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https://www.albert.io/ie/ap-calculus-ab-bc/graphical-interpretation-intervals-of-concave-up | ?
Free Version
Difficult
# Graphical Interpretation: Intervals of Concave Up
APCALC-97D@8B
Given the graph of $f'(x)$ above (consisting of a triangle and semicircle), on which interval(s) are $f(x)$ concave up?
A
$(-4, 0)$
B
$(-2, 4)$
C
$(0, 8)$
D
$(-4, -2)$ and $(4,8)$ | 2016-12-04 00:02:04 | {"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": 0.6767863631248474, "perplexity": 8501.112907605508}, "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-2016-50/segments/1480698541142.66/warc/CC-MAIN-20161202170901-00173-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://qudt.org/vocab/quantitykind/AngleOfAttack | quantitykind:AngleOfAttack
Type
Description
Properties
Angle of attack is the angle between the oncoming air or relative wind and a reference line on the airplane or wing.
$$\alpha$$
Annotations
Angle Of Attack(en)
Generated 2021-09-16T16:16:32.967-07:00 by lmdoc version 1.1 with TopBraid SPARQL Web Pages (SWP) | 2021-09-23 14:38:08 | {"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": 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": 0.6214787364006042, "perplexity": 7249.826017103208}, "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-2021-39/segments/1631780057424.99/warc/CC-MAIN-20210923135058-20210923165058-00538.warc.gz"} |
https://stats.stackexchange.com/questions/444982/reduce-the-redundancy-in-features | Problem
I have a very large scale tabular dataset that have more than 200 features and more than 2 million records. I would like to run some machine learning models to predict some target $$y$$. However, it is difficult to put even 70% of the data (I was expecting 95% or 98%) into the memory no matter how I optimize my code (I am using Julia).
Except going on optimizing my code and trying to find some good libraries, I am now thinking maybe it is a good idea to look at the data itself.
I believe there are some redundancies within the data, there are two thoughts
• From the perspective of mapping $$f:\mathbf{x}\mapsto y$$. Not all features are predictive of $$y$$. Therefore, these features could be removed. A thread mentions this and points to a library to do this.
• From the perspective of feature $$\mathbf{x}$$. Some of the features could be predicted from others and these features are proxies. If we could somehow effectively remove proxies. Then hopefully we could largely reduce the number of features. I haven't seen people do this.
My question is: does the second thought look promising? If it does, how could I implement it?
• Your second point requires making 200 models. For anything more complex than a linear regression, this may be prohibitive, in which case just remove highly correlated features. – Demetri Pananos Jan 16 at 1:46 | 2020-04-03 11:56:35 | {"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": 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": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5223605632781982, "perplexity": 387.3659466047125}, "config": {"markdown_headings": false, "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-2020-16/segments/1585370510846.12/warc/CC-MAIN-20200403092656-20200403122656-00554.warc.gz"} |
http://www.ck12.org/geometry/SSS-Similarity/exerciseint/Similarity-Test-True-or-False/ | <meta http-equiv="refresh" content="1; url=/nojavascript/">
# SSS Similarity
## Triangles are similar if their corresponding sides are proportional.
0%
Progress
Practice SSS Similarity
Progress
0%
Similarity Test True or False
Teacher Contributed
When you compare two triangles, if the corresponding sides are in ratio of 510$\dfrac{5}{10}$ and 820$\dfrac{8}{20}$, then the triangles are similar by the SAS similarity test.
qid: 100274 | 2015-10-05 00:16:22 | {"extraction_info": {"found_math": true, "script_math_tex": 2, "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": 0.18400968611240387, "perplexity": 5794.195930658091}, "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-2015-40/segments/1443736676381.33/warc/CC-MAIN-20151001215756-00125-ip-10-137-6-227.ec2.internal.warc.gz"} |
http://prep2013.mosaic-web.org/Notes/Functions/WorldPopulation.html | # A Fitting Problem
A problem from Stewart Calculus: Concepts and Contexts 2/e p. 38. It appears in Chapter 1.
• Why use “estimate” rather than “interpolate”?
• What are the essential properties of the function needed to create a reasonable interpolation?
• What's the point of using a cubic?
• How would you decide whether to use a cubic or some other function?
## Using mosaic
• Start R
• Load the package (you need do this only at the start of a session)
require(mosaic)
• Read in the data and display a bit
pop = fetchData("PREP-Stewart-World-Population.csv")
## Retrieving from
## http://www.mosaic-web.org/go/datasets/PREP-Stewart-World-Population.csv
head(pop)
## Year Population
## 1 1900 1650
## 2 1910 1750
## 3 1920 1860
## 4 1930 2070
## 5 1940 2300
## 6 1950 2560
What skills are needed to do this? Careful spelling, attention to punctuation, use of quotes, understanding the structure of a file name, understanding the syntax for use of R functions, understanding assignment.
• Plot the data
plotPoints(Population ~ Year, data = pop)
What do you need to know to answer the questions posed earlier?
Some techniques:
quadf = fitModel(Population ~ a * Year^2 + b * Year + c, data = pop)
## [1] 1878
quadfResids = with(pop, Population - quadf(Year))
#### Fit a cubic
You figure it out!
#### Fit a Spline
splinef = spliner(Population ~ Year, data = pop)
splinef(1925)
## [1] 1956
splinefResids = with(pop, Population - splinef(Year))
You can also try a monotonic spline. Use help(spliner) to find out how.
#### Fit an Exponential
This is hard, for reasons that relate to the data and numerics. Here, a guess is being made for a doubling time of 30 years.
expf = fitModel(Population ~ A + B * 2^((Year - 1900)/30), data = pop)
According to this model, what's the population in 1925?
### QUESTIONS
• Why are there so many extra parameters in the functions? Why not just a*Year^3 for the cubic?
• Which function is right?
• How well do the various functions work for extrapolation? Look up the world population in 2010 and check. Also, look up the world population in 1500 and check.
• What factors might influence world population that might mean that the rules of growth in 2000 might be different than 1900? If the system is changing, why can a mathematical function that doesn't change in form over the years capture the dynamics of population?
• Suppose that your job is to predict the world population in 2020. How would you build a model for this purpose? | 2018-12-13 06:31:15 | {"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": 0.38511309027671814, "perplexity": 2286.6468821791786}, "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-2018-51/segments/1544376824525.29/warc/CC-MAIN-20181213054204-20181213075704-00180.warc.gz"} |
https://planetmath.org/VariantsOnCompassAndStraightedgeConstructions | variants on compass and straightedge constructions
Many variants on compass and straightedge constructions have been investigated. The first type that will be discussed here is the removal of one of these tools.
It is pretty clear that one cannot construct much with just a straightedge. With this tool, line segments can be extended in either direction as far as one likes, but that is about all.
A more interesting idea is getting rid of the straightedge and using only the compass. Unfortunately, no line segments can now be drawn, but using only the compass always enables us to find at least two points on the line that is needed. Thus, using the convention that determining two points on a line also determines the line, the compass is the only tool that is . It turns out that the straightedge is only useful for drawing lines, not for determining points. On the other hand, being able to use the straightedge makes such constructions easier, which is why it is still used in these constructions.
Another type of variants on compass and straightedge constructions is introducing new tools. Probably the best known example of this variant is the one in which a sufficiently long piece of string is allowed. (In the spirit of traditional constructions, marking lengths on the string with a writing utensil is not allowed; however, using it like a compass to measure lengths is allowed.) The ability to use this tool changes the nature of constructions drastically. For example, with a compass, a straightedge, and a piece of string, one can construct a line segment of length $\pi$ given a line segment of length $1$. Here is how:
1. 1.
Construct a circle with radius $1$.
2. 2.
Extend the line segment sufficiently far in one direction.
3. 3.
Use the string to measure the circumference of the circle. Keeping your fingers on the string in order to preserve the length of the circumference, straighten the string out and mark this length off on the ray from the previous step.
4. 4.
Construct the perpendicular bisector of the line segment constructed in the previous step in order to find its midpoint. Each half of the bisected line segment has a length of $\pi$.
Another variant on compass and straightedge constructions is the one in which a marked ruler (that is, a ruler with marks on it) is allowed as a tool. Of course, with a marked ruler available, the straightedge is completely extraneous. In such constructions, it is interesting to determine the minimum number of marks needed on the ruler to perform the construction. Archimedes proved that an angle can be trisected using a compass and a ruler with one mark on it. See the entry titled trisection of angle for more details.
References
• 1 Rotman, Joseph J. A First Course in Abstract Algebra. Upper Saddle River, NJ: Prentice-Hall, 1996.
Title variants on compass and straightedge constructions VariantsOnCompassAndStraightedgeConstructions 2013-03-22 17:14:52 2013-03-22 17:14:52 Wkbj79 (1863) Wkbj79 (1863) 12 Wkbj79 (1863) Feature msc 01A20 msc 51M15 TrisectionOfAngle marked ruler | 2021-01-20 07:40:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 4, "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": 0.7178568840026855, "perplexity": 372.3264092600605}, "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-2021-04/segments/1610703519923.26/warc/CC-MAIN-20210120054203-20210120084203-00016.warc.gz"} |
http://techkumar.com/standard-error/how-to-interpret-standard-error-in-regression.html | Home > Standard Error > How To Interpret Standard Error In Regression
# How To Interpret Standard Error In Regression
## Contents
That in turn should lead the researcher to question whether the bedsores were developed as a function of some other condition rather than as a function of having heart surgery that Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the As a special case for the estimator consider the sample mean. Porter, this model identifies and analyzes 5 competitive forces ... http://techkumar.com/standard-error/how-to-interpret-standard-error.html
ISBN 0-8493-2479-3 p. 626 ^ a b Dietz, David; Barr, Christopher; Çetinkaya-Rundel, Mine (2012), OpenIntro Statistics (Second ed.), openintro.org ^ T.P. standard-error share|improve this question asked Jan 8 '13 at 16:53 setudent 612 What do you mean by "How exactly do statistical packages choose regression models (in particular ordinal regression)?"? It is particularly important to use the standard error to estimate an interval about the population parameter when an effect size statistic is not available. For the age at first marriage, the population mean age is 23.44, and the population standard deviation is 4.72. http://www.biochemia-medica.com/content/standard-error-meaning-and-interpretation
## How To Interpret Standard Error In Regression
You can vary the n, m, and s values and they'll always come out pretty close to each other. Moreover, this formula works for positive and negative ρ alike.[10] See also unbiased estimation of standard deviation for more discussion. Indeed, if you had had another sample, $\tilde{\mathbf{x}}$, you would have ended up with another estimate, $\hat{\theta}(\tilde{\mathbf{x}})$. Standard error: meaning and interpretation.
• Will I encounter any problems as a recognizable Jew in India?
• estimate – Predicted Y values close to regression line Figure 2.
• The obtained P-level is very significant.
The determination of the representativeness of a particular sample is based on the theoretical sampling distribution the behavior of which is described by the central limit theorem. Available at: http://www.scc.upenn.edu/čAllison4.html. up vote 1 down vote favorite Suppose we have a regression model. Difference Between Standard Error And Standard Deviation That in turn should lead the researcher to question whether the bedsores were developed as a function of some other condition rather than as a function of having heart surgery that
Thank you once again. Standard Error Example Taken together with such measures as effect size, p-value and sample size, the effect size can be a very useful tool to the researcher who seeks to understand the reliability and Oracle flashback query syntax - all tables to same timestamp C++11 - typeid uniqueness What would be the value of gold and jewelry in a post-apocalyptic society? http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/hypothesis-tests/tests-of-means/what-is-the-standard-error-of-the-mean/ A second generalization from the central limit theorem is that as n increases, the variability of sample means decreases (2).
Executing Sitecore logic from a Windows Scheduled Task Why can't the second fundamental theorem of calculus be proved in just two lines? Standard Error Of Estimate Formula The standard deviation is most often used to refer to the individual observations. Lane DM. Your cache administrator is webmaster.
## Standard Error Example
menuMinitab® 17 SupportWhat is the standard error of the mean?Learn more about Minitab 17 The standard error of the mean (SE of the mean) estimates the variability between sample means that you would http://stats.stackexchange.com/questions/47245/high-standard-errors-for-coefficients-imply-model-is-bad Consider, for example, a researcher studying bedsores in a population of patients who have had open heart surgery that lasted more than 4 hours. How To Interpret Standard Error In Regression For some statistics, however, the associated effect size statistic is not available. Can Standard Error Be Greater Than 1 When the S.E.est is large, one would expect to see many of the observed values far away from the regression line as in Figures 1 and 2. Figure 1.
Use the standard error of the mean to determine how precisely the mean of the sample estimates the population mean. his comment is here A more precise confidence interval should be calculated by means of percentiles derived from the t-distribution. Was there something more specific you were wondering about? However, one is left with the question of how accurate are predictions based on the regression? Standard Error Vs Standard Deviation
The standard error of a statistic is therefore the standard deviation of the sampling distribution for that statistic (3) How, one might ask, does the standard error differ from the standard Are you asking how the models are fit? –Macro Jan 9 '13 at 13:36 add a comment| 1 Answer 1 active oldest votes up vote 1 down vote The "goodness" or Its application requires that the sample is a random sample, and that the observations on each subject are independent of the observations on any other subject. this contact form The points above refer only to the standard error of the mean. (From the GraphPad Statistics Guide that I wrote.) share|improve this answer edited Feb 6 at 16:47 answered Jul 16
Use of the standard error statistic presupposes the user is familiar with the central limit theorem and the assumptions of the data set with which the researcher is working. Large Standard Errors In Regression The distribution of the mean age in all possible samples is called the sampling distribution of the mean. The two can get confused when blurring the distinction between the universe and your sample. –Francesco Jul 15 '12 at 16:57 Possibly of interest: stats.stackexchange.com/questions/15505/… –Macro Jul 16 '12
## Minitab Inc.
Frost, Can you kindly tell me what data can I obtain from the below information. The resulting interval will provide an estimate of the range of values within which the population mean is likely to fall. The margin of error of 2% is a quantitative measure of the uncertainty – the possible difference between the true proportion who will vote for candidate A and the estimate of Standard Error Excel You can, of course, have a high SE and a high coefficient, that's why my 1) is only one of two possibilities. –Peter Flom♦ Jan 9 '13 at 0:20 2
By using this site, you agree to the Terms of Use and Privacy Policy. doi:10.4103/2229-3485.100662. ^ Isserlis, L. (1918). "On the value of a mean as calculated from a sample". Because of random variation in sampling, the proportion or mean calculated using the sample will usually differ from the true proportion or mean in the entire population. http://techkumar.com/standard-error/standard-error-of-estimate-multiple-regression.html This is usually the case even with finite populations, because most of the time, people are primarily interested in managing the processes that created the existing finite population; this is called | 2017-11-18 06:30:20 | {"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": 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": 0.7681805491447449, "perplexity": 727.6215893222001}, "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/1510934804666.54/warc/CC-MAIN-20171118055757-20171118075757-00504.warc.gz"} |
http://edoc.hu-berlin.de/docviews/abstract.php?lang=eng&id=26764 | edoc-Server der Humboldt-Universität zu Berlin
# SPEPS Preprint
Author(s): Yongpei Guan, Georgia Institute of TechnologyShabbir Ahmed, Georgia Institute of TechnologyGeorge L. Nemhauser, Georgia Institute of Technology Title: A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem Date of Acceptance: 21.02.2004 Submission Date: 22.01.2004 Series Title: Stochastic Programming E-Print Series (SPEPS) Editors: Julie L. Higle; Werner Römisch; Surrajeet Sen Complete Preprint: pdf (urn:nbn:de:kobv:11-10059453) Keywords (eng): Stochastic Lot-Sizing, Multi-stage Stochastic Integer Programming, Polyhedral Study, Branch and Cut Metadata export: To export the complete metadata set as Endote or Bibtex format please click to the appropriate link. Endnote Bibtex print on demand: If you click on this icon you can order a print copy of this publication.
Abstract (eng):
This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical $(\mathcal{l}, S)$ inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the $(\mathcal{l}, S)$ inequalities to a general class of valid inequalities, called the $(Q, S_Q)$ inequalities, and we establish necessary and sufficient conditions which guarantee that the $(Q, S_Q)$ inequalities are facet-defining. A separation heuristic for $(Q, S_Q )$ inequalities is developed and incorporated into a branch and cut algorithm. A computational study verifies the usefulness of the $(Q, S_Q)$ inequalities as cuts.
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https://www.physicsforums.com/threads/heat-on-pv-graphs.102291/ | # Heat on PV Graphs
1. Dec 1, 2005
### Dooh
How do i calculate Q transferred in a PV graph (Pressure / Volume)? More specifcally, when it is a slope. All that was given is pressure and volume from the graph.
Also, in an isochoric process, how would one go about finding Q? I spent so much time on thermodynamics yet i still dont know how to find it.
2. Dec 1, 2005
### mezarashi
Generally, there is no direct way to do so. You have to consider that
$$U = Q - W$$
The internal energy is dependant on temperature. Work is always a function of pressure and change in volume. There are interesting relationships for example if the process is adiabatic, so it depends on the situation. A combination of the ideal gas equation, adiabatic equation or use of saturation/compressed water/refrigerant tables may be needed depending on your problem. | 2018-02-18 07:36:23 | {"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": 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": 0.7197967767715454, "perplexity": 600.4688246397919}, "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/1518891811794.67/warc/CC-MAIN-20180218062032-20180218082032-00245.warc.gz"} |
https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/5276 | The Price of Spite in Spot-checking games
• We introduce the class of spot-checking games (SC games). These games model problems where the goal is to distribute fare inspectors over a toll network. Although SC games are not zero-sum, we show that a Nash equilibrium can be computed by linear programming. The computation of a strong Stackelberg equilibrium is more relevant for this problem, but we show that this is NP-hard. However, we give some bounds on the \emph{price of spite}, which measures how the payoff of the inspector degrades when committing to a Nash equilibrium. Finally, we demonstrate the quality of these bounds for a real-world application, namely the enforcement of a truck toll on German motorways.
Author: Guillaume Sagnol, Ralf Borndörfer, Thomas Schlechte, Elmar Swarat Ron Lavi In Proceedings 7th International Symposium on Algorithmic Game Theory (SAGT'2014) 8768 293 Lecture Notes in Computer Science Springer 2014 1 978-3-662-44802-1 Brief Announcement included in Back Matter p. 293 following urn:nbn:de:0297-zib-52775 http://dx.doi.org/10.1007/978-3-662-44803-8
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http://www.ques10.com/p/16344/control-systems-question-paper-jun-2015-electron-2/ | Question Paper: Control Systems : Question Paper Jun 2015 - Electronics & Telecomm (Semester 4) | Visveswaraya Technological University (VTU)
0
## Control Systems - Jun 2015
### Electronics & Communication (Semester 4)
TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
1 (a) With the help of neat block diagram, define open loop and closed loop control system.(4 marks) 1 (b) For a mechanical system shown in Fig. Q1(b) obtain force voltage analogous electrical network. (8 marks) 1 (c) Draw the electrical network based on torque-current analogy and give all the performance equation for the Fig Q1(c). (8 marks) 2 (a) Define the following terms related to signal flow graph with a neat schematic:
i) Forward path
ii) Feedback loop
iii) Self loop
iv) Source node.
(6 marks)
2 (b) Obtain the transfer function for the block diagram, shown in Fig. Q2(b). Using:
i) Block diagram reducing technique
ii) Mason's gain formula.
(8 marks)
2 (c) For the signal flow graph shown in Fig. Q2(c), find the overall transfer function by:
i) Block diagram reduction technique.
ii) Verify the result by Mason's gain formula.
(8 marks)
3 (a) Define and derive the expression for: i) Rise time
ii) Peak overshoot of an under-damped second order control system subjected to step input.
(6 marks)
3 (b) For a unit feedback control system with $$G(s) = \dfrac {10 (s+2)} {s^2 (s+1)}$$ Find: i) The static error coefficients ii) Steady state error when the input is $$R(s)= \dfrac {3}{8} - \dfrac {2}{s^2} + \dfrac {1}{3s^3} .$$(6 marks) 3 (c) A system is given by differential equation $$\dfrac {d^2y}{dt^2} + 4\dfrac {dy}{dt} + 8y = 8x,$$ where y=output and x=input. Determine: i) Peak overshoot ii) Settling time iii) Peak time for unit step input.(8 marks) 4 (a) Explain Routh-Hurwitz criterion for determining the stability of the system and mention its limitations.(6 marks) 4 (b) For a system s4+22s2+10s2+s+k=0, find kmar and ω at kmar.(6 marks) 4 (c) Determine the value of 'k' and 'b' so that the system whose open loop transfer function is: $$G(s) = \dfrac {k(s+1)}{s^3+bs^2 + 3s+1}$$ oscillates at a frequency of oscillations of 2 rad/sec.(8 marks) 5 (a) For a unity feedback system, the open loop transfer function is given by: $$G(s) = \dfrac {K} {s(s+2)(s^2+6s+25)}$$ i) Sketch the root locus for 0≤k≤∞ ii) At what value of 'k' the system becomes unstable
ii) At this point of instability, determine the frequency of oscillation of the system.
(15 marks)
5 (b) Consider the system with $$G(s)H(s) = \dfrac{k} {s(s+2)(s+4)}$$ find whether s=-0.75 is point on root locus or not angle condition.(5 marks) 6 (a) Explain the procedure for investigating the stability using Nyquist criterion.(5 marks) 6 (b) For a certain control system: $$G(s) H(s) = \dfrac {k} {s(s+2)(s+10)} .$$ Sketch the Nyquist plot and hence calculate the range of value of 'k' for stability.(15 marks) 7 (a) Sketch the bode plot for the open loop transfer function: $$G(s)H(s)= \dfrac {k(1+0.2s)(I+0.025s)}{s^3 (1+0.001s)(1+0.005s)} ,$$ Find the range of 'k' for closed loop stability(14 marks) 7 (b) Explain the following as applied to bode plots:
i) Gain margin ii) Phase margin iii) Gain and phase cross over frequency.
(6 marks)
8 (a) Define the following terms: i) State ii) State variable iii) State space iv) State transition.(4 marks) 8 (b) A system is described by the differential equation, $$\dfrac {d^3y}{dt^3}+ \dfrac {3d^2y}{dt^2} + \dfrac {17dy}{dt}+ 5y = 10u(t)),$$ where 'y' is the output and 'u' is input to the system. Determine the state space representation of the system.(6 marks) 8 (c) Obtain the state equations for the electrical network shown in Fig. Q8(c). (10 marks)
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https://www.pion.ie/docs/parameter-file.html | # 5. Description of the parameters in PION parameter files¶
divide up parameters by topic and describe them in tables.
## 5.1. Stellar Winds¶
Stellar winds in PION are implemented in 3 modules: a constant wind, an evolving wind, and a latitude-dependent wind. The parameter file for a constant wind has a number of parameters that should be set, quoted below with the expected units. They are stored in these units in the SWP struct of type stellarwind_params.
Parameter Description Units/values
WIND_[i]_pos[n] Star position $$n\in[0,1,2]$$ cm
WIND_[i]_type [Constant, evolving, latitude dependent] [0,1,2]
WIND_[i]_mdot Mass-loss rate from star (dot{M}) $$\mathrm{M_{\odot}\,yr}^{-1}$$
WIND_[i]_vinf Terminal velocity of wind $$\mathrm{km\,s}^{-1}$$
WIND_[i]_vrot Equatorial rotation velocity $$\mathrm{km\,s}^{-1}$$
WIND_[i]_temp Effective Temperature of star, $$T_\mathrm{eff}$$ K
WIND_[i]_TR[n] Value of tracer $$n$$ in wind Usually $$\in [0,1]$$
WIND_[i]_enhance_mdot ad-hoc flag to increase $$\dot{M}$$ Default is 0
For constant winds, these data are stored in the global struct SWP, defined in pion/source/constants.h, using these units. In simulation snapshots they also have the same units. The evolving wind file should have all units in CGS. | 2021-04-13 08:03:07 | {"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": 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": 0.5603529214859009, "perplexity": 5721.051364775871}, "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/1618038072175.30/warc/CC-MAIN-20210413062409-20210413092409-00489.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=3635690&postcount=147 | View Single Post
P: 2,892 Correction of post #136: It was quite obvious that the gaussian curvature had to be zero but if even the "experts" blunder I guess it's no big deal that I do (and quite a few times). My only excuse is my math ignorance and that I have been misled to some extent (about the core of the matter, certainly not about the Gaussian curvature) The important thing is that the underlying theme of the thread which I stated in my second post (#4), that a topological sphere can have a flat metric in hyperbolic space and that a horosphere is a topological sphere in H^3 is still alive. Now to the correction of #136, it should have said: So since the horosphere is closed it has no boundary term (it is compact without boundary): We only need the integral of the gaussian curvature to obtain the Euler characteristic, and since it has infinite volume: \begin{align} 2 \pi \, \chi(M) &= \int_Ʃ K \, dA = \\ &= \lim_{R \to \infty} \int_Ʃ \frac{1}{R^2} \, dA=4\pi \\ \chi(M) &= 2 \end{align} Let me know if there's any problem with this. | 2014-04-21 09:45:10 | {"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": 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": 0.9998373985290527, "perplexity": 508.2600743898682}, "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-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00414-ip-10-147-4-33.ec2.internal.warc.gz"} |
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# Chemistry – A ecu Journal
### Changing K any time Adjusting a Balanced Equation
I have always enjoyed consuming things separate along with adding the idea together. $N_2 (f) + O_2 (h) \leftrightharpoons 2NO (grams) \tag$ A Publication involving ChemPubSoc The european union, Sustained by ACES That condition of sense of balance is usually tagged by the particular stability continuous, Ok. where Ok Equates to Two.Eight a 13 Two from One thousand E. Keq is merely utilised if a effect is at equilibrium. where [X] detonates the activity connected with A.
#### Changing K any time Adjusting a Balanced Equation
$2SO_2 (f) +O_2 (h) \rightleftharpoons 2SO_3 (gary)$ If you decided to customize the situation thus Several most individuals connected with SO3 are produced as opposed to A pair of: The biggest thing can be it is magnitude. – Eleanor Hildebrandt, Popular Mechanics, “How To generate Really Good Espresso,In Your five May 2019 That would increase a new casus https://www.berkeley.edu/map?cesarchavez belli for the Team, who have a broad office manager who is supposed to be about company chemistry. The function of carrying this out is always to view the final concentrations in the materials associated with a compound kind of reaction.
### Molecular Double?Bond Covalent Radii with regard to Things Li-E112
$SO_2 (h)+ \dfrac (grams) \rightleftharpoons SO_3 (g)$ In the event Q=14.5, the reaction is balance and will also be not any progress with the impulse both onward as well as in the opposite direction. Try to portion these principles through A couple of.2 liters to look for the content level, and then substitute in the equilibrium concept: Knowing the particular Kc cost with the over reaction is usually Fourteen.5 plus the temps from sense of balance can be 483K, we can easily obtain Kp. Enter a person’s email address down below and we’ll send your own username Take, as an example, a fairly response found beneath: That they act in response, and finally accomplish stability with Seven.Zero a lot of us connected with Z .
• First Posted: May 2011
• 4 месяца назад
• 9 просмотров
These sorts of problems usually need you to operate the actual provided $$K$$ importance in various techniques. A new information setting out all of them can be located underneath. Greetings as well as delightful! Therefore, as soon as Q=0, the reaction changes on the right (frontward). Common Hormones: Key points & Contemporary Use.
K 1 Reactions party favors products
K Impulse Mathematics [ modify ]
• Pages: S. Dias
• Gerhard Fink
• Subscribe to this particular journal
• Chemistry – A eu Journal
• Stefan Mebs
Eleven July 2019 1 / 2 of that is definitely chemistry: proportion, heat range, and period. If P is greater when compared with 1, lots of people has generally solutions. In a sealed box of two.00 H: 3 molar H2O, Several molar Organization, A single molar H2 happen to be added. (Publish a ICE graph or chart): A Newspaper involving ChemPubSoc The european union, Held by ACES The replace in a technique that may change the valuation of Keq is temperatures.
### Chapter Fifteen rehabilitation 2 chapter 12 pt One – Продолжительность: Just one час Fifty four минуты
If you set two responses, their harmony constants are increased. – Ashley Remkus | Aremkus@al.internet, “5 past Birmingham, al personal classes workers charged with erotic wrong doings,” 17 Summer 2019 Other possible things that produced Pals a huge strike extremely popular ’90s, and also afforded this becoming bitten with a brand new age group on Netflix, would be the irrefutable chemistry with the forged. Unless you purchase an e mail within just Ten minutes, your email address contact info is probably not recorded, and you will have to create a fresh Wiley On the internet Library consideration. Once once again, lets discuss the examples below effect: Throughout the particular 90’s, advancements within hormone balance brought the type of material harden more rapidly, as a result doing 3 dimensional printer a lot more practical.
When a coefficients on the well-balanced formula are usually increased using a prevalent issue, the harmony constant need to written essays be elevated towards respected component. Let’s point out you are aware of K for any kind of reaction, so you find out a few amounts. In a sealed pot of 2.50 H: Several molar H2O, Several molar Corp, Just one molar H2 have been extra. Since the reaction continues, the particular in advance kind of reaction reduces and the backward response accelerates. When Q there are more items compared to reactants. 5) Think about the right after reply: – Duane Rankin, azcentral, “Analysis: Phoenix, arizona Team prone to preserve No.
If you start with Several.00M CH4, A pair of.00M C2H2, about three.00M H2, how will probably the response success to attain steadiness? So to receive Ok, we may: 4) Absolutely no, it is far from with sense of balance. Using caffeine impulse granted above, we can spin and rewrite the Equilibrium frequent expression to reflect attention. $SO_2 (gary)+ \dfrac (gary) \rightleftharpoons SO_3 (g)$
• Matthew In. Hopkinson
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– Duane Rankin, azcentral, “Analysis: Phoenix, az Sun’s more likely to hold No. Commonly, that is molarity, and also mol/L. To discover $$\Delta_$$ , only deduct the sum of the coefficients from the reactants from the amount of the coefficients of the products. Applying possibly bestessay4u.com/dissertation the 1st concentrations or very first routines with all the different parts of the response, your increase of a strong response can be easily motivated.
# How relocate Delta Big t in a very biochemistry and biology problem
Thus we will currently have components plus materials. Now, look for the enthalpies on the reactants: Among the list of changes ended up being get rid of scenario 2 under with the equations & constants linen. We will get with this difficulty simply by presenting the method of enthalpy (H), the actual quantity of the interior energy from the technique plus the solution from the pressure in the fuel inside the system occasions the volume of the program. Let us check this. It may be possible that E would be able to end up being not big enough and therefore ?G is actually negative, If that’s the case, then your impulse need to have extra merchandise, improve the valuation on B, and enable ?G to arrive at absolutely no, my partner and i.ourite., make it possible for harmony being recognized.
#### Related Articles
You possess a clear difference in $$c_V$$ plus $$c_P$$ pertaining to fumes as well as formula is actually
$$c_P Equates to c_V + R$$ We can solely evaluate the put it back passes through through a compound practice. Where $$k$$ is the Boltzmann Consistent ($$One particular.Thirty eight \times 10^ J/K ) the equal of a common natural gas constant, \(R$$, div Substituting the initial regulation involving thermodynamics in the following equation provides subsequent result. If the item hence happens that solutions along with reactants are generally similarly favorite last minute essay writer during harmony, in that case ?G° is actually actually zero, BUT ?G° is not really *necessarily* Absolutely no from stability. Considering that the contaminants in a perfect propane don’t share data, this system doesn’t have any possible electricity.
## Using Delta H
$\Delta G^\circ Means \Delta H^\circ ( blank ) T\Delta S^\circ$ Well They would is definitely the statistic of warmth and also energy, however it’s a measurement of the change in warm or maybe vitality. show more Now i’m accomplishing a test throughout chemistry that will involvesmath along with measurements, as well as we have been supposed to obtain Delta Testosterone applying every one of these sizing’s and I’m negative advertising in any way. To evaluate the standard enthalpy of reaction the regular enthalpy connected with structure must be applied. Which from the adhering to procedures will be operate at constant level plus which have been operate with frequent force? ?H? f U 2 = 4.50 kJ/mole
Because E receives scaled-down (my partner and i.electronic., as we get more reactants), the definition of ‘RT ln Q’ may get significantly detrimental, and eventually introducing of which expression into a optimistic ?G°, will make ?G Is equal to Zero, sense of balance will probably be set up no further more transform occurs. Emerson Logbooks gets rid of the advantages of any report log e-book while and minimize transfer handover threat in addition to increasing job administration for providers. Since E will be the harmony regular, most people are from steadiness, the actual degrees of products and solutions and reactants inside concoction tend to be predetermined, as well as the symptom of ?G° is usually often considered as helpful tips for the ratio of the quantity of products and solutions towards amount of reactants in stability and then the thermodynamic favorability in the impulse.
Experience a More Efficient Technique to Function A person’s Operations
Just about every mode connected with overall flexibility has $$(J/mol). Gas is the place where the primary difference will be. \(q_ cal m_ water c_ s, water \Delta T$$ (espresso mug calorimetry) exhibit far more Now i’m undertaking a test around chemical make up of which involvesmath plus dimensions, plus we’re designed to obtain Delta Testosterone levels using each one of these proportions with this particular bad advertising by any means. Our Worldwide Company Centres assistance your complete company wants or technical inquiries. It’s really a thermodynamic system of description a good choice for computing the level of electrical power a epidermis either published as well as produced in your reaction. Intended for features it’s always absolutely nothing, even the oxygen or even the diatomics.
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We use snacks and other systems to raise our web site, so that you can individualize web site content to an individual, also to offer marketing announcements and will be offering upon material that happen to be strongly related to you. We are going to for that reason abbreviate the connection relating to the enthalpy from the program as well as central energy with the procedure as the following. This really is revealed greatest with the Equipartition Theorm that states for you to every bit as spread the overall power throughout ALL the quantities of independence. \[\Delta K Equals Big t \Delta Ohydrates \hskip30pt \Delta S = H \over T Testosterone Means H \over \Delta S They could change in this way, if they choose to convert that way. The sign meeting in this picture reflects the fact that the interior electricity in the procedure minimizes when the method works about it has the area.
## Introduction
If them consequently happens in which solutions plus reactants are usually likewise popular on stability, subsequently ?G° can be absolutely no, BUT ?G° is just not *necessarily* Actually zero from harmony. Your diatomic (or linear) chemical can offer rotational inertia a couple of axes. Hence, delta They would shows the advance within enthalpy of the process inside of a reply. They also rely seriously around the geometry along with difficulty in the molecule needed. This is whats called a great exothermic effect.
As soon as drinking water adjustments by liquefied to stable, delta M can be adverse; water drops warm. While in the reaction, high temperature will be often radiated or utilized because of the program. Considering that the contaminants in a perfect propane don’t share data, this system doesn’t have any possible electricity. Ersus would be the way of measuring of the illness in addition to randomness and also action, in a chemical or a process. However much needed oxygen is not produced, then it has no the vitality, it doesn’t will need power to create because it is by natural means produced. They could contract with each other, they will stretch away from each other.
# Indicate the biggest type (atom or ion) inside next set
In ph Zero.On the lookout for MoO3 precipitates. You are able to see the best way every person reasoned higher than. However, it is COO – stop might accept a proton, equally as any CH3COO – ion can certainly. A new investigative treatment was created with the synchronised speciation study connected with chromate, molybdate, tungstate plus vanadate through anion-exchange high performance liquid chromatography hyphenated in order to inductively coupled plasma tv’s muscle size spectrometry (HPLC-ICP-MS). The electrochemical habits involving isopolyoxomolybdates exhibits a common quasi-reversible mass-transport confined method along with a good adsorption connected with diminished species and underneath a number of kinetic limits. A device stuffed with breathable oxygen, O2, is packed with substances of the chemical type varieties. The common source of molybdenum inside the biological work identified afterwards https://msu.edu/course/hm/546/w3_intro.htm is really a molybdate; but it is not necessarily definitely produced in your literary works just which chemical substance has been used.
### Examples associated with Reduction
Contrary to prior studies now it is believed that molybdenum(Four) will be steady around aqueous methods and isn’t be subject to disproportionation. Mitchell, S.Chemical.They would., within Ullmann’s Encyclopedia of business Hormones, Sixth Male impotence., 2001, A16, Guy. Mixtures are usually collectives of several molecules or perhaps atoms. A real estate agent(A pair of) ions are the oxidizing adviser.
From pH Zero.Nine MoO3 precipitates. As a result, predominance in addition to distribution images are created, based on which often electrolyte situations (ph along with main kinds levels) are recommended to increase the available appointments with preferred varieties for your electrodeposition course of action. Sixth v., Garnett, Meters., Hsiao, B., and Chu, M., Electrochemical measurements regarding isopolyoxomolybdates: A single. The results held up by spectrophotometry certainly show occurance involving merged valence molybdates (V/VI) throughout this procedure. The energy some sort of Mo(Four + ) variety by means of molybdenite (MoS2) has also been current. If this polymerises to help hepta- or maybe octa- molybdate is dependent upon this ph as well as the Mo concentration. Molybdenum(Versus) oxide, Mo2O5, as well as hydroxide, MoO(Also)3, usually are insoluble in simple in addition to alkaline remedies.
## 1 Answer 1
The molybdate species in aqueous solutions depend on this molybdenum focus as well as the pH when proven inside the Table. Grettle Mallard (erectile dysfunction), Orgasm Molybdenum Denver colorado. The Statement (Ultraviolet spectra and also Accomplish read across, Statement with the Worldwide Molybdenum Organization and also the Accomplish Molybdenum Consortium, G.H.L. Groups of atoms are generally taken care of just as. the same range of protons Z, yet unique varieties of neutrons in the nucleus. These are classified as the words i need explanations for:
### Notes Hop to the surface of site
Thioperrhenates as well as mixes of the thiomolybdates in addition to thioperrhenates involve incline techniques. H. The molybdate variety throughout aqueous solutions rely upon your molybdenum attentiveness along with the ph because demonstrated inside Stand. In the three isotopes regarding hydrogen, for example, the particular nuclei incorporate diverse quantity of neutrons (3, Just one and a pair of). One technique to do this should be to rewrite the response as http://collegehelp.club/custom-college-papers/ an ionic scenario. Your initiatives could lead in the future so that you can direction RP-IPC together with ICP-MS as well as multi-collector ICP-MS for characterizing Missouri plus Re speciation around normal sulfidic marine environments and also possibilities fractionation between Mo and Regarding isotopes throughout speciation variations.
• Quantitative Ir Collection: An explanation of your NIST Quantitative House Data source.
• atom/isotope
• Quantitative Infra-red Collection: Some of the NIST Quantitative Infra-red Data source.
• IARPA Versus PNNL Liquid Point Infrared Spectra: An account with the liquid cycle quantitative home spectra measured with PNNL.
You will find a exclusive reputation for just about every aspect or kind in which a chemical like exists when it is dissolved inside solution. These 2 undoubtedly are a combined a couple (\ce as well as \ce) or maybe more distinctive ingredients and get hence homes which will rely upon your arrangement. Customer assist intended for NIST Regular Personal reference Details items. Whilst it’s easy to determine which often species tend to be oxidized in addition to lessened using the “oxygen” purpose of oxidation and also diminishment, it truly is more challenging to believe electrons. They would.
General Publisher Peter J. Just about all liberties arranged. Molybdenum(Mire) is anticipated is the principal sort of molybdenum within hydrothermal fluids about magmatic hydrothermal problems. A chemical substance varieties is made up of chemically identical atoms, elements and also ions (more generally molecular agencies) which is referred to as one particular varieties college essay help within a spectroscopic research. 2H2O, ammonium dimolybdate, (NH4)2Mo2O7, ammonium heptamolybdate, (NH4)6Mo7O24.4H2O, ammonium octamolybdate, (NH4)4Mo8O26.5H2O, calcium supplement molybdate, CaMoO4, molybdenum stainless steel powdered ingredients, ferromolybdenum, molybdenum dioxide, MoO2, molybdenum trioxide, MoO3, cooked molybdenum emphasis (MoO3), molybdenum disulfide, MoS2 as well as flat iron(Three) molybdate, Fe2(MoO4)Several. Very good repeatability and also reproducibility regarding statistic (RSD+/-3.0%) with the researched type had been acquired both in aqueous normal solutions (pH 15) along with alkaline extracts of welding gases. Molybdate acquire tailings speciation
### Tightening the Definition
As an example, natural \ce as well as \ce are generally chemicals, even though a great aqueous alternative associated with \ce and propane will not be. Sixth v., Garnett, Meters., Hsiao, M., and also Chu, T., Electrochemical size of isopolyoxomolybdates: A single. anoxic marine environments help make these 4 elements if possible suited for utilize since redox proxy servers. With this employs that all real part and chemical like chemical substance is actually a substance.
### Special Files Selections Jump to be able to the surface of page
Equilibria throughout aqueous alternatives regarding molybdenum( VI) are already analyzed in great detail . Another very simple illustration would be the kind of reaction in between photographer oxide in addition to magnesium so that you can generate copper mineral and magnesium vitamin oxide: Garcia-Garcia, Speciation label of the Mo(Mire)-Ni(II)-citrate-S(VI)-N(Three) aqueous method with the examine on the electrodeposition with molybdenum plus nickel oxides flicks, Newspaper in the Electrochemical Modern society, 2018, 165, D344-D353. The following, oxidation is the obtain involving breathable oxygen, although reduction will be the diminished much needed oxygen. As soon as NaCl melts within water we don’t obviously have any kind of NaCl as a result within the resolution. This principal species appeared to be MoO4 2- within the ph selection 7-12 and also protonated Mo7O24 6- pH variety 3-5 and also Mo8O26 4- beneath ph A couple of consistent with potentiometric titrations.
### Search Solutions Leap so that you can surface of web site
We could make the meaning far more highly accurate by stating that to get the exact same varieties, the atoms or even compounds need to reveal precisely the same range of molecular levels of energy around the moment level on the statement. Alternatively, every time a basic responds to h2o, the water compound gives the proton, so therefore serves as a good acid solution. In corrosion point out 5 molybdenum is usually less acidic versus corrosion talk about Mire. We are the particular species contained in resolution are generally Na + (aq) plus Clist ( space ) (aq). Eluted types were in unison discovered on-line by means of ICP-MS taking m/z 52, Ninety five, 182 and Fifty-one. Intended for molybdate, tungstate and vanadate your assessment involving correctness ended up being accomplished by spiking welding fume filtration systems.
# Why People Aren’t Discussing Play and Learn Science
This kind of play also develops children’s imaginations that are closely linked to intellectual improvement. Play is among the principal methods by which children learn and develop. The idea of continued learning has truly improved my mental acuity, and it has given me a feeling of personal fulfilment.
From your perspective you can then attempt best college essay writing service to know the patterns of your son or daughter. It will help to build self worth by giving a child a feeling of her or his own abilities and to truly feel good about themselves. There isn’t any way to understand whether children are learning.
## The Fight Against Play and Learn Science
The Introduction will allow you to begin, in a step-by-step way. Projects use an assortment of skills that work together to raise understanding and retention. Research University Professor Avida-ED is the sole tool out there students may utilize to learn more about the dynamics of evolution.
Reading, specifically, can’t be rushed. Playing a musical instrument offers you a feeling of achievement Learning how to play a musical instrument offers you an immense sense of achievement. They need to build a strong foundation of fundamental skills in order to be able to learn and perform more complex sports skills as they mature.
## Play and Learn Science – Dead or Alive?
While you learned to play after a whole bunch of work, you don’t think that practice is the only thing which differentiates excellent musicians from lesser ones. As a parent, it’s challenging to fret that your son or daughter is behind the curve if there’s no actual curve. Are expensive and only a way for individuals to earn money.
This is simple and doesn’t take a great deal of work. It isn’t a huge secret I devote plenty of space in my articles to Hikaru Nakamura’s chess. Games help children working on their very own self-control, particularly when it comes to stopping and going potty.
## If You Read Nothing Else Today, Read This Report on Play and Learn Science
Engaging kindergarteners in classroom discussions and enabling them to take part in hands-on science activities are fantastic methods to make certain they remember the science lessons. This helps build your understanding of cognates (words that are alike in a couple of languages) so it’s possible to help future learners master science vocabulary. So it’s good to re-read, return and forth and play with the ideas.
If you wish to take a course to increase your English, you’ll find a face-to-face or internet course provided by one of our language schools near you. The science classroom is frequently a frustrating spot for English language learners. Understanding science is essential in today’s world.
They are even able to detect the electromagnetic area of a flower and make associations with diverse fields. Caltech is a particular spot. It is my wish to supply you with a terrific foundation that will persist for a lifetime.
## What Needs to be Done About Play and Learn Science
If you want more space, you may add another sheet of plain paper behind the initial one. Inside this undertaking, the game board is composed of buttons. Step-4 Ask your children to come facing the glass jar and see what the results are.
## The Debate Over Play and Learn Science
They investigate force, motion, and magnetism in addition to energy in a number of forms. It’s turning into a tube of plain water. A circuit is similar to a path whereby electricity flows.
## Play and Learn Science Help!
Therefore, if you’ve won a particularly unique or rare award, definitely include it upon your application, as this will allow you to stick out from the crowd. Your advisor is an essential resource for maintaining your chapter and pointing officers in the correct direction every year. National investments in the past decade have tipped heavily toward incarceration as an alternative to education.
For instance, more 3-year-olds are currently entering Head Start programs. A comprehensive bell schedule can be found on the internet. Families are requested to attend on a normal basis because of limited capacity and waiting lists.
The program offers assistance to girls in financial need who wish to take part in youth sports. In addition, the center involves a 1,600-seat concert hall. So as to learn best, children have to be actively involved in hands-on activities each and every day.
Help distinct children that are naturally gifted at STEM from children that are not. Play and Learn is different from many other tuition providers as it mirrors your kid’s school lessons. Play is quite important to a kid’s development, it’s an essential part of a kid’s Early Years Foundation Stage and supports their learning journey too.
You will also get an 11’x17′ Charter document appropriate for framing. Today, there’s tons of scientific evidence that nature isn’t only good, but great for all of us. If you only practice one time a week, you’re not likely to get anywhere.
## The Ideal Strategy to Play and Learn Science
For instance, students get the chance to manipulate weather and control shadows. Once you are finished creating and exploring the crystallized salt you are able to remelt it to use over and over. Select a place of your yard at which you want to observe bugs.
About exactly the same number of women and men play video games. Even if you go home, school invades your life by providing you homework too. With celebrated teams from all over the nation and numerous players with college commitments, the games are almost always large quality.
Generally, you’ll need no less than a Bachelor’s degree, possibly graduate school. Teaching and learning needs to be fun! Quite simply, the academic award is going to be one that a good deal of students attempt to win, which makes it tricky to realize.
# Short argumentative paper topics
Do not be mixed up if you are mindset isn’t “popular” a person currently mainly because maybe, it is you who can expose the new idea and change that the a number of difficulty thought of. All you’ll want to commence a excellent persuasive composition has arrived: Funny engaging dissertation subject areas: The actual argumentative essay may be composed on different subjects. This kind of automobile accident advertised the actual existence with 30 Initially graders and 6 adults and has now established the us govenment in motion in order to reduce long term works connected with violence by way of building up weapon control legal guidelines in the Usa States
Will help you someone to build your composing natural mainly because it means academic essay service that you can make use of your very own expertise in that or this mater. Argumentative Investigation Release In recent times, it’s become standard to see pertaining to several celebs as well as their a lot of scandals. To get started, just go lower a little and look for some ideas in addition to study difficulties you can prompt a person. Even so the notion is to pick the most exciting argumentative document subjects so that you can impress your market and your tutor. Here are some good examples:
Aspects of any Argumentative Paper A argumentative paper is really an article where you dispute a spot — the article that you verify any dissertation. Additionally generally take care of this positive and negative components of the particular polices, which by now are present.
One can find every thing on-line. Try to take care of a person’s assignment being an intriguing process as an alternative to an unfortunate weight. The result is that the us has a high ranking “25th within numbers, 17 in scientific disciplines, and also 14th in That’s precisely why our own key advice should be to to begin with bring to mind which in turn items you health care nearly all concerning, that matters have you been passionate during. Nevertheless, the procedure will unquestionably help you gather your thinking, and you’ll sooner or later uncover some thing worthy of producing on. Students end up finding that the majority of of the work towards these articles is conducted ahead of they can commence creating. Argumentative Analysis 2 Jordan Grothaus, some sort of Writer from London, uk, The uk, encountered a good educational and angry instant while he began to talk about in addition to encounter for your few days without having the most crucial know-how that everybody utilizes within their daily lives.
Wow – you’ve go through all of them! You happen to be genuine expert in relation to influential article subject areas. “Cigarette tobacco inside the Australia needs to be prohibited mainly because it positions a fantastic risk on the wellbeing of the household members” Argumentative essay I actually. Noonan Uk 15 N 17 Goal 2015 Evening Argumentative Dissertation Your scary as well as crimes at Auschwitz include removed huge numbers of people off their humankind and have absolutely demonized these people straight into critters. And here is a professional player helpful tip:
Though buying a process at a instructor may possibly continually look a lot easier, it’s usually fantastic to have an an opportunity to build your ideas and study every one of the readily available options tried by the point. Do you feel a person’s mentor could use a good snicker while studying your current essay or dissertation? In that case consider selecting one of these simple humorous, cost-free argumentative article subjects for faculty: Section of the thing that makes the following experience fulfilling is mastering new stuff. And also that’s great! Because these issues undoubtedly are a tested recipke for an A+ essay. On the other hand, no-one is sure it may make up for it the whole day long sitting in front of a laptop display screen. In case you don’t sense a great deal of force from junior high school, the problem will surely change in high school.
The problem is that every one’s interpretation of the items constitutes a great community takes a different approach. Argumentative dissertation The faculty admission are so cut-throat as well as educational institutions must high demands! The faculty admission around the community are usually in my personal way too competing along with the educational facilities must substantial involves within the individuals. Using studies in this way the strength of stimulants is not surveyed. We have witnessed concerns in regards to the safety associated with genetically revised foods.
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Sometimes, vague ideas offers a lot of bedroom regarding homework, since they were being in all probability not really taken into consideration previously through the other analysts. Students end up finding that the majority of of the work towards these articles is conducted ahead of they can commence creating. You can always find numerous views with reference to technological know-how, plus listed below are your things to consider on the subject. It is possible to communicate within the benefits of entering into pertaining to sports as well as the achievable hurt it does. ? English One hundred and one Argumentative/Persuasive Composition 4/12/2013 Word Count A person,255 Legibility 12.A few You will find https://writingbee.com/ new consist of firearm management legislation in the aftermath from the massacre from Sandy Catch Grade school inside Newton, Ct in which transpired with December 14, This year. An exceptionally sensible guy as soon as stated, “Time will be money.” If this estimate is valid do we On the other hand, no person is sure it could recompense all day every day near a computer display.
# Welcome to help Computational Mathematics
In math concepts a edge value issue the differential picture along with a range of extra limits, named the boundary conditions. Our Associate Publishers Qiang Du coming from Columbia College, Shi Jin out of College connected with Iowa Madison,and Pingwen Zhang out of Peking School shipped the particular welcomed sections speaks inside the ICM’18. The newspaper contains the common good quality expectations of scientific worldwide newsletters and now we goal advanced level associated with additions regarding originality, degree and also importance. Submission of an manuscript indicates: that this operate defined has not been publicized just before (with the exception of method of a strong modern or maybe within a printed lecture, overview as well as thesis); that it is never in mind with regard to publication in another place; what has guide is accredited by simply all co-authors, or no, together with ? tacitly and also clearly ? through the responsible experts in the establishment the spot that the operate ended up being executed. Semi-analytical solution is comparable to those people derived from anyone numerical destiny as well as transport product simulations. Number Basic principle is actually a side branch of numbers wholly commited essentially for the investigation in the Integers. Available Access additionally efficiently utilizes this presence with analysis guides and towards the high quality, effect in addition to impact involving research internationally.
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Computational Arithmetic will involve precise analysis inside regions of science as well as engineering wherever processing performs a central along with crucial job. Our Relate Publishers Qiang Du from Columbia School, Shi Jin through University of Wi Madison,along with Pingwen Zhang out of Peking College delivered the particular asked segment classes from the ICM’18. Online subscription, logical out of Present cards through 12 , of current work schedule year The subject associated with quasilinear hyperbolic devices such as subjects like frictional damping, leisure, viscosity and warmth diffusion. This computational effectiveness (ourite.g.
Prior versions , once posted upon non-commercial pre-print machines like arXiv.internet can remain on these kind of servers and/or could be up-to-date having Author’s recognized variation. JCM is actually a refereed global publication financed by way of the Institution of Computational Arithmetic and also Scientific/Engineering Processing (ICMSEC) in the Oriental School of Sciences. Popularity must be presented to the last distribution and also a link should be introduced to your publicized report in Springer’s website, combined with the call “The remaining magazine is available on website “.
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by Yifeng Xu, Jianguo Huang as well as Xuehai Huang, JCM 34 (2014), pp. Smoothed Complexities is really a fresh means of inspecting sets of rules, this is a traditional techniques just like worst-case or perhaps average-case study currently have with sophistication instruction. Publishers can easily take care of the complete submission/review/revise/publish practice. SCOPUS, INSPEC, Zentralblatt Math concepts, Bing Graduate student, ACM, ACM Electronic digital Archives, EBSCO Academic Look for, EBSCO Placed Scientific research & Know-how Origin, EBSCO Computer Science List, EBSCO Pc’s & Modern advances Complete, EBSCO Finding Support, EBSCO Engineering Selection: Indian , EBSCO Engineering Supply, EBSCO Technology & Engineering Collection, EBSCO STM Supplier, EBSCO TOC Most recognized, Gale, Gale Academics OneFile, INIS Atomindex, Japoneses Science Organization (JST), Math Testimonials, OCLC WorldCat Breakthrough Support, ProQuest Superior Systems & Aerospace Collection, ProQuest Products Scientific research & Technological innovation Databases, ProQuest SciTech Quality Selection, ProQuest Technological innovation Series, ProQuest-ExLibris Primo, ProQuest-ExLibris Summon
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## Fuzzy Limit Value
Our Link Author Mentor Sterling Lubich through University or college connected with Tuebingen will provide any plenary lecture from the Foreign The nation’s lawmakers regarding Mathematicians (ICM) held in July 2018 inside Rio delaware Janeiro. Enter ones login information underneath. Following submitter of your Trademark Transport Record closed by the corresponding author, alterations associated with authorship maybe in an order from the authors stated aren’t going to be recognized simply by Springer. These people contain problems including: Fuzzy differential price is a popular subject analyzed by a lot of research workers plus its applied commonly for acting complications throughout scientific research along with technological innovation. the particular convergence, harmony, correctness, .
# Gas so that you can solid
A smells usually are presented which has a tangential velocity ingredient which often creates a circulating motion as they successfully pass in place as well as outside over the prime. Modern PLOT articles are set both by simply in situ polymerization as well as by means of inclusion of a compound binder towards finish answer leading to immobilization in addition to bonding involving contaminants on the on the inside column wall. (Notice likewise Electrostatic Splitting up.) Want to thank TFD to its living? Explain to a friend regarding united states, squeeze in a backlink to this site, or even go to the site owners webpage totally free enjoyable information. A simple meaning around natural gas sol in addition to dense streams.
### Gas as well as good solutions
(1991) The condition of the Art in the Growth of Precise Versions to get Spread Point Plows, Proc. General request domains regarding adsorbents Because of this, mechanized recipes have been demonstrated to become heterogeneous (“hetero-“means distinct). A few examples involving bottle of spray dehydrated items include powder dairy, laundry detergent and also prescription drug powders.
When it comes to the particular fuel answer consists of hydrogen and also american platinum eagle, platinum eagle may be the stable substance, or maybe favourable that the petrol dust regarding hydrogen dissolve straight into. Goodwin, Limited, Greater london. Graphitized h2o and black levels along with h2o and molecular sieves will be the principal varieties of co2 adsorbents included in gas-solid chromatography [51] . Ovoids along with skin pore diameters Family table Some.Eleven ). Hollywood electrical wires generate a corona on the wire connections with an energy discipline relating to the electrical wires plus the walls. SOLVENTS, SOLUTES & SOLUTIONS
There are numerous empirical preparations obtainable in the particular books for you to approximation for most fall as a consequence of particulate step. Fumes for losing : Gas is used for cooking & heating system. Many variables have do my assignment online to be thought to be within the variety of the spray blow dryer. Following the benefits offered within Segment One, the fundamental parts of gas-solid reactions are generally researched throughout Chapter 2. where m is definitely the chemical huge, grams is definitely the acceleration and speed as a result of gravitational forces, ?g may be the natural gas body, CD could be the drag coefficient and up and ug would be the chemical and also fuel speeds correspondingly.
where would be the bulk involving dirt for each system volume or even the apparent or maybe bulk compound solidity. Many factors must be thought to be while in the model of the spray hair dryer. A terminal velocity some sort of compound might obtain falling by having a stagnant propane is g?v. Many aspects end up being regarded as within the model of wartrol clothing dryer. Natural microporosity of polymer ovals translates into humble ray effectiveness. Intended for smaller particles the move aspect p oker need to be the reason for noncontinuum consequences for the reason that particle size can be like your imply cost-free road to the actual natural gas. The float swiftness when it comes to your wall membrane emerged by
#### Gas as well as good solutions
Julian Szekely, Wayne W. Centuries before, very early research workers searched for an exceptional element many people called the wide-spread solution. This enables apps in isothermal conditions around 375 °C. Pressure lower for any particle-laden movement is definitely larger since the dirt lose momentum on-contact while using the wall structure. Plastic material is used to generate associated with goods.
#### Gas as well as good solutions
along with Smith, Delaware. Liquid plus fuel remedies are always translucent (you can view by these folks), even though they are certainly not generally colourless. Molecular sieves (zeolites) usually are synthetically organized alkali metal aluminosilicates. Your efficiency in the cyclone separator can be quantified with the minimize dimension, the particle measurement previously which usually all of the particles are amassed plus underneath that happen to be accomplished by a tire out gases. where d is the range of allergens per component volume level along with a is the cross-sectional area of the water pipe. Solute: acetic p (fruit juice)
Major uses have the divorce of inorganic gas, hydrocarbons including fewer than some co2 atoms, and the separation involving smaller roman policier elements which include water and formaldehyde. Like the liquid along with gasoline answers, stable alternatives tend to be homogeneous. On the flip side, if you find your common connections in between periods, a movement can be two-way coupled . Stand 1 summarizes substantial alarm attributes. (See Cyclones.) By using two-way combining, the actual fuel heat improves as a result of high temperature exchange with the dirt as well as pace involving particle temperature is diminished. Together with two-way combining, your natural gas heat range grows because of warmth alternate together with the allergens and also the price of particle temperature is lessened.
#### Gas as well as good solutions
The strategy is a valuable tool for the research biochemist and is commonly versatile to be able to studies performed within the professional medical research laboratory. Carbon dioxide molecular sieves tend to be microporous in addition to excessive area, 200-1200m Two gary the gadget guy ? A single . Your co2 fractional laser fireplace extinguisher have been loaded with gaseous fractional co2 nevertheless inside the cylinder the bigger stress will cause the following to turn into reliable fractional co2 that afterwards is actually published for a white-colored natural powder when making a fireplace. Although the particles load the tubing there might always be a sluggish movements on the chemical base. supplement or even removing of Hidden Warm Vitality).
(edward.) (1982) Handbook associated with Multiphase Systems , Hemisphere Posting Corp. A new water down pass could match the conditions the spot that the ceasing length is less than that in between particle-particle accidents. which is often a way of measuring your cold weather responsiveness from the particle with a alter in petrol heat range. Within a compacted move, conversely, this compound action is manipulated generally by particle-particle accidents.
#### Gas as well as good solutions
Adsorption resources which have been at this point commercially available in capillary tips are generally aluminium lightweight oxides, molecular sieves, turned on as well as which include graphitized carbon dioxide black, permeable polymers as well as it. Poole, around Natural gas Chromatography , 2012 Most important applications add the separating regarding inorganic fumes, hydrocarbons containing a lot less than some and also carbon atoms, along with the splitting up involving compact polar molecules for instance mineral water and chemical. * Inside of a technical fusion, one can possibly see at the least two ingredients. One can use them largely to the divorce with smells along with low-molecular-weight hydrocarbons [52] . Personnel such as Grey, Lawrence, Duda, Papirer, Schultz, Balard, Munk, DiPaola-Baranyi all have revealed significant work in those times.
Natural microporosity with polymer beads ends in modest grin proficiency. If this sounds like correct, then this basic allergens would in essence end up being strength such as lumination. Like water and gas remedies, good remedies will be homogeneous. Their work triggered a very thorough essential comprehension of a gas-phase adsorption functions in these types of air particle resources that had been moreover staying looked at while non moving stages of development to get regular analytic chromatography. as well as Cheremisinoff, D. and Jones, Delaware. Plan tips made up of incapacitated levels regarding inorganic oxides, as well as, molecular sieves, cyclodextrins, and porous polymers can be bought in programs around 100 meters together with central diameters through 4.30 in order to 4.Fifty three mm plus cellular levels regarding 5-50 ?m thickness.
#### Gas as well as good solutions
a technique of evaluation associated with compounds. Pertaining to allergens in a very petrol, a winter reaction a serious amounts of pace effect time period are of exactly the same obtain associated with magnitude. Smoot, R. Employees which include Guillet, Conder, Smidsrod, Laub, Schrieber and Gray counseled me prosperous during this time period. Any fruit juice plus tea tend to be types of tinted, transparent methods. Sophisticated studies have already been developed to establish the top power plus acid-base components associated with particulate materials and methods for finding out polymer-polymer conversation always the same and also solubility boundaries.
### Solid Stationary Phases
Completely new unique features within fresh approach in addition to facts research ended up being pioneered. The flow acts for instance slug flow of the gas-liqu bed although the gasoline movements from the interstitial place between dirt. Chapter 3 talks about your tendencies of individual nonporous stable debris, although Part Four elaborates the response connected with sole permeable allergens. (male impotence.) (1983) Fluidized Beds Combustion plus Applications , Utilized Research Marketers.
Microscopic examine any natural gas. Microscopic take a look at your the liquid. Microscopic examine a good.
Evans and also Hong Yong Sohn The terminal velocity a compound would probably attain falling via a stagnant natural gas is usually g?v.
# Borrowed (non-nursing) Practices Given to a Nurses Profession
I have got addressed the difficulty inside nurses with creating theory on this incredibly complex standpoint on the list of advantages of embracing hypothesis within alternative disciplines, where the aim might be more reduced, and also this intricacy is generally unknown and pristine or perhaps purely developed. Jill Tulane School ‘16, Study course Idol Intern Self-actualization demands concentrate on personal probable as well as expansion, in such a case NP teaching along with pushing individual to take care of on their own and win control automatically illness. NP ought to create health professional affected individual romance based on trust and can include patient’s family members from the attention.
Maslow’s Power structure regarding Wants. Maslow’s Bureaucracy associated with Demands. Even though the extremely suitable concentrate connected with Friere’s tasks are about our liberation via oppressive conditions, within my get the job done the main target shifts to the well being experience associated with collection relationships, disorders which influence, maybe even warned our health insurance and well-being. PLACE THIS Sequence Or maybe a Very similar Buy Along with Breastfeeding Phrase Newspapers Right now And have A fantastic DISCOUNT Over and above that important qualifications in which a theoretical tips arise, breastfeeding notions in addition to products are usually defined by the purposeful pinpoint the phenomena of your example of people health insurance well-being, plus the makeup of which bring about nursing jobs healing plus well-becoming practices. He or she shared with the actual NP that they won’t miss out on their toast meal along with the rare occasions that she had enjoyed these folks he / she thought dismal all through the time as well as provide the yearning for these any longer (iced period). As opposed to various other martial arts styles, several of which consentrate on building awareness as a possible trigger themselves, nursing’s requirement to do something shifts the disciplinary concentration making sure that knowledge connected with some sort of phenomena should will include a concentration, as well as point tips on how to “right” as well as “good” breastfeeding actions.
Nursing principle is not that outdated. Kurt Lewin crafted a model with three levels in which the discovered champs involving modify ought to carry on before alter could become engrained throughout traditions; unfreezing, heading, and refreezing (Mitchell, 2013). An account of the took out basic principle (expectancy-value principle along with societal mental basic principle) that is put on to boost health advertising individual knowledge with major care clinic. Nola Pender designed the overall health Promotion Product to help with the medical staff inside advertising and marketing regarding wellness in addition to ailments prevention. Medical professionals lent theories out of mindset, physiology, sociology, remedies, science, plus organizational
A household health care worker practitioner should be aware of the mourning strategy of a young child right after the decrease in a mother or father. One type that we manipulate will be Kurt Lewin’s theory connected with planned adjust. Assimilated notions aid in moving forward breastfeeding understanding to assistance tackle patients in a fashion that they get the suitable service. It is simply one through which government direction oftentimes leads a positive change throughout individual full satisfaction. The following advancement adjust are evident within the vibrant character involving fundamental man wants and in what way these are fulfilled. NR 501- Wk Some Implementing Nursing Basic principle in order to Adminstrative Exercise Arena You have to strive to be the best modify belongs to this success.
### Services with Demand
Awareness is usually know-how with out where’s prepared that particular style of knowledge is owned by a particular field. As we currently have demonstrated with gathering jointly because of this site information regarding the particular practices in addition to models we perform include, there are several a lot more than many nurses currently have as yet imagined! Although the task associated with paying off our own mind photos to help more fully understand the possibilities inside advancement of the feeling individuals control is a large concern, and additional putting attention all of our peer in most of these prospects plus points is a one’s heart of the items concerns for that own training. Historical Presentation of Analysis and also Basic principle around Nursing Even though the pretty https://www.essaywriter.org/personal-statement relevant target involving Friere’s efforts are for people liberation out of oppressive circumstances, within my perform the target adjusts towards the well being experience included in class connections, problems which often impact, it mat be endanger man health insurance well-being. Borrowed (non-nursing) practices employed in breastfeeding exercise composition.
### What scholars are saying
We all delightful your opinions, problems, substitute points of view, and important questions! Don’t be afraid to discuss this as well as other posting at any time! You’re our own “peer reviewers” and your views give rise to almost all within our neighborhood! Borrowed (non-nursing) practices employed in breastfeeding exercise composition. When practices will be borrowed inside developing information within nursing, they can be designed or employed in the direction they are usually. This may aid in building a few approaches regarding how to approach supplied cases that’s why produce a few common sense about accomplishing procedures in a part. A lot like some other jobs, nursing jobs has its own theories which have been frequently embraced or perhaps coppied by the pros. Health-promoting group relationships around my perform use particularly with Friere’s freedom basic principle, but you are specifically aimed at building set steps in addition to connections that are life-affirming, nurturing, along with aid people well-becoming.
• SCIENCE – Consequence of the link involving analysis & theory
• 7. b Generating Forces usually are allows of which propel inside a route that causes switch the signal from come about. e That they start a change in the steadiness on the way to switch.
• It enables the particular researchers so that you can integration the main points alongside one another.
• NR 501- Wk Some Impression regarding Nursing jobs Basic principle After Health Organization
NR 501 Wk Your five Discussion A Sally Thorne (2014) has addressed this kind of anxiety frequently in the operate, nearly all specially in the girl section that seems while in the text “Philosophies in addition to Practices regarding Emancipatory Medical.” With this part called “A Scenario regarding Emancipatory Disciplinary Theorizing” (internet pages 79-90), Medical professional. Decide on a nursing jobs apply spot (my partner and i.ourite. To be able to get the arena with nursing jobs, asking for essay help writer online ideas needs to be made it possible for but not seen as damages however a method of strengthening functions in neuro-scientific nursing. NP treat affected individual with respect continually and also impowering them to command on their own disease in order that they can care for independently if they are in your own home. The important thing for me is where a new principle as well as style works on a eyes – just what exactly phenomena are generally core, and are generally these key thoughts in conjuction with the defining emphasis of the willpower. Want your own speech to help add up inside? Send you your own assessment considering the details
### Borrowed idea are nurses idea obtained off their specialities.
A brief history on the lent (expectancy-value concept in addition to societal mental concept) theory’s root base. They claim that “The FNP has a special opportunity to care for the household while they move through agony. Below is really an evaluation on the way critical funding notions within nursing assist industry experts. My selected caregiving exercise is Nurse Practitioner or healthcare provider (NP). Health-promoting party relationships at my get the job done draw on the ones connected with Friere’s liberation idea, but you’re specially directed toward making party activities as well as affairs which can be life-affirming, looking after, along with assistance human well-becoming. In progressing caregiving know-how, you have to find guidance out of domains that have developed their particular base.
## Wednesday, Feb 6, 2013
Grounded Idea Research Start, nurse practitioners ought to employ any sort of information that provides route towards top quality overall health. Lent (non-nursing) theories utilized in nursing exercise essay or dissertation. He designed a style using a few levels where the actual determined winners with switch ought to proceed prior to alter could become engrained around traditions; unfreezing, going, in addition to refreezing (Mitchell, 2013). Going would be the measure the spot where you may establish ones change winners, put into action and try out the change, and then make the desired modifications (Mitchell, The year 2013). One reason why queries about the character connected with breastfeeding hypothesis keep appearing is a lot of nurses who launched into routines linked to enhancing nursology (caregiving research) were taught to end up being historians (investigators, basic principle builders) within career fields away from, however in connection with caregiving. You will find there’s huge personnel of educational writers, such as local loudspeakers on the Us, the united kingdom, Nova scotia, and Quarterly report.
Thorne directed to the behaviors involving “false dichotomizing” and also the allure involving applying for notions utilizing disciplines, both of which lead to valorizing constructions utilizing procedures, whilst ignoring this specific concentration regarding nurses. Apart from Florencia Nightingale’s Paperwork on Caregiving with 1860, caregiving ideas solely has become printed in the 1950s. Articles released on this internet site depend on the personal references created by the authors. Intellectual principle will depend on assumptions along with overnight accommodation regarding how an individual feels about along with adapts in order to fresh info. Nursing principle is certainly not older. Is it coppied concept suitable? Transition Stage- This particular step is the interior mobility that any of us create in answer to switch.
#### Borrowed idea are nurses idea obtained off their specialities.
Diet at the same time enjoy part within sufferer health insurance must be deal with through NP as soon as patient avoiding to eat or not getting enough source of nourishment. Below is really an investigation how crucial credit practices with nursing jobs assistance authorities. Inside content, the copy writers provide unhappy mishaps connected with loss of young children, good friends or littermates inside an elementary school. Theories via Nurses or any other Disciplines In your medical location concentrate connected with concept commonly contributes to people who are usually medical relevant idea.
### Borrowed Theories
Unfreezing is how the established order is screened so enabling a person determine just what exactly must be changed (Mitchell, The year 2013). Articles printed members derived from a personal references made by the particular publishers. These types of coppied concepts enjoy a similarly part a lot like medical hypotheses. PLACE That Buy Or even a Equivalent Buy Using Breastfeeding Phrase Documents These days And find A fantastic DISCOUNT When your lover went back, the client seemed to be telling her what exactly nutritious dishes your dog was going to cook and exactly how fired up he / she would be to move shopping pertaining to a good diet (change step). Types regarding idea and also similar research
#### Borrowed Theories
Belonginess center on have confidence in, camaraderie, devotion as well as love. As a result, we could identify the best 1 to your particular buy. In medical exercise, all of us employ quite a few took out notions. Finish The primary incredible importance of coppied theories as well as discussed ideas is usually to provide a basis for a provided scenario. I have got sorted out the challenge with nursing jobs involving creating principle using this particularly complex point of view among the list of factors behind looking towards concept throughout additional procedures, the place that the emphasis is a lot more constrained, and this also sophistication is usually unacknowledged and also undeveloped or even purely developed. Lewin’s basic principle of structured transform as a tactical useful resource.
## Middle Assortment Caregiving Theory-Part Three or more(Borrowed Theory)
This is where management management oftentimes leads a positive change within affected individual pleasure. Diary associated with Medical Supervision, 43(Only two), 69-72. Below is really an investigation how crucial credit practices with nursing jobs assistance authorities. Please interact with this kind of Coppied (non-nursing) practices included in nurses exercise dissertation submit almost like it had been everyone.
Lent (non-nursing) theories utilized in nursing exercise essay or dissertation. So as to provide the subject regarding nursing, funding theories ought to be allowed without viewed as damages but a method of strengthening operations in the field of nursing jobs. Your second phase in the overview yet still in period one of the adjust design is where command will going for walks fits during the entire capability and also timepieces this communications going down concerning treatment professional and also client. Impact regarding Took out Theories with Medical Launch Breastfeeding profession has become the number of vocations which require expertise progression to be able to achieve targets in the collection goals. Articles posted on this internet site are based on the individual references created by the actual publishers. Bookmark NR 501- Wk A few Lent Theories . Jim is actually a 58-year-old affected person in which entered a healthcare facility which has a person suffering from diabetes foot or so ulcer in addition to boost A1c.
# The Insider Secrets of Modern Physics Textbook
## The Supreme Strategy for Modern Physics Textbook
The range of context-rich problems is increased to facilitate the increased learning gains they can provide. This topic teaches you how to fix an equation symbolically employing the symbolic solver solve. Solve precisely the same equation for the complete solution.
## The Start of Modern Physics Textbook
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Choose the Relation There are 3 crucial relations for Special Relativity. The most important Physics Degree gives a very good grounding in classical and contemporary physics, and includes an extensive research project in the last year. Universe represents all of the he has a good point matter and space.
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## Details of Modern Physics Textbook
As the renowned scientist and educator Carl Sagan states, we’re star stuff. The battle for Copernicanism was fought in the world of mechanics along with astronomy. And although it is really hard to say if that’s true or not, 1 thing is undeniable these lectures are invaluable when it has to do with teaching conceptual comprehension of basic physics.
Applied physics is an overall term for physics research that is intended for a specific use. The MIT Physics Department is among the largest in the country, in part because it has astronomy and astrophysics. The study of quantum physics has caused numerous groundbreaking inventions like the personal computer and the web.
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Review the whole course to make sure that you understand what is going to be required https://www.physics.harvard.edu/ of you. These texts are costly, and we strongly recommend that you make sure you are comfortable reading them before you go right ahead and purchase them. Phrase Searching You can use double quotes to hunt for a succession of words in a specific order.
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A Pfund color grader is only a conventional amber-colored glass wedge which goes from light to dark. If you go for a quicker-developing formula, observe the clock and when you want to shower (usually a couple of hours after the treatment). You may observe some surprising flashes. | 2019-10-22 22:37:34 | {"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": 0.2237813025712967, "perplexity": 3955.1191801276855}, "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-2019-43/segments/1570987824701.89/warc/CC-MAIN-20191022205851-20191022233351-00238.warc.gz"} |
https://destevez.net/2018/05/dslwp-bs-journey-to-the-moon-part-i/ | # DSLWP-B’s journey to the Moon: part I
As you may well know, on May 20 a CZ-4C rocket launched from Xichang, China, to deliver Queqiao, the Chang’e 4 relay satellite, to the Moon. Queqiao is a communications relay satellite designed to orbit the L2 point of the Earth-Moon system, supporting the future Chang’e 4 rover that will land on the far side of the Moon. From the L2 point, Queqiao has a good view of both the Earth and the far side of the Moon.
