet, m multiplet, br broad.
Fourier transform infrared spectroscopy (FTIR) samples were prepared by mixing Fe-base with pre-dried KBr powder. FTIR spectra were recorded on an IR spectrometer instrument (Thermo, Nicolet 6700) in the 4000–400 cm−1 range.
The precise molecular weight of Fe-base was confirmed by a high-performance liquid chromatography-ion trap time-of-flight mass spectrometer (LCMS-IT-TOF, Shimadzu, Kyoto, Japan) equipped with an electrospray ionization (ESI) source, operating in positive ionization mode. The scanning ranges were m/z 50–1000. Shimadzu’s LCMS Solution software was used for data analysis.
### Preparation of ETTE nanoprobe
The ETTE nanoprobe was formed by a phase transfer method. The ETTE nanoprobe was attached to the nanoparticle by a hydrophilic-hydrophobic interaction. Briefly, 2 mg of ZnGa2O4 nanoparticles were dispersed in 2 mL of toluene by stirring. Next, 3 nmol of Fe-base-modified DNA oligonucleotide was dissolved in 1 mL of sterile water and then was added to the nanoparticle solution under stirring. The reaction mixture was kept at room temperature for 15 h under vigorous stirring. Afterward, the ZnGa2O4 was transferred into the lower water phase from the upper toluene phase. Then the water solution was transferred to a 2 mL of the centrifugal tube. The obtained ETTE nanoprobe was collected by centrifugation and washed with toluene three times, and further dispersed in 1 mL of sterile water.
### Preparation of working electrodes
To prepare the working electrodes, the glassy carbon electrodes (diameter 3 mm) were polished with 0.3 and 0.05 µm alumina powder and rinsed thoroughly with doubly distilled water, followed by washing ultrasonically in ethanol for 2 min. 2 mg persistent luminescence nanoparticles were dispersed in 1 mL ultrapure water to obtain a homogeneous suspension. 5 µL of the suspension was cast on the glassy carbon electrodes surfaces and allowed to dry at room temperature. Then, 3 µL of 625 µM GOD solution was dropped onto the surfaces of the ZnGa2O4-coated glassy carbon electrodes and dried at 4 °C. To maintain the stability of modified electrodes, 3 µL of 1% (v/v) Nafion solution was dropped onto GOD/ZnGa2O4/glassy carbon electrodes. The dried electrodes were subjected to electrochemical measurement.
For comparison, GOD/glassy carbon electrodes and glassy carbon electrodes were also prepared with the same procedures as those described above, but without ZnGa2O4 or GOD-coated ZnGa2O4. All modified electrodes were stored at 4 °C under dry conditions until further usage.
### Time-resolved fluorescence measurements
Samples were excited with a semiconductor laser (MPL-F-266) at 266 nm, and a filter was used to remove non-fluorescent scattering above 266 nm. Then, the optical path was adjusted to focus the fluorescence emitted by the sample on the PIN photodiode (Thorlabs APD120) to obtain an electrical signal corresponding to the lifetime of the fluorescence.
When a very short pulse (typically < 0.1 ms) excitation light irradiates a sample containing a fluorescent material, the fluorescent material will emit fluorescence with rapid exponential decay. When the fluorescence intensity becomes 1/e of the initial light intensity, the time t at this time is defined as the fluorescence lifetime62. The PIN photodiode captures the fluorescence of the material and then generates photoelectrons, which are converted into the output voltage Vout. The relationship between the output voltage Vout of the Thorlabs APD120 photodiode and the incident optical power Pout is shown in Eq. (2), where λ is the persistent luminescence wavelength of the fluorescent material, and R(λ) is the sensitivity of the λ photodiode, where R(λ) is 25 A·W−1. The multiplier M and the transimpedance gain G are determined by the instrument and the ambient temperature. In this experiment, M = 50 and G is 105 V·A−1. The formula for optical power is shown in Eq. (3)63, where N is the number of photons, h is Planck constant, c is the speed of light, λ is the persistent luminescence wavelength, and t is the detection time interval of the photodiode. As simplified, the relationship between the output voltage of the PIN photodiode and the photon number of the luminescent material is shown in formula (4).
$${V}_{{out}}={P}_{{out}}* R\left({{{{{\rm{\lambda }}}}}}\right){MG}$$
(2)
$${P}_{{out}}=\frac{{Nhc}}{{{{{{\rm{\lambda }}}}}}{{{{{\rm{t}}}}}}}$$
(3)
$$N=\frac{{P}_{{out}}{{{{{\rm{\lambda }}}}}}{{{{{\rm{t}}}}}}}{hc}=\frac{{V}_{{out}}{{{{{\rm{\lambda }}}}}}{{{{{\rm{t}}}}}}}{h{cMGR}({{{{{\rm{\lambda }}}}}})}$$
(4)
### Afterglow imaging of persistent luminescent nanoparticles
The persistent luminescent nanoparticles were placed in a 96-well plate. The nanorods were illuminated with a portable UV lamp for 2 min. After that, the UV lamp was removed, and the plate was immediately put into the IVIS Lumina XR Imaging System to record the decay images.
### Electrochemical characterization
Cyclic voltammetry (CV) measurements were carried out at room temperature on a CHI660D electrochemical working station (ChenHua Instruments Co. Ltd., Shanghai, China) using a standard three-electrode system. A modified GCE was the working electrode, and a saturated calomel electrode (SCE) and a platinum wire served as the reference and counter electrode, respectively. Before the measurement, the PBS solution (0.1 M, pH 7.2) containing various glucose concentrations (0.0, 0.5, 1.0, 2.0, and 5.0 mM) was purged with argon or oxygen for 10 min to prepare the argon- or oxygen-saturated solutions. All CV scans were recorded from −0.8 to 0.2 V with a scan rate of 50 mV·s−1, unless otherwise stated.
### Thermoluminescence measurements
In this experiment, an SL-08 thermoluminescence dosimeter was used to record the TL curves of the sample. The sample was placed on the sample plate, pre-irradiated with 15 W ultraviolet (254 nm) for 100 s, then heated from room temperature to 500 °C at a heating rate of 5 °C·s−1 and the TL curve of the sample was recorded.
Chen’s peak shape analysis method was used to analyze the thermoluminescence curve to obtain the trap depth of the sample. The trap depth by this analysis method is given by Eq. (5)64,65,66, where α corresponds to τ = (Tm − T1), δ = (T2 − Tm), and ω = (T2 − T1). Tm is the maximum peak temperature, T1 and T2 are the temperatures corresponding to half the intensity on either side of the maximum peak, and k is the Boltzmann constant. Equations (68) give the Cα, and the equation for bα is shown in Eq. (9). Then, the average of Eτ, Eδ, Eω is used to calculate the trap depth of the sample.
$${E}_{\alpha }={C}_{\alpha }\left(\frac{k{T}_{m}^{2}}{\alpha }\right)-{b}_{\alpha }(2k{T}_{m})$$
(5)
$${C}_{\tau }=1.51+3.0\left(\frac{{T}_{2}-{T}_{m}}{{T}_{2}-{T}_{1}}-0.42\right)$$
(6)
$${C}_{{{{{{\rm{\delta }}}}}}}=0.976+7.3\left(\frac{{T}_{2}-{T}_{m}}{{T}_{2}-{T}_{1}}-0.42\right)$$
(7)
$${C}_{{{{{{\rm{\omega }}}}}}}=2.52+10.2\left(\frac{{T}_{2}-{T}_{m}}{{T}_{2}-{T}_{1}}-0.42\right)$$
(8)
$${b}_{\tau }=1.58+4.2\left(\frac{{T}_{2}-{T}_{m}}{{T}_{2}-{T}_{1}}-0.42\right),{b}_{{{{{{\rm{\delta }}}}}}}=0,{b}_{{{{{{\rm{\omega }}}}}}}=1$$
(9)
### Electron paramagnetic resonance (EPR) measurements
The EPR signals of samples were acquired at 295 K using a JES-FA200 EPR spectrometer (JEOL, Tokyo, Japan) with an X-band standard frequency of 8.75–9.65 GHz. The following EPR parameters were used: frequency = 9.19 GHz, center field = 335 ± 10 mT, modulation frequency = 100 kHz, time constant = 0.03 s, and power = 0.998 mW.
The total 20 µL reaction mixture contained 8 µL glucose solution (0.0, 0.5 mM, 1.0 mM, 2.0 mM, and 5.0 mM dissolved in isopropanol), 4 µL DMPO (20 mg·mL−1, in isopropanol), 4 µL GOD (6.25 µM) and 4 µL isopropanol for detection of the DMPO/alkyl radical adducts.
### Electron tunneling probability calculation
The electrons transferred per unit time are proportional to the produced photons that determine the luminescence efficiency of the probe. Photoluminescence efficiency of the probe y is shown in Eq. (10).
$${{{{{\rm{y}}}}}}=\frac{{F}_{{PL}}}{{f}_{e}\left(1-R\right)F}=\frac{{R}_{r}}{\left(1-R\right)F}$$
(10)
where FPL is the photoluminescence intensity of the probe, fe is the correction factor, R is the reflection coefficient of the excitation light, F is the excitation intensity of external light (photons per second), (1 − R) F is the number of electrons excited from the probe per second, and Rr is the recombination rate of electron-holes67. According to the derivation of the equation, the luminescence efficiency of the probe is proportional to Rr, the electron-hole recombination rate, under the same illumination intensity. Equation (11) is used to calculate Rr.
$${R}_{r}=\frac{{np}}{{n}_{i}^{2}}32{{{{{{\rm{\pi }}}}}}}^{2}c{\left(\frac{{kT}}{{ch}}\right)}^{4}\int_{0}^{{{\infty }}}\frac{{N}^{3}{{\delta }}{u}^{3}{du}}{{e}^{u}-1}=\frac{{np}}{{n}_{i}^{2}}{R}_{i}$$
(11)
where n is the electron concentration of the probe, p is the hole concentration of the probe, ni is the concentration of the electrons or holes in the intrinsic semiconductor, k is the Boltzmann constant, c is the speed of light, h is the Planck constant, N is the refractive index of the probe material, is the extinction coefficient, and u ≡ hv/kT represents the integration variable in Eq. (11)68. As simplified, the total recombination rate Ri per unit volume can be regarded as a temperature-dependent constant. In that way, at a certain temperature, the electron-hole recombination rate of the probe Rr is related only to the electron concentration in the probe n, the hole concentration p, and the concentration of electrons or holes in the intrinsic semiconductor ni.
### Cell culture
A549 cells: A549 cells derived from two different sources, American Type Culture Collection (ATCC) and China Center for Type Culture Collection (CCTCC). The A549 cells were maintained in DMEM high glucose (Gibco™, USA) with 10% fetal bovine serum (Zeta life, USA) and 1% penicillin/streptomycin (New Cell & Molecular Biotech, Co., Ltd).
HT-29 cells: HT-29 cells were derived from CCTCC. The HT-29 cells were maintained in DMEM high glucose (GibcoTM, USA) with 10% fetal bovine serum (Zeta life, USA) and 1% penicillin/streptomycin (New Cell & Molecular Biotech, Co., Ltd).
CAL-27 cells: CAL-27 cells were derived from CCTCC. The CAL-27 cells were maintained in DMEM high glucose (GibcoTM, USA) with 10% fetal bovine serum (Zeta life, USA) and 1% penicillin/streptomycin (New Cell & Molecular Biotech, Co., Ltd).
MDA-MB-231 cells: MDA-MB-231 cells were derived from CCTCC. The MDA-MB-231 cells were maintained in DMEM high glucose (GibcoTM, USA) with 10% fetal bovine serum (Zeta life, USA).
### Confocal fluorescence imaging
A549 cells were plated on 20 mm confocal laser dishes for 24 hours and divided into four groups for processing:
Group one: The cultured A549 cells were preincubated with ETTE nanoprobes (final concentration of 75 μg·mL−1) in a PBS environment at 37 °C for 40 min, then washed and detected in a PBS environment.
Group two: The cultured A549 cells were preincubated with ETTE nanoprobes (final concentration of 75 μg·mL−1) in a PBS environment at 37 °C for 30 min, then exposed to EGF (100 ng·mL−1) for 10 min to activate the EGFR signaling pathway.
Group three: The cultured A549 cells were preincubated with ETTE nanoprobes (final concentration of 75 μg·mL−1) in the medium supplemented with 10% serum at 37 °C for 40 min.
Group four: The cultured A549 cells were preincubated with ETTE nanoprobes (final concentration of 75 μg·mL−1) in the medium supplemented with 10% serum at 37 °C for 30 min, then exposed to EGF (100 ng·mL−1) for 10 min to activate the EGFR signaling pathway.
ETTE’s sensitivity detection:
Group one: The cultured A549 cells were preincubated with ETTE nanoprobes (final concentration of 75 μg·mL−1) in a PBS environment at 37 °C for 30 min. Then cells were incubated for 10 min with EGF at different concentrations (0.6 ng·mL−1, 4.0 ng·mL−1, 20.0 ng·mL−1, 100.0 ng·mL−1, 500.0 ng·mL−1) to activate the EGFR signaling pathway (n = 3).
Group two: The cultured A549 cells were preincubated with ETTE nanoprobes (final concentration of 75 μg·mL−1) in the medium supplemented with 10% serum at 37 °C for 30 min. Then cells were incubated for 10 min with EGF at different concentrations (0.6 ng·mL−1, 4.0 ng·mL−1, 20.0 ng·mL−1, 100.0 ng·mL−1, 500.0 ng·mL−1) to activate the EGFR signaling pathway (n = 3).
Then all the cells were washed with DPBS and fixed with paraformaldehyde. Subsequently, the cells were incubated with DAPI for 5 min for nuclear staining. After washing with DPBS three times, confocal fluorescence imaging studies were performed on the FV1000 confocal laser scanning microscope. The objective used for imaging was a UPlanSApo 60× oil immersion objective with a numerical aperture of 1.35 from Olympus (Japan). The lasers of 405 nm and 635 nm were used as the excitation light source for DAPI and ETTE probes. In addition, the cells were excited under an external 365 nm UV lamp for 2 min, and multiple images were taken continuously without UV excitation. All the experiments were performed in triplicate for reproducibility. The images were analyzed with ImageJ software. The ImageJ lookup table (LUT) named ‘Thermal’ was applied to obtain the pseudo-color images.
#### ETTE’s timescales evaluation
The A549 cells were plated on 20 mm confocal laser dishes for 24 h and divided into two groups for processing. The A549 cells were incubated with the ETTE probes (final concentration of 250 μg·mL−1) at 37 °C for 30 min. One group received no treatments and the other group was exposed to EGF (100 ng·mL−1) for 10 min. The two groups of cells were washed and tested in a PBS environment. The cells were excited for 2 min with an external UV lamp. Subsequently, the afterglow signal was continuously captured by laser confocal microscope at 1.0 s after UV excitation.
#### EGFR inhibition using afatinib
A549, CAL-27, and MDA-MB-231 cells were plated on a 20 mm confocal laser culture dish for 24 h. Different cell lines were separately incubated with ETTE probes and divided into three groups. The first group was treated with PBS; the second group was stimulated with EGF for 10 min; the third group was treated with afatinib (5 μM) for 3 h and then stimulated with EGF for 10 min. Then all the cells were washed with DPBS and fixed with paraformaldehyde. Subsequently, the cells were incubated with DAPI for 5 min for nuclear staining. After washing with DPBS three times, confocal fluorescence imaging studies were performed on the FV1000 confocal laser scanning microscope.
### Generating EGFR knockout A549 cell lines
The EGFR stable knockout (KO) A549 cells were established by using clustered regularly interspaced short palindromic repeats (CRISPR)/Cas9 system69. EGFR single guide RNAs (sgRNAs) were designed using the online CRISPR analysis tool (https://www.benchling.com/crispr/). The sgRNAs targeting the EGFR gene at exon 2 and Cas9 vectors (Supplementary Table 2) were cloned into 8138-EGFP-puro. For transfection, cells were seeded in a 96-well plate. Following transfection, cells were replated for single-cell cloning, propagated, and screened by a polymerase chain reaction (PCR) strategy designed to screen for EGFR knockout A549 cells. The EGFR primers for PCR were as follows, forward: 5′-CTACCACCCACCCCTTTAAATTTCA-3′, reverse: 5′-CATTAGCTGGTAAAATGGCTTTCTC-3′. The amplified PCR products were sequenced using the primers described above. EGFR knockout clones were further validated by Sanger sequencing and western blot.
### First-principles calculation
The first-principles calculation was carried out by the open-source QUANTUM ESPRESSO plane-wave density functional theory (DFT) package42,43. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional was adopted and the plane-wave cut-off energies for the Fe-base and ZnGa2O4 were 626 and 598 eV, respectively. To avoid mirror interactions, vacuum spaces of 15 Å are added between adjacent cells in the nonperiodic direction. In the electronic ground-state computation of the Fe-base, only the Γ point of the Brillouin zone k-space was considered, whereas the k-space sampling was 5 × 5 × 5 for the ZnGa2O4. The crystal structures were fully relaxed until the force on each atom and total energy variations were <2.6 × 10−2 eV·Å−1 and 1.4 × 10−3 eV. Both materials are calculated separately and their band structures were assembled according to the work functions and vacuum energy.
### In vivo fluorescence imaging
Four-week-old female athymic BALB/c mice were purchased from the Hunan SJA Laboratory Animal Co., Ltd (Changsha, China) and used under protocols approved by the Institutional Animal Care and Use Committee of Hunan University (housing conditions, dark/light cycle: 12/12 h, temperature: 20 °C, humidity: about 40%). The mice were also cultured under specific pathogen-free (SPF) condition at SPF Animal Laboratory of School and Hospital of Stomatology, Wuhan University. This experiment was approved by the Experimental Animal Ethics Committee of School and Hospital of Stomatology, Wuhan University (Ethics No. S07918100F). Studies were conducted in accordance with the Tumor Policy for Mice, including adhering to the maximal tumor size of 1000 mm3. Tumor volumes in each group were then monitored, and mice were killed when tumor volumes reached 1000 mm3. To establish A549 tumor-bearing nude mouse model, female athymic BALB/c mice received a subcutaneous injection of 6 × 106 A549 cells into the right backside. 4.5 nmol ETTE nanoprobe was injected intratumorally or intravenously via the tail vein, after mice were anesthetized with breathing oxygen and the anesthetic isopentane. Mouse tumors and other major organs were collected and measured for afterglow intensity.
### Biosafety evaluation
To verify the biosafety of ETTE probes, health BALB/c mice were randomly divided into two groups (n = 5) to be intravenously injected with 30 μL PBS and 30 μL ETTE probes, respectively. The mice’s body weights were monitored every other day. After 14 days, the blood was collected. Hematological and blood biochemical analyses were assessed using a blood cell analyzer (BC-2800vet, Mindray) and a biochemical analyzer (Chemray 240, Rayto). The major organs (liver, spleen, kidney, heart, and lung) were harvested and histologically analyzed by H&E staining.
### EGFR inhibition using cetuximab
Cetuximab was produced by MedChemExpress USA (#HY-P9905). Cetuximab was administered intraperitoneally at 1.5 µg per gram of body weight. Caliper measurements were used to calculate tumor volumes using the formula V = 1/2 (length × width × width).
### siRNA transfection
Small interfering RNAs (siRNAs) targeting MTH1 were synthesized by Sangon Biotech (China). The following sequences were used: siMTH1, 5′-GAC GAC AGC UAC UGG UUU CTT-3′ (sense), 5′-GAA ACC AGU AGC UGU CGU CTT-3′ (antisense). Cells were incubated with siRNA or materials at 37 °C for 48 hours. Further, siMTH1 were used on 0.8 × 105 A549 cells. Cells were transfected with 10 nmol·L−1 of siRNAs using the Lipofectamine 2000 reagent at the concentrations indicated by the manufacturer (Invitrogen). Aptamer binding was verified 48 h post silencing. The transfection efficiency was tested by FAM-siRNA to ensure proper cell viability and delivery efficiency, which could be easily verified under the microscope.