This launch was shared by the DSLWP-A and -B microsatellites, also called Longjiang 1 and 2. These two satellites are designed to be put on a 200 x 9000km lunar orbit and their main scientific mission is a proof of concept of the Discovering the Sky at Longest Wavelengths experiment, a radioastronomy HF interferometer that uses the Moon as a shield from Earth’s interferences.
The DSLWP satellites carry an Amateur radio payload which consists of a 250 baud (or 500 baud) GMSK transmitter which uses $$r=1/2$$ or $$r=1/4$$ turbo codes, a JT4G beacon, and a camera allowing open telecommand (such as the camera on BY70-1 and LilacSat-1). A year ago, while the radio system was being designed, I wrote a post about DSLWP’s SSDV downlink, which transmits the images taken by the camera.
Wei Mingchuan BG2BHC, who is part of the DSLWP team, has been posting updates on Twitter about the status of the mission. If you’ve been following these closely, you’ll already know that unfortunately radio contact with DSLWP-A was lost on the UTC afternoon of May 22. Since then, all tries to contact the spacecraft have failed (the team will publicly release more information about its fate soon). On the other hand, DSLWP-B has been successfully injected into lunar orbit and is now orbiting the Moon since the UTC afternoon of May 25.
More posts will follow about the radio communications of DSLWP, but this series of posts will deal with the orbital dynamics part of the mission. In this first post, I will look at the tracking files released so far by Wei, which can be used to compute the spacecraft’s position and Doppler.
The main tool I will be using is GMAT R2018a, which is an open-source tool designed to plan, analyse and model spacecraft trajectories. It is a very comprehensive tool developed by NASA and other partners. I have been introduced to GMAT by Nico Janssen PA0DLO, who has proposed it for tracking of Amateur deep space missions. See Nico’s page for some useful resources regarding GMAT and deep space tracking, including some scripts for DSLWP.
While I know the basics of orbital dynamics (dating back to my years of playing Orbiter), I’m far from an expert on the field, and I had never used GMAT before, so I’m learning as I go. I hope that these series of posts can help others to get introduced to this very interesting topic.
GMAT has a very nice series of tutorials, which get you from the very basics to rather complex topics. I suggest you to go at some point through the first tutorial to get a hang of GMAT’s workflow and general concepts.
So far, three tracking files have been released by Wei, the first one is included in gr-dslwp, the GNU Radio decoder for the GMSK telemetry, and was released on May 21. The second and third have been released today.
These tracking files are plain text files which contain a listing of the spacecraft’s position and velocity at specific moments in time. Each line contains the data for a different instant. There are seven columns in each line. The first column contains the date and time in Unix timestamp format (seconds since January 1 1970). The three next columns contain the position in ECEF coordinates. The units are km. The last three columns contain the velocity in ECEF coordinates, in units of km/s. There is a line for each second (the timestamp increases one second per line).
ECEF coordinates are a cartesian reference system with origin on the Earth’s centre and which is “fixed” to Earth, so it rotates together with Earth’s rotation. In this manner, points fixed on the Earth’s surface have constant ECEF coordinates (i.e., their coordinates do not vary with time, as Earth rotates), so it is possible to pass from latitude and longitude to ECEF. The Z axis of ECEF coordinates points to the north pole, the X axis points to the point with latitude 0º and longitude 0º, and the Y axis is chosen to form a right-handed system.
There is a different type of coordinates, which are known as ECI or inertial coordinates. Their origin is also the Earth’s centre, but they do not rotate with the Earth. The Z axis of ECI coordinates also points to the north pole, but the X axis points to the vernal equinox (the intersection of Earth’s ecliptic and equatorial planes, in the direction of Aries).
In many applications, ECI coordinates are more useful than ECEF. For instance, in the case of DSLWP, ECI coordinates can be easier to interpret. In ECEF coordinates it seems that DSLWP orbits the Earth roughly one time per day, while in reality it is the Earth the thing that rotates. On the other hand, ECEF coordinates simplify somewhat the calculation of azimuth, elevation, distance to the spacecraft and Doppler for an observer on a fixed point on Earth, so probably that’s the reason why DSLWP tracking files are given in ECEF coordinates, since they are used in the GNU Radio decoder for that purpose. ECI coordinates are also useful because Newton’s laws can only be applied in an inertial reference frame.
Now that we’ve understood DSLWP tracking files, the main goal of this post is to interpret and validate them using GMAT. If you’ve already taken a look at the first tutorial, you’ll have seen that GMAT’s most basic functionality is trajectory propagation. Essentially, you give a spacecraft’s orbital elements, which are its position and velocity at a particular instant in time (for instance using ECEF or ECI coordinates, but other ways are possible) and tell GMAT to propagate the trajectory of the spacecraft for a certain time. GMAT will calculate the forces acting on the spacecraft (mainly gravity, but also others such as solar radiation pressure and atmospheric drag) and will update and propagate the position of the spacecraft by using Newton’s second law.
Thus, we will take each of the tracking files and use the first listed position and velocity as the spacecraft’s orbital elements in GMAT. Then we will use GMAT to propagate the trajectory of the spacecraft and compute its position every second. Finally we compare GMAT’s results with the tracking file. To do so, we can use GMAT to generate a file similar to the tracking file, with the position and velocity in ECEF coordinates. The only difference is that GMAT doesn’t use the Unix timestamp format, so we output time in UTC Modified Julian Days (note that GMAT’s MJD convention is different from the usual MJD convention). To compare GMAT’s propagation with the tracking file, we use Python to calculate and the position error (the distance between the position as predicted by GMAT and the position given by the tracking file).
This serves two purposes. First, we can validate that the published tracking files do not contain any planned manoeuvres, where the spacecraft fires its thrusters to change course (in which case we would also need to inform GMAT of those manoeuvres). Second, we can compare the force models used by GMAT with those used to compute the tracking files. As we shall see, there are many subtle details regarding the forces acting on a spacecraft.
There are two ways of running calculations with GMAT. The first one is to create a mission, where one can add and edit simulation objects using the GUI. The second one is to create is a script file, which is a text file that sets up variables and uses commands to set up the simulation. The nice thing is that both ways are linked: a mission is really saved as a script file, and a script file can be opened and edited using the GUI (however, some advanced features are not available in the GUI and are only accessible by writing a script file).
Since I’m going to process three tracking files (and perhaps more that get published in the future), I have wanted to automate the procedure as much as possible. Therefore, I use Python to read the tracking file and create a GMAT script file using some data read from the tracking file (the spacecraft’s orbital elements, for instance). While this can seem hard, it is actually not that difficult. I have first created a mission using the GUI and then have used the corresponding script file as a template.
The Python code used in this post can be found in this Jupyter notebook.
The first tracking file starts at 20 May 21:54:51 UTC and last 48 hours. It contains data from shortly after trans-lunar injection to before mid-course correction, which was programmed at 22 May 22:55 UTC, but was presumably delayed. The image below shows the orbit as propagated by GMAT (click on the image to view it in full size). Note that this orbit is plotted using ECI coordinates. The orbit in ECEF coordinates would look rather weird.
Below we can see the error between GMAT’s propagation and the tracking file. We see that it is small. This error is due to different modelling of the forces acting on the spacecraft. In GMAT I have used a very detailed model (probably an overkill) which simulates Earth’s non-spherical gravity using spherical harmonics up to degree and order 10 (see below for a discussion about spherical harmonics), point mass gravities for the Sun, Moon, and all the planets except for Mercury and Pluto (which are very small and distant), solar radiation pressure, relativistic effects, and Earth’s atmosphere (even though it is of little significance, since the spacecraft gets away from the atmosphere really soon).
The second tracking file starts at 26 May 00:00:00 UTC and lasts 48 hours. It contains data after DSLWP-B was established on a lunar elliptical orbit.
Below we can see the error between GMAT’s propagation and the tracking file. Again, I have used a very detailed force model in GMAT. The model is is the same as before, but using the Moon’s spherical harmonics instead of those of Earth’s, since now the Moon is the nearest object. Also Earth’s atmosphere is not included now.
Now we see that the error is rather large, peaking up to 50km at times. These peaks corresponds to the times when DSLWP-B passes the periapsis (the point of the orbit which is nearest to the Moon). Recall that an object on an elliptical orbit moves faster at the periapsis and slower at the apoapsis (the point of the orbit which is farthest from the Moon), because of Kepler’s second law. Since DSLWP-B is on a highly elliptical orbit, it passes the periapsis very quickly and spends most of its time near the apoapsis.
Therefore, if there is a small difference between the orbital calculation on GMAT and in the calculations used to obtain the tracking file, that difference will cause DSLWP-B to pass the periapsis at slightly different times. This causes a large error in the position of the spacecraft when passing near the periapsis.
Another effect that we see in the graph is that there is an error that accumulates with every orbit. We will study this in more detail later.
The third tracking file starts at 28 May 00:00:00 UTC and lasts 48 hours. It is just a continuation of the data in the second tracking file.
The figure below shows the position error. We have used the same force model in GMAT as for the second tracking file. We see the same behaviour as before.
The large errors seen in the second and third tracking files seem to indicate that a different force model has been for the calculation of the tracking files. To investigate this, I have run again the second file in GMAT using a simplified model that only includes the gravity of the Moon with a variable number of spherical harmonics and the gravity of Earth as a point mass.
Without going into much detail about spherical harmonics, I can say the following. An object which is a perfect sphere generates the same gravitational field as a point mass (meaning a mass concentrated at a single point). This gravitational field is symmetric and only depends on the distance to the object, which is what Kepler’s laws and all other simple theories of orbital mechanics assume. However, planets and moons are not perfectly spherical, and also they do not have a uniform density. Therefore, their gravitational field deviates from the ideal symmetric field. At a large distance, all these deviations somehow even out and the gravitational field seems almost perfect. However, at shorter distances, the effects of these deviations can be more significant.
Spherical harmonic coefficients are a mathematical way to measure these deviations. They are numbers with an associated degree and order. The coefficients with the lower order play the largest role, while the effect of coefficients of higher order is very mild and can usually be ignored. Hundreds or thousands of spherical harmonic coefficients have been measured for the Earth’s gravitational field and other astronomic bodies, including the Moon. The most important coefficient is the one having degree 2 and order 0. This coefficient describes the oblateness of the body, and it is the coefficient that plays the largest role in the non-spheric gravitational field.
It has been necessary to include the Earth in this simple model, since otherwise the error quickly grows and becomes very large, due to the fact that the Earth’s gravity plays a very important role in lunar orbits. This is not surprising since the Earth is a rather massive and near object. Checking this with a simulation in GMAT is left as an exercise for the reader.
Below we can see the errors obtained using models with a different number of spherical harmonics for the Moon’s gravity, and also the error obtained above using the complete model.
The error is much lower when no spherical harmonics are considered. This indicates that the calculations in the tracking files haven’t even taken the Moon’s oblateness into account. This will give an error in the tracking files that accumulates with each orbit.
The error when passing the periapsis is large in all the models I’ve used. I do not know what is the reason for this. Probably there is some peculiarity about the model used for the tracking files that I haven’t considered. It would be interesting to know how they are calculated. Wei tells me that the team uses STK for orbit calculations.
Update: People interested in using the GMAT output reports generated by this Jupyter notebook as tracking files for gr-dslwp can use the following Python script. It reads a GMAT report from the standard input and writes the corresponding tracking file to the standard output.
1. Mai says:
Fun-tastic Go Moon !
2. Elther says:
Hello, gmat and other soft are superb but confusing
I just need
to know when the probe is visible from my location,
at minimum distance from good old planet earth…
Some website ny2o style?
1. Brief answer if you don’t want to get into technical details: DSLWP-B is visible whenever the Moon is visible from your location (except for the brief time it spends in eclipse).
3. Elther says:
Nice…!
but the actual distance must be calculatted I presume ?
how much time, the moon orbit ?
(I’m a total newb into space!)
This site uses Akismet to reduce spam. Learn how your comment data is processed. | 2021-10-24 00:51:16 | {"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": 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": 0.6881378293037415, "perplexity": 739.4765429363499}, "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/1634323585828.15/warc/CC-MAIN-20211023224247-20211024014247-00664.warc.gz"} |
https://cs.paperswithcode.com/latest | # HyperLogLogLog: Cardinality Estimation With One Log More
23 May 2022
We present HyperLogLogLog, a practical compression of the HyperLogLog sketch that compresses the sketch from $O(m\log\log n)$ bits down to $m \log_2\log_2\log_2 m + O(m+\log\log n)$ bits for estimating the number of distinct elements~$n$ using $m$~registers.
Data Structures and Algorithms
4
23 May 2022
23 May 2022
Where dual-numbers forward-mode automatic differentiation (AD) pairs each scalar value with its tangent derivative, dual-numbers /reverse-mode/ AD attempts to achieve reverse AD using a similarly simple idea: by pairing each scalar value with a backpropagator function.
Programming Languages
0
23 May 2022
# CIRCLE: Continual Repair across Programming Languages
22 May 2022
However, we observe that existing DL-based APR models suffer from at least two severe drawbacks: (1) Most of them can only generate patches for a single programming language, as a result, to repair multiple languages, we have to build and train many repairing models.
Software Engineering
0
22 May 2022
# ALITA: A Large-scale Incremental Dataset for Long-term Autonomy
22 May 2022
This dataset includes a campus-scale track and a city-scale track: 1) the campus-track focuses the long-term property, we record LiDAR device and an omnidirectional camera on 10 trajectories, and each trajectory are repeatly recorded 8 times under variant illumination conditions.
Robotics
5
22 May 2022
# Shared-Control Robotic Manipulation in Virtual Reality
21 May 2022
In this paper, we present the implementation details of a Virtual Reality (VR)-based teleoperation interface for moving a robotic manipulator.
Robotics
3
21 May 2022
# Learning to Dynamically Select Cost Optimal Schedulers in Cloud Computing Environments
21 May 2022
The operational cost of a cloud computing platform is one of the most significant Quality of Service (QoS) criteria for schedulers, crucial to keep up with the growing computational demands.
Distributed, Parallel, and Cluster Computing Performance
2
21 May 2022
# AGA: An Accelerated Greedy Additional Algorithm for Test Case Prioritization
20 May 2022
Moreover, we conducted an industrial case study on 22 subjects, collected from Baidu, and find that the average speedup ratio of AGA over GA is 44. 27X, which indicates the practical usage of AGA in real-world scenarios.
Software Engineering
1
20 May 2022
# A Fully Implicit Method for Robust Frictional Contact Handling in Elastic Rods
20 May 2022
Accurate frictional contact is critical in simulating the assembly of rod-like structures in the practical world, such as knots, hairs, flagella, and more.
Graphics
0
20 May 2022
# Subset Node Anomaly Tracking over Large Dynamic Graphs
19 May 2022
Thanks to recent advances in dynamic representation learning based on Personalized PageRank, \textsc{DynAnom} is 1) \textit{efficient}: the time complexity is linear to the number of edge events and independent on node size of the input graph; 2) \textit{effective}: \textsc{DynAnom} can successfully track topological changes reflecting real-world anomaly; 3) \textit{flexible}: different type of anomaly score functions can be defined for various applications.
Social and Information Networks
0
19 May 2022
# Collision Detection Accelerated: An Optimization Perspective
19 May 2022
Collision detection between two convex shapes is an essential feature of any physics engine or robot motion planner.
Robotics
2
19 May 2022 | 2022-05-25 14:19:31 | {"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": 0.22942374646663666, "perplexity": 6130.130532380671}, "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/1652662587158.57/warc/CC-MAIN-20220525120449-20220525150449-00667.warc.gz"} |
https://letianzj.github.io/portfolio-management-two.html | # Quantitative Investing
Letian Wang Blog on Quant Trading and Portfolio Management
0%
## Introduction
In previous post we reviewed the basics of mean-variance optimization (MVO), and portfolios such as minimum variance and maximmum sharpe. This post continues to discuss some popular practices in asset allocation, namely risk parity and maximum diversification. Then we evaluate these allocation strategies using historical market data.
It is known that all these portfolios are special cases of MVO under some conditions. For example, MVO becomes minimum variance if expected reutrns are equal; it coincides with maximum diversification if return-risk ratios are the same across assets ([2]). The portfolio selection depends on our information and knowledge about the market. If expected return and risks are known with certainty, a maximum sharpe ratio is good choice. If expected return is hard to measure, covariance knowledge can be leveraged to construct minimum variance or risk parity portfolios. If we have no knowledge at all about the market, a naive equal-weighting portfolio will be a default option, which is also served as benchmark in our backtest.
## Risk Parity and Maximum Diversification
Risk parity was coined by Edward Qian ([1]) and popularized by Bridgewater all weather fund. It focuses on budgeting and allocation of risks rather than capitals. The essence is equal/parity risk contribution (ERC) defined as
\begin{aligned} MRC_i&=\frac{\partial \sigma_p}{\partial w_i}=\frac{\sum_{j=1}^nw_j\sigma_i\sigma_j}{\sigma_p}=\frac{(\Sigma w)_i}{\sigma_p}\\ RC_i&=w_i * MRC_i \end{aligned}
Note that ERC weights are not necessarily proportional to inverse volatililty. In fact, [3] gives the solution as to minimize
$f(x)=\sum_{i=1}^{n}\sum_{j=1}^{n}\left( w_i (\Sigma \cdot w)_i - w_j(\Sigma \cdot w)_j\right)^2$
Risk parity is embedded with momemtum features that may cause a problem when its followers stampede in the market. If stock market suddenly collapses, stock's risk will spike which in turn demands less asset allocation on it. The liquidation will then reinforce the sell-off.
Maximum diversification portfolio tries to maximize the ratio of weighted average volatility over portfolio volatility as $\underset{w}{\max} \frac{w^T \sigma}{\sqrt{w^T\Sigma w}}$
which can also be solved numerically.
## Assessments
Let's compare long-only portfolios consist of five stocks, SPY, EFA, TIP, GSG, and VNQ, to represent US, international, treasury, commodity, real estates, respectively, between 2011-2019. The portfolios include minimum variance, maximum sharpe, risk parity, and maximum diversification, rebalanced at month end. Benchmark is equal-weights. The code can be found on Github.
First of all, results below show none of them have CAGR above equal weights. It may well be due to the hostorical long bull market after 2008-2009 financial crisis. GMV has the lowest risk, which is expected, and the highest Sharpe ratio. The maximum diversified portfolio has the second lowest volatility, and second highest Sharpe. Controlling risk is a feasible way to achieve high Sharpe, but may comes with the cost of low returns, especially in bull markets. Equal weights has the biggest drawdown among the five, dragged by commodity underperformance, but then fully recovered. Risk parity follows closely maximum diversified in the beginning, and then outperform marginally in Elliott wave 3 and 5.
The portfolo values over time and monthly returns are shown as blows. During this period, domestic stocks went up the most, followed by real estates. Commodities is the only one that went south. GMV essentially dodged the market pull back at the end of 2018, yet it also missed the market recovery afterwards.
The stock holdings below explains why they are all behind equal weights, because they hold too much Treasuries in the bull markets. GMV has predominantly treasuries and S&P500. The maxinum sharpe portfolio is somehow agressive and unstable. It has the biggest rotation costs, and sometimes it holds all stocks, sometimes it holds all cash.
## Reference
• Qian, Edward. "Risk parity portfolios: Efficient portfolios through true diversification." Panagora Asset Management (2005).
• Roncalli, Thierry. Introduction to risk parity and budgeting. CRC Press, 2013.
• Maillard, Sébastien, Thierry Roncalli, and Jérôme Teïletche. "The properties of equally weighted risk contribution portfolios." The Journal of Portfolio Management 36.4 (2010): 60-70.
• Spinu, Florin. "An algorithm for computing risk parity weights." Available at SSRN 2297383 (2013).
• Bender, Jennifer, Thomas Blackburn, and Xiaole Sun. "Clash of the Titans: Factor Portfolios versus Alternative Weighting Schemes." The Journal of Portfolio Management 45.3 (2019): 38-49.
• The Quant MBA
DISCLAIMER: This post is for the purpose of research and backtest only. The author doesn't promise any future profits and doesn't take responsibility for any trading losses. | 2020-08-13 13:45:23 | {"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": 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": 0.5212750434875488, "perplexity": 6800.615685846269}, "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-2020-34/segments/1596439739046.14/warc/CC-MAIN-20200813132415-20200813162415-00305.warc.gz"} |
https://www.profillic.com/s/Yu%20Xiong | Models, code, and papers for "Yu Xiong":
##### Unifying Identification and Context Learning for Person Recognition
Jun 08, 2018
Qingqiu Huang, Yu Xiong, Dahua Lin
Despite the great success of face recognition techniques, recognizing persons under unconstrained settings remains challenging. Issues like profile views, unfavorable lighting, and occlusions can cause substantial difficulties. Previous works have attempted to tackle this problem by exploiting the context, e.g. clothes and social relations. While showing promising improvement, they are usually limited in two important aspects, relying on simple heuristics to combine different cues and separating the construction of context from people identities. In this work, we aim to move beyond such limitations and propose a new framework to leverage context for person recognition. In particular, we propose a Region Attention Network, which is learned to adaptively combine visual cues with instance-dependent weights. We also develop a unified formulation, where the social contexts are learned along with the reasoning of people identities. These models substantially improve the robustness when working with the complex contextual relations in unconstrained environments. On two large datasets, PIPA and Cast In Movies (CIM), a new dataset proposed in this work, our method consistently achieves state-of-the-art performance under multiple evaluation policies.
* CVPR 2018
##### Growing a Brain: Fine-Tuning by Increasing Model Capacity
Jul 18, 2019
Yu-Xiong Wang, Deva Ramanan, Martial Hebert
CNNs have made an undeniable impact on computer vision through the ability to learn high-capacity models with large annotated training sets. One of their remarkable properties is the ability to transfer knowledge from a large source dataset to a (typically smaller) target dataset. This is usually accomplished through fine-tuning a fixed-size network on new target data. Indeed, virtually every contemporary visual recognition system makes use of fine-tuning to transfer knowledge from ImageNet. In this work, we analyze what components and parameters change during fine-tuning, and discover that increasing model capacity allows for more natural model adaptation through fine-tuning. By making an analogy to developmental learning, we demonstrate that "growing" a CNN with additional units, either by widening existing layers or deepening the overall network, significantly outperforms classic fine-tuning approaches. But in order to properly grow a network, we show that newly-added units must be appropriately normalized to allow for a pace of learning that is consistent with existing units. We empirically validate our approach on several benchmark datasets, producing state-of-the-art results.
* CVPR
##### Encrypted Speech Recognition using Deep Polynomial Networks
May 11, 2019
Shi-Xiong Zhang, Yifan Gong, Dong Yu
The cloud-based speech recognition/API provides developers or enterprises an easy way to create speech-enabled features in their applications. However, sending audios about personal or company internal information to the cloud, raises concerns about the privacy and security issues. The recognition results generated in cloud may also reveal some sensitive information. This paper proposes a deep polynomial network (DPN) that can be applied to the encrypted speech as an acoustic model. It allows clients to send their data in an encrypted form to the cloud to ensure that their data remains confidential, at mean while the DPN can still make frame-level predictions over the encrypted speech and return them in encrypted form. One good property of the DPN is that it can be trained on unencrypted speech features in the traditional way. To keep the cloud away from the raw audio and recognition results, a cloud-local joint decoding framework is also proposed. We demonstrate the effectiveness of model and framework on the Switchboard and Cortana voice assistant tasks with small performance degradation and latency increased comparing with the traditional cloud-based DNNs.
* ICASSP 2019, slides@ https://www.researchgate.net/publication/333005422_Encrypted_Speech_Recognition_using_deep_polynomial_networks
##### Learning Compositional Representations for Few-Shot Recognition
Dec 21, 2018
Pavel Tokmakov, Yu-Xiong Wang, Martial Hebert
One of the key limitations of modern deep learning based approaches lies in the amount of data required to train them. Humans, on the other hand, can learn to recognize novel categories from just a few examples. Instrumental to this rapid learning ability is the compositional structure of concept representations in the human brain - something that deep learning models are lacking. In this work we make a step towards bridging this gap between human and machine learning by introducing a simple regularization technique that allows the learned representation to be decomposable into parts. We evaluate the proposed approach on three datasets: CUB-200-2011, SUN397, and ImageNet, and demonstrate that our compositional representations require fewer examples to learn classifiers for novel categories, outperforming state-of-the-art few-shot learning approaches by a significant margin.
##### Towards Good Practices for Very Deep Two-Stream ConvNets
Jul 08, 2015
Limin Wang, Yuanjun Xiong, Zhe Wang, Yu Qiao
Deep convolutional networks have achieved great success for object recognition in still images. However, for action recognition in videos, the improvement of deep convolutional networks is not so evident. We argue that there are two reasons that could probably explain this result. First the current network architectures (e.g. Two-stream ConvNets) are relatively shallow compared with those very deep models in image domain (e.g. VGGNet, GoogLeNet), and therefore their modeling capacity is constrained by their depth. Second, probably more importantly, the training dataset of action recognition is extremely small compared with the ImageNet dataset, and thus it will be easy to over-fit on the training dataset. To address these issues, this report presents very deep two-stream ConvNets for action recognition, by adapting recent very deep architectures into video domain. However, this extension is not easy as the size of action recognition is quite small. We design several good practices for the training of very deep two-stream ConvNets, namely (i) pre-training for both spatial and temporal nets, (ii) smaller learning rates, (iii) more data augmentation techniques, (iv) high drop out ratio. Meanwhile, we extend the Caffe toolbox into Multi-GPU implementation with high computational efficiency and low memory consumption. We verify the performance of very deep two-stream ConvNets on the dataset of UCF101 and it achieves the recognition accuracy of $91.4\%$.
##### Differentiating Features for Scene Segmentation Based on Dedicated Attention Mechanisms
Nov 19, 2019
Zhiqiang Xiong, Zhicheng Wang, Zhaohui Yu, Xi Gu
Semantic segmentation is a challenge in scene parsing. It requires both context information and rich spatial information. In this paper, we differentiate features for scene segmentation based on dedicated attention mechanisms (DF-DAM), and two attention modules are proposed to optimize the high-level and low-level features in the encoder, respectively. Specifically, we use the high-level and low-level features of ResNet as the source of context information and spatial information, respectively, and optimize them with attention fusion module and 2D position attention module, respectively. For attention fusion module, we adopt dual channel weight to selectively adjust the channel map for the highest two stage features of ResNet, and fuse them to get context information. For 2D position attention module, we use the context information obtained by attention fusion module to assist the selection of the lowest-stage features of ResNet as supplementary spatial information. Finally, the two sets of information obtained by the two modules are simply fused to obtain the prediction. We evaluate our approach on Cityscapes and PASCAL VOC 2012 datasets. In particular, there aren't complicated and redundant processing modules in our architecture, which greatly reduces the complexity, and we achieving 82.3% Mean IoU on PASCAL VOC 2012 test dataset without pre-training on MS-COCO dataset.
##### Unsupervised Feature Learning via Non-Parametric Instance-level Discrimination
May 05, 2018
Zhirong Wu, Yuanjun Xiong, Stella Yu, Dahua Lin
Neural net classifiers trained on data with annotated class labels can also capture apparent visual similarity among categories without being directed to do so. We study whether this observation can be extended beyond the conventional domain of supervised learning: Can we learn a good feature representation that captures apparent similarity among instances, instead of classes, by merely asking the feature to be discriminative of individual instances? We formulate this intuition as a non-parametric classification problem at the instance-level, and use noise-contrastive estimation to tackle the computational challenges imposed by the large number of instance classes. Our experimental results demonstrate that, under unsupervised learning settings, our method surpasses the state-of-the-art on ImageNet classification by a large margin. Our method is also remarkable for consistently improving test performance with more training data and better network architectures. By fine-tuning the learned feature, we further obtain competitive results for semi-supervised learning and object detection tasks. Our non-parametric model is highly compact: With 128 features per image, our method requires only 600MB storage for a million images, enabling fast nearest neighbour retrieval at the run time.
* CVPR 2018 spotlight paper. Code: https://github.com/zhirongw/lemniscate.pytorch
##### From Trailers to Storylines: An Efficient Way to Learn from Movies
Jun 14, 2018
Qingqiu Huang, Yuanjun Xiong, Yu Xiong, Yuqi Zhang, Dahua Lin
The millions of movies produced in the human history are valuable resources for computer vision research. However, learning a vision model from movie data would meet with serious difficulties. A major obstacle is the computational cost -- the length of a movie is often over one hour, which is substantially longer than the short video clips that previous study mostly focuses on. In this paper, we explore an alternative approach to learning vision models from movies. Specifically, we consider a framework comprised of a visual module and a temporal analysis module. Unlike conventional learning methods, the proposed approach learns these modules from different sets of data -- the former from trailers while the latter from movies. This allows distinctive visual features to be learned within a reasonable budget while still preserving long-term temporal structures across an entire movie. We construct a large-scale dataset for this study and define a series of tasks on top. Experiments on this dataset showed that the proposed method can substantially reduce the training time while obtaining highly effective features and coherent temporal structures.
##### Low-Shot Learning from Imaginary Data
Apr 03, 2018
Yu-Xiong Wang, Ross Girshick, Martial Hebert, Bharath Hariharan
Humans can quickly learn new visual concepts, perhaps because they can easily visualize or imagine what novel objects look like from different views. Incorporating this ability to hallucinate novel instances of new concepts might help machine vision systems perform better low-shot learning, i.e., learning concepts from few examples. We present a novel approach to low-shot learning that uses this idea. Our approach builds on recent progress in meta-learning ("learning to learn") by combining a meta-learner with a "hallucinator" that produces additional training examples, and optimizing both models jointly. Our hallucinator can be incorporated into a variety of meta-learners and provides significant gains: up to a 6 point boost in classification accuracy when only a single training example is available, yielding state-of-the-art performance on the challenging ImageNet low-shot classification benchmark.
##### Taylorized Training: Towards Better Approximation of Neural Network Training at Finite Width
Feb 10, 2020
Yu Bai, Ben Krause, Huan Wang, Caiming Xiong, Richard Socher
We propose \emph{Taylorized training} as an initiative towards better understanding neural network training at finite width. Taylorized training involves training the $k$-th order Taylor expansion of the neural network at initialization, and is a principled extension of linearized training---a recently proposed theory for understanding the success of deep learning. We experiment with Taylorized training on modern neural network architectures, and show that Taylorized training (1) agrees with full neural network training increasingly better as we increase $k$, and (2) can significantly close the performance gap between linearized and full training. Compared with linearized training, higher-order training works in more realistic settings such as standard parameterization and large (initial) learning rate. We complement our experiments with theoretical results showing that the approximation error of $k$-th order Taylorized models decay exponentially over $k$ in wide neural networks.
##### LiteEval: A Coarse-to-Fine Framework for Resource Efficient Video Recognition
Dec 03, 2019
Zuxuan Wu, Caiming Xiong, Yu-Gang Jiang, Larry S. Davis
This paper presents LiteEval, a simple yet effective coarse-to-fine framework for resource efficient video recognition, suitable for both online and offline scenarios. Exploiting decent yet computationally efficient features derived at a coarse scale with a lightweight CNN model, LiteEval dynamically decides on-the-fly whether to compute more powerful features for incoming video frames at a finer scale to obtain more details. This is achieved by a coarse LSTM and a fine LSTM operating cooperatively, as well as a conditional gating module to learn when to allocate more computation. Extensive experiments are conducted on two large-scale video benchmarks, FCVID and ActivityNet, and the results demonstrate LiteEval requires substantially less computation while offering excellent classification accuracy for both online and offline predictions.
* NeurIPS 2019
##### Patchy Image Structure Classification Using Multi-Orientation Region Transform
Dec 02, 2019
Xiaohan Yu, Yang Zhao, Yongsheng Gao, Shengwu Xiong, Xiaohui Yuan
Exterior contour and interior structure are both vital features for classifying objects. However, most of the existing methods consider exterior contour feature and internal structure feature separately, and thus fail to function when classifying patchy image structures that have similar contours and flexible structures. To address above limitations, this paper proposes a novel Multi-Orientation Region Transform (MORT), which can effectively characterize both contour and structure features simultaneously, for patchy image structure classification. MORT is performed over multiple orientation regions at multiple scales to effectively integrate patchy features, and thus enables a better description of the shape in a coarse-to-fine manner. Moreover, the proposed MORT can be extended to combine with the deep convolutional neural network techniques, for further enhancement of classification accuracy. Very encouraging experimental results on the challenging ultra-fine-grained cultivar recognition task, insect wing recognition task, and large variation butterfly recognition task are obtained, which demonstrate the effectiveness and superiority of the proposed MORT over the state-of-the-art methods in classifying patchy image structures. Our code and three patchy image structure datasets are available at: https://github.com/XiaohanYu-GU/MReT2019.
* Accepted by AAAI 2020
##### An Intelligent Extraversion Analysis Scheme from Crowd Trajectories for Surveillance
Sep 27, 2018
Wenxi Liu, Yuanlong Yu, Chun-Yang Zhang, Genggeng Liu, Naixue Xiong
In recent years, crowd analysis is important for applications such as smart cities, intelligent transportation system, customer behavior prediction, and visual surveillance. Understanding the characteristics of the individual motion in a crowd can be beneficial for social event detection and abnormal detection, but it has rarely been studied. In this paper, we focus on the extraversion measure of individual motions in crowds based on trajectory data. Extraversion is one of typical personalities that is often observed in human crowd behaviors and it can reflect not only the characteristics of the individual motion, but also the that of the holistic crowd motions. To our best knowledge, this is the first attempt to analyze individual extraversion of crowd motions based on trajectories. To accomplish this, we first present a effective composite motion descriptor, which integrates the basic individual motion information and social metrics, to describe the extraversion of each individual in a crowd. The social metrics consider both the neighboring distribution and their interaction pattern. Since our major goal is to learn a universal scoring function that can measure the degrees of extraversion across varied crowd scenes, we incorporate and adapt the active learning technique to the relative attribute approach. Specifically, we assume the social groups in any crowds contain individuals with the similar degree of extraversion. Based on such assumption, we significantly reduce the computation cost by clustering and ranking the trajectories actively. Finally, we demonstrate the performance of our proposed method by measuring the degree of extraversion for real individual trajectories in crowds and analyzing crowd scenes from a real-world dataset.
##### Knowledge Guided Disambiguation for Large-Scale Scene Classification with Multi-Resolution CNNs
Feb 21, 2017
Limin Wang, Sheng Guo, Weilin Huang, Yuanjun Xiong, Yu Qiao
Convolutional Neural Networks (CNNs) have made remarkable progress on scene recognition, partially due to these recent large-scale scene datasets, such as the Places and Places2. Scene categories are often defined by multi-level information, including local objects, global layout, and background environment, thus leading to large intra-class variations. In addition, with the increasing number of scene categories, label ambiguity has become another crucial issue in large-scale classification. This paper focuses on large-scale scene recognition and makes two major contributions to tackle these issues. First, we propose a multi-resolution CNN architecture that captures visual content and structure at multiple levels. The multi-resolution CNNs are composed of coarse resolution CNNs and fine resolution CNNs, which are complementary to each other. Second, we design two knowledge guided disambiguation techniques to deal with the problem of label ambiguity. (i) We exploit the knowledge from the confusion matrix computed on validation data to merge ambiguous classes into a super category. (ii) We utilize the knowledge of extra networks to produce a soft label for each image. Then the super categories or soft labels are employed to guide CNN training on the Places2. We conduct extensive experiments on three large-scale image datasets (ImageNet, Places, and Places2), demonstrating the effectiveness of our approach. Furthermore, our method takes part in two major scene recognition challenges, and achieves the second place at the Places2 challenge in ILSVRC 2015, and the first place at the LSUN challenge in CVPR 2016. Finally, we directly test the learned representations on other scene benchmarks, and obtain the new state-of-the-art results on the MIT Indoor67 (86.7\%) and SUN397 (72.0\%). We release the code and models at~\url{https://github.com/wanglimin/MRCNN-Scene-Recognition}.
* To appear in IEEE Transactions on Image Processing. Code and models are available at https://github.com/wanglimin/MRCNN-Scene-Recognition
##### Learning Generalizable Representations via Diverse Supervision
Nov 29, 2019
Ziqi Pang, Zhiyuan Hu, Pavel Tokmakov, Yu-Xiong Wang, Martial Hebert
The problem of rare category recognition has received a lot of attention recently, with state-of-the-art methods achieving significant improvements. However, we identify two major limitations in the existing literature. First, the benchmarks are constructed by randomly splitting the categories of artificially balanced datasets into frequent (head), and rare (tail) subsets, which results in unrealistic category distributions in both of them. Second, the idea of using external sources of supervision to learn generalizable representations is largely overlooked. In this work, we attempt to address both of these shortcomings by introducing the ADE-FewShot benchmark. It stands upon the ADE dataset for scene parsing that features a realistic, long-tail distribution of categories as well as a diverse set of annotations. We turn it into a realistic few-shot classification benchmark by splitting the object categories into head and tail based on their distribution in the world. We then analyze the effect of applying various supervision sources on representation learning for rare category recognition, and observe significant improvements.
##### Predicting microRNA-disease associations from knowledge graph using tensor decomposition with relational constraints
Nov 13, 2019
Feng Huang, Zhankun Xiong, Guan Zhang, Zhouxin Yu, Xinran Xu, Wen Zhang
Motivation: MiRNAs are a kind of small non-coding RNAs that are not translated into proteins, and aberrant expression of miRNAs is associated with human diseases. Since miRNAs have different roles in diseases, the miRNA-disease associations are categorized into multiple types according to their roles. Predicting miRNA-disease associations and types is critical to understand the underlying pathogenesis of human diseases from the molecular level. Results: In this paper, we formulate the problem as a link prediction in knowledge graphs. We use biomedical knowledge bases to build a knowledge graph of entities representing miRNAs and disease and multi-relations, and we propose a tensor decomposition-based model named TDRC to predict miRNA-disease associations and their types from the knowledge graph. We have experimentally evaluated our method and compared it to several baseline methods. The results demonstrate that the proposed method has high-accuracy and high-efficiency performances.
##### A Graph-Based Framework to Bridge Movies and Synopses
Oct 24, 2019
Yu Xiong, Qingqiu Huang, Lingfeng Guo, Hang Zhou, Bolei Zhou, Dahua Lin
Inspired by the remarkable advances in video analytics, research teams are stepping towards a greater ambition -- movie understanding. However, compared to those activity videos in conventional datasets, movies are significantly different. Generally, movies are much longer and consist of much richer temporal structures. More importantly, the interactions among characters play a central role in expressing the underlying story. To facilitate the efforts along this direction, we construct a dataset called Movie Synopses Associations (MSA) over 327 movies, which provides a synopsis for each movie, together with annotated associations between synopsis paragraphs and movie segments. On top of this dataset, we develop a framework to perform matching between movie segments and synopsis paragraphs. This framework integrates different aspects of a movie, including event dynamics and character interactions, and allows them to be matched with parsed paragraphs, based on a graph-based formulation. Our study shows that the proposed framework remarkably improves the matching accuracy over conventional feature-based methods. It also reveals the importance of narrative structures and character interactions in movie understanding.
* Accepted by ICCV 2019 (oral)
##### Double Anchor R-CNN for Human Detection in a Crowd
Sep 22, 2019
Kevin Zhang, Feng Xiong, Peize Sun, Li Hu, Boxun Li, Gang Yu
Detecting human in a crowd is a challenging problem due to the uncertainties of occlusion patterns. In this paper, we propose to handle the crowd occlusion problem in human detection by leveraging the head part. Double Anchor RPN is developed to capture body and head parts in pairs. A proposal crossover strategy is introduced to generate high-quality proposals for both parts as a training augmentation. Features of coupled proposals are then aggregated efficiently to exploit the inherent relationship. Finally, a Joint NMS module is developed for robust post-processing. The proposed framework, called Double Anchor R-CNN, is able to detect the body and head for each person simultaneously in crowded scenarios. State-of-the-art results are reported on challenging human detection datasets. Our model yields log-average miss rates (MR) of 51.79pp on CrowdHuman, 55.01pp on COCOPersons~(crowded sub-dataset) and 40.02pp on CrowdPose~(crowded sub-dataset), which outperforms previous baseline detectors by 3.57pp, 3.82pp, and 4.24pp, respectively. We hope our simple and effective approach will serve as a solid baseline and help ease future research in crowded human detection. | 2020-02-20 21:28:56 | {"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": 0.3347778022289276, "perplexity": 2577.8844350604377}, "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/1581875145282.57/warc/CC-MAIN-20200220193228-20200220223228-00165.warc.gz"} |
https://www.jobilize.com/course/section/using-the-law-of-sines-to-solve-ssa-triangles-by-openstax?qcr=www.quizover.com | # 8.1 Non-right triangles: law of sines (Page 2/10)
Page 2 / 10
## Solving for two unknown sides and angle of an aas triangle
Solve the triangle shown in [link] to the nearest tenth.
The three angles must add up to 180 degrees. From this, we can determine that
$\begin{array}{l}\begin{array}{l}\hfill \\ \beta =180°-50°-30°\hfill \end{array}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=100°\hfill \end{array}$
To find an unknown side, we need to know the corresponding angle and a known ratio. We know that angle $\alpha =50°$ and its corresponding side $a=10.\text{\hspace{0.17em}}$ We can use the following proportion from the Law of Sines to find the length of $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$
Similarly, to solve for $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ we set up another proportion.
Therefore, the complete set of angles and sides is
$\begin{array}{l}\begin{array}{l}\hfill \\ \alpha =50°\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a=10\hfill \end{array}\hfill \\ \beta =100°\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b\approx 12.9\hfill \\ \gamma =30°\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}c\approx 6.5\hfill \end{array}$
Solve the triangle shown in [link] to the nearest tenth.
$\begin{array}{l}\alpha ={98}^{\circ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a=34.6\\ \beta ={39}^{\circ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=22\\ \gamma ={43}^{\circ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}c=23.8\end{array}$
## Using the law of sines to solve ssa triangles
We can use the Law of Sines to solve any oblique triangle, but some solutions may not be straightforward. In some cases, more than one triangle may satisfy the given criteria, which we describe as an ambiguous case . Triangles classified as SSA, those in which we know the lengths of two sides and the measurement of the angle opposite one of the given sides, may result in one or two solutions, or even no solution.
## Possible outcomes for ssa triangles
Oblique triangles in the category SSA may have four different outcomes. [link] illustrates the solutions with the known sides $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and known angle $\text{\hspace{0.17em}}\alpha .$
## Solving an oblique ssa triangle
Solve the triangle in [link] for the missing side and find the missing angle measures to the nearest tenth.
Use the Law of Sines to find angle $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ and angle $\text{\hspace{0.17em}}\gamma ,\text{\hspace{0.17em}}$ and then side $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ Solving for $\text{\hspace{0.17em}}\beta ,\text{\hspace{0.17em}}$ we have the proportion
$\begin{array}{r}\hfill \frac{\mathrm{sin}\text{\hspace{0.17em}}\alpha }{a}=\frac{\mathrm{sin}\text{\hspace{0.17em}}\beta }{b}\\ \hfill \frac{\mathrm{sin}\left(35°\right)}{6}=\frac{\mathrm{sin}\text{\hspace{0.17em}}\beta }{8}\\ \hfill \frac{8\mathrm{sin}\left(35°\right)}{6}=\mathrm{sin}\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}\\ \hfill 0.7648\approx \mathrm{sin}\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}\\ \hfill {\mathrm{sin}}^{-1}\left(0.7648\right)\approx 49.9°\\ \hfill \beta \approx 49.9°\end{array}$
However, in the diagram, angle $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ appears to be an obtuse angle and may be greater than 90°. How did we get an acute angle, and how do we find the measurement of $\text{\hspace{0.17em}}\beta ?\text{\hspace{0.17em}}$ Let’s investigate further. Dropping a perpendicular from $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ and viewing the triangle from a right angle perspective, we have [link] . It appears that there may be a second triangle that will fit the given criteria.
The angle supplementary to $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ is approximately equal to 49.9°, which means that $\text{\hspace{0.17em}}\beta =180°-49.9°=130.1°.\text{\hspace{0.17em}}$ (Remember that the sine function is positive in both the first and second quadrants.) Solving for $\text{\hspace{0.17em}}\gamma ,$ we have
$\gamma =180°-35°-130.1°\approx 14.9°$
We can then use these measurements to solve the other triangle. Since $\text{\hspace{0.17em}}{\gamma }^{\prime }\text{\hspace{0.17em}}$ is supplementary to $\text{\hspace{0.17em}}\gamma ,$ we have
${\gamma }^{\prime }=180°-35°-49.9°\approx 95.1°$
Now we need to find $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{c}^{\prime }.$
We have
Finally,
To summarize, there are two triangles with an angle of 35°, an adjacent side of 8, and an opposite side of 6, as shown in [link] .
However, we were looking for the values for the triangle with an obtuse angle $\text{\hspace{0.17em}}\beta .\text{\hspace{0.17em}}$ We can see them in the first triangle (a) in [link] .
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas | 2020-07-05 11:14:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 28, "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": 0.7823665142059326, "perplexity": 803.780708524153}, "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/1593655887319.41/warc/CC-MAIN-20200705090648-20200705120648-00301.warc.gz"} |
https://www.jobilize.com/statistics/section/references-test-of-independence-by-openstax?qcr=www.quizover.com | # 11.3 Test of independence (Page 3/20)
Page 3 / 20
De Anza College is interested in the relationship between anxiety level and the need to succeed in school. A random sample of 400 students took a test that measured anxiety level and need to succeed in school. [link] shows the results. De Anza College wants to know if anxiety level and need to succeed in school are independent events.
Need to succeed in school vs. anxiety level
Need to Succeed in School High
Anxiety
Med-high
Anxiety
Medium
Anxiety
Med-low
Anxiety
Low
Anxiety
Row Total
High Need 35 42 53 15 10 155
Medium Need 18 48 63 33 31 193
Low Need 4 5 11 15 17 52
Column Total 57 95 127 63 58 400
a. How many high anxiety level students are expected to have a high need to succeed in school?
a. The column total for a high anxiety level is 57. The row total for high need to succeed in school is 155. The sample size or total surveyed is 400.
$E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}=\frac{155\cdot 57}{400}=22.09$
The expected number of students who have a high anxiety level and a high need to succeed in school is about 22.
b. If the two variables are independent, how many students do you expect to have a low need to succeed in school and a med-low level of anxiety?
b. The column total for a med-low anxiety level is 63. The row total for a low need to succeed in school is 52. The sample size or total surveyed is 400.
c. $E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}$ = ________
c. $E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}=8.19$
d. The expected number of students who have a med-low anxiety level and a low need to succeed in school is about ________.
d. 8
## Try it
Refer back to the information in [link] . How many service providing jobs are there expected to be in 2020? How many nonagriculture wage and salary jobs are there expected to be in 2020?
12,727, 14,965
## References
DiCamilo, Mark, Mervin Field, “Most Californians See a Direct Linkage between Obesity and Sugary Sodas. Two in Three Voters Support Taxing Sugar-Sweetened Beverages If Proceeds are Tied to Improving School Nutrition and Physical Activity Programs.” The Field Poll, released Feb. 14, 2013. Available online at http://field.com/fieldpollonline/subscribers/Rls2436.pdf (accessed May 24, 2013).
Harris Interactive, “Favorite Flavor of Ice Cream.” Available online at http://www.statisticbrain.com/favorite-flavor-of-ice-cream (accessed May 24, 2013)
“Youngest Online Entrepreneurs List.” Available online at http://www.statisticbrain.com/youngest-online-entrepreneur-list (accessed May 24, 2013).
## Chapter review
To assess whether two factors are independent or not, you can apply the test of independence that uses the chi-square distribution. The null hypothesis for this test states that the two factors are independent. The test compares observed values to expected values. The test is right-tailed. Each observation or cell category must have an expected value of at least 5.
## Test of independence
• The number of degrees of freedom is equal to (number of columns - 1)(number of rows - 1).
• The test statistic is $\underset{\left(i\cdot j\right)}{\Sigma }\frac{{\left(O–E\right)}^{2}}{E}$ where O = observed values, E = expected values, i = the number of rows in the table, and j = the number of columns in the table.
• If the null hypothesis is true, the expected number $E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}$ .
what is standard deviation?
It is the measure of the variation of certain values from the Mean (Center) of a frequency distribution of sample values for a particular Variable.
Dominic
Yeah....the simplest one
IRFAN
what is the number of x
10
Elicia
Javed Arif
Jawed
how will you know if a group of data set is a sample or population
population is the whole set and the sample is the subset of population.
umair
if the data set is drawn out of a larger set it is a sample and if it is itself the whole complete set it can be treated as population.
Bhavika
hello everyone if I have the data set which contains measurements of each part during 10 years, may I say that it's the population or it's still a sample because it doesn't contain my measurements in the future? thanks
Alexander
Pls I hv a problem on t test is there anyone who can help?
Peggy
Dominic
Bhavika is right
Dominic
what is the problem peggy?
Bhavika
hi
Sandeep
Hello
hi
Bhavika
hii Bhavika
Dar
Hi eny population has a special definition. if that data set had all of characteristics of definition, that is population. otherwise that is a sample
Hoshyar
three coins are tossed. find the probability of no head
three coins are tossed consecutively or what ?
umair
umair
or .125 is the probability of getting no head when 3 coins are tossed
umair
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Simone
what is two tailed test
if the diameter will be greater than 3 cm then the bullet will not fit in the barrel of the gun so you are bothered for both the sides.
umair
in this test you are worried on both the ends
umair
lets say you are designing a bullet for thw gun od diameter equals 3cm.if the diameter of the bullet is less than 3 cm then you wont be able to shoot it
umair
In order to apply weddles rule for numerical integration what is minimum number of ordinates
excuse me?
Gabriel
why?
didn't understand the question though.
Gabriel
which question? ?
We have rules of numerical integration like Trapezoidal rule, Simpson's 1/3 and 3/8 rules, Boole's rule and Weddle rule for n =1,2,3,4 and 6 but for n=5?
John
geometric mean of two numbers 4 and 16 is:
10
umair
really
iphone
quartile deviation of 8 8 8 is:
iphone
sorry 8 is the geometric mean of 4,16
umair
quartile deviation of 8 8 8 is
iphone
can you please expalin the whole question ?
umair
mcq
iphone
h
iphone
can you please post the picture of that ?
umair
how
iphone
hello
John
10 now
John
how to find out the value
can you be more specific ?
umair
yes
KrishnaReddy
what is the difference between inferential and descriptive statistics
descriptive statistics gives you the result on the the data like you can calculate various things like variance,mean,median etc. however, inferential stats is involved in prediction of future trends using the previous stored data.
umair
if you need more help i am up for the help.
umair
Thanks a lot
Anjali
Inferential Statistics involves drawing conclusions on a population based on analysis of a sample. Descriptive statistics summarises or describes your current data as numerical calculations or graphs.
fred
my pleasure😊. Helping others offers me satisfaction 😊
umair
for poisson distribution mean............variance.
both are equal to mu
Faizan
mean=variance
Faizan
what is a variable
something that changes
Festus
why we only calculate 4 moment of mean? asked in papers.
why we only 4 moment of mean ? asked in BA exam
Faizan
Hello, can you please share the possible questions that are likely to be examined under the topic: regression and correlation analysis.
Refiloe
for normal distribution mean is 2 & variance is 4 find mu 4?
repeat quastion again
Yusuf
find mu 4. it can be wrong but want to prove how.
Faizan
for a normal distribution if mu 4 is 12 then find mu 3?
Question hi wrong ha
Tahir
ye BA mcqs me aya he teen he. 2dafa aya he
Faizan
if X is normally distributed. (n,b). then its mean deviation is?
Faizan
The answer is zero, because all odd ordered central moments of a normal distribution are Zero.
nikita
which question is zero
Faizan
sorry it is (5,16) in place of (n,b)
Faizan
I got. thanks. it is zero.
Faizan
a random variable having binomial distribution is?
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http://hermes.roua.org/hermes-pub/arxiv/05/04/220/article.xhtml | ## The Weyl Integration Model for KAK decomposition of Reductive Lie Group
### HU Zhi-guang 1 YAN Kui-hua 1 , 2 (1. School of Mathematical Sciences, Peking University, Beijing, 100871; 2. School of Mathematics and Physics, Zhejiang Normal University, Zhejiang Jinhua, 321004) (Email: zhiguang_hu@eyou.com; yankh@zjnu.cn)
Abstract
The Weyl integration model presented by An and Wang can be effectively used to reduce the integration over $G$ -space. In this paper, we construct an especial Weyl integration model for KAK decomposition of Reductive Lie Group and obtain an integration formula which implies that the integration of ${L}^{1}$ -integrable function over reductive Lie group $G$ can be carried out by first integrating over each conjugacy class and then integrating over the set of conjugacy classes.
Key Words: Weyl Integration Model; Reductive Lie Group; KAK Decomposition; Restricted roots Mathematics Subject Classification 2000: 22E15; 58C35 CLC number: O152.5; O186.1
1 Introduction
For a compact connected Lie group $G$ , the classical Weyl integration formula (e.g.
ref. Knapp[1) indicates that an integration of any continuous function over $G$ can be reduced to one over its maximal torus. That is to say the integration can be carried out by first integrating over each conjugacy class and then integrating over the set of conjugacy classes.
Then for a general Lie group, how to reduce the integration of functions over it ?
In Helgason[2, the author presents some integral formulas related to the Cartan, Iwasawa and Bruhat decompositions for semisimple Lie groups. In this paper, we will use the Weyl integration model, which is first introduced by An and Wang in [3and can be used to generalize the reducing integration idea to integrations over G-spaces, to obtain an integral formula for KAK decomposition of reductive Lie groups. It is worthy of indicating that this reducible idea of integrations can be effectively used to calculate the eigenvalue distribution of random matrices in random matrix ensemble theory. But here we do not plan to study those applications in random matrix theory.
Let $G$ be a Lie group, $X$ a $G$ -space, $Y$ a imbedding submanifold of $X$ and $\sigma$ the $G$ -action on $X$ . Suppose that $dx,dy$ and $d\mu$ are proper invariant measures over $X,Y$ and $G/K$ respectively. The Weyl integration model (see section 2 for its details) is a system $\left(G,\sigma ,\left(X,dx\right),\left(Y,dy\right),\left(G/K,d\mu \right)\right)$ satisfying some proper conditions (see 7 - 9 ). Indeed, for a Weyl integration model, the following formula holds for all $f\in {L}^{1}\left(X,dx\right)$ ,
$\begin{array}{c}{\int }_{X}f\left(x\right)dx=\frac{1}{d}{\int }_{Y}\left({\int }_{G/K}f\left(\sigma \left(g,y\right)\right)d\mu \left(\left[g\right]\right)\right)J\left(y\right)dy,\end{array}$ (1)
where $J\in {C}^{\infty }\left(Y\right)$ and $d$ is the multiplicity of the model. That is to say that the integration over a $G$ -space $X$ can be converted to the integration by first integrating over each orbit and then integrating over the orbits space $Y$ . Moreover, under some orthogonal conditions, $J\left(y\right)$ can be effectively calculated, i.e. there is a constant $C$ such that
$\begin{array}{c}J\left(y\right)=C|det{\Psi }_{y}|,\text{for all}y\in {Y}^{\prime },\end{array}$ (2)
where ${\Psi }_{y}$ is a mapping induced by G-action $\sigma$ (see 11 ). In this paper, we will construct a Weyl integration model $\left(K×K,\sigma ,\left(G,dg\right),\left(A,da\right),\left(\left(K×K\right)/M,d\mu \right)\right)$ . The meanings of its members are as follow. $G$ is a reductive Lie group with Lie algebra $\mathfrak{g}$ . Let $\theta$ be the Cartan involution on $\mathfrak{g}$ . $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ is the Cartan decomposition of $\mathfrak{g}$ , where $\mathfrak{k}$ and $\mathfrak{p}$ are the $+1$ and $-1$ eigenspaces of $\theta$ respectively. Let $K$ be the associated maximal compact subgroup of $G$ with Lie algebra $\mathfrak{k}$ .
Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$ . Set $A=exp\left(\mathfrak{a}\right)$ , then $A$ is a closed subgroup of $G$ . It is known that $G=KAK$ in the sense that every element in $G$ has a decomposition as ${k}_{{}_{1}}a{k}_{{}_{2}}$ with ${k}_{{}_{1}},{k}_{{}_{2}}\in K$ and $a\in A$ .
To consider the following group $K×K$ -action
$\begin{array}{c}\sigma :\left(K×K\right)×A\to G,\left(\left({k}_{{}_{1}},{k}_{{}_{2}}\right),a\right)↦{k}_{{}_{1}}a{k}_{{}_{2}}^{-1}.\end{array}$ (3)
Let
$\begin{array}{c}M=\left\{\left({k}_{{}_{1}},{k}_{{}_{2}}\right)\in K×K|{k}_{{}_{1}}a{k}_{{}_{2}}^{-1}=a,\forall a\in A\right\}.\end{array}$ (4)
Then the $K×K$ -action $\sigma$ can be naturally reduced to a map $\phi$ , i.e.
$\begin{array}{c}\phi :\left(K×K\right)/M×A\to G,\left(\left[\left({k}_{{}_{1}},{k}_{{}_{2}}\right)\right],a\right)↦{k}_{{}_{1}}a{k}_{{}_{2}}^{-1}.\end{array}$ (5)
Let $dg,da$ be the Haar measures on $G$ and $A$ respectively. There has a $K×K$ -invariant measure $d\mu$ on $\left(K×K\right)/M$ since $K×K$ is compact. Then we can prove that $\left(K×K,\sigma ,\left(G,dg\right),\left(A,da\right),\left(\left(K×K\right)/M,d\mu \right)\right)$ is a Weyl integration model with finite multiplicities.
Thus by 1 , we obtain an integral formula (see theorem 2.6 for details) which reduces the integration over reductive Lie group $G$ to one first over $\left(K×K\right)/M$ and then over its subgroup $A$ . Moreover, by 2 , we can calculate that the according $J\left(y\right)$ has the following formula
$\begin{array}{c}J\left(a\right)=C{\prod }_{\lambda \in {\Sigma }^{+}}|sinh\left(\lambda \left(H\right)\right){|}^{{\beta }_{{}_{\lambda }}},\end{array}$ (6)
where $a={e}^{H},H\in \mathfrak{a}$ , ${\beta }_{{}_{\lambda }}$ is the dimension of the restricted root space ${\mathfrak{g}}_{{}_{\lambda }}$ with respect to the restricted root $\lambda$ in the positive restricted root system ${\Sigma }^{+}$ and $C$ is a proper constant.
By comparison, it is formally analogous to the integration formula for Cartan decomposition of noncompact semisimple Lie groups (see [2).
The structure of the paper is as follow. In section 2, we briefly introduce the Weyl integration model and the Restricted root system and KAK decomposition for a reductive Lie group. The main results of the paper are presented at the end of this section. In section 3, we prove the main results.
2 Preliminaries and Main Results
2.1 Weyl integration model
In this subsection, we briefly introduce the Weyl integration model. One may refer to [3for details.
Let $G$ be a Lie group which acts on a $n$ -dimensional smooth manifold $X$ . The action is denoted by $\sigma :G×X\to X$ . Let $dx$ be a $G$ -invariant admissible measure on $X$ . $Y$ is an imbedding submanifold of $X$ . Suppose that there is an admissible measure $dy$ on $Y$ , and ${X}_{0}\subset X$ , ${Y}_{0}\subset Y$ are closed zero measure subsets of $X$ and $Y$ respectively. Set ${X}^{\prime }=X\{X}_{0}$ , ${Y}^{\prime }=Y\{Y}_{0}$ and $K=\left\{g\in G|\sigma \left(g,y\right)=y,\forall y\in Y\right\}$ . Let ${K}_{x}=\left\{g\in G|\sigma \left(g,x\right)=x\right\}$ be the isotropic subgroup associated with $x\in X$ and ${O}_{y}=\left\{\sigma \left(g,y\right)|g\in G\right\}$ the orbit of $y\in Y$ . Then $K\subset {K}_{y},\forall y\in Y$ . In the following text we suppose that
$\begin{array}{c}{X}^{\prime }={\bigcup }_{y\in {Y}^{\prime }}{O}_{y}.\end{array}$ (7)
$\begin{array}{c}{T}_{y}X={T}_{y}{O}_{y}\oplus {T}_{y}Y,\forall y\in {Y}^{\prime },\end{array}$ (8)
which is called the transversal condition.
$\begin{array}{c}dim{K}_{y}=dimK,\forall y\in {Y}^{\prime }.\end{array}$ (9)
The G-action $\sigma :G×X\to X$ can be reduced to a map $\phi :G/K×Y\to X$ by $\phi \left(\left[g\right],y\right)=\sigma \left(g,y\right)$ and furthermore to a map which is still denoted by $\phi$ , i.e. $\phi :G/K×{Y}^{\prime }\to {X}^{\prime }$ by restriction. By the above assumption, $\phi$ is surjective. Suppose that there is a $G$ -invariant admissible measure $d\mu$ on $G/K$ .
Proposition 2.1. Suppose the conditions 7 , 8 and 9 hold, then $\phi :G/K×{Y}^{\prime }\to {X}^{\prime }$ is a local diffeomorphism.
Let $G,X,Y,K,\sigma ,\phi ,dx,dy,d\mu$ be the above objects. A Weyl integration model is a system $\left(G,\sigma ,\left(X,dx\right),\left(Y,dy\right),\left(G/K,d\mu \right)\right)$ , in which we can choose ${X}_{0}$ such that the map $\phi :G/K×{Y}^{\prime }\to {X}^{\prime }$ is a finite-sheeted covering map. The number of sheets of the covering map is called the multiplicity of the model. About a Weyl integration model, we have the following basic theorem.
Theorem 2.2. If $\left(G,\sigma ,\left(X,dx\right),\left(Y,dy\right),\left(G/K,d\mu \right)\right)$ is a Weyl integration model with multiplicity $d$ , then the formula 1 holds for all $f\in {C}^{\infty }\left(X\right)$ with $f\ge 0$ or $f\in {L}^{1}\left(X,dx\right)$ .
Let $\mathfrak{s}$ be a linear subspace of the Lie algebra $\mathfrak{g}$ of $G$ , such that $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{s}$ , where $\mathfrak{k}$ is the Lie algebra of $K$ . Then $\mathfrak{s}$ can be identified with the tangent space ${T}_{\left[e\right]}\left(G/K\right)$ of $G/K$ at point $\left[e\right]$ in a natural way:
${T}_{\left[e\right]}\left(G/K\right)\sim =\mathfrak{s}.$ By the transversal condition 8 , the tangent map $d\phi :{T}_{\left[e\right]}\left(G/K\right)\oplus {T}_{y}Y\to {T}_{y}X$ of $\phi$ at point $\left(\left[e\right],y\right)$ ( $y\in {Y}^{\prime }$ ) can be regarded as a linear transformation $d\phi :\mathfrak{s}\oplus {T}_{y}Y\to {T}_{y}{O}_{y}\oplus {T}_{y}Y.$ It can be proved that (see [3)
$\begin{array}{c}d\phi =\left(\begin{array}{cc}{\Psi }_{y}& 0\\ 0& id\end{array}\right),\end{array}$ (10)
where ${\Psi }_{y}:\mathfrak{s}\to {T}_{y}{O}_{y}$ is given by
$\begin{array}{c}{\Psi }_{y}\left(\xi \right)=\frac{d}{dt}{|}_{t=0}\sigma \left(expt\xi ,y\right),\forall \xi \in \mathfrak{s}.\end{array}$ (11)
Suppose that there is a Riemannian structure on $X$ such that the following orthogonal condition holds
$\begin{array}{c}{T}_{y}Y\perp {T}_{y}{O}_{y},\text{for all}y\in {Y}^{\prime }.\end{array}$ (12)
Let $dx$ and $dy$ are the Riemannian measures on $X$ and $Y$ respectively. Then there is a constant $C$ such that
$\begin{array}{c}J\left(y\right)=C|det{\Psi }_{y}|,\text{for all}y\in {Y}^{\prime }.\end{array}$ (13)
2.2 Restricted root system and KAK decomposition for reductive Lie group
In the sense of Knapp ([1, Sec.7.2), a reductive Lie group is a 4-tuple $\left(G,K,\theta ,B\right)$ consisting of a Lie group $G$ , a compact subgroup $K$ of $G$ , a Lie algebra involution $\theta$ of the Lie algebra $\mathfrak{g}$ of $G$ and a nondegenerate $Ad\left(G\right)$ -invariant $\theta$ -invariant bilinear form $B$ on $\mathfrak{g}$ such that (i) $\mathfrak{g}$ is a reductive Lie algebra, (ii) the decomposition of $\mathfrak{g}$ into $+1$ and $-1$ eigenspaces under $\theta$ is $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ , where $\mathfrak{k}$ is the Lie algebra of K, (iii) $\mathfrak{k}$ and $\mathfrak{p}$ are orthogonal under $B$ and $B$ is positive definite on $\mathfrak{p}$ and negative definite on $\mathfrak{k}$ , (iv) multiplication, as a map from $K×exp\left(\mathfrak{p}\right)$ into $G$ , is a diffeomorphism onto, and (v) every automorphism $Ad\left(g\right)$ of ${\mathfrak{g}}^{\mathbb{C}}$ is inner for $g\in G$ , i.e., is given by some $x\in Int\left({\mathfrak{g}}^{\mathbb{C}}\right)$ . $K$ is called the associated maximal compact subgroup, $\theta$ the Cartan involution and $B$ the invariant bilinear form. The decomposition of $\mathfrak{g}$ (according $G$ ) in property (iii) (according (iv)) is called (global) Cartan decomposition.
Now let $G$ is a reductive Lie group with Lie algebra $\mathfrak{g}$ . Let $\mathfrak{a}$ be a maximal abelian subspaces of $\mathfrak{p}$ , then $\mathfrak{p}={\bigcup }_{k\in K}Ad\left(k\right)\mathfrak{a}$ . Set ${\mathfrak{g}}_{{}_{\lambda }}=\left\{X\in \mathfrak{g}|\left(adH\right)X=\lambda \left(H\right)X,\text{for all}H\in \mathfrak{a}\right\}$ .
A nonzero $\lambda \in {\mathfrak{a}}^{*}$ is called a restricted root of $\mathfrak{g}$ if ${\mathfrak{g}}_{{}_{\lambda }}$ is nonzero. Accordingly ${\mathfrak{g}}_{{}_{\lambda }}$ is called a restricted root space. The set of restricted roots is denoted by $\Sigma$ and Let ${\Sigma }^{+}$ be the set of positive restricted roots.
Reflections in the restricted roots generate the Weyl group $W\left(\Sigma \right)$ of $\Sigma$ . Denoted by ${N}_{K}\left(\mathfrak{a}\right)$ and ${Z}_{K}\left(\mathfrak{a}\right)$ the normalizer and centralizer of $\mathfrak{a}$ in $K$ respectively, then the Weyl group $W={N}_{K}\left(\mathfrak{a}\right)/{Z}_{K}\left(\mathfrak{a}\right)$ and the Lie algebras of ${N}_{K}\left(\mathfrak{a}\right)$ and ${Z}_{K}\left(\mathfrak{a}\right)$ are $\mathfrak{m}={Z}_{\mathfrak{k}}\left(\mathfrak{a}\right)$ .
The restricted root space ${\mathfrak{g}}_{{}_{\lambda }}$ satisfies the following basic properties
Proposition 2.3. (see [1) (i) $\mathfrak{g}={\mathfrak{g}}_{{}_{0}}\oplus {\oplus }_{\lambda \in \Sigma }{\mathfrak{g}}_{{}_{\lambda }}$ .
(ii) $\left[{\mathfrak{g}}_{{}_{\lambda }},{\mathfrak{g}}_{{}_{\gamma }}\right]\subset {\mathfrak{g}}_{{}_{\lambda +\gamma }}$ and if $\lambda \ne \gamma$ , then ${\mathfrak{g}}_{{}_{\lambda }}\perp {\mathfrak{g}}_{{}_{\gamma }}$ in the sense of $B$ .
(iii) $\theta {\mathfrak{g}}_{{}_{\lambda }}={\mathfrak{g}}_{{}_{-\lambda }}$ , so if $\lambda \in \Sigma$ , then $-\lambda \in \Sigma$ .
(iv) ${\mathfrak{g}}_{{}_{0}}=\mathfrak{a}\oplus \mathfrak{m}$ orthogonally.
Let $A=exp\mathfrak{a}$ . For any reductive Lie group $G$ , it has the following decomposition
Theorem 2.4. (KAK decomposition [1) Every element in G has a decomposition as ${k}_{{}_{1}}a{k}_{{}_{2}}$ with ${k}_{{}_{1}},{k}_{{}_{2}}\in K$ and $a\in A$ . In this decomposition, $a$ is uniquely determined up to conjugation by a member of $W$ . If $a$ is fixed as $expH$ with $H\in \mathfrak{a}$ and $\lambda \left(H\right)\ne 0$ for all $\lambda \in \Sigma$ , then ${k}_{{}_{1}}$ is unique up to right multiplication by a member of ${Z}_{K}\left(\mathfrak{a}\right)$ .
2.3 Main result
Now let us present the main result of the paper. Let $\left(G,K,\theta ,B\right)$ be any reductive Lie group. Let $dg$ and $da$ be the left-invariant measures on $G$ and $A$ respectively corresponding to the Riemannian structure induced by $B$ and $d\mu$ a $K×K$ -invariant measure on $\left(K×K\right)/M$ , where the set $M$ are defined by 4 .
Theorem 2.5. $\left(K×K,\phi ,\left(G,dg\right),\left(A,da\right),\left(\left(K×K\right)/M,d\mu \right)\right)$ is a Weyl integration model with $d$ -sheeted multiplicities, where $d=|W|$ and the map $\phi$ is defined by 5 .
Theorem 2.6. For any $f\in {L}^{1}\left(G,dg\right)$ ,
$\begin{array}{c}{\int }_{G}f\left(g\right)dg={\int }_{A}\left({\int }_{\left(K×K\right)/M}f\left({k}_{{}_{1}}a{k}_{{}_{2}}^{-1}\right)d\mu \right)J\left(a\right)da,\end{array}$ (14)
where $J\left(a\right)$ has the formula 6 .
3 Proof of Main Results
In this section, we will prove the main results by constructing a Weyl integration model and using the theorem 2.2 .
Now Let $\left(G,K,\theta ,B\right)$ be any reductive Lie group. $\mathfrak{a},A,\mathfrak{m},\Sigma ,{\Sigma }^{+}$ and $W$ are defined in the above section. We come to consider the G-action $\sigma$ defined by 3 and its reduced map $\phi$ defined by 5 . Let ${A}^{\prime }$ be the set of regular elements in $A$ and ${G}^{\prime }=K{A}^{\prime }K$ . It naturally has the following map by restriction of $\phi$ , which is still denoted by $\phi$ .
$\begin{array}{c}\phi :\left(K×K\right)/M×{A}^{\prime }\to {G}^{\prime }.\end{array}$ (15)
Lemma 3.1. (i) $M$ is isomorphic to ${Z}_{K}\left(\mathfrak{a}\right)$ .
(ii) $\phi$ is a surjective $d$ -sheeted map, where $d=|W|$ .
Proof. (i) Note that the unit element $e\in A$ . If $\left({k}_{{}_{1}},{k}_{{}_{2}}\right)\in M$ , then ${k}_{{}_{1}}e{k}_{{}_{2}}^{-1}=e$ , i.e. ${k}_{{}_{1}}={k}_{{}_{2}}$ .
Thus $M=\left\{\left(k,k\right)\in K×K|ka=ak,\forall a\in A\right\}$ is isomorphic to ${Z}_{K}\left(\mathfrak{a}\right)$ .
(ii) By the conclusion $1$ , it is the direct corollary of theorem 2.4 .
Now for any $x\in G$ , set ${M}_{x}=\left\{\left({k}_{{}_{1}},{k}_{{}_{2}}\right)\in K×K|{k}_{{}_{1}}x{k}_{{}_{2}}^{-1}=x\right\}$ be the isotropic subgroup associated with $x\in G$ and ${O}_{a}=\left\{{k}_{{}_{1}}a{k}_{{}_{2}}^{-1}|\left({k}_{{}_{1}},{k}_{{}_{2}}\right)\in K×K\right\}$ the orbit of $a\in {A}^{\prime }$ . Then $M\subset {M}_{a}$ , for all $a\in {A}^{\prime }$ .
By the definition of reductive group, the invariant bilinear $B$ determines an inner product on $\mathfrak{p}$ , and $-B$ determines an inner product on $\mathfrak{k}$ . We write $\mathfrak{b}={\mathfrak{a}}^{\perp }$ in $\mathfrak{p}$ , and $\mathfrak{l}={\mathfrak{m}}^{\perp }$ in $\mathfrak{k}$ .
By proposition 2.3 , it is obvious that
$\begin{array}{c}\mathfrak{g}=\mathfrak{m}\oplus \mathfrak{l}\oplus \mathfrak{a}\oplus \mathfrak{b},\mathfrak{b}\oplus \mathfrak{l}={\oplus }_{\lambda \in \Sigma }{\mathfrak{g}}_{{}_{\lambda }},\mathfrak{l}=\mathfrak{k}\bigcap {\oplus }_{\lambda \in \Sigma }{\mathfrak{g}}_{{}_{\lambda }},\mathfrak{b}=\mathfrak{p}\bigcap {\oplus }_{\lambda \in \Sigma }{\mathfrak{g}}_{{}_{\lambda }}.\end{array}$ (16)
For all $\lambda \in {\Sigma }^{+}$ , we choose a normal orthogonal basis $\left\{{\xi }_{{}_{\lambda ,1}},\cdots ,{\xi }_{{}_{\lambda ,{\beta }_{{}_{\lambda }}}}\right\}$ in ${\mathfrak{g}}_{{}_{\lambda }}$ , where ${\beta }_{{}_{\lambda }}=dim{\mathfrak{g}}_{{}_{\lambda }}$ which may be larger than one. Then for all ${\xi }_{{}_{\lambda ,j}}$ , we have
$\begin{array}{c}\begin{array}{}\text{(17)}& \theta \left({\xi }_{{}_{\lambda ,j}}+\theta {\xi }_{{}_{\lambda ,j}}\right)={\xi }_{{}_{\lambda ,j}}+\theta {\xi }_{{}_{\lambda ,j}}\in \mathfrak{l},\theta \left({\xi }_{{}_{\lambda ,j}}-\theta {\xi }_{{}_{\lambda ,j}}\right)=-\left({\xi }_{{}_{\lambda ,j}}-\theta {\xi }_{{}_{\lambda ,j}}\right)\in \mathfrak{b}.\end{array}\end{array}$ (18)
Indeed, $\left\{\left({\xi }_{{}_{\lambda ,j}}+\theta {\xi }_{{}_{\lambda ,j}}\right)|\lambda \in {\Sigma }^{+},j=1,\cdots ,{\beta }_{{}_{\lambda }}\right\}$ composes of a basis of $\mathfrak{l}$ and $\left\{\left({\xi }_{{}_{\lambda ,j}}-\theta {\xi }_{{}_{\lambda ,j}}\right)|\lambda \in {\Sigma }^{+},j=1,\cdots ,{\beta }_{{}_{\lambda }}\right\}$ a basis of $\mathfrak{b}$ . Then we get
$\begin{array}{c}dim\mathfrak{l}=dim\mathfrak{b}={\sum }_{\lambda \in {\Sigma }^{+}}{\beta }_{{}_{\lambda }}.\end{array}$ (19)
Therefore,
$\begin{array}{cc}dim\left(\left(K×K\right)/M×A\right)& =dim\left(\left(\mathfrak{k}\oplus \mathfrak{k}\right)/\mathfrak{m}\oplus \mathfrak{a}\right)=dim\mathfrak{k}+dim\mathfrak{l}+dim\mathfrak{a}\end{array}$
$\begin{array}{cc}& =dim\mathfrak{k}+dim\mathfrak{b}+dim\mathfrak{a}=dim\mathfrak{k}+dim\mathfrak{p}=dimG.\end{array}$
$\begin{array}{}\end{array}$
So $\phi$ is a map between the same dimension manifolds.
Lemma 3.2. (i) For $a\in {A}^{\prime }$ , denoted by ${L}_{a}$ the left translation, then
$\begin{array}{c}{T}_{a}{O}_{a}=\left\{d{L}_{a}\left(Ad\left({a}^{-1}\right){\zeta }_{{}_{1}}-{\zeta }_{{}_{2}}\right)|\left({\zeta }_{{}_{1}},{\zeta }_{{}_{2}}\right)\in \left(\mathfrak{k},\mathfrak{k}\right)\right\}.\end{array}$ (20)
(ii) The following set is composed of a basis of ${T}_{a}{O}_{a}$ ,
$\begin{array}{c}\begin{array}{cc}F\equiv \left\{d{L}_{a}\left({\eta }_{{}_{i}}\right),d{L}_{a}\left({\xi }_{{}_{\lambda ,j}}^{+}\right),d{L}_{a}\left({\xi }_{{}_{\lambda ,j}}^{-}\right)|i=1,2,\cdots ,m,& \lambda \in {\Sigma }^{+},j=1,2,\cdots ,{\beta }_{{}_{\lambda }}\right\}.\end{array}\end{array}$ (21)
Proof. (i) Note that ${T}_{a}{O}_{a}$ is exactly the set composed of those tangent vectors of the smooth curves $exp\left(t{\zeta }_{{}_{1}}\right)\cdot a\cdot exp\left(-t{\zeta }_{{}_{2}}\right)$ at $t=0$ by the definition of orbit ${O}_{a}$ , where $\left({\zeta }_{{}_{1}},{\zeta }_{{}_{2}}\right)\in \left(\mathfrak{k},\mathfrak{k}\right)$ and $t<|\varepsilon |$ . But
$\begin{array}{cc}\frac{d}{dt}{|}_{t=0}exp\left(t{\zeta }_{{}_{1}}\right)\cdot a\cdot exp\left(-t{\zeta }_{{}_{2}}\right)& =\frac{d}{dt}{|}_{t=0}a\cdot exp\left(t\left(Ad\left({a}^{-1}\right){\zeta }_{{}_{1}}-{\zeta }_{{}_{2}}\right)+o\left({t}^{2}\right)\right)\end{array}$ (22)
$\begin{array}{cc}& =d{L}_{a}\left(Ad\left({a}^{-1}\right){\zeta }_{{}_{1}}-{\zeta }_{{}_{2}}\right).\end{array}$ (23)
$\begin{array}{}\end{array}$
Hence the conclusion is obtained.
(ii) Now suppose that $\left\{{\eta }_{{}_{1}},{\eta }_{{}_{2}},\cdots ,{\eta }_{{}_{m}}\right\}$ is a basis of $\mathfrak{m}$ and denote ${\xi }_{{}_{\lambda ,j}}^{±}\equiv {\xi }_{{}_{\lambda ,j}}±\theta {\xi }_{{}_{\lambda ,j}}$ .
By $\mathfrak{k}=\mathfrak{m}\oplus \mathfrak{l}$ , the following set is composed of a basis of $\left(\mathfrak{k},\mathfrak{k}\right)$ ,
$\begin{array}{c}\begin{array}{c}\left\{\left({\eta }_{{}_{i}},0\right),\left({\xi }_{{}_{\lambda ,j}}^{+},0\right),\left(0,{\eta }_{{}_{i}}\right),\left(0,{\xi }_{{}_{\lambda ,j}}^{+}\right)|i=1,2,\cdots ,m,\lambda \in {\Sigma }^{+},j=1,2,\cdots ,{\beta }_{{}_{\lambda }}\right\}.\end{array}\end{array}$ (24)
Set $a=expH,H\in \mathfrak{a}$ . Then by $\left[H,{\eta }_{{}_{i}}\right]=0$ , $\left[H,{\xi }_{{}_{\lambda ,j}}^{±}\right]=\lambda \left(H\right){\xi }_{{}_{\lambda ,j}}^{\mp }$ and 22 , it can be calculated that
$\begin{array}{cc}Ad\left({a}^{-1}\right)\left({\eta }_{{}_{i}}\right)-0& ={\eta }_{{}_{i}},\end{array}$ (25)
$\begin{array}{cc}Ad\left({a}^{-1}\right)\left({\xi }_{{}_{\lambda ,j}}^{+}\right)-0& =-sinh\left(\lambda \left(H\right)\right){\xi }_{{}_{\lambda ,j}}^{-}+cosh\left(\lambda \left(H\right)\right){\xi }_{{}_{\lambda ,j}}^{+},\end{array}$ (26)
$\begin{array}{cc}Ad\left({a}^{-1}\right)\left(0\right)-{\eta }_{{}_{i}}& =-{\eta }_{{}_{i}},\end{array}$ (27)
$\begin{array}{cc}Ad\left({a}^{-1}\right)\left(0\right)-{\xi }_{{}_{\lambda ,j}}^{+}& =-{\xi }_{{}_{\lambda ,j}}^{+}.\end{array}$ (28)
$\begin{array}{}\end{array}$
Thus by the first conclusion (i), the set $F$ is exactly composed of a basis of ${T}_{a}{O}_{a}$ .
Lemma 3.3. (i) $dim{M}_{a}=dimM$ , for all $a\in {A}^{\prime }$ .
(ii) ${T}_{a}G={T}_{a}A\oplus {T}_{a}{O}_{a}$ , for all $a\in {A}^{\prime }$ .
Proof. (i) Since $M\sim ={Z}_{{}_{K}}\left(\mathfrak{a}\right)$ in lemma 3.1 , the Lie algebra of $M$ is $\mathfrak{m}={Z}_{\mathfrak{k}}\left(\mathfrak{a}\right)$ . By lemma 3.2 , it is obvious that
$\begin{array}{c}{T}_{a}{O}_{a}=d{L}_{a}\left(\mathfrak{k}\oplus \mathfrak{b}\right).\end{array}$ (29)
Then by 19 ,
$\begin{array}{cc}dim{M}_{a}& =dim\left(K×K\right)-dim{O}_{a}\end{array}$
$\begin{array}{cc}& =dim\mathfrak{k}+dim\mathfrak{k}-\left(dim\mathfrak{k}+dim\mathfrak{b}\right)\end{array}$
$\begin{array}{cc}& =dim\mathfrak{k}-dim\mathfrak{l}\end{array}$
$\begin{array}{cc}& =dim\mathfrak{m}\end{array}$
$\begin{array}{cc}& =dimM.\end{array}$
$\begin{array}{}\end{array}$
(ii) By 29 , it is directly obtained that ${T}_{a}G=d{L}_{a}\left(\mathfrak{g}\right)=d{L}_{a}\left(\mathfrak{k}\oplus \mathfrak{p}\right)=d{L}_{a}\left(\mathfrak{a}\oplus \left(\mathfrak{b}\oplus \mathfrak{k}\right)\right)={T}_{a}A\oplus {T}_{a}{O}_{a}.$ Proof of theorem 2.5 . By Lemma 3.1 - 3.3 , proposition 2.1 and the definition of Weyl integration model, it is obvious.
Proof of theorem 2.6 . By theorem 2.2 and 2.5 , it only needs to calculate $J\left(a\right)$ in 3 . It is obvious that ${T}_{a}A\perp {T}_{a}{O}_{a}$ for any $a\in {A}^{\prime }$ . Then by the second conclusion in lemma 3.3 and 13 , we need to consider the map
$\begin{array}{c}{\Psi }_{a}:{T}_{\left[\left(e,e\right)\right]}\left(\left(K×K\right)/M\right)\to {T}_{a}{O}_{a},\left({\zeta }_{{}_{1}},{\zeta }_{{}_{2}}\right)↦\frac{d}{dt}{|}_{t=0}exp\left(t{\zeta }_{{}_{1}}\right)\cdot a\cdot exp\left(-t{\zeta }_{{}_{2}}\right).\end{array}$ (30)
Note that the tangent space ${T}_{\left[\left(e,e\right)\right]}\left(\left(K×K\right)/M\right)$ of $\left(K×K\right)/M$ at its unit element $\left[\left(e,e\right)\right]$ is exactly isomorphic to
$\begin{array}{cc}\left(\mathfrak{k},\mathfrak{l}\right)& \equiv \left\{\left({\zeta }_{{}_{1}},{\zeta }_{{}_{2}}\right)|{\zeta }_{{}_{1}}\in \mathfrak{k},{\zeta }_{{}_{2}}\in \mathfrak{l}\right\}.\end{array}$ (31)
$\begin{array}{}\end{array}$
Then the set
$\begin{array}{c}\begin{array}{c}\left\{\left({\eta }_{{}_{i}},0\right),\left({\xi }_{{}_{\lambda ,j}}^{+},0\right),\left(0,{\xi }_{{}_{\lambda ,j}}^{+}\right)|i=1,2,\cdots ,m,\lambda \in {\Sigma }^{+},j=1,2,\cdots ,{\beta }_{{}_{\lambda }}\right\}\end{array}\end{array}$ (32)
is composed of a basis of ${T}_{\left[\left(e,e\right)\right]}\left(\left(K×K\right)/M\right)$ .
It is completely analogous to the calculation in 22 and 25 -(27), then we obtain
$\begin{array}{cc}{\Psi }_{a}\left(\left({\eta }_{{}_{i}},0\right)\right)& =d{L}_{a}\left({\eta }_{{}_{i}}\right),\end{array}$ (33)
$\begin{array}{cc}{\Psi }_{a}\left(\left({\xi }_{{}_{\lambda ,j}}^{+},0\right)\right)& =d{L}_{a}\left(-sinh\left(\lambda \left(H\right)\right){\xi }_{{}_{\lambda ,j}}^{-}+cosh\left(\lambda \left(H\right)\right){\xi }_{{}_{\lambda ,j}}^{+}\right),\end{array}$ (34)
$\begin{array}{cc}{\Psi }_{a}\left(\left(0,{\xi }_{{}_{\lambda ,j}}^{+}\right)\right)& =d{L}_{a}\left(-{\xi }_{{}_{\lambda ,j}}^{+}\right).\end{array}$ (35)
$\begin{array}{}\end{array}$
Thus by the second conclusion in lemma 3.2 and the invariance of those chosen measures, there is a constant $C$ such that
$J\left(a\right)=C{\prod }_{\lambda \in {\Sigma }^{+}}|sinh\left(\lambda \left(H\right)\right){|}^{{\beta }_{{}_{\lambda }}}.$
It completes the proof of theorem 2.6 .
References
1. KNAPP, A. W., Lie Groups—Beyond an Introduction, 2nd edition, Boston, Birkhäuser,2002.
2. HELGASON, S., Groups and Geometric Analysis—Integral Geometry, Invariant Differetial Operators and Spherical Functions, Mathematical Surveys and Monographs, Vol. 83, New York, AMS, 2000.
3. An, J. P. and Wang, Z. D., A Generalization of Weyl Integration Formula, Institute of Mathematics, Peking University, Research Report No.18, 2004.
4. HELGASON, S., Differential Geometry, Lie Groups, and Symmetric Spaces, New tork, Acadymic Press, 1978. | 2022-09-29 02:27:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 367, "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": 0.9714914560317993, "perplexity": 1284.6200442114427}, "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-2022-40/segments/1664030335303.67/warc/CC-MAIN-20220929003121-20220929033121-00496.warc.gz"} |
http://math-mprf.org/journal/articles/id1482/ | A Spatial Epidemic Model with Site Contamination
#### T. Britton, M. Deijfen, F.M. Lopes
2018, v.24, Issue 1, 25-38
ABSTRACT
We introduce the effect of site contamination in a model for spatial epidemic spread and show that the presence of site contamination may have a strict effect on the model in the sense that it can make an otherwise subcritical process supercritical. Each site on $\mathbb{Z}^d$ is independently assigned a random number of particles and these then perform random walks restricted to bounded regions around their home locations. At time 0, the origin is infected along with all its particles. The infection then spread in that an infected particle that jumps to a new site causes the site along with all particles located there to be infected. Also, a healthy particle that jumps to a site where infection is present, either in that the site is infected or in the presence of infected particles, becomes infected. Particles and sites recover at rate $\lambda$ and $\gamma$, respectively, and then become susceptible to the infection again. We show that, for each given value of $\lambda$, there is a positive probability that the infection survives indefinitely if $\gamma$ is sufficiently small, and that, for each given value of $\gamma$, the infection dies out almost surely if $\lambda$ is large enough. Several open problems and modifications of the model are discussed, and some natural conjectures are supported by simulations.
Keywords: spatial epidemic, interacting particle system, phase transition, critical value | 2021-12-05 09:05:32 | {"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": 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": 0.5193189382553101, "perplexity": 462.53879308705467}, "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/1637964363149.85/warc/CC-MAIN-20211205065810-20211205095810-00171.warc.gz"} |
https://intelligencemission.com/free-electricity-in-which-country-free-electricity-how-to-make.html | But I will send you the plan for it whenever you are ready. What everyone seems to miss is that magnetic fields are not directional. Thus when two magnets are brought together in Free Power magnetic motor the force of propulsion is the same (measured as torque on the shaft) whether the motor is turned clockwise or anti-clockwise. Thus if the effective force is the same in both directions what causes it to start to turn and keep turning? (Hint – nothing!) Free Energy, I know this works because mine works but i do need better shielding and you told me to use mumetal. What is this and where do you get it from? Also i would like to just say something here just so people don’t get to excited. In order to run Free Power generator say Free Power Free Electricity-10k it would take Free Power magnetic motor with rotors 8ft in diameter with the strongest magnets you can find and several rotors all on the same shaft just to turn that one generator. Thats alot of money in magnets. One example of the power it takes is this.
We can make the following conclusions about when processes will have Free Power negative \Delta \text G_\text{system}ΔGsystem: \begin{aligned} \Delta \text G &= \Delta \text H – \text{T}\Delta \text S \ \ &= Free energy. 01 \dfrac{\text{kJ}}{\text{mol-rxn}}-(Free energy \, \cancel{\text K})(0. 022\, \dfrac{\text{kJ}}{\text{mol-rxn}\cdot \cancel{\text K})} \ \ &= Free energy. 01\, \dfrac{\text{kJ}}{\text{mol-rxn}}-Free energy. Free Power\, \dfrac{\text{kJ}}{\text{mol-rxn}}\ \ &= -0. Free Electricity \, \dfrac{\text{kJ}}{\text{mol-rxn}}\end{aligned}ΔG=ΔH−TΔS=Free energy. 01mol-rxnkJ−(293K)(0. 022mol-rxn⋅K)kJ=Free energy. 01mol-rxnkJ−Free energy. 45mol-rxnkJ=−0. 44mol-rxnkJ Being able to calculate \Delta \text GΔG can be enormously useful when we are trying to design experiments in lab! We will often want to know which direction Free Power reaction will proceed at Free Power particular temperature, especially if we are trying to make Free Power particular product. Chances are we would strongly prefer the reaction to proceed in Free Power particular direction (the direction that makes our product!), but it’s hard to argue with Free Power positive \Delta \text GΔG! Our bodies are constantly active. Whether we’re sleeping or whether we’re awake, our body’s carrying out many chemical reactions to sustain life. Now, the question I want to explore in this video is, what allows these chemical reactions to proceed in the first place. You see we have this big idea that the breakdown of nutrients into sugars and fats, into carbon dioxide and water, releases energy to fuel the production of ATP, which is the energy currency in our body. Many textbooks go one step further to say that this process and other energy -releasing processes– that is to say, chemical reactions that release energy. Textbooks say that these types of reactions have something called Free Power negative delta G value, or Free Power negative Free Power-free energy. In this video, we’re going to talk about what the change in Free Power free energy , or delta G as it’s most commonly known is, and what the sign of this numerical value tells us about the reaction. Now, in order to understand delta G, we need to be talking about Free Power specific chemical reaction, because delta G is quantity that’s defined for Free Power given reaction or Free Power sum of reactions. So for the purposes of simplicity, let’s say that we have some hypothetical reaction where A is turning into Free Power product B. Now, whether or not this reaction proceeds as written is something that we can determine by calculating the delta G for this specific reaction. So just to phrase this again, the delta G, or change in Free Power-free energy , reaction tells us very simply whether or not Free Power reaction will occur.
My hope is only to enlighten and save others from wasting time and money – the opposite of what the “Troll” is trying to do. Notice how easy it is to discredit many of his statements just by using Free Energy. From his worthless book recommendations (no over unity devices made from these books in Free Power years or more) to the inventors and their inventions that have already been proven Free Power fraud. Take the time and read ALL his posts and notice his tactics: Free Power. Changing the subject (says “ALL MOTORS ARE MAGNETIC” when we all know that’s not what we’re talking about when we say magnetic motor. Free Electricity. Almost never responding to Free Power direct question. Free Electricity. Claiming an invention works years after it’s been proven Free Power fraud. Free Power. Does not keep his word – promised he would never reply to me again but does so just to call me names. Free Power. Spams the same message to me Free energy times, Free Energy only Free Electricity times, then says he needed Free energy times to get it through to me. He can’t even keep track of his own lies. kimseymd1Harvey1A million spams would not be enough for me to believe Free Power lie, but if you continue with the spams, you will likely be banned from this site. Something the rest of us would look forward to. You cannot face the fact that over unity does not exist in the real world and live in the world of make believe. You should seek psychiatric help before you turn violent. jayanth 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 done so far! | 2021-01-27 01:15:59 | {"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": 0.8078935146331787, "perplexity": 1273.0718668248508}, "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-2021-04/segments/1610704804187.81/warc/CC-MAIN-20210126233034-20210127023034-00578.warc.gz"} |
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# According to a certain estimate, the depth N(t), in centimeters, of
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According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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According to a certain estimate, the depth N(t), in centimeters, of the water in a certain tank at t hours past 2:00 in the morning is given by N(t)= -20(t - 5)^2 + 500 for 0 ≤ t ≤ 10. According to this estimate, at what time in the morning does the depth of the water in the tank reach its maximum?
a) 5:30
b) 7:00
c) 7:30
d) 8:00
e) 9:00
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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24 Jan 2012, 01:45
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Baten80 wrote:
According to a certain estimate, the depth N(t), in centimeters, of the water in a certain tank at t hours past 2:00 in the morning is given by N(t)= -20(t-5)^²+500 for 0≤t≤10. According to this estimate, at what time in the morning does the depth of the water in the tank reach its maximum?
a) 5:30
b) 7:00
c) 7:30
d) 8:00
e) 9:00
Don't get bogged down by the dirty N(t) expression. Just think of it this way:
N(t) is a combination of two terms: a positive term (500) and a negative term ($$-20(t-5)^2$$).
To maximize N(t), I need to make the positive term as large as possible (It is a constant here so I cannot do much with it) and the absolute value of the negative term as small as possible. The smallest absolute value is 0. Can I make it 0? Yes, if I make t = 5, the negative term becomes 0 and N(t) is maximized. My answer must be 2:00 + 5 hrs i.e. 7:00.
Most of the maximum minimum questions on GMAT will require you to only think logically. The calculations involved will be minimum.
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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24 Jan 2012, 00:35
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Baten80 wrote:
puneetkr wrote:
the expression is N(t)= -20(t-5)^²+500
(of course valid after 2:00 in the morning)
"the depth would be maximum" means the value of the above expression should be maximum
or the value of square term (which has a negative 20 attached to it) should be minimum i.e. zero
the square part is zero at t=5
so the time at which the depth is maximum is 2:00 + 5 hrs= 7:00 (B)
if -20(t-5)^²=0
then t = 5
But Why is the 500 of the equation is not considered? Show your complete calculation.
According to a certain estimate, the depth N(t), in centimeters, of the water in a certain tank at t hours past 2:00 in the morning is given by N(t)= -20(t-5)^²+500 for 0≤t≤10. According to this estimate, at what time in the morning does the depth of the water in the tank reach its maximum?
A. 5:30
B. 7:00
C. 7:30
D. 8:00
E. 9:00
Consider this: $$-20(t-5)^2\leq{0}$$ hence $$500-20(t-5)^2$$ reaches its maximum when $$-20(t-5)^2=0$$, thus when $$t=5$$. 2:00AM+5 hours=7:00AM.
Hope it helps.
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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28 Nov 2015, 15:07
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Hi All,
These specific types of "limit" questions are relatively rare on Test Day, although you'll likely be tested on the concept at least once. Whenever you're asked to minimize or maximize a value, you should look to do something with the other "pieces" of the equation (usually involving maximizing or minimizing those pieces).
In the given equation, notice how you have two "parts": the -20(something) and a +500. Here, to MAXIMIZE the value of N(t), we have to minimize the "impact" that the -20(something) has on the +500. By making that first part equal 0, we'll be left with 0 + 500. Mathematically, we have to make whatever is inside the parentheses equal 0....
(T-5) = 0
T = 5
Since T represents the number of hours past 2:00am, we know that at 7:00am, the water will reach 500cm (the maximum value).
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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23 Jan 2012, 21:44
2
the expression is N(t)= -20(t-5)^²+500
(of course valid after 2:00 in the morning)
"the depth would be maximum" means the value of the above expression should be maximum
or the value of square term (which has a negative 20 attached to it) should be minimum i.e. zero
the square part is zero at t=5
so the time at which the depth is maximum is 2:00 + 5 hrs= 7:00 (B)
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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28 Jan 2012, 02:48
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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28 Nov 2015, 08:25
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Baten80 wrote:
According to a certain estimate, the depth N(t), in centimeters, of the water in a certain tank at t hours past 2:00 in the morning is given by N(t)= -20(t - 5)^2 + 500 for 0 ≤ t ≤ 10. According to this estimate, at what time in the morning does the depth of the water in the tank reach its maximum?0
N(t)= $$-20(t - 5)^2$$ + 500 ;
The red highlighted portion of the equation is the most critical part, further t = 2am + time past 2am
Now why $$-20(t - 5)^2$$ is important ;
Any value of t less than 5 will result in a negative value of $$-20(t - 5)^2$$
Lets check :
Time : 3am , t = 1 { t = 2am + time past 2am }
$$-20(t - 5)^2$$
$$-20(1 - 5)^2$$
$$-20(-4)^2$$
$$-20(16)$$
$$-320$$
When put in the final equation : $$-20(t - 5)^2$$ the result will be -320 + 500 => 180
Check a few more any value of t less than 5 will result in -ve value of $$-20(t - 5)^2$$ and will ultimately lead to a depth of water less than 500.
Our target is getting t = 5 ; since at t = 5
$$-20(t - 5)^2$$ will be 0 and the final equation $$-20(t - 5)^2$$ + 500 will be maximum ; ie 500
We already know t = 2am + time past 2am
So, 7 = 2am + 5 hours.
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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14 Apr 2018, 16:49
1
A useful trick to solve Maximum of any equation,with a single vraiable:
Just differentiate the equation with the variable and equate it to zero to find the solution. That solution is the value at which the euqation will have optimum value.
N(t)= -20(t-5)^²+500
d/dt N(t)= -40(t-5)=0
-40t=-200 => t=5
So, 7 = 2am + 5 hours.
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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23 Jan 2012, 21:52
puneetkr wrote:
the expression is N(t)= -20(t-5)^²+500
(of course valid after 2:00 in the morning)
"the depth would be maximum" means the value of the above expression should be maximum
or the value of square term (which has a negative 20 attached to it) should be minimum i.e. zero
the square part is zero at t=5
so the time at which the depth is maximum is 2:00 + 5 hrs= 7:00 (B)
if -20(t-5)^²=0
then t = 5
But Why is the 500 of the equation is not considered? Show your complete calculation.
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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23 Jan 2012, 22:01
this is the complete calculation
N(t)= -20(t-5)^²+500
now for any value of t we put in; we get some negative value of -20(t-5)^² (say -x)
so our expression is now N(t)=500-x
this expression would have maximum value only when x is minimum
we know the minimum value for a square term is "zero" and (x has a square term)
and that comes when t=5
i.e. when we put t=5 here we get N(t) = 500-0 = 500
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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27 Nov 2015, 21:58
Because if I plug in t = 3.5 (which is 5.30 (3.5 after the 2:00) , it gives me the maximum value of -20(t-5)^2 expression, which will be positive. Hence, the height will be at its max of 545. Hence, shouldn't the answer be A?
Thank you
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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28 Nov 2015, 07:33
herein wrote:
Because if I plug in t = 3.5 (which is 5.30 (3.5 after the 2:00) , it gives me the maximum value of -20(t-5)^2 expression, which will be positive. Hence, the height will be at its max of 545. Hence, shouldn't the answer be A?
Thank you
Also, if t = 3.5, then 500 - 20(3.5 - 5)^2 = 455, not 545.
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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04 Jun 2018, 02:38
valardohaeris wrote:
A useful trick to solve Maximum of any equation,with a single vraiable:
Just differentiate the equation with the variable and equate it to zero to find the solution. That solution is the value at which the euqation will have optimum value.
N(t)= -20(t-5)^²+500
d/dt N(t)= -40(t-5)=0
-40t=-200 => t=5
So, 7 = 2am + 5 hours.
Hi,
I used that method too.
I would like to know if there are always questions like this on every exam.
BR
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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05 Jun 2018, 09:32
Baten80 wrote:
According to a certain estimate, the depth N(t), in centimeters, of the water in a certain tank at t hours past 2:00 in the morning is given by N(t)= -20(t - 5)^2 + 500 for 0 ≤ t ≤ 10. According to this estimate, at what time in the morning does the depth of the water in the tank reach its maximum?
a) 5:30
b) 7:00
c) 7:30
d) 8:00
e) 9:00
Since -20(t - 5)^2, will be a nonpositive number, its maximum value is 0 when t = 5, and the maximum value of the function will then be:
N(5) = -20(5 - 5)^2 + 500 = 500
Thus, the maximum depth is at 2am + 5 hours = 7am.
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According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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14 Feb 2019, 06:05
Can someone explain why / how you would know not to simplify the original provided equation (-20(t-5)^2 + 500)? When you simplify the equation, you seems to get a different answer. Please see work below:
-20(t-5)^2 + 500
-20(t^2 - 10t + 25) + 500 <-- foil
-20t^2 + 200t - 500 + 500 <-- distributed the -20
-20t^2 + 200t
20t(t + 10t) <-- simplified formula
Given the 20t(t+10t) is simplified correctly, then the water level will continue to grow with every hour with the latest hour being the maximum. This is different than the answer provided, where t is at the max once 5 hours past.
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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14 Feb 2019, 13:42
rjacobson99 wrote:
Can someone explain why / how you would know not to simplify the original provided equation (-20(t-5)^2 + 500)? When you simplify the equation, you seems to get a different answer. Please see work below:
-20(t-5)^2 + 500
-20(t^2 - 10t + 25) + 500 <-- foil
-20t^2 + 200t - 500 + 500 <-- distributed the -20
-20t^2 + 200t
20t(t + 10t) <-- simplified formula
Given the 20t(t+10t) is simplified correctly, then the water level will continue to grow with every hour with the latest hour being the maximum. This is different than the answer provided, where t is at the max once 5 hours past.
Hi rjacobson99,
In the last "step" of you work, you have not properly accounted for the 'minus' sign. If you want to factor out "20t", then that's fine, but here's what you would be left with:
-20t^2 + 200t
(20t)(-t + 10)
(20t)(10 - t)
The maximum result will occur when t = 5.
GMAT assassins aren't born, they're made,
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According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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15 Feb 2019, 08:07
EMPOWERgmatRichC wrote:
rjacobson99 wrote:
Can someone explain why / how you would know not to simplify the original provided equation (-20(t-5)^2 + 500)? When you simplify the equation, you seems to get a different answer. Please see work below:
-20(t-5)^2 + 500
-20(t^2 - 10t + 25) + 500 <-- foil
-20t^2 + 200t - 500 + 500 <-- distributed the -20
-20t^2 + 200t
20t(t + 10t) <-- simplified formula
Given the 20t(t+10t) is simplified correctly, then the water level will continue to grow with every hour with the latest hour being the maximum. This is different than the answer provided, where t is at the max once 5 hours past.
Hi rjacobson99,
In the last "step" of you work, you have not properly accounted for the 'minus' sign. If you want to factor out "20t", then that's fine, but here's what you would be left with:
-20t^2 + 200t
(20t)(-t + 10)
(20t)(10 - t)
The maximum result will occur when t = 5.
GMAT assassins aren't born, they're made,
Rich
Hi Rich,
In the correct version of the formula (20t)(10-t), is there a way to determine the inflection point of t = 5 without having to plug in values from t = 1 to 6? I.e. is there a shortcut?
RJ
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
### Show Tags
15 Feb 2019, 11:55
rjacobson99 wrote:
EMPOWERgmatRichC wrote:
rjacobson99 wrote:
Can someone explain why / how you would know not to simplify the original provided equation (-20(t-5)^2 + 500)? When you simplify the equation, you seems to get a different answer. Please see work below:
-20(t-5)^2 + 500
-20(t^2 - 10t + 25) + 500 <-- foil
-20t^2 + 200t - 500 + 500 <-- distributed the -20
-20t^2 + 200t
20t(t + 10t) <-- simplified formula
Given the 20t(t+10t) is simplified correctly, then the water level will continue to grow with every hour with the latest hour being the maximum. This is different than the answer provided, where t is at the max once 5 hours past.
Hi rjacobson99,
In the last "step" of you work, you have not properly accounted for the 'minus' sign. If you want to factor out "20t", then that's fine, but here's what you would be left with:
-20t^2 + 200t
(20t)(-t + 10)
(20t)(10 - t)
The maximum result will occur when t = 5.
GMAT assassins aren't born, they're made,
Rich
Hi Rich,
In the correct version of the formula (20t)(10-t), is there a way to determine the inflection point of t = 5 without having to plug in values from t = 1 to 6? I.e. is there a shortcut?
RJ
Hi rjacobson99,
Unfortunately, manipulating the equation in the way that you did places the variable "t" in both pairs of parentheses, so there isn't an 'obvious' solution to maximize the value. However, in the original equation, there IS an obvious Number Property that you can use...
N(t) = -20(t - 5)^2 + 500 for 0 ≤ t ≤ 10.
In the given equation, notice how you have two "parts": the -20(something) and a +500. Here, to MAXIMIZE the value of N(t), we have to minimize the "impact" that the negative term - the -20(something) - has on the +500. By making that first part equal 0, we'll be left with 0 + 500. Mathematically, we have to make whatever is inside the parentheses equal 0....
(T-5) = 0
T = 5
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Rich
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
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08 Jun 2019, 08:48
This is beyond the scope of what's tested on the GMAT, but using derivatives, we can answer this in 15 seconds. The derivative will always be where the quadratic is at its maximum.
Given: $$N(t) = -20 (t-5)^2 + 500$$
Take the derivative: (t-5) = 0
Solve: t = 5
Plug into prompt: 2:00 am + 5 hours = 7:00 am
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Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink]
### Show Tags
16 Sep 2019, 07:40
Baten80 wrote:
According to a certain estimate, the depth N(t), in centimeters, of the water in a certain tank at t hours past 2:00 in the morning is given by N(t)= -20(t - 5)^2 + 500 for 0 ≤ t ≤ 10. According to this estimate, at what time in the morning does the depth of the water in the tank reach its maximum?
a) 5:30
b) 7:00
c) 7:30
d) 8:00
e) 9:00
According to a certain estimate, the depth N(t), in centimeters, of the water in a certain tank at t hours past 2:00 in the morning is given by N(t)= -20(t - 5)^2 + 500 for 0 ≤ t ≤ 10. According to this estimate, at what time in the morning does the depth of the water in the tank reach its maximum?
For t = 5 ; N(t) = 500 max
5 hours past 2:00 = 7:00
IMO B
Re: According to a certain estimate, the depth N(t), in centimeters, of [#permalink] 16 Sep 2019, 07:40
Display posts from previous: Sort by | 2020-03-31 11:09:38 | {"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": 0.664681077003479, "perplexity": 3039.557637759591}, "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/1585370500426.22/warc/CC-MAIN-20200331084941-20200331114941-00153.warc.gz"} |
https://www.chegg.com/homework-help/questions-and-answers/solve--please-include-detailedsteps-find-vertices-foci-asymptopes-x2-4-y2-16-1-apprently-h-q583067 | How do you solve this. Please include detailedsteps.
Find the vertices, foci, and asymptopes
x2/4 - y2/16 = 1
Apprently the hyperbola opens left and right. And theformula is x2/a2 -y2/b2 = 1, I am confused; don't you have torearrange this problem to match this formula? How do youknow if it is vertical or sideways hyperbola? I thougha2 makes the hyperbola vertical or horizontal. ?? | 2019-04-22 06:05:51 | {"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": 0.8076303005218506, "perplexity": 3615.970109708442}, "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/1555578544449.50/warc/CC-MAIN-20190422055611-20190422081611-00516.warc.gz"} |
http://math.stackexchange.com/questions/61774/how-do-i-solve-this-equation-involving-a-logarithm?answertab=active | # How do I solve this equation involving a logarithm?
I'm running in circles and I don't understand how to do this.
$$x\log(x) = 100$$
Where the $\log$ is in base $10$, I understand that $\log(y)=x$ is $10^x = y$. So is it the same for $x\log(x) = 100$? Would it be $10^{100}=x\cdot x$? It doesn't come out right when I do it, and it's clear that I have holes in my knowledge on logarithms, could someone please tell me my flaws and explain this to me?
-
Wait, suddenly I'm not so sure it is something for [algebra-precalculus] anymore. – Asaf Karagila Sep 4 '11 at 9:24
You will need the services of the Lambert function $W(x)$ to solve this equation. Briefly, the Lambert function is the inverse of the function $xe^x$: if $x=ye^y$, then $y=W(x)$.
To turn your equation into a form where the Lambert function's appearance becomes transparent, let's first turn everything into natural logarithms:
$$x\ln\,x=100\ln\,10$$
and then we make the left side a "little" complicated:
$$(\ln\,x)e^{\ln\,x}=100\ln\,10$$
We now recognize the Lambert form, and thus perform the inversion:
$$\ln\,x=W(100\ln\,10)$$
from which
$$x=e^{W(100\ln\,10)}\approx 56.961248432\dots$$
-
WOW. This is a mouth full. I did not think it would lead to so much math. – Doug Sep 4 '11 at 9:48
@J. M.: Do you think the [special-functions] tag should be added to the question? Or even create one for the Lambert function. – Américo Tavares Sep 4 '11 at 12:42
@Américo: I'm not quite sure if there should be a special-functions tag, much less a more specialized lambert-w tag. On the other hand, we now have quite a pile of questions where the Lambert function shows up in the solution... tell you what, let's create a tag for Lambert questions if at least five people upvote your comment. How's that sound? – Guess who it is. Sep 4 '11 at 12:47
It sounds good. – Américo Tavares Sep 4 '11 at 12:49
Here is the link for a meta thread about adding a tag for the Lambert W function: meta.math.stackexchange.com/q/2908/622 – Asaf Karagila Sep 4 '11 at 15:52
There is no way to solve $x\log_{10}x=100$ exactly using the methods of school algebra. There is a way, called Newton's Method, to get a solution to as many decimals as you want, using Calculus. Newton's Method is in a thousand intro Calculus textbooks, also a thousand websites. If you haven't done Calculus yet, you have something to look forward to.
-
If $x \log_b (y) = z$ then taking anti-logarithms you get $y^x = b^z$.
So in this case with $y=x$ and $b=10$ you get $x^x = 10^{100}$.
You will not find it easy to solve this explicitly for $x$; try reading about the Lambert W function or use numerical methods to get something just over 56.96.
-
Why is x the exponent for y? I don't find that really intuitive, could you please explain? – Doug Sep 4 '11 at 9:16
You forgot some braces there, I hope you did not mind. – Asaf Karagila Sep 4 '11 at 9:16
@Asaf: thank you – Henry Sep 4 '11 at 9:42
@Doug: it is a basic property of logarithms: see the power formula here – Henry Sep 4 '11 at 9:44
@Asaf: To the meta-cave! – The Chaz 2.0 Sep 4 '11 at 15:54 | 2015-05-27 10:27:18 | {"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": 0.9032915234565735, "perplexity": 747.6988373251327}, "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-2015-22/segments/1432207928923.85/warc/CC-MAIN-20150521113208-00240-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://codegolf.stackexchange.com/questions/37851/string-prototype-isrepeated/37879 | # String.prototype.isRepeated
UPDATE : isaacg's Pyth submission is the winner!
Many of you must have heard that there is a cooler version of JavaScript in town (read ES6) which has a method String.prototype.repeat so that you can do
"Hello, World!".repeat(3)
and get
"Hello, World!Hello, World!Hello, World!"
as the output.
Your job is to write a function or a program in a language of your choice which detects if a string has been gone under such transformation.
i.e. The input string can be represented as an exact n times repetition of a smaller string. The output (as function's return statement or STDOUT) should be truthy if the string can be or falsy if the string cannot be represented as a repetition of smaller string.
Some sample input:
"asdfasdfasdf" // true
"asdfasdfa" // false
"ĴĴĴĴĴĴĴĴĴ" // true
"ĴĴĴ123ĴĴĴ123" // true
"abcdefgh" // false
Note that the last input is false, thus n should be greater than 1
Complete rules
• Write a function/program in any language to input (via function argument/command line args/STDIN) a string
• Return/Print truthy value if the given string is formed via an exact repetition of a smaller string, repeating at least twice.
• Maximum size of the input string is ideally Infinity
• String can have all possible ASCII characters
• This is a so smallest code in characters wins.
• What should "" - the empty string - return? (It contains an infinite number of copies of the empty string.) – billpg Sep 17 '14 at 13:29
• @billpg falsy value – Optimizer Sep 17 '14 at 13:42
• Are you tie-breaking by votes? The common practice is earlier submission I think (well, the first one that got golfed down to the tying score). But I'm not sure that's written down as the default tie-breaker anywhere, so ultimately it's up to you. – Martin Ender Sep 17 '14 at 16:54
• Time between their posting is only 30 minutes. I will not consider that to be enough for winning :) . Since that time won't change now, but votes can, I went with votes – Optimizer Sep 17 '14 at 17:03
• This question should be renamed into xnor :) He is the man! – Silviu Burcea Sep 19 '14 at 8:52
# Pyth, 9
/:+zz1_1z
Or
}z:+zz1_1
These are both close translations of @xnor's python answer, except that they take input from STDIN and print it. The first is equivalent to:
z = input()
print((z+z)[1:-1].count(z))
0 for False, 1 for True.
The second line is equivalent to:
z = input()
print(z in (z+z)[1:-1])
False for False, True for True.
Pyth's official compiler had a bug related to the second one, which I just patched, so the first is my official submission.
• I was just searching for a way to inform you about that bug (the Boolean doesn't get printed). Didn't think of the first and using x was too long... – Dennis Sep 16 '14 at 0:13
• Yeah, the bug is fixed now. Also, if you want to report bugs, a good way might be to open an issue on the github site, here: github.com/isaacg1/pyth/issues – isaacg Sep 16 '14 at 0:31
• Oh, there it is. I don't know my way around GitHub, and I never noticed the navigation panel on the right... – Dennis Sep 16 '14 at 0:33
## Python (24)
lambda s:s in(s+s)[1:-1]
Checks if the string is a substring of itself concatenated twice, eliminating the first and last characters to avoid trivial matches. If it is, it must be a nontrivial cyclic permutation of itself, and thus the sum of repeated segments.
• A trivial translation into Golfscript yields 10 chars: ..+);(;\?) – Justin Sep 15 '14 at 22:49
• I don't quite understand how this works. Can you give a manually explained example of how this would handle a string? – Nzall Sep 16 '14 at 8:37
• @NateKerkhofs take abcabc. s+s turns it into abcabcabcabc. the [1:-1] chops of the two ends to yield bcabcabcabcab. and then s in ... tries to find abcabc as a substring of that. This substring can't be found in either of the original half, because they have both been shortened, so it must span both halves. In particular, it must have its own end before its start, which implies that it must be made up of identical (repeated) substrings. – Martin Ender Sep 16 '14 at 9:26
• You chop it after you double it. ab becomes abab becomes ba, so it returns false, while aa becomes aaaa becomes aa, which returns true. – histocrat Sep 16 '14 at 16:53
• @SargeBorsch It works just the same: qweqweqwe in weqweqweqweqweqw is True. – xnor Sep 17 '14 at 20:24
## Regex (ECMAScript flavour), 11 bytes
Sounds like a job for regex!
^([^]+)\1+$ Test it here. I've chosen ECMAScript, because it's the only flavour (I know) in which [^] matches any character. In all others, I'd either need a flag to change the behaviour of . or use [\s\S] which is three characters longer. Depending on how we're counting the flag, that could of course be a byte shorter. E.g. if we're counting pattern + flags (e.g. ignoring delimiters), the PCRE/Perl equivalent would be /^(.+)\1+$/s
Which is 10 bytes, ignoring the delimiters.
Test it here.
This matches only strings which consist of at least two repetitions of some substring.
Here is a full 26-byte ES6 function, but I maintain that regular expression submissions are generally valid:
f=s->/^([^]+)\1+$/.test(s) • ^(.+)\1+$ works for me, which is 9 bytes. It doesn't work for you ? – Optimizer Sep 16 '14 at 8:51
• @Optimizer Try a string with line breaks. – Martin Ender Sep 16 '14 at 8:53
• I tried asd\nasd\nasd\n . It works – Optimizer Sep 16 '14 at 8:55
• @Optimizer refiddle.com/refiddles/5417fb2475622d4df7e70a00 doesn't seem to work for me (and it shouldn't) – Martin Ender Sep 16 '14 at 8:56
• Yup, that doesn't work. Maybe it escapes the \ when I write \n manually – Optimizer Sep 16 '14 at 9:07
# CJam, 9
q__+)@+#)
Similar to xnor's idea.
q " Read input. ";
__+ " Duplicate twice and concatenate them together. ";
) " Remove the last character of the longer string. ";
@+ " Insert that character at the beginning of the shorter string. ";
#) " Find the shorter string in the longer string, and increase by one. ";
• +1 obligated to upvote this ahead of my own CJam answer – Digital Trauma Sep 15 '14 at 23:36
• Why the need for the final )? I think its reasonable to have -1 mean FALSE and >=0 mean TRUE – Digital Trauma Sep 15 '14 at 23:39
• @DigitalTrauma I think 0 is falsy in CJam... for operators like g and ?. – jimmy23013 Sep 15 '14 at 23:42
• @DigitalTrauma: Whether it's needed ultimately depends on the OP, but strictly speaking only zero is considered falsy in CJam. – Dennis Sep 15 '14 at 23:43
• @user23013 @Dennis But what about the # find operator? Surely the result of that is also "truthy" from the success vs failure perspective? – Digital Trauma Sep 15 '14 at 23:45
## APL, 11
2<+/x⍷,⍨x←⍞
Explanation
⍞ takes string input from screen
x← assigns to variable x
,⍨ concatenates the string with itself
x⍷ searches for x in the resulting string. Returns an array consisting of 1's in the starting position of a match and 0's elsewhere.
+/ sums the array
2< check if the sum is greater than 2 (as there will be 2 trivial matches)
# CJam, 10 bytes
I caught the CJam bug. My first answer, so probably can be golfed some more:
q__+(;);\#
Outputs -1 for FALSE and a number >=0 for TRUE
• Welcome to the club! – Dennis Sep 15 '14 at 23:36
# GolfScript, 10 bytes
..+(;);\?)
Yet another implementation of xnor's clever idea.
• Hahaha, I just posted this a minute ago: codegolf.stackexchange.com/questions/37851/… . I thought about posting it as an answer, but I thought that trivial translations are uninteresting. – Justin Sep 15 '14 at 22:53
• I even checked for new answers this time, but not for new comments... Your code is missing the ) though; when there's not match, it will print -1. If you're going to post that as an answer, I'll gladly delete mine. – Dennis Sep 15 '14 at 22:54
• I added the ) just before you posted your answer (I edited the comment) – Justin Sep 15 '14 at 22:55
• Improved version (in CJam): q__+)@+#). It doesn't work in GolfScript. – jimmy23013 Sep 15 '14 at 23:19
• @user23013: Not again. I was just going to post that! There are too many CJammers out there by now... :P – Dennis Sep 15 '14 at 23:21
# Python - 59 57
lambda s:any([s*n==s[:n]*len(s)for n in range(2,len(s))])
# Pure bash, 30 bytes
Simple port of @xnor's clever answer:
[[ ${1:1}${1:0: -1} =~ "$1" ]] Exit code is 0 for TRUE and 1 for FALSE: $ for s in 'Hello, World!Hello, World!Hello, World!' 'asdfasdfasdf' 'asdfasdfa' 'ĴĴĴĴĴĴĴĴĴ' 'ĴĴĴ123ĴĴĴ123' 'abcdefgh'; do echo "./isrepeated.sh "\"$s\"" returns$(./isrepeated.sh "$s"; echo$?)"; done
./isrepeated.sh "Hello, World!Hello, World!Hello, World!" returns 0
./isrepeated.sh "asdfasdfasdf" returns 0
./isrepeated.sh "asdfasdfa" returns 1
./isrepeated.sh "ĴĴĴĴĴĴĴĴĴ" returns 0
./isrepeated.sh "ĴĴĴ123ĴĴĴ123" returns 0
./isrepeated.sh "abcdefgh" returns 1
$ Note =~ within [[ ... ]] is the regex operator in bash. However "Any part of the pattern may be quoted to force it to be matched as a string". So as ai often the case with bash, getting quoting right is very important - here we just want to check for a string submatch and not a regex match. ## TI-BASIC - 32 I thought I'd try a tokenized language. Run with the string in Ans, returns 0 if false and the length of the repeated string if true. inString(sub(Ans+Ans,1,2length(Ans)-1),sub(Ans,length(Ans),1)+Ans Amazing how it's a one-liner. • But... but... I was going to use TI-BASIC :P +1 – Timtech Sep 16 '14 at 22:57 • @Timtech Well, note to anyone trying string manipulation in TI-BASIC: Don't try string manipulation in TI-BASIC. :P That was so hard to make and optimize. – Josiah Winslow Sep 17 '14 at 1:18 • Good idea. String manipulation is one of the hardest things to do. However, I've posted several answers like this, so I guess now you have a competitor ;) – Timtech Sep 17 '14 at 11:03 • Bring it on! :P – Josiah Winslow Sep 17 '14 at 23:39 # ECMAScript 6 (189) (function(){var S=String.prototype,r=S.repeat;S.isRepeated=function(){return!1};S.repeat=function(c){var s=new String(r.call(this,c));if(c>1)s.isRepeated=function(){return!0};return s}}()); < console.log("abc".isRepeated(),"abc".repeat(10).isRepeated()); > false true Surely this is the only valid solution? For example, the word (string) nana isn't necessarily created from "na".repeat(2) • "nana" isn't, but the question is not testing whether .repeat was used or not. Rather, whether the string is a repeated one or not – Optimizer Sep 18 '14 at 11:00 • I know, I was just trying to be a smart-arse :P – Mardoxx Sep 18 '14 at 11:02 # ECMAScript 6 (34 36) Another ES6 answer, but without using repeat and using xnor's trick: f=i=>(i+i).slice(1,-1).contains(i) Must be run in the console of a ES6-capable browser such as Firefox. # C 85 l,d;f(s){return l=strlen(s),strstr(d,strcpy(strcpy(d=alloca(l*2+1),s)+l,s)-1)-d-l+1;} It turned out to be quite long but external functions are always like that. It came to my mind that I could rewrite every string function replacing them by a loop or a recursive one. But in my experience it would turn out longer and frankly I don't want to try that out. After some research I saw solutions on high performance but not as clever (and short) as xnor's one. just to be original... i rewrote the same idea in c. explanation: int length, duplicate; int is_repetition(char *input) { // length = "abc" -> 3 length = strlen(input); // alloca because the function name is as long as "malloc" // but you don't have to call free() because it uses the stack // to allocate memory // duplicate = x x x x x x + x duplicate = alloca(length*2 + 1); // duplicate = a b c 0 x x + x strcpy(duplicate, input); // duplicate = a b c a b c + 0 strcpy(duplicate + length, input); if (strstr(duplicate,duplicate + length - 1) != duplicate + length - 1) // repetition // e.g. abab -> abababab -> aba[babab] // -> first occurence of [babab] is not aba[babab] // but a[babab]ab -> this is a repetition return 1; else // not repetition // e.g. abc -> abcabc -> ab[cabc] // -> first occurence of [cabc] is ab[cabc] // it matches the last "cabc" return 0; } # ECMAScript 6 (59 626773) Not a winner, but seems like there should at least be one answer actually in ES6 for this question that actually uses the repeat function: f=i=>[...i].some((_,j)=>i.slice(0,j).repeat(i.length/j)==i) Must be run in the console of a ES6-capable browser such as Firefox. It does a lot of unnecessary iterations, but why make it longer just to avoid that, right? • Edit #1: Saved a few bytes by converting it into a function. Thanks to Optimizer! • Edit #2: Thanks to hsl for the spread operator trick to save more bytes! • Edit #3: And thanks to Rob W. for another 3 bytes! • You can just convert it into a function to save more bytes there – Optimizer Sep 16 '14 at 18:24 • @Optimizer True, I guess it doesn't have to be "stdin". Your live up to your name :) – Ingo Bürk Sep 16 '14 at 18:27 • I haven't tested this, but you should be able to use the spread operator for [...i] instead of i.split('') – NinjaBearMonkey Sep 16 '14 at 18:37 • @hsl Crazy, that works. I didn't know the spread operator works like that. Originally I desperately tried to use it to create an array with the range 0..N. Thanks! – Ingo Bürk Sep 16 '14 at 18:39 • .slice(0,j) is one character shorter than .substr(0,j). Further, the conversion to an integer seems unnecessary |0 can be removed (using |0 actually reduces the usefulness of the method because it will fail for repetitions that exceed 2^31). – Rob W Sep 19 '14 at 18:08 # Jelly, 3 bytes ṙJċ Try it online! Same as this answer (maybe the later challenge is a generalization of this one?). # Java 8, 28 bytes s->s.matches("(?s)(.+)\\1+") Try it online. Explanation: Checks if the input-String matches the regex, where String#matches implicitly adds ^...$ to match the entire String.
Explanation of the regex itself:
^(s?)(.+)\1+$^ Begin of the string (s?) Enable DOTALL-mode, where . also matches new-lines ( Open capture group 1 .+ One or more characters ) Close capture group 1 \1+ Plus the match of the capture group 1, one or more times$ End of the string
So it basically checks if a substring is repeated two or more times (supporting new-lines). | 2019-08-23 18:51:57 | {"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": 0.3435145914554596, "perplexity": 3270.559492006059}, "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-2019-35/segments/1566027318952.90/warc/CC-MAIN-20190823172507-20190823194507-00129.warc.gz"} |
http://bodhiandmindyoga.org/cyclohexanone-sds-butca/define-roots-math-is-fun-bf1779 | Definition of 'radical' as used in math. Definition of the roots of a polynomial. Definition of Factor explained with real life illustrated examples. As far as fun math-sex jokes go (and who doesn’t love those), there are many. Math conjugates are a simple concept, but are valuable when simplifying some types of fractions. Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory), structure (), space (), and change (mathematical analysis). The opening line of any book should say, in the words of Stephen King, “Listen. The length of the horizontal bar is important. SplashLearn is an award winning math learning program used by more than 30 Million kids for fun math practice. The root of a number x is another number, which when multiplied by itself a given number of times, equals x. 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While we’re focusing a lot on imperfect square roots, we can extend that by looking at imperfect cube roots and by pushing kids to make the connection. Radical. About this unit . 4 questions. This is not meant to be a formal definition of square root like most terms we define on Dictionary.com, but is rather an informal word summary that hopefully touches upon the key aspects of the meaning and usage of square root that will help our users expand their word mastery. Math Open Reference. For example, to find the roots of We are trying find find what value (or values) of x will make it come out to zero. Definition of Extraneous Root - Math is Fun Solution Definition (Illustrated Mathematics Dictionary) Basic Math Definitions - Math is Fun A theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. Specifically, it describes the nature of any rational roots the polynomial might possess. Home Contact About Subject Index. Here are a couple of easy rules to begin with: But you knew that, right? Some of the most common irrational numbers are roots, such as the square root of 5 or the cube root of 7. Math Open Reference. Definition of Equation explained with real life illustrated examples. Square root definition is - a factor of a number that when squared gives the number. Audience. Pupils can be encouraged to extend the 9 times table further and so they might look at 135 558 etc. See more. Unit test. Root (of a number) The root of a number x is another number, which when multiplied by itself a given number of times, equals x. Test your understanding of Radical equations & functions with these 9 questions. Learning Objectives. It includes unlimited math lessons on number counting, addition, subtraction etc. For example, the third root (also called the cube root) of 64 is 4, because if you multiply three fours together you get 64: 4 × 4 × 4 = 64 . How to use preceding in a sentence. When using math root rules, first note that you can’t have a negative number under a square root or any other even number root — at least, not in basic calculus. See more. Orthogonal Matrix Definition We know that a square matrix has an equal number of rows and columns. The positive number square root is the principal square root. Definition of Simplify explained with real life illustrated examples. To pass out of sight, especially quickly; disappear. The three components of a radical expression are Radicand The thing you are finding the root of. Root (of a polynomial) The roots of a polynomial are those values of the variable that cause the polynomial to evaluate to zero. Math Open Reference. • a number that results from multiplying an integer by itself. The opening line of any book should say, in the words of Stephen King, “Listen. Synonym Discussion of preceding. Start test. Year 1 to Year 12 students . Definition of the root of a number as used in math. Ss; square number • a number which can be represented in the shape of a square. Quadratic definition, square. You can think of it as the "root" of the square or the number that was used to make the square. Learn its types, formulas, tree diagram, events, terms used and examples, solved problems along with video lessons. SplashLearn is an award winning math learning program used by more than 30 Million kids for fun math practice. b. Definition of Angle explained with real life illustrated examples. Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn. It has no generally accepted definition.. Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. Radical symbol The √ symbol that means "root of". I have these 1-inch cubes that I love to use. Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn. Topics: Active | Unanswered; Index » Help Me ! It includes unlimited math lessons on number counting, addition, subtraction etc. Mathematics definition is - the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations. For example the second root of … Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn. » square roots; Pages: 1 #1 2007-05-30 08:16:54. shocamefromebay Member Registered: 2007-05-30 Posts: 103. square roots. Section 5-2 : Zeroes/Roots of Polynomials. An expression that uses a root, such as square root, cube root. SplashLearn is an award winning math learning program used by more than 30 Million kids for fun math practice. Our mission is to provide a free, world-class education to anyone, anywhere. are added together they come to 9. 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Remember, when a negative number is multiplied by itself (also a negative number) the product is positive. Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn. The square root is just the opposite of the square. See Synonyms at disappear. Plus, get practice tests, quizzes, and personalized coaching to help you succeed. How many times a particular number is a zero for a given polynomial.For example, in the polynomial function f(x) = (x – 3) 4 (x – 5)(x – 8) 2, the zero 3 has multiplicity 4, 5 has multiplicity 1, and 8 has multiplicity 2.Although this polynomial has only three zeros, we say that it … A square matrix with real numbers or elements is said to be an orthogonal matrix, if its transpose is equal to its inverse matrix or we can say, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix. Preceding definition is - existing, coming, or occurring immediately before in time or place. For example, using function in the sense of multivalued functions, just as the square root function y = √ x could be defined from y 2 = x, the function y = arcsin(x) is defined so that sin(y) = x. It includes unlimited math lessons on number counting, addition, subtraction etc. SplashLearn is an award winning math learning program used by more than 30 Million kids for fun math practice. Learn common math terms starting with letter C We say that $$x = r$$ is a root or zero of a polynomial, $$P\left( x \right)$$, if $$P\left( r \right) = 0$$. Home Contact About Subject Index. Roots. Discussion about math, puzzles, games and fun. Practice. They have to build a representation of their numbers’ cube root. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ ° Index; User list; Rules; Search; Register; Login; You are not logged in. Correlation Analysis. See definition of root. How to use square root in a sentence. The rational root theorem describes a relationship between the roots of a polynomial and its coefficients. Correlation analysis is applied in quantifying the association between two continuous variables, for example, an dependent and independent variable or … SplashLearn is an award winning math learning program used by more than 30 Million kids for fun math practice. This topic covers: - Solving radical equations - Graphing radical functions. Should say, in the shape of a polynomial is that was used to make the square root is principal... Used by more than 30 Million define roots math is fun for fun math practice as used in math try yourself & functions these... Three components of a polynomial and its coefficients roots, such as square root of '' - radical! T love those ), there are many about math, puzzles, and. Further and so they might look at 135 558 etc. math, puzzles, games fun! 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That a square when multiplied by itself ( also a negative number ) the product is positive get practice,! A random number of rows and columns Active | Unanswered ; Index » Help Me and! Work through some examples followed by problems to try yourself of Angle explained with real illustrated... They often notice that when squared gives the number most common irrational numbers are roots, such the... Irrational numbers are roots, such as the root of '' education to anyone, anywhere the roots a... Terms used and examples, solved problems along with video lessons and its coefficients polynomial is (,! Used in math integer by itself a given number of cubes between 1 and 30 the root of -. Math, puzzles, games and fun ’ ll start off this section by defining just what root... Of each multiple ( 9, 18, 27, 36, 45, etc... | 2021-05-16 01:41:12 | {"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": 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": 0.5585157871246338, "perplexity": 1189.5951285862031}, "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-2021-21/segments/1620243991659.54/warc/CC-MAIN-20210516013713-20210516043713-00165.warc.gz"} |
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## G = C3×C18.D4order 432 = 24·33
### Direct product of C3 and C18.D4
Series: Derived Chief Lower central Upper central
Derived series C1 — C18 — C3×C18.D4
Chief series C1 — C3 — C9 — C18 — C2×C18 — C6×C18 — C6×Dic9 — C3×C18.D4
Lower central C9 — C18 — C3×C18.D4
Upper central C1 — C2×C6 — C22×C6
Generators and relations for C3×C18.D4
G = < a,b,c,d | a3=b18=c4=1, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b9c-1 >
Subgroups: 342 in 134 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C18, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, Dic9, C2×C18, C2×C18, C2×C18, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C3×C18, C3×C18, C3×C18, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C3×Dic9, C6×C18, C6×C18, C6×C18, C18.D4, C3×C6.D4, C6×Dic9, C2×C6×C18, C3×C18.D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, D9, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, Dic9, D18, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C3×D9, C2×Dic9, C9⋊D4, C6×Dic3, C3×C3⋊D4, C3×Dic9, C6×D9, C18.D4, C3×C6.D4, C6×Dic9, C3×C9⋊D4, C3×C18.D4
Smallest permutation representation of C3×C18.D4
On 72 points
Generators in S72
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 21 53 64)(2 20 54 63)(3 19 37 62)(4 36 38 61)(5 35 39 60)(6 34 40 59)(7 33 41 58)(8 32 42 57)(9 31 43 56)(10 30 44 55)(11 29 45 72)(12 28 46 71)(13 27 47 70)(14 26 48 69)(15 25 49 68)(16 24 50 67)(17 23 51 66)(18 22 52 65)
(1 55 10 64)(2 72 11 63)(3 71 12 62)(4 70 13 61)(5 69 14 60)(6 68 15 59)(7 67 16 58)(8 66 17 57)(9 65 18 56)(19 37 28 46)(20 54 29 45)(21 53 30 44)(22 52 31 43)(23 51 32 42)(24 50 33 41)(25 49 34 40)(26 48 35 39)(27 47 36 38)
G:=sub<Sym(72)| (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,21,53,64)(2,20,54,63)(3,19,37,62)(4,36,38,61)(5,35,39,60)(6,34,40,59)(7,33,41,58)(8,32,42,57)(9,31,43,56)(10,30,44,55)(11,29,45,72)(12,28,46,71)(13,27,47,70)(14,26,48,69)(15,25,49,68)(16,24,50,67)(17,23,51,66)(18,22,52,65), (1,55,10,64)(2,72,11,63)(3,71,12,62)(4,70,13,61)(5,69,14,60)(6,68,15,59)(7,67,16,58)(8,66,17,57)(9,65,18,56)(19,37,28,46)(20,54,29,45)(21,53,30,44)(22,52,31,43)(23,51,32,42)(24,50,33,41)(25,49,34,40)(26,48,35,39)(27,47,36,38)>;
G:=Group( (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,21,53,64)(2,20,54,63)(3,19,37,62)(4,36,38,61)(5,35,39,60)(6,34,40,59)(7,33,41,58)(8,32,42,57)(9,31,43,56)(10,30,44,55)(11,29,45,72)(12,28,46,71)(13,27,47,70)(14,26,48,69)(15,25,49,68)(16,24,50,67)(17,23,51,66)(18,22,52,65), (1,55,10,64)(2,72,11,63)(3,71,12,62)(4,70,13,61)(5,69,14,60)(6,68,15,59)(7,67,16,58)(8,66,17,57)(9,65,18,56)(19,37,28,46)(20,54,29,45)(21,53,30,44)(22,52,31,43)(23,51,32,42)(24,50,33,41)(25,49,34,40)(26,48,35,39)(27,47,36,38) );
G=PermutationGroup([[(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,61,67),(56,62,68),(57,63,69),(58,64,70),(59,65,71),(60,66,72)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,21,53,64),(2,20,54,63),(3,19,37,62),(4,36,38,61),(5,35,39,60),(6,34,40,59),(7,33,41,58),(8,32,42,57),(9,31,43,56),(10,30,44,55),(11,29,45,72),(12,28,46,71),(13,27,47,70),(14,26,48,69),(15,25,49,68),(16,24,50,67),(17,23,51,66),(18,22,52,65)], [(1,55,10,64),(2,72,11,63),(3,71,12,62),(4,70,13,61),(5,69,14,60),(6,68,15,59),(7,67,16,58),(8,66,17,57),(9,65,18,56),(19,37,28,46),(20,54,29,45),(21,53,30,44),(22,52,31,43),(23,51,32,42),(24,50,33,41),(25,49,34,40),(26,48,35,39),(27,47,36,38)]])
126 conjugacy classes
class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6AE 9A ··· 9I 12A ··· 12H 18A ··· 18BK order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 1 1 2 2 1 1 2 2 2 18 18 18 18 1 ··· 1 2 ··· 2 2 ··· 2 18 ··· 18 2 ··· 2
126 irreducible representations
dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Dic3 D6 D9 C3×S3 C3×D4 C3⋊D4 Dic9 D18 C3×Dic3 S3×C6 C3×D9 C9⋊D4 C3×C3⋊D4 C3×Dic9 C6×D9 C3×C9⋊D4 kernel C3×C18.D4 C6×Dic9 C2×C6×C18 C18.D4 C6×C18 C2×Dic9 C22×C18 C2×C18 C2×C62 C3×C18 C62 C62 C22×C6 C22×C6 C18 C3×C6 C2×C6 C2×C6 C2×C6 C2×C6 C23 C6 C6 C22 C22 C2 # reps 1 2 1 2 4 4 2 8 1 2 2 1 3 2 4 4 6 3 4 2 6 12 8 12 6 24
Matrix representation of C3×C18.D4 in GL4(𝔽37) generated by
1 0 0 0 0 1 0 0 0 0 26 0 0 0 0 26
,
10 0 0 0 5 26 0 0 0 0 28 0 0 0 0 4
,
31 3 0 0 0 6 0 0 0 0 0 1 0 0 36 0
,
31 3 0 0 13 6 0 0 0 0 0 1 0 0 36 0
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,26,0,0,0,0,26],[10,5,0,0,0,26,0,0,0,0,28,0,0,0,0,4],[31,0,0,0,3,6,0,0,0,0,0,36,0,0,1,0],[31,13,0,0,3,6,0,0,0,0,0,36,0,0,1,0] >;
C3×C18.D4 in GAP, Magma, Sage, TeX
C_3\times C_{18}.D_4
% in TeX
G:=Group("C3xC18.D4");
// GroupNames label
G:=SmallGroup(432,164);
// by ID
G=gap.SmallGroup(432,164);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^18=c^4=1,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^9*c^-1>;
// generators/relations
×
𝔽 | 2021-07-25 17:57:30 | {"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": 0.9819709062576294, "perplexity": 4743.065003665581}, "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-2021-31/segments/1627046151760.94/warc/CC-MAIN-20210725174608-20210725204608-00460.warc.gz"} |
http://mathhelpforum.com/pre-calculus/19675-graph-equation-symmetric.html | # Thread: Graph of the equation is symmetric
1. ## Graph of the equation is symmetric
I need help solving this problem, can anyone do a step by step guide on how to do this?
Determine whether the graph of the equation is symmetric with respect to the line Y = -X
1) Y = 2X - 1
2. One way of checking it is by slopes, because both x = -y and y = 2x +1 are straight lines only.
If y = 2x +1 were symmetrical with respect to y = -x, then y = 2x +1 should be perpendicular to y = -x. If not, then there is no symmetry.
y = -x
slope, m1 = -1
y = 2x +1
slope, m2 = 2
To be perpendicular, their slopes must be the negative reciprocals.
m2 = -1/m1
2 = -1 / -1
2 = 1
False, so the two lines are not perpendicular, and so y = 2x +1 is not symmetrical wirh respect to y = -x.
-------------------------
Another way.
y = -x -------------axis of symmetry.
y = 2x +1 --------(i)
If (i) is symmetrical, when we swap the x and y into (i), we should get an equation similar or equal to (i).
-x = 2(-y) +1
-x = -2y +1
2y = x +1
y = (1/2)(x +1) -------not the same as (i), so, no symmetry.
3. Originally Posted by ticbol
y = -x -------------axis of symmetry.
y = 2x +1 --------(i)
If (i) is symmetrical, when we swap the x and y into (i), we should get an equation similar or equal to (i).
-x = 2(-y) +1
-x = -2y +1
How come the -x remained as a -x. While the 2y turned into a -2y and the -1 turned into a 1?
4. Originally Posted by Brazuca
How come the -x remained as a -x. While the 2y turned into a -2y and the -1 turned into a 1?
what are you talking about? he replaced y with -x and x with -y and solved for y. that's it.
5. Originally Posted by Jhevon
what are you talking about? he replaced y with -x and x with -y and solved for y. that's it.
I don't understand.
If Y = -X then wouldn't the -X have to turn into a X for the Y to turn into a -Y?
-x = 2(-y) +1
-x = -2y +1
Wouldn't the problem have to turn into this?
-x = 2y + 1
x = 2(-y) +1
x = -2y + 1
6. Originally Posted by Brazuca
I don't understand.
we have
$y = -x$ which is also saying that $x = -y$, this is line (1)
then we are given $y = 2x - 1$ ........line (2)
to find if $y = 2x - 1$ is symmetric with respect to line one, plug in $y = -x$ and $x = -y$ as line (1) directed. if we simplify and get the original formula for line (2), then line (2) is symmetric with respect to line (1)
${\color {red}y } = 2 {\color {blue}x} - 1$
Plug in $y = -x$ and $x = -y$, we get:
${\color {red}-x } = 2({\color{blue}-y}) - 1$
solving for $y$, we get:
$y = \frac 12x - \frac 12$
which is not the original formula for line (2), so line (2) is NOT symmetric with respect to line (1)
7. Originally Posted by Jhevon
we have
$y = -x$ which is also saying that $x = -y$, this is line (1)
Thanks, I had to read everything you posted step by step to understand that y = -x then x = -y.
I always looked at it as y = -x then -x = y. I was not switching the signs.
8. Originally Posted by Brazuca
Thanks, I had to read everything you posted step by step to understand that y = -x then x = -y.
I always looked at it as y = -x then -x = y. I was not switching the signs.
if you multiply both sides of the equation y = -x by -1 you get -y = x | 2016-09-28 00:57:37 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 14, "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": 0.5908393263816833, "perplexity": 932.9517905947403}, "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-2016-40/segments/1474738661289.41/warc/CC-MAIN-20160924173741-00081-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://www.maplesoft.com/support/help/Maple/view.aspx?path=worksheet/reference/contextmaplein | Context Bar for Math Input - Maple Programming Help
Home : Support : Online Help : Create Maple Worksheets : Enter Expressions : worksheet/reference/contextmaplein
Context Bar for Math Input
Switch the input display between 2-D Math and Maple Input notation.
At a Maple prompt, you can toggle the input mode by selecting Math or Text from the context bar.
Note: Set the input display mode from the Display tab of the Options dialog. See Options>Display.
In document mode, the context bar lets you switch between entering plain text and math. See Select Entry Mode. | 2017-03-29 21:03:44 | {"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": 0.9180895090103149, "perplexity": 7283.729033985641}, "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-2017-13/segments/1490218191396.90/warc/CC-MAIN-20170322212951-00418-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://fabricebaudoin.wordpress.com/2012/09/28/lecture-24-burkholder-davis-gundy-inequalities/ | ## Lecture 24. Burkholder-Davis-Gundy inequalities
In this section, we study some of the most important martingale inequalities: The BurkholderDavisGundy inequalities. Interestingly, the range of application of these inequalities is very large and they play an important role in harmonic analysis and the study of singular integrals (see for instance the nice survey by my colleague Pr. Bañuelos). These inequalities admit several proofs. We present here a proof using Itō’s formula and an interesting domination inequality which is due to Lenglart. For an alternative proof, you may refer to the original approach by Burkholder-Davis-Gundy.
We admit without proof, the following domination inequality which is is due to Lenglart.
Proposition.(Lenglart) Let $(N_t)_{t \ge 0}$ be a positive adapted right-continuous process and $(A_t)_{t \ge 0}$ be an increasing process. Assume that for every bounded stopping time $\tau$, $\mathbb{E} (N_\tau \mid \mathcal{F}_0 ) \le \mathbb{E} (A_\tau \mid \mathcal{F}_0 )$. Then, for every $k \in (0,1)$, $\mathbb{E} \left( \left(\sup_{0 \le t \le T} N_t \right)^k \right) \le \frac{2-k}{1-k} \mathbb{E} \left( A_T^k\right).$
We shall use this lemma to prove the following
Theorem. (Burkholder-Davis-Gundy inequalities) Let $T > 0$ and $(M_t)_{ 0 \le t \le T}$ be a continuous local martingale such that $M_0=0$. For every $0 < p < \infty$, there exist universal constants $c_p$ and $C_p$, independent of $T$ and $(M_t)_{ 0 \le t \le T}$ such that
$c_p \mathbb{E} \left( \langle M\rangle_T^{\frac{p}{2} } \right)\le \mathbb{E}\left(\left(\sup_{0 \le t \le T} |M_t|\right)^p \right) \le C_p \mathbb{E} \left( \langle M\rangle_T^{\frac{p}{2} } \right).$
Proof. By stopping it is enough to prove the result for bounded $M$. Let $q \ge 2$. From Itō’s formula we have
$d |M_t|^q =q|M_t|^{q-1} \mathbf{sgn}(M_t) dM_t +\frac{1}{2} q(q-1) | M_t |^{q-2} d \langle M \rangle _t$
$=q \mathbf{sgn}(M_t)|M_t|^{q-1}dM_t+ \frac{1}{2} q(q-1) | M_t |^{q-2}d\langle M \rangle_t.$
As a consequence of the Doob’s stopping theorem, we get that for every bounded stopping time $\tau$,
$\mathbb{E} \left( |M_\tau|^q \mid \mathcal{F}_0 \right) \le \frac{1}{2} q(q-1) \mathbb{E} \left( \int_0^\tau | M_t |^{q-2} d\langle M \rangle_t \mid \mathcal{F}_0 \right).$
From the Lenglart’s domination inequality, we deduce then that for every $k \in (0,1)$,
$\mathbb{E} \left( \left(\sup_{0 \le t \le T} |M_t|^q \right)^k \right) \le \frac{2-k}{1-k} \left( \frac{1}{2} q(q-1)\right)^k \mathbb{E} \left(\left( \int_0^T | M_t |^{q-2} d\langle M \rangle_t\right)^k \right).$
We now bound
$\mathbb{E} \left(\left( \int_0^T | M_t |^{q-2} d \langle M \rangle _t\right)^k \right)$
$\le \mathbb{E} \left(\left(\sup_{0 \le t \le T} |M_t| \right)^{k(q-2)}\left( \int_0^T d\langle M \rangle _t\right)^k \right)$
$\le \mathbb{E} \left(\left(\sup_{0 \le t \le T} |M_t| \right)^{kq} \right)^{1-\frac{2}{q}} \mathbb{E} \left( \langle M \rangle _T^{\frac{kq}{2}} \right)^{\frac{2}{q}}.$
As a consequence, we obtain:
$\mathbb{E} \left( \left(\sup_{0 \le t \le T} |M_t|^q \right)^k \right) \le \frac{2-k}{1-k} \left( \frac{1}{2} q(q-1)\right)^k \mathbb{E} \left(\left(\sup_{0 \le t \le T} |M_t| \right)^{kq} \right)^{1-\frac{2}{q}} \mathbb{E} \left(\langle M \rangle_T^{\frac{kq}{2}} \right)^{\frac{2}{q}}.$
Letting $p=qk$ yields the claimed result, that is
$\mathbb{E}\left(\left(\sup_{0 \le t \le T} |M_t|\right)^p \right) \le C_p \mathbb{E} \left( \langle M\rangle_T^{\frac{p}{2} } \right).$
We proceed now to the proof of the left hand side inequality. We have,
$M_t^2 =\langle M \rangle_t +2\int_0^t M_s dM_s.$
Therefore, we get
$\mathbb{E} \left( \langle M\rangle_T^{\frac{p}{2} } \right) \le A_p \left( \mathbb{E}\left(\left(\sup_{0 \le t \le T} |M_t|\right)^p \right) + \mathbb{E}\left(\sup_{0 \le t \le T}\ \left| \int_0^t M_s dM_s\right|^{p/2} \right) \right).$
By using the previous argument, we now have
$\mathbb{E}\left(\sup_{0 \le t \le T}\ \left| \int_0^t M_s dM_s\right|^{p/2} \right) \le B_p \mathbb{E}\left( \left( \int_0^T M^2_s d\langle M\rangle_s\right)^{p/4} \right)$
$\le B_p \mathbb{E}\left(\left(\sup_{0 \le t \le T} |M_t|\right)^{p/2} \langle M \rangle_T^{p/4} \right)$
$\le B_p \mathbb{E}\left(\left(\sup_{0 \le t \le T} |M_t|\right)^{p}\right)^{1/2} \mathbb{E} \left( \langle M \rangle_T^{p/2} \right)^{1/2}.$
As a conclusion, we obtained
$\mathbb{E} \left( \langle M\rangle_T^{\frac{p}{2} } \right) \le A_p \left( \mathbb{E}\left(\left(\sup_{0 \le t \le T} |M_t|\right)^p \right) + B_p \mathbb{E}\left(\left(\sup_{0 \le t \le T} |M_t|\right)^{p}\right)^{1/2} \mathbb{E} \left( \langle M \rangle_T^{p/2} \right)^{1/2} \right).$
This is an inequality of the form $x^2 \le A_p \left( y^2 +B_p xy\right)$, which easily implies
$c_p x^2 \le y^2$, thanks to the inequality $2xy \le \frac{1}{\delta} x^2+\delta y^2$, with a conveniently chosen $\delta$ $\square$
This entry was posted in Stochastic Calculus lectures. Bookmark the permalink. | 2017-09-20 03:49:02 | {"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": 40, "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": 0.9776404500007629, "perplexity": 234.10724779110467}, "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/1505818686169.5/warc/CC-MAIN-20170920033426-20170920053426-00511.warc.gz"} |
https://knowen.org/nodes/890?path=19-829-887-889-890 | Now you are in the subtree of Special Topics in Many-Body Theory, Spring 2016 project.