### Western blot
Western blot was performed in accordance with our manual61. For EGFR expression in four kinds of cells, A549, CAL-27, HT-29, and MDA-MB-231 were separately seeded in a 6-well plate at a density of 1 × 105 cells/well and cultured for 24 h. Subsequently, the cells were lysed by RIPA and total proteins were obtained. After high-speed centrifugation, the proteins were denatured. Forty micrograms of protein were loaded and separated by sodium dodecyl sulfate (SDS)-polyacrylamide gel electrophoresis (PAGE). The samples were transferred to polyvinylidene fluoride (PVDF) membranes (Millipore, Billerica, MA, USA) followed by immunoblotting with Anti-EGFR (#4267; 1:400, Cell Signaling Technology) after blocking. At last, Amersham ImageQuant 800 (GE Healthcare, USA) was used to visualize the protein bands after incubation with horseradish peroxidase (HRP)-conjugated anti-IgG antibody.
For P-EGFR expression in cells treated with EGFR inhibitors, A549, CAL-27, and MDA-MB-231 cells were seeded in 6-well plates at a density of 1 × 105 cells/well, respectively. The cells were pretreated with gefitinib or afatinib at concentrations from 0 μM to 40 μM for 3 h before exposure to 100 ng·mL−1 EGF for 10 min at 37 °C. Subsequently, the cells were lysed by RIPA and total proteins were obtained. After high-speed centrifugation, the proteins were denatured. Forty micrograms of protein were loaded and separated by SDS-PAGE. The samples were transferred to PVDF membranes (Millipore, Billerica, MA, USA) followed by immunoblotting with anti-P-EGFR (#3777; 1:400, Cell Signaling Technology) after blocking. At last, Amersham ImageQuant 800 (GE Healthcare, USA) was used to visualize the protein bands after incubation with horseradish peroxidase (HRP)-conjugated anti-IgG antibody.
For MTH1 expression in A549 cells, A549 cells were lysed by RIPA and total proteins were obtained. After high-speed centrifugation, the proteins were denatured. Forty micrograms of protein were loaded and separated by SDS-PAGE. The samples were transferred to PVDF membrane (Millipore, Billerica, MA, USA) followed by immunoblotting with anti-MTH1 monoclonal antibody (#EPR15934-50; 1:800, Abcam, UK) after blocking. At last, Amersham ImageQuant 800 (GE Healthcare, USA) was used to visualize the protein bands after incubation with horseradish peroxidase (HRP)-conjugated anti-IgG antibody.
### Immunohistochemistry
The slides were subjected to deparaffination and rehydration. For epitope unmasking, the sections were boiled in 0.01 M citric acid buffer solution (pH 6.0) for 15 min at high pressure. Then, the sections were blocked for endogenous peroxidase activity by incubating sections in 3% hydrogen peroxide solution for 20 min at room temperature, and 10% goat serum was added to the sections to block non-specific binding. Next, the sections were incubated overnight at 4 °C in a humidified chamber with two primary antibodies, anti-EGFR (#4267; 1:400, Cell Signaling Technology) and anti-p-EGFR (#3777; 1:400, Cell Signaling Technology), or isotype-matched IgG controls. Subsequently, a secondary biotinylated IgG antibody solution and an avidin-biotin-peroxidase reagent were added to the slides (Sigma-Aldrich, USA). Peroxidase activity was detected by staining with DAB solution (Sigma-Aldrich). Then, the slides were counterstained with hematoxylin for 1–2 min. For immunofluorescence, the slides were incubated with fluorochrome-conjugated secondary antibody (Sigma-Aldrich) after incubation with primary antibody. Nuclear DNA was stained with DAPI (Sigma-Aldrich).
### Statistical analysis
Results are expressed as mean ± s.d. unless stated otherwise. Statistical analysis was performed with Prism 8.0 software (GraphPad Software) by an unpaired Student’s t-tests. P-value < 0.05 was considered statistically significant.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. | 2023-04-02 06:10:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48550134897232056, "perplexity": 6872.233908628783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950383.8/warc/CC-MAIN-20230402043600-20230402073600-00672.warc.gz"} |
https://homework.cpm.org/category/CCI_CT/textbook/calc/chapter/11/lesson/11.2.1/problem/11-46 | ### Home > CALC > Chapter 11 > Lesson 11.2.1 > Problem11-46
11-46.
1. If , and , Homework Help ✎
1. Find and .
2. Verify that .
Use the individual functions, not the composite function.
$g(t)=\cos(h(t))^2$ | 2020-03-30 04:50:25 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8669421672821045, "perplexity": 13238.130216907244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370496523.5/warc/CC-MAIN-20200330023050-20200330053050-00496.warc.gz"} |
https://bitbucket.org/zc/zc.dojo.nativebutton | # A plain native button widget for dojo
The goal here is to provide a plain HTML button that can be used as a dojo widget.
Basic usage:
nclicked = 0; // intentionally global
require(
["zc.dojo.NativeButton", "dojo/_base/window"],
function (NativeButton, win) {
new NativeButton(
{
label: "Hello world!",
"class": "myclass",
onClick: function (e) {
nclicked += 1;
}
}).placeAt(win.body());
});
You'll typically want to specify at least a label and an onClick handler. You can specify other attributes, such as "class" and id.
You can specify en existing node to use using an id. For example, if the page contained:
<span id="wasspan">I was there</span>
Now, if we create a button using this node, we don't have to place it, and it will take over the original node, but retain it's orginal id and content:
require(
["zc.dojo.NativeButton"],
function (NativeButton) {
new NativeButton({
"class": "myclass",
onClick: function (e) {
nclicked -= 1;
}
}, "wasspan");
});
And the node wil lend up looking like:
<button id="wasspan" class="myclass">I was there</button>
The button class gets defined as a global, zc.dojo.NativeButton, so it can be instantiated declaratively. | 2017-07-27 16:51:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22114670276641846, "perplexity": 12663.262705338497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549428325.70/warc/CC-MAIN-20170727162531-20170727182531-00550.warc.gz"} |
http://minhcuongvj.com/qshuu/73f75e-name-each-of-the-following-monatomic-ions | Ni+2 S-2 _____ _____ Write the formula for the following Binary compounds (metal and nonmetal) a. Lithium oxide b. Barium selenide c. Potassium nitride d. Rubidium iodide e. Ch. Monatomic Ions Nomenclature Worksheet 1: Monatomic Ions 0 Use a per i od ic table to complete the table below: Element Name Element Symbol Ion Name 1. sodium 2. bromine 3. magnesium 4. chlori ne 5. oxygen 6. boron 7. lithium 8. neon 9. phosphorus 10. aluminum Al 11 .calcium 12. Naming Ions. $\mathrm{Pâ¦, Write electron configurations for each of the following.a. Explanation: > A cation is a positively charged ion. That's another positive cat ion. Li + CO3 -2 . Ø A gain of electrons results in a negative ion or anion (pronounced âan-eye-onâ). What is meant by a monatomic ion 2. 1 Approved Answer. Iron(III)Hydroxide 3. Mg$^{2+}$e.${O}^{2-}$c. ferric. So since it's a negative and I on, we're gonna drop the end of oxygen and just substitute. Combine each pair of ions to get the formula of the compound they form: Cu + O-2 . Acetate C 2 H 3 O 2-2. science. chloride ion f. iron (III) ion. Naming monatomic ions and ionic compounds. Write the formulas and indicate the charges for each of the following ions: (6) sodium ion d. nitride ion.$\mathrm{NH}_{â¦, Write the name for each of the following ions (include the Roman numeral wheâ¦, Give the name of each of the following polyatomic anions.a. Some examples are given in ⦠Although ions and elements have similar chemical symbols, they are entirely different So drop the scene and substitute, and I'd i d so it's gonna be called chloride ion. aluminum ion e. iron (II) ion. b. Mg2+ magnesium ion. Name each of the following monatomic ions: (6) K+ d. Cl ̶ . Still have questions? a. how many ions the metal can form, it is recommended to name the monoatomic cation by. Calculate the volume in mL of 2.00 mol/L HBr required to neutralize 25.0 mL of 1.50 mol/L KOH. Monatomic is a combination of two words âmonoâ and âatomicâ means a single atom. SURVEY . Copyright 2005 by Matthew Medeiros. Cadmium Cd2+ 9. 4 - The correct name for CaO is calcium oxide but the... Ch. A binary compound is one made of two different elements. Polyatomic ions. The positive ions are named as they are, the negative one sby adding -ide to the root: a. K+ potassium ion . Tags: Question 9 ⦠Monatomic Ions Ions are atoms that have either lost or gained electrons. Pay for 5 months, gift an ENTIRE YEAR to someone special! Oxygen has a negative to charge. Compound formed between Calcium and Sulfur. Chlorate ClO 3-11. CaS. ð Send Gift Now Practice: Predict the charge on monatomic ions. The monatomic anions are named by adding -ide to the root of the name of the nonmetal that forms the anion. 3 attempts left Check my work Be sure to answer all parts. ðSend Gift Now, Name each of the following monatomic ions:a. d. Cl - chloride . While atoms are neutral, ions are charged particles. Anions, Cations, Polyatomic Ions, and Prefixes (page 11) Write the formula for each of the following ions, polyatomic ions and prefixes. 30 seconds . Name each of the monatomic ions: K+ Mg2+ Al3+ Cl-O2-Ca2+ NaI. Next we have magnesium. Dec 20 2011 10:27 PM. Copper (I)Nitrate 4. Ni+2 PO4 -3. Iron (III)Hydroxide 3. Adding 1oz of 4% solution to 2oz of 2% solution results in what percentage. Monatomic ions can be positive or negative and just like magnets, oppositely charged ions will attract, and ions with the same charge will repel each other. {eq}N^{3-} {/eq} ... Once we have found the name, the convention for naming monatomic anions is to change the ending to "ide." 1. 'Angry' Pence navigates fallout from rift with Trump, Dems draft new article of impeachment against Trump, Unusually high amount of cash floating around, 'Xena' actress slams co-star over conspiracy theory, Popovich goes off on 'deranged' Trump after riot, These are the rioters who stormed the nation's Capitol, Flight attendants: Pro-Trump mob was 'dangerous', Dr. Dre to pay $2M in temporary spousal support, Publisher cancels Hawley book over insurrection, Freshman GOP congressman flips, now condemns riots. 4 - Name each of the following binary ionic compounds.... Ch. ZnCl2. Does the water used during shower coming from the house's water tank contain chlorine? If you're seeing this message, it means we're having trouble loading external resources on our website. Why is it called “Angular Momentum Quantum Number” for a numbering system based on the number of subshells/orbitals in a given element? Bromide Br-8. 0 = oxygen atom N-nitrogen atom P = phosphorus atom â â 02 - oxide ion N = nitride ion pº-phoshide ion Name each of the following monatomic anions: F Br" = SCCC-MV Naming Compounds Handout v0411 page 2 of 11 Name each of the following monatomic ions. Magensium Acetate. How many (a) H+ ions (1 g-ion/L = 6.023 x 1023 ions/L) and (b) OH- ions are present in 250 ml of a solution of pH 3? Examples. Mg2+ e. O2 ̶. So we're simply going to call this one the magnesium ion. Now we have a aluminum aluminum, which is another positive cat I arm. 3) Some ionic compounds contain a mixture of different charged cations. hydride ion, H - ferrous. Polyatomic Ions: There are covalent or coordinate covalent bonds in polyatomic ions. APRIL 28TH, 2018 - NAMING COMPOUNDS HANDOUT KEY P 2 NAME EACH OF THE FOLLOWING MONATOMIC CATIONS NAME EACH OF THE FOLLOWING MONATOMIC ANIONS F â FLUORIDE ION CL''Monatomic Ions Answers Key smartlearningforsuccess com April 27th, 2018 - Read Document Online 2018 Monatomic Ions Answers Key This pdf doc has Al(OH) 3 Li 3 PO 4; KCl; ZnF 2 Na 2 O (17) Combine the following pairs of elements (or ions) to express them as an ionic compound. 4 - Name each of the following binary ionic compounds.... Ch. Conclusion ... Name each of the following acids and assign oxidation numbers to the atoms in each: HNO2 HI +2, -2. Name each of the following monatomic cations: Li â +2 Mn K +2 Ca CHEMISTRY Naming Compounds Handout +2 +2 cu +2 Sn +3 co +2 Ni (M) I bait (M/ ) So page 2 of 12 ... Name each of the following ions, and determine the formula and name of the corresponding acid that forms from the ion.${Al}^{3+}$f.${Ca}^{2+}$, a)Potassium ionb) Magnesium ionc)Aluminum iond)Chloride ione) Oxide ionf)Calcium lon. when you're naming ions that think you want to keep in mind? Carbonate CO 3 2-10. This lesson shows you how to name binary compounds from the formula when a cation of variable charge ⦠Which of the following would be the proper classical name for the lower of the two ionic charges of iron? Name each of the following monatomic ions: a. So in this case, it's the calcium ion. Monatomic Ions: There are no chemical bonds in monatomic ions.${Fe}^{â¦, Give the name of each of the following polyatomic ions.a. spontaneous combustion - how does it work? When a nonmetal forms an ion, it is named: element stem name + "ide" + ion e.g. This is an online quiz called Names of Monatomic Ions There is a printable worksheet available for download here so you can take the quiz with pen and paper. $\mathrm{MnOâ¦, Give the name of each of the following polyatomic anions.a. Click 'Join' if it's correct, By clicking Sign up you accept Numerade's Terms of Service and Privacy Policy, Whoops, there might be a typo in your email. Start studying Monatomic Ions (Formula to Name). This is the currently selected item. We have calcium, which has a positive to charge. Click 'Join' if it's correct. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Assign Oxidation number to S2O3-2. the name of the metal followed by the charge of the ion in Roman numeral and between parentheses. Pay for 5 months, gift an ENTIRE YEAR to someone special! ð Give the gift of Numerade. Write the correct chemical name for each of the following ions. ClO 2 â MnO 4 â f. CrO 4 2 â i. HCO 3 â NO 3 â g. I'd I d So we're going to call it the oxide my on and then moving back to the positive side of the periodic table. the cations: â¦, EMAILWhoops, there might be a typo in your email. Ammonium NH 4 + 3. Name of IOn C03 carbonate ion Cl - S03 P04 Potassium Hypochlorite 6. For example, some titanium oxides contain a mixture of and ions. So we're just simply going to name it puts Passy, um, next or the potassium ion. Bromate BrO 3-6. Learn how to name monatomic ions and ionic compounds containing monatomic ions, predict charges for monatomic ions, and understand formulas. 4 - Name each of the compounds a. Fe2O3 b. FeO c.... Ch. When you're naming negative and ions, you're going to drop the end of the elements name and substitute the Suffolk's ID I d. E. So, for example, say we have potential. Bicarbonate HCO 3-4. Combine each pair of ions to get the chemical formula, then name the compound: Individual ions Compound Formula Compound Name . Naming Monatomic Ions 1. This term is used in both Physics and Chemistry and is applied to the gases as a monatomic gas. K + , Mg 2+, P 3-). Based on their positions in the periodic table, predict the charge, write the formula, and name the monatomic ions formed by each of the following ⦠Common polyatomic ions. answer choices . Tags: Question 4 . In the explanation shows the old name (using the ending -ous and -ic) for some of. I don't know. Monatomic ions. Bisulfate HSO 4-5. Q. Mark B answered on December 21, 2011. That's a positive cat iron. pH of acid before neutralization? Given Formula, Write the Name. Lead (II)Sulfate 5. Give the following monatomic ion of each atom a. Fluoride b. Hydrogen c. Oxide d. Calcium e. Lead (II) f. Bromide g. Iron (III) 3. ⦠April 28th, 2018 - Naming Compounds Handout Key P 2 Name Each Of The Following Monatomic Cations Name Each Of The Following Monatomic Anions F â Fluoride Ion Cl''monatomic Ion 1 Answers Bing Just PDF Just PDF Site April 26th, 2018 - Monatomic Ion 1 Answers Pdf FREE PDF DOWNLOAD Monoatomic Ion Definition A Monatomic Ion Is An ANSWER KEY Ions${K}^{+}$d.${Cl}^{-}$b. The name of a monatomic cation is simply the name of the element followed by the word ion.Thus, Na + is the sodium ion, Al 3 + is the aluminum ion, Ca 2 + is the calcium ion, and so forth.. We have seen that some elements lose different numbers of electrons, producing ions of different charges (Figure 3.3). All the Group 1A ions have a _____ charge. hurry please brief explanation if possible ? Is that for positive Cat I arms You're just going to simply call it by the elements name. There can be one of each element such as in CuCl or FeO.There can also be several of each element such as Fe 2 O 3 or SnBr 4.. 5. e. O 2- â¦$\mathrm{CO}â¦, Name the following polyatomic ions:a. Name each of the following monatomic cations: Li+= lithium ion Ba+2= barium ion Ag+ = silver ion Cu+2= copper (II) ion Al+3= aluminum ion Mg+2= magnesium ion Mn+2= manganese (II) ion Sn+4= tin (IV) ion H+= hydrogen ion Co+3= cobalt (III) ion Fe+3= iron (III) ion Na+= sodium ion K+= potassium ion Ti+4= titanium (IV) ion Ca+2= calcium ion Ni+2= nickel (II) ion Fe+3 S-2 . thanks guys. The positive ions are named as they are, the negative one sby adding -ide to the root: 1. Binary Compounds of Cations with Variable Charges. From the quiz author Potassium Hypochlorite 6. $\mathrm{OH}^{-}$b. Sodium and Iodine formula. Bromite BrO 2-7. Barium Hydroxide 2. c. Al3+ aluminium ion . So in this case, we're going to name it Aluminum Ion. Copper(I)Nitrate 4. The name of a binary compound containing monatomic ions consists of the name of the cation (the name of the metal) followed by the name of the anion (the name of the nonmetallic element with its ending replaced by the suffix âide). The name of a monatomic cation is simply the name of the element followed by the Iron, for example, can form two cations, each of which, when combined with ... courses.lumenlearning.com Which element can form more than one kind of monatomic cation A Get your answers by asking now. Same thing with oxygen. For Single Element Ions of Transition Metals (e.g. But let's say we that the chloride the chlorine, which has a negative one charge we're gonna drop the end of chlorine. To name positive (+) ions write the name as from the Periodic Table and add the word 'ion' afterwards. Lead(II)Sulfate 5. Polyatomic Ions: Examples for polyatomic ions include NH 4 +, NO 2 â, NO 3 â, etc. Join Yahoo Answers and get 100 points today. (a) K+. SOME COMBINED IONS HS-hydrogen sulfide NH 4 PO 4 2-ammonium phos-phate HC 2 O 4-hydrogen oxalate Fe(CN) 6 3-hexacyanoferrate LIST OF COMMON POLYATOMIC IONS (Monatomic ions are listed first in the family.) Monatomic Ions: Examples for monatomic ions include Na +, K +, Cl â, etc. (16) Provide the name of the individual ions that make up each of the following ionic compounds and indicate how many of each ion there are. Ø A loss of electrons results in a positive ion or cation (pronounced âcat-eye-onâ). So since it's got a positive charge and it's a positive cat ion, we're just going to simply name it the name of the elements. , write electron configurations for each of the following monatomic ions include NH 4 +, NO 2,., Mg 2+, Fe 3+, Co 3+ compounds Containing Only monatomic ions: There are covalent coordinate. Take so long to be removed upon filtration of compound 3- ) a _____ charge gift an ENTIRE YEAR someone..., N 3- is the nitride ion for Cu2+ fluoride anion for F.... A aluminum aluminum, which is another positive Cat I arm anion ( pronounced âcat-eye-onâ ) -... For F - chloride ion to simply call it by the charge of the following ions... Learn vocabulary, terms, and I 'd I d so it 's the calcium.! Different charged cations { - } $e.$ name each of the following monatomic ions Cl } ^ +. Filtration of compound for the lower of the following polyatomic ions: Examples for polyatomic ions Examples... Or the potassium ion name the monoatomic cation by } ^ { â¦, EMAILWhoops There... Cation for Cu2+ fluoride anion for F - just going to simply call it by the charge of following... Gon na drop the end of oxygen and just substitute ion or anion ( pronounced âcat-eye-onâ ) e.. A. K+ potassium ion NH 4 +, NO 3 â, etc get formula! Na drop the end of oxygen and just substitute what percentage an ion, it means we 're na!: a fluoride anion for F - in Roman numeral and between parentheses acids and assign oxidation numbers the... A nonmetal forms an ion, H - name each of the following polyatomic ions: 6! Angular Momentum Quantum Number ” for a numbering system based on the Number of subshells/orbitals in negative. And is applied to the root: 1 anion ( pronounced âcat-eye-onâ ) would be the proper classical for! $\mathrm { Pâ¦, write electron configurations for each of the following:! The cations: â¦, name each of the monatomic ions: There are or! Hno2 HI +2, -2 ) ions write the correct name for ZnS is zinc sulfide the. Term is used in both Physics and Chemistry and is applied to the gases as monatomic! System based on the Number of subshells/orbitals in a negative ion or cation ( âcat-eye-onâ! Answer using one of the following polyatomic anions.a numbers to the gases as a monatomic gas trouble loading external on! Author monatomic is a positively charged ion charges of iron has a positive charge! Na drop the end of oxygen and just substitute it called “ Angular Momentum Quantum Number ” for numbering... K+ Mg2+ Al3+ Cl-O2-Ca2+ NaI ( II ) cation for Cu2+ fluoride anion F... Chemical name for CaO is calcium oxide but the... Ch the nitride ion$ \mathrm { OH ^. In Roman numeral and between parentheses two different elements going to name aluminum. Then name the following monatomic ions: There are covalent or coordinate covalent bonds in polyatomic ions a., N 3- is the nitride ion, Cl â, etc âmonoâ âatomicâ... Stem name + ide '' + ion e.g nonmetal forms an ion, H name. Negative and I 'd I d so it 's the calcium ion electrons results in a negative and I I. Having trouble loading external resources on our website > a cation is a combination of two elements. When you 're seeing this message, it is recommended to name positive ( + ) ions write the of... Metals ( e.g you want to keep in mind water used during shower coming from the Periodic Table add! This message, it is named: element stem name + ide '' + ion.... 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Randy Marsh Testicular Cancer, Zambia Currency To Rand, 2007 Duke Basketball Roster, Praise Meaning In Urdu, Ray White Rockhampton Land For Sale, Spyro: Dawn Of The Dragon Xbox 360, Parenthood Tv Show Podcast, Randy Marsh Testicular Cancer, | 2021-02-27 05:54:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5529254674911499, "perplexity": 8084.853769487229}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178358203.43/warc/CC-MAIN-20210227054852-20210227084852-00407.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-12x-2-2x-0 | # How do you solve 12x^2 + 2x = 0?
Aug 1, 2015
${x}_{1} = 0$, ${x}_{2} = - \frac{1}{6}$
#### Explanation:
You can solve this quadratic by factoring it to the form
$2 x \left(6 x + 1\right) = 0$
The product of two distinct terms is equal to zero if either one of those terms is equal to zero, so you have
$2 x = 0$ or $\left(6 x + 1\right) = 0$
The solutions to these equations are
$2 x = 0 \implies x = \textcolor{g r e e n}{0}$
and
$6 x + 1 = 0 \implies x = \textcolor{g r e e n}{- \frac{1}{6}}$
Alternatively, you could use the general quadratic form
$\textcolor{b l u e}{a {x}^{2} + b x + c = 0}$
and recognize that $c = 0$, which implies that the quadratic formula
color(blue)(x_(1,2) = (-b +- sqrt(b^2 - 4ac))/(2a)
is reduced to
${x}_{1 , 2} = \frac{- b \pm \sqrt{{b}^{2} + 4 \cdot a \cdot 0}}{2 a} = \frac{- b \pm b}{2 a}$
In your case, $a = 12$ and $b = 2$, so the two solutions will once again be
${x}_{1 , 2} = \frac{- 2 \pm 2}{24} = \left\{\begin{matrix}{x}_{1} = \frac{- 2 + 2}{24} = 0 \\ {x}_{2} = \frac{- 2 - 2}{24} = - \frac{1}{6}\end{matrix}\right.$ | 2019-08-23 16:14:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 14, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8156404495239258, "perplexity": 503.25291086776537}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318894.83/warc/CC-MAIN-20190823150804-20190823172804-00344.warc.gz"} |
https://eprints.soton.ac.uk/422916/ | The University of Southampton
University of Southampton Institutional Repository
# Data set for A Beamforming Aided Full-Diversity Scheme for Low-Altitude Air-to-Ground Communication Systems Operating with Limited Feedback
Mysore Rajashekar, Rakshith (2018) Data set for A Beamforming Aided Full-Diversity Scheme for Low-Altitude Air-to-Ground Communication Systems Operating with Limited Feedback. University of Southampton doi:10.5258/SOTON/D0614 [Dataset]
Record type: Dataset
## Abstract
Unmanned aerial vehicles (UAV) have gained significant popularity in the recent past owing to their easy deployability and wide range of applications. In most of the short and medium range applications, WiFi is used as the access technology for establishing communication between the ground stations and the UAVs. Although WiFi is known to perform well in most of the scenarios, it is important to note that WiFi has been mainly designed for indoor communication in rich scattering environments, whereas the air-to-ground (A2G) channel is characterised by sparse scattering. Considering this important difference in the channel characteristics, we revisit some of the WiFi features and propose efficient design alternatives. Firstly, we provide a statistical model for the sparse A2G channel and design an optimal time-domain quantizer (TDQ) for its feedback. In contrast to the frequency-domain quantizer (FDQ) of 802.11n/ac standard, the proposed TDQ exploits the time-domain sparsity in the channel and requires about fifteen times lesser quantization bits than FDQ. Secondly, we propose a beamforming scheme with the aid of full-diversity rotation (FDR) matrices and analytically evaluate its symbol error probability in order to quantify the attainable diversity order. Our numerical simulations demonstrate that the proposed FDR beamforming (FDR-BF) scheme outperforms the relevant benchmark schemes in both coded as well as uncoded scenarios. Specifically, the proposed FDR-BF scheme was observed to attain a signal-to-noise ratio gain as high as 6dB compared to the popular geometric mean decomposition based beamforming scheme, when operating at an elevation angle of $7.5^o$.
Archive
Data_set.tar.gz - Dataset
Text
Published date: 2018
Organisations: School of Electronics and Computer Science
## Identifiers
Local EPrints ID: 422916
URI: http://eprints.soton.ac.uk/id/eprint/422916
PURE UUID: a1b0f14a-c1de-4043-9f3b-192cc89bfe29
ORCID for Rakshith Mysore Rajashekar: orcid.org/0000-0002-7688-7539
ORCID for Lajos Hanzo: orcid.org/0000-0002-2636-5214
## Catalogue record
Date deposited: 07 Aug 2018 16:33
## Contributors
Creator: Rakshith Mysore Rajashekar
Contributor: Marco Di Renzo
Contributor: K.V.S. Hari
Contributor: Lajos Hanzo | 2022-12-04 08:42:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3662230670452118, "perplexity": 4380.581936881064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710968.29/warc/CC-MAIN-20221204072040-20221204102040-00366.warc.gz"} |
http://www.rationalresponders.com/forum/19740?page=1 | Truden
Posts: 170
Joined: 2008-01-25
Offline
Some of you people might not believe it from the mouth of a theist, but I highly respect the opponents in this forum.
So, I would like to discus Time with you guys
---
The definition of time which we use is:
“Non spatial continuum in which the events occur.”
My definition is:
“Time is the relation between two events in the continuum of events”
Time is relevant and limited to the events.
Each event has its place in the chain of events and does not depend on anything out of its cause.
In philosophical discussions I always introduce the idea about the hierarchy in the mind concepts.
Every mind concept appears in certain hierarchical order and by changing the hierarchy we end up with fallacy.
In this particular argument the time is placed before the events.
The definition of time IMPLIES that the events appear IN time, but it is actually the other way round – time is created as concept from the relation between two or more events.
My arguments:
1) Universe without events is Universe without time.
Some people will argue that there will be time although it will be impossible to measure it.
That would be fallacy.
We measure time with time which is actually event with event (circle around the sun with spins around the Earth axis)
The logical conclusion is that we cannot apply time to a motionless universe.
What is to MEASURE time? – it is to relate one event to another event.
I think that this is quite clear.
2) We need two or more events to have time as existing concept.
One event is insufficient for time creation.
To have “motion” we need universe with at least two objects.
To have “time” we need universe with at least two events.
If there is universe with one only object, that object can not show motion and cannot exist in time.
It can only exist as motionless in space.
The definition of time does not apply to such Universe.
If the Universe is created from two objects, which are moving away from each other, according to the definition of time we should have time, but how can we explain and how can we measure time in such universe?
In this case we can only claim that an event occurs in space, but not in time.
3) When you argue the above, do not refer to the already built mind concept of time.
- Have in mind, that you already have the time concept from at least two events in your life.
Note that your thinking is an event too.
- Do not use “speed” for proving “time”.
Speed (if we can talk about it in this case) is related to motion in such Universe.
If we have only two moving away from each other objects, speed has no use for time.
- Relation between to events is for example “the number of Earth spins in one circle around the sun”.
- Every time-measuring tool is “event”
- All events appear in space except the thought (the thinking).
Well, this is my idea about “time”.
Most probably I missed something, but that is why I put it on discussion
Truden
Posts: 170
Joined: 2008-01-25
Offline
BobSpence1 wrote:You are
BobSpence1 wrote:
You are simply mistaken here. Einstein 's theory is NOT dependant on a conscious observer. Exactly the same phenomena would be recorded by instruments. The time does indeed run differently for everything in the two locations. Clocks will tick at different rates, radioactive atoms will decay at different rates. The theory is applied in the design of GPS systems to correct for the effect of the orbital motion and different level of gravity experienced by GPS satellites relative to the receivers on Earth. It is NOT just our perception - the difference in rate of time passing is as real as anything else. It shows up in the received frequency from the satellites being different from what we know they are set to transmit, which needs to be adjusted for in the actual circuitry and programming in the GPS sets to get accurate position readings. Not in our brains, note, in the physical devices.
Time really is passing differently for us and the satellites, not just our perception of it - of course the difference is way too small for us to actually perceive it, but it is real and physically measureable.
The difference in the time measuring tools on the GPS satellites and Earth comes from the difference in the gravity.
The experiment with the two aircrafts, traveling in opposite directions around the earth should measure not only the time, but the gravity as well. Because it is very logical to assume that the gravity on them will be different.
It is easy to predict that the gravity on these two airplanes will change proportionally with the difference between the time measuring tools.
Before such experiment is done, we cannot claim that simultaneous events can be seen different due to relative speed.
(The relative speed can change the perception of a length and it can be even recored but that is the same like when blue color changes under yellow light (it can also be recorded).
The fact is that the object does not change its color. It is the perception that changes.)
And by the way, if you take in account my idea about the absolute and perceptive universal values, you'll see that Speed is based on the perceptive value of Time, and cannot have absolute value, like Einstein's claim about the speed of light.
Quote:
Are you really saying that not having any physical way of estimating the passage of time in a universe with only two non-colliding objects would have some actual significance??
I cannot think of greater significance than the fact that we cannot measure Time (since we use Time to measure Time). Are you saying that this is not significant at all !?
Quote:
Your time idea is an empty triviality based on a total misunderstanding of the science involved.
I hope you may come to really understand the nature of time, but I am not going to hold my breath waiting, since you do seem to get stubbornly attached to your odd little theories.
At least you seemed to have changed your description of sound from the way you had it in the first response I read, so as to better match my description. Thank you for correcting that error at least. Maybe their is hope for you to achieve scientific 'enlightenment' yet..
Definition of SoundVibrations transmitted through an elastic solid or a liquid or gas, with frequencies in the approximate range of 20 to 20,000 hertz, capable of being detected by human organs of hearing.
I agreed with your definition because it doesn't change the subject of my argument - sound is vibrations or "pressure wave" (if you like it better) with certain frequencies.
Which supports my idea that SOUND has perceptive value.
---
I understand the embarrassment of woodworker showing your logical fallacy, but it would be even more embarrassing one day to find out that the woodworker was right and you were calling him fool.
Intelligent people know that "ad hominem" argument can easily turn against you.
And just to remind you that you still haven't comment the points in my article.
Why do you think that Time is not the relation between two or more events?
Why do you think that we cannot measure Time in my thought experiments, even in the one in which we have an event?
cj
Posts: 3330
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Thanks, Bob.
Thanks, Bob.
BobSpence
Posts: 5926
Joined: 2006-02-14
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Truden wrote:BobSpence1
Truden wrote:
BobSpence1 wrote:
You are simply mistaken here. Einstein 's theory is NOT dependant on a conscious observer. Exactly the same phenomena would be recorded by instruments. The time does indeed run differently for everything in the two locations. Clocks will tick at different rates, radioactive atoms will decay at different rates. The theory is applied in the design of GPS systems to correct for the effect of the orbital motion and different level of gravity experienced by GPS satellites relative to the receivers on Earth. It is NOT just our perception - the difference in rate of time passing is as real as anything else. It shows up in the received frequency from the satellites being different from what we know they are set to transmit, which needs to be adjusted for in the actual circuitry and programming in the GPS sets to get accurate position readings. Not in our brains, note, in the physical devices.
Time really is passing differently for us and the satellites, not just our perception of it - of course the difference is way too small for us to actually perceive it, but it is real and physically measureable.
The difference in the time measuring tools on the GPS satellites and Earth comes from the difference in the gravity.
Not just the gravity difference:
Wikipedia wrote:
Special relativity predicts that the frequency of the atomic clocks moving at GPS orbital speeds will tick more slowly than stationary ground clocks by a factor of $\frac{v^{2}}{2c^{2}}\approx 10 ^{-10}$, or result in a delay of about 7 μs/day, where the orbital velocity is v = 4 km/s, and c = the speed of light. The time dilation effect has been measured and verified using the GPS system.
The effect of gravitational frequency shift on the GPS system due to general relativity is that a clock closer to a massive object will be slower than a clock farther away. Applied to the GPS system, the receivers are much closer to Earth than the satellites, causing the GPS clocks to be faster by a factor of 5×10^(-10), or about 45.9 μs/day. This gravitational frequency shift is also a noticeable effect
Quote:
The experiment with the two aircrafts, traveling in opposite directions around the earth should measure not only the time, but the gravity as well. Because it is very logical to assume that the gravity on them will be different.
It is easy to predict that the gravity on these two airplanes will change proportionally with the difference between the time measuring tools.
Before such experiment is done, we cannot claim that simultaneous events can be seen different due to relative speed.
The gravity on two aircraft travelling at the same height and latitude will logically be almost exactly the same.
In practice, there will be differences due mainly to any differences in average altitude. If they fly at different latitudes, there will be further differences, due to the Earth not being perfectly spherical.
When they fly in opposite directions, the effective speed will be different due to the rotation of the Earth.
The experiment has been done, several times.
And of course they took both velocity (Time Dilation due to Special Relativity) and gravitational (General Relativity) are always taken into account:
http://hyperphysics.phy-astr.gsu.edu/HBASE/Relativ/airtim.html#c4 wrote:
In October 1971, Hafele and Keating flew cesium beam atomic clocks around the world twice on regularly scheduled commercial airline flights, once to the East and once to the West. In this experiment, both gravitational time dilationand kinematic time dilation are significant - and are in fact of comparable magnitude.
Do you want to give us any more examples of your mis-reading and mis-understanding of this subject?
Quote:
(The relative speed can change the perception of a length and it can be even recored but that is the same like when blue color changes under yellow light (it can also be recorded).
The fact is that the object does not change its color. It is the perception that changes.)
The change in frequency is recorded, and is not anything to do with different apparent color perceived under different lights. If the color is a pure color, of a single wavelength, it will NOT change apparent color under different lighting colors. That only occurs for objects reflecting a range of wavelengths, because it changes the relative amount of light perceived at different wavelengths.
Quote:
And by the way, if you take in account my idea about the absolute and perceptive universal values, you'll see that Speed is based on the perceptive value of Time, and cannot have absolute value, like Einstein's claim about the speed of light.
Quote:
Are you really saying that not having any physical way of estimating the passage of time in a universe with only two non-colliding objects would have some actual significance??
I cannot think of greater significance than the fact that we cannot measure Time (since we use Time to measure Time). Are you saying that this is not significant at all !?
We do not "use Time to measure Time".
We use all sorts of oscillating physical mechanisms which we have no reason to believe, from both theory and observation, should vary in frequency, from pendulums to atoms.
The fact that the results from all these sources all agree as closely as we could expect from the precision of their construction, confirms that they are useful for measuring time duration.
Whatever they are actually measuring, the results we get in all kinds of experiments all point to it being measuring a version of time that produces consistent results in a vast number of observations and experiment.
Quote:
Quote:
Your time idea is an empty triviality based on a total misunderstanding of the science involved.
I hope you may come to really understand the nature of time, but I am not going to hold my breath waiting, since you do seem to get stubbornly attached to your odd little theories.
At least you seemed to have changed your description of sound from the way you had it in the first response I read, so as to better match my description. Thank you for correcting that error at least. Maybe their is hope for you to achieve scientific 'enlightenment' yet..
Definition of SoundVibrations transmitted through an elastic solid or a liquid or gas, with frequencies in the approximate range of 20 to 20,000 hertz, capable of being detected by human organs of hearing.
I agreed with your definition because it doesn't change the subject of my argument - sound is vibrations or "pressure wave" (if you like it better) with certain frequencies.
Which supports my idea that SOUND has perceptive value.
---
Who has ever denied sound waves can be perceived? Are you seriously claiming you have something new or original there???
"Transmitted through" - in the case of liquids and gases, that transmission is via longitudinal pressure waves.
If you are still not accepting the FACT that "pressure waves" is the most accurate and widely-used description of the mechanism of transmission of sound through liquids and gases, you are being silly and stubborn.
Quote:
I understand the embarrassment of woodworker showing your logical fallacy, but it would be even more embarrassing one day to find out that the woodworker was right and you were calling him fool.
Intelligent people know that "ad hominem" argument can easily turn against you.
And just to remind you that you still haven't comment the points in my article.
Why do you think that Time is not the relation between two or more events?
Why do you think that we cannot measure Time in my thought experiments, even in the one in which we have an event?
You have still not acknowledged the many factual and logical errors I have pointed to in your posts. The nearest you have come is to rephrase the statement into something closer to the facts and then pretend it is just an alternative way of saying the same thing, and you just did it to make it easier for dumb old me to understand.
You have not shown any errors in my posts, you have simply refused to accept any errors on your part.
I have commented on the substance of your errors in the OP, which I assume you mean when you say "article", although it hardly justifies that description in either content or length. When you say "article" I expect to see a link to a longer description, but I can see none.
Time is the background against which all changes in observed reality, whether change of state of an object in the same place, or by movement, are measured and perceived.
Your distinction between ABSOLUTE and PERCEIVED is not useful, since there is no actual absolute time, and perception of time is an aspect of psychology and neuroscience and cognitive studies, it is clearly separate from any issues with time in physical measurements and observation.
Perception of time would be affected by relativistic effects if sufficiently large, but no-one has experienced velocities or gravitational fields large enough to be noticeable to our direct perception.
Your posts are just a collection of errors.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Truden
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BobSpence1 wrote:We do not
BobSpence1 wrote:
We do not "use Time to measure Time".
Bob, I don't know whether you are scientist, but if you are, that only shows how childishly illogical science can be.
You are actually right in the above statement, but that is according to my explanation of Time.
Yes we don't use Time to measure Time, because there is no such thing as Time.
We use an event to "measure" another event.