# Ten facts to remember about BCS
(There was once a late-night comedian named David Letterman who was known for his "Top Ten" lists. This is like that but any humor is accidental.)
1. The variational ground state we found for a model pairing Hamiltonian,
$$$$|\Psi_{BCS}\rangle = \prod_k (u_k + v_k b^\dagger_k) |0\rangle, \quad {u_k}^2 = \frac{1}{2}\left(1 + {\epsilon_k - \mu \over E_k}\right), \quad {v_k}^2 = \frac{1}{2}\left(1 - {\epsilon_k - \mu \over E_k}\right),$$$$
has a gap to fermionic excitations. The excitation at momentum $k$ has
energy
$$$$E_k =\sqrt{(\epsilon_k - \mu)^2 + {\Delta_k}^2}.$$$$
Trying to add or subtract an electron gives
$$$$c^\dagger_{p\uparrow} |\Psi_{BCS}\rangle = u_p \prod_{k \not = p} (u_k + v_k b^\dagger_k) |0\rangle = u_p |\psi_{p^\uparrow}\rangle,\quad c_{-p\downarrow} |\Psi_{BCS}\rangle = -v_p \prod_{k \not = p} (u_k + v_k b^\dagger_k) |0\rangle = -v_p |\psi_{p^\uparrow}\rangle.$$$$
Hence we defined normalized, fermionic creation and annihilation operators
$$$$\gamma_{p\uparrow}^\dagger = {u_p} c_{p\uparrow}^\dagger - {v_p} c_{-p \downarrow}, \quad \gamma_{-p\downarrow} = {v_p} c_{p\uparrow}^\dagger + {u_p} c_{-p \downarrow}.$$$$
Note that each of these adds momentum $p$ and spin up.
2. The particle number is uncertain in the BCS ground state as written above, and the Bogoliubov excitations do not have well-defined particle number (or charge). Schrieffer argues that one shouldn't be troubled by the uncertainty in the ground state because the relative fluctuations are small. A different interpretation is that the particle number of an isolated system would be well-defined: the particle number is only uncertain when a supercurrent flows, i.e., when it is possible to measure the superconducting phase.
3. The Ginzburg-Landau equation
$$$${\hbar^2 n^* \over 2 m^*} \left[ \nabla + {i e^* \over \hbar c} {\bf A}({\bf r})\right]^2 \psi({\bf r}) + a(T) \psi({\bf r}) + b(T) |\psi({\bf r})|^2 \psi({\bf r}) = 0.$$$$
describes the center-of-mass eigenstate of the Cooper pairs. In the BCS form written above, there are $N/2$ Cooper pairs that all have total momentum zero and zero spin. In a vector potential, say, the center-of-mass wavefunction would obey the Ginzburg-Landau eqn.
4. The main error in the unrealistic interaction model (the pairing form for the Coulomb term) is the neglect of long-range interactions, which will turn out to be responsible for the phonon gap in an s-wave superconductor in three dimensions (the Anderson-Higgs effect).
5. With a specific, even more unrealistic model for the pair potential,
$$$$V_{kk^\prime} = \begin{cases} V_0 & {\rm if}\ |\epsilon_k - \mu|\ {\rm and}\ |\epsilon_{k^\prime}-\mu| < \omega_c \cr 0&{\rm otherwise} \end{cases}$$$$
we could exactly solve the gap equation and found $\Delta \approx 2 \omega_c e^{-1/(N(0) V)}$.
6. The gap, which is nonperturbative in the coupling strength $V$ even in more realistic models of the potential, is of the same order of magnitude as the superconducting transition temperature $T_c \approx (2 \Delta) / 3.5$. There is really just one "small" energy scale in the theory.
7. The gap is not necessary for superconductivity. For instance, suppose we tried to argue that $T_c$ and $\Delta$ had to be of the same order because once the temperature was larger than $2 \Delta$, it would be easy to thermally excite quasiparticles. The counterexample is that $p$-wave and $d$-wave superconductors, which have gapless quasiparticles, can have nonzero superconducting transition temperatures. Another way to create gapless superconductivity is in a disordered system. Magnetic impurities (which break time-reversal symmetry) tend to destroy superconducting order much more rapidly than nonmagnetic impurities.
8. Suppose we try to make a length scale associated with the gap. A choice with the right units is
$$$$\xi \sim {\hbar v_F \over \Delta}.$$$$
The penetration depth is rather different and has no $\Delta$:
$$$$\lambda = \left({m c^2 \over 4 \pi n_s e^2}\right)^{1/2}.$$$$
The physical significance of $\xi$, known as (Pippard's) coherence length, is that it corresponds to the size of a Cooper pair, in the sense that the BCS wavefunction can be rewritten in the real-space form
$$$$|\Psi_{BCS}\rangle = {\cal A} \phi(x_1 - x_2) \phi(x_3 - x_4) \phi(x_5 - x_6)\ldots,$$$$
and the pair wavefunction $\phi$ falls off exponentially beyond the scale $\xi$ (with a considerable amount of algebra, it can be shown that $\phi(r) = \sum_k g_k e^{i k r}$.
(This rewriting was noticed by Dyson some time after the original BCS theory.) You should remind yourself why the coherence length and penetration depth are very different scales (cf. Ashcroft and Mermin or some similar text): note in particular that the coherence length gets shorter as the gap gets larger and $T_c$ increases. The penetration depth is essentially a measure of the superfluid density, and can be determined from the Ginzburg-Landau equation, while the coherence length cannot. Note that high-temperature superconductors have short coherence lengths.
9. The pair wavefunction introduced in the previous item has a symmetry related to that of the gap $\Delta_k$. For instance, if $\Delta_k$ has p-like or d-like symmetry, then the pair wavefunction vanishes when the two points come together. For this reason, $p$-like and $d$-like symmetries are favored when the two-particle potential is repulsive at short distances but attractive at long distances, as for the neutral atoms of He$^3$.
10. The Ginzburg-Landau equation, like single-particle quantum mechanics, is invariant under the gauge transformation
$$$$V \rightarrow V - {1 \over c} {\partial \chi \over \partial t},\quad A \rightarrow A + \nabla \chi,\quad \Phi \rightarrow \Phi + {2 e \chi \over \hbar c}.$$$$
This has fundamental consequences like the Josephson effect and the quantization of flux
through a superconducting ring. The notion of Cooper pairing explained the known experimental fact that $2e$ appears here rather than $e$.
Recall that a superconductor is type I (only one critical field) essentially if $\xi > \lambda$, and type II if $\lambda > \xi$. A simple way to express the difference is that the penetration depth depends on the energy cost to vary the magnetic field, while the coherence length depends on the energy cost to bend the phase. For example, around a vortex core the wavefunction must twist by $2 \pi$, so the core size (the region of no superconductivity) is at least of order $\xi$.
Because there are two lengths, in a type II superconductor there are two critical fields: $H_{c1}$, when the field begins to penetrate macroscopically into the superconductor (leading to a "mixed state" of vortices), and $H_{c2}$, when the vortex cores begin to overlap and superconductivity is destroyed. We can estimate $H_{c2}$ by asking when there is one flux quantum through an area the size of the coherence length (here the numerical factor requires a more detailed calculation:
$$$$H_{c2} = {\phi_0 \over 2 \pi \xi^2}, \phi_0 = {2 \pi \hbar c \over 2 |e|} = 2 \times 10^{-7} \rm{G\ cm^2}.$$$$
Note that the magnetic field in a type II superconductor near $H_{c2}$ is more uniform than the superfluid density. A good introductory reference for you to review is the superconductivity chapter of Ashcroft and Mermin.
Bonus example: The structure of a vortex in a Type II superconductor is interesting. The GL current density is (here $n$ is the total density of electrons)
$$$${j_x \over n} = - {i e \hbar \over 2 m} \left(\psi_1^* {\partial \psi \over \partial x} - \psi_1 {\partial \psi^* \over \partial x} \right) - {2 e^2 \over m c} A_x \psi_1^* \psi_1.$$$$
and similarly for $j_y$ and $j_z$. Consider a single hole containing magnetic flux in a large superconducting body. Far away from the hole, the supercurrent should be zero and the magnitude $|\psi|$ should be constant. For constant $|\psi|$, the current can be rewritten
$$$$j = {e \hbar n_s \over 2 m} \left(\nabla \phi - {2 e \over \hbar c} {\bf A}\right)$$$$.
Setting this equal to zero gives
$$$$\nabla \phi = {2 e \over \hbar c} {\bf A}.$$$$
Now, around any closed loop the phase $\Phi$ should change by a multiple of $2 \pi$ for the wavefunction in the GL equation to be single-valued.
Then the line integral of $A$ around the loop must be a multiple of ${2 \pi \hbar c \over 2 e} = \Phi_0$. Finally, the line integral of $A$ around the loop is just the integrated magnetic flux through the loop
$$$$\Phi = \int {\bf B} dS = n \Phi_0 = n {\pi \hbar c \over e} = {h c \over 2 e}.$$$$
In real units the flux quantum is $2 \times 10^{-7}$ G cm$^2$ as mentioned above.
The above vortex is a simple example of a "topological defect" in a field theory: on a length scale much larger than the vortex, the vortex appears as a point singularity around which the phase wraps by $2 \pi n$. Similar topological defects occur in magnets, in liquid crystals (described by a classical field theory) and in high-energy physics (for example, magnetic monopoles in gauge theories).
A key difference between superfluids and superconductors is that, in a superconductor, the winding of the gauge field compensates the winding of the condensate phase $\theta$ so that the supercurrent goes to zero, which is what we used above. In a superfluid, there is no such gauge field so the supercurrent decays to zero only slowly (there is a quantization of vorticity''). As a result, the interaction between superfluid vortices is long-ranged, which leads to a famous phase transition in thin films: the Kosterlitz-Thouless transition. | 2021-02-28 19:13:02 | {"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": 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": 0, "wp-katex-eq": 0, "align": 0, "equation": 15, "x-ck12": 0, "texerror": 0, "math_score": 0.9842842221260071, "perplexity": 1012.7467345383984}, "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-2021-10/segments/1614178361723.15/warc/CC-MAIN-20210228175250-20210228205250-00039.warc.gz"} |
https://www.ijidonline.com/article/S1201-9712(20)32579-0/fulltext | Full length article| Volume 104, P132-138, March 01, 2021
# Transmission dynamics of the COVID-19 epidemic in England
Open AccessPublished:December 22, 2020
## Highlights
• A Bayesian susceptible–exposed–infected–removed model was used to infer the transmission dynamics of coronavirus disease 2019 in nine regions of England.
• The basic reproduction number for each region was estimated and found to be significantly correlated with the population size of each region.
• The temporally varying effective reproduction number was estimated; this showed that the control measures were effective.
• Based on data prior to June 2020, the model predicted that several regions have the possibility to experience a second wave of outbreaks.
## Abstract
### Background
The ongoing coronavirus disease 2019 (COVID-19) pandemic has caused a tremendous health burden and impact on the world economy. The UK Government implemented the biggest lockdown of society during peacetime in British history at the end of March 2020, aiming to contain the rapid spread of the virus. The UK lockdown was maintained for 7 weeks, but the effectiveness of the control measures in suppressing disease transmission remains incompletely understood.
### Methods
A Bayesian SEIR (susceptible–exposed–infected–removed) epidemiological model was used to rebuild the local transmission dynamics of the spread of COVID-19 in nine regions of England.
### Results
The basic reproduction number (R0) in England was found to be relatively high compared with China. The estimate of the temporally varying effective reproduction number (Rt) suggests that the control measures, especially the forced lockdown, were effective to reduce transmissibility and curb the COVID-19 epidemic. Although the overall incidence rate in the UK has declined, forecasting highlights the possibility of a second epidemic wave in several regions.
### Conclusion
This study enhances understanding of the current outbreak and the effectiveness of control measures in the UK.
## Introduction
The unexpected emergence and outbreak of coronavirus disease 2019 (COVID-19), caused by severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) (
• Lam T.T.
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• Zhu H.C.
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Identifying SARS-CoV-2 related coronaviruses in Malayan pangolins.
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Genomic characterisation and epidemiology of 2019 novel coronavirus: implications for virus origins and receptor binding.
,
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A pneumonia outbreak associated with a new coronavirus of probable bat origin.
), has caused a tremendous health burden and impact on the world economy. Early cases were reported in Wuhan, China in late December 2019 (
• Huang C.
• Wang Y.
• Li X.
• Ren L.
• Zhao J.
• Hu Y.
• et al.
Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China.
,
• Li Q.
• Guan X.
• Wu P.
• Wang X.
• Zhou L.
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• et al.
Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia.
,
• Wang D.
• Hu B.
• Hu C.
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• Zhang J.
• et al.
Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus-infected pneumonia in Wuhan, China.
). Subsequently, geographical spread of the disease was expedited by the return-to-home migration during Chinese New Year, which led to reports of numerous successive outbreaks in other provinces of China (
• Jia J.S.
• Lu X.
• Yuan Y.
• Xu G.
• Jia J.S.
• Christakis N.A.
Population flow drives spatio-temporal distribution of COVID-19 in China.
,
• Kang D.
• Choi H.
• Kim J.H.
• Choi J.
Spatial epidemic dynamics of the COVID-19 outbreak in China.
).
The World Health Organization (WHO) declared a Public Health Emergency of International Concern on 30 January 2020. Although great efforts were made to contain the disease, with only a few imported cases initially reported in Europe and North America in early February 2020, the outbreak soon became global. Outbreaks were reported almost simultaneously in Lombardy, Italy (
• Grasselli G.
• Pesenti A.
• Cecconi M.
Critical care utilization for the COVID-19 outbreak in Lombardy, Italy: early experience and forecast during an emergency response.
); Daegu, South Korea (
• Shim E.
• Tariq A.
• Choi W.
• Lee Y.
• Chowell G.
Transmission potential and severity of COVID-19 in South Korea.
); and Qom, Iran (
• Safavi F.
• Nourbakhsh B.
• Azimi A.R.
B-cell depleting therapies may affect susceptibility to acute respiratory illness among patients with multiple sclerosis during the early COVID-19 epidemic in Iran.
), and quickly spread to neighbouring countries. This resulted in WHO characterizing COVID-19 as a pandemic on 11 March 2020.
The first confirmed case in the UK was identified in York on 31 January 2020. This was followed by several cases reported sporadically between 1 and 27 February 2020. Most of these early cases had a clear overseas travel history, and they were quarantined and received immediate supportive care. As of 14 February 2020, eight of the nine confirmed cases had recovered. However, the number of confirmed cases in the four nations (England, Scotland, Wales and Northern Ireland) of the UK began to increase rapidly from 28 February 2020.
Initially, London was the most severely affected, where the confirmed number of cases accounted for almost one-third of the total in England by 31 March 2020. Local transmission chains were identified between large cities and neighbouring towns and rural areas in almost every region of the UK by the end of May 2020. The death toll increased along with the number of cases, resulting in the UK overtaking Italy as the country with the highest death toll in Europe and the second highest in the world on 5 May 2020. As of 24 June 2020, there had been over 306,000 confirmed cases of COVID-19 and almost 43,000 deaths in the UK.
Although the majority of infected patients show mild symptoms (
• Chen J.
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• Ling Y.
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• et al.
Clinical progression of patients with COVID-19 in Shanghai, China.
), including fever and cough (
• Guan W.J.
• Ni Z.Y.
• Hu Y.
• Liang W.H.
• Ou C.Q.
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• et al.
Clinical characteristics of coronavirus disease 2019 in China.
), some patients develop critical symptoms following hospital admission with likely immune-mediated and aggravated disease (
• Ye Q.
• Wang B.
• Mao J.
The pathogenesis and treatment of the ‘cytokine storm’ in COVID-19.
). No effective treatment options have been identified definitively, as demonstrated by well-designed randomized controlled trials (
• Zhai P.
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• Wu X.
• Long J.
• Zhong Y.
• Li Y.
The epidemiology, diagnosis and treatment of COVID-19.
), and clinical trials of candidate SARS-CoV-2 vaccines are still in their early stages (
• Zhu F.C.
• Li Y.H.
• Guan X.H.
• Hou L.H.
• Wang W.J.
• Li J.X.
• et al.
Safety, tolerability, and immunogenicity of a recombinant adenovirus type-5 vectored COVID-19 vaccine: a dose-escalation, open-label, non-randomised, first-in-human trial.
). Thus, the lack of pharmaceutical interventions, together with the high transmissibility of the virus, was clearly exacerbating the COVID-19 pandemic in the UK and elsewhere (
• Huang L.
• Zhang X.
• Zhang X.
• Wei Z.
• Zhang L.
• Xu J.
• et al.
Rapid asymptomatic transmission of COVID-19 during the incubation period demonstrating strong infectivity in a cluster of youngsters aged 16–23 years outside Wuhan and characteristics of young patients with COVID-19: a prospective contact-tracing study.
).
In addition, it is still unclear whether and how SARS-CoV-2 will circulate and interact with other seasonal human coronaviruses, in the UK and globally, and to what extent it may become seasonal (HCoV-229E, HCoV-NL63, HCoV-OC43 and HCoV-HKU1) (
• Liu Y.
• Lam T.T.Y.
• Lai F.Y.L.
• Krajden M.
• Drews S.J.
• Hatchette T.F.
• et al.
Comparative seasonalities of influenza A, B and’ common cold’ coronaviruses – setting the scene for SARS-CoV-2 infections and possible unexpected host immune interactions.
,
• Shi P.
• Dong Y.
• Yan H.
• Zhao C.
• Li X.
• Liu W.
• et al.
Impact of temperature on the dynamics of the COVID-19 outbreak in China.
,
• Xie J.
• Zhu Y.
Association between ambient temperature and COVID-19 infection in 122 cities from China.
,
• Yao Y.
• Pan J.
• Liu Z.
• Meng X.
• Wang W.
• Kan H.
• et al.
No association of COVID-19 transmission with temperature or UV radiation in Chinese cities.
). Nevertheless, given the ongoing COVID-19 activities in tropical regions, it is now very unlikely that the current UK epidemic will end naturally during the summer. Therefore, identifying effective, practical and economic public health interventions, both for now and in the future, will be critical to contain the spread of the virus and alleviate the pressure on healthcare systems.
The Chinese Government banned all transportation to and from Wuhan on 23 January 2020 and subsequently closed the border of remaining cities in Hubei Province. Similar measures aimed to reduce human mobility were issued in other Chinese cities, and have been shown to mitigate the spread of infection (
• Kraemer M.U.G.
• Yang C.H.
• Gutierrez B.
• Wu C.H.
• Klein B.
• Pigott D.M.
• et al.
The effect of human mobility and control measures on the COVID-19 epidemic in China.
,
• Tian H.
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• Wu C.H.
• Chen B.
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• et al.
An investigation of transmission control measures during the first 50 days of the COVID-19 epidemic in China.
). In Europe, Italy was the first country to implement a national lockdown on 11 March 2020. This was followed by Spain on 15 March and France on 17 March, and finally the UK on 23 March (https://www.gov.uk/government/speeches/pm-address-to-the-nation-on-coronavirus-23-march-2020). These measures eventually proved to be effective in curtailing local COVID-19 outbreaks in these countries by reducing the effective reproduction number to <1 (
• Aleta A.
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Evaluation of the potential incidence of COVID-19 and effectiveness of containment measures in Spain: a data-driven approach.
,
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• et al.
Spread and dynamics of the COVID-19 epidemic in Italy: effects of emergency containment measures.
,
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Comparing the impact of various interventions to control the spread of COVID-19 in 11 countries.
).
Therefore, exploring the transmission dynamics of SARS-CoV-2 and investigating the effectiveness of various control measures is important to acquire better understanding of this ongoing pandemic to develop and improve public health intervention policies. Studies focusing on China, Continental Europe and North America (e.g.
• Kucharski A.J.
• Russell T.W.
• Diamond C.
• Liu Y.
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• Funk S.
• et al.
Early dynamics of transmission and control of COVID-19: a mathematical modelling study.
,
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• Wu J.T.
• Liu D.
• Leung G.M.
First-wave COVID-19 transmissibility and severity in China outside Hubei after control measures, and second-wave scenario planning: a modelling impact assessment.
,
• Liang K.
Mathematical model of infection kinetics and its analysis for COVID-19, SARS and MERS.
,
• Peirlinck M.
• Sahli Costabal F.
• Kuhl E.
Outbreak dynamics of COVID-19 in Europe and the effect of travel restrictions.
,
• Pan A.
• Liu L.
• Wang C.
• Guo H.
• Hao X.
• Wang Q.
• et al.
Association of public health interventions with the epidemiology of the COVID-19 outbreak in Wuhan, China.
,
• Wu J.T.
• Leung K.
• Leung G.M.
Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study.
,
• Yang Z.
• Zeng Z.
• Wang K.
• Wong S.S.
• Liang W.
• Zanin M.
• et al.
Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions.
,
• Zhang J.
• Litvinova M.
• Wang W.
• Wang Y.
• Deng X.
• Chen X.
• et al.
Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: a descriptive and modelling study.
) have been performed, but few studies have analysed the UK COVID-19 epidemic specifically in terms of local transmission dynamics and evaluation of control measures.
This study applied a Bayesian SEIR (susceptible–exposed–infected–removed) epidemiological model that incorporates internal migration data and the regional daily number of laboratory-confirmed cases to reveal local epidemic progression of COVID-19 in nine regions of England: East Midlands, East of England, London, North East, North West, South East, South West, West Midlands, and Yorkshire and the Humber. The regional basic reproduction number (R0) and temporally varying effective reproduction number (Rt) were estimated by a sequential Monte Carlo method to identify the effectiveness of control measures. In addition, forecasts are provided for the number of daily cases for these nine regions.
## Methods
### Data sources
Two datasets were used in this study to rebuild the transmission dynamics of the COVID-19 epidemic in England: the daily number of laboratory-confirmed cases between 27 February and 31 May 2020, collected from the publicly available dashboard provided by Public Health England (PHE; https://coronavirus.data.gov.uk/); and the internal migration data collected from the UK Office of National Statistics (https://www.ons.gov.uk/).
PHE publishes a case dataset that comprises the number of laboratory-confirmed cases of COVID-19 within different types of administrative areas of England (i.e. region, upper-tier local authority and lower-tier local authority).
The laboratory-confirmed cases were identified in local National Health Service laboratories by testing specimens from people eligible for SARS-CoV-2 testing, according to the national guidance active at that time. The geographical location of each specimen was tracked by the home postcode of the person being tested. If repeat tests were conducted, the date when the first positive test occurred was recorded. Redundant tests from the same person were removed so there was no double record. Cases were aggregated according to the corresponding administrative area. Not all local authorities had complete records, and some administrative regions were too small and did not seem to have significant or continuous outbreaks. Therefore, this study focused on regional level data.
The model used in this study only considered local transmissions; therefore, cases reported prior to 27 February 2020 (the date when local transmission was considered to commence) were excluded. It should be noted that data were not always up-to-date as some community test results would have been delayed (including care home figures). Therefore, the daily numbers of laboratory-confirmed cases from all nine regions were collated on 7 June 2020 (1 week after the last date of collected data) and are shown in Figure 1. It is likely that the epidemic peak in England was reached on 8 April 2020. By 1 June 2020, the highest number of cumulative cases had occurred in London (18.4%), followed by the North West (17.6%).
Annual mid-year internal migration (i.e. residential moves across the boundaries of the nine English regions) data were used to account for movement of the population between regions when modelling disease transmission. The latest available annual data, from 2018, were used. Inflow and outflow data were aggregated across sex and age. In order to have a constant population size in the model, the inflow and outflow data involved in the model were transformed so that they were both equal to the mean of the observed inflow and outflow data.
### Mathematical model
A SEIR compartmental model, which is widely used in infectious disease modelling to describe transmission dynamics within a community, was applied in this study. The model divides the population into susceptible (S), exposed (E; but not infectious), infected (I) and removed (R) compartments, and people progress between these disease states which have been clinically described elsewhere (
• Guan W.J.
• Ni Z.Y.
• Hu Y.
• Liang W.H.
• Ou C.Q.
• He J.X.
• et al.
Clinical characteristics of coronavirus disease 2019 in China.
,
• Huang C.
• Wang Y.
• Li X.
• Ren L.
• Zhao J.
• Hu Y.
• et al.
Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China.
,
• Wang D.
• Hu B.
• Hu C.
• Zhu F.
• Liu X.
• Zhang J.
• et al.
Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus-infected pneumonia in Wuhan, China.
).
The early advice from PHE was targeted at those who exhibited COVID-19 symptoms, and recommended that they should stay at home and self-isolate. However, several studies had already reported the existence of asymptomatic COVID-19 cases (
• Cai X.
• Ma Y.
• Li S.
• Chen Y.
• Rong Z.
• Li W.
Clinical characteristics of 5 COVID-19 cases with non-respiratory symptoms as the first manifestation in children.
,
• Lu S.
• Lin J.
• Zhang Z.
• Xiao L.
• Jiang Z.
• Chen J.
• et al.
Alert for non-respiratory symptoms of coronavirus disease 2019 (COVID-19) patients in epidemic period: a case report of familial cluster with three asymptomatic COVID-19 patients.
), so it was very likely that an unknown proportion of the population in England had been infected and recovered from COVID-19 without ever being diagnosed or tested. Hence, people in the infected (I) compartment were divided into two groups: a diagnosed group (Ia), representing people who had severe symptoms and were subsequently diagnosed; and an undiagnosed group (Ib), representing people who may have recovered without being diagnosed due to mild symptoms. The proportions of hospitalized and community patients are significant (
• Bird P.
• Fallon K.
• Kwok K.O.
• Tang J.W.
High SARS-CoV-2 infection rates in respiratory staff nurses and correlation of COVID-19 symptom patterns with PCR positivity and relative viral loads.
,
• Tang J.W.
• Young S.
• May S.
• Bird P.
• Bron J.
• Mohamedanif T.
• et al.
Comparing hospitalised, community and staff COVID-19 infection rates during the early phase of the evolving COVID-19 epidemic.
). It was assumed that a proportion δ of the exposed population will enter Ia, and a uniform prior distribution U[0,1] is used to account for uncertainty about δ. Equations of changes in each compartment are set as follows:
$dS(t)dt=−β(t)I(t)NS(t)+Lin−LoutS(t)N−Ia(t),dE(t)dt=β(t)I(t)NS(t)−ηE(t)−LoutE(t)N−Ia(t),dIa(t)dt=δηE(t)−μIa(t),dIb(t)dt=(1−δ)ηE(t)−μIb(t)−LoutIb(t)N−Ia(t),dR(t)dt=μ(Ia(t)+Ib(t))−LoutR(t)N−Ia(t),$
(1)
where it is assumed that there are no imported cases. In addition to the compartment model, a testing module was added to account for the reporting delay:
$dW(t)dt=δσE(t)−ρW(t),dC(t)dt=ρW(t).$
(2)
The model is depicted in Figure 2. In this model, N is the total population size of the region of interest. S(t), E(t), I(t) and R(t) are the numbers of susceptible, exposed, infectious and removed people at time t, respectively. Lin and Lout are the inflow and outflow inferred from the internal migration dataset, respectively, and it was assumed that inflow and outflow stopped after the national lockdown. It was assumed that people show symptoms once they enter Ia. W(t) is the number of people who are waiting for their test result after showing symptoms at time t. C(t) is the estimated cumulative number of cases, η is the rate of being infectious (i.e. inverse of the incubation period), μ is the rate of recovery (i.e. inverse of the infectious period, which equals the mean serial interval minus the incubation period) (
• Lipsitch M.
• Cohen T.
• Cooper B.
• Robins J.M.
• Ma S.
• James L.
• et al.
Transmission dynamics and control of severe acute respiratory syndrome.
), and 1/ρ is the number of days between showing symptoms and receiving test results. 1/η was set as 5.2 days and 1/μ was set as 2.3 days according to estimates from a comprehensive study of the early transmission dynamics of COVID-19 (
• Li Q.
• Guan X.
• Wu P.
• Wang X.
• Zhou L.
• Tong Y.
• et al.
Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia.
). 1/ρ was set as 4 based on estimates from
• Chen J.
• Qi T.
• Liu L.
• Ling Y.
• Qian Z.
• Li T.
• et al.
Clinical progression of patients with COVID-19 in Shanghai, China.
. Following
• Kucharski A.J.
• Russell T.W.
• Diamond C.
• Liu Y.
• Edmunds J.
• Funk S.
• et al.
Early dynamics of transmission and control of COVID-19: a mathematical modelling study.
, a temporally varying transmission rate β(t) that follows a log-normal sequential update was assumed [i.e. $log(β(t))∼N(log(β(t−1)),σ)$, given standard deviation σ], and the effective reproduction number Rt at time t was approximated by (see online Supplementary material):
$Rˆt=β(t)S(t)μN.$
(3)
A Poisson generating model was assumed for the observed daily number of laboratory-confirmed cases Y(t), given R0 and transmission rate β1:t from the start to time t:
$P(Y(t)|R0,β1:t)∼Pois(ρW(t)).$
(4)
To account for the early period when the virus started to seed in each region of England before the public became aware, a preliminary model similar to Eq. (1) was run, except with a constant β derived from R0. The preliminary model assumes that a single infected person enters the region of interest 14 days before the date of the first confirmed local case after 27 February 2020.
Analysis was conducted using R Version 4.0. The sequential Monte Carlo method was used to draw samples of R0, and β1:t from the posterior distribution, where the optimal standard deviation σ that gives the highest likelihood was selected by an exhaustive search. Sensitivity analysis is available in the online Supplementary material.
## Results and discussion
### Basic reproduction number
R0 is an important parameter that quantifies disease transmissibility at the start of an epidemic. Estimated R0 numbers for each of the nine regions are listed in Table 1 (histograms of the posterior samples are shown in Figure S1, see online Supplementary material). All regions have R0 between 2.8 and 3.9, which is significantly higher than 1.
Table 1Estimated basic reproduction number (R0) and corresponding 95% credible interval (CI) in each English region.
RegionMedian R095% CI (lower)95% CI (upper)
East Midlands3.22.45.3
East of England3.22.74.6
London3.93.45.3
North East2.82.14.7
North West3.63.05.2
South East3.93.05.5
South West3.93.44.7
West Midlands3.52.85.2
Yorkshire and the Humber3.02.54.5
Notably, estimated R0 numbers were found to be positively correlated with population size in each region. Spearman’s rank correlation was 0.77, which is significantly higher than 0 (P < 0.05). These estimates were relatively high (seven of the nine regions had median values >3) compared with the early estimate in China by WHO (1.4–2.5), although a high R0 (3.8–8.9) was also reported in China more recently (
• Sanche S.
• Lin Y.T.
• Xu C.
• Romero-Severson E.
• Hengartner N.
• Ke R.
High contagiousness and rapid spread of severe acute respiratory syndrome coronavirus 2.
).
Compared with other major airborne viruses, SARS-CoV-2 in England has an estimated R0 similar to that of SARS-CoV-1 but significantly higher than that of Middle East respiratory syndrome coronavirus (MERS-CoV) and other human influenza viruses (
• Chen J.
Pathogenicity and transmissibility of 2019-nCoV – a quick overview and comparison with other emerging viruses.
). This indicates that SARS-CoV-2 has very strong transmissibility in the early stages, and the infected population size will expand rapidly without human intervention in all nine regions. The estimates of R0 in England are largely consistent with estimates from other major European countries (e.g. Italy: 3.49–3.84; France: 3.1–3.3) (
• Distante C.
• Piscitelli P.
• Miani A.
Covid-19 outbreak progression in Italian regions: approaching the peak by the end of March in Northern Italy and first week of April in Southern Italy.
,
• Gatto M.
• Bertuzzo E.
• Mari L.
• Miccoli S.
• Carraro L.
• Casagrandi R.
• et al.
Spread and dynamics of the COVID-19 epidemic in Italy: effects of emergency containment measures.
,
• Roques L.
• Klein E.K.
• Papaix J.
• Sar A.
• Soubeyrand S.
Using early data to estimate the actual infection fatality ratio from COVID-19 in France.
,
• Yuan J.
• Li M.
• Lv G.
• Lu Z.K.
Monitoring transmissibility and mortality of COVID-19 in Europe.
). This might explain the rapid spread and high pandemic potential of COVID-19 when relatively few control measures were implemented in Europe.
### Effectiveness of control measures
On 25 February 2020, the UK Government announced its general strategy, which aimed to reduce the impact of the disease by four successive phases: contain; delay; research; and mitigate. The respective aims of these four phases were to: detect, trace and isolate early cases; slow the spread and delay the peak until warmer months; develop diagnostic tests, drugs and vaccines; and save lives and maintain nationwide order once the disease is widespread. This announcement was followed by advice that travellers from heavily hit countries should self-isolate. On 12 March, the UK Government started to issue policies for local residents, advising those with respiratory symptoms to self-isolate at home.
After the release of the controversial herd immunity strategy, the UK Government advised people against ‘non-essential’ travel and use of public entertainment venues on 16 March 2020. This was followed by the closure of all pubs, cafes, restaurants, bars, gyms, etc. on 20 March 2020. On 23 March 2020, a restrictive national lockdown was announced by the UK Government, and the police force was provided with powers to ensure compliance on 26 March 2020.
From this time, people were only allowed to leave their homes for limited reasons, gatherings of more than two people were forbidden, and social distancing was required in shops. Use of public transport declined significantly during the lockdown period. Control measures began to ease gradually from 10 May 2020 when the UK Government allowed certain groups of people to work and encouraged outdoor exercise, yet people were warned to ‘stay alert’. Given these measures, the control period was defined in this paper as 12–26 March 2020 and the lockdown period was defined as 26 March–10 May 2020.
Estimated Rt numbers are shown in Figure 3. For all nine English regions, Rt exhibited an overall decreasing trend during the control period. London, West Midlands and South East showed a mostly decreasing Rt during the control period. In the remaining regions, although Rt increased or oscillated at the beginning of the control period when measures were largely mild suggestions, it started to decrease when the more restrictive and forceful national lockdown was implemented. This reveals the effectiveness of issuing forceful control measures to contain an epidemic. Although Rt numbers in East Midlands, North East and North West rose after the control period, it subsequently decreased towards 1 within 1 week for all regions. These transmission dynamics patterns are also consistent with some country-level estimates in the literature (
• Kwok K.
• Lai F.
• Wai V.
• Tsoi M.
• Wong S.
• Tang J.
Comparing the impact of various interventions to control the spread of COVID-19 in 11 countries.
).
In general, the control measures issued in England in March 2020 were effective to contain the spread of COVID-19 in all nine English regions. This finding is consistent with a previous study which investigated the effect of the reduction of social contacts on transmissibility based on surveys (
• Jarvis C.I.
• Van Zandvoort K.
• Gimma A.
• Prem K.
• CMMID COVID-19 Working Group
• Klepac P.
• et al.
Quantifying the impact of physical distance measures on the transmission of COVID-19 in the UK.
). It is noteworthy that Rt remained slightly <1 during most of the lockdown period, rather than diminishing to 0. This suggests that local transmissions were maintained, possibly in small towns and communities, although the overall epidemic was contained, and the virus might have circulated between different areas within each English region.
### Estimation of the daily number of new cases
To verify the model, the estimated daily number of cases was compared with the observed daily number of confirmed cases. The result is shown in Figure 4 ( estimates of the number of infected population and susceptible population are shown in Figure S2, see online Supplementary material). Although the observed number fluctuates over time, the overall trend in daily cases is well captured by the model.
Broadly, the daily number of confirmed cases in the nine English regions peaked in early April 2020. Notably, London was the first region where the daily number of new cases rose rapidly in early March 2020. The study findings highlight the serious epidemic in London during the early phase. This may have been because London receives the highest proportion of inbound visitors from other English regions, and is the most densely populated area among all English regions. In addition, a constantly high daily number of cases was maintained in Yorkshire and the Humber in April 2020, as more infected population may have been seeded prior to lockdown, resulting in transmission to their household contacts during lockdown. This may explain the delay in the reduction of Rt. The daily number of cases in two regions of the Midlands increased briefly in late April 2020, but this started to decline from the beginning of May 2020. The daily number of cases in the remaining regions declined during late April and May 2020. These results correspond with the authors’ previous estimate that Rt was generally <1 during the lockdown period. This indicates the effectiveness of the control measures in all regions.
### Forecasting potential second wave outbreaks
In model forecasting, it was assumed that Rt stayed constant after 31 May 2020, and the posterior samples of Rt on that day were used to forecast the daily number of confirmed cases in June and July 2020. As the lockdown was lifted on 1 June 2020, it was assumed that inflow and outflow of the population would restart. Relevant results are shown in Figure 4.
Although the daily number of new cases is expected to decrease in most regions, it is estimated that, except in the East of England, North West and South East where Rt remains significantly <1, other regions may witness a second wave of outbreaks.
In particular, East Midlands, South West, West Midlands, and Yorkshire and the Humber may experience a rebound in incidence after June 2020, as projected by the upper 95% credible interval of the daily number of new cases (which may go up to ≥40% of the number at the March/April peak).
Notably, Leicester (East Midlands:
BBC News. Coronavirus: BAME community fearful over Leicester COVID surge. BBC News, 19 June 2020a. Available at: https://www.bbc.co.uk/news/uk-england-leicestershire-53105336. [Accessed 28 June 2020].
) and Cleckheaton (Yorkshire:
BBC News. Coronavirus: ‘secrecy’ claims over Cleckheaton outbreak. BBC News, 19 June 2020b. Available at: https://www.bbc.co.uk/news/uk-england-leeds-53105266. [Accessed 28 June 2020].
) recently reported surges in cases, which corroborates the model forecast. Moreover, as the estimated Rt in the South West includes high values on 31 May 2020, it will likely maintain a relatively high median number of infected population until August 2020 (>100 individuals). While the UK Government has been considering lifting some control measures to restore the economy, particular attention should be paid to regions at risk of a second wave.
## Conclusion
This study demonstrated the use of a Bayesian SEIR model to reconstruct the transmission dynamics of COVID-19 in nine English regions. Although the true dynamics of transmission of COVID-19 is a complex process, the estimated daily number of cases closely follows the trend of the observed daily number of cases, indicating the validity of the model. The findings show that R0 in England is generally higher compared with China but is in line with some major European countries. The effective reproduction number estimates present a temporally varying trend of transmissibility. The present results suggest that transmissibility of COVID-19 was reduced effectively by the control measures adopted by the UK Government. This led to a decline in the number of infected population in May 2020. Notably, although critics may argue that restriction of the free movement of people violates basic human rights when milder measures such as social distancing can be equally useful, such strict measures within a national lockdown were efficient to contain transmission in some regions. The forecasting data highlight the possibility of early secondary outbreaks, so close monitoring of the rate of transmission and Rt will be required after the lockdown measures are lifted.
## Ethical approval
Not required.
## Conflict of interest
None declared.
## Acknowledgements
The authors wish to thank Robert J.B. Goudie for helpful discussions of this work. Yang Liu was supported by a Cambridge International Scholarship from the Cambridge Commonwealth, European and International Trust . Tommy T.Y. Lam was supported by the N ational Natural Science Foundation of China Excellent Young Scientists Fund (Hong Kong and Macau) ( 31922087 ).
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• Chen X.
• et al.
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Lancet Infect Dis. 2020; 20: 793-802
• Zhou P.
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• Wang X.G.
• Hu B.
• Zhang L.
• Zhang W.
• et al.
A pneumonia outbreak associated with a new coronavirus of probable bat origin.
Nature. 2020; 579: 270-273
• Zhu F.C.
• Li Y.H.
• Guan X.H.
• Hou L.H.
• Wang W.J.
• Li J.X.
• et al.
Safety, tolerability, and immunogenicity of a recombinant adenovirus type-5 vectored COVID-19 vaccine: a dose-escalation, open-label, non-randomised, first-in-human trial.
Lancet. 2020; 395: 1845-1854 | 2022-12-07 23:51:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "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": 0.33352693915367126, "perplexity": 14542.194261218641}, "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/1669446711221.94/warc/CC-MAIN-20221207221727-20221208011727-00329.warc.gz"} |
https://math.stackexchange.com/questions/728998/what-is-the-weakest-notion-of-set-that-we-need-so-that-we-can-say-the-yoneda | # What is the weakest notion of "set" that we need, so that we can say the Yoneda lemma implies something about sets?
We set up a set theory axiomatically by fixing certain statements about "$\in$". There are many different set theories. The Yoneda lemma (the theorem about functors to Set) is a early but central result in category theory and relates very much to sets (the functor is to Set, and often the Hom-sets are indeed sets and the collection of natural transformations.)
Now if you pull up category theory, maybe in purely logical terms, or within a type theoretic background, or maybe using some small set theory, then you will somehow get to some variant of the Yoneda lemma and indirectly say something about sets.
My question is: What is the least restrictive notion of "set" so that we can justify to say the Yoneda indeed tells us something about sets or embeddings in "the world of sets"? For exmaple, I hardly imagine we need to assume the axiom schema of replacement to end up with the Yoneda lemma, but conversely I expect the "sets" talked about in the Yoneda lemma must be subject of some sort of comprehension principle.
• You don't need very much comprehension, because we're not really forming any new sets. But we do need to be able to define maps. Replacement can be avoided provided you have disjoint unions of certain sets. Mar 27 '14 at 14:43
[EDIT: I miscalculated local exponents in the previous revision of this answer. Local cartesian closedness is, obviously, unnecessary.]
We have a Yoneda lemma whenever we can speak of categories relative to any "sets" that form a category. In particular, it suffices that the category of "sets" has pullbacks. If $\mathbb{C}$ has pullbacks, then for every category $A$ internal to $\mathbb{C}$ there is a full and faithful functor:
$$y_A \colon A \rightarrow \mathbb{C}^{A^{op}}$$
where $\mathbb{C}^{A^{op}}$ is thought of as a locally $\mathbb{C}$-internal category of internal presheaves on $A$, and $y_A$ is the adjoint transposition of:
$$\hom(=, -) \colon A^{op} \times A \rightarrow \mathbb{C}$$
which is defined on generalized objects $X, Y \in_K A$ as the local exponent $X^Y$. In more details, let $X, Y \colon K \rightarrow A_0$ be two morphisms in $\mathbb{C}$ (i.e. objects in the fibre over $K$ in the externalisation of $A$), where $A_0$ is the object of objects of category $A$. The local exponent $X^Y$ is defined as the representation of:
$$K' \overset{f}\rightarrow K \mapsto \hom_{\mathbf{Cat}(\mathbb{C})}(K', A)(X \circ f, Y \circ f)$$
Morphisms $X \circ f \rightarrow Y \circ f$ in $\hom_{\mathbf{Cat}(\mathbb{C})}(K', A)$ are internal natural transformations. Spelling out the definition, we have a morphism $\alpha \colon K' \rightarrow A_1$ in $\mathbb{C}$ that commutes with $X \circ f$ and $Y \circ f$ --- i.e. $X \circ f = \mathit{dom} \circ \alpha$ and $Y \circ f = \mathit{cod} \circ \alpha$, where $\mathit{dom}, \mathit{cod} \colon A_1 \rightarrow A_0$ are the usual projections from objects of morphisms $A_1$ to the object of objects $A_0$. In other words: $$\langle X, Y \rangle \circ f = \langle \mathit{dom}, \mathit{cod} \rangle \circ \alpha$$ and for given $f$, morphisms $\alpha$ are tantamount to morphisms to the pullback of $\langle X, Y \rangle$ with $\langle \mathit{dom}, \mathit{cod} \rangle$, which we denote by $\hom(X, Y)$. By the universal property of pullbacks, this operation extends to the locally internal functor $A^{op} \times A \rightarrow \mathbb{C}$.
Moreover, if $\mathbb{C}$ has finite colimits and is additionally locally cartesian closed, then $\mathbb{C}^{A^{op}}$ is an internal monoidal free (small) cocompletion of $A$. See also this question on MO: | 2021-09-20 07:49:58 | {"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": 0.9589893221855164, "perplexity": 229.2317197354506}, "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-2021-39/segments/1631780057033.33/warc/CC-MAIN-20210920070754-20210920100754-00266.warc.gz"} |
https://terrytao.wordpress.com/2008/12/09/an-airport-inspired-puzzle/?like=1&_wpnonce=4cdcfef5a4 | I was recently at an international airport, trying to get from one end of a very long terminal to another. It inspired in me the following simple maths puzzle, which I thought I would share here:
Suppose you are trying to get from one end A of a terminal to the other end B. (For simplicity, assume the terminal is a one-dimensional line segment.) Some portions of the terminal have moving walkways (in both directions); other portions do not. Your walking speed is a constant $v$, but while on a walkway, it is boosted by the speed $u$ of the walkway for a net speed of $v+u$. (Obviously, given a choice, one would only take those walkways that are going in the direction one wishes to travel in.) Your objective is to get from A to B in the shortest time possible.
1. Suppose you need to pause for some period of time, say to tie your shoe. Is it more efficient to do so while on a walkway, or off the walkway? Assume the period of time required is the same in both cases.
2. Suppose you have a limited amount of energy available to run and increase your speed to a higher quantity $v'$ (or $v'+u$, if you are on a walkway). Is it more efficient to run while on a walkway, or off the walkway? Assume that the energy expenditure is the same in both cases.
3. Do the answers to the above questions change if one takes into account the various effects of special relativity? (This is of course an academic question rather than a practical one. But presumably it should be the time in the airport frame that one wants to minimise, not time in one’s personal frame.)
It is not too difficult to answer these questions on both a rigorous mathematical level and a physically intuitive level, but ideally one should be able to come up with a satisfying mathematical explanation that also corresponds well with one’s intuition.
[Update, Dec 11: Hints deleted, as they were based on an incorrect calculation of mine.] | 2015-03-02 23:07:12 | {"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": 0, "mathjax_asciimath": 0, "img_math": 44, "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": 0.6469785571098328, "perplexity": 438.5010327625433}, "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-2015-11/segments/1424936463093.76/warc/CC-MAIN-20150226074103-00054-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/cathode-ray-tubes.54960/ | # Cathode Ray Tubes
How does a CRT focus an electron beam? I get the basics of it. But I'm a little confused on the details.
For electric focusing, how does accelerating the electrons cause it to focus?
I'm looking for a more mathematical treatment of the subject. Everything I've found is just descriptive. For example, I supposed I'd like to know what the electric field looks like between the plates, grids, anodes, etc?
What are the shapes of the plates, grides and anodes? How is the potential difference set up and measured?
I know there are lots of variations of CRTs, but I am interested in the specifics of how they work (so pick your favorite type to discuss).
Any information you could provide would be greatly appreciated!
## Answers and Replies
Astronuc
Staff Emeritus
Science Advisor
See this on Scanning electron microscope basics
http://www.uleth.ca/emf/2001/lecture_notes/lecture_1.pdf
Also check the library for the following reference or others like it:
P. W. Hawkes and E. Kasper, Principles of Electron Optics, vol. 3: Wave Optics, Academic Press, London, 1994.
krab
Science Advisor
Euclid said:
For electric focusing, how does accelerating the electrons cause it to focus?
Electrons pass thru apertures which are held at various electric potentials. This can be thought of as causing a potential V(z) on the axis joining the aperture centres. If you solve Laplaces equation in cylindrical coordinates for this axial symmetry, you find the potential can be expanded as follows:
$$\Phi(r,z)=V(z)-(V''(z)/4)r^2+...$$
Differentiate this and you will find that the electric field has a radial component, and so there is a radial force. This varies along z, being sometimes positive and sometimes negative, but the dynamics is such that the net effect is to focus.
krab said:
If you solve Laplaces equation in cylindrical coordinates for this axial symmetry, you find the potential can be expanded as follows:
$$\Phi(r,z)=V(z)-(V''(z)/4)r^2+...$$
.
What considerations lead you to the Laplace equation?
pervect
Staff Emeritus
Science Advisor
In the absence of any charges, maxwell's equations implies that the electric potential must satisfy Laplace's equation.
$$E = -\nabla V$$
$$\nabla \cdot E = \frac{\rho}{\epsilon} = 0$$
therefore
$$\nabla^2 V = 0$$
See for instance this link
I don't know anything specifically about electron optics, though. It's not as obvious as one would think that the charge density is zero - space charge is a real possibility in vacuum tubes. I'd assume from the reference to Laplace's equation that it's avoided when designing electron optics - I'm not surprised, I would expect that it would be hard to control precisely.
pervect said:
In the absence of any charges, maxwell's equations implies that the electric potential must satisfy Laplace's equation.
$$E = -\nabla V$$
$$\nabla \cdot E = \frac{\rho}{\epsilon}+\frac{{\partial^2 V}{\partial z^2}} = 0$$
therefore
$$\nabla^2 V = 0$$
I guess I have two questions.
(1) Wouldn't the fact that there are electrons in a CRT imply the the change density is nonzero.
(2) How do you go about solving this equation in the case of a CRT?
In plane cylindrical coordinates,
$$\nabla^2 V= \frac{\partial ^2 V}{\partial r^2}+\frac{1}{r}\frac{\partial V}{\partial r}=0$$
The solution is V=a+blnr.
How do I determine the constants a and b?
Last edited:
I'd also like to solve it in the 3d case, but the plane view will show how the beam focuses. I want to know E along the z axis however to estimate how much energy the electrons gain in passing through the anodes.
krab
Science Advisor
Euclid said:
I guess I have two questions.
(1) Wouldn't the fact that there are electrons in a CRT imply the the change density is nonzero.
(2) How do you go about solving this equation in the case of a CRT?
In plane cylindrical coordinates,
$$\nabla^2 V= \frac{\partial ^2 V}{\partial r^2}+\frac{1}{r}\frac{\partial V}{\partial r}=0$$
The solution is V=a+blnr.
How do I determine the constants a and b?
(1) Yes, but in general, the fields due to the electrons themselves are much smaller than those due to the electrodes. You can estimate it from Gauss' law. It will probably only be a few volts compared with kV for the electrodes. In any case, it is something that can be added later.
(2) You forgot
$$\partial^2V/\partial z^2$$
V is a function of z, not just r. That's the whole point of using electrodes to focus the electrons. Your formula is applicable to the case where the electrons are between coaxial cylinders, with a potential difference between the cylinders. It is not applicable to the case at hand.
krab said:
Your formula is applicable to the case where the electrons are between coaxial cylinders, with a potential difference between the cylinders. It is not applicable to the case at hand.
My thought is that I want to ignore the z part of V. In any event, the z part of V has no effect on the focusing of the beam. The focusing effect is mainly from the E field in the radial direction. I was hoping to show that E(r)=-b/r in the radial direction (ignoring the z). This way I can give an explanation for why the beam focuses. The problem is showing that b>0 (it seems to me that b will depend on z, however, for what I'm doing I think it's sufficient to find b at some z).
krab
Science Advisor
The radial fields and the z-variation are related through Laplace's equation. Read post #3 again. In a very real sense, it is the z-variation that causes the radial fields. You cannot show E=-b/r because it is not true. You were expecting an infinite field for r=0?
pervect
Staff Emeritus
Science Advisor
Euclid said:
I guess I have two questions.
(1) Wouldn't the fact that there are electrons in a CRT imply the the change density is nonzero.
(2) How do you go about solving this equation in the case of a CRT?
In plane cylindrical coordinates,
$$\nabla^2 V= \frac{\partial ^2 V}{\partial r^2}+\frac{1}{r}\frac{\partial V}{\partial r}=0$$
The solution is V=a+blnr.
How do I determine the constants a and b?
Basically, the solution is determined by the boundary condtions. The boundary conditions are that the potential on the focusing electrodes must be constant. Also, the potential at infinity must be zero, but that will be mainly a concern for the function of the potential outside the focusing electrodes.
To have a unique solution to Laplaces equation within a region, you have to specify the potential or the electric field (the normal derivative of the potential) everywhere on a closed boundary.
I don't quite see how to setup the boundary conditoins for this problem though, it's easy enough to specify the potential on the electrodes is constannt, but if the electrodes aren't solid I'm not sure what to do exactly. (What do the focusing electodes look like, anyway?)
Your solution is not general, as was pointed out. Your solution will work if the focusing electode is an infinitely long cylinder, or some approximation thereof (a cylinder much longer than it's length). If the focusing electrode doesn't have this shape, you need a more general solution.
As far as methods go, I'd suggest a computer solution, but you can also do some problems with conformal transforms (you should be able to find some info on this with a web search).
krab said:
Electrons pass thru apertures which are held at various electric potentials. This can be thought of as causing a potential V(z) on the axis joining the aperture centres. If you solve Laplaces equation in cylindrical coordinates for this axial symmetry, you find the potential can be expanded as follows:
$$\Phi(r,z)=V(z)-(V''(z)/4)r^2+...$$
Differentiate this and you will find that the electric field has a radial component, and so there is a radial force. This varies along z, being sometimes positive and sometimes negative, but the dynamics is such that the net effect is to focus.
Ok, what is the difference between phi and V? Are they both the potential? What are the terms after the '...'? Can they be neglected? If I have this right, does this mean that the electric field in the radial direction is
E=(-1/2)V''(z)r ?
How do I determine V''(z)? Can you point me to a source that might discuss this in more detail?
krab
Science Advisor
Euclid said:
Ok, what is the difference between phi and V? Are they both the potential? What are the terms after the '...'? Can they be neglected? If I have this right, does this mean that the electric field in the radial direction is
E=(-1/2)V''(z)r ?
How do I determine V''(z)? Can you point me to a source that might discuss this in more detail?
1. phi and V are related by the following:
$$V(z)=\Phi(0,z)$$
2. Yes, for a first pass, ignore the terms implied by "...". They are just a further expansion; terms are even powers of r. Their presence makes the optics nonlinear and so are undesirable, because they make it impossible to get a point focus. Generally, you can ignore those powers if the apertures in the electrodes are large compared with the size of the electron beam.
3. Yes, that is the radial electric field.
4. V'' is not easy to calculate. It is found numerically by solving Laplace's equation using a technique called over-relaxation. Some (complicated) formulas exist for specialized geometries.
5. Look up "electron optics", "einzel lens", "unipotential lens", etc. | 2021-06-13 18:08:05 | {"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": 0.7389328479766846, "perplexity": 454.0259810065357}, "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-2021-25/segments/1623487610196.46/warc/CC-MAIN-20210613161945-20210613191945-00060.warc.gz"} |
http://stackoverflow.com/questions/13407156/how-to-consume-a-webservice-using-wsdl-files-in-c-sharp/13407222 | # How to Consume a WebService using WSDL files in C#
So I have two wsdl files (xml based) and I need to use them to consume a web service. Not sure where to start? I'm trying to add them in my Visual Studio Project Solution by clicking on "Add Service Reference" but I need an URL.
Should I add them to a Virtual Directory? If so, how?
-
run svcutil yourfile.wsdl or wsdl yourfile.wsdl and add the resulting .cs file to your project. – L.B Nov 15 '12 at 22:15
You shouldn't use wsdl.exe if it's a WCF service. I recommend either using svcutil.exe or you can browse to the WSDL file through the add service UI. – Lee O. Nov 16 '12 at 2:06
You already found the solution. Use the "Add Service Reference" dialog and make sure your service is accessible by a URL. To do this either request the URL by the people offering the service or deploy the service in IIS.
Personally I would forget about svcutil.exe. If you have Visual Studio, it is much easier to add and update the service reference using the excellent integration of web services in Visual Studio.
-
How can I deploy it to the service? Like I said, I have two xml files with all the specs – franciscovalera Nov 15 '12 at 22:30
As far as I understand you only have the WSDL of the service you want to consume. In order to consume a service you need both, the URL of the actual service and the WSDL! – CodeZombie Nov 15 '12 at 22:35
@ZombieHunter thats not true. For security purposes a business may choose not to expose the wsdl through a URL. You can create a client using just the wsdl files. – Lee O. Nov 16 '12 at 1:23
@LeeO.: But you still need both of them, the WSDL (either as a file or a URL) and the URL of the actual service. – CodeZombie Nov 16 '12 at 8:18
You can use the wsdl.exe tool which comes with Visual Studio (in there you can specify a local file path to your wsdl file) -
http://msdn.microsoft.com/en-us/library/d2s8y7bs(VS.100).aspx
-
You can add a service reference by using the path to the wsdl files.
-
You can use svcutil.exe as such:
svcutil.exe /language:cs /out:MyServiceProxy.cs /config:app.config c:\path\to\my.wsdl
- | 2015-07-31 20:26:03 | {"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": 0.4174538254737854, "perplexity": 3506.3284202035115}, "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-2015-32/segments/1438042988311.72/warc/CC-MAIN-20150728002308-00233-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://mathematica.stackexchange.com/questions/27837/speeding-up-2d-histogram-plotting-perhaps-via-listdensityplot | # Speeding up 2D histogram plotting (perhaps via ListDensityPlot)? [on hold]
What would be the fastest way to take an array containing about $n=10^6$ to $10^7$ elements and plot a two-dimensional histogram with A x B bins? ListDensityPlot is taking upwards of half-an-hour. Should I be using compiled code?
-
## put on hold as off-topic by Jason is no longer a postdoc, m_goldberg, MarcoB, Louis, Öskåyesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – Jason is no longer a postdoc, m_goldberg, Öskå
If this question can be reworded to fit the rules in the help center, please edit the question.
One way is to use Interpolation on your histogram data and use DensityPlot. Some more useful tips here (possible duplicate?): mathematica.stackexchange.com/a/7556/5 – R. M. Jun 29 '13 at 5:20
Possible duplicate of Plotting large datasets – MarcoB 2 days ago | 2016-05-30 01:05:11 | {"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": 0.23069484531879425, "perplexity": 1682.5384510623037}, "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-2016-22/segments/1464049282327.67/warc/CC-MAIN-20160524002122-00169-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://www.astro.washington.edu/users/vanderplas/Astr599/notebooks/11_EfficientNumpy.html | This notebook was put together by Jake Vanderplas for UW's Astro 599 course. Source and license info is on GitHub.
# Efficient Numerical Computing with Numpy
In this session, we'll discuss how to make your programs as efficient as possible, mainly by taking advantage of vectorization in NumPy.
## Aside: the "Unladen Swallow"
In [2]:
from IPython.display import YouTubeVideo
Out[2]:
That video was the original reference behind the now-defunct unladen swallow project, which had the goal of making Python's C implementation faster.
Yes, Python is (unfortunately) a rather slow language.
Here is an example:
In [3]:
# A silly function implemented in Python
def func_python(N):
d = 0.0
for i in range(N):
d += (i % 3 - 1) * i
return d
In [4]:
# Use IPython timeit magic to time the execution
%timeit func_python(10000)
1000 loops, best of 3: 1.81 ms per loop
To compare to a compiled language, let's write the same function in fortran and use the f2py tool (included in NumPy) to compile it
In [5]:
%%file func_fortran.f
subroutine func_fort(n, d)
integer, intent(in) :: n
double precision, intent(out) :: d
integer :: i
d = 0
do i = 0, n - 1
d = d + (mod(i, 3) - 1) * i
end do
end subroutine func_fort
Overwriting func_fortran.f
In [6]:
!f2py -c func_fortran.f -m func_fortran > /dev/null
In [7]:
from func_fortran import func_fort
%timeit func_fort(10000)
100000 loops, best of 3: 20.3 µs per loop
Fortran is about 100 times faster for this task!
## Why is Python so slow?
We alluded to this yesterday, but languages tend to have a compromise between convenience and performance.
• C, Fortran, etc.: static typing and compiled code leads to fast execution
• But: lots of development overhead in declaring variables, no interactive prompt, etc.
• Python, R, Matlab, IDL, etc.: dynamic typing and interpreted excecution leads to fast development
• But: lots of execution overhead in dynamic type-checking, etc.
We like Python because our development time is generally more valuable than execution time. But sometimes speed can be an issue.
## Strategies for making Python fast
1. Use Numpy ufuncs to your advantage
2. Use Numpy aggregates to your advantage
5. Use a tool like SWIG, cython or f2py to interface to compiled code.
Here we'll cover the first four, and leave the fifth strategy for a later session.
## Strategy 1: Using UFuncs in Numpy
A ufunc in numpy is a Universal Function. This is a function which operates element-wise on an array. We've already seen examples of these in the various arithmetic operations:
In [8]:
import numpy as np
x = np.random.random(4)
print x
print x + 1 # add 1 to each element of x
[ 0.90839173 0.6614708 0.74607367 0.66114385]
[ 1.90839173 1.6614708 1.74607367 1.66114385]
In [9]:
print x * 2 # multiply each element of x by 2
[ 1.81678346 1.32294159 1.49214734 1.32228771]
In [10]:
print x * x # multiply each element of x by itself
[ 0.82517553 0.43754362 0.55662592 0.43711119]
In [11]:
print x[1:] - x[:-1]
[-0.24692093 0.08460287 -0.08492982]
These are binary ufuncs: they take two arguments.
There are also many unary ufuncs:
In [12]:
-x
Out[12]:
array([-0.90839173, -0.6614708 , -0.74607367, -0.66114385])
In [13]:
np.sin(x)
Out[13]:
array([ 0.78851565, 0.61427811, 0.67876066, 0.61402009])
### The Speed of Ufuncs
In [14]:
x = np.random.random(10000)
In [15]:
%%timeit
# compute element-wise x + 1 via a ufunc
y = np.zeros_like(x)
y = x + 1
10000 loops, best of 3: 23.7 µs per loop
In [16]:
%%timeit
# compute element-wise x + 1 via a loop
y = np.zeros_like(x)
for i in range(len(x)):
y[i] = x[i] + 1
100 loops, best of 3: 13.7 ms per loop
#### Why is NumPy so much faster?
Numpy UFuncs are faster than Python functions involving loops, because the looping happens in compiled code. This is only possible when types are known beforehand, which is why numpy arrays must be typed.
### Other Available Ufuncs
• Trigonometric functions (np.sin, np.cos, etc.)
• Scipy special functions (scipy.special.j0, scipy.special.gammaln, etc.)
• Element-wise minimum/maximum (np.minimum, np.maximum)
• User-defined ufuncs (read more here)
In [17]:
x = np.random.random(5)
print x
print np.minimum(x, 0.5)
print np.maximum(x, 0.5)
[ 0.03731972 0.94058919 0.79808336 0.45500933 0.03967752]
[ 0.03731972 0.5 0.5 0.45500933 0.03967752]
[ 0.5 0.94058919 0.79808336 0.5 0.5 ]
In [18]:
# contrast this behavior with that of min() and max()
print np.min(x)
print np.max(x)
0.0373197161493
0.940589193882
In [19]:
%matplotlib inline
# On older IPython versions, use %pylab inline
import matplotlib.pyplot as plt
In [20]:
x = np.linspace(0, 10, 1000)
plt.plot(x, np.sin(x))
Out[20]:
[<matplotlib.lines.Line2D at 0x1060badd0>]
In [21]:
from scipy.special import gammaln
plt.plot(x, gammaln(x))
Out[21]:
[<matplotlib.lines.Line2D at 0x106353610>]
### Some interesting properties of UFuncs
UFuncs have some methods built-in, which allow for some very interesting, flexible, and fast operations:
In [22]:
x = np.arange(5)
y = np.arange(1, 6)
Out[22]:
array([1, 3, 5, 7, 9])
In [23]:
np.add.accumulate(x)
Out[23]:
array([ 0, 1, 3, 6, 10])
In [24]:
np.multiply.accumulate(x)
Out[24]:
array([0, 0, 0, 0, 0])
In [25]:
np.multiply.accumulate(y)
Out[25]:
array([ 1, 2, 6, 24, 120])
In [26]:
np.add.identity
Out[26]:
0
In [27]:
np.multiply.identity
Out[27]:
1
In [28]:
np.add.outer(x, y)
Out[28]:
array([[1, 2, 3, 4, 5],
[2, 3, 4, 5, 6],
[3, 4, 5, 6, 7],
[4, 5, 6, 7, 8],
[5, 6, 7, 8, 9]])
In [29]:
# make a times-table
x = np.arange(1, 13)
print np.multiply.outer(x, x)
[[ 1 2 3 4 5 6 7 8 9 10 11 12]
[ 2 4 6 8 10 12 14 16 18 20 22 24]
[ 3 6 9 12 15 18 21 24 27 30 33 36]
[ 4 8 12 16 20 24 28 32 36 40 44 48]
[ 5 10 15 20 25 30 35 40 45 50 55 60]
[ 6 12 18 24 30 36 42 48 54 60 66 72]
[ 7 14 21 28 35 42 49 56 63 70 77 84]
[ 8 16 24 32 40 48 56 64 72 80 88 96]
[ 9 18 27 36 45 54 63 72 81 90 99 108]
[ 10 20 30 40 50 60 70 80 90 100 110 120]
[ 11 22 33 44 55 66 77 88 99 110 121 132]
[ 12 24 36 48 60 72 84 96 108 120 132 144]]
### Ufunc mini-exercises
Each of the following functions take an array as input, and return an array as output. They are implemented using loops, which is not very efficient.
1. For each function, implement a fast version which uses ufuncs to calculate the result more efficiently. Double-check that you get the same result for several different arrays.
2. use the %timeit magic to time the execution of the two implementations for a large array (say, 1000 elements).
In [30]:
# 1. computing the element-wise sine + cosine
from math import sin, cos
def slow_sincos(x):
"""x is a 1-dimensional array"""
y = np.zeros_like(x)
for i in range(len(x)):
y[i] = sin(x[i]) + cos(x[i])
return y
x = np.random.random(5)
print slow_sincos(x)
[ 1.41379464 1.24311597 1.39711558 1.1674103 1.41168674]
In [31]:
# write a fast_sincos function
In [32]:
# 2. computing the difference between adjacent squares
def slow_sqdiff(x):
"""x is a 1-dimensional array"""
y = np.zeros(len(x) - 1)
for i in range(len(y)):
y[i] = x[i + 1] ** 2 - x[i] ** 2
return y
x = np.random.random(5)
print slow_sqdiff(x)
[-0.11965629 -0.25739054 0.01840098 0.2762583 ]
In [33]:
# write a fast_sqdiff function
In [34]:
# 3. computing the outer-product of each consecutive pair
def slow_pairprod(x):
"""x is a 1-dimensional array"""
if len(x) % 2 != 0:
raise ValueError("length of x must be even")
N = len(x) / 2
y = np.zeros((N, N))
for i in range(N):
for j in range(N):
y[i, j] = x[2 * i] * x[2 * j + 1]
return y
x = np.arange(1, 9)
print slow_pairprod(x)
[[ 2. 4. 6. 8.]
[ 6. 12. 18. 24.]
[ 10. 20. 30. 40.]
[ 14. 28. 42. 56.]]
In [35]:
# write a fast_pairprod function
## Strategy 2. Using Numpy Aggregates
Aggregates are functions over arrays which return smaller arrays.
Numpy has several built-in
In [36]:
# 10 x 10 array drawn from a standard normal
x = np.random.randn(10, 10)
In [37]:
print x.mean()
0.00323045925586
In [38]:
print np.mean(x)
0.00323045925586
In [39]:
print x.std()
1.0801379167
In [40]:
print x.var()
1.1666979191
In [41]:
print x.sum()
0.323045925586
In [42]:
print x.prod()
-5.72610473992e-23
In [43]:
print np.median(x)
0.00389134929792
In [44]:
print np.percentile(x, 50)
0.00389134929792
In [45]:
print np.percentile(x, (25, 75))
[-0.78582974895964608, 0.7374780554465159]
### Aggregates along certain dimensions
In [46]:
x = np.random.rand(3, 5)
print x
[[ 0.20461757 0.63148584 0.87816143 0.60442118 0.79228768]
[ 0.40654761 0.97038926 0.96468488 0.27046867 0.17623957]
[ 0.08734478 0.38353736 0.16706956 0.17744179 0.14886722]]
In [47]:
print x.sum(0) # sum along rows
[ 0.69850996 1.98541246 2.00991587 1.05233165 1.11739447]
In [48]:
print x.sum(1) # sum along columns
[ 3.11097371 2.78832999 0.96426071]
In [49]:
print np.median(x, 1)
[ 0.63148584 0.40654761 0.16706956]
In [50]:
print np.mean(x, 1)
[ 0.62219474 0.557666 0.19285214]
### Binary ufuncs as aggregates
Any binary ufunc (a ufunc taking two arguments) can be turned into an aggregate using the reduce() method:
In [51]:
np.sum(x, 1)
Out[51]:
array([ 3.11097371, 2.78832999, 0.96426071])
In [52]:
np.add.reduce(x, 1)
Out[52]:
array([ 3.11097371, 2.78832999, 0.96426071])
In [53]:
np.prod(x, 1)
Out[53]:
array([ 0.05433798, 0.01814108, 0.00014784])
In [54]:
np.multiply.reduce(x, 1)
Out[54]:
array([ 0.05433798, 0.01814108, 0.00014784])
In [55]:
np.divide.reduce(x, 1)
Out[55]:
array([ 0.77051726, 9.11086414, 51.60324064])
A caution: for reduce methods, the default axis is 0:
In [56]:
np.add.reduce(x)
Out[56]:
array([ 0.69850996, 1.98541246, 2.00991587, 1.05233165, 1.11739447])
In [57]:
np.sum(x)
Out[57]:
6.8635644110229004
### Beware the built-in Python aggregates!
Python has a min, max, and sum aggregate built-in. These are much more general than the versions in NumPy:
In [58]:
x = np.random.random(10000)
%timeit np.sum(x)
%timeit sum(x)
100000 loops, best of 3: 16.4 µs per loop
100 loops, best of 3: 4 ms per loop
Dynamic type-checking is slow.
Make sure to use Numpy's sum, min, and max.
### Aggregate Mini-exercises
Take the following functions, and convert them into an efficient form using aggregates. Each function expects a 1-dimensional array as input. Double-check that your function returns the same result as the original
In [59]:
def slow_cubesum(x):
"""x is a 1D array"""
result = 0
for i in range(len(x)):
result += x[i] ** 3
return result
x = np.random.random(100)
print slow_cubesum(x)
28.3126613072
In [60]:
# implement fast_cubesum
In [61]:
def slow_rms(x):
"""x is a 1D array"""
m = np.mean(x)
rms = 0
for i in range(len(x)):
rms += (x[i] - m) ** 2
rms /= len(x)
return np.sqrt(rms)
x = np.random.random(100)
print slow_rms(x)
0.298246500479
In [62]:
# implement fast_rms
Now we return to our silly function from the beginning of this section. Can you implement a fast version using ufuncs and aggregates?
In [63]:
def slow_sillyfunc(N):
"""N is an integer"""
d = 0.0
for i in range(N):
d += (i % 3 - 1) * i
return d
print slow_sillyfunc(100)
-33.0
In [64]:
# Implement fast_sillyfunc using ufuncs & aggragates
## Strategy 3: Using Numpy Broadcasting
We've taken a look at broadcasting previously. But it's important enough that we'll review it quickly here:
1. If the two arrays differ in their number of dimensions, the shape of the array with fewer dimensions is padded with ones on its leading (left) side.
2. If the shape of the two arrays does not match in any dimension, the array with shape equal to 1 in that dimension is stretched to match the other shape.
3. If in any dimension the sizes disagree and neither is equal to 1, an error is raised.
### Some Broadcasting examples...
In [65]:
x = np.arange(10)
print x ** 2
[ 0 1 4 9 16 25 36 49 64 81]
In [66]:
Y = x * x[:, np.newaxis]
print Y
[[ 0 0 0 0 0 0 0 0 0 0]
[ 0 1 2 3 4 5 6 7 8 9]
[ 0 2 4 6 8 10 12 14 16 18]
[ 0 3 6 9 12 15 18 21 24 27]
[ 0 4 8 12 16 20 24 28 32 36]
[ 0 5 10 15 20 25 30 35 40 45]
[ 0 6 12 18 24 30 36 42 48 54]
[ 0 7 14 21 28 35 42 49 56 63]
[ 0 8 16 24 32 40 48 56 64 72]
[ 0 9 18 27 36 45 54 63 72 81]]
In [67]:
print Y + 10 * x
[[ 0 10 20 30 40 50 60 70 80 90]
[ 0 11 22 33 44 55 66 77 88 99]
[ 0 12 24 36 48 60 72 84 96 108]
[ 0 13 26 39 52 65 78 91 104 117]
[ 0 14 28 42 56 70 84 98 112 126]
[ 0 15 30 45 60 75 90 105 120 135]
[ 0 16 32 48 64 80 96 112 128 144]
[ 0 17 34 51 68 85 102 119 136 153]
[ 0 18 36 54 72 90 108 126 144 162]
[ 0 19 38 57 76 95 114 133 152 171]]
In [68]:
print Y + 10 * x[:, np.newaxis]
[[ 0 0 0 0 0 0 0 0 0 0]
[ 10 11 12 13 14 15 16 17 18 19]
[ 20 22 24 26 28 30 32 34 36 38]
[ 30 33 36 39 42 45 48 51 54 57]
[ 40 44 48 52 56 60 64 68 72 76]
[ 50 55 60 65 70 75 80 85 90 95]
[ 60 66 72 78 84 90 96 102 108 114]
[ 70 77 84 91 98 105 112 119 126 133]
[ 80 88 96 104 112 120 128 136 144 152]
[ 90 99 108 117 126 135 144 153 162 171]]
In [69]:
Y = np.random.random((2, 3, 4))
x = 10 * np.arange(3)
print Y + x[:, np.newaxis]
[[[ 0.86229625 0.64617272 0.81395431 0.57150281]
[ 10.72496019 10.95945758 10.0420753 10.37791557]
[ 20.70117792 20.47498435 20.92576569 20.26448722]]
[[ 0.98847073 0.84611754 0.62249531 0.97384757]
[ 10.29122012 10.81186259 10.9161937 10.14179008]
[ 20.65886012 20.99372243 20.88986833 20.12829632]]]
### Quick Broadcasting Exercise
Now, assume you have $N$ points in $D$ dimensions, represented by an array of shape [N, D].
1. Compute the mean of the distribution of points efficiently using the built-in np.mean aggregate (that is, find the D-dimensional point which is the mean of the rest of the points)
2. Compute the mean of the distribution of points efficiently using the np.add ufunc.
3. Compute the standard error of the mean $\sigma_{mean} = \sigma N^{-1/2}$, where $\sigma$ is the standard-deviation, using the np.std aggregate.
4. Compute this again using the np.add ufunc.
5. Construct the matrix M, the centered and normalized version of the X array: $M_{ij} = (X_{ij} - \mu_j) / \sigma_j$ This is one version of whitening the array.
In [70]:
X = np.random.random((1000, 5)) # 1000 points in 5 dimensions
In [71]:
# 1. Compute the mean of the 1000 points in X
In [72]:
# 2. Compute the mean using np.add
In [73]:
# 3. Compute the standard deviation across the 1000 points
In [74]:
# 4. Compute the standard deviation using np.add only
In [75]:
# 5. Compute the whitened version of the array
## Strategy 4: Fancy Indexing and Masking
The last strategy we will cover is fancy indexing and masking.