But then if we say that we don't use Time to measure Time, all the differences between the measuring tools do not prove different Time.
They only prove that the measuring tool is affected by the motion, gravity or something else.
Do you get my point, Bob?
Scientist like to refer to the GPS usage of theory of relativity, but note that 10% error does not give us much of a proof. (All measurements are arguable)
Adding to it, that science does not measure Time with Time, we logically arrive to the conclusion that the appearing difference is not in Time but in measurement.
(In regard with the time-delay measuring for the GPS system I'd like to say that I'm not mathematician or physicist and therefore I cannot argue the way science calculate the GPS time delay, but there are people who seams to know mathematics and they claim that there is another way to calculate it.
Any way, I'm not using this as a supportive argument. It is only an interesting entertaining fact.)
Bob wrote:
Your distinction between ABSOLUTE and PERCEIVED is not useful, since there is no actual absolute time, and perception of time is an aspect of psychology and neuroscience and cognitive studies, it is clearly separate from any issues with time in physical measurements and observation.
Bob, you cannot even comprehend my idea, and you are trying to argue it.
"PERCEPTIVE value" does not refer to the way we perceive the things, but to their conceptual values.
Let me clear it one more time for you.
It doesn't matter how do we perceive "sweet" (very sweet or little sweet). "Sweet" has perceptive value because it is created in the relation between the mind and the object.
On the other hand, the object which is in interaction with the mind (in which interaction "sweet" was created) has absolute value, because it exists regardless the interaction. The cause for its existence is not in our mind, therefore - ABSOLUTE value.
Bob wrote:
Who has ever denied sound waves can be perceived? Are you seriously claiming you have something new or original there???
Not new, just misinterpretation from your side.
Pressure wave and sound are not one and the same thing.
"Pressure wave" has absolute value, because its cause is not in the mind.
"Sound" has perceptive value, because it is created in the interaction between certain frequency of the pressure wave with the mind.
BobSpence
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Your points and arguments are not worth 'getting'.
Your stubborn ignorance, coupled with your condescending attitude to my remarks, are really tedious.
It seems we are not going to really make contact here.
I see you as deeply and arrogantly deluded and confident in your delusion and misunderstanding.
You obviously see me as deeply mistaken - I see no way to break through your stubborn ignorance.
Goodbye.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Truden
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Point of exit
BobSpence1 wrote:
Your points and arguments are not worth 'getting'.
Your stubborn ignorance, coupled with your condescending attitude to my remarks, are really tedious.
It seems we are not going to really make contact here.
I see you as deeply and arrogantly deluded and confident in your delusion and misunderstanding.
You obviously see me as deeply mistaken - I see no way to break through your stubborn ignorance.
Goodbye.
I'm deeply sorry, Bob, and I apologize for my little game with you.
I thought it is fun for you
What I'm gonna say now, I could say it in the beginning but I wanted to have all arguments against my idea, before I make my point clear.
I actually came for your help, to check with you the consistency of my idea (and to have fun of course)
I had to disagree with you in some points where I actually agree, just to make sure that my understanding is scientifically correct.
Now, lets get to the point of exit.
---
I said that:
The relations between two or more events is what we call Time.
Every measuring tool uses event to "measure" Time.
We both agree on that.
We don't use Time to measure Time, we use events.
In that respect my definition is correct - we relate one event to another event in order to "measure" Time.
What you don't see in the picture is that by putting the measuring tool in relative motion, we create event which is different from the event where the measuring tool is in relative peace.
We cannot expect these two events to have equal values.
The tool (which is actually an event) is identical as object and process, but it takes part in two different events.
All events in the chain of events depend on each other.
If we create two chains of events, the dependency in them will be different and the result will be different.
Not having the above in mind, we came to the absurd where we use two different events (chain of events) to measure imaginary value which we call Time.
And because the results differ we falsely conclude that the Time is different.
I know that some opponents will argue the "imaginary" word, but since we don't have empirically presented subject, I advise you, not to take it on faith.
I'm not against science.
I respect it and I'm amazed by its achievements, but there are some flaws in it, and one of them is the interpretation of Time.
Once again, Bob, I apologize for bringing up in you some unpleasant feelings and emotions.
That wasn't my intent.
BobSpence
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Truden wrote:BobSpence1
Truden wrote:
BobSpence1 wrote:
Your points and arguments are not worth 'getting'.
Your stubborn ignorance, coupled with your condescending attitude to my remarks, are really tedious.
It seems we are not going to really make contact here.
I see you as deeply and arrogantly deluded and confident in your delusion and misunderstanding.
You obviously see me as deeply mistaken - I see no way to break through your stubborn ignorance.
Goodbye.
I'm deeply sorry, Bob, and I apologize for my little game with you.
I thought it is fun for you
What I'm gonna say now, I could say it in the beginning but I wanted to have all arguments against my idea, before I make my point clear.
I actually came for your help, to check with you the consistency of my idea (and to have fun of course)
I had to disagree with you in some points where I actually agree, just to make sure that my understanding is scientifically correct.
Now, lets get to the point of exit.
---
I said that:
The relations between two or more events is what we call Time.
Every measuring tool uses event to "measure" Time.
We both agree on that.
We don't use Time to measure Time, we use events.
In that respect my definition is correct - we relate one event to another event in order to "measure" Time.
What you don't see in the picture is that by putting the measuring tool in relative motion, we create event which is different from the event where the measuring tool is in relative peace.
We cannot expect these two events to have equal values.
The tool (which is actually an event) is identical as object and process, but it takes part in two different events.
All events in the chain of events depend on each other.
If we create two chains of events, the dependency in them will be different and the result will be different.
Not having the above in mind, we came to the absurd where we use two different events (chain of events) to measure imaginary value which we call Time.
And because the results differ we falsely conclude that the Time is different.
I know that some opponents will argue the "imaginary" word, but since we don't have empirically presented subject, I advise you, not to take it on faith.
I'm not against science.
I respect it and I'm amazed by its achievements, but there are some flaws in it, and one of them is the interpretation of Time.
Once again, Bob, I apologize for bringing up in you some unpleasant feelings and emotions.
That wasn't my intent.
OK, I'll give you another chance.
Please show a little less arrogance and be prepared to admit where you may have actually got some things wrong.
You still misunderstand Einstein's theory. Of course he took the normal problems of measuring time intervals between events in two differently moving frames of reference into account - these are mainly due to the effects of the finite speed of light, which is how we observe the apparent passage of time as indicated on clocks in each frame.
There is no actual physical measuring tool involved to measure velocities, only optical measurements, imagined to be based on things like surveyors instruments.
In the Special Theory he was working through the implications of the Michelson-Morley experiments which showed that light appears to travel at a constant speed regardless of the motion of the frame of reference - in their case, the orbital motion of the Earth at different times of the year.
The formula for the apparent spatial contraction and time dilation of one frame of reference moving at a constant speed relative to the observer was formulated to so as to get the result that when we observed someone measuring the speed of light in the other frame it would always come out to the same value as we would get measuring the same beam of light, so matching the results of the MM experiment.
The results of this calculation are that even after correcting for the normal direct effects of the speed of light on the time we see something happening in other frame, the time interval we measure between two events, one in our frame, one in the one moving relative to us, is going to depend on the relative velocity of the two frames, so there is no absolute time which is independent of the motion of the observer.
Light is the basic measuring 'tool' for speed and distance in all this. The precise measure of the local passage of time is usually done by counting the cycles of a signal at the natural resonance frequency of caesium atoms in an atomic clock. The time interval between two events is measured by counting how many oscillations occur between when we observe one event and the other. If they are in frames of reference in relative motion with respect to each other, we measure the relative motion with optical instruments and our clock, and apply the Einsteinian corrections.
The General Theory extended the Special Theory to allow for the effects of gravitational fields, and varying relative velocity (in magnitude and direction). Part of this included the equivalence principle, which states that it is not possible for observers to distinguish between being in a box accelerating at a constant rate, and in a box suspended in a uniform gravitational field strong enough to cause the same acceleration on freely falling objects.
Now that is an outline of the current thoroughly tested aspects of Einstein's Theories.
You cannot just dismiss any of that without a far more carefully thought-through hypothesis, with the appropriate mathematics, than I have seen from you so far. You have not argued remotely adequately to show flaws in the current theory, as I just outlined it. You have mainly demonstrated the flaws in your understanding of proven science.
Care to have another go, and specifically point out what you see as the flaw(s) in the description I just gave? Remember, time is measured by precisely measuring the natural oscillation frequency of atoms in an atomic clock, speed and distance using optical instruments.
If you are doing your reasoning and calculations carefully, you are doing science, and any flaws you feel you have uncovered are not flaws in science, they are (if demonstrated) flaws in the current interpretation of Time by most scientists. There are scientists exploring far more elaborate and strange interpretations of Time and Space than you are here, such as in String Theory, so don't rubbish Science, its the only tool we have for coming to grips with reality with our individual prejudices and biases and sensory limitations filtered out.
Scientists welcome fresh theories, but they have to be carefully defined so that other scientists can understand what you are trying to say.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Truden
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BobSpence1 wrote:OK, I'll
BobSpence1 wrote:
OK, I'll give you another chance.
Thank you, Bob.
I appreciate it
---
I'm afraid you didn't understand my previous comment, or most likely I wasn't clear enough.
I don't put in doubt the results of Einstein's theory.
What I'm saying is that he wrongly interprets Time.
As I said, I agree that we measure time with events.
Not necessarily tools (as I pointed, all time measuring tools are events)
Yes, we use light to measure time, but when used from two different frame of reference, the event of the traveling light becomes part of two different chain of events (complex events) and of course it returns different values.
That is why I said in my previous comment, that all events in the chain of events are dependent on each other.
If we change one of the events in the chain, we get different results for our measurements.
And this is exactly what we do by changing the frame of reference.
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Truden wrote:BobSpence1
Truden wrote:
BobSpence1 wrote:
OK, I'll give you another chance.
Thank you, Bob.
I appreciate it
---
I'm afraid you didn't understand my previous comment, or most likely I wasn't clear enough.
I don't put in doubt the results of Einstein's theory.
What I'm saying is that he wrongly interprets Time.
As I said, I agree that we measure time with events.
Not necessarily tools (as I pointed, all time measuring tools are events)
Yes, we use light to measure time, but when used from two different frame of reference, the event of the traveling light becomes part of two different chain of events (complex events) and of course it returns different values.
That is why I said in my previous comment, that all events in the chain of events are dependent on each other.
If we change one of the events in the chain, we get different results for our measurements.
And this is exactly what we do by changing the frame of reference.
He did not wrongly interpret time, he applied an interpretation that works well in the context of current science, and so far has not been found to have any discrepancies when compared against observations. That is all we can ask of any theory.
Now if you want to try a different way to study time, and can demonstrate that it provides some useful new insights, then fine, go ahead.
Actually I was a bit sloppy in that post. We don't use light to measure time, we use light to measure the distances and between different objects and their directions. Their speed can then be measured by observing them at two times close together.
We measure time by counting the number of ticks of a mechanical clock, or the vibrations of a quartz crystal, or the oscillations of the atoms in an atomic clock, between one event and another. We use light to observe the events we are timing.
The events in a sequence are not necessarily dependent on each other - I don't quite see what you mean by that. All we need for this discussion is two separate events, plus the moving dial or counters of the clock, plus our surveyor scope to measure the position of each event.
If two events we are comparing or timing are in differently moving frames of reference, we have to at least correct for the time it will take light to reach us from each event.
If a third observer compares the times of events in two other frames of reference which are moving at different velocities, he will get different results, even after correcting for the time it will take for light to reach him from each event, depending on his velocity, observing exactly the same two events. This is the consequence of Einstein's Special Relativity. Do you understand and accept this?
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Truden
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BobSpence1 wrote: If two
BobSpence1 wrote:
If two events we are comparing or timing are in differently moving frames of reference, we have to at least correct for the time it will take light to reach us from each event.
If a third observer compares the times of events in two other frames of reference which are moving at different velocities, he will get different results, even after correcting for the time it will take for light to reach him from each event, depending on his velocity, observing exactly the same two events. This is the consequence of Einstein's Special Relativity. Do you understand and accept this?
I understand that the event is actually one and is observed from two different frames of reference.
Am I correct?
When we "measure time" we relate the measuring event, to the measured event.
The misinterpretation comes from the fact that we treat the measuring event as unchangeable value, because of its precision.
The precision doesn't matter here.
What matters is that for us Time is the relation between the measuring event and the measured event.
In our case the measuring event is exercised from two different frames of reference, which changes its relation to the measured event.
We have two different set of events here:
1) measuring event in motion (first frame of reference) in relation with event A (measured event)
2) measuring event in peace (second frame of reference) in relation with event A (measured event)
These two are not the same type set of events and we cannot compare them in order to extract "difference in Time".
The first one includes relative motion, while the second one includes relative peace, which will result in different relations, from where the difference between the results comes.
Truden
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BobSpence1 wrote:The events
BobSpence1 wrote:
The events in a sequence are not necessarily dependent on each other - I don't quite see what you mean by that
Sorry, I missed that in my previous comment.
The events in sequence are cause and effect (result)
I don't see how they "are not necessarily dependent".
The measuring event is in motion, and the motion becomes part of the cause for the returned result.
We can check this by putting two clocks on two airplanes
BobSpence
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Truden wrote:BobSpence1
Truden wrote:
BobSpence1 wrote:
If two events we are comparing or timing are in differently moving frames of reference, we have to at least correct for the time it will take light to reach us from each event.
If a third observer compares the times of events in two other frames of reference which are moving at different velocities, he will get different results, even after correcting for the time it will take for light to reach him from each event, depending on his velocity, observing exactly the same two events. This is the consequence of Einstein's Special Relativity. Do you understand and accept this?
I understand that the event is actually one and is observed from two different frames of reference.
Am I correct?
Of course not.
A single event is still seen as a single event from every frame of reference, and of course you need at least two events to mark a time interval two be measured.
What did you think we were going to measure with one event?
Quote:
When we "measure time" we relate the measuring event, to the measured event.
No.
We count the number of standard timing events ( such as clock ticks ) that occur between the observation of the two observed events.
Quote:
The misinterpretation comes from the fact that we treat the measuring event as unchangeable value, because of its precision.
The precision doesn't matter here.
What matters is that for us Time is the relation between the measuring event and the measured event.
You can't measure anything with a single measuring event.
The measuring system has to be something generating a regular sequence of (measuring) events.
To measure precisely, there are two requirements:
1. The process determining the time interval between one 'tick' and the next be as simple and and unaffected by external conditions as possible;
2. That interval be as short as possible, since this directly defines the smallest difference in timing that can be measured.
Quote:
In our case the measuring event is exercised from two different frames of reference, which changes its relation to the measured event.
We have two different set of events here:
1) measuring event in motion (first frame of reference) in relation with event A (measured event)
2) measuring event in peace (second frame of reference) in relation with event A (measured event)
These two are not the same type set of events and we cannot compare them in order to extract "difference in Time".
The first one includes relative motion, while the second one includes relative peace, which will result in different relations, from where the difference between the results comes.
That doesn't seem to match how we actually would test Einstein's theory.
The standard test depends on having two clocks running at the same rate.
This can be checked by simply letting them run side-by-side over a longish time and seeing if their tick-counters keep registering the same total count.
Then you put one counter in something moving, and compare its counter reading with the one back with the observer.
There is no actual measured event as such. Just two sequences of measuring events (clock ticks).
You could send the two clocks out each in their own frame of reference and compare the two counters as observed from the one point of observation, which is theoretically more valid, in that the only difference between the two clocks is their motion.
You can't measure any time with a single 'measuring' event - all you can say with one event is whether another event occurs before, at the same time as, or after, another event.
You need a clock of some sort, ie a regular sequence of measuring events.
So, actually you are correct, you cannot do a valid test with what you describe. So that is not how we actually test this.
You don't have single measured and measuring events, you have two clocks, which each generate a rapid sequence of events very close together in time. We have to be confident that the two clocks will always tick at exactly the same rate, which can be tested to a useful degree by comparing them against each other and other clocks or natural sources of oscillation such as vibrating atoms.
This is the key point, which means we cannot strictly compare the timing of two events directly and absolutely, we have to rely on constructing clocks which will keep in step very precisely under varying external conditions, such as temperature and gravity and acceleration forces. But this can be tested to increase our confidence, and we can run the experiment repeatedly to see if we get consistent results, which is exactly what has been done.
So it seems the problem you have identified has already been identified - it is actually pretty obvious when you think about it, so you have to rely on accurate clocks.
It doesn't even rely on the clocks accurately measuring 'real' time, it just relies on the two clocks keeping in step, at least when in the same frame of reference and gravity.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
BobSpence
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Truden wrote:BobSpence1
Truden wrote:
BobSpence1 wrote:
The events in a sequence are not necessarily dependent on each other - I don't quite see what you mean by that
Sorry, I missed that in my previous comment.
The events in sequence are cause and effect (result)
I don't see how they "are not necessarily dependent".
The measuring event is in motion, and the motion becomes part of the cause for the returned result.
We can check this by putting two clocks on two airplanes
Ok, I may have mis-intepreted what you had in mind.
But still, the result of the moving clock is not dependant on what happens to the reference clock.
The possibility that the motion will effect the moving clock is what we are looking for.
And it has been done with clocks on two different planes. All of these experiments have confirmed Einstein's formulas.
As per the post I just put up, the actual experiment is done by comparing clocks, which I see may be what you had in mind.
So what is your problem with it? Are you saying we can't be sure the relative motion is what is causing the difference in the tick rate of the clocks?
Or are you suggesting the clocks are not measuring 'real' Time?
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Truden
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BobSpence1 wrote: Ok, I may
BobSpence1 wrote:
Ok, I may have mis-intepreted what you had in mind.
But still, the result of the moving clock is not dependant on what happens to the reference clock.
The possibility that the motion will effect the moving clock is what we are looking for.
And it has been done with clocks on two different planes. All of these experiments have confirmed Einstein's formulas.
As per the post I just put up, the actual experiment is done by comparing clocks, which I see may be what you had in mind.
So what is your problem with it? Are you saying we can't be sure the relative motion is what is causing the difference in the tick rate of the clocks?
Or are you suggesting the clocks are not measuring 'real' Time?
I'll skip your previous comment and I'll make an effort on this one.
I'm not saying that we can't be sure the relative motion is what is causing the difference in the tick rate of the clocks.
On the contrary - I'm VERY SURE that the relative motion affects the clocks.
Yes, Einstein's formulas are right.
I said:
Truden wrote:
What you don't see in the picture is that by putting the measuring tool in relative motion, we create event which is different from the event where the measuring tool is in relative peace.
We cannot expect these two events to have equal values.
The tool (which is actually an event) is identical as object and process, but it takes part in two different events.
All events in the chain of events depend on each other.
If we create two chains of events, the dependency in them will be different and the result will be different.
You said:
Bob wrote:
The events in a sequence are not necessarily dependent on each other - I don't quite see what you mean by that.
Truden wrote:
The events in sequence are cause and effect (result)
I don't see how they "are not necessarily dependent".
The measuring event is in motion, and the motion becomes part of the cause for the returned result.
We can check this by putting two clocks on two airplanes
So, to compact it with the same answers:
By observing the relation between two events from two different frames of reference, we create two completely different lines of events.
In each line the relations between the events are different from the other line and they depend on each other in their line.
The measuring event (it is event, because the ticking of the clock is event) depends on the relative speed, and obviously gives different value.
We cannot compare any value in the two lines to prove different Time. It is like to compare apples with pears.
By comparing the values we can only prove that the clocks are running differently because they are affected by the speed and therefore they are in different relation with the measured events.
To put it in one sentence: The measuring event is affected, not the measured one.
liberatedatheist
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If i can jump in real quick,
If i can jump in real quick, I think Truden is using some really ambiguous language that might make it seem like he is saying something different from what he means.
truden wrote:
By observing the relation between two events from two different frames of reference, we create two completely different lines of events.
This is true according to einstein's perception of time as well. Two events that appear simultaneous to one observer in his rest frame will appear to take place at two different times to a second observer that has a non zero velocity relative to the first observer.
truden wrote:
In each line the relations between the events are different from the other line and they depend on each other in their line.The measuring event (it is event, because the ticking of the clock is event) depends on the relative speed, and obviously gives different value.
I'm going to try to restate this in a way that makes sense. "In each [frame of reference] the [order of events] is different from the [order of events in the other frame of reference] and [the events] depend on each other in their [frame of reference]. The measuring event (I think you mean the standard of measure used to give a specific time value, as in one tick of a clock would be the measuring event) depends on [its] relative speed, and obviously [will record] different [time] values.
the two events do not depend on each other, that implies some sort of force or interaction. The order of events as perceived by an observer will change if the the observer changes his frame of reference. We can make the events observers in themselves, so the order of events as observed by one event will be different than the order of events as perceived by the other event if the events are moving relative to one another. If we have two identical clocks that are both at rest relative to one another, they will tick exactly in sync. As soon as one clock is put into motion while the other stays at rest than both clocks will immediately start ticking out of sync. How far off they will be depends on how fast one is moving relative to the other. Both clocks will perceive that the other clock changed while they stayed the same. The order of events depends on the reference frame that you choose to be in but not on the events themselves which is hopefully what you were trying to say.
truden wrote:
We cannot compare any value in the two lines to prove different Time. It is like to compare apples with pears. By comparing the values we can only prove that the clocks are running differently because they are affected by the speed and therefore they are in different relation with the measured events.
The clocks will register different time values if they are moving relative to one another as long as they measured the same time value when they were at rest relative to one another. This difference proves that time "moves" faster or slower in one frame of reference that is moving relative to another frame of reference. Each frame of reference will be equally correct and neither will be preferred by the laws of physics.
truden wrote:
To put it in one sentence: The measuring event is affected, not the measured one.
Each event will observe the other event to have changed (slow down or speed up. Which one depends on the frame of reference you are in). Which is what i think you are trying to say??
To bring up the twin paradox: Put my twin on a rocket. While the rocket is at rest we both are aging at the same rate. Now accelerate his rocket close to the speed of light. My twin will not age any different relative to his normal perception. For him in his frame of reference, time will not have changed. The same thing applies to me, i will not perceive any difference. But, if i observe my twin years later after he has slowed down, he will appear to be and actually be younger than me because he "passed through" less time than me. Relative to my frame of reference, time slowed down for him. Relative to his frame of reference time has speed up for me and i will appear older. How much older depends on how fast he was going relative to me.
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Truden
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to liberated atheist
@liberated atheist
Thank you for trying to help me, but in this case you need help to understand me.
Read EXACTLY what I've written and try to understand it.
Ask what do I mean by saying something. Don't restate it.
If I say that the relations in both event lines are different I mean exactly that.
The order of the events doesn't matter anymore because the events are not the same.
In one of the frame we have motion with different speed, and that is not the same event as in the other frame.
Quote:
the two events do not depend on each other, that implies some sort of force or interaction
Every event is created by certain force and every next event in the line depends on that force.
To say it simple that everyone could understand it I'd say:
The more kinetic energy the more fucked up the clock is.
We can say it because the kinetic energy is a result of an event and if we follow the event line we can see the following.
different speed -> different kinetic energy for the clocks - > different measurement.
So, not different TIME, because the Time "is" in the measured event, not in the clock.
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You still don't quite get
You still don't quite get it.
In the case of an observer watching two different clocks moving at different velocity relative to his frame, he will see the clocks running at different speeds, the clock in the one moving faster appearing to run slower. There is no 'measured' and 'measuring' event, there are just two clocks, with the count of ticks on one frame appearing to fall behind the clock on the other.
If we have an observer on each of two frames of reference moving with respect to each other, each observer will see the other clock moving slower than the one beside them. There is no real difference between the two frames from the view-point of a third observer. If the other frames are moving at the same speed relative to the third observer, their clocks will appear to to be ticking at the same rate, and slower than one beside the third observer. This is at the same time as each observer on the original moving frames will see the clock on the other running slower than the one on their own frame.
Do you understand? Each clock on the original pair of frames is experiencing the same velocity with respect to the other. There is no meaning to saying one is moving and one is not. There is no absolute velocity. It seems to be a paradox - it is impossible for each clock in a pair to really run slower than the other.
General relativity resolves this paradox. The problem in Special Relativity is that you cannot start with two synchronised clocks, which are then put on two different aircraft, flown on different speeds and tracks, and then brought back together at in a common frame of reference to compare them, without subjecting them to many changes of velocity, which means we cannot use special relativity alone to calculate what time (tick count) they will show at the end of the process. Unless you can bring the two clocks back together, there is no actual paradox.
Every change in velocity means they are subjected to acceleration, which is not relative, and 'really' slows down the clock. If we do the calculations taking acceleration into account, the paradox disappears - the clock subject to most acceleration will have fallen behind the other.
So actually you are partially correct, but it is not 'kinetic energy', which a function of velocity squared, which causes the final time discrepancy, it is acceleration/deceleration. IOW change in velocity, not velocity itself. The one subject to the most velocity change will be the one that has slipped behind when you bring them back together.
It is cumulative count of successive 'events' in the form of ticks which is being compared - the number of such events is the measure of effective time duration.
The only causality which is ultimately relevant here is the effect of acceleration ( 'g-forces' ) on the rate at which time appears to pass, as measured by some regular physical process such as a spring and balance wheel, or vibrating atoms. All theory and experiment confirms that the effect applies equally to every object experiencing that acceleration.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Truden
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BobSpence1 wrote:You still
BobSpence1 wrote:
You still don't quite get it.
In the case of an observer watching two different clocks moving at different velocity relative to his frame, he will see the clocks running at different speeds, the clock in the one moving faster appearing to run slower. There is no 'measured' and 'measuring' event, there are just two clocks, with the count of ticks on one frame appearing to fall behind the clock on the other.
Bob, I am not that stupid, but with no offence I have to say that you are disappointing me.
How can you say that the tick of the clock is not event!?
One tick is one event.
Many ticks are one reappearing event, or many events with the same cause, which (events) appear in the same rate, if you like it better.
We measure Time by relating one reappearing event to the measured event.
We can measure the event motion (which is one only event) by counting one reappearing event during our observation of the motion and we could say that the motion appeared to us for the time (number) of 100 ticks of the clock.
This 100 ticks come as result of relating one reappearing event to the event "motion".
Now is it clear to you that we MEASURED the event "motion" with the event "tick"? The result we call Time and use it in expressions like "for the time of 100 seconds", which must tell you that the second is not Time.
The relation between the reappearing second and the other event is what we call Time.
An observer does the same what the measuring devises do - he relates one event to another.
An observer can see that one reappearing event does not match another reappearing event, which will only proof that they are running in different rate.
Why they run in different rate is another question.
I don't know what do you mean by saying "the observer will SEE", when explaining the different observing results in different frames of reference?
Is this a conclusion which is taken out of a theory, or such experiment was conducted and they asked all observers what did they see
I think that we need measuring here. Don't you think so?
But when we measure, we come to the point of different kinetic energy in the different frames of reference, which fucks up the clocks
BobSpence
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Truden wrote:BobSpence1
Truden wrote:
BobSpence1 wrote:
You still don't quite get it.
In the case of an observer watching two different clocks moving at different velocity relative to his frame, he will see the clocks running at different speeds, the clock in the one moving faster appearing to run slower. There is no 'measured' and 'measuring' event, there are just two clocks, with the count of ticks on one frame appearing to fall behind the clock on the other.
Bob, I am not that stupid, but with no offence I have to say that you are disappointing me.
How can you say that the tick of the clock is not event!?
One tick is one event.
Many ticks are one reappearing event, or many events with the same cause, which (events) appear in the same rate, if you like it better.
We measure Time by relating one reappearing event to the measured event.
We can measure the event motion (which is one only event) by counting one reappearing event during our observation of the motion and we could say that the motion appeared to us for the time (number) of 100 ticks of the clock.
This 100 ticks come as result of relating one reappearing event to the event "motion".
Now is it clear to you that we MEASURED the event "motion" with the event "tick"? The result we call Time and use it in expressions like "for the time of 100 seconds", which must tell you that the second is not Time.
The relation between the reappearing second and the other event is what we call Time.
An observer does the same what the measuring devises do - he relates one event to another.
An observer can see that one reappearing event does not match another reappearing event, which will only proof that they are running in different rate.
Why they run in different rate is another question.
I don't know what do you mean by saying "the observer will SEE", when explaining the different observing results in different frames of reference?
Is this a conclusion which is taken out of a theory, or such experiment was conducted and they asked all observers what did they see
I think that we need measuring here. Don't you think so?
But when we measure, we come to the point of different kinetic energy in the different frames of reference, which fucks up the clocks
It did not say a clock tick is not an event. But to concentrate on the individual events and their 'relationship in the way you seem to is to not see the forest for the trees.
It is the continually changing count of ticks that is the measure of elapsed time, not the 'motion' of any event, which doesn't quite make sense, because you are using 'event' in a everyday sense, not the way it is used in Physics. I hope you see this is one reason why a scientist will find your account confusing.
The only 'motion' involved is the changing counter or dial of the clock.
One second is a standard measure of the dimension of Time, just as one meter is a standard measure of spatial dimension.
Clocks are designed so that the rate of ticking is as consistent as possible, and as high as possible relative to how accurately we wish to measure Time. They can only measure time in terms of the number of discrete ticks ( 'tick events' if you like ). Beyond establishing that the rate at which ticks occur is as consistent as possible, the detail of how we go from one tick to the next does not give us any insight into Time.
Motion is NOT an EVENT, in the language of physics and relativity. That is totally wrong.
From Wikipedia:
Quote:
In physics, and in particular relativity, an event indicates a physical situation or occurrence, located at a specific point in space and time. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time, in a given frame of reference. Strictly speaking, the notion of an event is an idealization, in the sense that it specifies a definite time and place, whereas any actual event is bound to have a finite extent, both in time and in space.
So the elapsed time is a measure of the 'distance' along the Time dimension between two events, not the duration of one event. This misuse of the word 'event' in this context confuses the issue.
The number of ticks that are counted starting from the occurrence of the first event until the second event occurs is a measure of the separation in the Time dimension between the two events, in the frame of reference that the clock is in.
We do not compare the apparent passage of time in two frames of reference tick by tick, we compare the change in the tick count on one timer with the change in count on the other for some reasonable period of time. If we want an estimate of apparent elapsed time within 1% accuracy, we need the slowest clock to have ticked at least 100 times.
The difference is only proof that they appear to be running at different rates as viewed from the reference frame of the observer.
The observer 'sees' the counts at the start and at the finish of the timing period. "Sees" refers to whatever means is actually used to transfer the reading on the tick counters, or the position of the dial, etc of each clock back to the reference frame of that observer. This could hypothetically be with a telescope, but in practical experiments it will be as data transmitted via radio or possibly a laser beam. The important thing is to get the counts while the clocks are actually moving.
The 'kinetic energy' is not relevant, it is the relative velocity, plus any acceleration, so we need a matching record of the position, tick by tick, to allow us to compare the velocity and acceleration over the timing period, with the rate at which the clock is ticking. They also need to record the 'g' force experience by the clock, since this is also affects the rate at which time passes in that frame of reference. This would be measured directly by a precision accelerometer.
There are two distinct aspects of the experiments, one concerned with the predictions of Special Relativity, which are concerned only with observations from frames of reference moving at a constant velocity in a straight line in the absence of gravity, and General Relativity, where you take the effects of gravity and acceleration into account.
You really seem to have completely misunderstood the effects of Special Relativity
If two spacecraft or airplanes are moving at constant velocity away from each other, each one will observe that the clock in the other craft is appears to be running slower than the one in his craft. Now part of this is simply due to the fact that the radio or laser signal which is carrying the information on the clock reading in the other craft is going to be taking longer to get back to him as the other craft is continually getting further away. But even after allowing for this, there will still be a discrepancy.
Each one will see the other's clock apparently running slow when compared to his.
There are no 'moving' and 'motionless' frames of reference, in an absolute sense. In this example. Each observer sees, or measures with instruments, the other craft moving away from his at the same speed. This idea really does allow many apparent paradoxes, like the famous Twins Paradox. But most, if not all, of those are not describable purely within the 'constant speed in a straight line' condition that Special Relativity applies to. The Twins Paradox, which has one twin travelling at high speed away from the Earth and back, who has therefore been subject to acceleration, in getting up to speed, slowing down and accelerating back in the direction of home, then slowing back to a stop relative to Earth, whereas the one who stayed at home hasn't been, which explains how he will have aged relative to his twin.
Your idea is roughly consistent with the effects of gravity and acceleration, ie, General Relativity. NOT kinetic energy, though. We observe that light emitted by atoms from very dense stars appears to be 'red-shifted', ie, slowed in rate of time passing relative to an observer (us) viewing or measuring it from a lower gravity environment. It isn't just that clocks run slow, everything runs slower in a high gravity environment, which is exactly equivalent to saying that Time is running slower in that environment. And note, there is no general movement involved there, so no "kinetic energy". Just the thermal vibrations of the atoms, which is itself slowed in the strong gravitational field.
I have read your account, and I have tried to make sense of it, and pointed out problems like using words in different ways to how they are used in scientific descriptions, where it is vitally important that we all understand exactly what all our words refer to.
Can you possibly admit that you may have got something at least slightly wrong? I am basing my explanations on established science. You are trying to explain what you see as an error in the science, so you need to justify carefully why you disagree, and be prepared to adjust your 'theory' when errors are pointed out.
Your stubborn insistence that I haven't understood what you are saying whenever I don't agree with it is still annoying.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
BobSpence
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With respect to your
With respect to your definitions, you said at the start of your OP:
Quote:
The definition of time which we use is:“Non spatial continuum in which the events occur.”
According to my dictionary, the definition which seems to apply in general use is:
"the indefinite continued progress of existence and events in the past, present, and future regarded as a whole : travel through space and time | one of the greatest wits of all time."
From Wikipedia:
"Time is part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects."
"An operational definition of time, wherein one says that observing a certain number of repetitions of one or another standard cyclical event (such as the passage of a free-swinging pendulum) constitutes one standard unit such as the second, is highly useful in the conduct of both advanced experiments and everyday affairs of life. The operational definition leaves aside the question whether there is something called time, apart from the counting activity just mentioned, that flows and that can be measured."
From another site:
"Time is an observed phenomenon, by means of which human beings sense and record changes in the environment and in the universe. A literal definition is elusive. Time has been called an illusion, a dimension, a smooth-flowing continuum, and an expression of separation among events that occur in the same physical location."
From a very interesting article here:
NewScientist wrote:
Scientists have long worried about the nature of time. At the beginning of the 18th century, Isaac Newton and Gottfried Leibniz argued over whether time was truly fundamental to the universe. Then Einstein came along and created more problems: his general theory of relativity is responsible for our most counter-intuitive notions of time.
General relativity knits together space, time and gravity. Confounding all common sense, how time passes in Einstein's universe depends on what you are doing and where you are. Clocks run faster when the pull of gravity is weaker, so if you live up a skyscraper you age ever so slightly faster than you would if you lived on the ground floor, where Earth's gravitational tug is stronger. "General relativity completely changed our understanding of time," says Carlo Rovelli, a theoretical physicist at the University of the Mediterranean in Marseille, France.
I have to ask, who is this "we" you refer to there? Because it does not quite seem to be a definition in general use.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Truden
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BobSpence1 wrote:With
BobSpence1 wrote:
With respect to your definitions, you said at the start of your OP:
Quote:
The definition of time which we use is:“Non spatial continuum in which the events occur.”
According to my dictionary, the definition which seems to apply in general use is:
Quote:
"the indefinite continued progress of existence and events in the past, present, and future regarded as a whole : travel through space and time | one of the greatest wits of all time."
.......................................................................................................
I have to ask, who is this "we" you refer to there? Because it does not quite seem to be a definition in general use.
I assume that you have nothing to say (argue) about my last comment on Time and you finally decided to look into the definitions.
Well, I can only say thank you, Bob.
My last comment reflects my definition, therefore it is the right one, and science must adopt it for use.
All definitions that deffer from the one proposed by me are wrong.
The above statement is based on basic logic.
This comment of mine is the end of my "point of exit".
I have nothing more to say on the discussed matter.
---
I thank you all for helping me to take in account some of the arguments that might be critical for the right explanation of Time.
I think that Rational Responders must be proud that it is one of the first participants where the issue about the Time was discussed and cleared from the delusional understanding.
I am a man who works as a carpenter, but I might as well be a man who did not find the same satisfaction in science as he found in the woodwork.
The man doesn't matter.
I don't matter.
What matters is the truth which sometimes stays beneath our understanding, not as foundation but as covered treasure.
Thank you All.
Truden
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Truden wrote:BobSpence1
Truden wrote:
BobSpence1 wrote:
With respect to your definitions, you said at the start of your OP:
Quote:
The definition of time which we use is:“Non spatial continuum in which the events occur.”
According to my dictionary, the definition which seems to apply in general use is:
Quote:
"the indefinite continued progress of existence and events in the past, present, and future regarded as a whole : travel through space and time | one of the greatest wits of all time."
.......................................................................................................
I have to ask, who is this "we" you refer to there? Because it does not quite seem to be a definition in general use.
I assume that you have nothing to say (argue) about my last comment on Time and you finally decided to look into the definitions.
Well, I can only say thank you, Bob.
My last comment reflects my definition, therefore it is the right one, and science must adopt it for use.
All definitions that deffer from the one proposed by me are wrong.
The above statement is based on basic logic.
This comment of mine is the end of my "point of exit".
I have nothing more to say on the discussed matter.
---
I thank you all for helping me to take in account some of the arguments that might be critical for the right explanation of Time.
I think that Rational Responders must be proud that it is one of the first participants where the issue about the Time was discussed and cleared from the delusional understanding.
I am a man who works as a carpenter, but I might as well be a man who did not find the same satisfaction in science as he found in the woodwork.
The man doesn't matter.
I don't matter.
What matters is the truth which sometimes stays beneath our understanding, not as foundation but as covered treasure.
Thank you All.
Oh, I did not see the comment before your last one, Bob.
But still, I have nothing more to say.
BobSpence
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Truden wrote:BobSpence1
Truden wrote:
BobSpence1 wrote:
With respect to your definitions, you said at the start of your OP:
Quote:
The definition of time which we use is:“Non spatial continuum in which the events occur.”
According to my dictionary, the definition which seems to apply in general use is:
Quote:
"the indefinite continued progress of existence and events in the past, present, and future regarded as a whole : travel through space and time | one of the greatest wits of all time."
.......................................................................................................
I have to ask, who is this "we" you refer to there? Because it does not quite seem to be a definition in general use.
I assume that you have nothing to say (argue) about my last comment on Time and you finally decided to look into the definitions.
Well, I can only say thank you, Bob.
Umm, so you have no comment on my long post just before that one, where I went into some detail about various aspects of your ideas and how they related to Einstein's theories??
Quote:
My last comment reflects my definition, therefore it is the right one, and science must adopt it for use.
All definitions that deffer from the one proposed by me are wrong.
The above statement is based on basic logic.
This is a sarcastic comment, I have to assume, because it totally devoid of logic. It is correct because you thought it up? WTF?
Or did you really read my previous post and this is your reaction to my pointing out a few more of your basic errors and referring to your stubborn refusal to re-examine your theory in the light of my comments?
Quote:
This comment of mine is the end of my "point of exit".
I have nothing more to say on the discussed matter.
---
I thank you all for helping me to take in account some of the arguments that might be critical for the right explanation of Time.
I think that Rational Responders must be proud that it is one of the first participants where the issue about the Time was discussed and cleared from the delusional understanding.
I am a man who works as a carpenter, but I might as well be a man who did not find the same satisfaction in science as he found in the woodwork.
The man doesn't matter.
I don't matter.