For example, imagine you have an array of data where negative values indicate some kind of error.
In [76]:
x = np.array([1, 2, 3, -999, 2, 4, -999])
How might you clean this array, setting all negative values to, say, zero?
In [77]:
for i in range(len(x)):
if x[i] < 0:
x[i] = 0
print x
[1 2 3 0 2 4 0]
A faster way is to construct a boolean mask:
In [78]:
x = np.array([1, 2, 3, -999, 2, 4, -999])
mask = (x < 0)
[False False False True False False True]
And the mask can be used directly to set the value you desire:
In [79]:
x[mask] = 0
print x
[1 2 3 0 2 4 0]
Typically this is done directly:
In [80]:
x = np.array([1, 2, 3, -999, 2, 4, -999])
x[x < 0] = 0
print x
[1 2 3 0 2 4 0]
### Useful masking functions
In [81]:
x = np.random.random(5)
print x
[ 0.97053187 0.72813759 0.95240005 0.04301341 0.77007516]
In [82]:
x[x > 0.5] = np.nan
print x
[ nan nan nan 0.04301341 nan]
In [83]:
x[np.isnan(x)] = np.inf
print x
[ inf inf inf 0.04301341 inf]
In [84]:
np.nan == np.nan
Out[84]:
False
In [85]:
x[np.isinf(x)] = 0
print x
[ 0. 0. 0. 0.04301341 0. ]
In [86]:
x = np.array([1, 0, -np.inf, np.inf, np.nan])
print "input ", x
print "x < 0 ", (x < 0)
print "x > 0 ", (x > 0)
print "isinf ", np.isinf(x)
print "isnan ", np.isnan(x)
print "isposinf", np.isposinf(x)
print "isneginf", np.isneginf(x)
input [ 1. 0. -inf inf nan]
x < 0 [False False True False False]
x > 0 [ True False False True False]
isinf [False False True True False]
isnan [False False False False True]
isposinf [False False False True False]
isneginf [False False True False False]
### Boolean Operations on Masks
Use bitwise operators (and make sure to use parentheses!)
In [87]:
x = np.arange(16).reshape((4, 4))
print x
[[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[12 13 14 15]]
In [88]:
print (x < 5)
[[ True True True True]
[ True False False False]
[False False False False]
[False False False False]]
In [89]:
print ~(x < 5)
[[False False False False]
[False True True True]
[ True True True True]
[ True True True True]]
In [90]:
print (x < 10) & (x % 2 == 0)
[[ True False True False]
[ True False True False]
[ True False False False]
[False False False False]]
In [91]:
print (x > 3) & (x < 8)
[[False False False False]
[ True True True True]
[False False False False]
[False False False False]]
### Counting elements with a mask
Sum over a mask to find the number of True elements:
In [92]:
x = np.random.random(100)
print "array is length", len(x), "and has"
print (x > 0.5).sum(), "elements are greater than 0.5"
array is length 100 and has
50 elements are greater than 0.5
In [93]:
# clip is a useful function:
x = np.clip(x, 0.3, 0.6)
print np.sum(x < 0.3)
print np.sum(x > 0.6)
0
0
In [94]:
# works for 2D arrays as well
X = np.random.random((10, 10))
print (X < 0.1).sum()
9
### where function: Turning a mask into indices
In [95]:
x = np.random.random((3, 3))
print x
[[ 0.82518495 0.49805524 0.00840461]
[ 0.85186368 0.98583616 0.77922605]
[ 0.52795826 0.20786051 0.49627606]]
In [96]:
print np.where(x < 0.3)
(array([0, 2]), array([2, 1]))
In [97]:
print x[x < 0.3]
[ 0.00840461 0.20786051]
In [98]:
print x[np.where(x < 0.3)]
[ 0.00840461 0.20786051]
When you index with the result of a where function, you are using what is called fancy indexing: indexing with tuples
### Fancy Indexing (indexing with sequences)
In [99]:
X = np.arange(16).reshape((4, 4))
print X
[[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[12 13 14 15]]
In [100]:
X[(0, 1), (1, 0)]
Out[100]:
array([1, 4])
In [101]:
X[range(4), range(4)]
Out[101]:
array([ 0, 5, 10, 15])
In [102]:
X.diagonal()
Out[102]:
array([ 0, 5, 10, 15])
In [103]:
X.diagonal() = 100
File "<ipython-input-103-3064a6b8dbd8>", line 1
X.diagonal() = 100
SyntaxError: can't assign to function call
In [104]:
X[range(4), range(4)] = 100
In [105]:
print X
[[100 1 2 3]
[ 4 100 6 7]
[ 8 9 100 11]
[ 12 13 14 100]]
#### Randomizing the rows
In [106]:
X = np.arange(24).reshape((6, 4))
print X
[[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
[20 21 22 23]]
In [107]:
i = np.arange(6)
np.random.shuffle(i)
print i
[0 5 3 1 4 2]
In [108]:
print X[i] # X[i, :] is identical
[[ 0 1 2 3]
[20 21 22 23]
[12 13 14 15]
[ 4 5 6 7]
[16 17 18 19]
[ 8 9 10 11]]
Fancy indexing also works for multi-dimensional index arrays
In [109]:
i2 = i.reshape(3, 2)
print X[i2]
[[[ 0 1 2 3]
[20 21 22 23]]
[[12 13 14 15]
[ 4 5 6 7]]
[[16 17 18 19]
[ 8 9 10 11]]]
In [110]:
print X[i2].shape
(3, 2, 4)
## Summary: Speeding up NumPy
It's all about moving loops into compiled code:
1. Use Numpy ufuncs to your advantage (eliminate loops!)
2. Use Numpy aggregates to your advantage (eliminate loops!)
4. Use Numpy slicing and masking to your advantage (eliminate loops!)
5. Use a tool like SWIG, cython or f2py to interface to compiled code.
## Homework: asteroid data
In the github repository, there is a file containing measurements of 5000 asteroid orbits, at notebooks/data/asteroids5000.csv.
These are compiled from a query at http://ssd.jpl.nasa.gov/sbdb_query.cgi
### Part 1: loading and exploring the data
1. Use np.genfromtxt to load the data from the file. This is like loadtxt, but can handle missing data.
• Remember to set the appropriate delimiter keyword.
2. genfromtxt sets all missing values to np.nan. Use the operations we discussed here to answer these questions:
• How many values are missing in this data?
• How many complete rows are there? i.e. how many objects have no missing values?
3. Create a new array containing only the rows with no missing values.
4. Compute the maximum, minimum, mean, and standard deviation of the values in each column.
In []:
### Part 2: Plotting the data
1. Use the bash head command to display the first line of the data file: this lists the names of the columns in the dataset. (remember that bash commands in the notebook are indicated by !, and that head -n displays the first n lines of a file)
2. Invoke the matplotlib inline magic to make figures appear inline in the notebook
3. Use plt.scatter to plot the semi-major axis versus the sine of the inclination angle (note that the inclination angle is listed in degrees -- you'll have to convert it to radians to compute the sine). What do you notice about the distribution? What do you think could explain this?
4. Use plt.scatter to plot a color-magnitude diagram of the asteroids (H vs B-V). You should see two distinct "families" of asteroids indicated in this plot. Over-plot a line that divides these.
5. Repeat the orbital parameter plot from above, but plot the two "families" from #4 in different colors. Note that this magnitude is undefined for many of the asteroids. Do you see any correlation between color and orbit?
6. Compare what you found to plots in Parker et al. 2008. Note that we're not using the same data here, but we're looking at a similar collection of objects.
In []: | 2014-10-25 21:08:06 | {"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": 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": 0.27119624614715576, "perplexity": 1967.0401635482392}, "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-2014-42/segments/1414119650516.39/warc/CC-MAIN-20141024030050-00058-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-find-the-length-of-the-missing-side-given-a-b-24-c-40 | # How do you find the length of the missing side given a? b=24 c=40?
Jun 29, 2016
a = 32
#### Explanation:
To answer this problem, you should use the Pythagorean Theorem:
The hypotenuse (c=40) and one of the legs (b=24) are known, so all we have to do is solve for a. We can do that by plugging in our known values:
${a}^{2} + {24}^{2} = {40}^{2}$
${24}^{2}$ or $24 \times 24$ = 576
${40}^{2}$ or $40 \times 40$ = 1600
Thus, ${a}^{2} + 576 = 1600$. Now subtract 576 from both sides of the equation to get ${a}^{2}$ by itself:
${a}^{2} + 576 = 1600$
-576 -576
You should end up with:
${a}^{2} = 1024$
Next, take the square root of both sides to find a. The square root
(sqrt) is the inverse of the square (${a}^{2}$)
$\sqrt{{a}^{2}} = \sqrt{1024}$
Therefore, a = 32
You can check your answer by plugging a and b into the equation and solve for c to see if your answer matches the given value of c:
${32}^{2} + {24}^{2} = {40}^{2}$ | 2020-08-14 17:29:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 13, "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": 0.8182908892631531, "perplexity": 348.94565907862375}, "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/1596439739347.81/warc/CC-MAIN-20200814160701-20200814190701-00326.warc.gz"} |
https://cstheory.stackexchange.com/questions/27206/how-high-are-the-higher-types-that-appear-in-practice/27208#27208 | # How high are the higher types that appear in practice?
This is admittedly a rather naively put and vague question, and I'm not sure how much more specific I want or can make it, but I'll try.
By "practice" I mean surely in actual programming practice (of which I embarrassingly don't know much), but also in mathematical practice, whenever certain higher-type objects are employed as examples or counterexamples within arguments.
By the "height" of a type, I don't mean to include the obvious and natural arbitrariness involved in objects like, say, the fixpoint functional (in fact, in the spirit of the question, I would prefer to understand this as a type $2$ object "up to parameter types", so to speak). A better example of what I mean would be the well-known (from mathematical practice, at least) fan functional, of type $((\mathbb{N} \to \mathbb{B}) \to \mathbb{N}) \to \mathbb{N}$, given by $$\lambda f. \mu m. \forall_{\alpha, \beta} \left( \forall_{n < m} \alpha(n) = \beta(n) \to f(\alpha) = f(\beta) \right) \ ,$$ where $f : (\mathbb{N} \to \mathbb{B}) \to \mathbb{N}$ and $\alpha, \beta : \mathbb{N} \to \mathbb{B}$.
My questions: Are there any objects of yet higher type than $3$ (possibly "up to parameter types") that are naturally used in the literature? In practice?
• It is also perhaps interesting to note that while the type might not be an higher than 3 or so (cody's example) that you might want to instantiate a quantified type varible in a 2nd or 3rd order function with other another 1st or 2nd order function type thus creating, subtly if we are talking about Hindly Milner, somthing as large as a 5th order function. Kinda rare but I am sure I have done this at some point. 6th order is pretty insane though. I've accidentally made some huge order functions playing around with Church numerals if I recall as well
– Jake
Oct 27 '14 at 5:26
Okasaki's Functional Pearl, "Even Higher-Order Functions for Parsing or Why Would Anyone Ever Want To Use a Sixth-Order Function?" answers this question with a type of order 6.
This is a very interesting question!
Andrej Bauer wrote a very nice blog post, Interesting Higher Order Functionals, which is precisely about this question. He defined a "genuine" function of order $n$ as one which is not expressible in terms of functions order $k < n$ plus typed lambda-calculus. He points out there are only a few known examples of order 3, and he knows of zero examples of order 4.
He also mentions that Paul Taylor does use some very unusual fourth-order functions in his program of Abstract Stone Duality. These are not "genuine" in Andrej's sense, in that they are definable as typed lambda terms, but they are definitely not trivial stackings of known constructions. (In some sense ASD exists to give them intellectually sensible types.)
• Right, I quickly read Bauer's post after cody mentioned it; it is indeed a nice one and I had somehow missed it, so thanks to both of you. Oct 27 '14 at 17:06
The continuation monad in Haskell has got some nice order 3 examples
withCont :: ((b -> r) -> a -> r) -> Cont r a -> Cont r b
There's some discussion on Andrej's blog, but they don't really get past order 3... | 2022-01-19 17:43:39 | {"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": 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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8319129347801208, "perplexity": 798.012065417404}, "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/1642320301475.82/warc/CC-MAIN-20220119155216-20220119185216-00466.warc.gz"} |
https://tex.stackexchange.com/questions/481127/sort-bibliography-with-the-order-of-appearance-into-bib-file-edited | # Sort bibliography with the order of appearance into .bib file (EDITED)
I work with natbib and \bibliographystyle{abstract}. I use this style because the ordering is done with respect to the key. Is it possible achieve a sorting with respect the order of appearance into the associated .bib file but without modifying anything other? Thanks in advance.
EDIT
I found this answer in the question here
Use BibTeX key as the cite key
relevant
\documentclass{article}
\usepackage[style=alphabetic,sorting=debug]{biblatex}
\DeclareFieldFormat{labelalpha}{\thefield{entrykey}}
\DeclareFieldFormat{extraalpha}{}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@misc{whatever,
author = {Author, A.},
year = {2001},
title = {Testing the effects of biblatex styles on bibliography formatting},
}
@misc{B02f,
author = {Buthor, B.},
year = {2002},
title = {First},
}
@misc{B02s,
author = {Buthor, B.},
year = {2002},
title = {Second},
}
\end{filecontents}
\nocite{*}
\begin{document}
\printbibliography
\end{document}
So basically how could I achieve the bibliography sorting with respect to the order of appearance in the bib file using natbib and \bibliographystyle{abstract}(so that a key of the form [CINV] comes before a key of the form [CINT])?
• The term "order of appearance" is slightly ambiguous: Do you mean order of first citation in the body of the document, or order of appearance in the bib file? – Mico Mar 23 at 21:53
• @Mico Sorry for not being clear. The latter one (in bib file!). – Dimitris Mar 23 at 21:54
• Could you please add a [minimal working example with bibliography (MWEB)][tex.meta.stackexchange.com/questions/4407/…) that show how you use the commands you describe in your question? – leandriis Mar 23 at 22:02
• The abstract bibliography style does not seem to cooperate well with the natbib package. (Package natbib Error: Bibliography not compatible with author-year citations.) – leandriis Mar 23 at 22:02
• @leandriis - The OP has (probably) figured out by now that the natbib citation management package needs to be loaded with the option numbers. – Mico Mar 23 at 22:25 | 2019-04-23 19:57:29 | {"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": 0.781782329082489, "perplexity": 3452.248285312953}, "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/1555578613603.65/warc/CC-MAIN-20190423194825-20190423220825-00152.warc.gz"} |
http://mathematica.stackexchange.com/questions/3966/arctan-expressed-as-a-radian-fraction?answertab=oldest | # ArcTan expressed as a radian fraction?
This type of answer is what I'm looking for:
In[58]:= ArcTan @ 1
Out[58]= π/4
This is what mathematica gives me:
In[59]:= ArcTan@2
Out[59]= ArcTan[2]
Is it possible to express ArcTan in terms of $\pi$? I understand some fractions would be hairy.
I am using Mathematica 8.
-
Maybe this
HoldForm[Pi] (1/Pi ArcTan@2.)
or if you want a nicer way
Rationalize /@ (HoldForm[Pi] N@(1/Pi ArcTan@Range[5]))
Edit
The latter method works well in cases when there is a rational fraction of $\pi$ :
Rationalize /@ (HoldForm[Pi] N @ (1/Pi ArcTan @ { Sqrt[1 - 2/Sqrt[5]], 2 - Sqrt[3],
1/Sqrt[3], Sqrt[3], 1}))
To sum up : Mathematica does what it should do, namely ArcTan[2] is not a rational fraction of $\pi$ and that's why it returns ArcTan[2] unlike in case ArcTan[1]. The above method is to express ArcTan[x] in terms of a real multiple of $\pi$.
If you want to get back what you have evaluated you shoud use ReleaseHold, e.g.
Tan @ ReleaseHold @ %
-
Short answer: no, ArcTan[2] is not fraction of $\pi$. But this is more of a mathematics question than pertaining to Mathematica.
If you want to “check” that the result is not expressable as a fraction of $\pi$, you can check for the continued fraction reprentation of $\arctan(2)/\pi$, and see that it does not seem to converge:
Table[FromContinuedFraction@ContinuedFraction[ArcTan[2]/\[Pi], n], {n, 20}]
- | 2015-05-26 16:05:49 | {"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": 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": 0.2080487608909607, "perplexity": 2033.3314223924435}, "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-2015-22/segments/1432207928864.73/warc/CC-MAIN-20150521113208-00259-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://math.meta.stackexchange.com/tags/answers/hot | # Tag Info
97
No you are not required to follow up in comments. At the same time I feel it is not unexpected and rather normal that an asker tries to engage somebody that answered in case something is open. I feel it is strange to get upset about this, doubly so given the sketchy nature of your answer. If you do not want to further engage, say so clearly and neutrally ...
59
I know for a fact that reputation does not reflect infallibility. As a general rule of politeness, when I see something that seems questionable, I ignore the reputation of the author and address them with respect. It is nothing more than the same that I would expect from someone else when I make a mistake. The comments suggesting phrases like "Perhaps I am ...
58
Yes, I do on occasion sit and wait for questions I can answer, although usually I will have on some background tv or news. I don't just intently stare at the screen hitting the refresh button every 5 seconds :) Here are some reasons why (for reference, I am starting grad school this fall): 1.) this gives me a chance to work on my teaching and exposition ...
55
Strictly speaking, to provide an answer to a question with bad formatting / lack of context does not break any rule. If you suggested such user to improve his/her question you did the right thing. The same applies to the downvote / closing vote, if you thought that was the right course of action. But. We all like MSE to be populated by good questions and ...
50
An answer with $0$ votes does not hurt your reputation. If it is a good answer, it helps the site and someone may discover that it is a good answer and upvote it in the future. If it is a good answer, leave it.
49
Please do not post joke answers in response to joke questions. My reasoning has three parts (the first two are SE specific and are linked, while the other is common internet protocol): Joke Questions are not on-topic for the site. Joke questions do not add anything to the site. Thus, they should be deleted: What are the criteria for deletion? For ...
47
I am strongly against downvoting complete solutions, and I believe such behavior would have a negative long-term impact on the site as a whole. I went over my reasons in detail in this past answer, A Consolidated Homework Policy, and I have pasted a large portion of that answer below. I also want to link to Professor Hamkins answer to the question What do ...
45
I have to disagree with you that the motivation for posting answers after others have done so is related to reputation hunt. I am a slow typer and a perfectionist ( others would say a nitpicker!), so that it takes me a long time to compose my answers since I do a lot of checking and try to write a well-formulated answer, of which I won't be ashamed in ...
44
While I agree with quid's1 answer, let me add a point I consider very important: You do volunteer work when you answer questions, and you are entitled to do it on your own terms. If you think a conversation is going in an unwanted direction, you can leave it. It is polite to tell the other party as much, but I would not consider you obliged to do anything. ...
43
No.
42
If I am working on an answer, and I see the "New Answer" display, I keep working until my answer is done and then submit it. If the new answer is correct, and especially if it is similar to mine, I upvote it. I never remove an answer of mine just because it is similar to another, and I see nothing wrong in similar answers if they were derived independently....
41
Gerry Myerson's reply is factually correct, and I do not want to dispute it, only add to it: The goal of the site is to have a repository of clear, complete, and correct answers. Those answers will last a long time and readers will continue to extract value from them far into the future. If some people are annoyed in the process of developing the archive, ...
39
I've wondered about this too. If it is a very simple answer that didn't take me too much time, I'll just delete it. But if I've invested a lot of time in my answer, and another answer just happens to beat mine by a short period of time, I'll post mine too and let the cards fall where they may. Incidentally, this topic is an interesting reflection of the ...
35
Your (hypothetical but not unrealistic) examples are essentially a symptom of the site having many users, which means that easy questions are likely to receive several answers at almost the same time. For more difficult questions, different answers are likely to be explained in different ways, even if they use the same techniques. That's potentially very ...
35
Sometimes the question explicitly only asks for hints. Rather more often, a hint will actually be more useful to the asker. This is often the case for "homework" questions (when they deserve to be answered at all). With many exercise-type questions, nobody (including the problem setter) actually cares about the answer, but the point of the exercise is to ...
34
Your answer is fine. I have no idea why it was downvoted so much. Even if the original person asking the question does not know about Cauchy sequences (we have no way to tell), it is OK for some answers to address more advanced viewpoints. And, surely, the most direct way to show that $\sum_{k \in \mathbb{N}} 1/k$ diverges is to verify that the sequence of ...
34
I don't really think that deleting your answers because no-one has reacted to them is the greatest use of your time, to be honest. There's little cost to the site to keep your answers available, and someone may come around later to whom they are helpful and who will provide you with some sort of feedback. And it does take some amount of your effort to ...
32
If you want to, I think it is good for you to post an answer to your question. You can even accept your own answer. To me, it seems like a waste to just delete the question because what if someone has a similar question later on? I agree. On the other hand, it may look like I am cheating the system somehow I don't think so. (although you don't ...
29
If the question is tagged homework, I will post hints as answers, since I hate questions showing up in the "unanswered" queue despite being fully resolved in the comments. If I strongly suspect the question is homework, I will post hints as answers to discourage others from spoiling full solutions. Otherwise, if the hint is very short and I expect others ...
29
A couple of things to answer your question: 1) Good hints belong here. Bad hints do not. If you want to make a good contribution to the site as an answerer, do the problem out always. It does not matter whether you plan to post a full, novella-like solution that details every step as far as $1+1=2$ or just enough to give a taste of the path one needs to ...
29
There are many possible reasons for down-voting that have nothing to do with correctness in a narrow sense, including: DVer thinks the answer is too sloppy. DVer thinks the answer is too terse. DVer thinks the answer is redundant. DVer thinks the answer is too clumsy. DVer thinks the answer uses needlessly advanced tools. Or, DVer thinks the question just ...
28
Mathematics, unlike most other branches of human knowledge and experience, is relatively universal in the sense that everyone has to take some mathematics classes in their academic career. This means that a very large number of people either are currently, or have in the past, worked through the kind of basic algebra and calculus problems that show up here ...
27
You have several options: If you can fix the error right away, use the "edit" button to do so. If you cannot fix the error right away, but want to work on it, and think the answer might be useful to others in the meantime, you can edit it to add a sentence at the top saying something like: Note. This answer has an error (explain what it is); I am ...
27
I understand that the consensus of the community about downvoting is that each one has its own personal reasons to downvote, and should not have to justify that. Yes, but there is one major exception, which is that one must not vote based on the identity of the post-owner. Shortly after I explained the downvote, my answer was downvoted with no reasoning....
27
As others have mentioned, bounties are Stack Exchange's tool for this purpose. Placing a bounty costs some reputation, but it will have the following effect: The question is immediately bumped to the top of the active question list. The question gets placed in the home page's featured tab for seven days. both of which attract more visitors who (...
26
The idea of this automatic message is that answers should be good, relatively self-contained, objective responses to a question. When the guidelines suggest avoiding asking for clarification or responding to other answers, they don't mean to avoid asking for clarification at all costs. Instead, they mean don't ask for clarification in your answer. There is ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2019-12-16 06:17:47 | {"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": 0.49985232949256897, "perplexity": 520.1855205239602}, "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/1575541317967.94/warc/CC-MAIN-20191216041840-20191216065840-00257.warc.gz"} |
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/12%3A_Group_Theory_-_The_Exploitation_of_Symmetry/12.01%3A_The_Exploitation_of_Symmetry | # 12.1: The Exploitation of Symmetry Can Help Simplify Numerical Calculations
To a fully understand the math behind group theory one needs to take a look at the theory portion of the Group Theory topic or refer to one of the reference text listed at the bottom of the page. Never the less as Chemist the object in question we are examining is usually a molecule. Though we live in the 21st century and much is known about the physical aspects that give rise to molecular and atomic properties. The number of high level calculations that need to be performed can be both time consuming and tedious. To most experimentalist this task is takes away time and is usually not the integral part of their work. When one thinks of group theory applications one doesn't necessarily associated it with everyday life or a simple toy like a Rubik's cube. A Rubik's cube is an a cube that has a $$3 \times 3$$ array of different colored tiles on each of its six surfaces, for a total of 54 tiles. Since the cube exist in 3D space, the three axis are $$x$$, $$y$$, $$z$$. Since the rubik's cube only allows rotation which are called operations, there are three such operations around each of the $$x$$, $$y$$, $$z$$ axis.
Of course the ultimate challenge of a Rubik's cube is to place all six colors on each of the six faces. By performing a series of such operations on the Rubik's cube one can arrive at a solution (A link of a person solving a Rubik's cube1 in 10.4s with operations performed noted, the operations performed will not translate to chemistry applications but it is a good example of how symmetry operations arrive at a solution). The operations shown in the Rubik's cube case are inherent to the make up of the cube, i.e., the only operations allowed are the rotations along the x, y, z axis. Therefore the Rubik's cube only has x,y,z rotation operations. Similarly the operations that are specific to a molecule are dependent on its symmetry.
Using group theory, we can exploit the symmetry of molecules to give us a rich amount of information on the molecular orbitals, rotations, and vibrations of bonds, to name a few. Making symmetry arguments, we can skip complicated quantum calculations to gain qualitatively accurate information.
In section 10.7, we used Hückel theory to explore the $$\pi$$ bonding network of benzene by constructing linear combinations of $$2p_x$$ atomic orbitals on the carbon atoms. In doing so, the roots of the secular equations were found via solving the $$6 \times 6$$ secular determinant.
$\left|\begin{array}{cccccc}x&1&0&0&0&1\\1&x&1&0&0&0\\0&1&x&1&0&0\\0&0&1&x&1&0\\0&0&0&1&x&1\\1&0&0&0&1&x\end{array}\right|=0\label{31}$
Since the secular determinant is a $$6 \times 6$$ matrix, there are six solutions or values of $$x$$ that can be determined after expanding the determinant into the resulting (6th-order) polynomial.
$x^6-6x^4 + 9x^2 -4 =0 \label{poly1}$
Secular determinants are formulated in terms of a specific basis set; i.e., a set of functions that describe the wavefunctions. For the determinnat in Equation $$\ref{31}$$, that basis set is the the $$\{|2p_z \rangle \}$$ orbitals on the carbons. However, any basis set can be used to represent the determinant (long as it span the same space). For example, the following linear combination of $$\{|2p_z \rangle \}$$ orbitals could also be used:
\begin{align} & | \phi_1 \rangle = \dfrac{1}{\sqrt{6}} \left[ | 2p_{z1} \rangle+ | 2p_{z2} \rangle + | 2p_{z3} \rangle + | 2p_{z4} \rangle + | 2p_{z5} \rangle + | 2p_{z6} \rangle \right] \nonumber \\ & | \phi_2 \rangle = \dfrac{1}{\sqrt{4}} \left[ | 2p_{z2} \rangle + | 2p_{z3} \rangle - | 2p_{z4} \rangle - | 2p_{z5} \rangle \right] \nonumber \\ & | \phi_3 \rangle = \dfrac{1}{\sqrt{3}} \left[ | 2p_{z1} \rangle + \dfrac{1}{2}| 2p_{z2} \rangle - \dfrac{1}{2} | 2p_{z3} \rangle - | 2p_{z4} \rangle - \dfrac{1}{2} | 2p_{z5} \rangle + \dfrac{1}{2} | 2p_{z6} \rangle \right] \nonumber \\ & | \phi_4 \rangle = \dfrac{1}{\sqrt{4}} \left[ | 2p_{z2} \rangle - | 2p_{z3} \rangle + | 2p_{z4} \rangle - | 2p_{z5} \rangle \right] \nonumber \\ & | \phi_5 \rangle = \dfrac{1}{\sqrt{3}} \left[ | 2p_{z1} \rangle - \dfrac{1}{2}| 2p_{z2} \rangle - \dfrac{1}{2} | 2p_{z3} \rangle + | 2p_{z4} \rangle - \dfrac{1}{2} | 2p_{z5} \rangle - \dfrac{1}{2} | 2p_{z6} \rangle \right] \nonumber \\ & | \phi_6 \rangle = \dfrac{1}{\sqrt{6}} \left[ | 2p_{z1} \rangle- | 2p_{z2} \rangle + | 2p_{z3} \rangle - | 2p_{z4} \rangle + | 2p_{z5} \rangle - | 2p_{z6} \rangle \right] \nonumber \end{align}
In this new basis set $$\{\phi \rangle \}$$, the secular determinant Equation $$\ref{31}$$ is represented as
$\left|\begin{array}{cccccc} x+2&0&0&0&0&0 \\0&x-2&0&0&0&0 \\0&0&x+1& \dfrac{x+1}{2}&0&0 \\0&0& \dfrac{x+1}{2} &x+1&0&0 \\0&0&0&0&x-1& \dfrac{x-1}{2} \\0&0&0&0& \dfrac{x-1}{2} &x-1\end{array}\right|=0\label{32}$
This is the determinant into a bock diagonal form; which can be expanded into a product of smaller determinants to give the polynomial
$\dfrac{9}{16} ( x +2)(x-2)(x+1)^2(x-1)^2=0 \nonumber$
The roots to this equation are $$\pm2$$, $$\pm1$$ and $$\pm 1$$. This is not surprising since these are the same roots obtained from expanding the determinant in the original basis set (Equation $$\ref{poly1}$$). You may remember that the selection of a specific basis set to represent a function does not change the fundamental nature of the function (e.g., a parabola in 2D space is the same curve if represented in terms of Cartesian coordinates ($$x$$ and $$y$$) or polar coordinates ($$\theta$$ and $$r$$), which both span 2-D space).
As you recall, Hückel theory (irrespective of the basis set ) was used to simplify the general secular determinant (e.g., for benzene)
$\left|\begin{array}{cccccc} H_{11} - ES_{11} & H_{12} - ES_{12} & H_{13} - ES_{13} & H_{14} - ES_{14} & H_{15} - ES_{15} & H_{16} - ES_{16} \\ H_{21} - ES_{21} & H_{22} - ES_{22} & H_{23} - ES_{23} & H_{24} - ES_{24} & H_{25} - ES_{25} & H_{26} - ES_{26} \\ H_{31} - ES_{31} & H_{32} - ES_{32} & H_{33} - ES_{33} & H_{34} - ES_{34} & H_{35} - ES_{35} & H_{36} - ES_{36} \\ H_{41} - ES_{41} & H_{42} - ES_{42} & H_{43} - ES_{43} & H_{44} - ES_{44} & H_{45} - ES_{45} & H_{46} - ES_{46} \\ H_{51} - ES_{51} & H_{52} - ES_{52} & H_{53} - ES_{53} & H_{54} - ES_{54} & H_{55} - ES_{55} & H_{56} - ES_{56} \\ H_{61} - ES_{61} & H_{62} - ES_{62} & H_{63} - ES_{63} & H_{64} - ES_{64} & H_{65} - ES_{65} & H_{66} - ES_{6} \end{array}\right|=0\label{33}$
where $$H_{ij}$$ are the Hamiltonian matrix elements
$H_{ij} = \langle \phi_i | \hat{H} | \phi_j \rangle = \int \phi _{i}H\phi _{j}\mathrm {d} v \nonumber$
and $$S_{ij}$$ are the overlap integrals.
$S_{ij}= \langle \phi_i | \phi_j \rangle = \int \phi _{i}\phi _{j}\mathrm {d} v \nonumber$
In general, this involves solving 36 Hamiltonian matrix elements ($$H_{ij}$$) and 36 overlap integrals ($$S_{ij}$$), which can be a daunting task to do by hand without the assumptions of Hückel theory to help out. As with the application of Hückel theory, which was used to set most of these integrals to zero, solving for the energies from Equation $$\ref{33}$$ can be simplified by using the intrinsic symmetry of the benzene system to demonstrate (rigorously) that many of these integrals are zero. This is the subject of group theory. Group theory is used to exploit the symmetry of molecules to quickly gain insights into their properties, such as vibrations and molecular orbitals.
12.1: The Exploitation of Symmetry Can Help Simplify Numerical Calculations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Delmar Larsen & Jerry LaRue. | 2023-03-23 10:16:04 | {"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": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9751015305519104, "perplexity": 542.7085170353201}, "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/1679296945144.17/warc/CC-MAIN-20230323100829-20230323130829-00388.warc.gz"} |
https://landonlehman.com/2013/06/03/g-2/ | ### g-2
At tree level, the magnetic moment of the electron, $g$, is equal to 2 (this can be obtained from the Dirac equation). However, higher order corrections due to quantum fluctuations slightly change the value from 2.
The latest experimental measurement from Harvard gave:
$a_e \equiv \frac{g-2}{2} = 1 \; 159 \; 652 \; 180.73 \; (0.28) \times 10^{-12} .$
This is a precision of 0.24 parts per billion!
A group calculated the $10^{\text{th}}$ order theoretical correction and got the following value:
$a_e = 1 \; 159 \; 652 \; 181.78 \; (0.77) \times 10^{-12}.$
A couple things to note: first, this is amazing theoretical precision. Second, it took a lot of work. The group had to calculate 12,672 diagrams! I am amazed that experimentalists can design experiments with such incredible exactness, and that the Standard Model (specifically QED) matches up this well with the experiment. If someone tries to tell you that the math behind theoretical particle physics is just “pie in the sky” that doesn’t really mean anything, point them to this measurement.
Here is the reference:
Aoyama, Tatsumi and Hayakawa, Masashi and Kinoshita, Toichiro and Nio, Makiko,
"Tenth-Order QED Contribution to the Electron $g\mathbf{-}2$
and an Improved Value of the Fine Structure Constant,"
Phys. Rev. Lett. 109 (2012)
Advertisements | 2017-10-18 14:58:23 | {"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": 5, "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": 0.7522889375686646, "perplexity": 2116.2458033990056}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2017-43/segments/1508187822992.27/warc/CC-MAIN-20171018142658-20171018162658-00134.warc.gz"} |
https://physics.stackexchange.com/questions/390742/are-there-places-in-space-completely-devoid-of-light | # Are there places in space completely devoid of light?
I'd be interested to know how far from the sun you'd have to be for it to feel 100% dark, as the human eye sees it.
But also, are there places in space so far removed that there is nothing visible there even if you had very fine measuring instruments. I guess I'm asking if there are places in space where there is nothing at all.
The answer to the second question is a qualified no (unless you find a really dark and opaque cloud or box to hide in). The reason is that the largest voids in the cosmic web are a few hundred megaparsec wide. That means that you will see light from galaxies on the outside since there has been enough time for it to reach you there.
The apparent magnitude of the milky way seen at 300 megaparsec is $−20.8 + 5\log_{10}(300\cdot 10^6)-5=16.59$. Now this is too faint to see with the naked eye or binoculars (even when ignoring redshift). But if you have a telescope this should be visible.
The first question is tricky since as you move away from the sun will see a lot of stars. The point where it gets naked eye dark will presumably be where the milky way reaches the limit $M_{lim}\approx 7$ magnitude, which would be 3.7 Mpc (just invert the above formula: $d=10^{(M_{MW}+M_{lim}+5)/5}$). At that point you are somewhere just outside the Local Cluster, and there will be other galaxies present. So I think the nearest really naked eye dark point is in the Local Void.
Here is a crude sketch, using the Karentchev catalog of nearby galaxies, assuming each is as bright as the Milky Way and plotting an isosurface where there is no galaxy with more apparent magnitude than 7. Red regions are dark places. The closest dark point is 4.6 Mpc away from us.
In order to receive randomly a single photon from the sun at a time you would have to place the sun ~1000 light years away from you.
The universe is relatively homogeneous and isotropic. No matter where you go in our observable universe you will have about the same quantity of light from the stars. There are places that are more devoid of matter, but light can travel with almost no dissipation to these places too.
If you want no light it is better to hide in a bunker, but you would have thermal radiation from the walls. But even if you have no visible light somewhere in outer space, you'll still have blackbody radiation from the cosmic microwave background (microwave radiation) from the Big Bang that you cannot escape because it fills the universe. | 2019-07-23 22:42:40 | {"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": 0.582126796245575, "perplexity": 380.0833336285286}, "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-2019-30/segments/1563195529737.79/warc/CC-MAIN-20190723215340-20190724001340-00299.warc.gz"} |
https://www.hpmuseum.org/forum/thread-13559-post-120608.html | Possible HHC door prize?
08-31-2019, 08:06 PM
Post: #1
telemachos Junior Member Posts: 13 Joined: Apr 2017
Possible HHC door prize?
I've picked up, for free, a TI Programmable 57. It's in very nice shape, but uses a funny battery pack that would likely need a jury-rig replacement to allow it to run.
I'd be happy to mail it to Reno for HHC: is there interest in such a thing?
08-31-2019, 08:39 PM
Post: #2
rprosperi Senior Member Posts: 3,637 Joined: Dec 2013
RE: Possible HHC door prize?
(08-31-2019 08:06 PM)telemachos Wrote: I've picked up, for free, a TI Programmable 57. It's in very nice shape, but uses a funny battery pack that would likely need a jury-rig replacement to allow it to run.
I'd be happy to mail it to Reno for HHC: is there interest in such a thing?
Yes. A TI-57 is of course not the most sought-after item in this crowd, but I can assure you it would NOT be left on the table when prize-drawing is completed.
--Bob Prosperi
09-03-2019, 07:31 PM (This post was last modified: 09-06-2019 08:57 PM by Jim Horn.)
Post: #3
Jim Horn Member Posts: 173 Joined: Dec 2013
RE: Possible HHC door prize?
Is there someone who will be at HHC2019 to whom I could send a few items for the door prize table? Since it looks like I may not be there, it would be nice to be able to contribute some small help.
09-03-2019, 10:12 PM
Post: #4
Sylvain Cote Senior Member Posts: 1,141 Joined: Dec 2013
RE: Possible HHC door prize?
(09-03-2019 07:31 PM)Jim Horn Wrote: Is there someone who will be at HHC2019 to whom I could send a few items for the doorprize table?
Euh! Joseph Horn ?
09-04-2019, 03:56 AM
Post: #5
Joe Horn Senior Member Posts: 1,507 Joined: Dec 2013
RE: Possible HHC door prize?
(09-03-2019 10:12 PM)Sylvain Cote Wrote:
(09-03-2019 07:31 PM)Jim Horn Wrote: Is there someone who will be at HHC2019 to whom I could send a few items for the doorprize table?
Euh! Joseph Horn ?
Good idea! Since I'm driving there, I can bring more stuff than people who are flying there, so yes, you can send any door prizes to me, and I'll put them on the door prize table. N.B. Make sure I get them before September 17, because that morning my road trip begins. If you don't have my shipping address, PM me.
<0|ɸ|0>
-Joe-
09-04-2019, 12:20 PM
Post: #6
Sylvain Cote Senior Member Posts: 1,141 Joined: Dec 2013
RE: Possible HHC door prize?
(09-04-2019 03:56 AM)Joe Horn Wrote:
(09-03-2019 10:12 PM)Sylvain Cote Wrote: Euh! Joseph Horn ?
Good idea! Since I'm driving there, I can bring more stuff than people who are flying there, so yes, you can send any door prizes to me, and I'll put them on the door prize table. N.B. Make sure I get them before September 17, because that morning my road trip begins. If you don't have my shipping address, PM me.
Oups! I got caught at my own game. It was intended as a joke.
I was finding it funny that Jim was asking if there was someone on the forum who could bring stuff to HHC while his brother was driving there.
Lesson learned, never try to make a joke in a language that you are not mastering, the backfiring potential is significant.
Sorry Joe.
Sylvain
09-04-2019, 04:00 PM
Post: #7
telemachos Junior Member Posts: 13 Joined: Apr 2017
RE: Possible HHC door prize?
I'm mailing my TI-57 to the Hyatt itself, addressed as follows:
Hyatt Place Reno-Tahoe Airport
1790 East Plumb Lane, Reno, NV 89502
Attn: Sherry Howell/HHC 21-22 Sept.
I am assured that that will work; Sherry is meetings coordinator and she knows what HHC means.
09-04-2019, 05:59 PM
Post: #8
rprosperi Senior Member Posts: 3,637 Joined: Dec 2013
RE: Possible HHC door prize?
Thanks very much for supporting HHC, even if you can't make it yourself. Hopefully, you can attend a future HHC and attendees can thank you in person.
This note is from the Datamath Calculator Museum on the TI-57 page:
Quote:Early in production (about wk 36 year 1977) TI changed the battery pack and the related connector and charger from the SR-51-II style to the TI-55 style. The battery packs are not compatible, don't use a BP6 with a AC9132 charger !
Maybe this is related to the unusual battery pack you mentioned? Anhow, I just wanted to add the note to be sure some lucky winner doesn't kill it, assuming all TI's use the same power adapter.
--Bob Prosperi
09-06-2019, 07:29 AM
Post: #9
ggauny@live.fr Senior Member Posts: 487 Joined: Nov 2014
RE: Possible HHC door prize?
Hello,
What is exactely the meaning of "The door prize" ?
I guess it it a table with gifts for winners ?
Good day and good meeting in Reno !
Gérard.
09-06-2019, 05:34 PM
Post: #10
Jake Schwartz Member Posts: 213 Joined: Dec 2013
RE: Possible HHC door prize?
(09-06-2019 07:29 AM)ggauny@live.fr Wrote: Hello,
What is exactely the meaning of "The door prize" ?
I guess it it a table with gifts for winners ?
Good day and good meeting in Reno !
In this case, attendees have traditionally brought calc-related (and some not calc related) items which they are willing to donate, and they are all placed on tables around the conference room. With the conference registration, a numbered prize ticket is given to each attendee. Then at the end of the conference, the ticket numbers are drawn randomly and each person is then able to choose a prize from the prize tables. The object has been to re-distribute all the items back to the group, so frequently, the prize-ticket drawing continues over and over until all the prizes are gone. I think in one conference, we had to go through all the tickets randomly around four times. It takes a relatively long time (approaching an hour), but is usually fun and there is typically SOMETHING that each person desires. In addition, we all get to also take home other peoples' junk <grin>.
Jake
09-07-2019, 09:53 AM
Post: #11
ggauny@live.fr Senior Member Posts: 487 Joined: Nov 2014
RE: Possible HHC door prize?
Hi,
Thank you Jake for this clear explanaition (explication).
Now I see what it is.
Good meeting.
PS : I have already the CD Rom #6 HP Museum, what can I do to have
Gérard.
09-07-2019, 10:50 AM (This post was last modified: 09-07-2019 10:51 AM by Thomas Okken.)
Post: #12
Thomas Okken Senior Member Posts: 887 Joined: Feb 2014
RE: Possible HHC door prize?
(09-07-2019 09:53 AM)ggauny@live.fr Wrote: I have already the CD Rom #6 HP Museum, what can I do to have the latest version ?
Order the latest version, 8.0, from here. People who have bought an earlier version pay the upgrade price of $20 instead of the full price of$38. Details on the upgrade policy are here -- in a nutshell, you qualify for the discount if you purchased the older version yourself, from the HP Museum.
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http://ieeexplore.ieee.org/ieee_pilot/articles/06/ttg2009061571/article.html | • Abstract
# Volume Ray Casting with Peak Finding and Differential Sampling
Direct volume rendering and isosurfacing are ubiquitous rendering techniques in scientific visualization, commonly employed in imaging 3D data from simulation and scan sources. Conventionally, these methods have been treated as separate modalities, necessitating different sampling strategies and rendering algorithms. In reality, an isosurface is a special case of a transfer function, namely a Dirac impulse at a given isovalue. However, artifact-free rendering of discrete isosurfaces in a volume rendering framework is an elusive goal, requiring either infinite sampling or smoothing of the transfer function. While preintegration approaches solve the most obvious deficiencies in handling sharp transfer functions, artifacts can still result, limiting classification. In this paper, we introduce a method for rendering such features by explicitly solving for isovalues within the volume rendering integral. In addition, we present a sampling strategy inspired by ray differentials that automatically matches the frequency of the image plane, resulting in fewer artifacts near the eye and better overall performance. These techniques exhibit clear advantages over standard uniform ray casting with and without preintegration, and allow for high-quality interactive volume rendering with sharp C0 transfer functions.
SECTION 1
## Introduction
Volume rendering is an indispensable tool in visualization, with applications ranging from simulation data analysis to imaging medical and biological scan data. The principal means of visualizing a volume consist of choosing an isosurface or interpreting the volume in its entirety via direct volume rendering (DVR). Traditionally, these modalities have been implemented using separate algorithms, and were employed with different visualization and application goals. Isosurfaces are commonly generated by extracting a triangle mesh, and are useful in understanding topological and geometric behavior of a scalar field at implicitly defined boundaries. Direct volume rendering is a more expressive method of visualizing volume data, in which the user supplies a transfer function mapping scalar values to colors. As opposed to a single isovalue corresponding to a 2-manifold surface, a transfer function allows for rendering 3-manifold segments of the volume.
In principle, an isosurface can be defined by a transfer function with a Dirac impulse at the chosen isovalue. However, rendering such a transfer function poses problems for most conventional volume rendering algorithms. Methods involving uniform sampling and postclassification will invariably miss an infinitely fine impulse, entirely omitting the desired surface features. Preintegrated transfer functions remedy this, but introduce new artifacts due to their discretization of scalar values into a 2D lookup table, and weighting assumptions on the volume rendering integral. Most commonly, the solution to rendering isosurfaces within a volume rendering framework has been to increase the sampling rate and to smooth the transfer function. Nonetheless, doing so can be computationally wasteful and limits classification.
Spatial traversal strategies for GPU isosurface ray casting closely mirror those for DVR sampling. We pair these processes by identifying isovalues of interest at peaks of a 1D transfer function, using the uniform volume ray casting process to isolate these roots, and sampling directly at the desired isovalues. While numerous applications allow for multi-modal DVR and isosurface visualization, to the best of our knowledge our approach of sampling isosurfaces directly within the volume rendering integral has not previously been employed. Perhaps this is because standard techniques employing smooth transfer functions were considered sufficient. Nonetheless, definition and accurate rendering of sharp transfer functions is desirable, not only in terms of overall image quality but in the ability to classify features flexibly and render accurately with a fixed sampling budget. To further ensure samples are spent wisely, we devise a novel approach to volumetric sampling using a quadratic function for incrementing samples based on ray differential propagation. This helps in sufficiently sampling features close to the viewpoint, and is particularly useful when employing higher-order filters for which samples are expensive. While orthogonal, these techniques work well together, particlarly in rendering nearby high-frequency features with high fidelity. We compare both methods with standard techniques, and show how they offer higher quality imaging and better classification for various data sets.
SECTION 2
## Related Work
Levoy [16] employed ray casting in the first implementation of direct volume rendering. The advent of z-buffer hardware and built-in texture interpolation units allowed for interactive performance with slice-based rasterization approaches [2], [4]. Similarly, rasterization methods employing splatting [32] proved to be efficient, particularly for applications involving unstructured data and higher-order reconstruction filters [34], [35]. While optimized CPU algorithms are capable of interactive volume rendering [11], [15], GPU approaches gained popularity, due to improved computational throughput and built-in texture fetching and interpolation. With programmable shader support for branching and looping, volume ray casting methods experienced resurgence on the GPU [14], [26],
The conventional means of rendering discrete isosurfaces from volume data has been to extract a mesh using marching cubes [20]. Mesh extraction methods can be combined with min-max spatial subdivision structures [33], as well as view dependent [18] approaches for further efficiency. Marching cubes only approximates the implicit surface on a coarse scale, and more sophisticated methods [28] are generally not suited for dynamic extraction. However, it is possible to combine extraction with splatting [19] for efficient rendering, or to employ splat-ting directly on isosurfaces [3].
Ray casting methods were first applied towards volumetric isosurfacing by Sramek [31]. Parker et al. [24], [25] implemented a tile-based parallel ray tracer and achieved interactive rendering of isosurfaces from large structured volumes, employing a hierarchical grid as a min-max acceleration structure and an analytical cubic root solving technique for trilinear patches. Hadwiger et al. [7] combined rasterization of min-max blocks with adaptive sampling and a secant method solver to ray cast discrete isosurfaces on the GPU. Our peak finding method is close in spirit to this approach; however we employ our solving method not only in rendering isosurfaces but in handling potentially sharp unidimensional transfer functions. Ray differentials were introduced by Igehy [8] as a way of calculating image-space derivatives of pixels as rays are transmitted, reflected and refracted in world-space, and using these values for filtering. While similar concepts have been used in multiresolution isosurface ray casting [13], to our knowledge no approach has used ray differentials for volumetric sampling.
A large body of volume rendering literature deals with transfer functions, both in how to construct them and employ them in classification. To limit artifacts when sampling high-frequency features of a transfer function, the best existing approaches are preintegration [5], [21], [27] and analytical integration of specially constructed transfer functions [10]. Hadwiger et al. [6] analyze the transfer function for discontinuities to generate a pre-compressed visibility function employed in volumetric shadow mapping. Our approach is similar except that we search for local maxima, and use these directly in enhancing classification.
SECTION 3
## Background and Overview
Direct volume rendering is the process of modeling a volume as a participating optical medium, and estimating the emission and absorption of these media according to a discrete approximation of the radiative transport equation. On a segment of a ray, irradiance is formulated as TeX Source $$I(a,b) = \int_a^b {\rho _E (f(s))\rho \alpha (f(s))e^{ - \int_a^s {\rho \alpha (f(t))dt} } ds}$$ where ρE is the emissive (color) term and ρα is the opacity term of the transfer function; a, b are the segment endpoints, and is the scalar field function evaluated at a distance t along the ray. To compute this integral, we must approximate it discretely. The conventional approach of Levoy [16] is to break up the ray into equally spaced segments, approximating the opacity integral as a Riemann Sum, TeX Source $$e^{ - \int_a^s {\rho \alpha (f(t))dt} } = \prod\limits_{i = 0}^n {e^{ - \Delta t\rho \alpha (f(i\Delta t))} } = \prod\limits_{i = 0}^n {(1 - \alpha _i)}$$ where Δ t is the uniform sampling step n = (s – a)/Δ t, and TeX Source $$\alpha _i \approx 1 - e^{ - \Delta t\rho \alpha (f(i\Delta t))}$$ Discretizing the integral on [a, b] in Equation 1 as a summation, we have the following discrete approximation for I, TeX Source $$I \approx \sum\limits_{i = 0}^n {\rho _E (i)\prod\limits_{j = 0}^{i - 1} {(1 - \alpha _j)} }$$ where ρE(i, = ρE(iΔ t)) is given by the transfer function. Evaluating the transfer function after reconstruction is known as postclassification. Typical sampling behavior of postclassification with uniform sampling along the ray is illustrated in Figure 1(a). When high-frequency features are present in ρα(f(t)), many samples are required to accurately integrate along the ray.
To eliminate artifacts and achieve high-quality volume rendering, we must adequately sample with respect to the Nyquist limits of all component functions contributing to the signal. The principal signal sources consist of the scalar field function and the transfer function . Engel et al. [5] note that this frequency can be either the maximum Nyquist frequency of all separate sources, or the product of the Nyquist frequencies of these sources. By discretizing the transfer function and scalar field integrals separately, preintegration can achieve greater fidelity for high-frequency transfer functions with fewer samples, as illustrated in Figure 1(b).
Fig. 1. Integration methods for direct volume rendering.
Separately integrating the transfer and field functions via preintegration presents separate issues, however. Problems occur when the scalar field or transfer function are undersampled by their respective discrete integrations. Like postclassification, preintegration is susceptible to undersampling, though artifacts are manifested differently. Preintegration assumes the scalar field function varies piecewise-smoothly between entry and exit samples fi = f(t) and . Depending on the frequency of the field function, this is often not the case. Specifically, computing the opacity integral on a segment uses the trapezoid rule (or similar numerical integration), which scales the opacity summation by Δ f = | fifo| to approximate ρα. When ρα(f) is smooth (specifically, Lipschitz) this approximation behaves nicely. However, sharp features in the transfer function break this assumption, leading to bias and improperly scaled opacity. Though blending the integrals of front and back samples smoothens results [21], it does not accurately capture sharp peaks. In addition, preintegration relies on a fixed quantization of entry and exit opacities into a table. Permitting dynamic changes in the transfer function limits the size of this table, hence the minimum width Δ f between two field values used to query the transfer function integral. Nonetheless, visualizing features with higher precision can be desirable for more accurate classification.
This paper describes two techniques that overcome deficiencies of existing methods. The main contribution is peak finding, which overcomes many limitations of postclassification and preintegration by sampling directly at sharp features of a transfer function. This consists of analyzing a transfer function for local maxima, and explicitly solving for roots of the filter function to render isosurfaces at these peaks. As with preintegration, peak finding employs a 2D lookup table; however rather than querying an approximation of the integral itself, we query which peaks possibly lie within that range of field values. The general concept is illustrated in Figure 1(c), and its implementation is described in detail in Section 4.
In Section 5 we present differential sampling. We note that transformations on the ray from world-space to image-space convolve the volume rendering integral, and provide a new sampling method respecting the Nyquist frequency of the image plane. Our method borrows from the ray differentials formulations of Igehy [8] in developing its sampling strategy. This is discussed in Section 5.
Our system consists of a straightforward volumetric ray caster, employing a grid acceleration structure traversed per-ray in a GLSL fragment shader, and classifying via a 1D transfer function specified as a piecewise-linear set of points. Section 6 discusses how to integrate differential sampling and peak finding into this framework. The end goal of this work is to enable interactive high-fidelity volume rendering with sharp transfer function features using fewer samples than conventional methods. We show how these algorithms help to accomplish that in Section 7.
SECTION 4
## Peak Finding
Peak finding is motivated by the shortcomings of both standard post-classified (Figure 1(a)) and preintegrated (Figure 1(b)) volume rendering with transfer functions containing sharp features approaching Dirac impulses. The general approach is similar to isosurface ray casting in that we solve directly for roots. Ray-isosurface intersection consists of solving the continuous reconstruction filter function as a 1D implicit function of t at an isovalue: .
Numerous numerical methods exist for solving roots of this equation; interactive ray tracing algorithms commonly employ a combination of Descartes's rule of signs and an iterative solver [7], [22], [30], or more robust recursive methods such as interval arithmetic [12]. The substantial difference is that in these systems, the isovalue is given explicitly by the user; whereas in ours the isovalue must be inferred from the transfer function. By employing these root-finding methods in searching for peaks of the transfer function, we have far lesser chance of missing them, allowing for smoother reconstruction and better shading of isosurface features within our volume rendering framework. The general concept is illustrated in Figure 1(c).
### 4.1 Determining peaks and building the lookup table
Peak finding is similar to preintegration in that we query a 2D lookup table for each segment along the ray. However, rather than storing a preintegrated radiance approximation, our table stores an isovalue ν or set of isovalues νi; that possibly exist within this segment, sorted from the first to last peak value encountered on a given segment defined by the entry and exit values of the scalar field function, [fi,f0].
Before building the lookup table, we analyze our transfer function ρα and search for peaks. Specifically, we consider whether a given point is a local maximum (i.e. greater than both its immediate neighbors) with respect to the opacity component. The set of peaks consists of at most half the number of actual data points in our piecewise-linear transfer function, but typically it is far less. Smooth ID functions such as splines would have relatively fewer peaks, existing at the critical points of these functions. As we are interested in sharp features, we consider piecewise-linear functions. It is equally possible to use this technique to search for local minima; however due to their low radiance contribution the impact of doing so is not generally noticeable.
Having computed the array of peaks, we construct the lookup table. For a range of values [i,j] corresponding to lookup entries from our volume . If i < j, we search our transfer function for the next peak point (or in the case of multiple peaks, next 4 points) such that the opacity ρα(ν) > i and ρα(ν) ≤ j. If i > j, we search in descending order for peaks with ρα(ν)≤ i and ρα(ν) > i. When necessary, a segment spanning multiple peaks will reverse the sorting order to register all possible peaks within that segment. This process is again similar to preintegration, except that separate discrete peak values are stored instead of a single integral approximation. In each table entry, we store the domain isovalue(s ν corresponding to each peak. When no peak exists, we use a flag outside of the range of scalar values in the volume. Building the lookup table is relatively undemanding, and proceeds in O(N2) time, similarly to the algorithm of [21] for preintegration. In practice, building a peak-finding table is roughly twice as fast as building a preintegrated table at the same resolution. Moreover, in many cases a coarser discretization (128 bins) is sufficient for peak finding, whereas preintegration would require a larger table for comparable quality when rendering near-discrete isosurfaces.
### 4.2 Root solving and classification
Peak finding occurs between samples in the main ray casting loop. Before sampling at the next step , we fetch the nearest peak value from a 2D texture using the same . If the peak exists, we subtract that isovalue from the entry and exit values, and employ Descartes' rule of signs. If this test succeeds, we assume the segment contains a root. Bracketed by , we use three iterations of a secant method (also employed by [7], [22]) to solve the root: TeX Source $$t_1 = t_0 - f(t_0){{t_1 - t_0 } \over {f(t_1) - f(t_0)}}$$ When the secant method completes, we have an estimate for the root t along the ray segment. We now sample at this position and perform postclassification. However, sampling at the peak requires two subtle choices. First, we do not evaluate our field , but rather assume that the value at this point is our desired isovalue. This works because we are solving for the root position, not its value; moreover for sharp transfer functions it is crucial in avoiding Moire patterns. Second, we do not not scale ρα by the segment distance Δ t (in Equation 3) but instead use a constant Δ t = 1. Although this may seem counterintuitive, the scaled extinction coefficient is itself a correction mechanism for the inherently discrete approximation of the volume rendering integral. Moreover, an unsealed opacity assumes that we always sample at this isovalue regardless of the sampling rate or local behavior of ρα(f) along the ray segment. This is precisely our goal with peak finding. While the resulting approach arguably biases the volume rendering integration towards these peaks, it is critical in detecting them without excessively increasing the sampling rate. In practice this strategy does not greatly bias our integral, as the relative contribution of values outside the peak is small.
Finding multiple peaks can be useful when the step size Δ t is large, or when peaks are spaced closely together. Our implementation handles up to four multiple peaks within a single segment with a straightforward extension, which can be enabled at runtime as necessary. As described in Section 4.1 we construct the peak finding table with four sequential peaks contained within the given segment [fi,f0]. Since isosurfaces are encountered in precomputed order between the minimum and maximum field values, we can simply perform peak finding sequentially on all four values in that order.
### 4.3 Algorithm integration and usage
Peak finding is equivalent to volume rendering with a discrete isosurface-finding step in between. One can trivially modify the algorithm to support different rendering modalities. We allow for:
• Sampling from both uniformly/differentially sampled DVR and peak finding (default).
• Sampling from either uniformly/differentially sampled DVR or peak finding (peak_xor_dvr).
• Transparent isosurfacing of peaks only (peak_only).
• DVR only, disabling peak finding (dvr_only).
These options can be invoked with small switches to the shader code and incur no performance penalty or code overhead. The uppercase flags above correspond to macros in the GLSL pseudocode provided in the Appendix.
Peak finding is attractive in that its algorithm is not significantly different from either volume rendering or isosurface ray casting. Both algorithms employ regular sampling, in the case of DVR to compute the volume rendering integral and in the case of isosurfacing to isolate roots. Peak finding takes advantage of this and does both. As a result, this technique can be implemented quickly by extending existing Tenderers. Although we propose peak finding in conjunction with differential sampling, the two techniques are orthogonal. It is equally possible to employ peak finding in a uniform sampling ray caster, a slice-based volume Tenderer, or a shear-warp system.
Overall, peak finding and preintegration are similar, but make different assumptions about the integral over a given segment. Preintegration assumes this integral can be accurately approximated by piecewise summation. This works well when the transfer function and convolved field are smooth, but encounters difficulties when they are not. Peak finding assumes this integral can be approximated by one or several discrete impulses. This introduces bias, but is better suited for noisy data and sharp C0 transfer functions for which standard techniques fail.
SECTION 5
## Differential Sampling for Volume Rendering
Uniform sampling ignores an important component of the convolved volume rendering integral and its resulting Nyquist limit. With a pin-hole camera, the projective transformation on the image plane is itself a signal convolution. Thus, regular sampling in world-space under-samples features close to the viewpoint relative to those further away. To remedy this, we can employ a sampling strategy that uses the ray distance itself as a sampling metric. This can be accomplished with a new function T whose derivative varies linearly with distance, i.e. TeX Source $$\Delta T = {{\partial T} \over {\partial t}} = at + b\quad T(t) = {a \over 2}t^2 + bt + c$$ Then we sample along the ray at . The question remains how to choose a, b and c so that the sampling step is proportional to pixel width. We turn to the concept of ray differentials [8], which quantifies world-space transformations in image-space derivatives. Specifically, we use the ray differential transfer equation to formulate T as a function of image-space.
### 5.1 Ray differentials
With ray differentials [8], the general goal is to compute the image-space derivatives of a series of functions convolving the image plane, beginning with generation of rays in a pinhole camera, TeX Source $$\vec d(x,y) = \vec w + x\vec u + y\vec v$$ where is the central view direction are the right and up vectors. Unitizing a ray comprises another transformation: TeX Source $$\vec O(x,y) = \vec o\quad \vec D(x,y) = (\vec d)(\vec d \cdot \vec d)^{ - 1/2}$$ Then the unit-parameterized ray has the image-space partial with respect to x (and similarly for y): TeX Source $${{\partial \vec R} \over {\partial x}}(t) = {{\partial \vec O} \over {\partial x}} + t{{\partial \vec D} \over {\partial x}} + {{\partial t} \over {\partial x}}\vec D$$ As , this holds for any discrete difference Δ t as well. For our purposes of choosing a constant image-space measure, it suffices to consider only x differentials. Lastly, the differential of the unitized with respect to the x image-space coordinate is: TeX Source $${{\partial \vec D} \over {\partial x}} = {{(\vec d \cdot \vec d)\vec u - (\vec d \cdot \vec u)\vec d} \over {(\vec d \cdot \vec d)^{3/2} }}$$ Derivations are given in more detail in the original paper [8],
### 5.2 Differential sampling construction
Our general strategy is to define a base sampling rate proportional to an image-space quantity, and use the ray differential transfer equation (Equation 9) to derive our sampling function T. To accomplish this, we use the image-space x as our discretization, and construct a sampling scheme where is proportional to the differential quantity . As world-space Δ t is proportional to x, for some scalar k, TeX Source $$\displaylines{ {{\partial \vec R} \over {\partial x}}(\Delta t) = \Delta t{{\partial \vec D} \over {\partial x}} + {{\partial \Delta t} \over {\partial t}}\vec D = kx\left| {{{\partial \vec D} \over {\partial x}}} \right|\vec D + k\vec D \cr = \vec D\left({k\left| {{{\partial \vec D} \over {\partial x}}} \right|x - k} \right) \cr}$$ Since is normalized and our discrete step Δ t is arbitrary, the user can choose any k and preserve a correlation between the distance-based sampling step Δ t and x. Similarly, to use x as unit of measure along the ray, we project so that it is collinear with , i.e . Then from Equation 9, we have: TeX Source $$\displaylines{ {{\partial \vec R} \over {\partial x}}(\Delta t) = \Delta t{{\partial \vec D} \over {\partial x}} + {{\partial \Delta t} \over {\partial x}}\vec D = kx|{{\partial \vec D} \over {\partial x}}|\vec D + k\vec D \cr = \vec D(k|{{\partial \vec D} \over {\partial x}}|x + k) \cr}$$
Fig. 2. Geometric construction of our differential sampling approach.
Since , this gives us TeX Source $$|{{\partial \vec R} \over {\partial x}}(\Delta t)| = k|{{\partial \vec D} \over {\partial x}}|x + k$$ From Figure 2, notice that . Since θ between any two rays is constant, tan(θ) is also constant (its computation is left as an exercise). This can be incorporated into a new constant k′ = k tan(θ); or if k is arbitrarily chosen we can omit this step and use k′ = k. We then employ the differential construction of T in Equation 6 but in terms of image-space x, TeX Source $${{\partial T} \over {\partial x}} = \Delta t = k'|{{\partial \vec D} \over {\partial x}}|x + k'$$ For convenience let and b = k′. The antiderivative yields our differential sampling function T: TeX Source $${{\partial ^2 T} \over {\partial x^2 }} = a\quad {{\partial T} \over {\partial x}} = ax + b\quad T(x) = {a \over 2}x^2 + bx + c$$ When we begin sampling at t = T(x, = 0, we can assume c = 0.
### 5.3 Computing and incrementing samples
Differential sampling is simple to implement in a volume ray casting framework. We first compute from Equation 10. While the user can choose any k, we ensure it is some multiple of world-space pixel footprint at the image plane, e.g . From this we compute k′ (if necessary a and b. Theoretically sk < 1/2 is required to satisfy the Nyquist limit of the image plane. In practice this rate is excessive, and sk = 4 is a good conservative default.
From the ray origin, the sampling process begins at x = 0, where TeX Source $${{\partial T} \over {\partial x}} = b\quad T(x) = 0$$ Then at each ray casting iteration, we sample at , and perform the following increments, where Δ t is our discretization of , TeX Source $$\vec P_1 = \vec P_0 + \Delta t_0 \vec D\quad \Delta t_1 = \Delta t_0 + a$$ Thus, incrementing the position from one sample to the next consists only of an extra vector multiplication and addition, on top of the vector addition for uniform sampling. This is also outlined in the pseudocode in the Appendix.
SECTION 6
## Implementation
We implemented our ray casting framework in OpenGL and GLSL. The pinhole camera vectors , and are computed on the CPU and then sent to the fragment shader, where a ray is generated from the pixel x and y values according to Equation 7. The 1D transfer function is given as a set of points {v, {r, g,b, a}}, then processed into a fairly wide (8K elements) ID texture, allowing for rapid access on the GPU and generally sufficient transfer function precision Δf > 1e − 4. We implemented a tricubic B-spline filter using the method of [29], with the BC smoothing (B = 2 C = 1) kernel of [23]. We optionally employ this for both DVR sampling and root solving.
Fig. 3. Simulated temperature of a heptane fire, with a transfer function consisting of a near-Dirac peak (width Δ f < 1 e – 4) in red, and a smoother feature for contrast in blue. Peak finding, postclassification and preintegration render at 7.1, 11.6 and 8.8 fps, respectively, at 1024x900.
### 6.1 Space skipping
Even with fairly dense transfer functions, most data sets are sparse enough to warrant an empty space skipping mechanism. We choose a simple uniform grid with a 3DDDA algorithm [1] where each grid cell stores min-max values of enclosed voxels. Fairly coarse grids (643 cells) work best on the GPU, and this structure can be updated interactively when the transfer function changes. The fragment shader then traverses the macrocell grid using the 3DDDA algorithm in an outer loop. When a macrocell is nonempty, we enter the volume rendering loop, with peak finding tests taken between samples. To begin sampling, we find the first t at which to sample when entering a macrocell. With differential sampling, we solve for the maximum x after Tenter, TeX Source $$ax^2 {\rm{/}}2 + bx = T_{enter} \quad x_{tenter} = (- b + \sqrt {b^2 + 2aT_{enter})} {\rm{/}}a$$ We then compute the floor values ⌊xtenter⌋, T(⌊xtenter⌋) and , which can be simplified significantly from Equation 15; and subsequently sample and increment as in Equation 17. To avoid duplicate samples, we store the greatest t at which we already sampled, and use the maximum of that and Tenter.
As discussed in [7], purely adaptive methods (for example based on local gradient) perform poorly on GPUs due to poor thread coherence. However, we do achieve better performance by varying the sampling rate on a per-macrocell basis. In this scheme, each macrocell computes a metric based on the ratio of the maximum standard deviation of its voxels to that of the entire volume . As this represents a multiplier for the frequency, its inverse can be used to vary the sampling step size Δ t. In practice we wish this to be a positive integer, and a multiplier M = 2m1 + 1 delivers good results. With uniform sampling one simply employs MΔ t as the new sampling rate. With differential sampling M modifies our increments as follows: TeX Source $${{\partial ^2 T_M } \over {\partial x^2 }} = \sum\limits_{i = 1}^M a = {{M(M + 1)} \over 2}a\quad {{\partial T_M } \over {\partial x}} = Max + {{\partial ^2 T_M } \over {\partial x^2 }} + b$$ No modifications to T(x) are required, since the initial x for that macrocell can be any integer.
SECTION 7
## Results
Unless otherwise stated, all results were collected on a 2.5 GHz Intel Xeon and an NVIDIA 285 GTX GPU, with trilinear filtering, differential sampling (sk = 4) and the exclusive-or peak finding modality. For each scene we plot the (f, ρα(f)), scaled to the maximum opacity of the transfer function. To evaluate complexity, we count the total number of filter evaluations (including peak finding) or DVR-only samples (without peak finding), and divide these by the number of pixels. As with any DVR system, performance varies widely with the number of samples taken. Opaque isosurfaces and low-frequency scenes are simplest and render at real-time rates. The focus of our work is in handling sharp features, which requires higher sampling rates. Overall, image quality is excellent and our system is generally interactive (Table 1). While analysis of macrocells falls outside the scope of this paper, they usually deliver 1.2x to 5x performance improvement depending on the scene. Although other approaches have greater total sample throughput, our system is competitive in how it spends samples and resulting quality.
TABLE 1 Overall performance in frames per second and average samples per ray for selected scenes with differential sampling sk = 4. The right three columns show average samples (filter function evaluations) per ray, average DVR-only samples per ray, and fps with peak finding, postclassification, and preintegration.
### 7.1 Peak finding
Peak finding is useful when the combined frequency of the volume and transfer function is too high for effective regular sampling. In such cases, postclassification would require near-infinite sampling to accurately reproduce features. Preintegration succeeds in detecting high-frequencies of the transfer function, but integrates and shades them incorrectly when undersampling the scalar field.
An obvious scenario in which conventional sampling methods fail is a transfer function containing one or more Dirac-like features, as shown in Figure 3. Peak finding succeeds in reproducing these features as semi-transparent isosurfaces, and rendering smoother volumetric features in the correct order. While postclassification misses peak features outright, preintegration detects and reproduces a surface. However, with preintegration the range Δ f along a given segment can significantly skew the opacity integral; two segments with different Δ f may sample the same impulse but have different irradiances. With peak finding, this is not the case. In addition, preintegration shades at the segment endpoint, as opposed to locally at the hit position of the isosurface, resulting in Moire patterns. Finally, when an impulse is defined with a discretization smaller than that of the preintegrated table, peak finding with a smaller table can reproduce features that preintegration misses. In practice, this is less a concern than the aforementioned integration and shading issues with preintegration.
Peak finding is an intriguing method for rendering noisy or entropic data, for example from scanned sources in medicine or biology. Here, even when the transfer function is sampled adequately, the filtered field function of the volume (hence the convolved signal) is not. While artifacts are not as noticeable due to the noisy nature of renderings, high-frequency features are again omitted. Due to convolution of the high data frequency, features can be lost even with moderate-frequency transfer functions. Simply increasing opacity at peaks does not correct the problem, and widening the transfer function broadens the classification. Choosing a higher sampling rate can remedy this, but at high performance cost. Meanwhile, at sampling rates well below the Nyquist limit, peak finding successfully reproduces sharp features with the desired opacity and color, as shown in Figure 4. The fireset in Figure 6 also illustrates this phenomenon.
Fig. 4. Zebrafish optic tract acquired through electron microscopy [9] rendered with differential sampling and peak finding at 1600x512 resolution. Peak finding (top, 2.1 fps) enables better classification of narrow-band segments in entropic data. Preintegration (bottom left, 2.0 fps) has difficulty accurately reproducing such features, and semi-transparent isosurfacing (bottom right, 4.3 fps) lacks the depth cues provided by volume rendering.
Finding multiple peaks is typically not necessary unless several sharp features are close together in the transfer function. This option better ensures peaks are rendered in the correct order, and costs roughly 20% performance (Figure 5 (left)). More significantly, we find that bias from always sampling at peaks is manageable Figure 5 (right) considers a smooth transfer function that looks nearly identical with peak finding and postclassification (Figure 5c,e). Peaks with opacity magnified by 16 (Figure 5d) and peak isosurfaces only (Figure 5f) are shown for contrast. The only disadvantage of peak finding in such cases is that it is not necessary and more costly. While it is possible to construct transfer functions for which peak regions have relatively higher contribution to the radiance and show greater bias, for the most part peak finding accentuates isosurface-like features as desired.
Fig. 5. Peak finding behavior (800x1024 resolution). Left: finding single and multiple peaks, at 10.7 and 8.2 fps, respectively. Right: bias from always sampling at peaks is generally subtle. At full frame resolution, (c-f) render at 11.7, 13.6, 20.5 and 15.6 fps, respectively.
### 7.2 Differential sampling
Differential sampling delivers better results close to the viewpoint, and not noticeably worse quality in the distance. A major appeal of this method is that the sampling rate is view-dependent; it automatically and locally matches sampling to the frequency of the image plane, thus requiring less work on the part of the user. In evaluating differential sampling, it is difficult to enforce a constant average sampling rate, so we use frame rate as the control variable and compare the results in Figure 6. Exact performance figures are given in Table 2. At similar frame rate, uniform sampling undersamples nearby features, and differential sampling remedies this, yielding consistently better quality and surpsingly little quality loss further away. Peak finding amplifies undersampling artifacts at silhouettes; as a result differential sampling in conjunction with peak finding is particularly desirable up close.
More subjectively, we can choose a single converged image as the control, and compare frame rates required for each scheme to achieve comparable quality. We use Figure 7 and the differential sampling halves of Figure 6 as reference; results are given in Table 2 (bottom). Adequately sampled, these scenes look generally similar with uniform and differential schemes. However, differential sampling can deliver up to 3x better frame rate, particularly when overall frequency is low. In Figure 7(a,b), converged images of the aneurism with postclassiflcation and B-spline filtering look nearly identical, but run at 1.0 and 2.7 fps with uniform and differential sampling, respectively (0.86 and 1.8 fps with peak finding). Conversely, in cases where data is entropic and classified with multiple peaks, differential sampling is less effective, requiring a smaller sk to adequately sample faraway regions, while oversampling nearby features. This is more noticeable with peak finding, where adequate sampling is necessary for robust root isolation of isosurfaces. Overall, differential sampling seldom delivers worse quality than uniform at the same frame rate. The backpack in Figure 7(c,d), a noisy scanned volume classified with peak finding and multiple peaks, still renders at 1.6 fps with both sampling methods and similar quality.
As evident in Table 2, differential sampling often requires half or less as many uniform samples for equivalent visual quality. Ideally, half as many samples would correspond to exactly double the frame rate. In practice this is not the case, due to the parallel nature of GPUs and worse memory coherence at far-away samples when using differential sampling. With tricubic B-spline filtering, the higher cost of computing samples outweighs this penalty, yielding relatively better performance with differential sampling than with uniform (1.5-3x as opposed to l-2x). Nonetheless, differential sampling remains clearly worthwhile with trilinear filtering.
Fig. 6. Close-up scenes with uniform (left) and differential sampling (right) at similar frame rates, rendered at 1280x800. Columns show postclassification, peak finding, and peak finding with higher-order B-spline filtering. Aneurism and fireset scenes are shown in the top and bottom rows, respectively.
Fig. 7. Far views with differential sampling (b,d) render 1-3× taster than uniform sampling (a,c) at similar quality. Left: aneurism (postclassified with B-spline filtering). Right: noisy backpack data. (a,b,c,d) render at 1.0, 2.7, 1.6 and 1.6 fns. respectively at 10242.
TABLE 2 Differential sampling performance for images in Figs. 6 and 7.