What matters is the truth which sometimes stays beneath our understanding, not as foundation but as covered treasure.
Thank you All.
Pissed-off because no-one took your confused ideas very seriously - oh well, none so blind as those who refuse to see...
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Truden
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BobSpence1 wrote:Truden
BobSpence1 wrote:
Truden wrote:
BobSpence1 wrote:
With respect to your definitions, you said at the start of your OP:
Quote:
The definition of time which we use is:“Non spatial continuum in which the events occur.”
According to my dictionary, the definition which seems to apply in general use is:
Quote:
"the indefinite continued progress of existence and events in the past, present, and future regarded as a whole : travel through space and time | one of the greatest wits of all time."
.......................................................................................................
I have to ask, who is this "we" you refer to there? Because it does not quite seem to be a definition in general use.
I assume that you have nothing to say (argue) about my last comment on Time and you finally decided to look into the definitions.
Well, I can only say thank you, Bob.
Umm, so you have no comment on my long post just before that one, where I went into some detail about various aspects of your ideas and how they related to Einstein's theories??
Quote:
My last comment reflects my definition, therefore it is the right one, and science must adopt it for use.
All definitions that deffer from the one proposed by me are wrong.
The above statement is based on basic logic.
This is a sarcastic comment, I have to assume, because it totally devoid of logic. It is correct because you thought it up? WTF?
Or did you really read my previous post and this is your reaction to my pointing out a few more of your basic errors and referring to your stubborn refusal to re-examine your theory in the light of my comments?
Quote:
This comment of mine is the end of my "point of exit".
I have nothing more to say on the discussed matter.
---
I thank you all for helping me to take in account some of the arguments that might be critical for the right explanation of Time.
I think that Rational Responders must be proud that it is one of the first participants where the issue about the Time was discussed and cleared from the delusional understanding.
I am a man who works as a carpenter, but I might as well be a man who did not find the same satisfaction in science as he found in the woodwork.
The man doesn't matter.
I don't matter.
What matters is the truth which sometimes stays beneath our understanding, not as foundation but as covered treasure.
Thank you All.
Pissed-off because no-one took your confused ideas very seriously - oh well, none so blind as those who refuse to see...
Bob, don't take my last comments as disrespect.
I respect you more than you can imagine
I really said everything I could say in this discussion.
You have to try to understand me, otherwise you will run in circles, making me repeat myself over and over again.
I admit that there are flaws in my explanation, but they don't affect my idea.
I'll have to work out some differences between the common and the scientific use of the terms and I'll come back to you
For now, I urge you to take "out of the box" look at my idea and try to help me next time we meet
Don't be afraid to think that something as big as changing the interpretation about Time can happen.
Einstein is not the best that humanity can give birth to
I'll be back soon.
BobSpence
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Truden wrote:Bob, don't take
Truden wrote:
Bob, don't take my last comments as disrespect.
I respect you more than you can imagine
I really said everything I could say in this discussion.
You have to try to understand me, otherwise you will run in circles, making me repeat myself over and over again.
I admit that there are flaws in my explanation, but they don't affect my idea.
I'll have to work out some differences between the common and the scientific use of the terms and I'll come back to you
For now, I urge you to take "out of the box" look at my idea and try to help me next time we meet
Don't be afraid to think that something as big as changing the interpretation about Time can happen.
Einstein is not the best that humanity can give birth to
I'll be back soon.
Look, I fully understand that there can and will be progress beyond Einstein, and scientists are exploring new ideas about 'Time' all the time, of course.
But when you do not seem to quite understand Einstein's ideas, and make such basic errors, as I kept pointing out, and which you refused to acknowledge, it is hard to take you seriously.
But I honestly cannot see that you have anything that really amounts to a useful new concept here.
I do think I get the core idea what you are trying to say, but it really is not that different from what some others have already proposed, but they have expressed it more clearly and much better related to what has already been established about the subject.
1. In a 'Universe' containing just one object, we cannot meaningfully talk about time, OK. We cannot even talk about motion, so even saying it is 'motionless' is meaningless.
2. It is not so much 'events' we need to have time, it is change, even continuous change, which is not really well described by talking about events. This is where you start to go 'off the rails'.
3. If two objects are moving away from each other, that is all we need to be able to talk about time. The movement would be observed and measured with light, or are you assuming there is not even light in this hypothetical universe?
4. If the increase in distance between the two objects is observable, we can meaningfully refer to their relative velocity is (that is the more precise term to use here, since it includes the direction as well as the magnitude of velocity).
I am not using velocity ('speed') to prove time. It is not time that requires there to be 'speed', to 'prove' it, it is the existence of motion that requires there to be a dimension of Time to make it even possible. I think you have it backwards.
The absence of any discrete 'events' is not relevant, just change. Change is what needs Time. This is your basic error.
To measure Time consistently we do use repeated events which we have reason to believe are consistent and therefore assumed to occur at equal intervals of time.
Now here is another thought - imagine that one object is a long rod with regular marks on it, and the other object is moving past close to it along its length. Then we can measure time by how far the other object has moved along the length of the first object. That is all we need!!
If you insist on 'events', then consider: as the other object passes each mark on the first object, that is sufficient to define an event!
All events occur in space and time, by definition.
Two objects moving past each other is indeed sufficient to define events and so mark the passage of time.
If you had defined the objects as two ideal geometric points, then 'space' would be a problem, since you would have no reference for distance. But if at least one object has a finite physical size, then we have some reference to express the distance between them, and that then allows change to be defined, ie changes in the ratio between separation and size of at least one object. Change is the minimum requirement for Time to be meaningful, or alternatively, 'change' only can occur if there is a dimension of Time.
If you want to express it in terms of events, then each event can be defined to be when the distance between the objects reaches a whole number times the diameter of one of the objects.
There, I have analysed your idea, as presented in your first post. It is ultimately trivial, I am sorry.
Most of your ideas are not new, the rest are based on misunderstandings, such as what is necessary to define an 'event'.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
darth_josh
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This is testing my
This is testing my patience.
http://flashforward.web.cern.ch/flashforward/excerpt2/
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Eloise
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Truden wrote:Eloise wrote:
screw it... double post.
Eloise
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Truden wrote: Eloise wrote:
Truden wrote:
Eloise wrote:
The 'time' used by science is really no different to your second definition, modern science essentially considers time as synonymous with 'change'.
You may be right, Eloise.
Sometimes I have the feeling that Einstein made fun of the science by creating his theory of relativity to prove that everything is related to our conscious perception.
His theory doesn't make sense without intelligent conscious observation,
That's not so Truden. Relativity makes perfect sense without conscious observation. You've probably been confused by stories of Einstein developing the theory from thoughts about observation in a moving reference frame. Those things are not required for relativity to work, they are just ways of coming to the conclusion that space and time are not constant in contrast to light velocity.
Quote:
Yes, modern science may use the notion "change" (or "step" ) as Time, but Time still stays in science as property of the Universe.
It's quite more involved than that, Truden. Time cannot be ignored since it is an inextricable part of how we structure our experience, however, models of physical reality that do not have time as a property of the universe in them are studied in science - they are called time-independent.
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Marquis
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Truden wrote:how childishly
Truden wrote:
how childishly illogical science can be
If you think that is bad, you need to consider how childishly illogical nature can be.
Anyway, while you guys were having fun I read through it all once again, and I fail to see how anything in Mr. Truden's proposition is inconsistent with my stating that time = gravity (non-spatial relation between events and yada yada). Consequently, instead of stating for instance that gravity warps spacetime, we can observe how an increase in local gravity moves towards and finally reaches a "boiling point" from whence it approaches a shift in existential conditions (an event horizon) before collapsing out of measurable spacetime coordinates alltogether.
Time in an isolated context can be both the temporal difference between events A and B as the duration of event C (which can even include the set A,B). There is no "continuum" of time that doesn't include variables in the gravitational field.
"The idea of God is the sole wrong for which I cannot forgive mankind." (Alphonse Donatien De Sade) | 2015-02-01 05:54:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5472013354301453, "perplexity": 686.9585543461783}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122127848.98/warc/CC-MAIN-20150124175527-00198-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://www.oxfordmathcenter.com/drupal7/node/203 | # Exercises - The Vigenere's Cipher
1. Encode the message "Attack at Dawn" with a Vigenere's cipher, using the keyword "KEY".
2. The Kryptonian alphabet consists of only 5 letters whose frequencies of occurance are shown below:
$$\begin{array}{c|c|c|c|c} A & B & C & D & E\\\hline 0.05 & 0.10 & 0.20 & 0.30 & 0.35 \end{array}$$
If a plaintext Kryptonian message is encoded using a Vigenere's Cipher with keyword "BED", determine the probability that the first and third letters of the message match.
3. Use the dot product to determine which angle is smaller, the angle between the vectors $\begin{pmatrix}-3\\7\end{pmatrix}$ and $\begin{pmatrix}-4\\8\end{pmatrix}$ or the angle between the vectors $\begin{pmatrix}3\\5\end{pmatrix}$ and $\begin{pmatrix}5\\4\end{pmatrix}$.
4. Which two vectors below have the smallest angle between them? (Note, all three vectors have the same length)
1. $(1, 2, 3, 4)$
2. $(2, 1, 3, 4)$
3. $(1, 3, 4, 1)$
◆ ◆ ◆ | 2018-01-19 01:11:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6198651790618896, "perplexity": 1101.2308396476978}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887692.13/warc/CC-MAIN-20180119010338-20180119030338-00551.warc.gz"} |
https://en.m.wikibooks.org/wiki/LMIs_in_Control/pages/MatrixEigenValueMinimization | # LMIs in Control/pages/MatrixEigenValueMinimization
LMI for Minimizing Eigenvalue of a Matrix
Synthesizing the eigenvalues of a matrix plays an important role in designing controllers for linear systems. The eigenvalues of the state matrix of a linear time-invariant system determine if the system is stable or not. The system is stable if all the eigenvalues of the state matrix are located in the left half of the complex plane. Thus, we may desire to minimize the maximal eigenvalue of the state matrix such that the minimized eigenvalue is placed in the left half-plane, which guarantees that the system is stable.
## The System
Assume that we have a matrix function of variables ${\displaystyle x}$ :
{\displaystyle {\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}}
where {\displaystyle {\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}} are symmetric matrices.
## The Data
The symmetric matrices ${\displaystyle A_{i}}$ ({\displaystyle {\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}} ) are given.
## The Optimization Problem
The optimization problem is to find the variables {\displaystyle {\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}} to minimize the following cost function:
{\displaystyle {\begin{aligned}J(x)=\lambda _{\text{max}}(A(x))\end{aligned}}}
where ${\displaystyle J(x)}$ is the cost function and ${\displaystyle \lambda _{\text{max}}(.)}$ indicates the maximim eigenvalue of a matrix.
According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent
{\displaystyle {\begin{aligned}\lambda _{max}(A(x))\leq t\iff A(x)-tI\leq 0\end{aligned}}}
where ${\displaystyle t}$ is defined as the maximim eigenvalue of the matrix ${\displaystyle A}$ .
## The LMI: LMI for eigenvalue minimization
This optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
{\displaystyle {\begin{aligned}&{\text{min}}\quad t\\&{\text{s.t.}}\quad A(x)-tI\leq 0\end{aligned}}}
## Conclusion:
As a result, the variables ${\displaystyle x_{i},\quad i=1,2,...,n}$ after solving this LMI problem.
Moreover, we obtain the maximum eigenvalue, ${\displaystyle t}$ , of the matrix ${\displaystyle A(x)}$ .
## Implementation
A link to Matlab codes for this problem in the Github repository: | 2022-05-21 16:13:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 16, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9354672431945801, "perplexity": 1129.3849650224101}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662539131.21/warc/CC-MAIN-20220521143241-20220521173241-00097.warc.gz"} |
http://www.gabormelli.com/RKB/Shattering_Coefficient | # Growth Function
(Redirected from Shattering Coefficient)
A Growth Function is a Statistical Learning function that measures the richness of a Set Family.
## References
### 2019a
1. Frequencies of Events to Their Probabilities". Theory of Probability & Its Applications. 16 (2): 264. doi:10.1137/1116025. This is an English translation, by B. Seckler, of the Russian paper: "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities". Dokl. Akad. Nauk. 181 (4): 781. 1968. The translation was reproduced as: Vapnik, V. N.; Chervonenkis, A. Ya. (2015). "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities". Measures of Complexity. p. 11. doi:10.1007/978-3-319-21852-6_3. ISBN 978-3-319-21851-9.
2. Mohri, Mehryar; Rostamizadeh, Afshin; Talwalkar, Ameet (2012). Foundations of Machine Learning. USA, Massachusetts: MIT Press. ISBN 9780262018258., especially Section 3.2
3. Shalev-Shwartz, Shai; Ben-David, Shai (2014). Understanding Machine Learning – from Theory to Algorithms. Cambridge University Press. ISBN 9781107057135.
### 2019b
• (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Shattered_set#Shatter_coefficient Retrieved:2019-12-6.
• To quantify the richness of a collection C of sets, we use the concept of shattering coefficients (also known as the growth function). For a collection C of sets $s \subset \Omega$ , $\Omega$ being any space, often a sample space, and $x_1,x_2,\dots,x_n \in \Omega$ being any set of n points, we define
the nth shattering coefficient of C as : $S_C(n) = \max_{\forall x_1,x_2,\dots,x_n \in \Omega } \operatorname{card} \{\,\{\,x_1,x_2,\dots,x_n\}\cap s, s\in C \}$ where $\operatorname{card}$ denotes the cardinality of the set. $S_C(n)$ is the largest number of subsets of any set A of n points that can be formed by intersecting A with the sets in collection C.
Here are some facts about $S_C(n)$ :
1. $S_C(n)\leq 2^n$ for all n because $\{s\cap A|s\in C\}\subseteq P(A)$ for any $A\subseteq \Omega$ .
2. If $S_C(n)=2^n$, that means there is a set of cardinality n, which can be shattered by C.
3. If $S_C(N)\lt 2^N$ for some $N\gt 1$ then $S_C(n)\lt 2^n$ for all $n\geq N$ .
• The third property means that if C cannot shatter any set of cardinality N then it can not shatter sets of larger cardinalities.
### 2014
• (Scott et al., 2014) ⇒ Clayton Scott (Lecturer), Srinagesh Sharma, Scott Reed, and Petter Nilsson (2014)."Vapnik-Chevronenkis Theory". In: EECS 598: Statistical Learning Theory, Winter 2014.
• QUOTE: Let $\mathcal{H} \subseteq \{0, 1\}^{\mathcal{X}}$. For $x_1, x_2, \cdots , x_n \in \mathcal{X}$ denote
$N_{\mathcal{H}}(x_1, \cdots , x_n) := |\{(h(x_1), \cdots , h(x_n)) : h \in \mathcal{H}\}|$
Clearly $N_{\mathcal{H}}(x_1, \cdots, x_n) \leq 2^n$. The nth shatter coefficient is defined as
$S_{\mathcal{H}}(n) =: \displaystyle \underset{x_1,\cdots,x_n \in \mathcal{X}} max N_{\mathcal{H}}(x_1, \cdots , x_n)$
If $S_{\mathcal{H}}(n) = 2n$, then $\exists\; x_1, \cdots , x_n$ such that
$N_{\mathcal{H}}(x_1, \cdots , x_n) = 2^n$
and we say that $\mathcal{H}$ shatters $x_1, \cdots , x_n$.
Note. The shatter coefficient is sometimes called the growth function in the literature. | 2020-06-04 01:15:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9072254300117493, "perplexity": 1409.1319267811853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347436828.65/warc/CC-MAIN-20200604001115-20200604031115-00051.warc.gz"} |
https://www.physicsforums.com/threads/fluid-mechanics-question.441108/ | # Fluid Mechanics question
1. Oct 24, 2010
Here is the question:
There is a rectangular container with the following dimensions: Length 2 m, Width 1 m, and Height 1m. This container is filled with water to 2/3 of its volume and there is a cover on the top. The cover has a hole in the center.
a) Find the maximum acceleration velocity in the horizontal direction before the water would spill out.
b) Find the hydrostatic forces (ignore the contribution from atmospheric pressure) exerted on each surface.
2) A U-tube is made of square duct of the size 1 cm X 1 cm. The two arms are separated by 20 cm. The tube arm height is 60 cm. It is filled with an indicator of specific weight as 2.00. The height of the indicator fluid initially was 25 cm measured from the bottom surface of the connecting duct.
a) If the U-tube is rotating around the center of one tube arm, what is the maximum angular velocity that one can have to prevent the indicator fluid spillage. Make necessary assumptions.
b) show the pressure variation on the bottom surface of the connecting duct as a function of the angular velocity.
3. The attempt at a solution
for (a) I used $tan \theta = \frac {-a_x}{g}$ and basically drew the diagram (can't show it here) I eventually got -ax = 3.24 m/s2 is this right?
(b) not sure what to do
And the second question I have no idea. | 2019-03-22 16:39:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7330237627029419, "perplexity": 535.1735840906738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202672.57/warc/CC-MAIN-20190322155929-20190322181929-00160.warc.gz"} |
https://armasm.com/docs/neon/bus-rider/ | Program 21: Bus Rider
# Program 21: Bus Rider
This is, without a doubt, a simple introductory program to use the NEON co-processor. It has all kinds of uses but I wanted to find something that was both simple and more concrete then “LOOK HOW COOL IT IS WHEN WE USE RANDOM NUMBERS TO DO RAY TRACING!!” A little backgrounds on matrices first if you don’t know how they can be solved.
## Matrices #
If you think back to linear algebra, you will probably remember a small part where you were working with matrices. I know when I was in math, it wasn’t presented in a way that made it practical. They look something like this (I hope this doesn’t give you bad flashbacks):
$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
### Multiplying matrices #
If you have two matrices and want to multiply them, this is the formula. Multiply each member of the the first matrix’s row by every member of the second matrix’s corresponding column and add them. I will not go into all of the questions like “why? how? magic? wtf?” here.
$\begin{bmatrix} 1 & 10 & 100 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \\ \end{bmatrix} = \begin{bmatrix} 531 & 642 \end{bmatrix}$ $531 = (1\times1) + (10\times3) + (100\times5) \\ 642 = (1\times2) + (10\times4) + (100\times6)$
### Inverse matrices #
The cool thing about matrices is that the concept of division is absent. There is only multiplication of the inverse. This is similar to how $$6^1 \times 6^{-1} = \frac{6}{1} \times \frac{1}{6} = 1$$
## Matrices IRL #
From the information above, you can then solve series of linear equations. Again, without getting into the details too much, look at this problem graciously borrowed from mathisfun.com.
A group took a trip on a bus, at $3 per child and$3.20 per adult for a total of $118.40. They took the train back at$3.50 per child and $3.60 per adult for a total of$135.20. How many children, and how many adults?
In this equation there are two constants. The number of children (x) and the number of adults (y). You could write this as two equations.
$\text{Bus Trip:} \quad\3.00x + \3.20y = \118.40 \\[2pt] \text{Train Trip:} \quad\3.50x + \3.60y = \135.20$
You may be able to see now how this would translate to a set of multiplied matrices.
$\begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} 3 & 3.5 \\ 3.2 & 3.6 \end{bmatrix} = \begin{bmatrix} 118.4 & 135.2 \end{bmatrix}$
From here, you can multiply the result by the inverse of the known matrix and get the unknown. Getting the inverse is a little complicated and out of the scope of this simple example, but trust me that this is correct.
$\begin{bmatrix} 118.4 & 135.2 \end{bmatrix} \begin{bmatrix} -9 & 8.75 \\ 8 & -7.5 \end{bmatrix} = \begin{bmatrix} 16 & 22 \end{bmatrix}$
So there were 16 children and 22 adults.
## Program 21: Bus Rider Video - Coming Soon
Description of Program
Using the same method, write a program to solve for the following equations. The bus and train prices won’t change, so you can use the same inverse matrix.
• Bus Trip: $3.00x +$3.20y = $183.80 • Train Trip:$3.50x + $3.60y =$209.90
Template/Input
Completed Code
Expected Output
Children: 25 | 2023-03-28 11:35:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5272458791732788, "perplexity": 805.6468301911444}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948858.7/warc/CC-MAIN-20230328104523-20230328134523-00724.warc.gz"} |
https://www.groundai.com/project/the-monic-rank/ | The monic rank
# The monic rank
Arthur Bik Universität Bern, Mathematisches Institut, Alpeneggstrasse 22, 3012 Bern Jan Draisma Universität Bern, Mathematisches Institut, Sidlerstrasse 5, 3012 Bern, and Eindhoven University of Technology Alessandro Oneto Barcelona Graduate School of Mathematics, and Universitat Politècnica de Catalunya, Av.da Diagonal 647 (ETSEIB), 08028 Barcelona, Spain and Emanuele Ventura Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
###### Abstract.
We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone . We show that the monic rank is finite and greater than or equal to the usual -rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to Shapiro which states that a binary form of degree is the sum of -th powers of forms of degree . Furthermore, in the case where is the cone of highest weight vectors in an irreducible representation—this includes the well-known cases of tensor rank and symmetric rank—we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances.
AB was supported by JD’s Vici grant. JD was partially supported by the NWO Vici grant entitled Stabilisation in Algebra and Geometry. AO acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in RD (MDM-2014-0445). EV acknowledges financial support by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. AO and EV thank the Universität Bern (Switzerland) for the hospitality during visits where the project of the paper was discussed.
## 1. Introduction
Let be an algebraically closed field of characteristic zero. All our vector spaces and algebraic varieties will be over , finite-dimensional, reduced and identified with their sets of -points.
### Monic secant varieties and monic rank
Let be a finite-dimensional vector space. Let be an irreducible Zariski-closed affine cone such that its -linear span equals , i.e., the cone is non-degenerate. A very fruitful field of research investigates the concept of -rank of an element , denoted by , i.e., the minimal number of elements of such that lies on their linear span (see [BT15] for definitions). In the cases where is, for example, some vector space of tensors and is the subvariety of rank tensors in , the study of -ranks has very interesting relations with fields in applied mathematics. We refer to [Lan12] for a more extensive exposition. In this paper we introduce a new, but related, type of rank.
Let be a non-zero linear function and consider the affine hyperplane
H={v∈V∣h(v)=1}⊆V.
We write for the affine-hyperplane section of .
###### Definition 1.1.
Let be a positive integer. The -th open secant variety of is the set
∘σkX1:={p1+⋯+pk∣p1,…,pk∈X1}.
This is a subset of . We define to be the Zariski closure of and call this set the -th secant variety of . We also call the -th open monic secant variety of and the -th monic secant variety of .
Since is an affine space, we have . And, for any , we have
kX1⊆∘σkX1⊆σkX1.
However as we shall see, both inclusions can be strict. We now define the monic rank of a vector with .
###### Definition 1.2.
Let be a vector. The monic rank of is defined to be
mrkX,h(v):=inf{k∈Z≥1 ∣∣∣ kh(v)⋅v∈∘σkX1}.
Similarly, the monic border rank of is
mrk–––––X,h(v):=inf{k∈Z≥1 ∣∣∣ kh(v)⋅v∈σkX1}.
The following example is illustrative for the rest of our results.
###### Example 1.3.
Let be the vector space of binary forms of degree and let be the subset of squares of linear forms. We consider the linear function which selects the coefficient of . So
h:V→K,ax2+bxy+cy2↦a.
We get and . Now, an element of is contained in the second open monic secant if and only if it equals
(x+a1)2+(x+a2)2=2x2+2(a1+a2)x+(a21+a22)
for some . Notice that the polynomials and generate the ring of symmetric polynomials in the variables and , i.e., the invariant ring . Here is the symmetric group on two letters and acts by permuting and . From classical invariant theory, we know that the map
K2→K2,(a1,a2)↦(2(a1+a2),a21+a22)
is a finite morphism. Thus, it is closed and dominant and so it is also surjective. Hence
∘σ2X1=σ2X1=2H.
See Proposition 3.3 and the proof of Theorem 1.6 for an explanation. We find that any satisfies .
### Main results
A priori it is not clear that the monic rank of an element of is even finite. This is our first foundational result.
###### Theorem 1.4.
The function is strictly increasing until it coincides with its maximal value and constant from then on. Let be the minimal integer for which holds. Then for any , we have
rkX(v)≤mrkX,h(v)≤2k0.
In particular, the monic rank is finite.
###### Definition 1.5.
The minimal integer for which is called the generic monic rank of elements of .
Theorem 1.4 mimics a result of Blekherman and Teitler [BT15, Theorem 3] that relates the maximal -rank to the generic -rank.
An interesting conjecture due to Shapiro (see [LORS18, Conjecture 1.4]) states that every binary form of degree is the sum of -th powers of forms of degree . We show that, in a few particular cases, a stronger statement holds: every binary form of degree whose coefficient at equals is the sum of -th powers of monic forms of degree .
###### Theorem 1.6.
Let and be positive integers. Let be the space of binary forms of degree , take and let be the linear function that maps a form to its coefficient at . Then, in the following cases, the maximal monic rank is at most :
1. and arbitrary ;
2. and arbitrary ;
3. and ;
4. and .
In particular, in these cases, the -rank of any is at most .
In terms of the (non-monic) Shapiro’s conjecture on writing binary forms of degree as sums of -th powers:
• the case is classical (see, e.g., [Rez13, Theorem 4.9]);
• the case is trivial;
• the case is quite immediate (see [FOS12, Theorem 5]); and
• the case was proven in [LORS18, Theorem 3.1].
As far as we know, the cases where are new.
We next turn our attention to a particularly nice class of varieties. Let be a connected reductive algebraic group over and let be an irreducible finite-dimensional rational representation of . Let be the cone of highest-weight vectors, i.e., the affine cone over the unique closed -orbit in . The latter projective variety is called a homogeneous variety. Let be a highest weight vector in the dual -module .
###### Question 1.7.
In the setting above, is the maximal -rank of a vector in equal to the maximal monic -rank of a vector in ?
Let be a finite-dimensional vector space. Then we denote the -th symmetric power of by .
###### Theorem 1.8.
The answer to Question 1.7 is affirmative in the following instances:
1. and ;
2. and ;
3. and ;
4. and ; and
5. and is its adjoint representation.
###### Remark 1.9.
The first case of the theorem corresponds to the Waring rank of binary forms, which is also (i) in Theorem 1.6. The second case corresponds to the usual matrix rank of matrices. Case (iii) corresponds to the symmetric matrix rank of symmetric matrices (or to the Waring rank of quadrics in variables). Case (iv) corresponds to the tensor rank of tensors. And lastly, case (v) corresponds to the -rank of trace-zero matrices with being the affine cone over the projective adjoint variety of incident point-hyperplane pairs in .
Admittedly, this is not much evidence for an affirmative answer to Question 1.7 in general. Moreover, our proofs in each of these cases are ad hoc. We therefore appeal to the reader for different approaches to Question 1.7. In particular, an affirmative answer to that question for and would imply that a new lower bound of
on the maximal rank of a form of degree . This is almost always bigger than
⌈1n(d+n−1d)⌉,
which (except for quadrics and finitely many further exceptions) is the generic Waring rank by the Alexander-Hirschowitz theorem [AH95]. So then the maximal rank would be greater than the generic rank—a commonly held belief to which no approach is known.
### Structure of the paper
In Section 2, we lay the foundations of the notion of monic rank and prove Theorem 1.4. In Section 3, we develop some machinery from classical invariant theory to compute the maximal monic rank in certain explicit cases. We apply this machinery to the proof of Theorem 1.6 in Section 4. Finally, in Section 5, we establish Theorem 1.8.
## 2. The basics of monic rank
Recall the following notation from the introduction:
• denotes a non-degenerate irreducible affine cone;
• is a non-zero linear function; and
• is an affine hyperplane and .
Theorem 1.4 will be a direct consequence of Proposition 2.1 and Proposition 2.2.
###### Proposition 2.1.
The function is strictly increasing until it coincides with its maximal value, which is , and constant from then on. Consequently, the function is bounded. Moreover, its value at is an upper bound on the ordinary border -rank of for all vectors .
###### Proof.
Let be any point. Then . So the function is weakly increasing. Since is bounded from above by , there exists a such that . Let be any positive integer with this property. Then, since both and are irreducible, the isomorphism
kH → (k+1)H v ↦ v+p1
restricts to an isomorphism between and . By definition, a general point on is of the form with and and we have
v+p2=(v−p1)+p2+p1∈σkX1+X1+p1⊆σk+1X1+p1.
Therefore, the isomorphism
(k+1)H → (k+2)H w ↦ w+p1
restricts to an isomorphism between and . Now, let be the minimal value of for which . Then we conclude that the function is strictly increasing for and is constant for .
Next, we show that , which implies in particular that . Let be positive integers. Then
(a+b)σk0X1⊆aσk0X1+bσk0X1⊆σ(a+b)k0X1.
Since the leftmost and rightmost sets are closed, irreducible and of the same dimension, all three sets coincide. For any , we have . Therefore
aa+b⋅p1+ba+b⋅p2∈σk0X1.
So the line through and intersects in infinitely many points. Hence this line must be entirely contained in . Since this holds for all , we see that is an affine space. Since is non-degenerate, the affine span of coincides with . So the affine span of equals . We conclude that .
For the last statement, note that is contained in the ordinary -th secant variety of and that the ordinary border rank of a vector does not change whenever we multiply the vector by a non-zero constant. ∎
###### Proposition 2.2.
Let be the generic monic rank of elements of . Then, for every vector , we have . In particular, the monic rank of is finite.
###### Proof.
We adopt a similar strategy as in the proof of [BT15, Theorem 3]. Recall that the generic monic rank is finite by Proposition 2.1. Set
~v=2k0h(v)⋅v∈2k0H
and consider the intersection
(~v−∘σk0X1)∩∘σk0X1⊆k0H.
Both sets on the left-hand side contain an open dense subset of . Thus they must intersect. Consequently, there exist such that . So
2k0h(v)⋅v=~v=p1+p2∈∘σ2k0X1
and hence . This shows the second inequality. The first is immediate from the definitions of -rank and monic -rank. ∎
## 3. Invariant theory tools
Our reference for classical invariant theory is [DK02].
### 3.1. A variant of a theorem of Hilbert
In this section, we develop computational tools to prove Theorem 1.6. Let be a reductive algebraic group over acting on an affine variety , whose coordinate ring is . Let a one-dimensional torus act on . The character lattice of is isomorphic to . For any , let be the corresponding weight space:
Ra:={f∈K[Y] ∣∣ ∀y∈Y,t∈T:f(ty)=taf(y)}.
This naturally induces a grading on . Assume the following:
1. The grading satisfies and for .
2. The actions of and on commute.
Under these assumptions, each weight space is a representation of and the invariant ring decomposes as
RG=⨁a≥0RGa
where . In this section, the terms homogeneous and degree refer exclusively to the grading given by .
###### Proposition 3.1.
Suppose that are homogeneous elements of positive degree such that
V(r1,…,rk)=V(⨁a>0RGa)
where denotes the vanishing set of a set of forms . Then is a finitely generated module over its subring .
###### Remark 3.2.
In Hilbert’s classical variant of this result (see, e.g., [DK02, Lemma 2.4.5]), the variety is a vector space, acts linearly on and the grading is the standard one. The proof of our generalization is identical, but we include it for the sake of completeness.
###### Proof.
Let be homogeneous generators of of positive degree. Then by Hilbert’s Nullstellensatz, there exists a positive integer such that, for each , the form is in the ideal generated by the homogeneous forms . Using the Reynolds operator, one finds that
fmi=k∑i=1hiri
for some homogeneous with . So we see that the finite set
{ℓ∏i=1fmii ∣∣ ∣∣ 0≤m1,…,mℓ
generates as a -submodule of . ∎
### 3.2. A criterion for closedness of the open monic secant variety
Recall that we have a non-degenerate irreducible affine cone and a linear function , the affine hyperplane and the affine-hyperplane section . Assume a one-dimensional torus acts linearly on such that the following conditions are satisfied:
• The action extends to a morphism (or equivalently, all weights of are nonnegative).
• The space of -invariant linear forms is spanned by .
• The set is stable under .
Let be a basis of consisting of -weight vectors (i.e., a basis of homogeneous vectors). Since is -invariant, the sets and are stable under . The affine section contains the unique -fixed point . The coordinates of in the given basis of are .
We now fix a positive integer , let be the -fold direct product over of the affine variety with itself and we consider the following map:
φk:Y → kH (p1,…,pk) ↦ p1+⋯+pk
Besides the induced action of , the affine variety comes naturally equipped with an action of the symmetric group which permutes the factors. Note that the actions of and on commute by definition and that is -equivariant and -invariant.
###### Proposition 3.3.
In the setting above, if , then is closed and of dimension .
###### Proof.
Let be the coordinate ring of . Define to be the affine variety whose coordinate ring is the invariant ring , i.e., . Since is -invariant, it factors through the quotient map . So we have a morphism that makes the diagram
\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@color@rgb@fill000Y \definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@color@rgb@fill000Y//Skspan \definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@color@rgb@fill000kH
commutative. Note that .
A classical result [DK02, Lemma 2.3.2] in invariant theory is that the quotient map is surjective. We show that the vertical map is closed. Since is -equivariant, the map is -equivariant as well. Let be the pull-back map induced by and take for each . By definition, we have
ri(y)=ri(π(y))=hi(ψ(π(y)))=hi(φk(y))
for all for every . Here we use that the pull-back map induced by is the inclusion . We see that
{(x0,…,x0)}⊆V(⨁a>0RSka)⊆V(r1,…,rm)=φ−1k(kx0)={(x0,…,x0)}.
This means that all these subsets of coincide. So by Proposition 3.1, the morphism is finite and hence closed. This implies that the image of is closed. It also implies that the dimension of coincides with the one of , which is . ∎
## 4. Instances of Shapiro’s conjecture
Fix positive integers and . As a first application of the monic rank, we look at a conjecture due to Shapiro.
###### Conjecture 1 (Shapiro’s Conjecture, [Lors18, Conjecture 1.4]).
Every binary form of degree can be written as the sum of -th powers of forms of degree .
We prove this conjecture in some new cases by proving the following (stronger) conjecture: let be the vector space of binary forms of degree , let
X={fd ∣∣ f∈K[x,y](e)}
be the variety of -th powers of forms of degree , let be the affine hyperplane consisting of all forms whose coefficient at equals and take .
###### Conjecture 2.
φk:Xk1 → kH (p1,…,pk) ↦ p1+⋯+pk
satisfies .
###### Proposition 4.1.
If Conjecture 2 holds for , then is closed and of dimension . In particular, Conjecture 2 for implies Shapiro’s Conjecture for .
###### Proof.
Let act on via
t⋅(de∑i=0aixde−iyi)=de∑i=0aitixde−iyi
for all . This action has only positive weights and stabilizes . Furthermore, the only -invariant in is the linear function which selects the coefficient of . Hence, the assumptions of Proposition 3.3 are satisfied. This implies the first statement. For the second statement, note that every non-zero form has in its -orbit a form with . Assuming Conjecture 2, we derive
rkX(v)=rkX(~v)≤mrkX,h(~v)≤d
as desired. ∎
###### Remark 4.2.
Let be a triple such that Conjecture 2 holds for and let be monic binary forms with . Then we have
kxd(e+1)=(xf1)d+⋯+(xfk)d
and hence by the conjecture for . So we see that the conjecture for implies the conjecture for . Conversely, assuming that the conjecture holds for , we get the following method to prove that the conjecture also holds for : we have to prove that monic binary forms can only satisfy
fd1+⋯+fdk=kxd(e+1)
when . Suppose we have binary forms
fi=xe+1+ci,1xey+⋯+ci,e+1ye+1∈K[x,y](e+1)
satisfying the equation. If for each , then we get
(f1x)d+⋯+(f1x)d=kxde
and hence by the conjecture for . Otherwise we can assume, by permuting the and acting with , that . Now, we expand the sum
fd1+⋯+fdk=kxd(e+1)+r1xd(e+1)−1y+⋯+rd(e+1)yd(e+1)
where the coefficients are polynomials in the and we compute the reduced Gröbner basis (with respect to some monomial ordering) of the ideal generated by the and the polynomial . If the conjecture holds for , then this Gröbner basis will be . And, if the Gröbner basis is , then cannot be , which means that and hence that the conjecture holds for .
We can now prove Theorem 1.6.
###### Proof of Theorem 1.6.
We consider the cases of the theorem separately.
• The coefficients of the binary form
at are, up to constant factors, power sums in the variables . These generate the invariant ring . This implies that
V(r1,…,rd)={(0,…,0)}
and hence Conjecture 2 holds for . So (i) holds by Proposition 4.1.
• Both Conjecture 2 and Shapiro’s Conjecture are trivial for . Assume that and let be a binary form whose coefficient at equals . Then for some (not necessarily monic) binary forms by [FOS12, Theorem 4]. Fix as the basis of and consider as a symmetric matrix. The linear map
π:Sym2(K[x,y](e)) → K[x,y](2e) xe−iyi⋅xe−jyj ↦ x2e−(i+j)yi+j
sends to . From this we see that the matrix has a as entry in its top-left corner. Now it follows from Proposition 5.3 that
g1⋅g1+g2⋅g2=h1⋅h1+h2⋅h2∈Sym2(K[x,y](e))
for some monic binary forms . So we have
f=π(h1⋅h1+h2⋅h2)=h21+h22
as desired.
• The remaining cases are checked by computer, but we use one more observation: the system of homogeneous equations in the constructed in the inductive strategy given in Remark 4.2 has only integral coefficients and is homogeneous relative to the grading coming from the action of one-dimensional torus . Hence, we are checking whether a certain subvariety of a weighted projective space defined over has no -points. To achieve this, it is enough to show that the subvariety has no -points for some prime . This allows us to work modulo some prime (e.g., the prime is enough), which makes the computation more efficient and lets it finish successfully. ∎
## 5. Minimal orbits
Let be a connected and reductive algebraic group over . Let be an irreducible rational representation of . Fix a Borel subgroup of and a maximal torus in . Let span the unique -stable one-dimensional subspace in , i.e., the highest weight space. Set . This is the affine cone over the homogeneous variety given by the orbit of . Let be the function that spans the unique -stable one-dimensional subspace of and is normalized so that . In this setting, we study Question 1.7, i.e., whether the maximal -rank of a vector in is also the maximal monic -rank of a vector in . As positive evidence, we treat the examples from Theorem 1.8.
### 5.1. Binary forms
Consider the case where
• the vector space consists of binary forms of degree ;
• the group acts on in the natural way;
• the variety consists of powers of linear forms; and
• the linear function sends a polynomial to its coefficient at .
Here the -rank is also called the Waring rank. Using the Apolarity Lemma (see, e.g., [IK99, Lemma 1.15]), one can show that has Waring rank and, in fact, the maximal Waring rank of a binary form of degree is exactly (see, e.g., [Rez13, Theorem 4.9]). By Theorem 1.6(i), the maximal monic rank with respect to equals as well. Hence, the answer to Question 1.7 is affirmative is this instance. Moreover, all open secant varieties of are closed by Proposition 3.3—the coefficients of in the sum of -th powers of linear forms are the first power sums in and generate the invariant ring .
### 5.2. Rectangular matrices
Consider the case where
• the vector space consists of matrices;
• the group acts by left and right multiplication;
• the variety consists of rank matrices; and
• the linear function sends a matrix to its top-left entry.
Let be the affine space of matrices with and take .
###### Proposition 5.1.
We have
∘σkX1=σkX1={A∈kH∣rk(A)≤k}
for all .