SECTION 8
## Discussion
Our proposed techniques advance the state-of-the-art in high-quality volume ray casting. Peak finding allows for near-discrete isosurfaces to be specified within a volume rendering transfer function, and provides a new tool in the classification arsenal. It yields viable classification of entropic and noisy data, handles pathological cases that are unadressed by postclassification and preintegration, and is not significantly slower than those techniques. Differential sampling allows for better quality rendering of features closer to the camera, with less overall sampling and correspondingly higher frame rate.
The main drawback of peak finding is that it is more costly than preintegration, and unnecessary when the transfer function and data are smooth. Again, an argument can be made that introducing discrete isosurfaces into the volume rendering integral is inherently biased. In addition, the rule of signs is not a robust root isolation method, and surfaces can be missed near sharp silhouettes. The main limitation of differential sampling is that it would be difficult to implement outside of a ray casting framework. When sk is very small, differential sampling encounters numerical problems resulting in worse artifacts at greater sampling rates, shown in the close-up in Figure 7(c,d). This is rarely an issue in practice, and could be remedied with double-precision GPU arithmetic. The chief drawback of our implementation is that it traverses an acceleration structure in the fragment shader, which is likely slower than rasterized bricking or slicing. Most of our chosen scenes are costly to sample regardless of space skipping, but we could employ a proxy rasterization technique such as [17] for better performance.
Several extensions to this work are worth pursuing. Differential sampling could be used in more traditional applications of ray differentials such as multiscale filtering and level of detail, which could improve quality and allow efficient rendering of large data. Peak finding could be extended to handle multidimensional and multifield transfer functions, which could use topological methods to find peaks in higher dimensions. We are also interested in combining preintegration and peak finding for better classification.
### Acknowledgments
This work was supported by the German Research Foundation (DFG) through the University of Kaiserslautern International Research Training Group (IRTG 1131); as well as the National Science Foundation under grants CNS-0615194, CNS-0551724, CCF-0541113, IIS-0513212, and DOE VACET SciDAC, KAUST GRP KUS-C1-016-04. Additional thanks to Liz Jurrus and Tolga Tasdizen for the zebrafish data, and to the anonymous reviewers for their comments.
## Footnotes
Aaron Knoll is with the University of Kaiserslautern, E-mail: knolla@rhrk.uni-kl.de.
Younis Hijazi is with LSIIT at the University of Strasbourg. E-mail: hijazi@lsiitu-strasbg.fr.
Rolf Westerteiger is with the University of Kaiserslautern. E-mail: rolfwesterteiger@googlemail.com.
Mathias Schott is with the SCI Institute, University of Utah. E-mail: mschott@cs.utah.edu.
Charles Hansen is with the SCI Institute, University of Utah. E-mail: hansen@cs.Utah.edu.
Hans Hagen is with the University of Kaiserslautern. E-mail: hangen@informatik.uni-kl.de.
Manuscript received 31 March 2009; accepted 27 July 2009; posted online 11 October 2009; mailed on 5 October 2009.
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This paper appears in:
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
Issue Date:
November/December 2009
On page(s):
1571 - 1578
ISBN:
1077-2626
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10930764
Digital Object Identifier:
10.1109/TVCG.2009.204
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01 Nov, 2009
Date of Original Publication:
23 Sep, 2009 | 2015-08-30 05:56:48 | {"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": 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": 0.6485274434089661, "perplexity": 2572.460375772692}, "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-2015-35/segments/1440644064919.24/warc/CC-MAIN-20150827025424-00322-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://as.yazd.ac.ir/article_1353.html | # Characterizing some groups with nilpotent derived subgroup
Document Type : Research Paper
Authors
1 Department of Mathematics, University of Birjand, Birjand, Iran.
2 Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
10.29252/as.2019.1353
Abstract
In this paper, groups with trivial intersection between Frattini and derived subgroups are considered. First, some structural properties of these groups are given in an important special case. Then, some family invariants of each $n$-isoclinism family of such groups are stated. In particular, an explicit bound for the order of each center factor group in terms of the order of its derived subgroup is also provided.
Keywords
#### References
[1] X. Guo and L. Gong, A note on the size of the nilpotent residual in nite groups, Arch. Math. Vol. 99
(2012), pp. 413-416. DOI: 10.1007/s00013-012-0452-5
[2] Z. Halasi and K. Podoski, Bounds in groups with trivial Frattini subgroup, J. Algebra Vol. 319 (2008), pp.
893-896. DOI: 10.1016/j.jalgebra.2007.02.053
[3] P. Hall, The classi cation of prime power groups, J. Reine Angew. Math. Vol. 182 (1940), pp. 130-141.
[4] P. Hall, The construction of soluble groups, Journal fr die reine und angewandte Mathematik Vol. 182
(1940), pp. 206-214,
[5] N.S. Hekster, On the structure of n-isoclinism classes of groups, J. Pure Appl. Algebra Vol. 40 (1986), pp.
63-85. DOI.10.1016/0022-4049(86)90030-7
[6] M. Hezog, G. Kaplan and A. Lev, On the commutator and the center of nite groups, J. Algebra Vol. 278
(2004), pp. 494-501. DOI.10.1016/j.jalgebra.2004.03.021
[7] A.Y.E. Ol'shanskii, Varieties of nitely approximable groups, Izvestiya Rossiiskoi Akademii Nauk. Vol. 33
(1969), pp. 915-927.
[8] D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, (1982).
[9] D.R. Taunt, On A-groups, Mathematical proceedings of the cambridge philosophical society Vol. 45 (1949),
pp. 24-42. DOI: 10.1017/S0305004100000414 Published online: 24 October 2008
[10] J.H. Walter, The characterization of nite groups with abelian Sylow 2-subgroups, Annals of Mathematics
Vol. 89 No. 3 (1969), pp. 405-514. | 2021-09-16 18:46:11 | {"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": 0.5132419466972351, "perplexity": 1532.2348366484011}, "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-2021-39/segments/1631780053717.37/warc/CC-MAIN-20210916174455-20210916204455-00629.warc.gz"} |
https://www.physicsforums.com/threads/current-using-kirchhoffs-loop-rule.401791/ | # Current using kirchhoff's loop rule
## Homework Statement
By applying kirchhoffs rules, show that the in the RL circuit is given by: $$I=\frac{\epsilon}{R}\left(1-e^{-\frac{Rt}{L}}\right)$$
## Homework Equations
kirchhoff's second rule: $$\epsilon-IR-L\frac{dI}{dt}=0$$
## The Attempt at a Solution
$$\epsilon=IR+L\frac{dI}{dt}$$
$${\epsilon}dt=IRdt+LdI$$
$$\frac{\epsilon}{I}dt-Rdt=L\frac{1}{I}dI$$
then i thought to integrate both sides, LHS w.r.t t and RHS w.r.t I, but doing this on paper doesnt get to the right answer, am i going wrong somewhere with the rearranging?? | 2021-09-19 13:10:56 | {"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": 0.8012216687202454, "perplexity": 2328.9827975912967}, "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-2021-39/segments/1631780056890.28/warc/CC-MAIN-20210919125659-20210919155659-00511.warc.gz"} |
http://wikieducator.org/PCF5:Inputbox-instructions | # PCF5:Inputbox-instructions
Jump to: navigation, search
## Abstract
Abstract text goes here ......
## Subsitute_this_text_with_your_first_heading
This is where you type your text that goes under the first heading ....
## Second heading goes here
This is where you type your text for the second heading .....
Work in progress, expect frequent changes. Help and feedback is welcome. See discussion page. | 2017-02-21 03:07:10 | {"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": 0.9058910608291626, "perplexity": 10073.227580205194}, "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-2017-09/segments/1487501170624.14/warc/CC-MAIN-20170219104610-00314-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://mathoverflow.net/revisions/84604/list | ## Return to Question
4 added 2 characters in body; added 7 characters in body
In the Bayesian version of (binary) hypothesis testing one has to decide which of two hypotheses $A$ and $B$ holds true. The two hypotheses are given prior probability $p(A)$ and $p(B)$, summing up to 1. $A$ and $B$ induce two probability distributions on a set of possible observations $X$, say $p(x|A)$ and $p(x|B)$. One has two decide between $A$ and $B$ after looking at one observation $x$.
It is known that the best strategy, that is the one that minimizes the probability of uncorrect guess, is to choose the hypothesis $H\in{A,B}$ that maximizes the $p(H|x)$, where $x$ is the observation. The probability of error (averaged on all $x$ and $H$) can be expressed
$$P_e = 1-\sum_x \max( p(x|A)p(A), p(x|B)p(B))\tag{1}$$
The expression (1) is often regarded as 'intractable' due to the presence of the max operator. Hence tractable bounds are seeked. An example is the harmonic lower-bound
$$P_e \geq E_x[P(A|x)P(B|x)]\tag{2}$$
($E_x$ is expectation over $x$; see e.g. : Routtenberg, Tabrikian, "A General Class of Lower Bounds on the Probability of Error in Multiple Hypothesis Testing", http://arxiv.org/abs/1005.2880, May 2010, and references therein).
My questions:
1) In what sense are expressions like the rhs of (2) "more tractable" than (1)? Computationally, they still require integration over (functions of the) PMF's $p(x|A)$ and $p(x|B)$. Maybe they are more convenient from an analytical point of view?
2) An exact and simple expressions for $P_e$ is:
$$P_e = \frac{||p(\cdot|A)p(A) 1-\frac{||p(\cdot|A)p(A) - p(\cdot|B)p(B)||_1 + 1}2$$1}2\tag{3}$$(Here p(\cdot|H) is viewed as a vector in R^{|X|}, and ||\cdot||_1 denotes the norm-1). This is conceptually interesting because it relates probability of error to a distance between PMF's. Is this expression regarded as "intractable" in the same sense as (1)? M. 3 displayed TeX In the Bayesian version of (binary) hypothesis testing one has to decide which of two hypotheses A and B holds true. The two hypotheses are given prior probability p(A) and p(B), summing up to 1. A and B induce two probability distributions on a set of possible observations X, say p(x|A) and p(x|B). One has two decide between A and B after looking at one observation x. It is known that the best strategy, that is the one that minimizes the probability of uncorrect guess, is to choose the hypothesis H\in{A,B} that maximizes the p(H|x), where x is the observation. The probability of error (averaged on all x and H) can be expressed P_e P_e = 1-\sum_x \max( p(x|A)p(A), p(x|B)p(B)) (1)p(x|B)p(B))\tag{1}$$
The expression (1) is often regarded as 'intractable' due to the presence of the max operator. Hence tractable bounds are seeked. An example is the harmonic lower-bound
$P_e$P_e \geq E_x[P(A|x)P(B|x)]$(2)E_x[P(A|x)P(B|x)]\tag{2}$$(E_x is expectation over x; see e.g. : Routtenberg, Tabrikian, "A General Class of Lower Bounds on the Probability of Error in Multiple Hypothesis Testing", http://arxiv.org/abs/1005.2880, May 2010, and references therein). My questions: 1) In what sense are expressions like the rhs of (2) "more tractable" than (1)? Computationally, they still require integration over (functions of the) PMF's p(x|A) and p(x|B). Maybe they are more convenient from an analytical point of view? 2) An exact and simple expressions for P_e is: P_e P_e = \frac{||p(\cdot|A)p(A) - p(\cdot|B)p(B)||_1 + 1}21}2$$ (Here$p(\cdot|H)$is viewed as a vector in$R^{|X|}$, and$||\cdot||_1$denotes the norm-1). This is conceptually interesting because it relates probability of error to a distance between PMF's. Is this expression regarded as "intractable" in the same sense as (1)? M. 2 edited body; deleted 6 characters in body In the Bayesian version of (binary) hypothesis testing one has to decide which of two hypotheses$A$and$B$holds true. The two hypotheses are given prior probability$p(A)$and$p(B)$, summing up to 1.$A$and$B$induce two probability distributions on a set of possible observations$X$, say$p(x|A)$and$p(x|B)$. One has two decide between$A$and$B$after looking at one observation$x$. It is known that the best strategy, that is the one that minimizes the probability of uncorrect guess, is to choose the hypothesis$H\in{A,B}$that maximizes the$p(H|x)$, where$x$is the observation. The probability of error (averaged on all$x$and$H$) is can be expressed$P_e = 1-\sum_x \max( p(x|A)p(A), p(x|B)p(B))$(1) The expression (1) is often regarded to be as 'intractable' due to the presence of the max operator. Hence tractable bounds are seeked. An example is the harmonic lower-bound$P_e \geq E_x[P(A|x)P(B|x)]$(2) ($E_x$is expectation over$x$; see e.g. : Routtenberg, Tabrikian, "A General Class of Lower Bounds on the Probability of Error in Multiple Hypothesis Testing", http://arxiv.org/abs/1005.2880, May 2010, and references therein). My questions: 1) In what sense are expressions like the rhs of (2) "more tractable" than (1)? Computationally, they still require integration over (functions of the) PMF's$p(x|A)$and$p(x|B)$. Maybe they are more convenient from an analytical point of view? 2) An exact and simple expressions for$P_e$is:$P_e = \frac{||p(\cdot|A)p(A) - p(\cdot|B)p(B)||_1 + 1}2$(Here$p(\cdot|H)$is viewed as a vector in$R^{|X|}$, and$||\cdot||_1\$ denotes the norm-1).
This is conceptually interesting because it relates probability of error to a distance between PMF's. Is this expression regarded as "untractable" intractable" in the same sense as (1)?
M.
1 | 2013-05-24 23:29:17 | {"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": 0.9363950490951538, "perplexity": 876.3216527651522}, "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-2013-20/segments/1368705284037/warc/CC-MAIN-20130516115444-00067-ip-10-60-113-184.ec2.internal.warc.gz"} |
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# If 2/x^3 < 0.02 what is the least positive integer value of x?
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If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
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27 Jun 2018, 21:51
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If $$\frac{2}{x^3}< 0.02$$ what is the least positive integer value of x?
A. No such least value exists.
B. 101
C. 100
D. 5
E. 4
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Posts: 2269
Re: If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
### Show Tags
27 Jun 2018, 22:08
Solution
Given:
• $$\frac{2}{x^3} < 0.02$$
To find:
• The least integer value of x
Approach and Working:
We can rewrite the given expression in the following manner:
• $$\frac{2}{x^3} < 0.02$$
Or, $$\frac{2}{x^3} < \frac{2}{100}$$
If this is true, then $$x^3$$ must be greater than 100
• If x = 4, then $$4^3$$ = 64 < 100
• If x = 5, then $$5^3$$ = 125 > 100
Therefore, the least integer value of x must be 5
Hence, the correct answer is option D.
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Re: If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
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28 Jun 2018, 02:28
Bunuel wrote:
If $$\frac{2}{x^3}< 0.02$$what is the least integer value of x?
A. No such least value exists.
B. 101
C. 100
D. 5
E. 4
Note: we are asked to determine the least value of x.
what if $$x^3$$ = 100
$$\frac{2}{100} = 0.02$$
then we can take care x = 5
$$x^3$$ = 125
it is clear that
$$\frac{2}{125}$$ < 0.02..........as we know$$\frac{2}{100}= 0.02$$
Thus the best answer is D.
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Re: If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
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28 Jun 2018, 09:51
By substitution,
2/64 = 1/32 > 0.02 - Incorrect
2/125 < 0.02 - Correct.
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Re: If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
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02 Jul 2018, 08:44
Bunuel wrote:
If $$\frac{2}{x^3}< 0.02$$what is the least integer value of x?
A. No such least value exists.
B. 101
C. 100
D. 5
E. 4
We can test the values given in the answer choices, starting with 4 (choice E):
2/4^3 < 0.02 ?
2 < 0.02 x 4^3 ?
2 < 0.02 x 64 ?
2 < 1.28 ? → No!
Now let’s test 5:
2/5^3 < 0.02 ?
2 < 0.02 x 5^3 ?
2 < 0.02 x 125 ?
2 < 2.5 ? → Yes!
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Re: If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
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13 Jul 2018, 07:02
1
What if x is negative?
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Re: If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
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14 Jul 2018, 00:02
1
What if we take the value of X to be negative ?
Can anyone please clear my doubt?
Thanks!
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Re: If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
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14 Jul 2018, 00:52
Insiya wrote:
What if we take the value of X to be negative ?
Can anyone please clear my doubt?
Thanks!
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You are right. The word "positive" was missing in the stem.
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If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
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14 Jul 2018, 02:13
Since x must be a positive integer, both sides of the inequality can be multiplied by $$x^3$$
$$\frac{2}{x^3} < 0.02$$
$$2 < 0.02x^3$$
$$200 < 2x^3$$
$$100 < x^3$$
$$x^3 > 100$$
In the resulting inequality, the least positive integer value for x is 5.
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Re: If 2/x^3 < 0.02 what is the least positive integer value of x? [#permalink]
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14 Jul 2018, 04:20
Bunuel wrote:
If $$\frac{2}{x^3}< 0.02$$ what is the least positive integer value of x?
A. No such least value exists.
B. 101
C. 100
D. 5
E. 4
$$\frac{2}{x^3}< 0.02...............2<x^3*0.02......100<x^3$$
Now x is integer, 4^3=64 and 5^3=125
So minimum is 5
D
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Re: If 2/x^3 < 0.02 what is the least positive integer value of x? &nbs [#permalink] 14 Jul 2018, 04:20
Display posts from previous: Sort by | 2018-12-11 09:21:05 | {"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": 0.6547055840492249, "perplexity": 7132.527533804527}, "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-2018-51/segments/1544376823614.22/warc/CC-MAIN-20181211083052-20181211104552-00191.warc.gz"} |
https://plainmath.net/82613/we-have-ab-bc-ac-cd-angle-acd-circ | # We have AB=BC, AC=CD, angle ACD=90^circ. If the radius of the circle is 'r' , find BC in terms of r .
We have $AB=BC$, $AC=CD$, $\mathrm{\angle }ACD={90}^{\circ }$. If the radius of the circle is 'r' , find BC in terms of r .
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Step 1
Angle ABC is $135°$ because it is half the concave angle $COA=270°$.
Step 2
Thus angle $BAC=22.5°$ and $BC=2r\mathrm{sin}22.5°=2r\cdot \frac{\sqrt{2-\sqrt{2}}}{2}=r\sqrt{2-\sqrt{2}}$ | 2022-12-04 05:35:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 29, "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": 0.9519575238227844, "perplexity": 1111.7834201813882}, "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/1669446710962.65/warc/CC-MAIN-20221204040114-20221204070114-00578.warc.gz"} |
https://www.physicsforums.com/threads/infinite-union-of-sigma-algebras.987679/ | # Infinite union of sigma algebras
Homework Statement:
Let ##(X, \mathcal{A})## be a measurable space and ##(A_n)_{n\in\mathbb{N}}## be a strictly increasing sequence of ##\sigma## algebras. Show $$\mathcal{A}_\infty := \bigcup_{n\in\mathbb{N}} A_n$$ is never a ##\sigma## algebra.
Relevant Equations:
A sigma algebra on ##X## is a subset of ##\mathcal{P}(X)## that contains the identity, is closed under complements, and closed under countable union.
For all ##n\in\mathbb{N}## we have ##\emptyset \in A_n##. Hence, ##\emptyset \in \mathcal{A}_\infty##. Let ##A \in \mathcal{A}_\infty##. Then ##A \in A_k## for some ##k\in\mathbb{N}##. So ##A^c \in A_k##. Hence, ##A^c \in \mathcal{A}_\infty##. Thus, ##\mathcal{A}_\infty## is closed under complements. So ##\mathcal{A}_\infty## must fail countable union.
We have ##A_1 \subsetneq A_2 \subsetneq A_3 \subsetneq \dots##. Let us define a sequence ##(B_n)_{n\in \mathbb{N}^{\ge 2}}## where ##B_n## is some set in ##A_n## but not in ##A_{n-1}##.
Also, if ##\mathcal{A}_\infty## is closed under countable union, then ##\mathcal{A}_\infty## is closed under countable intersection. So maybe that's how to get a contradiction?
Consider ##\bigcap_{n\ge 2} B_n##. Is this on the right track?...
This is a difficult problem. You will need to work a lot harder than your attempt shows.
Here is a solution: https://math.stackexchange.com/ques...igma-algebras-is-not-a-sigm?noredirect=1&lq=1
Obviously don't click it if you don't want to get spoiled. If you want to, I can give you some hints.
Thanks for the reply. I tried working with the sequence ##(B_n)## where ##B_n## is an element of ##\mathcal{A}_{n+1} \setminus \mathcal{A}_n##. We want to show ##\bigcup_{n} B_n \not\in \mathcal{A}_\infty##. We can observe that ##B_1 \in \mathcal{A}_2\setminus\mathcal{A}_1, B_2 \in \mathcal{A}_3\setminus\mathcal{A_2} \supset \mathcal{A}_3 \setminus \mathcal{A}_1 \dots## and so for all ##n## we have ##B_n \not\in \mathcal{A}_1##. Similarly, for all ##n \ge 2## we have ##B_n \not\in \mathcal{A}_2## and continuing in this way, for all ##n \ge k## we have ##B_n \not\in \mathcal{A}_k##.
Assume by contradiction that ##\bigcup_n B_n \in \mathcal{A}_m## for some ##m##. Then, ##B_1, B_2, \dots, B_{m-1} \in \mathcal{A}_m##. Since ##\mathcal{A}_m## is closed under complements and countable intersection, we have ##\bigcup_{k=m}^{\infty}B_k \in \mathcal{A}_m##. By construction, ##B_m \not\in \mathcal{A_m}##. Under these assumptions, can we show ##B_m \in \mathcal{A}_m## to get a contradiction?
Thanks for the stack exchange link (I haven't clicked on it yet but maybe if this problem turns out too be too hard I will...) its funny, my homework was too hard so I found this problem in text book and thought it'd be be a fun one to do as a warm up.... if you have time, would you be able to give me a hint, please?
Last edited:
member 587159
It's definitely no warm-up problem haha. It seems to be a statement that was published (together with its proof) in
A. Broughton and B. W. Huff: A comment on unions of sigma-fields. The American Mathematical Monthly, 84, no. 7 (1977), 553-554
I don't think the approach you suggest works. It was my first idea too when I saw the question this morning but I don't think you can make it work like that, as you don't know how the sets relate.
I underestimated the problem. I don't think I can give a "good" hint that will make you solve the problem, as the solution in the link gets quite technical, so I suggest reading the answer in the link. I think it will already be a good exercise to make sure you understand that answer! This is the kind of question on which you spend several days to solve.
Now, I'm curious. What were the "hard" questions in your measure theory exercise session?
fishturtle1
It's definitely no warm-up problem haha. It seems to be a statement that was published (together with its proof) in
A. Broughton and B. W. Huff: A comment on unions of sigma-fields. The American Mathematical Monthly, 84, no. 7 (1977), 553-554
I don't think the approach you suggest works. It was my first idea too when I saw the question this morning but I don't think you can make it work like that, as you don't know how the sets relate.
I underestimated the problem. I don't think I can give a "good" hint that will make you solve the problem, as the solution in the link gets quite technical, so I suggest reading the answer in the link. I think it will already be a good exercise to make sure you understand that answer! This is the kind of question on which you spend several days to solve.
Now, I'm curious. What were the "hard" questions in your measure theory exercise session?
Wow, ok! I'll do as you suggested then and go through the proof.
As to the hard questions i'd prefer not to say, since they're homework but it's on L^p spaces as well as Construction of measures. Sorry I know that's not a satisfying answer =\...
Thank you for your time on this.
member 587159
Wow, ok! I'll do as you suggested then and go through the proof.
As to the hard questions i'd prefer not to say, since they're homework but it's on L^p spaces as well as Construction of measures. Sorry I know that's not a satisfying answer =\...
Thank you for your time on this.
I understand. Have a good day! | 2021-07-31 09:39:11 | {"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": 0.9813584685325623, "perplexity": 1066.180725633002}, "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/1627046154085.58/warc/CC-MAIN-20210731074335-20210731104335-00275.warc.gz"} |
https://mathoverflow.net/tags/harmonic-analysis/hot | # Tag Info
3
The asymptotics for large $n$ is $$J(n,\kappa) := \Big(\frac{2}{\pi}\Big)^2 \int_{-\pi}^\pi \int_{-\pi}^\pi \exp{(i\,n(x+y))}\frac{\sin^2x\,\sin^2y} {2\kappa - (\cos{x}+\cos{y}) }\, dx \,dy \sim$$ $$\sim \frac{8}{\sqrt{\pi \kappa n}}(\kappa^2-1)^{7/4} (\kappa - \sqrt{\kappa^2-1})^{2n}\quad, \quad (\kappa>1)$$ The proof consists of 5 parts. The first ...
3
Yes. see:The symplectic camel and phase space quantization, by Maurice De Gosson. Journal of Physics A: Mathematical and General, Volume 34, Number 47
2
the Plancherel density is derived from the Plancherel measure, see arXiv:1812.00047 for the precise definition:
2
You have $I(\lambda, x)=x\cdot\int_{\mathbb S^{n-1}} ye^{i \lambda x\cdot y} d\sigma(y)=x\cdot J(x,\lambda)$ and you claim that for $\vert x\vert \lambda \ge 1$, you have $$J(x,\lambda)=O((\vert x\vert \lambda)^{-\frac{n-1}{2}}).$$ Indeed, using coordinate charts and a finite partition of unity, you are reduced to the case where J(x,\lambda)=\int_{\...
1
Let $u$ be a smooth function on $\mathbb R^n\backslash\{0\}$ homogeneous with degree $\lambda$ (on $\mathbb R^n\backslash\{0\}$). If $\lambda$ is not an integer $\le -n$, then $u$ can be uniquely extended to a tempered distribution homogeneous with degree $\lambda$. Moreover, the Fourier transform of an homogeneous distribution with degree $\lambda$ is an ...
1
The answer is: Weyl, H. 1919, Annalen der Physik, 365, 481 doi: 10.1002/andp.19193652104 Though it's difficult to find in there if you don't understand German
Only top voted, non community-wiki answers of a minimum length are eligible | 2020-04-07 08:03:20 | {"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": 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": 0.9546523094177246, "perplexity": 331.19074400777606}, "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-2020-16/segments/1585371675859.64/warc/CC-MAIN-20200407054138-20200407084638-00458.warc.gz"} |
https://asa2.silverchair.com/anesthesiology/article/96/1/1/39842/Two-Examples-of-How-to-Evaluate-the-Impact-of-New?searchresult=1 | DURING the past 15 yr, there has been increasing interest in using newer technologies to enhance the education and training of medical personnel. In this issue of Anesthesiology, Morgan et al. 1and Birnbach et al. 2give two examples of how to evaluate the impact of new approaches to teaching.
Morgan et al. conducted a careful study comparing faculty-led sessions using either an “exemplar” video of proper practice or demonstrations with a high-fidelity patient simulator to teach final-year Canadian medical students some key points of the medical responses to specific intraoperative events. Although the students preferred the simulation sessions, there was no difference between the groups in the ability of students to respond to the events when tested in the simulator. Although the authors are circumspect in their claims, others might view this as proof that simulators are not worth their nontrivial price. However, at most, such a view would find justification only for a very restricted set of questions asked in this study.
To assess the value of an educational or training modality we must consider various factors, including the target population, the goal, and the overall costs of the intervention. Typical target populations for simulation activities have ranged from outreach programs involving children and lay adults to preclinical and clinical students in medicine, nursing, and allied health professions to highly experienced physicians and nurses. Not all purposes and goals are equally applicable to all target populations. We should distinguish between education and training. The goal of education is typically to teach or improve conceptual understanding or to introduce individuals to skills. For training, the goal is to implement or improve specific skills and behaviors needed to accomplish a real-world job. Medicine especially has emphasized education, leaving training largely to an apprenticeship model.
Morgan et al. chose a target population of final-year medical students. The goal of the intervention must be inferred to be education about intraoperative critical events rather than training because no one would expect these students to be able to perform this task adequately in real patient care. This is reflected in the substantial simplification of the task in the demonstrations and test relative to that encountered in real clinical situations. Given such a restricted goal and task, it may not be surprising that the students who had intensive faculty teaching using either the exemplar videos or the exemplar simulations improved their understanding and abilities versus their baseline but did not differ in their performance depending on the modality used to teach them.
Further, was this really a comparison between a $100 intervention (the video) and a$150,000 intervention (the simulator)? Making a good training video can itself be expensive, and may require a simulator to create the clinical scenarios. Moreover, in assessing the costs of the simulator intervention, one cannot attribute to any single activity the capital costs of the simulator and the accompanying space and infrastructure. Nearly all simulation centers have a diverse set of users from different departments, for different target populations, and for different purposes. The fixed expenses of the center must be amortized over a number of years and across all the users. Although substantially greater than the cost of a video player, the simulator center usage costs attributable specifically to the intervention studied by Morgan et al. might not be that high. This is especially true because the major cost of simulation training is faculty time. Morgan et al. acknowledged that for both video- and simulation-based teaching, a roughly equal—and substantial—amount of faculty time was required.
Within the limits that they posed for themselves, Morgan et al. demonstrated that it is possible to conduct a careful test of different educational modalities. The conundrum is that measuring the results of the intervention requires the ability to assess performance. Although this proved feasible for the simplified tasks expected of students, it will be more difficult to do so for more complex tasks and behaviors expected of experienced personnel. However, in some cases, studying the details of even a restricted task has important ramifications for safe and efficient patient care. Birnbach et al. showed that most aspects of epidural catheter placement can be assessed robustly by reviewing videotapes of clinicians performing the task. Although it is a relatively simple act, successful placement of an epidural catheter is a crucial task in clinical domains, such as obstetric analgesia and anesthesia. Therefore, the results of the study by Birnbach et al. , though limited in scope, may be more relevant to clinical practice than those of Morgan et al. —although whether improvement of catheter placement skill through video analysis has practical outcome benefit for patients remains to be seen.
Another key lesson from both of these studies is that video can be a powerful teaching tool, especially when it is applied (as by Birnbach et al. ) to specific performances of those under instruction. Nonetheless, in both studies, the use of videotapes was coupled with expert teaching by motivated faculty. This reinforces a common belief that modern technologies provide tools that can enhance, but not substitute for, skilled and dedicated teachers.
By comparison to other industries, such as aviation, in my view, the greatest promise for the use of simulators and other training modalities to impact patient safety lies not in the education of target populations of early learners regarding simplified tasks, but rather with initial and recurrent training of advanced trainees and experienced practitioners regarding much more complex tasks. For these challenging settings, tests that are easy to score unambiguously will rarely replicate or capture the demands of real patient care. Tests that do address the complexity of real care will suffer from higher subjectivity. Therefore, it will be more difficult to make assessments of the impact of novel training for complex real-world job skills. Any such studies are likely to be expensive to conduct because of the high interindividual variability, the need for multiple experienced raters, and the imprecision of existing or proposed metrics of complex performance. Nonetheless, in a recent assessment of the evidence base for a variety of patient safety interventions sponsored by the Agency for Healthcare Research and Quality, the authors concluded the following regarding patient simulators 3:
Definitive experiments to improve our understanding of their effects on training will allow them to be used more intelligently to improve provider performance, reduce errors and ultimately, promote patient safety. Although such experiments will be difficult and costly, they may be justified to determine how this technology can best be applied.
For simulation and for video analysis, the work of Morgan et al. and Birnbach et al. are only the beginning of a very long road.
1.
Morgan PJ, Cleave-Hogg D, McIlroy J, Devitt JM: Simulation technology: A comparison of experiential and visual learning for undergraduate medical students. A nesthesiology 2002; 96: 10–6
2.
Birnbach CJ, Santos AC, Bourlier RA, Meadows WE, Datta S, Stein DJ, Kuroda MM, Thys DM: The effectiveness of video technology as an adjunct to teach and evaluate epidural anesthesia performance skills. A nesthesiology 2002; 96: 5–9
3.
Jha AK, Duncan BW, Bates DW: Simulator-based training and patient safety, Making Health Care Safer: A Critical Analysis of Patient Safety Practices (Evidence Report/Technology Assessment No. 43. AHRQ Publication 01-E058). Edited by Shojania KG, Duncan BW, McDonald KM, Wachter RM. Rockville, Agency for Healthcare Research and Quality, 2001, pp 510–7 | 2023-03-31 07:14:28 | {"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": 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": 0.3687908947467804, "perplexity": 2987.324044988614}, "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/1679296949573.84/warc/CC-MAIN-20230331051439-20230331081439-00049.warc.gz"} |
http://www.hindawi.com/isrn/appmath/2012/475781/ | `ISRN Applied MathematicsVolume 2012 (2012), Article ID 475781, 16 pagesdoi:10.5402/2012/475781`
Research Article
## An L-Moment-Based Analog for the Schmeiser-Deutsch Class of Distributions
1Section on Statistics and Measurement, Department of EPSE, Southern Illinois University Carbondale, 222-J Wham Building, Carbondale, IL 62901-4618, USA
2University of Texas at Arlington, 320B Science Hall, Arlington, TX 76019, USA
Received 28 June 2012; Accepted 7 August 2012
Academic Editors: I. Doltsinis and E. Yee
Copyright © 2012 Todd C. Headrick and Mohan D. Pant. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
#### Abstract
This paper characterizes the conventional moment-based Schmeiser-Deutsch (S-D) class of distributions through the method of L-moments. The system can be used in a variety of settings such as simulation or modeling various processes. A procedure is also described for simulating S-D distributions with specified L-moments and L-correlations. The Monte Carlo results presented in this study indicate that the estimates of L-skew, L-kurtosis, and L-correlation associated with the S-D class of distributions are substantially superior to their corresponding conventional product-moment estimators in terms of relative bias—most notably when sample sizes are small.
#### 1. Introduction
The conventional moment-based Schmeiser-Deutsch (S-D) [1] class of distributions has demonstrated to be useful for modeling or simulating phenomena in the contexts of operations research and industrial engineering. Some examples include modeling stochastic inventory processes and lead time distributions [27], unpaced line efficiency [8, 9], two-stage production systems [10], stochastic activity networks [11, 12], and the newsboy problem [13]. Further, it is also common practice for methodologists to investigate the Type I error and power properties associated with inferential statistics (e.g., [14]). In many cases, these investigations may only require an elementary transformation to produce distributions with specified values of conventional skew, kurtosis, and Pearson correlations (or Gaussian copulas). The S-D class of distributions is particularly well suited for this task as it is computationally efficient because it only requires the knowledge of four parameters and an algorithm that generates zero-one uniform pseudorandom deviates.
Specifically, the quantile function for generating S-D distributions can be succinctly described as in [1]: where is zero-one uniformly distributed. The values of and are the location and scale parameters, while and are the shape parameters that determine the skew and kurtosis. The system of equations for determining the parameters in (1.1) for a S-D distribution with prespecified values of mean, variance, skew, and kurtosis is given in the Appendix. Figure 1 gives an example of an S-D distribution.
Figure 1: A graph of an S-D distribution based on the pdf in (2.4). The values of skew and kurtosis were determined based on (A.2) for and in the Appendix. The values of and were determined based on (3.4) and (3.5). The estimates (; ) and bootstrap confidence intervals (C.I.s) were based on resampling 25,000 statistics. Each sample statistic was based on a sample size of .
There are problems associated with conventional moment-based estimators (e.g., skew and kurtosis in Figure 1) insofar as they can be substantially biased, have high variance, or can be influenced by outliers. For example, inspection of Figure 1 indicates, on average, that the estimates of attenuate 27.51% and 38.82% below their associated population parameters. Note that each estimate of in Figure 1 was calculated based on samples of size and the formulae currently used by most commercial software packages such as SAS, SPSS, and Minitab for computing skew and kurtosis.
However, -moment-based estimators, such as -skew (), -kurtosis (), and the -correlation, have been introduced to address the limitations associated with conventional moment-based estimators [1518]. Specifically, some of the advantages that -moments have over conventional moments are that they (i) exist whenever the mean of the distribution exists, (ii) are nearly unbiased for all sample sizes and distributions, and (iii) are more robust in the presence of outliers. For example, the estimates of in Figure 1 are relatively much closer to their respective parameters with smaller relative standard errors than their corresponding conventional moment-based analogs . More specifically, the estimates of that were simulated are, on average, 9.19% below and 5.7% above their respective parameters.
Thus, the present aim here is to characterize the S-D class of distributions through the method of -moments. The characterization will enable researchers to model or simulate nonnormal distributions with specified values of -skew, -kurtosis, and -correlation. The rest of the paper is outlined as follows. In Section 2, a summary of univariate -moment theory is provided as well as additional properties associated with S-D distributions. In Section 3, the derivation of the system of equations for specifying values of -skew and -kurtosis for the S-D class of distributions is subsequently provided. In Section 4, the coefficient of -correlation is introduced and the equations are developed for determining intermediate correlations for specified -correlations associated with the S-D class of distributions. In Section 5, the steps for implementing a simulation procedure are described. Numerical examples and the results of a simulation are also provided to confirm the derivations and compare the new methodology with its conventional moment-based counterparts. In Section 6, the results of the simulation are discussed and concluding comments are made.
#### 2. Preliminaries on Univariate L-Moments and the Schmeiser-Deutsch Class of Distributions
##### 2.1. Univariate L-Moments
Let be identically and independently distributed random variables each with continuous pdf , cdf , order statistics denoted as , and -moments defined in terms of either linear combinations of (i) expectations of order statistics or (ii) probability weighted moments (). For the purposes considered herein, the first four -moments associated with are expressed as [16, pages 20–22] where the are determined from where . The coefficients associated with in (2.2) are obtained from shifted orthogonal Legendre polynomials and are computed as shown in [16, page 20].
The -moments and in (2.1) are measures of location and scale and are the arithmetic mean and one-half the coefficient of mean difference, respectively. Higher order -moments are transformed to dimensionless quantities referred to as -moment ratios defined as for , and where and are the analogs to the conventional measures of skew and kurtosis. In general, -moment ratios are bounded in the interval as is the index of -skew () where a symmetric distribution implies that all -moment ratios with odd subscripts are zero. Other smaller boundaries can be found for more specific cases. For example, the index of -kurtosis () has the boundary condition for continuous distributions of [19] .
##### 2.2. The Schmeiser-Deutsch (S-D) Class of Distributions
The cdf and pdf associated with the S-D quantile function in (1.1) are expressed as in [1]: Setting in (2.4) produces symmetric S-D densities with a lower bound of kurtosis of as . Positive (negative) skew is produced for cases where () and , where reverses the direction of skew. For (), the unique mode (antimode) is located at and where produces uniform distributions. For example, the distribution in Figure 1 is bounded in the range of with mode, mean, and variance of 10, 10.042, and 0.0271, respectively. See the Appendix for the formulae for computing the moments associated with S-D distributions. In the next section, the system of -moments for the class of S-D distributions is derived.
#### 3. L-Moments for the Schmeiser-Deutsch (S-D) Class of Distributions
The derivation of the first four -moments associated with the S-D class of distributions begins by defining the probability weighted moments based on (2.2) in terms of (1.1) as where and are the zero-one uniform cdf and pdf. As such, integrating (3.1) for and simplifying using (2.1) give , , , and as Thus, given user-specified values of , , , and , (3.2)–(3.5) can be numerically solved to obtain the parameters for , , , and . Inspection of (3.4) and (3.5) indicates that the solutions to and are independent of the location and scale parameters ( and ). As with the conventional S-D class of symmetric distributions () the lower-bound of -kurtosis is obtained as .
Table 1 gives four examples of S-D distributions. These four distributions are used in the simulation portion of this study in Section 5. Distribution 1 was used by Lau [13] for stochastic modeling related to the newsboy problem and Distribution 2 was used by Aardal et al. [2] in the context of modeling inventory-control systems. The values of conventional skew and kurtosis for the four distributions were determined based on (A.2) in the Appendix. In the next section, we introduce the topic of the -correlation and subsequently develop the methodology for simulating S-D distributions with specified -correlations.
Table 1: Four S-D distributions and their associated -moments, conventional moments (C-moments), and S-D parameters used in the simulation. The values of -skew () and -kurtosis () are based on (3.4) and (3.5). The values of skew () and kurtosis () are based on the expressions in (A.2) of the Appendix.
#### 4. L-Correlations for the Schmeiser-Deutsch (S-D) Class of Distributions
The -correlation [17, 18] is introduced by considering two random variables and with distribution functions and , respectively. The second -moments of and can alternatively be expressed as The second -comoments of toward and toward are As such, the -correlations of toward and toward are expressed as The -correlation in (4.5) (or (4.6)) is bounded such that where a value of () indicates a strictly increasing (decreasing) monotone relationship between the two variables. In general, we would also note that .
In the context of the -moment-based S-D class of distributions, suppose it is desired to simulate a -variate distribution based on quantile functions of the forms in (1.1) with a specified -correlation matrix and where each distribution has its own specified values of and . Let denote standard normal variables where the distribution functions and bivariate density function associated with and are expressed as Using (4.7), it follows that the th S-D distribution associated with (1.1) can be expressed as since is zero-one uniformly distributed. As such, using (4.5), the -correlation of toward can be evaluated using the solved values of the parameters for , a specified intermediate correlation (IC) in (4.9), and the following integral generally expressed as where it is required that the location and scale parameters (, ) in (4.10) are to be solved such that will have the values of and in (3.2) and (3.3), that is, set to the values of and associated with the unit normal distribution. Note that this requirement is not a limitation as the -correlation is invariant to linear transformations [17]. Further, we would point out that the purpose of the IC () in (4.9) and (4.10) is to adjust for the effect of the transformation , which is induced by the parameters, such that has its specified -correlation () toward . Analogously, the -correlation of toward is expressed as Note for the special case that if in (4.10) and in (4.11) have the same parameters, that is, ;; ; , then . Provided in Algorithm 1 is source code written in Mathematica [20] that implements the computation of an IC () based on (4.10). The details for simulating S-D distributions with specified values of -skew, -kurtosis, and -correlations are described in the next section.
Algorithm 1: Mathematica source code for computing intermediate correlations for specified -correlations. The example is for distributions and () in Table 1. See Tables 2 and 4.
#### 5. The Procedure for Simulation and Monte Carlo Study
To implement a method for simulating S-D distributions with specified -moments and -correlations we suggest the following six steps.(1)Specify the -moments for transformations of the forms in (1.1), that is, and obtain the solutions for the parameters of , , , and by solving (3.2)–(3.5) with and using the specified values of -skew () and -kurtosis () for each distribution. Specify a matrix of -correlations () for toward , where .(2)Compute the (Pearson) intermediate correlations (ICs) by substituting the solutions of the parameters from Step (1) into (4.10) and then numerically integrate to solve for (see Algorithm 1 for an example). Repeat this step separately for all pairwise combinations of correlations.(3)Assemble the ICs into a matrix and decompose this matrix using a Cholesky factorization. Note that this step requires the IC matrix to be positive definite.(4)Use the results of the Cholesky factorization from Step (3) to generate standard normal variables correlated at the intermediate levels as follows: where are independent standard normal random variables and where represents the element in the th row and the th column of the matrix associated with the Cholesky factorization performed in Step (3).(5)Substitute from Step (4) into the following Taylor series-based expansion for the standard normal cdf [21]: where denotes the standard normal pdf and where the absolute error associated with (5.2) is less than .(6)Substitute the zero-one uniform deviates, , generated from Step (5) into the equations of the form of to generate the S-D distributions with the specified -moments and -correlations.To demonstrate the steps above, and evaluate the proposed method, a comparison between the proposed -moment and conventional product-moment-based procedures is subsequently described. Specifically, the parameters for the distributions in Table 1 are used as a basis for a comparison using the specified correlation matrix in Table 2. Tables 3 and 4 give the solved IC matrices for the conventional moment and -moment-based methods, respectively. See Algorithm 2 for an example of computing ICs for the conventional method. Tables 5 and 6 give the results of the Cholesky decompositions on the IC matrices, which are then used to create with the specified ICs by making use of the formulae given in (5.1) of Step (4) with . The values of are subsequently transformed to using (5.2) and then substituted into equations of the forms in (1.1) to produce for both methods.
Table 2: Specified correlation matrix for the distributions in Table 1.
Table 3: Intermediate correlations for the conventional moment procedure.
Table 4: Intermediate correlations for the -moment procedure.
Table 5: Cholesky decompositions for the conventional moment procedure.
Table 6: Cholesky decompositions for the -moment procedure.
Algorithm 2: Mathematica source code for computing intermediate correlations for specified conventional Pearson correlations. The example is for distributions and () in Table 1. See also Tables 2 and 3.
In terms of the simulation, a Fortran algorithm was written for both methods to generate 25,000 independent sample estimates for the specified parameters of (i) conventional skew (), kurtosis (), and Pearson correlation (); (ii) -skew (), -kurtosis (), and -correlation (). All estimates were based on sample sizes of and . The formulae used for computing estimates of were based on Fisher’s -statistics, that is, the formulae currently used by most commercial software packages such as SAS, SPSS, and Minitab, for computing indices of skew and kurtosis (where for the standard normal distribution). The formulae used for computing estimates of were Headrick’s equations and [22]. The estimate for was based on the usual formula for the Pearson product-moment of correlation statistic and the estimate for was computed based on (4.5) using the empirical forms of the cdfs in (4.1) and (4.3). The estimates for and were both transformed using Fisher’s transformation. Bias-corrected-accelerated-bootstrapped average (mean) estimates, confidence intervals (C.I.s), and standard errors were subsequently obtained for the estimates associated with the parameters (, , , ) using 10,000 resamples via the commercial software package Spotfire S+ [23]. The bootstrap results for the estimates of the means and C.I.s associated with and were transformed back to their original metrics (i.e., estimates for and ). Further, if a parameter () was outside its associated bootstrap C.I., then an index of relative bias (RB) was computed for the estimate () as: . Note that the small amount of bias associated with any bootstrap C.I. containing a parameter was considered negligible and thus not reported. The results of the simulation are reported in Tables 712 and are discussed in the next section.
Table 7: Skew () and kurtosis () results for distributions 2 and 4 in Table 1.
Table 8: -skew () and -kurtosis () results for distributions 2 and 4 in Table 1.
Table 9: Correlation results for the conventional moment procedure .
Table 10: Correlation results for the -moment procedure .
Table 11: Correlation results for the conventional moment procedure .
Table 12: Correlation results for the -moment procedure .
#### 6. Discussion and Conclusion
One of the primary advantages that -moments have over conventional moment-based estimators is that they can be far less biased when sampling is from distributions with more severe departures from normality (e.g. [16, 21]). Inspection of the simulation results in Tables 7 and 8 of this study clearly indicates that this is also the case for the S-D class of distributions. Specifically, the superiority that estimates of -moment ratios () have over their corresponding conventional moment based counterparts () is obvious. For example, with samples of size the estimates of skew and kurtosis for Distribution 4 (Table 7) were, on average, 85.65% and 74.74% of their associated population parameters, whereas the estimates of -skew and -kurtosis were 97.34% and 92.70% of their respective parameters. Similar results were also obtained for Distributions 1 and 3 and thus not reported. It is also evident from Tables 7 and 8 that -skew and -kurtosis are more efficient as their relative standard errors RSE = (standard error/estimate) × 100 are smaller than the conventional estimators of skew and kurtosis. For example, in terms of Distribution 4, inspection of Tables 7 and 8 () indicates RSE measures of RSE() = 0.1533% and RSE() compared with RSE() and RSE() . This demonstrates that -skew and -kurtosis have more precision because they have less variance around their estimates.
Presented in Tables 912 are the results associated with the conventional Pearson and -correlations. Inspection of Tables 9 and 10 indicates that the -correlation is substantially superior to the Pearson correlation in terms of relative bias for small sample sizes. For example, in terms of a moderate correlation (Table 9, , ) the relative bias for Distributions 3 and 4 was 23.95% for the Pearson correlation compared to 5.43% for the -correlation (Table 10, , ). For large sample sizes (Tables 11 and 12, ), both procedures performed adequately as their estimates were in close proximity with their respective population parameters.
In summary, the proposed -moment-based S-D class of distributions is an attractive alternative to the conventional moment-based S-D system. In particular, the -moment-based system has distinct advantages when leptokurtic distributions and small sample sizes are of concern. Finally, we would note that Mathematica Version 8.0 [20] source code is available from the authors for implementing the -moment-based method.
#### System of Conventional Moment-Based Equations for S-D Distributions
The moments () associated with the S-D class of distributions in (1.1) can be determined from where is the zero-one uniform pdf. The mean, variance, skew, and kurtosis are defined in general as in [24]: The moments associated with the location and scale parameters in (A.2) are The moments associated with the shape parameters of skew and kurtosis in (A.2) are
#### References
1. B. W. Schmeiser and S. J. Deutsch, “A versatile four parameter family of probability distributions suitable for simulation,” American Institute of Industrial Engineers (AIIE) Transactions, vol. 9, no. 2, pp. 176–182, 1977.
2. K. Aardal, Ö Jonsson, and H. Jönsson, “Optimal inventory policies with service-level contraints,” Journal of the Operations Research Society, vol. 40, no. 1, pp. 65–73, 1989.
3. O. Tang and R. W. Grubbström, “On using higher-order moments for stochastic inventory systems,” International Journal of Production Economics, vol. 104, no. 2, pp. 454–461, 2006.
4. A. Akcay and S. Tayur, “Improved inventory targets in the presence of limited historical demand data,” Manufacturing and Service Operations Management (M&SOM), vol. 13, no. 3, pp. 297–309, 2011.
5. J. F. Kottas and H. Lau, “A realistic approach for modeling stochastic lead time distributions,” American Institute of Industrial Engineers (AIIE) Transactions, vol. 11, no. 1, pp. 54–60, 1979.
6. J. F. Kottas and H. S. Lau, “Inventory control with general demand and lead time distributions,” International Journal of Systems Science, vol. 10, no. 5, pp. 485–492, 1979.
7. J. F. Kottas and H. S. Lau, “The use of versatile distribution families in some stochastic inventory calculations,” Journal of the Operations Research Society, vol. 31, no. 5, pp. 393–403, 1980.
8. H. Lau and G. E. Martin, “The effects of skewness and kurtosis of processing times in unpaced lines,” International Journal of Production Research, vol. 25, no. 10, pp. 1483–1492, 1987.
9. G. E. Martin, “Predictive formulae for unpaced line efficiency,” International Journal of Production Research, vol. 31, no. 8, pp. 1981–1990, 1993.
10. H. Lau, “The production rate of a two-stage system with stochastic processing times,” International Journal of Production Research, vol. 24, no. 2, pp. 401–412, 1986.
11. H. Lau, A. Lau, and C. Ho, “Improved moment-estimation formulas using more than three subjective fractiles,” Management Science, vol. 44, no. 3, pp. 346–351, 1998.
12. C. Somarajan and H.-S. Lau, “Stochastic activity networks: a four-parameter approach for completion times,” International Journal of Systems Science, vol. 25, no. 10, pp. 1607–1619, 1994.
13. H. S. Lau, “The newsboy problem under alternative optimization objectives,” Journal of the Operations Research Society, vol. 31, no. 6, pp. 525–535, 1980.
14. T. C. Headrick, Statistical Simulation: Power Method Polynomials and Other Transformations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2010.
15. J. R. M. Hosking, “$L$-moments: analysis and estimation of distributions using linear combinations of order statistics,” Journal of the Royal Statistical Society, vol. 52, no. 1, pp. 105–124, 1990.
16. J. R. M. Hosking and J. R. Wallis, Regional Frequency Analysis: An Approach Based on L-Moments, Cambridge University Press, Cambridge, UK, 1997.
17. R. Serfling and P. Xiao, “A contribution to multivariate $L$-moments: $L$-comoment matrices,” Journal of Multivariate Analysis, vol. 98, no. 9, pp. 1765–1781, 2007.
18. T. C. Headrick and M. D. Pant, “Simulating non-normal distributions with specified L-moments and L-correlations,” Statistica Neerlandica. In press.
19. M. C. Jones, “On some expressions for variance, covariance, skewness and $L$-moments,” Journal of Statistical Planning and Inference, vol. 126, no. 1, pp. 97–106, 2004.
20. Wolfram Research, Mathematica, Version 8.0, Wolfram Research, Champaign, Ill, USA, 2010.
21. G. Marsaglia, “Evaluating the normal distribution,” Journal of Statistical Software, vol. 11, no. 5, pp. 1–11, 2004.
22. T. C. Headrick, “A characterization of power method transformations through $L$-moments,” Journal of Probability and Statistics, vol. 2011, Article ID 497463, 22 pages, 2011.
23. TIBCO Software, TIBCO Spotfire S+ 8.1 for Windows, Palo Alto, Calif, USA, 2008.
24. T. C. Headrick, R. K. Kowalchuk, and Y. Sheng, “Parametric probability densities and distribution functions for Tukey $g$-and-$h$ transformations and their use for fitting data,” Applied Mathematical Sciences, vol. 2, no. 9–12, pp. 449–462, 2008. | 2013-05-20 08:14:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "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": 0.773295521736145, "perplexity": 1266.469244907149}, "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-2013-20/segments/1368698646863/warc/CC-MAIN-20130516100406-00084-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-to-multiply-in-scientific-notation-0-00100-mole-x-6-02-x-10-23-molecules-1-m | # How to multiply in scientific notation (0.00100 "mole") xx (6.02 xx 10^23 "molecules")/(1 "mole")?
Apr 23, 2015
Convert $0.00100 \text{mole}$ to scientific notation by moving the decimal to the right 3 places. This will give the number in scientific notation as $1.00 \times {10}^{- 3} \text{mole}$.
Now the question becomes 1.00xx10^(-3)"mole"*(6.022xx10^23"molecules")/(1 "mole")
Multiply the coefficients.
$1.00 \times 6.022 = 6.022$
Multiply the base $10$ for each number.
${10}^{- 3} \times {10}^{23} = {10}^{\left(- 3 + 23\right)} = {10}^{20}$
Put the coefficient together with the base $10$.
$6.022 \times {10}^{20}$
Rewrite the problem, canceling moles.
1.00xx10^(-3)cancel("mole")*(6.022xx10^23"molecules")/(1 cancel("mole"))=6.02xx10^20"molecules" | 2021-09-19 07:09:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "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": 0.7244455218315125, "perplexity": 4059.9373700721603}, "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-2021-39/segments/1631780056752.16/warc/CC-MAIN-20210919065755-20210919095755-00192.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-1-foundations-for-algebra-1-6-multiplying-and-dividing-real-numbers-standardized-test-prep-page-44/71 | ## Algebra 1
Published by Prentice Hall
# Chapter 1 - Foundations for Algebra - 1-6 Multiplying and Dividing Real Numbers - Standardized Test Prep - Page 44: 71
.6
#### Work Step by Step
We take the absolute value of -.2, which makes it positive, so we simplify the expression to: $3\times .2= .6$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-09-19 01:33:43 | {"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": 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": 0.7010329365730286, "perplexity": 1872.2528568884384}, "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-2018-39/segments/1537267155814.1/warc/CC-MAIN-20180919004724-20180919024724-00083.warc.gz"} |
https://www.nag.com/numeric/mb/nagdoc_mb/manual_25_1/html/g02/g02lcf.html | Integer type: int32 int64 nag_int show int32 show int32 show int64 show int64 show nag_int show nag_int
Chapter Contents
Chapter Introduction
NAG Toolbox
# NAG Toolbox: nag_correg_pls_fit (g02lc)
## Purpose
nag_correg_pls_fit (g02lc) calculates parameter estimates for a given number of factors given the output from an orthogonal scores PLS regression (nag_correg_pls_svd (g02la) or nag_correg_pls_wold (g02lb)).
## Syntax
[b, ob, vip, ifail] = g02lc(nfact, p, c, w, rcond, orig, xbar, ybar, iscale, xstd, ystd, vipopt, ycv, 'ip', ip, 'my', my, 'maxfac', maxfac)
[b, ob, vip, ifail] = nag_correg_pls_fit(nfact, p, c, w, rcond, orig, xbar, ybar, iscale, xstd, ystd, vipopt, ycv, 'ip', ip, 'my', my, 'maxfac', maxfac)
## Description
The parameter estimates $B$ for a $l$-factor orthogonal scores PLS model with $m$ predictor variables and $r$ response variables are given by,
$B=W PTW-1 CT , B∈ ℝm×r ,$
where $W$ is the $m$ by $k$ ($\ge l$) matrix of $x$-weights; $P$ is the $m$ by $k$ matrix of $x$-loadings; and $C$ is the $r$ by $k$ matrix of $y$-loadings for a fitted PLS model.
The parameter estimates $B$ are for centred, and possibly scaled, predictor data ${X}_{1}$ and response data ${Y}_{1}$. Parameter estimates may also be given for the predictor data $X$ and response data $Y$.
Optionally, nag_correg_pls_fit (g02lc) will calculate variable influence on projection (VIP) statistics, see Wold (1994).
## References
Wold S (1994) PLS for multivariate linear modelling QSAR: chemometric methods in molecular design Methods and Principles in Medicinal Chemistry (ed van de Waterbeemd H) Verlag-Chemie
## Parameters
### Compulsory Input Parameters
1: $\mathrm{nfact}$int64int32nag_int scalar
$l$, the number of factors to include in the calculation of parameter estimates.
Constraint: $1\le {\mathbf{nfact}}\le {\mathbf{maxfac}}$.
2: $\mathrm{p}\left(\mathit{ldp},{\mathbf{maxfac}}\right)$ – double array
ldp, the first dimension of the array, must satisfy the constraint $\mathit{ldp}\ge {\mathbf{ip}}$.
$x$-loadings as returned from nag_correg_pls_svd (g02la) and nag_correg_pls_wold (g02lb).
3: $\mathrm{c}\left(\mathit{ldc},{\mathbf{maxfac}}\right)$ – double array
ldc, the first dimension of the array, must satisfy the constraint $\mathit{ldc}\ge {\mathbf{my}}$.
$y$-loadings as returned from nag_correg_pls_svd (g02la) and nag_correg_pls_wold (g02lb).
4: $\mathrm{w}\left(\mathit{ldw},{\mathbf{maxfac}}\right)$ – double array
ldw, the first dimension of the array, must satisfy the constraint $\mathit{ldw}\ge {\mathbf{ip}}$.
$x$-weights as returned from nag_correg_pls_svd (g02la) and nag_correg_pls_wold (g02lb).
5: $\mathrm{rcond}$ – double scalar
Singular values of ${P}^{\mathrm{T}}W$ less than rcond times the maximum singular value are treated as zero when calculating parameter estimates. If rcond is negative, a value of $0.005$ is used.
6: $\mathrm{orig}$int64int32nag_int scalar
Indicates how parameter estimates are calculated.
${\mathbf{orig}}=-1$
Parameter estimates for the centered, and possibly, scaled data.
${\mathbf{orig}}=1$
Parameter estimates for the original data.
Constraint: ${\mathbf{orig}}=-1$ or $1$.
7: $\mathrm{xbar}\left({\mathbf{ip}}\right)$ – double array
If ${\mathbf{orig}}=1$, mean values of predictor variables in the model; otherwise xbar is not referenced.
8: $\mathrm{ybar}\left({\mathbf{my}}\right)$ – double array
If ${\mathbf{orig}}=1$, mean value of each response variable in the model; otherwise ybar is not referenced.
9: $\mathrm{iscale}$int64int32nag_int scalar
If ${\mathbf{orig}}=1$, iscale must take the value supplied to either nag_correg_pls_svd (g02la) or nag_correg_pls_wold (g02lb); otherwise iscale is not referenced.
Constraint: if ${\mathbf{orig}}=1$, ${\mathbf{iscale}}=-1$, $1$ or $2$.
10: $\mathrm{xstd}\left({\mathbf{ip}}\right)$ – double array
If ${\mathbf{orig}}=1$ and ${\mathbf{iscale}}\ne -1$, the scalings of predictor variables in the model as returned from either nag_correg_pls_svd (g02la) or nag_correg_pls_wold (g02lb); otherwise xstd is not referenced.
11: $\mathrm{ystd}\left({\mathbf{my}}\right)$ – double array
If ${\mathbf{orig}}=1$ and ${\mathbf{iscale}}\ne -1$, the scalings of response variables as returned from either nag_correg_pls_svd (g02la) or nag_correg_pls_wold (g02lb); otherwise ystd is not referenced.
12: $\mathrm{vipopt}$int64int32nag_int scalar
A flag that determines variable influence on projections (VIP) options.
${\mathbf{vipopt}}=0$
VIP are not calculated.
${\mathbf{vipopt}}=1$
VIP are calculated for predictor variables using the mean explained variance in responses.
${\mathbf{vipopt}}={\mathbf{my}}$
VIP are calculated for predictor variables for each response variable in the model.
Note that setting ${\mathbf{vipopt}}={\mathbf{my}}$ when ${\mathbf{my}}=1$ gives the same result as setting ${\mathbf{vipopt}}=1$ directly.
Constraint: ${\mathbf{vipopt}}=0$, $1$ or ${\mathbf{my}}$.
13: $\mathrm{ycv}\left(\mathit{ldycv},{\mathbf{my}}\right)$ – double array
ldycv, the first dimension of the array, must satisfy the constraint if ${\mathbf{vipopt}}\ne 0$, $\mathit{ldycv}\ge {\mathbf{nfact}}$.
If ${\mathbf{vipopt}}\ne 0$, ${\mathbf{ycv}}\left(\mathit{i},\mathit{j}\right)$ is the cumulative percentage of variance of the $\mathit{j}$th response variable explained by the first $\mathit{i}$ factors, for $\mathit{i}=1,2,\dots ,{\mathbf{nfact}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{my}}$; otherwise ycv is not referenced.
### Optional Input Parameters
1: $\mathrm{ip}$int64int32nag_int scalar
Default: the dimension of the arrays xbar, xstd and the first dimension of the arrays p, w. (An error is raised if these dimensions are not equal.)
$m$, the number of predictor variables in the fitted model.
Constraint: ${\mathbf{ip}}>1$.
2: $\mathrm{my}$int64int32nag_int scalar
Default: the dimension of the arrays ybar, ystd and the first dimension of the array c and the second dimension of the array ycv. (An error is raised if these dimensions are not equal.)
$r$, the number of response variables.
Constraint: ${\mathbf{my}}\ge 1$.
3: $\mathrm{maxfac}$int64int32nag_int scalar
Default: the second dimension of the arrays p, c, w. (An error is raised if these dimensions are not equal.)
$k$, the number of factors available in the PLS model.
Constraint: $1\le {\mathbf{maxfac}}\le {\mathbf{ip}}$.
### Output Parameters
1: $\mathrm{b}\left(\mathit{ldb},{\mathbf{my}}\right)$ – double array
${\mathbf{b}}\left(\mathit{i},\mathit{j}\right)$ contains the parameter estimate for the $\mathit{i}$th predictor variable in the model for the $\mathit{j}$th response variable, for $\mathit{i}=1,2,\dots ,{\mathbf{ip}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{my}}$.
2: $\mathrm{ob}\left(\mathit{ldob},{\mathbf{my}}\right)$ – double array
If ${\mathbf{orig}}=1$, ${\mathbf{ob}}\left(1,\mathit{j}\right)$ contains the intercept value for the $\mathit{j}$th response variable, and ${\mathbf{ob}}\left(\mathit{i}+1,\mathit{j}\right)$ contains the parameter estimate on the original scale for the $\mathit{i}$th predictor variable in the model, for $\mathit{i}=1,2,\dots ,{\mathbf{ip}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{my}}$. Otherwise ob is not referenced.
3: $\mathrm{vip}\left(\mathit{ldvip},{\mathbf{vipopt}}\right)$ – double array
If ${\mathbf{vipopt}}=1$, ${\mathbf{vip}}\left(\mathit{i},1\right)$ contains the VIP statistic for the $\mathit{i}$th predictor variable in the model for all response variables, for $\mathit{i}=1,2,\dots ,{\mathbf{ip}}$.
If ${\mathbf{vipopt}}={\mathbf{my}}$, ${\mathbf{vip}}\left(\mathit{i},\mathit{j}\right)$ contains the VIP statistic for the $\mathit{i}$th predictor variable in the model for the $\mathit{j}$th response variable, for $\mathit{i}=1,2,\dots ,{\mathbf{ip}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{my}}$.
Otherwise vip is not referenced.
4: $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).
## Error Indicators and Warnings
Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: if ${\mathbf{orig}}=1$, ${\mathbf{iscale}}=-1$ or $1$.
Constraint: ${\mathbf{ip}}>1$.
Constraint: ${\mathbf{my}}\ge 1$.
Constraint: ${\mathbf{orig}}=-1$ or $1$.
Constraint: ${\mathbf{vipopt}}=0$, $1$ or ${\mathbf{my}}$.
${\mathbf{ifail}}=2$
Constraint: $1\le {\mathbf{maxfac}}\le {\mathbf{ip}}$.
Constraint: $1\le {\mathbf{nfact}}\le {\mathbf{maxfac}}$.
Constraint: if ${\mathbf{orig}}=1$, $\mathit{ldob}\ge {\mathbf{ip}}+1$.
Constraint: if ${\mathbf{vipopt}}\ne 0$, $\mathit{ldvip}\ge {\mathbf{ip}}$.
Constraint: if ${\mathbf{vipopt}}\ne 0$, $\mathit{ldycv}\ge {\mathbf{nfact}}$.
Constraint: $\mathit{ldb}\ge {\mathbf{ip}}$.
Constraint: $\mathit{ldc}\ge {\mathbf{my}}$.
Constraint: $\mathit{ldp}\ge {\mathbf{ip}}$.
Constraint: $\mathit{ldw}\ge {\mathbf{ip}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
## Accuracy
The calculations are based on the singular value decomposition of ${P}^{\mathrm{T}}W$.
nag_correg_pls_fit (g02lc) allocates internally $l\left(l+r+4\right)+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2l,r\right)$ elements of double storage.
## Example
This example reads in details of a PLS model, and a set of parameter estimates are calculated along with their VIP statistics.
```function g02lc_example
fprintf('g02lc example results\n\n');
nfact = int64(2);
p = [-0.6708, -1.0047, 0.6505, 0.6169;
0.4943, 0.1355, -0.9010, -0.2388;
-0.4167, -1.9983, -0.5538, 0.8474;
0.3930, 1.2441, -0.6967, -0.4336;
0.3267, 0.5838, -1.4088, -0.6323;
0.0145, 0.9607, 1.6594, 0.5361;
-2.4471, 0.3532, -1.1321, -1.3554;
3.5198, 0.6005, 0.2191, 0.0380;
1.0973, 2.0635, -0.4074, -0.3522;
-2.4466, 2.5640, -0.4806, 0.3819;
2.2732, -1.3110, -0.7686, -1.8959;
-1.7987, 2.4088, -0.9475, -0.4727;
0.3629, 0.2241, -2.6332, 2.3739;
0.3629, 0.2241, -2.6332, 2.3739;
-0.3629, -0.2241, 2.6332, -2.3739];
c = [ 3.5425, 1.0475, 0.2548, 0.1866];
w = [-1.5764e-1 -1.5935e-1 1.7774e-1 5.4029e-2;
8.5680e-2 -1.5240e-4 -1.2179e-1 1.0989e-1;
-1.6931e-1 -3.7431e-1 9.4348e-2 3.1878e-1;
1.2153e-1 2.0589e-1 -1.8144e-1 -4.4610e-2;
7.1133e-2 5.5884e-2 -2.6916e-1 5.4912e-2;
6.5188e-2 2.4170e-1 2.3365e-1 -1.8849e-1;
-4.2481e-1 -1.8798e-3 -3.2413e-1 -1.1600e-1;
6.5370e-1 1.6725e-1 2.1908e-1 2.5461e-1;
2.8504e-1 3.6549e-1 -1.9244e-1 -1.5430e-1;
-2.9341e-1 5.0464e-1 -1.0952e-2 1.3881e-1;
2.9829e-1 -3.6979e-1 -4.9942e-1 -4.9355e-1;
-2.0313e-1 4.1952e-1 -2.5684e-1 -7.5647e-2;
5.6905e-2 -2.3197e-2 -3.0503e-1 3.9673e-1;
5.6905e-2 -2.3197e-2 -3.0503e-1 3.9673e-1;
-5.6905e-2 2.3197e-2 3.0503e-1 -3.9673e-1];
vipopt = int64(1);
ycv = [89.638060; 97.476270; 97.939839; 98.188474];
% Means and scalings
orig = int64(1);
xbar = [-2.6137; -2.3614; -1.0449; 2.8614; 0.3156;
-0.2641; -0.3146; -1.1221; 0.2401; 0.4694;
-1.9619; 0.1691; 2.5664; 1.3741; -2.7821];
ybar = [0.452];
iscale = int64(1);
xstd = [1.4956; 1.3233; 0.5829; 0.7735; 0.6247;
0.7966; 2.4113; 2.0421; 0.4678; 0.8197;
0.9420; 0.1735; 1.0475; 0.1359; 1.3853];
ystd = [0.9062];
% Calculate predictions
rcond = -1;
[b, ob, vip, ifail] = ...
g02lc( ...
nfact, p, c, w, rcond, orig, xbar, ybar, ...
iscale, xstd, ystd, vipopt, ycv);
% Display results
disp('Parameter estimates');
disp(b);
disp('Intercept values');
disp(ob);
disp('VIP statistics');
disp(vip);
```
```g02lc example results
Parameter estimates
-0.1383
0.0572
-0.1906
0.1238
0.0591
0.0936
-0.2842
0.4713
0.2661
-0.0914
0.1226
-0.0488
0.0332
0.0332
-0.0332
Intercept values
-0.4374
-0.0838
0.0392
-0.2964
0.1451
0.0857
0.1065
-0.1068
0.2091
0.5155
-0.1011
0.1180
-0.2548
0.0287
0.2214
-0.0217
VIP statistics
0.6111
0.3182
0.7513
0.5048
0.2712
0.3593
1.5777
2.4348
1.1322
1.2226
1.1799
0.8840
0.2129
0.2129
0.2129
``` | 2023-01-31 04:32:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 143, "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": 0.9969037771224976, "perplexity": 10104.593344248526}, "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-20230131023947-20230131053947-00658.warc.gz"} |
https://math.stackexchange.com/questions/2475295/find-the-remainder-when-528528528-up-to-528-digits-is-divided-by-27 | # Find the remainder when $528528528…$up to $528$ digits is divided by $27$?
Find the remainder when $528528528...$up to $528$ digits is divided by $27$?
Here's what I have done: The number can be written as $528\cdot 10^{525}+528\cdot 10^{522}+...+528$ which has $176$ terms and each term is $\equiv15 \mod 27$ thus the number should be $176*15 \mod 27$ hence $21$ should be the remainder. But book says it is $6$. I don't understand the flaw in my logic. Please correct me.
• you have $21+6=27$ perhaps you are off by a sign? – gt6989b Oct 16 '17 at 15:26
• I think that your answer is correct. – alexp9 Oct 16 '17 at 15:29
• Brute force in python gives 21. – Gribouillis Oct 16 '17 at 15:31
• Are you sure the book says $6$ and not $-6$? After all, $21\equiv -6 \pmod{27},$ so then both answers would agree. – David K Oct 16 '17 at 15:36
• @fleablood Also a good point about negative remainders. It looks like there's probably an error in the book. (Unless the transcription of the problem is very confused, it's only $176$ "copies" of the group of digits $528,$ which makes $528$ digits altogether since each group has three digits. Also, we don't need $10^k$ to be congruent to $1$; we only need $10^{3k}\equiv 1 \pmod{27},$ which is true.) – David K Oct 16 '17 at 17:00
Here is a python3 session
>>> s = '528' * 176
>>> len(s)
528
>>> int(s) % 27
21
• Is it possible to multiply a string by a number? My god...python is awesome. – Integral Oct 16 '17 at 15:35
• @Integral yes strings, lists and tuples can be repeated by multiplying them by a number. – Gribouillis Oct 16 '17 at 15:36
You can see that $6$ cannot be correct by casting out $9$'s: Since $5+2+8=5+5+5$, we have
$$528528\ldots528\equiv5+5+5+\cdots+5+5+5=5\cdot528\equiv5(5+2+8)\equiv5\cdot6\equiv3\mod 9$$
so the remainder mod $27$ must be either $3$, $12$, or $21$. Your approach gave the correct answer, $21$.
• Anyway it is also $-6$ so there is a typo in the book or Anuran has not have seen the minus sign. – Piquito Oct 16 '17 at 15:54
• @Piquito, I agree, a negative sign would fix things (as David K noted in comments). But remainders are usually understood to be nonnegative, so I'm inclined to think it's a typo. – Barry Cipra Oct 16 '17 at 15:57
Since $$3\mid111$$, we know that $$27\mid999$$, Therefore, $$1000\equiv1\pmod{27}$$ Thus, \begin{align} \sum_{k=0}^{175}528\cdot1000^k &\equiv528\cdot176\pmod{27}\\ &\equiv3\cdot176^2\pmod{27}\\ &\equiv3\cdot14^2\pmod{27}\\ &\equiv3\cdot7\pmod{27}\\ &\equiv21\pmod{27} \end{align}
You are incorrect in assuming $10^{k} \equiv 1 \mod 27$. As $10^k \not \equiv 1$ we do not have $528*10^k \equiv 15 \mod 27$.
What you need instead is $528528... = 528(1001001001......)$
And $1001001..... =\sum_{k=0}^{175} 10^{3k}$
$10^3 \equiv 1 \mod 27$... Oh... we do have that and you were not wronng after all.... so $\sum 10^{3k}\equiv 176 \equiv 14 \mod 27$.
So $528528....... \equiv 15*14 \equiv 21 \mod 27$.
And... the book is wrong.
Had it been 527 iterations of 528 the answer would be $6$.
Note that $$27 \times 37 = 999$$.
To find the remainder you get when you divide $$528528\cdots 528$$ by $$999$$, you can "cast out" $$999$$s.
$$\underbrace{528 + 528 + \cdots 528}_{\text{176 times} } \to 176 \times 528 \to 92928 \to 92+928 \to 1020 \to 21$$
So the remainder is $$21$$.
Note. If the remainder was bigger than 26, you would have had to divide it by 27 to get the correct remainder. | 2021-04-17 09:25:07 | {"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": 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": 10, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.802299439907074, "perplexity": 442.68319873287027}, "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/1618038118762.49/warc/CC-MAIN-20210417071833-20210417101833-00116.warc.gz"} |
https://catalyst.sciml.ai/dev/tutorials/basics/ | # The Reaction DSL
This tutorial covers some of the basic syntax for building chemical reaction network models. Examples showing how to both construct and solve ODE, SDE, and jump models are provided in Basic Chemical Reaction Network Examples.
#### Basic syntax
The @reaction_network macro allows the (symbolic) specification of reaction networks with a simple format. Its input is a set of chemical reactions, and from them it generates a ReactionSystem reaction network object. The ReactionSystem can be used as input to ODEProblem, SteadyStateProblem, SDEProblem, JumpProblem, and more. ReactionSystems can also be incrementally extended as needed, allowing for programmatic construction of networks and network composition.
The basic syntax is:
rn = @reaction_network begin
2.0, X + Y --> XY
1.0, XY --> Z1 + Z2
end
where each line corresponds to a chemical reaction. Each reaction consists of a reaction rate (the expression on the left hand side of ,), a set of substrates (the expression in-between , and -->), and a set of products (the expression on the right hand side of -->). The substrates and the products may contain one or more reactants, separated by +. The naming convention for these are the same as for normal variables in Julia.