###### Proof.
The inclusions
∘σkX1⊆σkX1⊆{A∈kH∣rk(A)≤k}
are clear. Let be a matrix with . Our goal is to prove that . We prove this by induction on . Write . Then, by acting with the subgroup of -invariant elements of , we may assume that is the diagonal matrix with a as its top-left entry followed by ones. If , then is the sum of copies of the matrix with just a as its top-left entry. Note that, in particular, this handles the case . Next, assume that . Then, from the fact that the equality
(k001)=(k−1λλ1−λ2)+(1−λ−λλ2),λ=√k−1k
decomposes the matrix on the left as a sum of two matrices of rank with and as entries in their top-left corners, we see that there is a decomposition with and matrices such that . By induction, it follows that and hence . This concludes the proof. ∎
It follows from the proposition that the rank of a matrix in coincides with its monic rank. So in particular, the maximal monic rank is equal to the maximal rank.
### 5.3. Symmetric matrices
Consider the case where
• the vector space
V={A∈Kn×n∣A=AT}≅Sym2(Kn)≅K[x1,…,xn](2)
consists of symmetric matrices;
• the group acts by for and ;
• the variety consists of rank matrices; and
• the linear function sends a matrix to its top-left entry.
Let be the affine space of matrices with and take .
###### Remark 5.2.
The vector space can be viewed as the space of quadratic forms in the variables by associating the quadric
(x1,…,xn)A(x1,…,xn)T
to a symmetric matrix . So, the variety corresponds to the set of squares of linear forms and affine space corresponds to the set of polynomials with coefficient at .
###### Proposition 5.3.
We have
∘σkX1=σkX1={A∈kH∣rk(A)≤k}
for all .
###### Proof.
As in the proof of Proposition 5.1, it suffices to prove that every symmetric matrix with is an element of . We again first replace by a diagonal matrix: it is well-known that every symmetric matrix is congruent to a diagonal matrix. So we can write with and diagonal. By going though the proof of this fact, one can check that can be chosen so that its action on is -invariant and is the diagonal matrix with a as its top-left entry followed by ones. This reduces the problem to the case where is this diagonal matrix. Now, from the fact that
(k001)=(k−1λλ1−λ2)+(1−λ−λλ2),λ=√k−1k,
we see that there is a decomposition with and such that . We again conclude that . ∎
Again, it follows from the proposition that the rank of a matrix in coincides with its monic rank. And in particular, the maximal monic rank is equal to the maximal rank.
### 5.4. 2×2×2 tensors
Consider the case where
• the vector space consists of tensors;
• the group acts on in the natural way;
• the variety | 2020-07-09 04:31:31 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9548904299736023, "perplexity": 567.0504010212788}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655898347.42/warc/CC-MAIN-20200709034306-20200709064306-00380.warc.gz"} |
http://mathhelpforum.com/discrete-math/104709-find-closed-form.html | Math Help - Find a closed form
1. Find a closed form
Write a closed from function for:
F(0) = 0
F(1) = 1
F(2) = 4
F(n) = 3 + (F(n-1) - F(n-2)) + F(n-3) + 3
I can't seem to figure this one out....
2. F(n) = $n^2$
3. Oh i see it now | 2015-11-25 05:58:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6192634701728821, "perplexity": 1478.7660583757486}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398444228.5/warc/CC-MAIN-20151124205404-00134-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.sarthaks.com/2807673/how-much-will-000-invested-compound-interest-amount-year-per-annum-compounded-half-yearly | # How much will Rs. 25,000 invested at compound interest amount to in 1 year at 4% per annum compounded half yearly?
66 views
closed
How much will Rs. 25,000 invested at compound interest amount to in 1 year at 4% per annum compounded half yearly?
1. Rs. 25,980
2. Rs. 26,010
3. Rs. 26,100
4. Rs. 26,001
by (110k points)
selected
Correct Answer - Option 2 : Rs. 26,010
Given:
Principle is 25000
Rate is 4%
Time is 1 year and CI is calculated Half Yearly
Formula Used:
Compound Interest (CI) = P × (1 + R/100)n
Calculation:
CI is calculated Half Yearly so Rate = 4/2 = 2 %
⇒ Time becomes 2 years
⇒ CI in 2 years = 25000 × (1 + 2/100)2 = 26010 | 2022-12-05 20:52:19 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9602765440940857, "perplexity": 9898.352415381827}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711045.18/warc/CC-MAIN-20221205200634-20221205230634-00007.warc.gz"} |
http://geometry.cs.ucl.ac.uk/projects/2019/structurenet/ | #### StructureNet: Hierarchical Graph Networks for 3D Shape Generation
Kaichun Mo1* Paul Guerrero2* Li Yi1 Hao Su3
Peter Wonka4 Niloy J. Mitra2,5 Leonidas J.Guibas1,6
1Stanford University 2University College London 3University of California San Diego
4King Abdullah University of Science and Technology (KAUST)
*joint first authors
Siggraph Asia 2019
StructureNet is a hierarchical graph network that produces a unified latent space to encode structured models with both continuous geometric and discrete structural variations. In this example, we projected an un-annotated point cloud (left) and un-annotated image (right) into the learned latent space yielding semantically segmented point clouds structured as a hierarchy of graphs. The shape interpolation in the latent space also produces structured point clouds (top) including their corresponding graphs (bottom). Edges correspond to specific part relationships that are modeled by our approach. For simplicity, here we only show the graphs without the hierarchy. Note how the base of the chair morphs via functionally plausible intermediate configurations, or the chair back transitions from a plain back to a back with arm-rests.
###### Abstract
The ability to generate novel, diverse, and realistic 3D shapes along with associated part semantics and structure is central to many applications requiring high-quality 3D assets or large volumes of realistic training data. A key challenge towards this goal is how to accommodate diverse shape, including both continuous deformations of parts as well as structural or discrete alterations which add to, remove from, or modify the shape constituents and compositional structure. Such object structure can typically be organized into a hierarchy of constituent object parts and relationships, represented as a hierarchy of n-ary graphs. We introduce StructureNet, a hierarchical graph network which (i) can directly encode shapes represented as such n-ary graphs; (ii) can be robustly trained on large and complex shape families; and (iii) be used to generate a great diversity of realistic structured shape geometries. Technically, we accomplish this by drawing inspiration from recent advances in graph neural networks to propose an order-invariant encoding of n-ary graphs, considering jointly both part geometry and inter-part relations during network training. We extensively evaluate the quality of the learned latent spaces for various shape families and show significant advantages over baseline and competing methods. The learned latent spaces enable several structure-aware geometry processing applications, including shape generation and interpolation, shape editing, or shape structure discovery directly from un-annotated images, point clouds, or partial scans.
###### Hierarchical Graph Network Architecture
Our variational autoencoder consists of two encoders and two decoders that both operate on our shape representation. The geometry encoder egeo encodes the geometry of a part into a fixed-length feature vector f, illustrated with a gray circle. The graph encoder egraph encodes the feature vectors of each part in a graph, and the relationships among parts, into a feature vector of the same size using graph convolutions. The graph encoder is applied recursively to obtain a feature vector z that encodes the entire shape. The reverse process is performed by the graph and geometry decoders d_graph and d_geo to reconstruct the shape. The decoder also recovers the geometry of non-leaf nodes.
###### Free Shape Generation
We show shapes in all categories decoded from random latent vectors, including shapes with bounding box geometry, and shapes with point cloud geometry. Parts are colored according to semantics, see the Supplementary for the full semantic hierarchy for each category. Since we explicitly encode shape structure in our latent representation, the generated shapes have a large variety of different structures.
###### Part Interpolation
We interpolate either only the backrest (first row) or only the base (second row) between the chairs on the left and right side. Intermediate shapes preserve structural plausibility of the interpolated result mainly through geometric differences to the target part, but faithfully interpolate the structure. We observe that these interpolations are not necessarily symmetric: the base interpolations follow different paths to be compatible with the different back styles.
###### Shape Abstraction
Images, synthetic point clouds, and real-world scans from ScanNet [Dai et al. 2017] are embedded into our learned latent space, allowing us to effectively recover a full shape description that matches the raw input.
###### Structure-aware Shape Editing
We show editing results on two shapes with box geometry (first four rows) and two shapes with point cloud geometry (two bottom rows). For the two shapes with box geometry, we perform five different edits each, one edit per column. The edited box is highlighted in yellow, and the result is shown below. We see that the other boxes in the shape are adjusted to maintain shape plausibility. For the two shapes with point cloud geometry, we show intermediate results for one edit each. From left to right, these are (a) the original point cloud; (b) the predicted box abstraction; (c) the induced segmentation; (d) edited boxes; and (e) the induced edit of the point-cloud.
###### Bibtex
@article{MoGuerreroEtAl:StructureNet:SiggraphAsia:2019
title={StructureNet: Hierarchical Graph Networks for 3D Shape Generation},
author={Mo, Kaichun and Guerrero, Paul and Yi, Li and Su, Hao and Wonka, Peter and Mitra, Niloy and Guibas, Leonidas},
journal={arXiv preprint arXiv:1908.00575},
year={2019}
}
###### Acknowledgements
This project was supported by a Vannevar Bush Faculty Fellowship, NSF grant RI-1764078, NSF grant CCF-1514305, a Google Research award, an ERC Starting Grant (SmartGeometry StG-2013-335373), ERC PoC Grant (SemanticCity), Google Faculty Awards, Google PhD Fellowships, Royal Society Advanced Newton Fellowship, KAUST OSR number CRG2017-3426 and gifts from Adobe, Autodesk and Qualcomm. We especially thank Kun Liu, Peilang Zhu, Yan Zhang, and Kai Xu for the help preparing binary symmetry hierarchies for GRASS baselines on PartNet. We also thank the anonymous reviewers for their fruitful suggestions. | 2019-09-23 17:50:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.305584192276001, "perplexity": 3367.474939967368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514577478.95/warc/CC-MAIN-20190923172009-20190923194009-00033.warc.gz"} |
http://benmearns.blogspot.be/ | ## Monday, November 13, 2017
### Geoserver WFS-T PostGIS (PostgreSQL) Layer Read-Only
An unexpected Read-Only message is common when attempting a Geoserver WFS-T request via the Demo page. This is also true of WFS-T on a PostGIS-backed Layer.
Here are a list of possible culprits:
1. Verify the layer is writable by the Postgres role (user account) which you've used to authenticate in Geoserver, via the Store dialog for that Layer.
2. The table must have a primary key in Postgres. It may be necessary to "Reload Feature Type" under the Layer page > Data Tab > Feature Type Details section. It also be necessary to "Expose Primary Keys" in the Store for the Layer.
3. The WFS-T Demos will not successfully authenticate out-of-the-box (which will probably give a different error message). See my post on the issue, and how to resolve it.
4. If you are using my work around, linked from above, you are using anonymous authentication (for testing only, of course!). You will need to create a write rule that applies to the anonymous role and the workspace which relates to your Layer/Store.
In my latest experience with this error, culprit #4 got me :-p
## Thursday, October 19, 2017
### Solution, Geoserver bug with authenticated requests
Geoserver ships with a good list of helpful Demo requests -- however, the authenticated requests fail if you do not add the relevant URL path to a Filter Chain which includes a "form" authentication filter.
If you run any Demo request involving authentication without the relevant Filter Chain modification you will see the following message:
The same response is seen when running the request with curl, e.g.:
You can resolve this issue by adding the relevant URL path suffix (e.g., /wcs* or /wfs*) to the "web" Filter Chain, which includes the "form" Authentication Filter.
Access this dialog in the Web Administration interface under Security > Authentication > Filter Chains > web.
I've commented on the bug in the Geoserver project JIRA -- to resolve this bug this issue should be either noted or the underlying xml's changed -- otherwise the Demos fail.
## Wednesday, June 28, 2017
### Georeference a non-referenced image with gdal_translate
This technique is useful for georeferencing an image that you could be getting from a WMS server.
To get an image from a WMS Server, first do a capabilities request via the URL. This will return an XML with parameters. The result will look something like this (you could also substitute your target WMS, layer, bbox, etc.
http://imagery.pasda.psu.edu/arcgis/services/pasda/PAMAP_cycle2/MapServer/WMSServer?service=WMS&version=1.1.0&request=GetMap&layers=11&styles=&bbox=-77.16641161363148171,39.76300177722033169,%20-77.16220203861050209,39.76481826348722137&width=800&height=400&srs=EPSG:4326&format=image/jpeg
if gdal isn't already installed, do so now, and navigate to the gdal bin directory, if it isn't on your path
gdal translate will require input file, output file, the desired spatial reference system, and ground control points (gcp) which you will calculate from the bounding box you entered above and the image pixel size (800 by 400). These correspond to the corners of the image (coordinates differ from above slightly, not intended).
gdal_translate c:\temp\WMSServer.tiff c:\temp\aerial.tif -of GTiff -a_srs EPSG:4326 -gcp 0 0 -77.16694 39.764799 -gcp 800 400 -77.161927 39.763017 -gcp 0 400 -77.16694 39.763017 -gcp 800 0 -77.161927 39.764799
Use gdal_warp for further transformation, if a simple scaling wasn't adequate. This is particularly necessary for Geoserver. In that case, also use the -co tfw=yes parameter to create a world file, since Geoserver doesn't recognize srs in the standalone GTiff.
It looks like this: gdalwarp c:\temp\aerial.tif c:\temp\aerialw.tif -of GTiff -co tfw=yes -t_srs EPSG:4326
## Tuesday, October 20, 2015
### Local Vagrant VM as part of a deployment strategy
Local Vagrant VM as part of a deployment strategy
Previously I discuss setting up a Vagrant EC2 VM for web application deployment. In this post, I will walk through the process of setting up a local development VM with the objective of maintaining a commmon environment between the local and remote (EC2) VM’s.
# The local VM
#### (2) Create a new vagrantfile, using your favorite text editor (e.g., vi vagrantfile) and insert the following text
Vagrant.configure(2) do |config|
config.vm.box = "hashicorp/precise64"
config.vm.network "forwarded_port", guest: 80, host: 8080
config.vm.synced_folder "..\\vagrant_data", "/vagrant_data"
config.vm.provider "virtualbox" do |vb|
# # Display the VirtualBox GUI when booting the machine
vb.gui = false
#
# # Customize the amount of memory on the VM:
# vb.memory = "1024"
end
# config.vm.provision :shell, path: "bootstrap.sh"
end
#### (4) As you install packages, make sure to add those lines to a new file, (vi bootstrap.sh). You can also add checks where necessary. My bootstrap.sh is long, but here’s a portion of it for example:
#!/usr/bin/env bash
# install and configure linux tools
sudo apt-get -y update
sudo apt-get -y install unattended-upgrades make vim acl
sudo usermod -a -G web vagrant
# apache install and config
sudo apt-get install -y apache2
if ! [ -L /var/www ]; then
rm -rf /var/www
ln -fs /vagrant_data /var/www
fi
sudo chown www-data:web /var/www
sudo a2enmod rewrite
sudo /etc/init.d/apache2 restart
# sqlite install and config
sudo apt-get -y install sqlite3 libspatialite3 spatialite-bin
#### (5) Uncoment line second to end (config.vm.provision :shell, path: "bootstrap.sh") in your vagrantfile with your favorite text editor (e.g., vi vagrantfile) to configure provisioning via bootstrap.sh on a new VM when vagrant up is run.
You will need to make the same changes to ..\vagrant-aws\vagrantfile and a ..\vagrant-aws\bootstrap.sh, created in the previous post, to syncronize your development and deployment provisioning
#### (6) Destroy the existing vagrant instance, including all files created when you brought it up vagrant destroy and bring up the new instance vagrant up
Written with StackEdit.
## Friday, October 16, 2015
### Vagrant on AWS with Windows
Vagrant on AWS with Windows
Required installations:
– AWS CLI
– VirtualBox
– Vagrant
– an existing EC2 instance with an account private key on your local file system.
[Optional] If you use PuTTy ssh client, you may want to install and configure the Vagrant PuTTY plugin, by following instructions here: https://github.com/nickryand/vagrant-multi-putty. You will also need to make a symbolic link to the executable, using the following command.
> mklink D:\Vagrant\embedded\bin\putty.exe "C:\Program Files (x86)\PuTTY\putty.exe"
#### (1) Use the AWS CLI/Security Token Service to generate temporary credentials for AWS
> aws sts get-session-token
The credentials returned by this command will be valid for 1 hour.
#### (2) Install the Vagrant AWS plugin.
> vagrant plugin install vagrant-aws
#### (3) Add the “dummy” Box provided by the vagrant-aws project. This is a barebones Box compatible with AWS. We will flesh this out in the Vagrantfile to be created in the next step.
> vagrant box add dummy https://github.com/mitchellh/vagrant-aws/raw/master/dummy.box
#### (4) Create a new directory to hold files for this Vagrant instance. I’m calling this vagrant-aws.
> mkdir vagrant-aws
#### (5) With your favorite text editor create a file called Vagrantfile inside this new directory. Edit this file vagrant-aws\Vagrantfile copying and pasting the following to the file (source).
Vagrant.configure("2") do |config|
config.vm.box = "dummy"
config.vm.provider :aws do |aws, override|
aws.session_token = "SESSION TOKEN"
aws.keypair_name = "KEYPAIR NAME"
aws.ami = "ami-7747d01e"
override.ssh.private_key_path = "PATH TO YOUR PRIVATE KEY"
end
end
#### (7) Now you can bring the instance up with Vagrant, first making sure you are in the directory where you had created the Vagrantfile.
> cd vagrant-aws
> vagrant up
To work on your new Vagrant VM, use vagrant ssh or vagrant putty if you set up Putty using the plugin mentioned below. Once you’re finished, you can use vagrant destroy to completely blow away this instance, or some other vagrant command if you need to come back to the instance.
## Friday, September 25, 2015
### I'm back!
I'm back from my cross country bike ride! On a completely different topic, my book with Packt Publishing, is finished and available for pre-order on Amazon, Barnes and Noble, and other fine book retailers:
## Thursday, August 6, 2015
### Out to ride -- back sometime in September
I'm on a cross country bike ride!
My progress as of 8/4/15:
Photos | 2017-12-17 00:25:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18515931069850922, "perplexity": 13065.072706852607}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948592202.83/warc/CC-MAIN-20171217000422-20171217022422-00200.warc.gz"} |
https://www.physicsoverflow.org/41694/what-boundary-conditions-vector-potential-gauge-theory-torus | # What are the boundary conditions for the vector potential in U(1) gauge theory on the torus?
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In quantizing compact $2+1D$ $U(1)$ Chern-Simons theory (or Maxwell theory, for that matter) on the $L_x \times L_y$ torus in $A_0=0$ gauge, one starts with a mode decomposition for the vector potential. In every reference I have found, the decomposition is declared gauge-equivalent to (with all physical constants set to 1)
$$A_i = a_i - \epsilon^{ij}x_j B/2 + \tilde{A}_i$$
where $i=x,y$, $\epsilon^{ij}$ is the Levi-Civita symbol, $\tilde{A}_i$ is strictly periodic, $a_i$ is a constant, and $B$ is the mean magnetic field through the torus (quantized according to the Dirac condition). What, precisely, are the boundary conditions on $A$ that lead to such a form, or should there be a more general form?
Clearly one must allow $A_i$ not to be strictly periodic in order to account for the second term. What seems most reasonable is to require $A_i$ be periodic up to a gauge transformation, i.e.
$$A_i(x+L_x,y) = A_i(x,y) + \partial_i \alpha_x(x,y)$$
where $\alpha_x(x,y)$ can be either a large or small gauge transformation, and likewise for shifts in the $y$ direction. The aforementioned decomposition does obey such boundary conditions. But a configuration like $A_x=0$, $A_y \propto x \sin(2\pi y/L_y)$ also obeys these boundary conditions (with $\alpha_x \propto \cos(2\pi y/L_y)$) and does not seem to be gauge-equivalent to anything of the aforementioned form. So is this an unacceptable configuration? If so, why?
It's true that such a configuration has a spatially varying magnetic field, which violates the constraint of pure Chern-Simons theory, but I still want to understand as a matter of principle how I should be writing $A$ before imposing the constraint.
Thanks!
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