The chemical reaction model is generated by the @reaction_network macro and stored in the rn variable (a normal Julia variable, which does not need to be called rn). The generated ReactionSystem can be converted to a differential equation model via
osys = convert(ODESystem, rn)
oprob = ODEProblem(osys, Pair.(species(rn),u0), tspan, Pair.(params(rn),p))
or more directly via
oprob = ODEProblem(rn, u0, tspan, p)
For more detailed examples, see the Basic Chemical Reaction Network Examples. The generated differential equations use the law of mass action. For the above example, the ODEs are then
$$$\frac{d[X]}{dt} = -2 [X] [Y]\\ \frac{d[Y]}{dt} = -2 [X] [Y]\\ \frac{d[XY]}{dt} = 2 [X] [Y] - [XY]\\ \frac{d[Z1]}{dt}= [XY]\\ \frac{d[Z2]}{dt} = [XY]$$$
#### Arrow variants
A variety of unicode arrows are accepted by the DSL in addition to -->. All of these work: >, → ↣, ↦, ⇾, ⟶, ⟼, ⥟, ⥟, ⇀, ⇁. Backwards arrows can also be used to write the reaction in the opposite direction. For example, these three reactions are equivalent:
rn = @reaction_network begin
1.0, X + Y --> XY
1.0, X + Y → XY
1.0, XY ← X + Y
end
Note, currently Julia's parser does not support <--, <-> or <-->, so that --> is the only supported plain text arrow.
#### Using bi-directional arrows
Bi-directional unicode arrows can be used to designate a reaction that goes two ways. These two models are equivalent:
rn = @reaction_network begin
2.0, X + Y → XY
2.0, X + Y ← XY
end
rn = @reaction_network begin
2.0, X + Y ↔ XY
end
If the reaction rates in the backward and forward directions are different, they can be designated in the following way:
rn = @reaction_network begin
(2.0,1.0) X + Y ↔ XY
end
which is identical to
rn = @reaction_network begin
2.0, X + Y → XY
1.0, X + Y ← XY
end
#### Combining several reactions in one line
Several similar reactions can be combined in one line by providing a tuple of reaction rates and/or substrates and/or products. If several tuples are provided, they must all be of identical length. These pairs of reaction networks are all identical:
rn1 = @reaction_network begin
1.0, S → (P1,P2)
end
rn2 = @reaction_network begin
1.0, S → P1
1.0, S → P2
end
rn1 = @reaction_network begin
(1.0,2.0), (S1,S2) → P
end
rn2 = @reaction_network begin
1.0, S1 → P
2.0, S2 → P
end
rn1 = @reaction_network begin
(1.0,2.0,3.0), (S1,S2,S3) → (P1,P2,P3)
end
rn2 = @reaction_network begin
1.0, S1 → P1
2.0, S2 → P2
3.0, S3 → P3
end
This can also be combined with bi-directional arrows, in which case separate tuples can be provided for the backward and forward reaction rates. These reaction networks are identical
rn1 = @reaction_network begin
(1.0,(1.0,2.0)), S ↔ (P1,P2)
end
rn2 = @reaction_network begin
1.0, S → P1
1.0, S → P2
1.0, P1 → S
2.0, P2 → S
end
#### Production and Destruction and Stoichiometry
Sometimes reactants are produced/destroyed from/to nothing. This can be designated using either 0 or ∅:
rn = @reaction_network begin
2.0, 0 → X
1.0, X → ∅
end
If several molecules of the same reactant are involved in a reaction, the stoichiometry of a reactant in a reaction can be set using a number. Here, two molecules of species X form the dimer X2:
rn = @reaction_network begin
1.0, 2X → X2
end
this corresponds to the differential equation:
$$$\frac{d[X]}{dt} = -[X]^2\\ \frac{d[X2]}{dt} = \frac{1}{2!} [X]^2$$$
Other numbers than 2 can be used, and parenthesis can be used to reuse the same stoichiometry for several reactants:
rn = @reaction_network begin
1.0, X + 2(Y + Z) → XY2Z2
end
#### Variable reaction rates
Reaction rates do not need to be constant, but can also depend on the current concentration of the various reactants (when, for example, one reactant can activate the production of another). For instance, this is a valid notation:
rn = @reaction_network begin
X, Y → ∅
end
and will have Y degraded at rate
$$$\frac{d[Y]}{dt} = -[X][Y]$$$
Note that this is actually equivalent to the reaction
rn = @reaction_network begin
1.0, X + Y → X
end
except that the latter will be classified as ismassaction and the former will not, which can impact optimizations used in generating JumpSystems. For this reason, it is recommended to use the latter representation when possible.
Most expressions and functions are valid reaction rates, e.g.:
rn = @reaction_network begin
2.0*X^2, 0 → X + Y
gamma(Y)/5, X → ∅
pi*X/Y, Y → ∅
end
but please note that user-defined functions cannot be called directly (see later section User defined functions in reaction rates).
#### Defining parameters
Parameter values do not need to be set when the model is created. Components can be designated as symbolic parameters by declaring them at the end:
rn = @reaction_network begin
p, ∅ → X
d, X → ∅
end p d
Parameters can only exist in the reaction rates (where they can be mixed with reactants). All variables not declared after end will be treated as a chemical species, and may lead to undefined behavior if unchanged by all reactions.
#### Pre-defined functions
Hill functions and a Michaelis-Menten function are pre-defined and can be used as rate laws. Below, the pair of reactions within rn1 are equivalent, as are the pair of reactions within rn2:
rn1 = @reaction_network begin
hill(X,v,K,n), ∅ → X
v*X^n/(X^n+K^n), ∅ → X
end v K n
rn2 = @reaction_network begin
mm(X,v,K), ∅ → X
v*X/(X+K), ∅ → X
end v K
Repressor Hill (hillr) and Michaelis-Menten (mmr) functions are also provided:
rn1 = @reaction_network begin
hillr(X,v,K,n), ∅ → X
v*K^n/(X^n+K^n), ∅ → X
end v K n
rn2 = @reaction_network begin
mmr(X,v,K), ∅ → X
v*K/(X+K), ∅ → X
end v K | 2021-01-24 08:35:38 | {"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": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7247605919837952, "perplexity": 3170.875900746975}, "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/1610703547475.44/warc/CC-MAIN-20210124075754-20210124105754-00362.warc.gz"} |
https://physics.stackexchange.com/questions/196458/is-the-mass-of-the-z-due-to-mixing-with-the-photon-precursor-b-or-to-interacti | # Is the mass of the Z due to mixing with the photon “precursor” B or to interaction with the Higgs?
I want to get something clear that I do not seem to understand. I used to read that the photon A and the Z boson are (different) linear combinations between the W^0 (neutral weak boson before SU(2) breaking) and the B ("photon precursor" before SU(2) symmetry breaking).
Nowadays it is more common to say that the Z gets its mass from the Higgs through Yukawa coupling.
Which is the right way to look at things? What is wrong or incomplete in the previous description? Where exactly does the Z mass come from - and is it due to the Higgs mass or not?
• no, the Z mass is not coming from Yukawa coupling (Yukawa coupling means a coupling between a scalar and 2 fermions, while the Z boson is a spin 1 boson). It comes from the gauge interaction, Higgs boson carrying both weak hypercharge and weak isospin quantum numbers. – Paganini Jul 28 '15 at 19:39
• Paganini, thank you. So what is the exact origin of the Z mass? And does the Higgs play a role in the mass of the Z or not? – Hans973 Jul 29 '15 at 4:06
It seems to me that there is nothing fundamentally wrong about your statements, thus I also don't see any contradictions. The electroweak unification states that, as you said, the $Z$ and the $\gamma$ are different linear combinations of $B^0$ and $W^0$. This all works very nicely, there is just the problem of the masses which is then fixed my the Higgs mechanism. | 2019-08-19 17:17:10 | {"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": 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": 0.6506098508834839, "perplexity": 359.0955431495111}, "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-2019-35/segments/1566027314852.37/warc/CC-MAIN-20190819160107-20190819182107-00161.warc.gz"} |
https://ai.stackexchange.com/questions/25985/factors-that-causing-totally-different-outcomes-from-an-exactly-same-model-and-d | # Factors that causing totally different outcomes from an exactly same model and datasets
Here is a model that trains time series data in (batch, step, features) way.
I have kept the random state for train test split function the same. Every parameter below the same, running the model training yields different outcomes every time and the outcomes are drastically different.
What may be the factors that led to this? Regularization?
X_train, X_val, y_train, y_val = train_test_split(X_train, y_train, test_size=0.2, random_state=666)
def attention_model(X_train, y_train, X_test, y_test,num_classes,dropout=0.2, batch_size=68, learning_rate=0.0001,epochs=20,optimizer='Adam'):
Dense_unit = 12
LSTM_unit = 12
attention_param = LSTM_unit*2
attention_init_value = 1.0/attention_param
u_train = np.full((X_train.shape[0], attention_param),
attention_init_value, dtype=np.float32)
u_test = np.full((X_test.shape[0],attention_param),
attention_init_value, dtype=np.float32)
with keras.backend.name_scope('BLSTMLayer'):
# Bi-directional Long Short-Term Memory for learning the temporal aggregation
input_feature = Input(shape=(X_train.shape[1],X_train.shape[2]))
x = Masking(mask_value=0)(input_feature)
x = Dense(Dense_unit,kernel_regularizer=l2(0.005), activation='relu')(x)
x = Dropout(dropout)(x)
x = Dense(Dense_unit,kernel_regularizer=l2(0.005),activation='relu')(x)
x = Dropout(dropout)(x)
x = Dense(Dense_unit,kernel_regularizer=l2(0.005),activation='relu')(x)
x = Dropout(dropout)(x)
x = Dense(Dense_unit,kernel_regularizer=l2(0.005), activation='relu')(x)
x = Dropout(dropout)(x)
y = Bidirectional(LSTM(LSTM_unit,activity_regularizer=l2(0.000029),kernel_regularizer=l2(0.027),recurrent_regularizer=l2(0.025),return_sequences=True, dropout=dropout))(x)
# y = Bidirectional(LSTM(LSTM_unit, kernel_regularizer=l2(0.01),recurrent_regularizer=l2(0.01), return_sequences=True, dropout=dropout))(y)
with keras.backend.name_scope('AttentionLayer'):
# Logistic regression for learning the attention parameters with a standalone feature as input
input_attention = Input(shape=(LSTM_unit * 2,))
u = Dense(LSTM_unit * 2, activation='softmax')(input_attention)
# To compute the final weights for the frames which sum to unity
alpha = dot([u, y], axes=-1) # inner prod.
alpha = Activation('softmax')(alpha)
with keras.backend.name_scope('WeightedPooling'):
# Weighted pooling to get the utterance-level representation
z = dot([alpha, y], axes=1)
# Get posterior probability for each emotional class
output = Dense(num_classes, activation='softmax')(z)
model = Model(inputs=[input_attention, input_feature], outputs=output)
optimizer = opt_select(optimizer,learning_rate)
model.compile(loss='categorical_crossentropy', metrics=['accuracy'], optimizer=optimizer)
hist = model.fit([u_train, X_train],
y_train,
batch_size=batch_size,
epochs=epochs,
verbose=2,
validation_data=([u_test, X_test], y_test))
#kernel_regularizer=l2(0.002),recurrent_regularizer=l2(0.002),
return hist
batch_size= 150
#217
epochs = 1000
learning_rate = 0.00081
optimizer = 'RMS'
num_classes = y_train.shape[1]
dropout=0.22
tf.keras.backend.clear_session()
history = attention_model(X_train, y_train, X_test, y_test, num_classes,dropout = dropout,batch_size=batch_size, learning_rate=learning_rate,epochs=epochs,optimizer=optimizer
)
• Hello. Welcome to AI SE. I don't know if this is "just" a programming issue or not, but I would like you to note that programming questions (like "Why am I getting this programming error/bug?") are generally off-topic here. These questions are better suited for Stack Overflow. I don't know if your question is a pure programming issue (because I didn't read it, but it contains code...), but just keep this in mind. You should read ai.stackexchange.com/help/on-topic for more info.
– nbro
Jan 26, 2021 at 15:05
• @nbro Hi nbro, I was not sure what was causing the problem. My initial thought was that the problem was caused by some randomness of models not because of codes so I post it here.
– Leo
Jan 27, 2021 at 1:10
## 1 Answer
By default, Keras sets shuffle argument True, so you should set the numpy seed before importing Keras.
CPU
from numpy.random import seed
seed(25)
from keras.models import Sequential
GPU
tf.random.set_seed(seed)
• Hi yahkyo, I set the random seed to a fixed value and rerun the whole model but the outcomes still changed every training. So I then set the shuffle = False, the outcomes were still not exactly the same but this time the trends were generally the same.
– Leo
Jan 26, 2021 at 5:50
• Hi Leo, I think random shuffle in your code is on GPU. However numpy works on CPU. Have you tried tf.random.set_seed(seed) ? Jan 26, 2021 at 5:58
• You are right, I was running GPU. tf.random.set_seed(seed) this code doesn't give the 100% same outcomes, but I think 95% similarity is good for parameter adjustment. Thanks.
– Leo
Jan 26, 2021 at 6:14 | 2022-08-16 10:49:12 | {"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": 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": 0.637718915939331, "perplexity": 7290.3398776277045}, "config": {"markdown_headings": true, "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-2022-33/segments/1659882572286.44/warc/CC-MAIN-20220816090541-20220816120541-00254.warc.gz"} |
https://www.amtoolbox.org/amt-1.1.0/doc/models/osses2021.php | # THE AUDITORY MODELING TOOLBOX
Applies to version: 1.1.0
Go to function
# OSSES2021 - Monaural perceptual similarity
## Usage
[outsig, fc] = osses2021(insig,fs);
[outsig, fc] = osses2021(insig,fs,...);
[outsig, fc, params] = osses2021(insig,fs,...);
## Input parameters
insig input acoustic signal. fs sampling rate.
## Description
osses2021(insig,fs) computes the internal representation of the signal insig sampled with a frequency of fs Hz.
[outsig,fc,mfc]=osses2021(...) additionally returns the center frequencies of the filter bank and the center frequencies of the modulation filterbank.
The model consists of the following stages:
1. an outer- and middle-ear filtering as used by Jepsen et al. 2008
2. a gammatone filter bank with 1-erb spaced filters.
3. an envelope extraction stage done by half-wave rectification followed by low-pass filtering to 770 Hz as used by Breebaart et al. 2001
4. an adaptation stage modelling nerve adaptation by a cascade of 5 loops using a limiter factor of 5 (Osses and Kohlrausch, 2021).
5. a modulation filterbank
Any of the optional parameters for auditoryfilterbank, ihcenvelope and adaptloop may be optionally specified for this function. They will be passed to the corresponding functions.
## References:
T. Dau, B. Kollmeier, and A. Kohlrausch. Modeling auditory processing of amplitude modulation. I. Detection and masking with narrow-band carriers. J. Acoust. Soc. Am., 102:2892--2905, 1997a.
T. Dau, B. Kollmeier, and A. Kohlrausch. Modeling auditory processing of amplitude modulation. II. Spectral and temporal integration. J. Acoust. Soc. Am., 102:2906--2919, 1997b. | 2023-01-31 00:18:58 | {"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": 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": 0.35797256231307983, "perplexity": 11905.323651082408}, "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/1674764499831.97/warc/CC-MAIN-20230130232547-20230131022547-00189.warc.gz"} |
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MeltingCaramel
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A Marvel- obsessed, crazy girl, that pukes color and passes it off as art. Most of her time is spent on acting, writing , gaming and, of course, drawing. She still doesn't have a constant artstyle.
Activity
2 deviations
posted a Journal
If I was a profession I'd be: Arcitecture
If I was a country I'd be: Italy
If I was a ocean or body of water I'd be: Great blue hole, Caribbean
If I was a piece of candy I'd be: Reese's or Grand Candy
If I was a famous building or piece of architecture I'd be: Leaning Tower of Pisa
If I was a bad habit I'd be: Rage Quitting
If I was a swear word I'd be: Shit
If I was a ice cream flavor I'd be: Mango
If I was a disease I'd be: Cancer
If I was a board game I'd be: Chess
If I was a feeling I'd be: Hatred
If I was a war I'd be: 100 years war
If I was a city I'd be: Rimini, Italy
If I was a celebrity I'd be: The les friction dude, or Melanie Martzines
If I was a brand of toothpaste I'd be: Oral-B
If I were a month, I'd be: February
If I were a day of the week, I'd be: Monday
If I were a time of day, I'd be: 00:01
If I were a planet, I'd be: Saturn
If I were a sea animal, I'd be: A seal
If I were a piece of furniture, I'd be: bookshelf
If I were a sin, I'd be: Wrath
If I were a liquid, I'd be: Magma
If I were a tree, I'd be: Oak
If I were a bird, I'd be: Raven
If I were a tool, I'd be: A dagger
If I were a flower/plant, I'd be: Devil's Claw
If I were a kind of weather, I'd be: Lighting
If I were a musical instrument, I'd be: Drums
If I were an animal, I'd be: Poodle Moth
If I were a vegetable, I'd be: Paprika
If I were a sound, I'd be: Claws on Chalk
If I were an element, I'd be: Fire
If I were a car, I'd be: Smart
If I were a song, I'd be: Louder than Words, les friction
If I were a book, I'd be: "How to manipulate and control people"
If I were a food, I'd be: Spring Rolls
If I were a material, I'd be: Copper
If I were a taste, I'd be:Sour
If I were a word, I'd be: F*ck off
If I were a body part, I'd be: Eye
If I were a facial expression, I'd be: Weird Smile
If I were a shape, I'd be a: Star
If I were a number, I'd be: 13
If I were a band, I'd be: Les Friction\ 21 pilots
If I were a mythical creature, I'd be: Chimera
If I were...
posted a Journal
If I was a profession I'd be: Arcitecture
If I was a country I'd be: Italy
If I was a ocean or body of water I'd be: Great blue hole, Caribbean
If I was a piece of candy I'd be: Reese's or Grand Candy
If I was a famous building or piece of architecture I'd be: Leaning Tower of Pisa
If I was a bad habit I'd be: Rage Quitting
If I was a swear word I'd be: Shit
If I was a ice cream flavor I'd be: Mango
If I was a disease I'd be: Cancer
If I was a board game I'd be: Chess
If I was a feeling I'd be: Hatred
If I was a war I'd be: 100 years war
If I was a city I'd be: Rimini, Italy
If I was a celebrity I'd be: The les friction dude, or Melanie Martzines
If I was a brand of toothpaste I'd be: Oral-B
If I were a month, I'd be: February
If I were a day of the week, I'd be: Monday
If I were a time of day, I'd be: 00:01
If I were a planet, I'd be: Saturn
If I were a sea animal, I'd be: A seal
If I were a piece of furniture, I'd be: bookshelf
If I were a sin, I'd be: Wrath
If I were a liquid, I'd be: Magma
If I were a tree, I'd be: Oak
If I were a bird, I'd be: Raven
If I were a tool, I'd be: A dagger
If I were a flower/plant, I'd be: Devil's Claw
If I were a kind of weather, I'd be: Lighting
If I were a musical instrument, I'd be: Drums
If I were an animal, I'd be: Poodle Moth
If I were a vegetable, I'd be: Paprika
If I were a sound, I'd be: Claws on Chalk
If I were an element, I'd be: Fire
If I were a car, I'd be: Smart
If I were a song, I'd be: Louder than Words, les friction
If I were a book, I'd be: "How to manipulate and control people"
If I were a food, I'd be: Spring Rolls
If I were a material, I'd be: Copper
If I were a taste, I'd be:Sour
If I were a word, I'd be: F*ck off
If I were a body part, I'd be: Eye
If I were a facial expression, I'd be: Weird Smile
If I were a shape, I'd be a: Star
If I were a number, I'd be: 13
If I were a band, I'd be: Les Friction\ 21 pilots
If I were a mythical creature, I'd be: Chimera
If I were...
posted a Journal
You want people to like you but won’t change yourself for them. [x]
You feel that there’s no point in gaining their approval if you can't be yourself. []
A lot of the time you feel alone and depressed even if you know you have people that care for you. [x]
Sometimes you just want a hug from someone that truly cares but don’t let people in on the secret. [x]
You don’t like to show your weakness (even if it may come out at times). [x]
You let the pain of others get the better of you and wish so badly to make their hurting end because you have so much pain in your life and don’t want anyone else to feel as broken inside as you do. []
You try to put on a smile and act like everything is fine but sometimes you just can't pretend any longer. [x]
~(030)~
posted a Journal
You want people to like you but won’t change yourself for them. [x]
You feel that there’s no point in gaining their approval if you can't be yourself. []
A lot of the time you feel alone and depressed even if you know you have people that care for you. [x]
Sometimes you just want a hug from someone that truly cares but don’t let people in on the secret. [x]
You don’t like to show your weakness (even if it may come out at times). [x]
You let the pain of others get the better of you and wish so badly to make their hurting end because you have so much pain in your life and don’t want anyone else to feel as broken inside as you do. []
You try to put on a smile and act like everything is fine but sometimes you just can't pretend any longer. [x]
~(030)~
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Thanks for the
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I really hope you'll enjoy my next drawings.
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I really appreciate it (^ω^) | 2017-02-28 03:38:20 | {"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": 0.909457266330719, "perplexity": 11170.08015204665}, "config": {"markdown_headings": false, "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-2017-09/segments/1487501174124.9/warc/CC-MAIN-20170219104614-00128-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://itectec.com/superuser/sha1sum-for-a-directory-of-directories/ | # Sha1sum for a directory of directories
bashhashing
sha1sum ./path/to/directory/* | sha1sum
the above was posted as a way to compute a sha1sum of a directory which contains files. This command fails if the directory includes more directories. Is there a way to recursively compute the sha1sum of a directory of directories universally (without custom fitting an algorithm to the particular directory in question)?
find . -type f $$-exec sha1sum "PWD"/{} \;$$ | sha1sum | 2021-09-20 22:21:47 | {"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": 0.721260130405426, "perplexity": 4949.277641550455}, "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-2021-39/segments/1631780057119.85/warc/CC-MAIN-20210920221430-20210921011430-00471.warc.gz"} |
https://lists.torproject.org/pipermail/tor-commits/2009-March/019122.html | # [or-cvs] r18860: {projects} finish my section 4 work (projects/performance)
arma at seul.org arma at seul.org
Tue Mar 10 12:44:48 UTC 2009
Author: arma
Date: 2009-03-10 08:44:48 -0400 (Tue, 10 Mar 2009)
New Revision: 18860
Modified:
projects/performance/performance.bib
projects/performance/performance.tex
Log:
finish my section 4 work
Modified: projects/performance/performance.bib
===================================================================
--- projects/performance/performance.bib 2009-03-10 12:43:31 UTC (rev 18859)
+++ projects/performance/performance.bib 2009-03-10 12:44:48 UTC (rev 18860)
@@ -154,7 +154,6 @@
www_pdf_url = {http://petworkshop.org/2007/papers/PET2007_preproc_Performance_comparison.pdf},
}
-
@Misc{economics-tor,
author = {Steven J. Murdoch},
title = {Economics of {Tor} performance},
@@ -164,3 +163,12 @@
note = {\url{http://www.lightbluetouchpaper.org/2007/07/18/economics-of-tor-performance/}},
}
+ at inproceedings{hs-attack,
+ title = {Locating Hidden Servers},
+ author = {Lasse {\O}verlier and Paul Syverson},
+ booktitle = {Proceedings of the 2006 IEEE Symposium on Security and Privacy},
+ year = {2006},
+ month = {May},
+ publisher = {IEEE CS},
+}
+
Modified: projects/performance/performance.tex
===================================================================
--- projects/performance/performance.tex 2009-03-10 12:43:31 UTC (rev 18859)
+++ projects/performance/performance.tex 2009-03-10 12:44:48 UTC (rev 18860)
@@ -1118,20 +1118,53 @@
we should solve this by reweighting at the clients, reweighting in the
directory status, or ignoring the issue entirely.
-\subsection{Entry guards might be overloaded}
+\subsection{Older entry guards are overloaded}
-make guard flag easier to get, so there are more of them. also would
-improve anonymity since more entry points into the network.
+While the load on exit relays is skewed based on having an unusual exit
+policy, load on entry guards is skewed based on how long they've been
+in the network.
-also, are old guards more overloaded than new guards, since there are
-more clients that have the old guards in their state file?
+Since Tor clients choose a small number of entry guards and keep them
+for several months, a relay that's been listed with the Guard flag for a
+long time will accumulate an increasing number of clients. A relay that
+just earned its Guard flag for the first time will see very few clients.
-\subsection{Two hops vs three hops.}
+To combat this skew, clients should rotate entry guards every so
+often. We need to look at network performance metrics and discern how
+long it takes for the skew to become noticeable -- it might be that
+rotating to a new guard after a week or two is enough to substantially
+resolve the problem. We also need to consider the added risk that
+higher guard churn poses versus the original attack they were designed
+to thwart~\cite{hs-attack}, but I think a few weeks should still be
+plenty high.
+At the same time, there are fewer relays with the Guard flag than there
+should be. While the Exit flag really is a function of the relay's exit
+policy, the required properties for entry guards are much more vague:
+we want them to be fast enough'', and we want them to be likely to
+be around for a while more''. I think the requirements currently are too
+strict. This scarcity of entry guards in turn influences the anonymity
+the Tor network can provide, since there are fewer potential entry points
+into the network.
+{\bf Impact}: High.
+{\bf Effort}: Low.
+{\bf Risk}: Low.
+{\bf Plan}: We should do it, early in Tor 0.2.2.x. We'll need proposals
+first, both for the dropping old guards'' plan (to assess the tradeoff
+from the anonymity risk) and for the opening up the guard criteria''
+plan.
+
+%\subsection{Two hops vs three hops.}
+
+% People periodically suggest two hops rather than three. The problem
+% is that the design we have right now is more like two hops plus an
+% entry guard'', so removing any of the hops seems like a bad move.
+% But YMMV.
+
\section{Better handling of high/variable latency and failures}
\subsection{Our round-robin and rate limiting is too granular} | 2020-10-27 19:23:20 | {"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": 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": 0.5637384057044983, "perplexity": 12356.515773432404}, "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-2020-45/segments/1603107894426.63/warc/CC-MAIN-20201027170516-20201027200516-00457.warc.gz"} |
https://socratic.org/questions/how-do-you-simplify-4sqrt-192x | # How do you simplify -4sqrt(192x)?
Apr 15, 2018
The answer is $- 32 \sqrt{3 x}$.
#### Explanation:
Note: when the variables a, b, and c are used, I am referring to a general rule that will work for every real value of a, b, or c.
Since $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$, you can rewrite this:
$- 4 \sqrt{192 x} \to - 4 \sqrt{64} \cdot \sqrt{3 x}$.
$\sqrt{64}$ is equal to $8$, so:
$- 4 \sqrt{64} \cdot \sqrt{3 x} \to - 4 \cdot 8 \cdot \sqrt{3 x} \to - 32 \sqrt{3 x}$, your final answer.
Hope this helps! | 2019-09-18 23:52:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.8398551344871521, "perplexity": 503.5038380862805}, "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-2019-39/segments/1568514573385.29/warc/CC-MAIN-20190918234431-20190919020431-00379.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/earth-possesses-electric-field-magnitude-150n-c-near-surface-q1531739 | ## the earth possesses an electric field of magnitude 150n/c near its surface
the earth possesses an electric field of magnitude 150n/c near its surface | 2013-06-20 07:54:29 | {"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": 0.8706408143043518, "perplexity": 2123.110721791032}, "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-2013-20/segments/1368711005723/warc/CC-MAIN-20130516133005-00077-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://brilliant.org/problems/an-algebra-problem-by-harsh-shrivastava/ | # Play with functions
Algebra Level 5
Let $f: \mathbb R \to \mathbb R$ be a function satisfying $f(x-f(y)) = f(f(y))+xf(y)+f(x)-1$ where $x$ and $y$ are real numbers.
Find the value of $f(1) - f(2)$.
× | 2020-08-11 06:56:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "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": 0.344594806432724, "perplexity": 441.61644665430754}, "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-2020-34/segments/1596439738735.44/warc/CC-MAIN-20200811055449-20200811085449-00214.warc.gz"} |
http://www.vinakadem.ru/evdzxpt/fl7ujei.php?spavraeep=epsilon-zero-dimensional-definition | • epsilon zero dimensional definition When one discusses supersymmetry, the non-zero modes typically come in pairs and it can be proved, but the zero modes - massless particles in 3+1 dimensions, as I mentioned Pseudospectrum: Let A be a square n-by-n matrix of complex numbers and . We first review the definition and properties of Gaussian distribution: Zero-mean is always possible by subtracting sample mean. a rigorous non-perturbative definition/generalisation of dimensional Epsilon-zero to the epsilon-zero / omega to the omega to the epsilon-zero-times-two ε 0 ε0 / ω ω ^( ε0*2) This is a particularly interesting ordinal, the result of raising epsilon-zero to its own power. Pick an arbitrary x {\displaystyle x} . where: $$X$$ is some function of $$x$$; $$T$$ is some function of $$t$$. $October 8, 2018 general-topology, proof-verification Math geek On a Yamabe Type Problem on Three Dimensional Thin Annulus By Mohamed Ben Ayed, Khalil El Mehdi, Mokhless Hammami and Mohameden Ould Ahmedou Download PDF (235 KB) The analysis is extended to show that when only bounded $$2+\epsilon$$ moments exist for $$\epsilon\in(0,2)$$, matrix estimators with satisfactory convergence rates are still attainable. The ε-pseudospectrum of A is by definition The ε-pseudospectrum of A is by definition where is a matrix norm. In general, the Electric Field due to a point charge will be reduced due to the molecules within a material. This system is identical to Eq. 1 I having trouble setting this up. 1. In general, permittivity is symbolized and is a constant of proportionality that exists between electric displacement and electric field intensity in a given medium. Re: Definition of the ordinal epsilon-nought Post by lightvector » Mon Apr 26, 2010 1:14 am UTC Deadcode, I think the place you had the fundamental misunderstanding was in thinking that the exponentiation operator was some how different than addition and multiplication and had to be reversed to be reasoned about. A multi-layer, feedforward, backpropagation neural network is composed of 1) an input layer of nodes, 2) one or more intermediate (hidden) layers of nodes, and 3) an output layer of nodes (Figure 1). Fields in fractional parallel plate D'B', DB' and D'B waveguides and Laplacian operator applying to the magnetic component Bx would result in zero: We consider a two-player, zero-sum differential game governed by an abstract nonlinear differential equation of accretive type in an infinite dimensional space. A delta-epsilon proof requires an arbitrary epsilon. In this section, we will explore what a limit is. Double click Results in the main Workbench window to open CFD Post, where we will be viewing the results. What is Volume Rendering? The term volume rendering is used to describe techniques which allow the visualization of three-dimensional data. ) Eddy viscosity is zero if the velocity gradients are zero Prove limit using epsilon /delta definition of limit Show transcribed image text Prove limit using epsilon /delta definition of limit (c) cn = n /2^ n Show transcribed image text Let V be an n-dimensional vector space, and U V a subspace of diamension If you have "just learned about" the$\delta-\epsilon$definition of a limit, then plunging into the limit of a two-variable function is a really big step, since there are subtleties that do not crop up in one variable. Although orthogonality is a concept from Linear Algebra, and it means that the dot-product is zero, the term is loosely used in statistics. Pi Mu Epsilon Undergraduate Math Society at USC Spring 2003 Semester January 17. Vectors *Definition & types of vectors… A to Z of Physics This blog will be useful for the students of Intermediate M. To start a Maplet, click on its name. e. Finding the limit using the de–nition is a long process which we will try to avoid whenever The inverse scattering method is used to determine the weak limit of solutions of the Korteweg-deVries equation as dispersion tends to zero. Volume rendering is a technique for visualizing sampled functions of three spatial dimensions by computing 2-D projections of a colored semitransparent volume. I think I got the answer for this question. 3. ) 3 Turbulence Intensity: urms/u (5) The subscript ‘rms’ stands for root-mean-square. The molecules of the gas can adsorb to specific sites on the surface. Keywords: Epsilon-near-zero materials, ENZ, flexible cavity, open cavity, quantum emitter, radiating mode, nonradiating mode INTRODUCTION Cavity quantum electrodynamics (QED) is the field of research that investigates the interaction between quantum emitters (QEs), such as atoms and quantum dots, and a resonant cavity ( 1 ). the number of vectors) of a basis of V over its base field. Results Depending on the selected lead, timing of the ECG-derived time markers changed considerably compared with mitral valve closure. Abstract. Question. Three distinct flavors: Many thanks to Bart Andrews for this contribution!. Welcome to Maplets for Calculus. Section 2-1 : Limits. epsilon],[delta]](*) by constructing asymptotic expansions of its probability densities, which are associated with the adjoint operator [L. You should recognize the definition of urms given in (4) as the standard deviation of the set of “random” velocity fluctuations, u′i. If the potential \alpha\Phi possesses a zero-energy resonance, then S_0 describes a non trivial point interaction at the origin. Jeff, y+ is a non-dimensional distance. Here u, v and w denoteappropriate precisely, shear strains are assumed to be zero everywhere so angles are preserved) but side lengths changeto x+ u, y+ v and z+ w , respectively. Suppose I have a function f(x) defined between a and b. Index notation provides a very powerful tool for proving many identities in vector calculus, or for manipulating formulae for multi-dimensional calculus. Plus, get practice tests, quizzes, and personalized coaching to help you succeed. The discovery of the laws of dynamics, or the laws of motion, was a dramatic moment in the history of science. So the initial volume before any deformation will be simply L1 times L2 times L3. Epsilon Theory is Dr. Read in 20 numbers, each of which is between 10 and 100, inclusive. [/math] These are the infi where L is the number of loops and d = 4 − 2 ϵ the number of space–time dimensions in the context of dimensional regularization. Section 7. A few are somewhat challenging. In mathematics, the dimension of a vector space V is the cardinality (i. where the potentials V and K satisfy rather generic conditions, allowing V even to vanish on the infinity. Furthermore, it is assumed that the limit in (1) exists and does not depend on how the grid is chosen. One-Dimensional Chirality: Strong Optical Activity in Epsilon-Near-Zero Metamaterials Carlo Rizza,1,2 Andrea Di Falco,3 Michael Scalora,4 and Alessandro Ciattoni2 1Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. two-dimensional phase space portraits of both records, we identify six regions of Lyapunov stability, which, in the case of the ice volume proxy, represent stable ice configurations that recur episodically throughout the Quaternary (figure 1). Material behavior may depart significantly from that shown in Figure 5. The denominators D i in are usually of the form p 2 − m 2, where p is a linear combination of loop momenta and external momenta and m some mass. two-dimensional surfaces. Indeed, consider a 1-dimensional wave ψ ( x ) = Ap 0 ( x; k ) eikx propagating in a 1-dimensional reciprocal medium with k = k (ω ) the Floquet constant, where Ap 0 is the periodic amplitude of the wave. When α > 1, counterclockwise flow varies with the value of θ, having a maximum at θ = -pi / 2, and a minimum at θ = pi / 2. For two hydrogen Fortran 90 ArraysFortran 90 Arrays Program testing can be used to show the presence of bugsProgram testing can be used to show the presence of bugs, A vector is a quantity or phenomenon that has two independent properties: magnitude and direction. In calculus, the (ε, δ)-definition of limit ("epsilon–delta definition of limit") is a formalization of the notion of limit. The obtained results are in good agreement with recent tunneling experiments on two-dimensional GaAs/AlGaAs heterostructures and quasi-one-dimensional doped multiwall Such an approximation is known by various names: Taylor expansion, Taylor polynomial, finite Taylor series, truncated Taylor series, asymptotic expansion, Nth-order approximation, or (when f is defined by an algebraic or differential equation instead of an explicit formula) a solution by perturbation theory. The expression iff " will Epsilon Theory is Dr. With the exception of the empty set, no open set of the real numbers has a length of zero. As a consequence, our asymptotic formula also describes the minimum three-dimensional anisotropic energy as$\epsilon$tends to zero Topics: Mathematics - Analysis of PDEs A convex body (i. Hopefully any self-learners out there can benefit from this. Epsilon itself could be considered in five dimensional space. The dimension of a physical quantity is the combination of the basic physical dimensions (usually mass, length, time, electric charge, and temperature) which The LibreText Project is fortunate to accept a$5 million Open Textbooks Pilot Program award from the Department of Education funded by Congress in the 2018 Fiscal Year omnibus spending bill. This function can have many zeros, but also many asymptotes. We have had the rotation operator $\Rop_y(\theta)$ which takes a state $\ket{\psi}$ and produces a new state, which is the old state as seen in a rotated coordinate system. Finding any solution to a problem is not nearly as good as finding the one "optimal solution" to the problem. P. The compatibility equations reduce to, The compatibility equations reduce to, Note that some references use engineering shear strain ( ) when referencing compatibility equations. (x, a, and L are two dimensional vectors, x->a . epsilon new times d nu T mu nu. If you derive an expression for velocity which has dimensions other than L/T you’ve made a mistake. Specifically, the low wave number provided by ENZ metamaterials forces the phase advance of light passing through an ENZ region to go to zero. We show theoretically that a properly chosen one-dimensional array of coupled photonic resonators (cavities) may possess localized zero-dimensional topological modes bearing resemblance with the corresponding edge modes/Majorana states in semiconductor nanowires atop a superconducting substrate. if this is not good, check the return value of this method. . One of the best uses for dimensional analysis is a quick check on your math. Preform a Dimensional analysis of Electric permittivity to show that the units of epsilon zero are C^2/(Nm^2). Issuu is a digital publishing platform that makes it simple to publish magazines, catalogs, newspapers, books, and more online. As each number is read, validate it and store it in the array only if it is not a duplicate of a number already read. Ben Hunt’s ongoing examination of the narrative machine driving human behavior, political policy and, ultimately, capital markets—an unconventional worldview best understood through the lenses of history, game theory and philosophy. A basis B of a vector space V over a field F is a linearly independent subset of V that spans V. In the one-dimensional case, the Dirac 'function' represents a distribution, that is, an object defined not by establishing the relation between the values of the function and those of its argument, but by establishing the rule for integrating its product with sufficiently regular functions. This post will be theorem, proof, algorithm, data. The physical constant ε 0 (pronounced as “epsilon nought” or “epsilon zero”), commonly called the vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. This problem will help you determine the chance that two random vectors are pointing in the same direction. To develop calculus for functions of one variable, we needed to make sense of the concept of a limit, which we needed to understand continuous functions and to define the derivative. n epsilon The fifth letter of the Greek alphabet, equivalent to short e. It exploits the numerical precision of the data to extrapolate the sequence to its limit. If you know this material you can always skip to the discussion of mean dimension by scrolling down or by clicking here. Vectors a 1, a 2,…, a n are called linearly dependent if there exist such numbers α 1, α 2,…, α n, of which at least one of them differs from zero, that the linear combination α 1 a 1 + · · · +α n a n of these vectors is equal to zero. In this note we utilize the approach of Sklar [] to have a specific form relative to an m-dimensional multivariate normal form combined with a flexible family of epsilon–skew–normal distributions [4,9]. and f is a function of two variables) slightly as, for any disk centred around L, there is a disk centred around a such that f maps every point of the disk into the disk centred around L. This function can be expressed as a Fourier series, This function can be expressed as a Fourier series, Permittivity and dielectric constant are two terms that are at the very heart of capacitor technology. C & Bi. More than 28 million people use GitHub to discover, fork, and contribute to over 85 million projects. g. With a small parameter $\epsilon$, Poisson-Nernst-Planck (PNP) systems over a finite one-dimensional (1D) spatial domain have steady state solutions, called 1D boundary layer solutions, which profiles form boundary layers near boundary points and become at in the interior domain as $\epsilon$ approaches zero. The Euler characteristic (the alternative sum of the dimensions of the homology groups) is then called the index of the Fredholm complex. Underlying all of calculus is the idea of a limit. It is important in turbulence modeling to determine the proper size of the cells near domain walls. An n-dimensional vector eld is described by a one-to-one correspondence between n-numbers and a point. For values at a finite set of points to be a fair reflection of the behaviour of the function elsewhere, the function needs to be well-behaved, for example differentiable except perhaps k] is of measure zero if and only if for every [epsilon] > 0 there exist countably many closed k-dimensional rectangles [R. Clearly, a squared term like . S. If you do not own Maple, click Use MapleNET 12 at the top-right corner of this page. For a student who hasn't had the rigorous $\epsilon - \delta$ definition (or who has merely been exposed to it, but not enough to really work with it), option 2 is probably the best (and is what Stewart does, if I recall correctly). Posts about Dimensional formula written by gyaunnrraje. A more elegant procedure would be to use a try-catch construction. A tensor may consist of a single number, in which case it is referred to as a tensor of order zero, or simply a scalar. Geophysics plays a critical role in the oil and gas industry. Mobile Task Forces (MTFs) are elite units comprised of personnel drawn from across the Foundation and are mobilized to deal with specific threats or situations that sometimes exceed the operational capacity or expertise of regular field personnel and — as their name suggests — may be relocated between facilities or locations as they are needed. exponentiation . The definition of a distribution is conceptually very simple, even if the technicalities are a bit scary at first. In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. In this case, the relationship is $\sigma = E \, \epsilon$ MOLECULAR PARTITION FUNCTIONS Introduction In the last chapter, we have been introduced to the three main ensembles used in statistical mechanics and some examples of calculations of partition functions were also Another important high dimensional system of coupled ordinary differential equations is an ensemble of N all-to-all coupled phase oscillators [9] . algebra, and differential equations to a rigorous real analysis course is a bigger step to- discontinuitiesform a set of Jordan content zero. To find expressions for [[Delta]][[epsilon]] / [[epsilon]] and for the sheath thickness d, we will perform derivations making use of the dimensionless collision rate, defined to be the number of collisions per Debye length, Note that the time-average of the product of the two fluctuations a' and b' cannot be set to zero; for instance, if b = a , the product of the two fluctuations would be a ' a ' = a '2 . Customary System of Units (USCS) stress is expressed in pounds per square inch (psi) or kilopounds (kips) per square inch (ksi) 1 kilopound 1000 pounds Stress 5 . We first review the definition and properties of Gaussian distribution: A Gaussian random variable $X\sim \mathcal{N}(\mu,\Sigma)$, where $\mu$ is the mean and Specifically, $$N(\epsilon)$$ is the number of d-dimensional cubes of edge length $$\epsilon$$ from the grid that are needed to cover the attractor. The fact that f(x) is a functional argument results in the third expression (upon calling zero ) never to be evaluated. Doesn't the definition itself say that you have a given epsilon and then from that fact generate a delta. Nanoparticles and quantum dots are the zero dimensional structures. The Epsilon itself is constantly moving in the absolute "n" dimensional space with speed equal or larger than light speed. So let's suppose that the solid originally has edges L1, L2, and L3 along x1, x2, and x3, respectively. By definition, at the plasma frequency of a Drude medium the real part of the permittivity effectively goes to zero and at the resonance of a Lorentzian medium, the permittivity may become very large. The main purpose of feature subset selection is to remove irrelevant and redundant features from data, so that learning algorithms can be trained by a subset of relevant features. There is a way to use this idea to take any column of a matrix and make those entries below the diagonal entry be zero. The ground state of a quantum-mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. LIMIT OF A SEQUENCE: THEOREMS 117 4. The epsilon algorithm is recommended as the best all-purpose acceleration method for slowly converging sequences. In this section we will take a look at limits involving functions of more than one variable. ) Here, we assume that x is a data structure that contains one training example per column (so, x is a \textstyle n-by-\textstyle m matrix). Geophysical data are used by exploration and production personnel to ascertain the presence, nature and size of subsurface rock layers and reservoirs contained therein. DIFFERENTIAL FORMS AND INTEGRATION 3 Thus if we reverse a path from a to b to form a path from b to a, the sign of the integral changes. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, . The modulus of toughness is the work done per unit volume of material when the simple tensile load is increased from zero until the material ruptures. The limit of f as x approaches (x_0,y_0) equals L if and only if for every epsilon>0 there exists a delta>0 such that f satisfies In the study of fractals, Minkowski dimension (a. requests that a zero be used as starting values during empirical Bayes estimation. 5. The data set we test on is a thousand-story CNN news data set. Here the perturbation $\xi_t$ is sampled uniformly from a ball centered at zero with a suitably small radius, and is added to the iterate when the gradient is The first step in getting to the full model is to start with simple uniaxial tension/compression. Consider a gas in contact with a solid surface. The term also denotes the mathematical or geometrical representation of such a quantity. We encourage you to check it out and subscribe to receive all of Ben’s Notes, plus a host of new content from an expanding cast of writers from the Salient family and beyond. Intuitive Topology - American Mathematical Society Home This book is an introduction to elementary topology presented in an intuitive way, emphasizing the visual aspect. Since Ω is evaluated at z ′ in (2), this map is a volume-preserving diffeomorphism for any smooth functions Ω and g. Plane stress is defined to be a state of stress in which the normal stress and the shear stresses directed perpendicular to the plane are assumed to be zero . 2 deals In the previous chapter, we saw two approaches to dealing with recovering marginal effects in panel data. We discussed problems A1-A5 from the 2002 Putnam exam. If this is true, we say has zero volume, or is negligible, or is a null-set, and we write it as . immediately from the definition of a smooth ring map but it also follows from the stronger Algebra, Lemma \ref {algebra-lemma-smooth-at-point}). We must show that there exists a delta for which the limit statement follows, and we claim this delta will suffice. The device generator and the simulator are controlled via input decks which contain the default settings and instructions. 85 x 10-12 farad per meter (F/m). The 1-dimensional case can easily be proved using the intermediate value theorem (IVT). 1. You can find further discussion in several places on the web. The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. The goal of bi-objective optimization is to find a small set of good compromise solutions. The permittivity of free space (a vacuum) is a physical constant equal to approximately 8. It is defined as It is defined as dφ k / dt = ω k + ε / N Σ j sin( φ j - φ k ) What is Mean Dimension? Yonatan Gutman Before we discuss mean dimension, we need to discuss other concepts. hpp. 5 and epsilon= 0. The wave equation is a linear second-order partial differential equation which describes the propagation of oscillations at a fixed speed in some quantity $$y$$: A solution to the wave equation in two dimensions propagating over a fixed region [1]. To have a uniform operator to put an element to zero, for scalar values and for objects. Note that this is the one dimensional form of Schrödinger's Equation, it does become more complex for higher It is used ƒ for measuring displacement, distance, position and elongation ƒ for in-process quality control and dimensional testing - The sensor may only be operated within the limits specified in the technical data, see Chap. berkeley. First, we showed how simple differencing in a two period example can effectively rid the model of the unobserved individual effects, albeit with some strong assumptions. You will find the definition of Newton's method in Quarteroni, Sacco, and Saleri on pp. In order to simplify the presentation, we will develop integration using the Riemann (or Jordan) approach rather than the more general theory of Lebesgue integration. 65 (=0. I’m just going to jump right into the definitions and rigor, so if you haven’t read the previous post motivating the singular value decomposition, go back and do that first. 1 in decimal doesn't have a simple binary representation, so when you declare a double as 0. 2898 that's the answer i got. The general idea is that it is possible to find an infinite number of these solutions to the PDE. We use MathJax. ; n epsilon In mathematics, a quantity which approaches zero when the independent variable approaches a certain limit fixed for it by the conditions of the particular problem or discussion. In this section we’re going to be taking a look at the precise, mathematical definition of the three kinds of limits we looked at in this chapter. 286ff, and in the lectures, along with convergence theorems and the like. An infinitesimal quantity is supposed to be a quantity that is infinitely small in size, yet not necessarily perfectly small (zero). Big news: Epsilon Theory has launched as an expanded media site at www. When you look through a hologram printed on a two-dimensional surface, a three-dimensional projection appears. An infinitesimal space is supposed to be a space whose extension is infinitely small, yet not necessarily perfectly small (pointlike). I need to retrieve all the zeros of this function. Two different structures are considered: the classical PEC antenna and the dielectric dipole covered by ENZ material antenna. The limit operator S_0 depends on the shape of \Phi and \Psi as well as on the limit of ratio \nu/\epsilon. So eccentricity is simply a measure of how elliptical an orbit is. Optimization problems are typically reformatted so they become minimization problems, which are well-studied problems in the field of The concept of loss-compensated broadband epsilon-near-zero metamaterials consisting of step-like metal-dielectric multilayer structures doped with gain media is proposed based on the combination of the Milton representation of the effective permittivity and the optical nonlocality due to the metal-dielectric multilayer structures. Numerical Results. These local epsilon constants are defined for one-dimensional it is a very difficult task to prove well-defined-ness of this definition of epsilon factors (see The EM crystals under study here are amenable to a photonic tight-binding description within the framework of the coupled-dipole method []. Singularly perturbed multi-scale switching diffusions It is worth noting, on the one hand, that Jaeger omits the derivation of the asymptotic formula (1), being this given in dimensionless form. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. 4. Micro-Epsilon offers one of the broadest product ranges of high-precision displacement sensors, 2D/3D laser scanners, IR temperature sensors, colour sensors and inspection systems in Europe. In two dimensional problems (e. The definition of differentiability in multivariable calculus formalizes what we meant in the introductory page when we referred to differentiability as the existence of a linear approximation. Normalizes this vector and returns it norm makes v a unitvector and returns the norm of v. The solution to the inverse problem via fitting of the parameters within the WKB approach is unique for arbitrary tideless wormholes and some wormholes with non-zero tidal effects, but this is not so for arbitrary wormholes. org) encyclopedia article. (The first Maplet may take a little longer to open because it needs to start Java. : Create an epsilon of room with the functions , where and is a constant greater than 2 (the diameter of ). The movement of the Epsilon itself could be considered as fifth dimension. Design of Matched Zero-Index Metamaterials Using Non-Magnetic Inclusions in Epsilon-Near-Zero (ENZ) Media Mário Silveirinha(1,2)and Nader Engheta(1)* (1) University of Pennsylvania, Department of Electrical and Systems Engineering, This is 1. Where epsilon is the eccentricity of the ellipse. Pi. (4. There is essentially zero probability that one attenuation model will give you an epsilon value of -1, while another model will give you +1. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. Ground state. 4/0. By default, the starting values are set equal to the estimates from the previous iteration (or zero for the first iteration). Definition – Zero volume set: We say a set has zero volume if for any , we can cover by open boxes such that and . In three dimensional Euclidean space the wedge product and the cross product of two vectors are each other's Hodge dual . An excited state is any state with energy greater than the ground state. State-of-the-art optical cavities are based 1 PART 1: INTRODUCTION TO TENSOR CALCULUS A scalar eld describes a one-to-one correspondence between a single scalar number and a point. plane strain), all z terms are set to zero. So it is a separate interesting storing to consider such coordinate transformation for which this guy doesn't vanish at the boundary. We have already shown you many examples of quantum-mechanical operators. zero stress components and non zero strain components and then we need to also know, what is the relationship between strains and displacements and also how the various stress components are related to various strain components. 14159265358979323846 Definition of . Definition at line 1062 of file frames. If it is too large, we have a high penalty for nonseparable points and we may store many support vectors and overfit. The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. The transformation may be non-linear and the transformed space high dimensional; thus though the classifier is a hyperplane in the high-dimensional feature space, it may be non-linear in the original input space. precisely, shear strains are assumed to be zero everywhere so angles are preserved) but side lengths changeto x+ u, y+ v and z+ w , respectively. The following definition and results can be easily generalized to functions of more than two variables. It can be calculated finding the area under the stress-strain curve from the origin to the failure strain, that is, A three-dimensional periodic function f is defined such that it has a constant value C inside the cubes and is zero outside the cubes. Here u, v and w denoteappropriate calc 1 lim x---> 2 4x+1/3x-4 illustrate definition 2 by finding values of delta that correspond to epsilon=0. with the same definition for the metric on is the value ofthe zero sumtwoperson game, which, in the radius necessary to insure finite epsilon entropy in the 7- The seventh consequence that follows from 1/c=0, and the definition of interval 'd(s)' in four dimensional space-time continuum, is that the time elapsed 'd(t)' between any two events occurring Suppose $\epsilon>0$ has been provided. Planck's constant: Planck’s constant, fundamental physical constant characteristic of the mathematical formulations of quantum mechanics, which describes the behavior of particles and waves on the atomic scale, including the particle aspect of light. Integrals. Stochastic Gradient Descent¶. One unsolved question is about problem A4 (determinant tic-tac-toe): can you deduce that zero always wins (with optimal play) from the fact that the regular tic-tac-toe is a draw (again, with optimal play)? What it shows is that rotations about an arbitrary axis can be written as an exponential that can be thought of as the infinite product of a series of infinitesimal transformations where each transformation has various nice properties. epsilontheory. The following problems require the use of the precise definition of limits of functions as x approaches a constant. then your data set consists In their definition, there appears a small positive parameter, usually called $\epsilon$, that was originally introduced in order to avoid a division by zero on constant states, but whose value was later shown to affect the convergence properties of the schemes. As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. The output layer can consist of one or more nodes, depending on the problem at hand. In this paper the design of Epsilon Near Zero (ENZ) antennas, working in the infrared and optical regime, is presented. For instance, the rotational modes correspond with Spinors while the vibrational modes correspond with Vectors. Abstract An improved zero-dimensional model which includes a coronal non-equilibrium treatment of impurities and which takes into account the ionization energies of deuterium and of the impurities in the energy balance for a Reversed Field Pinch (RFP) device, is applied to the study of the ZT-H device. This is the permittivity of a vacuum (no atoms present). So we have a term epsilon 11 zero zero zero zero; epsilon 22 zero zero zero; epsilon 33. 243) more density at the value zero meters than at the value one meter of that random variable, but the Probability at zero meter is equal to the Probability at one meter, equal to Zero (dimensionless). Units of Stress Stress has units of pressure which is a unit of force divided by unit of area. This blog is written keeping in mind the syllabus of Board of Intermediate,Andhrapradesh. It is symbolized o. Controlling the emission and interaction properties of quantum emitters (QEs) embedded within an optical cavity is a key technique in engineering light-matter interactions at the nanoscale, as well as in the development of quantum information processing. All the spinors $\psi$ on the 6-dimensional manifold may be written as a combination of the "modes", some of which are the zero modes but most of them are non-zero modes. Contrary to that definition, epsilon is actually a concept used to eliminate the ambiguity of conversion between binary and decimal representations of values. If epsilon is equal to zero, the two foci have no distances between them and the ellipse is now a circle. Easily share your publications and get them in front of Issuu’s The defining equation for the von Mises stress was first proposed by Huber in 1904, but apparently received little attention until von Mises proposed it again in 1913. These simple solutions called fundamental solutions of the form: $$u_n=X_n(x) \bullet T_n(t)$$ are the building blocks of the pr Optimization is an important concept in engineering. To do so, they use a modification of the penalization technique, originally presented in [], in such a way that compactness is recovered to the modified energy functional. The dielectric is the material that provides the insulation between the capacitor plates, and many of the characteristics of the capacitor will be dependent upon the properties of the dielectric In Equation [1], is the permittivity of Free Space, which is measured in Farads/meter. We will begin with the precise definition of the limit of a function as x approaches a constant. Re: Definition of the ordinal epsilon-nought Post by skeptical scientist » Mon May 03, 2010 2:11 pm UTC I'd look for definitions that don't involve $$\uparrow \uparrow$$ which the wikipedia article seems to apply to ordinals without ever carefully defining it on ordinals. For example, a Planet Math (planetmath. complete information about the dimensions, definition of an dimensions, examples of an dimensions, step by step solution of problems involving dimensions. Let f be a function of two variables that is defined in some circular region around (x_0,y_0). : 1 Several types of fractal dimension can be measured theoretically and empirically (). GitHub is where people build software. of Banach spaces and bounded operators is said to be a Fredholm complex if the images d i d_i are closed and the chain homology of the complex is finite dimensional. This increases the chance of a correct answer – any function whose integral over an infinite interval is finite must be near zero for most of that interval. Could you guys also clear up these questions, do you have a give E and choose a delta, or do you have a give delta and choose a E. Before Newton’s time, the motions of things like the planets were a mystery, but after Newton there was complete understanding. Dimensional Regularization and epsilon Page 2 of 2 but the success rate has been zero. calculus Need help with the following proof: prove that if lim x->c 1/f(x)= 0 then lim x->c f(x) does not exist. 1-D interpolation (interp1d) ¶The interp1d class in scipy. (Check the math yourself for correctness. 1, it is actually setting that value to an approximate representation in binary. 263f and systems of equations on pp. â ¦ (raised) to the power of â ¦. Idea. The concept is due to Augustin-Louis Cauchy, who never gave an (,) definition of limit in his Cours d'Analyse, but occasionally used , arguments in proofs. interpolate is a convenient method to create a function based on fixed data points which can be evaluated anywhere within the domain defined by the given data using linear interpolation. To see this, first note that every open set is measurable, and therefore the Lebesgue outer measure coincides with the regular Lebesgue measure, so it suff Section 2-10 : The Definition of the Limit. For reasons which will become apparent, a scalar may be thought of as an array of dimension zero (same as the order of the tensor). In that region the one-dimensional (1D) Hooke’s law is assumed to hold. U. Changing the definition of ED and ES resulted in significantly different ES-GLS and ES-SLS values in all subjects. A Spinor is a mathematical object which describe's a particle's Spin in a similar way that a Vector describes it's translation. By default, the epsilon factor and number of weight vectors for sigma are determined dynamically to be "optimal", while the epsilon factor for tau is taken to be 1/n and the number of weight vectors for tau is taken to be Ceiling(n/2), where n is the rank of I. The epsilon-near-zero response of matter can give rise to exciting properties in its interaction with the electromagnetic waves. 2) The nullclines of the FitzHugh-Nagumo model for zero input. Using the Delta Function in PDFs of Discrete and Mixed Random Variables In this section, we will use the delta function to extend the definition of the PDF to discrete and mixed random variables. Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The energy of the particle is given by \frac{h^2 n^2}{8 m L^2}, where h is the Planck constant, m is the mass of the particle, n is the energy state ( n = 1 corresponds to the ground-state energy), and L is the width of the well. do i set up two different functions then divide? 0. Epsilon Theory. 255f, with convergence discussed on pp. \medskip\noindent Use a one dimensional array to solve the following problem. A common problem for bi-objective evolutionary algorithms is the following subset selection problem (SSP): Given n solutions P ⊂ R 2 in the objective space, select k solutions P* from P that optimize an indicator function. Dimensions are used to describe the size and shape of an object. Let g {\displaystyle g} be an odd real-valued continuous function on a circle. I guarantee no correctness. Since, by definition, lower energy is more favorable, the - A/r6 part is the attractive part and the + B/r 12 part is the repulsive part of the interaction. Quantum Mechanics/Operators and Commutators. Formal Definition of a Limit [10/17/2001] Could you please explain the formal definition of a limit? I need help specifically with finding a delta for a given epsilon and using the epsilon-delta definiton of a limit. if v is smaller than eps, Vector(1,0,0) is returned with norm 0. C groups. The wave function of the ground state of a particle in a one-dimensional well is a half-period sine wave which goes to zero at the two edges of the well. A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale. This blog will be useful for the students of Intermediate M. Exercise 2 Use the maximum modulus principle to prove the Phragmén-Lindelöf principle: if f is complex analytic on the strip , is bounded in magnitude by 1 on the boundary of this strip, and obeys a growth condition on the interior of Roughly speaking, an attractor of a dynamical system is a subset of the state space to which orbits originating from typical initial conditions tend as time increases. All SVM Parameters C "However, it is critical here, as in any regularization scheme, that a proper value is chosen for C, the penalty factor. This is in contrast to the unsigned definite integral Definition of Stress Force Stress Area σ (sigma) = P/A Stress 4 . Prove that $\mathscr B=\{D(z,\epsilon)\} \cup \{E((x,0),\epsilon)\}$ be basis for a topology on $A. Also, even though it isn't a proof, you can show that on all lines through the origin the corresponding 1-dimensional limit is zero. It is often used to describe how coarse or fine a mesh is for a particular flow pattern. a box-counting dimension) is a notion of dimension for fractals, measuring how complexity of detail changes with the scale at which one views the fractal. However, Huber and von Mises' definition was little more than a math equation without physical interpretation until 1924 when Hencky [3] recognized that it is actually related to In my view dimensional analysis is a wonderful tool, but the risk is to take it too far. edu) September 16, 2001 The limit is formally de ned as follows: lim x!a f(x) = L From dimensional arguments and analogy with molecular transport Definition of L is different for each problem (boundary layes, mixing layers, etc. Looking at this projection carefully, you see that it is a little fuzzy. And this is a multi-dimensional thing, if you keep losing energy levels then these levels are going to, by the nature of this function are going to con strain you closer and closer and closer to being through the origin. The latter is an exact means of solving Maxwell’s equations in the presence of nonmagnetic scatterers. The ratio of a circle's circumference to its diameter In other words: all the way around a circle divided by all the way across it Equal to 3. In more detail, suppose that B = { v 1, …, v n} is a finite subset of a vector space V over a field F (such as the real or complex numbers R or C). Equation (16) is a one dimensional nonuniform oscillator, exhibiting the properties of a type one saddle-node bifurcation. , a compact convex set with non-empty interior) in $$\mathbb {R}^d$$ is called centered, if its center of mass is the origin. Choose$\delta=1$. Suppose X and Y are d-dimensional multivariate normal random vectors with mean zero and identity covariance matrix. 2] is the Laplacian operator and k = [omega][square root of [mu][epsilon]] is the wave number. I find the presentation of the$\epsilon$-$\delta$definition as a game tends to help beginning students: the players are Paul the prover, and Alice the adversary. Stochastic Gradient Descent (SGD) is a simple yet very efficient approach to discriminative learning of linear classifiers under convex loss functions such as (linear) Support Vector Machines and Logistic Regression. com. 2 It is exact volume preserving when g has zero average—or equivalently the zero Fourier component of g vanishes Lomelí and Meiss . For example, 0. This fuzziness is related to how small the pixels on the two-dimensional surface are. Most problems are average. In fact, we will concentrate mostly on limits of functions of two variables, but the ideas can be extended out to functions with more than two variables. This could perhaps build intuition, at the possible risk of students believing that the argument is in and of itself an adequate proof. This Pico function looks for a zero of a function f(x) between a and b given a precision epsilon. ojm: my biggest problem is that the *usual* definition of typical set is about *sequences* of independent draws from a single univariate generating process (thereby producing a high dimensional product space). The optimizer keeps inequality constraints above zero and equality constraints equal to zero, both conditions are met within a small epsilon range. My solutions to some of Spivak's exercises (I skip the few I found non-interesting). By definition, it separates the b b, and ϵ \epsilon. Further Examples of Epsilon-Delta Proof Yosen Lin, (yosenL@ocf. From a mathematical perspective, we introduce the space of test functions [math] \mathcal D(\mathbb{R}). The Limit Concept and its Definition. I think i need to use the delta epsilon definition i am not sure how to set it up. After the solution is complete, close the FLUENT window to return to the Workbench window. 0, while another gives you 1. The limit, valid for all time, is characterized in terms of a quadratic programming problem which can be solved with the aid of function theoretic methods. k. ) Eddy viscosity is zero if the velocity gradients are zero From dimensional arguments and analogy with molecular transport Definition of L is different for each problem (boundary layes, mixing layers, etc. A more likely situation would be for one model to give you an epsilon value of 1. At the Fermi level$\rho^{(d)}(\epsilon,T=0)\to 0\$, this indicates that interacting disordered two- and quasi-one-dimensional systems are in insulating state at zero temperature. We study the following problem. 2 Limit Laws The theorems below are useful when –nding the limit of a sequence. We prove that the value function of the game is the unique viscosity solution of the corresponding Hamilton-Jacobi-Isaacs equation in the sense of Crandall-Lions [12]. more The ratio of a circle's circumference to its diameter In other words: all the way around a circle divided by all the way across it The potential problem of zero division is here simply handled by the if test, meaning that if the denominator is too close to zero, that particular $$x$$ is skipped. epsilon zero dimensional definition | 2018-11-20 23:50:18 | {"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": 2, "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": 0.8161101937294006, "perplexity": 461.64980327838174}, "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/1542039746847.97/warc/CC-MAIN-20181120231755-20181121013755-00104.warc.gz"} |
https://cran.yu.ac.kr/web/packages/akmedoids/vignettes/akmedoids-vignette.html | # A guide to measuring long-term inequality in the exposure to crime at micro-area levels using 'Akmedoids' package
#### Date: 2021-04-13
Abstract
The 'akmedoids' package advances a set of R-functions for longitudinal clustering of long-term trajectories and determines the optimal solution based on the Caliński-Harabasz criterion (Caliński and Harabasz 1974). The package also includes a set of functions for addressing common data issues, such as missing entries and outliers, prior to conducting advance longitudinal data analysis. One of the key objectives of this package is to facilitate easy replication of a recent paper which examined small area inequality in the crime drop (see Adepeju et al. 2021). This document is created to provide a guide towards accomplishing this objective. Many of the functions provided in the akmedoids package may be applied to longitudinal data in general.
# Introduction
Longitudinal clustering analysis is ubiquitous in social and behavioral sciences for investigating the developmental processes of a phenomenon over time. Examples of the commonly used techniques in these areas include group-based trajectory modeling (GBTM) and the non-parametric kmeans method. Whilst kmeans has a number of benefits over GBTM, such as more relaxed statistical assumptions, generic implementations render it more sensitive to outliers and short-term fluctuations, which minimises its ability to identify long-term linear trends in data. In crime and place research, for example, the identification of such long-term linear trends may help to develop some theoretical understanding of criminal victimization within a geographical space (Weisburd et al. 2004; Griffith and Chavez 2004). In order to address this sensitivity problem, we advance a novel technique named anchored kmedoids ('akmedoids') which implements three key modifications to the existing longitudinal kmeans approach. First, it approximates trajectories using ordinary least square regression (OLS) and second, anchors the initialisation process with median observations. It then deploys the medoids observations as new anchors for each iteration of the expectation-maximization procedure (Celeux and Govaert 1992). These modifications ensure that the impacts of short-term fluctuations and outliers are minimized. By linking the final groupings back to the original trajectories, a clearer delineation of the long-term linear trends of trajectories are obtained.
We facilitate the easy use of akmedoids through an open-source package using R. We encourage the use of the package outside of criminology, should it be appropriate. Before outlining the main clustering functions, we demonstrate the use of a few data manipulation functions that assist in data preparation. The worked demonstration uses a small example dataset which should allow users to get a clear understanding of the operation of each function.
Table 1. Data manipulation functions
SN Function Title Description
1 data_imputation Data imputation for longitudinal data Calculates any missing entries (NA, Inf, null) in a longitudinal data, according to a specified method
2 rates Conversion of ‘counts’ to ‘rates’ Calculates rates from observed ‘counts’ and its associated denominator data
3 props Conversion of ‘counts’ (or ‘rates’) to ‘Proportion’ Converts ‘counts’ or ‘rates’ observation to ‘proportion’
4 outlier_detect Outlier detection and replacement Identifies outlier observations in the data, and replace or remove them
5 w_spaces Whitespace removal Removes all the leading and trailing whitespaces in a longitudinal data
6 remove_rows_n Incomplete rows removal Removes rows which contain ‘NA’ and ‘inf’ entries
# 1. Data manipulation
Table 1 shows the main data manipulation functions and their descriptions. These functions help to address common data issues prior to analysis, as well as basic data manipulation tasks such as converting longitudinal data from count to proportion measures (as per the crime inequality paper where akmedoids was first implemented). In order to demonstrate the utility of these functions, we provide a simulated dataset traj which can be called by typing traj in R console after loading the akmedoids library.
## (i) "data_imputation" function
Calculates any missing entries in a data, according to a chosen method. This function recognizes three kinds of data entries as missing. These are NA, Inf, null, and an option of whether or not to consider 0’s as missing values. The function provides a replacement option for the missing entries using two methods. First, an arithmetic method which uses the mean, minimum or maximum value of the corresponding rows or columns of the missing values. Second, a regression method which uses OLS regression lines to estimate the missing values. Using the regression method, only the missing data points derive values from the regression line while the remaining (observed) data points retain their original values. The function terminates if there is any trajectory with only one observation in it. Using the 'traj' dataset, we demonstrate how the 'regression' method estimates missing values.
#installing the akmedoids packages
install.packages("devtools")
devtools::install_github("manalytics/akmedoids")
#loading the package
library(akmedoids)
#import and preview the first 6 rows of 'traj' object
data(traj)
head(traj)
#> location_ids X2001 X2002 X2003 X2004 X2005 X2006 X2007 X2008 X2009
#> 1 E01012628 3 0 1 2 1 0 1 4 0
#> 2 E01004768 9 NA 2 4 7 5 1 3 1
#> 3 E01004803 4 3 0 10 2 3 6 6 8
#> 4 E01004804 7 3 9 3 2 NA 6 3 2
#> 5 E01004807 2 Inf 5 5 6 NA 3 5 4
#> 6 E01004808 8 5 8 4 1 5 6 1 1
#no. of rows
nrow(traj)
#> [1] 10
#no. of columns
ncol(traj)
#> [1] 10
The first column of the traj object is the id (unique) field. In many applications, it is necessary to preserve the id column in order to allow linking of outputs to other external datasets, such as spatial location data. Most of the functions of the akmedoids provides an option to recognise the first column of an input dataset as the unique field. The data_imputation function can be used to imput the missing data point of traj object as follows:
imp_traj <- data_imputation(traj, id_field = TRUE, method = 2,
replace_with = 1, fill_zeros = FALSE)
#> [1] "8 entries were found/filled!"
imp_traj <- imp_traj$CompleteData #viewing the first 6 rows head(imp_traj) #> location_ids X2001 X2002 X2003 X2004 X2005 X2006 X2007 X2008 X2009 #> 1 E01012628 3 0.00 1 2 1 0.00 1 4 0 #> 2 E01004768 9 6.44 2 4 7 5.00 1 3 1 #> 3 E01004803 4 3.00 0 10 2 3.00 6 6 8 #> 4 E01004804 7 3.00 9 3 2 3.90 6 3 2 #> 5 E01004807 2 3.92 5 5 6 4.36 3 5 4 #> 6 E01004808 8 5.00 8 4 1 5.00 6 1 1 The argument method = 2 refers to the regression technique, while the argument replace_with = 1 refers to the linear option (currently the only available option). Figure 1 is a graphical illustration of how this method approximates the missing values of the traj object. ### Estimating the population data using the 'data_imputation' function Obtaining the denominator information (e.g. population estimates to normalize counts) of local areas within a city for non-census years is problematic in longitudinal studies. This challenge poses a significant drawback to the accurate estimation of various measures, such as crime rates and population-at-risk of an infectious disease. Assuming a limited amount of denominator information is available, an alternative way of obtaining the missing data points is to interpolate and/or extrapolate the missing population information using the available data points. The data_imputation function can be used to perform this task. The key step towards using the function for this purpose is to create a matrix, containing both the available fields and the missing fields arranged in their appropriate order. All the entries of the missing fields can be filled with either NA or null. Below is a demonstration of this task with a sample population dataset with only two available data fields. The corresponding input matrix is constructed as shown. #import population data data(popl) #preview the data head(popl) #> location_id census_2003 census_2007 #> 1 E01004809 300 200 #> 2 E01004807 550 450 #> 3 E01004788 150 250 #> 4 E01012628 100 100 #> 5 E01004805 400 350 #> 6 E01004790 750 850 nrow(popl) #no. of rows #> [1] 11 ncol(popl) #no. of columns #> [1] 3 The corresponding input dataset is prepared as follows and saved as population2: #> location_ids X2001 X2002 X2003 X2004 X2005 X2006 X2007 X2008 X2009 #> 1 E01004809 NA NA 300 NA NA NA 200 NA NA #> 2 E01004807 NA NA 550 NA NA NA 450 NA NA #> 3 E01004788 NA NA 150 NA NA NA 250 NA NA #> 4 E01012628 NA NA 100 NA NA NA 100 NA NA #> 5 E01004805 NA NA 400 NA NA NA 350 NA NA #> 6 E01004790 NA NA 750 NA NA NA 850 NA NA The missing values are estimated as follows using the regression method of the data_imputation function: pop_imp_result <- data_imputation(population2, id_field = TRUE, method = 2, replace_with = 1, fill_zeros = FALSE) #> [1] "77 entries were found/filled!" pop_imp_result <- pop_imp_result$CompleteData
#viewing the first 6 rows
head(pop_imp_result)
#> location_ids X2001 X2002 X2003 X2004 X2005 X2006 X2007 X2008 X2009
#> 1 E01004809 350 325 300 275 250 225 200 175 150
#> 2 E01004807 600 575 550 525 500 475 450 425 400
#> 3 E01004788 100 125 150 175 200 225 250 275 300
#> 4 E01012628 100 100 100 100 100 100 100 100 100
#> 5 E01004805 425 412.5 400 387.5 375 362.5 350 337.5 325
#> 6 E01004790 700 725 750 775 800 825 850 875 900
Given that there are only two data points in each row, the regression method will simply generate the missing values by fitting a straight line to the available data points. The higher the number of available data points in any trajectory the better the estimation of the missing points. Figure 1 illustrates this estimation process.
## (ii) "rates" function
Given a longitudinal data ($$m\times n$$) and its associated denominator data ($$s\times n$$), the 'rates' function converts the longitudinal data to ‘rates’ measures (e.g. counts per 100 residents). Both the longitudinal and the denominator data may contain different number of rows, but need to have the same number of columns, and must include the id (unique) field as their first column. The rows do not have to be sorted in any particular order. The rate measures (i.e. the output) will contain only rows whose id's match from both datasets. We demonstrate the utility of this function using the imp_traj object (above) and the estimated population data (‘pop_imp_result’).
#example of estimation of 'crimes per 200 residents'
crime_per_200_people <- rates(imp_traj, denomin=pop_imp_result, id_field=TRUE,
multiplier = 200)
#view the full output
crime_per_200_people <- crime_per_200_people$rates_estimates #check the number of rows nrow(crime_per_200_people) #> [1] 9 From the output, it can be observed that the number of rows of the output data is 9. This implies that only 9 location_ids match between the two datasets. The unmatched ids are ignored. Note: the calculation of rates often returns outputs with some of the cell entries having Inf and NA values, due to calculation errors and character values in the data. We therefore recommend that users re-run the data_imputation function after generating rates measures, especially for a large data matrix. ## (iii) "props" function Given a longitudinal data, the props function converts each data point (i.e. entry in each cell) to the proportion of the sum of their corresponding column. Using the crime_per_200_people estimated above, we can derive the proportion of crime per 200 people for each entry as follows: #Proportions of crimes per 200 residents prop_crime_per200_people <- props(crime_per_200_people, id_field = TRUE, scale = 1, digits=2) #view the full output prop_crime_per200_people #> location_ids X2001 X2002 X2003 X2004 X2005 X2006 X2007 X2008 X2009 #> 1 E01012628 0.10 0.13 0.04 0.09 0.21 0.19 0.05 0.14 0.17 #> 2 E01004768 0.12 0.28 0.13 0.09 0.09 0.17 0.04 0.08 0.00 #> 3 E01004803 0.08 0.11 0.16 0.13 0.17 0.16 0.17 0.15 0.17 #> 4 E01004804 0.07 0.07 0.00 0.18 0.04 0.06 0.12 0.08 0.24 #> 5 E01004807 0.15 0.08 0.18 0.06 0.04 0.08 0.12 0.04 0.06 #> 6 E01004808 0.11 0.11 0.18 0.14 0.22 0.11 0.16 0.13 0.16 #> 7 E01004788 0.02 0.06 0.06 0.07 0.11 0.09 0.07 0.08 0.17 #> 8 E01004790 0.18 0.16 0.19 0.09 0.03 0.13 0.16 0.02 0.04 #> 9 E01004805 0.17 0.00 0.07 0.15 0.09 0.00 0.10 0.28 0.00 #A quick check that sum of each column of proportion measures adds up to 1. colSums(prop_crime_per200_people[,2:ncol(prop_crime_per200_people)]) #> X2001 X2002 X2003 X2004 X2005 X2006 X2007 X2008 X2009 #> 1.00 1.00 1.01 1.00 1.00 0.99 0.99 1.00 1.01 In line with the demonstration in Adepeju et al. (2021), we will use these proportion measures to demonstrate the main clustering function of this package. ## (iv) "outlier_detect" function This function is aimed at allowing users to identify any outlier observations in their longitudinal data, and replace or remove them accordingly. The first step towards identifying outliers in any data is to visualize the data. A user can then decide a cut-off value for isolating the outliers. The outlier_detect function provides two options for doing this: (i) a quantile method, which isolates any observations with values higher than a specified quantile of the data values distribution, and (ii) a manual method, in which a user specifies the cut-off value. The ‘replace_with’ argument is used to determine whether an outlier value should be replaced with the mean value of the row or the mean value of the column in which the outlier is located. The user also has the option to simply remove the trajectory that contains an outlier value. In deciding whether a trajectory contains outlier or not, the count argument allows the user to set an horizontal threshold (i.e. number of outlier values that must occur in a trajectory) in order for the trajectory to be considered as having outlier observations. Below, we demonstrate the utility of the outlier_detect function using the imp_traj data above. ## (v) "w_spaces" function Given a matrix suspected to contain whitespaces, this function removes all the whitespaces and returns a cleaned data. ’Whitespaces’ are white characters often introduced during data entry, for instance by wrongly pressing the spacebar. For example, neither " A" nor “A” equates “A” because of the whitespaces that exist in them. They can also result from systematic errors in data recording devices. ## (vi) "remove_rows_n" function This function removes any rows in which an ‘NA’ or an ‘Inf’ entry is found. #Plotting the data using ggplot library library(ggplot2) #> Warning: package 'ggplot2' was built under R version 4.0.3 #library(reshape2) #converting the wide data format into stacked format for plotting #doing it manually instead of using 'melt' function from 'reshape2' #imp_traj_long <- melt(imp_traj, id="location_ids") coln <- colnames(imp_traj)[2:length(colnames(imp_traj))] code_ <- rep(imp_traj$location_ids, ncol(imp_traj)-1)
d_bind <- NULL
for(v in seq_len(ncol(imp_traj)-1)){
d_bind <- c(d_bind, as.numeric(imp_traj[,(v+1)]))
}
code <- data.frame(location_ids=as.character(code_))
variable <- data.frame(variable=as.character(rep(coln,
each=length(imp_traj$location_ids)))) value=data.frame(value = as.numeric(d_bind)) imp_traj_long <- bind_cols(code, variable,value) #view the first 6 rows head(imp_traj_long) #> location_ids variable value #> 1 E01012628 X2001 3 #> 2 E01004768 X2001 9 #> 3 E01004803 X2001 4 #> 4 E01004804 X2001 7 #> 5 E01004807 X2001 2 #> 6 E01004808 X2001 8 #plot function p <- ggplot(imp_traj_long, aes(x=variable, y=value, group=location_ids, color=location_ids)) + geom_point() + geom_line() #options(rgl.useNULL = TRUE) print(p) Figure 2 is the output of the above plot function. Based on Figure 2 if we assume that observations of x2001, x2007 and x2008 of trajectory id E01004806 are outliers, we can set the threshold argument as 20. In this case, setting count=1 will suffice as the trajectory is clearly separable from the rest of the trajectories. imp_traj_New <- outlier_detect(imp_traj, id_field = TRUE, method = 2, threshold = 20, count = 1, replace_with = 2) #> [1] "1 trajectories were found to contain outlier observations and replaced accordingly!" #> [1] "Summary:" #> [1] "*--Outlier observation(s) was found in trajectory 10 --*" imp_traj_New <- imp_traj_New$Outliers_Replaced
#options(rgl.useNULL = TRUE)
print(imp_traj_New)
#> location_ids X2001 X2002 X2003 X2004 X2005 X2006 X2007 X2008 X2009
#> 1 E01012628 3.00 0.00 1.00 2 1 0.00 1.00 4.00 0
#> 2 E01004768 9.00 6.44 2.00 4 7 5.00 1.00 3.00 1
#> 3 E01004803 4.00 3.00 0.00 10 2 3.00 6.00 6.00 8
#> 4 E01004804 7.00 3.00 9.00 3 2 3.90 6.00 3.00 2
#> 5 E01004807 2.00 3.92 5.00 5 6 4.36 3.00 5.00 4
#> 6 E01004808 8.00 5.00 8.00 4 1 5.00 6.00 1.00 1
#> 7 E01004788 2.00 4.00 2.72 2 2 4.00 1.00 3.00 0
#> 8 E01004790 10.00 9.00 17.00 13 15 13.63 13.98 19.00 9
#> 9 E01004805 8.00 5.00 10.00 7 9 4.00 5.41 6.00 3
#> 10 E01004806 14.83 12.00 14.00 15 18 13.00 14.83 14.83 17
#imp_traj_New_long <- melt(imp_traj_New, id="location_ids")
coln <- colnames(imp_traj_New)[2:length(colnames(imp_traj_New))]
code_ <- rep(imp_traj_New$location_ids, ncol(imp_traj_New)-1) d_bind <- NULL for(v in seq_len(ncol(imp_traj_New)-1)){ d_bind <- c(d_bind, as.numeric(imp_traj_New[,(v+1)])) } code <- data.frame(location_ids=as.character(code_)) variable <- data.frame(variable=as.character(rep(coln, each=length(imp_traj_New$location_ids))))
value=data.frame(value = as.numeric(d_bind))
imp_traj_New_long <- bind_cols(code, variable,value)
#plot function
#options(rgl.useNULL = TRUE)
p <- ggplot(imp_traj_New_long, aes(x=variable, y=value,
group=location_ids, color=location_ids)) +
geom_point() +
geom_line()
#options(rgl.useNULL = TRUE)
print(p)
Setting replace_with = 2, that is to replace the outlier points with the ‘mean of the row observations’, the function generates outputs re-plotted in Figure 3.
## (vii) ‘Other’ functions
Please see the akmedoids user manual for the remaining data manipulation functions.
# 2. Data Clustering
Table 2 shows the two main functions required to carry out the longitudinal clustering and generate the descriptive statistics of the resulting groups. The relevant functions are akclustr and print_akstats. The akclustr function clusters trajectories according to the similarities of their long-term trends, while the print_akstats function extracts descriptive and change statistics for each of the clusters. The former also generates quality plots for the best cluster solution.
The long-term trends of trajectories are defined in terms of a set of OLS regression lines. This allows the clustering function to classify the final groupings in terms of their slopes as rising, stable, and falling. The key benefits of this implementation is that it allows the clustering process to ignore the short-term fluctuations of actual trajectories and focus on their long-term linear trends. Adepeju and colleagues (2021) applied this technique in crime concentration research for measuring long-term inequalities in the exposure to crime at find-grained spatial scales.
Their implementation was informed by the conceptual (inequality) framework shown in Figure 4. That said, akmedoids can be deployed on any measure (counts, rates) and is not limited to criminology, but rather, any field where the aim is to cluster longitudinal data based on long-term trajectories. By mapping the resulting trend lines grouping to the original trajectories, various performance statistics can be generated.
In addition to the use of trend lines, the akmedoids makes two other modifications to the expectation-maximisation clustering routines (Celeux and Govaert 1992). First, the akmedoids implements an anchored median-based initialisation strategy for the clustering to begin. The purpose behind this step is to give the algorithm a theoretically-driven starting point and try and ensure that heterogenous trend slopes end up in different clusters (Khan and Ahmad (2004); Steinley and Brusco (2007)). Second, instead of recomputing centroids based on the mean distances between each trajectory trend lines and the cluster centers, the median of each cluster is selected and then used as the next centroid. This then becomes the new anchor for the current iteration of the expectation-maximisation step (Celeux and Govaert 1992). This strategy is implemented in order to minimize the impact of outliers. The iteration then continues until an objective function is maximised.
Table 2. Data clustering functions
SN Function Title Description
1 akclustr Anchored k-medoids clustering Clusters trajectories into a k number of groups according to the similarities in their long-term trend and determines the best solution based on the Silhouette width measure or the Calinski-Harabasz criterion
2 print_akstats Descriptive (Change) statistics of clusters Generates the descriptive and change statistics of groups, and also plots the groups performances
3 plot_akstats Plots of cluster groups Generates different plots of cluster groups
#Visualizing the proportion data
#view the first few rows
head(prop_crime_per200_people)
#> location_ids X2001 X2002 X2003 X2004 X2005 X2006 X2007 X2008 X2009
#> 1 E01012628 0.10 0.13 0.04 0.09 0.21 0.19 0.05 0.14 0.17
#> 2 E01004768 0.12 0.28 0.13 0.09 0.09 0.17 0.04 0.08 0.00
#> 3 E01004803 0.08 0.11 0.16 0.13 0.17 0.16 0.17 0.15 0.17
#> 4 E01004804 0.07 0.07 0.00 0.18 0.04 0.06 0.12 0.08 0.24
#> 5 E01004807 0.15 0.08 0.18 0.06 0.04 0.08 0.12 0.04 0.06
#> 6 E01004808 0.11 0.11 0.18 0.14 0.22 0.11 0.16 0.13 0.16
#prop_crime_per200_people_melt <- melt(prop_crime_per200_people, id="location_ids")
coln <- colnames(prop_crime_per200_people)[2:length(colnames(prop_crime_per200_people))]
code_ <- rep(prop_crime_per200_people$location_ids, ncol(prop_crime_per200_people)-1) d_bind <- NULL for(v in seq_len(ncol(prop_crime_per200_people)-1)){ d_bind <- c(d_bind, prop_crime_per200_people[,(v+1)]) } prop_crime_per200_people_melt <- data.frame(cbind(location_ids=as.character(code_), variable = rep(coln, each=length(prop_crime_per200_people$location_ids)), value=d_bind))
#plot function
#options(rgl.useNULL = TRUE)
p <- ggplot(prop_crime_per200_people_melt, aes(x=variable, y=value,
group=location_ids, color=location_ids)) +
geom_point() +
geom_line()
#options(rgl.useNULL = TRUE)
print(p)
In the following sections, we provide a worked example of clustering with akclustr function using the prop_crime_per200_people object. The function will generate cluster solution over a set of k values, determine the optimal value of k. The print_akstats function will then be applied to generate the descriptive summary and the change statistics of the clusters. The prop_crime_per200_people object is plotted in 5.
## (i) akclustr function
Dataset:
Each trajectory in Figure 5 represents the proportion of crimes per 200 residents in each location over time. The goal is to first extract the inequality trend lines such as in Figure 4 and then cluster them according to the similarity of their slopes. For the akclustr function, a user sets the k value which may be an integer or a vector of length two specifying the minimum and maximum numbers of clusters to loop through. In the latter case, the akclustr function employs either the Silhouette (Rousseeuw (1987))) or the Calinski_Harabasz score (Caliński and Harabasz (1974); Genolini and Falissard (2010)) to determine the best cluster solution. In other words, it determines the k value that optimizes the specified criterion. The verbose argument can be used to control the processing messages. The function is executed as follows:
#clustering
akObj <- akclustr(prop_crime_per200_people, id_field = TRUE,
method = "linear", k = c(3,8), crit = "Calinski_Harabasz", verbose=TRUE)
#> [1] "Processing...."
#> [1] ".............."
#> [1] "solution of k = 3 determined!"
#> [1] "solution of k = 4 determined!"
#> [1] "solution of k = 5 determined!"
#> [1] "solution of k = 6 determined!"
#> [1] "solution of k = 7 determined!"
#> [1] "solution of k = 8 determined!"
In order to preview all the variables of the quality_plot object, type:
names(akObj)
#> [1] "traj" "id_field" "solutions" "qualitycriterion"
#> [5] "optimal_k" "qualityCrit.List" "qltyplot"
• The description of these variables are as follow:
• traj - returns the input data set used for the clustering.
• id_field - indicates whether the input data set included the id field.
• solutions - the list of cluster solutions by k values.
• qualitycriterion - the quality criterion specified.
• optimal_k - the optimal value of k as determined by the quality criterion.
• qualityCrit.List - the estimated quality of cluster solutions by k values.
• qltyplot - the plot of qualityCrit.List, with a red vertical line to indicate the optimal value of k.
Accessing the optimal solution*
The qualityCrit.List can be viewed graphically by setting the quality_plot argument as TRUE. Also, the plot may still be accessed after clustering by printing the variable akObj$qltyplot. From (Figure 6), the best value of k is the highest at k=5, and therefore determined as the best solution. It is recommended that the determination based on either of the quality criteria should be used complementarily with users judgment in relation to the problem at hand. Given a value of k, the group membership (labels) of its cluster solution can be extracted by entering k'= k - 2) into the variable akObj$solutions[[k']]. E.g.
#5-group clusters
akObj$solutions[[3]] #for k=5 solution #> [1] "D" "A" "D" "E" "B" "C" "E" "A" "C" #> attr(,"cluster labels for k =") #> [1] 5 Also, note that the indexes of the group memberships correspond to that of the trajectory object (prop_crime_per200_people) inputted into the function. That is, the membership labels, "D", "A", "A", .... are the group membership of the trajectories "E01012628","E01004768","E01004803",... of the object prop_crime_per200_people. ## (ii) print_akstats function: The properties (i.e. the descriptive and change statistics) of a cluster solutions (i.e. solution for any value of k) such as in k = 5 above can be generated by using the special print function print_akstats. The print function takes as input the akobject class, e.g. akObj. The descriptive statistics shows the group memberships and their performances in terms of their shares of the proportion measure captured over time. The change statistics shows the information regarding the direction variances of the groups in relation to reference direction. In trajectory clustering analysis, the resulting groups are often re-classified into larger classes based on the slopes, such as Decreasing, Stable, or Increasing classes (Weisburd et al. (2004); Andresen et al. 2017). The slope of a group is the angle made by the medoid of the group relative to a reference line ($$R$$). The reference argument is specified as 1, 2 or 3, representing the mean trajectory, medoid trajectory, or a horizontal line with slope = 0, respectively. Let $$\vartheta_{1}$$ and $$\vartheta_{n}$$ represent the angular deviations of the group medoids with the lowest slope (negative) and highest (positive) slopes, respectively. If we sub-divide each of these slopes into a specified number of equal intervals (quantiles), the specific interval within which each group medoid falls can be determined. This specification is made using the n_quant argument. Figure 7 illustrates the quantiles sub-divisions for n_quant = 4. In addition to the slope composition of trajectories found in each group, the quantile location of each group medoid can be used to further categorize the groups into larger classes. We refer users to the package user manual for more details about these parameters. Using the current example, the function can be ran as follows: #Specifying the optimal solution, output$optimal_k (i.e. k = 5) and using stacked type graph
prpties = print_akstats(akObj, k = 5, show_plots = FALSE)
#> Warning: fun.y is deprecated. Use fun instead.
prpties
#> $descriptive_stats #> group n n(%) %Prop.time1 %Prop.timeT Change %Change #> 1 A 2 22.2 30 4 -26 -650 #> 2 B 1 11.1 15 5.9 -9.1 -154.2 #> 3 C 2 22.2 28 15.8 -12.2 -77.2 #> 4 D 2 22.2 18 33.7 15.7 46.6 #> 5 E 2 22.2 9 40.6 31.6 77.8 #> #>$change_stats
#> group sn %+ve Traj. %-ve Traj. Qtl:1st-4th
#> 1 A 1 0 100 4th (-ve)
#> 2 B 2 0 100 3rd (-ve)
#> 3 C 3 100 0 1st (+ve)
#> 4 D 4 100 0 3rd (+ve)
#> 5 E 5 100 0 4th (+ve)
## (iii) plot_akstats function:
The above printouts represent the properties (i.e. the descriptive and change properties) of the clusters. Note: the show_plots argument of print_akstats function, if set as TRUE, will produce the plot of group trajectories, representing the group directional change over time. However, the plot_akstats has been designed to generate different performance plots of the groups. See below:
1. Group trajectories (directional change over time)
#options(rgl.useNULL = TRUE)
plot_akstats(akObj, k = 5, type="lines", y_scaling="fixed")
#> Warning: fun.y is deprecated. Use fun instead.
#> $cluster_plot 1. Proportional change of groups change over time #options(rgl.useNULL = TRUE) plot_akstats(akObj, k = 5, reference = 1, n_quant = 4, type="stacked") #>$cluster_plot
In the context of the long-term inequality study, broad conclusions can be made from both the statistical properties and the plots regarding relative crime exposure in the area represented by each group or class (Adepeju et al. 2021). For example, whilst relative crime exposure have declined in 33.3% (groups A and B) of the study area, the relative crime exposure have risen in 44.4% (groups D and E) of the area. The relative crime exposure can be said to be stable in 22.2% (group C) of the area, based on its close proximity to the reference line. The medoid of the group falls within the $$1^{st}(+ve)$$ quantile (see Figure ). In essence, we determine that groups A and B belong to the Decreasing class, while groups D and E belong to the Increasing class.
It is important to state that this proposed classification method is simply advisory; you may devise a different approach or interpretation depending on your research questions and data.
By changing the argument type="lines" to type="stacked", a quality plot is generated instead (see Figure ). Note that these plots make use of functions within the ggplot2 library (Wickham 2016). For a more customized visualization, we recommend that users deploy the ggplot2 library directly.
Table 3. field description of clustering outputs
SN field Description
1 group group membershp
2 n size (no.of.trajectories.)
3 n(%) % size
4 %Prop.time1 % proportion of obs. at time 1 (2001)
5 %Prop.timeT proportion of obs. at time T (2009)
6 Change absolute change in proportion between time1 and timeT
7 %Change % change in proportion between time 1 and time T
8 %+ve Traj. % of trajectories with positive slopes
9 %-ve Traj. % of trajectories with negative slopes
10 Qtl:1st-4th Position of a group medoid in the quantile subdivisions
# Conclusion
The akmedoids package has been developed in order to aid the replication of a place-based crime inequality investigation conducted in Adepeju, Langton, and Bannister (2021). Meanwhile, the utility of the functions in this package are not limited to criminology, but rather can be applicable to longitudinal datasets more generally. This package is being updated on a regular basis to add more functionalities to the existing functions and add new functions to carry out other longitudinal data analysis.
We encourage users to report any bugs encountered while using the package so that they can be fixed immediately. Welcome contributions to this package which will be acknowledged accordingly.
# References
Adepeju, M., S. Langton, and J. Bannister. 2021. “Anchored K-Medoids: A Novel Adaptation of K-Medoids Further Refined to Measure Inequality in the Exposure to Crime Across Micro Places.” Journal of Computational Social Science. https://doi.org/10.1007/s42001-021-00103-1.
Caliński, T., and J. Harabasz. 1974. “A Dendrite Method for Cluster Analysis.” Communications in Statistics-Theory and Methods 3(1): 1–27.
Celeux, G., and G. Govaert. 1992. “A Classification Em Algorithm for Clustering and Two Stochastic Versions.” Computational Statistics 14(3): 315–32.
Genolini, C., and B. Falissard. 2010. “Kml and Kml3d: R Packages to Cluster Longitudinal Data.” Computational Statistics 25(2): 317–28.
Griffith, E., and J. M. Chavez. 2004. “Communities, Street Guns and Homicide Trajectories in Chicago, 1980–1995: Merging Methods for Examining Homicide Trends Across Space and Time.” Criminology 42(4): 941–78.
Khan, S. S, and A. Ahmad. 2004. “Cluster Center Initialization Algorithm for K-Means Clustering.” Pattern Recognition Letters 25(11): 1293–1302.
Rousseeuw, P. J. 1987. “Silhouettes: A Graphical Aid to the Interpretation and Validation of Cluster Analysis.” Journal of Computational and Applied Mathematics, no. 20: 53–65.
Steinley, D., and M. J. Brusco. 2007. “Initializing K-Means Batch Clustering: A Critical Evaluation of Several Techniques.” Journal of Classification 24(1): 99–121.
Weisburd, D., S. Lum, C. Lum, and S. M. Lum. 2004. “Trajectories of Crime at Places: A Longitudinal Study of Street Segments in the City of Seattle.” Criminology 42(2): 283–322.
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https://study.com/academy/answer/when-recording-live-performances-sound-engineers-often-use-a-microphone-with-a-cardioid-pickup-pattern-because-it-suppresses-noise-from-the-audience-suppose-the-microphone-is-placed-4m-from-the-fron.html | # When recording live performances, sound engineers often use a microphone with a cardioid pickup...
## Question:
When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 4m from the front of the stage and the boundary of the optimal pickup region is given by the cardioid {eq}r = 8 + 8 \sin {/eq} where r is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microscope. Answer their question and round to two decimal places. _____ {eq}(m^2) {/eq}
## Area inside Cardioid in Polar Form:
{eq}\\ {/eq}
Application of Integration is used to find the area inside a cardioid.
The area inside the cardioid in polar form is given by the formula :-
{eq}\boxed{A=\displaystyle \int_{a }^{b }\dfrac{1}{2}r^2d\theta} {/eq}, where a, b are angles measured in radians and may be defined as the limits of integration.
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{eq}\\ {/eq}
Given : Equation of cardioid {eq}r= 8+8 \sin \theta {/eq}
As the sound receiving area covers the four quadrants, therefore the limits...
Cardioid in Math: Definition, Equation & Examples
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Chapter 1 / Lesson 13
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This lesson will cover a neat shape studied in upper-level mathematics called a cardioid. We will look at the basic shape, how it is constructed, its equation in polar form, and various examples of these equations and corresponding cardioids. | 2021-07-23 19:18:50 | {"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": 0.8812889456748962, "perplexity": 1652.2580097845346}, "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/1627046150000.59/warc/CC-MAIN-20210723175111-20210723205111-00312.warc.gz"} |
https://clubs.github.io/ | # CLUBS
## An RGB-D Dataset with Cluttered Box Scenes Containing Household Objects
Tonci Novkovic, Fadri Furrer, Marko Panjek, Margarita Grinvald, Roland Siegwart, and Juan Nieto
Autonomous Systems Lab, ETH Zurich, Switzerland
DATASET:
CLUBS is an RGB-D dataset that can be used for segmentation, classification and detetion of household objects in realistic warehouse box scenarios. The dataset contains the object scenes, the reconstructed models, as well as box scenes that contain mutliple objects packed in different configurations. Additionally, the raw scaning data, 2D object bounding boxes and pixel-wise labels in RGB images, 3D bounding boxes, and calibration data are also available.
The dataset contains 85 object scenes and 25 box scenes. Different box scenes have diffrent configurations of objects in them and vary in clutter level. More details about the specific box scenes can be found by clicking on the scene gif image in the Box scenes section below.
The data was collected using a robotic arm (UR10) with one RGB (Chameleon3) and three RGB-D (PrimeSense Carmine 1.09, Intel Realsense D415, Intel Realsense D435) cameras. All the cameras were calibrated for intrisic and extrinsic parameters. More details about the calibration and notation used can be found below.
Object scenes
The dataset contains 85 different objects. The path for scanning a single object scene contains 19 different poses at three different height levels. It covers the whole object from every side, except the bottom. Every object was, therefore, rotated and scanned a second time to also cover the bottom face.
The objects within the dataset are separated into 5 main categories by the type of the product, 3 categories based on shape, and 2 categories based on rigidity:
Box scenes
The robot path for box scenes is shorter and contains 9 poses. These poses were chosen such that the inside of a box of size 58x38x34 cm is covered from all sides including a top-view. The dataset contains 25 different box scenes, where 5 of them have 40 objects and rest 30 objects. Overall object distribution within the boxes is shown below:
To get more details about a specific box click on its image below:
Calibration
All the raw calibration data and results are available for download below. For the distortion, radial-tangential model was used, represented as:
$$[r_1, r_2, t_1, t_2, r_3]$$
All the frames and transformations between the cameras and the robot are displayed in the image below:
For the RealSense cameras, two different calibration files are provided, one containing depth intrinsic parameters read out from the device itself for the lower resolution depth *_device_depth.yaml, and the other containing higher-resolution depth intrinsic parameters obtained in the calibration step *_stereo_depth.yaml. If regular depth from the raw data is used, the former one should be used. In case the depth is generated from stereo using our Python script the latter calibration file should be used. The calibration folder is provided as: - calibration - primesense.yaml - realsense_d415_device_depth.yaml - realsense_d415_stereo_depth.yaml - realsense_d435_device_depth.yaml - realsense_d435_stereo_depth.yaml - chameleon3.yaml - calibration_raw_data MATLAB script for obtaining the calibration parameters is available on the clubs_dataset_tools github page.
Notation and data structure
Camera poses are represented by a translation vector and a Hamiltonian unit quaternion:
$$[stamp, t_x, t_y, t_z, q_x, q_y, q_z, q_w]$$
Scenes contain either 9 or 19 poses depending on the type of scene being scanned (object or box scene). Object scenes are labeled as: label_side_name, where label is a 3 digit number uniquely defining an object, side is either 0 or 1, since all objects are flipped in order to get full coverage, and the name describes the object. Box scenes have the following convention: box_label_iteration, where the label represents a box with a defined subset of N contained objects, and the iteration relates to the current number of objects in the box (N - iteration). In the first iteration all the objects are in the box, the box is scanned, one object is taken out and then the next iteration starts. This is repeated until there are no objects left in the box.
Each sensor folder contains the available raw data for that sensor and an additional folder with pixel-wise labeled images and .json files which include object 2D bounding boxes and polygon vertices used for generating the label image. Objects present in the scene, together with the 3D bounding box size and pose, are stored in the scene_objects.csv file:
$$[object\_id, x, y, z, qx, qy, qz, qw, size_x, size_y, size_z]$$
where [x, y, z] vector represents center coordinates, quaternion [qx, qy, qz, qw] represents orientation, and [sizex, sizey, sizez] represents the size of the 3D bounding box.
All the poses are also stored using the quaternion and translation format as described above. The robot's end-effector poses are stored in W_H_poses.csv, and for each sensor, the poses of the RGB camera and the IR cameras are stored in W_RGB_poses.csv, W_IR1_poses.csv and W_IR2_poses.csv respectively. The folder structure is as follows: - scene - scene_objects.csv - W_H_poses.csv - sensor - W_RGB_poses.csv - W_IR1_poses.csv - W_IR2_poses.csv - depth_images - stamp_depth.png - ir1_images - stamp_ir1.png - ir2_images - stamp_ir2.png - rgb_images - stamp_rgb.png - labels - rgb_images - stamp_rgb_label.json - stamp_rgb_label.png - scene_sensor_pointlcoud.ply For each sensor in one scene, we provide the reconstructed point cloud scene_sensor_pointcloud.ply obtained by integrating the point clouds with their corresponding camera poses into a TSDF volume. The pixel-wise labels and 2D bounding boxes are stored in json files timestamp_rgb_label.json as follows:
{"poly":
[[[x00, y00], ..., [x0L, y0L]],
...,
[[xN0, yN0], ..., [xNL, yNL]]],
"bbox":
[[bx0, by0, w0, h0],
...,
[bxN, byN, wN, hN]],
"labels":
["label0",
...,
"labelN"]}
where poly is a list of lists of image coordinates that define the surrounding polygon, bbox is a list of lists that contain 2D bounding box coordinates of the top left corner, width and height, and labels is a list of label names corresponding to the names of objects present in the scene. These json files are used to generate the label images timestamp_rgb_label.png. All the pixels in these images have values from 0 to 41, where 0 corresponds to the background, and rest of the values correspond to the labels in the json file, i.e. value 1 means it is the first label in the json file, label 2 is the second, etc.
By using the provided Python scripts for generating depth images from stereo, computing point clouds from RGB-D images, and registering depth images to the RGB image of the corresponding sensor, additional folders for each sensor are created. Namely, these are stereo_depth_images, point_clouds, and registered_depth_images.
PUBLICATION:
If you are using this dataset in your research, please cite the following publication:
@article{novkovic2019clubs,
author = {Novkovic, Tonci and Furrer, Fadri and Panjek, Marko and Grinvald, Margarita and Siegwart, Roland and Nieto, Juan},
journal = {The International Journal of Robotics Research (IJRR)},
title = {CLUBS: An RGB-D dataset with cluttered box scenes containing household objects},
year = {2019},
pages = {1538-1548},
volume = {38},
number = {14},
doi = {10.1177/0278364919875221}
}
Individual object scenes can be downloaded from the list in the Object scenes section. Furthermore, individual box scenes can be downloaded by clicking on the scene image in the Box scenes section. Finally, the whole dataset can be downloaded using the following links:
TOOLS:
This dataset comes with a set of tools which are avilable on our repo:
clubs_dataset_tools
These tools include: | 2021-05-06 01:39:57 | {"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": 0.22568774223327637, "perplexity": 3028.7302698717376}, "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-2021-21/segments/1620243988724.75/warc/CC-MAIN-20210505234449-20210506024449-00159.warc.gz"} |
https://chemistry.stackexchange.com/questions/9927/given-the-volumes-determine-the-ph-and-the-final-temperature-of-a-mixture-knowi?noredirect=1 | # Given the volumes: determine the pH and the final temperature of a mixture knowing only the initial pH and the temperature of the un-mixed components
I have a pretty basic question but the last time I took Chemistry was in 2007. I am studying the Navy's Nuclear study guide for their interviews and one of the question I am faced with is below.
Determine the final pH and temperature when these two solutions are mixed together in a 3L container.
Sol A: 2L, pH = 3, and 80F
Sol B: 1L, pH = 5, and 40F
Would this be as simple as $(2/3) \cdot 3 + (1/3)\cdot5$ for the pH and $(2/3)\cdot80 + (1/3)\cdot40$ for the temperature?
• I've edited tag,formatting and title let me know if you don't agree with my edit. – G M Apr 13 '14 at 11:57
For the temperature, this method of calculation will give you a rough estimate of the final temperature, but it will not be exactly right. The reason is that mixing two solutions comes at an energy cost/gain due to the interaction of the two solution molecules. For some mixtures this enthalpy of mixing yields a temperature rise (i.e. energy is released from the molecular potential energy), for others it yields a temperature decrease (i.e. energy is taken up to 'make the mixing possible'). For many mixtures this difference is not more than a few degrees, which makes your linear estimate reasonable, but it is good to keep in mind that it will not be exact.
For the pH, the shortcut you are taking is incorrect and will at best give you a very crude estimate. The reason is that the pH scale is logarithmic, it is correlated with the concentration of protons in the solution as $\ce{pH}=-\log_{10}\; [\ce{H+}]$, where $[\ce{H+}]$ is the proton concentration. (I will make the assumption here that you have simple liquids, which are not buffered.) Therefore, you would first need to calculate the proton concentration in both solutions from this equation, yielding $10^{-3}\,\ce{M}$ and $10^{-5}\,\ce{M}$ for sols A and B resp. This means that the amount of $\ce{H+}$ in sol A is $2\cdot10^{-3}\,\ce{mol}$ and that in sol B $10^{-5}\,\ce{mol}$. Mixing the two you get 3 liter of solution with $2.01\cdot10^{-3}\,\ce{mol}$ protons, which is $6.7\cdot10^{-4}\,\ce{M}$ which works out to $\ce{pH=}3.17$. Indeed, very far off from the 3.67 predicted in your linear method.
Short discussion on the pH: because the scale is logarithmic and your 2 $\ce{pH}$ values are actually not that far from each other the linear estimate is still ok-ish. If, for example, you would have used $\ce{pH}$ 0 and 7 you would have found $\ce{pH}=0.17$ vs. $2.33$ from your estimate.
• Where did 10^{-3} and 10^{-5} come from? Then why did you multiple by 2 on both of them? – dustin Apr 13 '14 at 15:04
• $\ce{pH}=-\log_{10}[\ce{H+}]$ gives $10^{-\ce{pH}}=[\ce{H+}]$ which shows how I got the $10^{-3}$ and $10^{-5}$, then I multiplied with 2 ONLY for the $10^{-3}$ because its units are $\ce{mol/l}$ and you indicated that you have 2 liters of sol A, which then makes it $2\cdot10^{-3} \ce{mol}$ – Michiel Apr 13 '14 at 16:18
• and the $\log$ is a $\log_{10}$, just to be clear. I edited that in for clarity – Michiel Apr 13 '14 at 16:21
• Minor mistake, $2\times 10^{-3} + 10^{-5} = 2.01 \times 10^{-3}$, not $2.05 \times 10^{-3}$. Mixed into 3 liters gives $6.7\times 10^{-4}$ M. Close enough. – LDC3 Apr 13 '14 at 16:25
• @LDC3 whoops, you are absolutely right, corrected it! – Michiel Apr 13 '14 at 16:28
The temperature would be correct. And the pH would be correct for this exercise, but not in reality. Since most solutions contain weak acid and base, it is difficult to determine the pH unless you knew which buffers and quantities are in the system.
• Can you explain what you mean by knowing which buffers and quantities are in the system? – dustin Apr 13 '14 at 0:03
• The pH portion of the answer is not correct. Dilution of species is linear in concentrations, but the pH scale is logarithmic in concentration. For example, 1 L of non-buffered pH 3 solution + 9 L of pure water (pH 7 solution) makes 10 L of pH 4 solution. – Nicolau Saker Neto Apr 13 '14 at 0:25
• @NicolauSakerNeto can you write an answer explaining how you solve that part so I can see? – dustin Apr 13 '14 at 1:48
• Does this answer give you enough of an idea of how to calculate pH after mixing? – Nicolau Saker Neto Apr 13 '14 at 1:57
• @NicolauSakerNeto that is too complicated. I need something simpler as an example. – dustin Apr 13 '14 at 2:14 | 2020-01-28 04:56:13 | {"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": 0.7796466946601868, "perplexity": 512.2686166179716}, "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/1579251773463.72/warc/CC-MAIN-20200128030221-20200128060221-00349.warc.gz"} |
https://www.askiitians.com/forums/Analytical-Geometry/find-the-equation-of-the-line-passing-through-the_109831.htm | Find the equation of the line passing through the inersection of lines x-y+1=0 and 2x-3y+5=0 and whose distance from the point (3,2) is 7/5.
Nishant Vora IIT Patna
7 years ago
Hello Student, Please find the solution
Here you can use the concept of family of lines
The equation of the line passing through the inersection of lines x-y+1=0 and 2x-3y+5=0 can be assumed as
Now the distance of this line from (3,2) is 7/5
so,
Solve the above equation and find out
Once you get , put it back in equation
And you will get the final answer | 2022-07-07 17:02:51 | {"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": 5, "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": 0.8010720014572144, "perplexity": 644.1029925995855}, "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-2022-27/segments/1656104495692.77/warc/CC-MAIN-20220707154329-20220707184329-00001.warc.gz"} |
https://www.semanticscholar.org/paper/On-the-minimal-model-theory-for-dlt-pairs-of-log-Gongyo/3f2eeed4dd913660f31463341f13697af2841677 | # On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero
@article{Gongyo2011OnTM,
title={On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero},
author={Yoshinori Gongyo},
journal={arXiv: Algebraic Geometry},
year={2011}
}
We prove the existence of good log minimal models for dlt pairs of numerical log Kodaira dimension 0.
On the finiteness of ample models
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https://jzus.zju.edu.cn/article.php?doi=10.1631/jzus.A1400191 | Full Text: <2995>
Summary: <2017>
CLC number: U238
On-line Access: 2014-12-04
Revision Accepted: 2014-11-03
Crosschecked: 2014-11-24
Cited: 9
Clicked: 7285
Citations: Bibtex RefMan EndNote GB/T7714
ORCID:
Xin ZHAO
http://orcid.org/0000-0002-0047-5925
Journal of Zhejiang University SCIENCE A 2014 Vol.15 No.12 P.946-963 http://doi.org/10.1631/jzus.A1400191
Modeling of high-speed wheel-rail rolling contact on a corrugated rail and corrugation development*
Author(s): Xin Zhao, Ze-feng Wen, Heng-yu Wang, Xue-song Jin, Min-hao Zhu Affiliation(s): . State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China Corresponding email(s): xinzhao@home.swjtu.edu.cn Key Words: Rail corrugation, Frictional rolling contact, Vehicle-track interaction, Friction exploitation level, Explicit finite element method Share this article to： More <<< Previous Article|Next Article >>>
Xin Zhao, Ze-feng Wen, Heng-yu Wang, Xue-song Jin, Min-hao Zhu. Modeling of high-speed wheel-rail rolling contact on a corrugated rail and corrugation development[J]. Journal of Zhejiang University Science A, 2014, 15(12): 946-963.
@article{title="Modeling of high-speed wheel-rail rolling contact on a corrugated rail and corrugation development",
author="Xin Zhao, Ze-feng Wen, Heng-yu Wang, Xue-song Jin, Min-hao Zhu",
journal="Journal of Zhejiang University Science A",
volume="15",
number="12",
pages="946-963",
year="2014",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.A1400191"
}
%0 Journal Article
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%A Xin Zhao
%A Ze-feng Wen
%A Heng-yu Wang
%A Xue-song Jin
%A Min-hao Zhu
%J Journal of Zhejiang University SCIENCE A
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%N 12
%P 946-963
%@ 1673-565X
%D 2014
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.A1400191
TY - JOUR
T1 - Modeling of high-speed wheel-rail rolling contact on a corrugated rail and corrugation development
A1 - Xin Zhao
A1 - Ze-feng Wen
A1 - Heng-yu Wang
A1 - Xue-song Jin
A1 - Min-hao Zhu
J0 - Journal of Zhejiang University Science A
VL - 15
IS - 12
SP - 946
EP - 963
%@ 1673-565X
Y1 - 2014
PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.A1400191
Abstract:
Short pitch rail corrugations were observed on a recently opened Chinese high-speed line. On the basis of field measurements and observations of corrugations occurred on the high-speed line, a 3D transient rolling contact model is developed using the explicit finite element (FE) method to investigate high-speed vehicle-track interactions in the presence of rail corrugations. The rotational and translational movements of the wheel are introduced as initial conditions in the model. The frictional rolling contact between the wheel and the corrugated rail is solved by a penalty method based surface-to-surface contact algorithm with Coulomb’s law of friction. The contact filter effect is considered automatically by the finite size of the contact patch. Through specifying a time-dependent driving torque applied to the wheel axle, the tangential vehicle-track interaction on the corrugated rail is analyzed in the time domain together with the normal one at different traction levels and at rolling speeds of up to 500 km/h. This analysis focuses on detailed contact solutions, such as distributions of the pressure, surface shear stress, Von Mises (V-M) stress, and frictional work. The corrugation dimensions, traction level, and rolling speed are varied to investigate their influences, building a solid basis for further studying the material damage mechanisms. A theory is proposed based on the simulations to explain the observed phenomenon that the corrugation gradually stabilizes. The traditional multi-body approach is found to overestimate the dynamic wheel-rail interaction on a corrugated rail.
## 1. Introduction
Rail corrugation is a long-standing problem observed worldwide on many kinds of railway tracks, including tram, metro, traditional railway, heavy haul, and high-speed tracks. Once present, corrugation can worsen the wheel-rail and vehicle-track interactions, leading to poor ride quality and an exacerbated rate of deterioration of the system, e.g., rail support failure (Zhou and Shen, ). Recently, rail corrugation, particularly short pitch rail corrugation (hereinafter referred to as corrugation for short), was observed on a recently opened high-speed line in China, causing great concern in the industry. Fig. 1 shows a corrugated rail section on the high-speed line.
Fig.1
A corrugated rail section observed on a Chinese high-speed line in 2011 (Its wavelength of about 80 mm can be estimated from the sleeper span of 0.65 m. The running speed on this section is about 300 km/h, currently the maximum commercial speed in China)
It is well known that corrugation should be a consequence of the accumulation of irregular wear and/or irregular plastic deformation (i.e., the material damage mechanism). Such irregular material damage is likely to be caused by certain eigen-modes of the vehicle-track system excited by some imperfections in the system (i.e., the wavelength fixing mechanism), and the unstable wheel-rail rolling contact resulting from those eigen-modes is the immediate cause. Considering different material damage and wavelength fixing mechanisms, Grassie and Kalousek () and Grassie () classified corrugation into six different categories based on engineering experience, which significantly increased the understanding of corrugation.
The corrugation mechanisms proposed by Grassie and Kalousek (), Grassie (), and Knothe and Groβ-Thebing () imply that the key to understanding and predicting corrugation initiation and development is to solve the dynamic vehicle-track interaction and the transient wheel-rail rolling contact. The present work employs a 3D transient rolling contact finite element (FE) model to solve the high-speed wheel-rail rolling contact and the vehicle-track interaction on a corrugated rail in the time domain. This FE model is valid for a rolling speed of up to 500 km/h. A Chinese high-speed railway system is considered. Emphasis of analysis is placed on detailed contact solutions. On the basis of numerical results and field measurements, a better understanding of the mechanism of corrugation development is achieved.
Traditionally, the vehicle-track interaction on corrugation sites was treated without detailed wheel-rail contact modeling. A simplified Hertz spring was usually employed to represent the contact, together with beams for rails and lumped masses for wheels, respectively (Knothe and Grassie, ; Hiensch et al., ; Wu and Thompson, ; Knothe and Groβ-thebing, ; Nielsen, ; Xie and Iwnicki, ; Iwnicki et al., ; Li et al., ; Xiao et al., ; Zhai et al., ). Moreover, most traditional models considered only the normal wheel-rail interaction (Knothe and Grassie, ; Hiensch et al., ; Wu and Thompson, ; Nielsen, ; Xie and Iwnicki, ; Iwnicki et al., ; Li et al., ). Nevertheless, the importance of the tangential interaction can be seen from the fact that both the plastic deformation and wear of rails are dominated by the tangential contact load on many corrugation sites, such as on corrugated curves. Taking into account the tangential interaction, Clark et al. () proposed a mechanism of slip-stick vibrations to explain the occurrence of corrugation. Knothe and Groβ-Thebing () and Groβ-Thebing et al. () treated the tangential interaction by using a viscous damper to simulate the steady rolling contact, and the combination of a viscous damper and a spring for the non-steady case. Kalker ()’s creep coefficient was employed to determine the characteristics of the viscous damper for the steady rolling contact, and a frequency-dependent creep coefficient (Gross-Thebing, ) for the corresponding parameters of the non-steady case. In addition, approaches have also been developed (Iwnicki et al., ) to derive the dynamic tangential contact force from the normal contact force and the geometric and material properties of the wheel and rail (Afshari and Shabana, ).
Recent studies have shown that the structural flexibility of the wheelset has a significant influence on the vehicle-track interaction (Ripke and Knothe, ; Chaar and Berg, ), and even the rotation of the wheel might also play an important role in high frequency vehicle-track dynamics under certain conditions (Baeza et al., ). Wen et al. (), Pang and Dhanasekar (), and Pletz et al. () considered the detailed geometries of the wheel and rail to take their flexibility into account, but only the normal wheel-rail interaction was solved for cases at joints or in crossings.
The FE modeling approach employed in this study origins from that published in (Li et al., ). This approach has been validated by Li et al. (; ) and Molodova et al. () for the high frequency vehicle-track interaction at squats (in the frequency range between a few hundred and about 2000 Hz), and by Zhao and Li () for the normal and tangential contact solutions. Li et al. () employed the FE modeling approach to study the wheel-rail rolling contact on corrugation and the resulting wear pattern at a speed of 108 km/h, for which a ballasted track was considered. However, the transient rolling contact of a wheel over a corrugated rail at high speeds has not been studied yet.
Following the same modeling approach, a 3D transient rolling contact FE model is developed for a slab track system of the Chinese high-speed line shown in Fig. 1 to study the corrugation phenomenon. The main advance of the model is that a rolling speed of up to 500 km/h can be simulated, while the maximum rolling speed considered in previous studies was 140 km/h on a ballasted track. In the model, the actual geometries of a wheelset and a rail are included by a mesh of solid elements, based on which a detailed surface-to-surface contact algorithm is employed to solve the transient rolling contact in the time domain. The flexibility of the vehicle and track subsystems and the wheel-rail continua are both included, and the rolling-sliding behavior of the wheel on the rail is simulated. The contact filter effect, which eliminates the corrugation components with a wavelength close to or less than the width of the contact patch (Knothe and Groβ-Thebing, ), is considered inherently. Hence, transient contact stresses, including both normal and tangential stresses, and their derivatives are obtained through the numerical simulation, together with the resultant contact forces. These ensure the applicability of the FE model to high-frequency vehicle-track interaction on corrugation sites. Furthermore, idealized corrugation models are applied and simulated to better understand the fundamentals of the corrugation phenomenon, even though measured corrugation profiles can be introduced. For clarity and ease of explanation, the case of a smooth rail (without any irregularities) is analyzed before considering corrugations.
## 2. Model descriptions
### 2.1.1. An overview
Fig. 2 illustrates a 3D transient rolling contact FE model developed with ANSYS/LS-DYNA, which considers a high-speed vehicle and a typical slab track on a Chinese high-speed line. The modeling approach for the vehicle is the same as that used by Li et al. () and Zhao and Li (). The track is composed of the rail, fastenings, slabs, and the mortar layer. For the investigated high-speed line, the minimum radius of curvature is 7000 m, on which the lateral movement of the wheelset is very well controlled, as predicted by multi-body simulations performed in the State Key Laboratory of Traction Power in Southwest Jiaotong University, China. Further considering that the rolled distance of the wheelset in a typical simulation in this study is less than 3.5 m, the lateral movement of the wheelset becomes negligible. Therefore, only a half wheelset and a half straight track are modeled in view of the symmetry of the system (Fig. 2b), and the lateral movement of the wheelset and the track are ignored. The simulated track is 15.2 m long and includes 24 fastenings.
Fig.2
A 3D transient FE model for wheel-rail rolling contact
(a) A schematic diagram; (b) The mesh. The slab layer is composed of pre-fabricated slabs of 6.5 m long and the gaps of 50 mm in between filled with concrete
A Lagrangian mesh of solid elements is applied to the wheelset and the rail. The minimum element size is 1.1 mm, used in the contact surface of the solution zone (BC in Fig. 2a), where irregularities such as corrugation are applied by modifying the coordinates of the nodes involved. The wheel profile is of the type LMA and the rail is the standard CN60 with an inclination of 1:40. A penalty method based surface-to-surface contact algorithm is employed to solve the wheel-rail rolling contact, in which Coulomb’s law of friction is used. A mesh of solid elements is also applied to the slabs and the mortar layer. A fastening system is simulated by 12 groups of parallel springs and dampers (three columns in the longitudinal and four rows in the lateral direction). In total, there are 1.46×106 elements and 1.29×106 nodes.
For solution, boundary conditions are applied as follows: symmetric boundary conditions are applied to the axle ends of the wheelset and to the rail ends; the bottom of the mortar layer is fixed; the fastenings, the slabs, and the mortar layer can only move vertically. Table 1 lists the values of the parameters involved. To simulate the worst case scenario, the sprung mass M c is determined by considering the weight of the loaded coach, the non-uniform distribution of the weight on different wheels, and the dynamic loads at low frequencies. A 3D right-handed Cartesian coordinate system (Oxyz) is defined, of which the origin O is located in the initial position of the contact patch center (position A in Fig. 2a, hereinafter referred to as the initial position of the wheel). The x axis is defined along the rolling direction (i.e., the longitudinal direction), and the z axis is in the vertical direction. Note that the difference between the vertical direction and the normal direction of the contact is negligible in this study since no lateral movement of the wheelset is considered. Such an FE model, developed specially for investigations into high frequency dynamics, is not suitable for low frequency vibrations such as vehicle hunting.
#### Table 1
Values of parameters involved in this study
Parameter Value Coefficient of friction, f 0.5 Lumped sprung mass, M c (kg) 8000 Wheel diameter, ϕ (m) 0.86 Wheelset mass, M w (kg) 586 Unsprung mass attached to wheelset, M a (kg) 340 Stiffness of primary suspension, K (kN/m) 880 Damping of primary suspension, C (kN·s/m) 4 Stiffness of fastenings, K f (MN/m) 22 Damping of fastenings, C f (kN·s/m) 200 Wheel & rail material Young’s modulus, E (GPa) 205.9 Poisson’s ratio, υ 0.3 Density, ρ (kg/m3) 7790 Damping constant, β (s) 1.0×10−4 Material of pre-fabricated slabs Young’s modulus, E s (GPa) 34.5 Poisson’s ratio, υ s 0.25 Density, ρ s (kg/m3) 2400 Material filled in slab gaps Young’s modulus, E g (GPa) 29.5 Poisson’s ratio, υ g 0.25 Density, ρ g (kg/m3) 2400 Mortar material Young’s modulus, E m (GPa) 8 Poisson’s ratio, υ m 0.2 Density, ρ m (kg/m3) 1600
### 2.1.2. A typical process of numerical simulation and the explicit time integration
The simulation is composed of two steps. Step 1: the static equilibrium state of the system under gravity is first solved by an implicit solver in the initial position of the wheel (position A in Fig. 2a). Step 2: an explicit solver is employed to simulate the transient rolling contact in the presence of friction, for which the displacement field obtained in Step 1 is used for stress initialization (at t=0), the predefined rotational and forward speeds of the wheelset are also applied at t=0 as initial conditions, and a specified acceleration or deceleration is further modeled by applying the time-dependent traction or braking torque to the wheel axle (M in Fig. 2b). The distance before the solution zone (i.e., AB in Fig. 2a) is designed to ensure that the wheelset achieves the steady state rolling approximately before entering the solution zone. When the wheelset passes by the solution zone, the transient results on forces, stresses, and strains are obtained. Such a process is sketched in Fig. 3. The determination of the distance AB is presented in Section 3.1. Note that the speed increase/decrease in the simulation caused by the torque M is negligible in comparison with the simulated speed because of the short time period simulated (typically 0.04 s) and, therefore, is ignored in later explanations.
Fig.3
A schematic diagram of the simulation process
A central difference method based explicit scheme is employed to treat the time integration in Step 2. A very small time step (e.g., 8.9×10−8 s for the model in this study) is required to meet the Courant stability condition of the explicit solver or to ensure the convergence of the method. Because of such a tiny time step, the high-frequency dynamic behavior of the vehicle-track system and the transient rolling contact phenomenon can be captured effectively. Due to the high computational costs, only one wheel passage is simulated at present.
### 2.1.3. Traction and creepage
The coefficient of friction (COF, f), defined by Eq. (1), is reported to be typically 0.4–0.65 for dry-clean wheel-rail contact and less than 0.3 in the presence of a thin film of water or oil (Cann, ). In this study, the COF is taken as 0.5 (Table 1) to simulate the dry-clean condition. $$f = {F_{\text{T}}}/{F_{\text{N}}}$$, where F T and F N are the tangential and the normal (vertical) contact forces, respectively, transmitted through a full-sliding contact. Obviously, the maximum tangential load that can be transmitted through a wheel-rail contact is fF N, i.e., in full-sliding contact, while a smaller load is transmitted in the case of rolling-sliding contact. Hereinafter, the tangential load actually transmitted is measured by a traction coefficient (μ) defined by $$\mu = {F_{\text{L}}}/{F_{\text{N}}} \leqslant f$$, where F L is the longitudinal component of the tangential wheel-rail contact force, being the traction force for a driving wheelset in acceleration or the braking force for a wheelset in braking. Different traction/braking loads or different friction exploitation levels are simulated by specifying the corresponding driving torque M in the model. The torque M is assumed first to increase linearly from zero to its maximum and then remain constant (Fig. 4). Note that F L is equivalent to F T in this study because the tangential plane coincides approximately with the horizontal plane and no lateral friction force is considered.
Fig.4
Variation of the driving torque M with time in the simulation
Referring to Fig. 5, the longitudinal creepage ξ corresponding to a specified traction load is calculated using $$\xi = \frac{{\omega R - v}}{{(\omega R + v)/2}}$$, where ω, R, and v are the angular speed around the axis, the radius of the wheel, and the translating speed, respectively, and ωR is the linear speed of a node in the wheel contact surface. Note that, traditionally, “creepage” is a concept defined by rigid motion, whereas elastic deformation is considered in this study. In Eq. (3) the linear speed of a node in the contact surface is taken as the linear speed of the wheel, through which the continuum vibrations excited by the contact load are considered in the calculated creepage.
Fig.5
A wheel rolls on a rail when only the longitudinal creepage exists
### 2.1.4. Material model
A linear elastic material model is used for the wheel and the rail for the following reasons: (1) for steels, known as Hookean solids, the assumption of linear elastic behavior is valid in many cases (Meyers and Chawla, ); (2) shakedown is expected for most cases because of the generation of residual stresses and work hardening, which ensures the relatively long service lives of wheels and rails; (3) even at locations where plastic deformation occurs, its magnitude must be very small in each wheel passage, leading to a continuous increase of the rail surface hardness during the first two years after installation (Olofsson and Telliskivi, ). More accurate material models, once ready and if necessary, can easily be introduced. The slab and mortar materials are also assumed to be linear elastic.
Material damping is considered by Rayleigh damping (C m): $${C_{\text{m}}} = \alpha m + \beta K$$, where α and β are the mass (m) and stiffness (K) proportional damping constants, respectively. Trial simulations confirmed that the mass proportional damping is more effective for low frequency vibrations, and also damps out rigid body motions, as stated in the keyword user’s manual of LS-DYNA. Therefore, the mass proportional damping is not suitable for this work and only the stiffness proportional damping is employed. The value of β is chosen based on the estimate made by Kazymyrovych et al. (), in which β for an FE model was determined by matching the measured stress level during testing with the level calculated using the FE method.
### 2.2. Frictional work and wear prediction
The frictional work at a point of the rail contact surface (W f) is calculated as $${W_{\text{f}}} = \int_0^t {\tau s{\text{d}}t} = \sum\limits_{i = 1}^{{n_{\text{t}}}} {{\tau _i}{s_i}\Delta {t_{\text{W}}}}$$, where τ and s are the surface shear stress and micro-slip (i.e., the relative speed between contacting particles) at the point, respectively, being functions of time. τi and si correspond to the instant iΔt W, the time step Δt W is taken as 1×10−5 s (the wheel translates 0.83 mm within Δt W at 300 km/h), and n t is the number of calculated time steps. Material wear, if assumed to be proportional to the frictional work (Clark et al., ), can directly be scaled from the frictional work. More complicated/realistic wear behavior is beyond the scope of this work.
### 2.3. Corrugation model
Fig. 6 shows an example of the 3D corrugation models. The distribution of the corrugation depth (d) is assumed to be sinusoidal in the vertical-longitudinal section and parabolic along the lateral direction, as determined by $${d_{\text{C}}} = - 0.5{d_{\text{m}}}(1 - \sin [2\pi (x - {x_{\text{s}}})/L + \pi /2])$$, $$d = {d_{\text{C}}}[1 - {(y/W)^2}]$$, where d m, L, and W are the maximum depth, wavelength, and width of the corrugation, being 0.14 mm, 80 mm, and 30 mm for corrugation D (default), respectively; d C is the maximum depth in the lateral direction and is located in the middle of the corrugation; x s is the longitudinal coordinate at the starting point of the corrugation and is equal to 2.4 m. To maximize the influence of the corrugation for better analysis, d C is applied to the position where the maximum contact pressure occurs in a case with smooth rail surfaces, i.e., in the vertical-longitudinal section of y=−2.8 mm (Fig. 17b). Note that the inclination of the rail is included in Fig. 6 and different scales are applied in Figs. 6a and 6b for clarity. The smooth case serves as a trial simulation for corrugation applications.
Fig.6
Geometry of corrugation D (default)
(a) In the vertical-longitudinal section of y=−2.8 mm (the deepest); (b) Along the lateral direction
## 3. Results of smooth contact surface
### 3.1. Dynamic relaxation
It is mentioned above that the static solution obtained by the implicit solver is employed to initialize the transient simulation. Fig. 7 shows that the vertical displacement field of the rail surface is symmetric in the vicinity of the contact patch in the static case, while it becomes asymmetric in the transient analysis due to the moving load. Hence, when the wheel rolls forward in a transient analysis, the displacement field gradually changes from the static to the transient case. Such a process inevitably introduces an oscillation (referred to later as the initial oscillation), as observed from the vertical (contact) forces plotted in Fig. 8. Note that Fig. 7 shows the results only along a longitudinal line where the maximum pressure is, i.e., the longitudinal axis of the contact patch when the contact patch is an ellipse. More studies have shown that the influence of the rolling speed on the pressure distribution is negligible in the case with a smooth rail contact surface. This is because the resultant speed at the contact point (assuming a rigid contact, the contact patch reduces to a point) is zero, being independent of the rolling speed, i.e., the instantaneous center of the wheel. Hereinafter, the longitudinal line is referred to as the longitudinal axis, although the contact patch is not necessarily an ellipse, e.g., on corrugated rails. Results shown later are all taken from the rail side.
Fig.7
Vertical displacement fields in the static and transient cases (v=300 km/h) (Only the results in the rail surface along the longitudinal axis are given and a longitudinal shift is applied to the transient solution for comparison)
Fig.8
Vertical force variations at different rolling speeds
From Fig. 8 it is seen that AB (Fig. 2a), designed for a dynamic relaxation, should be at least 2.4 m long for the system to relax to an acceptable oscillation level (10% of static load) in position B at 500 km/h. This is how AB was determined. For a speed under 300 km/h, the vertical force becomes very stable after a rolled distance of 2.4 m. Note that the abscissa of Fig. 8 is set to be the rolled distance for comparison.
Note that the displacement difference in Fig. 7 may not be the only reason behind the initial oscillation. Other factors, probably related to the initial conditions, may also play certain roles. Nevertheless, this is not discussed further because this study focuses on the contact solution after achieving an approximate rolling state.
### 3.2. Longitudinal and vertical forces
The initial increase rate of the driving torque is varied by setting T 1 (Fig. 4) at different values, from zero to sufficiently large. Fig. 9 shows the contact force results of a smooth case when T 1 is varied, for which a rolling speed of 300 km/h and a traction coefficient of 0.3 are assumed. For clarity, only representative results corresponding to three values of T 1, namely 0.0001, 0.005, and 0.01 s, are plotted. As expected, the vertical force does not change considerably with T 1 (a difference of much less than 1%). Hence, only one vertical force result scaled by the COF (i.e., fF N) is plotted in the figures to illustrate the limit of the longitudinal force.
Fig.9
Vertical and longitudinal forces as T 1 varies (μ=0.3, v=300 km/h)
The initial oscillation also influences the longitudinal (contact) forces considerably (Fig. 9). After the dynamic relaxation (t>0.028 s), the longitudinal force becomes very stable, like the vertical force, when T 1 is 0.005 or 0.01 s, but not in the case when T 1=0.0001 s. Moreover, the longitudinal force does not reach the specified value at T 1 but has a delay of about 0.005 s. This delay is the response time of the material in contact to the torque applied to the axle. Hereinafter, cases with T 1=0.005 s are used for analyses. Variations of the longitudinal force in the case with T 1=0.0001 s are not further studied here.
Contact force results at different rolling speeds and with different traction coefficients are shown in Figs. 10 and 11, respectively. A stable rolling contact is achieved after the dynamic relaxation in all cases.
Fig.10
Longitudinal forces at different rolling speeds (μ=0.3)
Fig.11
Vertical and longitudinal forces at different traction coefficients (v=300 km/h)
### 3.3. Contact stresses and frictional work
For the case of v=300 km/h and μ=0.3, distributions of the maximum contact stresses along the longitudinal axis are shown in Fig. 12 together with the corresponding frictional work. A maximum stress at a location means the maximum value reached at that location during the simulated wheel passage. The frictional work given in the figure indicates the calculated result in a longitudinal strip with a width of 1.1 mm (the width of the elements there). These results confirm that approximately steady rolling is achieved in the simulation.
Fig.12
Distributions of maximum pressure (P m), maximum surface shear stress (τ m), maximum V-M stress (max σ V-M), and frictional work along the longitudinal axis (μ=0.3, v=300 km/h, x=−0.00278 m)
As the traction coefficient becomes larger, the magnitudes of the stresses and frictional work shown in Fig. 13 all increase, as expected. Note that the Von Mises (V-M) stresses presented in this study are of the surface layer of elements (at their centers) because constant stress elements are employed. Pressure is not plotted in Fig. 13 because it is independent of the traction level.
Fig.13
Stresses and frictional work versus the traction coefficient at v=300 km/h
## 4. Corrugation
### 4.1. Measurements on a high-speed line
Corrugation occurred on a Chinese high-speed line was measured using a corrugation analysis trolley (CAT). Fig. 14a shows roughness level spectra in 1/3 octave bands measured at two sites where the running speeds were about 300 km/h. Two typical wavelengths, namely around 65 and 125 mm, are observed and the first one dominates. Note that the wavelength of 80 mm shown in Fig. 1 (a picture taken on site 2 illustrated in Fig. 14) is not obvious in the roughness level spectra. This may be because those spectra were obtained from measurements of rails of several hundred meters (about 300 and 1000 m at sites 1 and 2, respectively), in which the wavelength of the corrugation varied around 65 mm and the wavelength of 80 mm existed at only some locations. The physical essence behind such a wavelength variation is likely to be the inevitable variations of the vehicle-track parameters along the measured site. In addition, the discrete frequencies taken in the 1/3 octave bands may also cause some errors in the wavelength estimation. Fig. 14b shows five CAT measurements at site 2 conducted during a grinding cycle. It is seen that the corrugation re-occurred after grinding and gradually became stabilized in a short period.
Fig.14
Roughness level spectra in 1/3 octave bands measured on a Chinese high-speed line
(a) Typical measurements at two corrugation sites; (b) Corrugation development with time. Two measurements were conducted before and after a rail grinding on day 1
In this study, corrugation with a wavelength range of 65–95 mm is modeled to study its influence on transient wheel-rail interaction at high-speeds. All corrugation models are applied in the same way as corrugation D (with the default wavelength of 80 mm, Fig. 6), i.e., with the same location and phase in the longitudinal direction, and the same depth distribution in the lateral direction. Other corrugation models are later referred to by their wavelengths and maximum depths, without presenting their geometry in figures.
### 4.2. Transient wheel-rail interaction
The transient wheel-rail interaction on a corrugated rail is analyzed in this section, for which corrugation D is considered with v=300 km/h and μ=0.3. Fig. 15 shows variations of the contact forces together with the geometry of corrugation D, in which the vertical force is scaled by the COF (f) for comparison. As expected, both the vertical and longitudinal forces significantly fluctuate at the corrugated site. The maximum vertical force (F N) is 193.19 kN and the minimum is as low as 3.02 kN, i.e., the wheel and rail almost separate from each other. At several corrugation troughs fF N coincides with F L, i.e., full sliding occurs. The phenomenon that the force peaks do not occur exactly at the corresponding corrugation crests (referred to as the longitudinal shift hereinafter) will be discussed later.
Fig.15
Dynamic forces excited by corrugation D (v=300 km/h)
Furthermore, the vertical force is larger when the wheel is above the fastenings than in between, demonstrating the considerable influence of the discrete supports of the rail. Such a result is in line with authors’ observations that some corrugations are deeper in positions above fastenings than in between, which has also been reported in (Clayton and Allery, ; Jin et al., ). Note that the influence of the discrete supports on the longitudinal force is different from that on the vertical one. The relationships between the normal and the tangential wheel-rail interactions will be discussed later.
Pressure and surface shear stress distributions at 10 typical instants (i.e., t1–t10) are plotted in Fig. 16 to show the transient effects. As expected, the contact stresses vary greatly on the corrugated rail. Within the selected instants, the rolling contact changes from the rolling-sliding state to full sliding (at t4), and then back to rolling-sliding. It should be specified that the pressure reaches local maximums (within a wavelength) at t1 and t7, and local minimums at t4 and t10. At t1, t7, and t4 the surface shear stress also reach local maximums or minimums, but another local minimum occurs at t10′, being slightly different from t10. In a word, the typical instants are not completely the same in Figs. 16a and 16b for better illustration.
Fig.16
Transient contact stress distributions along the longitudinal axis
(a) Pressure; (b) Surface shear stress. The symbols represent nodes
It is further seen from Fig 16a that local maximums of pressure occur before the contact patch center reaches a corrugation crest. This is because the pressure is determined by both the contact geometry and the dynamic vertical force (as mentioned above, a longitudinal shift exists). Fig. 17a shows the 3D distribution of the maximum pressure in the same section as in Fig. 16. Comparing Fig. 17a with the corresponding results on the smooth rail shown in Fig. 17b, great influences of corrugation can be observed clearly.
Fig.17
3D distributions of the maximum pressure on corrugated rail (corrugation D) (a) and smooth rail (b)
Fig. 18 presents the creepage variation caused by corrugation D. From the creepage results calculated at node N cr (located in the contact patch at t=0, Fig. 5) it is seen that the maximum creepage reached on the corrugated rail is close to double the stabilized value on the smooth contact surface (before the wheel enters the corrugation, about 0.276%). Fluctuation of the creepage becomes obviously fiercer at x=2.69 m because the node N cr comes into contact again after a rolled distance of 2.69 m (a cycle), i.e., continuum (local) vibrations are excited by the contact load. To filter out the influence of the local vibrations, the calculated creepages at 28 selected nodes (distributed over the whole circumference of the wheel) are averaged and also plotted in Fig. 18. Little further explanation of the creepage results is given hereinafter because: (1) the transient contact stresses and the resulting irregular frictional work and/or irregular V-M stress, not the irregular creepage, are the immediate causes of corrugation development; (2) according to rolling contact theories, higher creepage usually means larger tangential contact force, higher frictional work, and larger V-M stress, which is valid for cases simulated in this study.
Fig.18
The creepage variation caused by corrugation D
Fig. 19 shows distributions of the maximum pressure, maximum surface shear stress, maximum V-M stress, and frictional work along the longitudinal axis in the corrugated section. It is observed that the stresses all vary following the pattern of corrugation, but with slightly different longitudinal phases, and the relative positions of the stress peaks with respect to the corresponding crests change slightly at different waves of corrugation. Moreover, the patterns of the maximum V-M stress and the frictional work are closer to that of the maximum surface shear stress than to that of the maximum pressure, as expected.
Fig.19
Distributions of the maximum stresses and frictional work along the longitudinal axis
### 4.3. Different wavelengths and depths
Keeping v=300 km/h and μ=0.3, the wavelength (L) and depth (d m) of corrugation are varied separately to study their influences. Considering the wavelength range reported in Section 4.1, three wavelengths, namely 65, 80, and 95 mm, are considered. Fig. 20 shows that among the simulated cases, the maximum vertical and longitudinal forces both occur in the case of L=80 mm. For L=65 mm, the longitudinal force does not follow the pattern of the simulated corrugation any more, but shows a shorter characteristic wavelength (Fig. 20b), and its magnitude is much smaller than in the other two cases. This may be explained as follows: the excitation frequency of corrugation (f ex=v/L, i.e., the passing frequency) monotonically decreases with the corrugation wavelength; for a wavelength of 80 mm, f ex is closest (among the three cases) to an eigen-frequency of the system related to corrugations, leading to the strongest response.
Fig.20
Dynamic forces caused by corrugations of different wavelengths
(a) Vertical force; (b) Longitudinal force. The depth remains constant, d m=0.14 mm
The mean value of the maximum V-M stress in the corrugated section is approximately constant for different wavelengths, while its fluctuation range is the largest at L=80 mm (Fig. 21). This is in line with the longitudinal force results in Fig. 20b. In contrast, the largest fluctuation of the frictional work occurs at L=95 mm (being 0.016, 0.023, and 0.024 J at L=65, 80, and 95 mm, respectively), because the frictional work is determined not only by the contact stresses (or the contact forces) but also by the micro-slip. Significant influences of the wavelength on the micro-slip in the corrugated section can be seen from the creepage (integration of the micro-slip over the contact patch) variations given in Fig. 22. No more detailed distributions of the maximum V-M stress and the frictional work in corrugated sections are given hereinafter, considering that their patterns are similar to that of the longitudinal force, as mentioned above.
Fig.21
Mean values (symbols) and fluctuation ranges (error bars) of the maximum V-M stress and the frictional work over the section in Fig. 19, as the wavelength varies
Fig.22
Creepage variations caused by corrugations of different wavelengths
Average of calculations at 28 nodes, d m=0.14 mm
When the corrugation depth increases, fluctuation ranges of the dynamic forces, the maximum V-M stress, and the frictional work all increase as expected (Fig. 23). Detailed variations of the dynamic forces are not illustrated here because their patterns remain constant for different depths.
Fig.23
Influences of the corrugation depth on mean values (symbols) and fluctuation ranges (error bars)
(a) Dynamic forces, over the section in Fig. 15; (b) Maximum V-M stress and frictional work, over the section in Fig. 19
### 4.4. Different traction coefficients
The traction coefficient is varied from 0.0 to 0.5 to examine its influence on the tangential wheel-rail interaction, for which corrugation D is considered and the rolling speed is kept at 300 km/h. From the dynamic forces shown in Fig. 24 it is found that the longitudinal force gradually becomes in phase with the vertical force scaled by the COF as the traction coefficient increases. This is determined by Coulomb’s law of friction employed in the model, i.e., the surface shear stress distribution approaches the scaled pressure distribution with the increase of traction coefficient. When the traction coefficient is very low, variation of the pressure distribution (i.e., variation of the upper limit of the surface shear stress distribution) has little influence on the surface shear stress distribution since the slip area in the contact patch (where the upper limit is reached) is very small. Obviously, such an influence becomes more significant as the slip area enlarges, i.e., as the traction coefficient increases. This is why the fluctuation range of the longitudinal force increases with the traction coefficient (Fig. 24). Fig. 25 presents the mean values and fluctuation ranges of the maximum V-M stress and the frictional work over the section in Fig. 19. It is seen that the mean values and the fluctuation ranges all increase with the traction coefficient, showing the same trend as the longitudinal force.
Fig.24
Dynamic forces excited by corrugation D at different traction levels. The vertical force does not vary with the traction coefficient
Fig.25
Mean values (symbols) and fluctuation ranges (error bars) of the maximum V-M stress and the frictional work at different traction levels, over the section of corrugation D in Fig. 19
### 4.5. Different rolling speeds
The dynamic forces caused by corrugation D at different rolling speeds are illustrated in Fig. 26, in which the traction coefficient remains 0.3. The maximum vertical and longitudinal forces first increase with the rolling speed when the speed is less than 300 km/h, and then decrease (i.e., the maximums at 500 km/h are lower than those at 300 km/h). This is a consequence of the following phenomena: (1) the excitation frequency of the corrugation changes with the rolling speed, leading to the strongest response at a certain speed (the same reason as that mentioned in Section 4.3); (2) among the simulated speeds, the influence of the discrete supports of the rail is the greatest at 300 km/h (Fig. 26).
Fig.26
Dynamic forces caused by corrugation D at different rolling speeds
(a) Vertical force; (b) Longitudinal force
Furthermore, the longitudinal shift of the vertical force decreases with the rolling speed. For example, as indicated in Fig. 26a, it varies from 0.55π at 100 km/h to 0.21π at 500 km/h at the first corrugation crest (a shift of 2π corresponds to a corrugation wavelength). Note that the relative positions of the force peaks with respect to the corresponding crests are not constant at different waves due to the transient effects.
Influences of the speed on the longitudinal shift of the longitudinal force are more complicated than those of the vertical force (comparing Figs. 26a and 26b). This is because the tangential interaction has its own eigen-modes, and in the meantime is limited by the normal one. As mentioned in Section 4.4, the tangential force will follow exactly the vertical force scaled by the COF in a full sliding case due to the application of Coulomb’s law of friction.
From Fig. 27 it is further seen that the fluctuation range of the maximum V-M stress caused by corrugation D is the largest at 300 km/h, while at 250 km/h the fluctuation range of the frictional work reaches its maximum. From the results in Figs. 26 and 27 it could be concluded that, for the simulated speed range, the responses of the system to corrugation D reach the maximum at a speed between 250 and 300 km/h. Moreover, Fig. 27 shows that the mean values vary considerably when the rolling speed changes from 180 to 250 km/h.
Fig.27
Mean values (symbols) and fluctuation ranges (error bars) of the maximum V-M stress and the frictional work at different rolling speeds, over the section of corrugation D in Fig. 19
### 4.6. Comparison with the multi-body approach
The vertical force obtained from the transient FE simulation is compared to that of the traditional multi-body approach (Jin et al., ; ; Xiao et al., ) in Fig. 28, for which a corrugation with a wavelength of 80 mm and a depth of d m=0.18 mm is considered at 500 km/h. It is seen that the transient FE result is significantly lower. Contact loss corresponding to the vertical force of zero is predicted by the multiple-body approach, but not by the transient FE simulation. Such a difference can be explained as follows: in the multi-body approach, the whole wheel is lumped into one mass particle, the rail is represented by Euler beams and a Hertz spring is used to model the wheel-rail contact, which significantly exaggerates the contact stiffness and assumes the contact patch is infinitesimal. In other words, the results from the transient FE model should be more reasonable and the multi-body approach overestimates the dynamic wheel-rail interaction under the excitation of corrugations. Note that the longitudinal force is not compared here because it cannot be obtained by the traditional multi-body approach.
Fig.28
Comparison between the vertical forces obtained by the traditional multi-body approach and the transient FE model at 500 km/h (L=80 mm, d m=0.18 mm)
It should be specified that with a modified Hertz spring the traditional multi-body approach may still provide accurate predictions of the contact forces caused by corrugations. Such an approach is appealing due to the low computational costs of multi-body simulations. To this end, the transient FE model developed in this study provides a potential calibration and validation tool for the suitable contact stiffness. Once calibrated and validated, more accurate contact force predictions may be realized without increasing the computational costs.
## 5. Discussion
Parameter variation analyses show that fluctuation ranges of the maximum V-M stress and the frictional work on a corrugated rail increase with the traction coefficient. This means that for the same excitation (e.g., a corrugation in this study) the irregular material response, namely the irregular plastic deformation and wear, probably becomes more severe as the friction exploitation level increases. Such results may explain why corrugations are more often observed on curves where the transmitted friction force is relatively high.
Moreover, for the simulated system, fluctuation ranges of the contact forces, the V-M stress, and the frictional work caused by a corrugation are found to reach their maximums at a speed between 250 and 300 km/h. As the wavelength varies, the maximum fluctuation ranges of the contact forces and the V-M stress all occur at the wavelength of 80 mm, whereas the frictional work fluctuation at the wavelength of 80 mm is slightly less than that at the wavelength of 95 mm, but significantly larger than that at the wavelength of 65 mm. Considering that components of the random rail roughness leading to higher dynamic responses than others may gradually develop, the above-mentioned results seem to explain why a corrugation with a wavelength of about 80 mm occurred in the rail section shown in Fig. 1 where the running speed was about 300 km/h.
The simulations also show that the longitudinal force variation excited by the corrugation with a wavelength of 65 mm becomes different from the corrugation in pattern and has a small magnitude, demonstrating the low possibility of occurrence of such a corrugation and of those with shorter wavelengths. This is in agreement with observations that a lower bound always exists for the corrugation on a track. Note that values of the system parameters, such as the wheel diameter and stiffness of the rail fastenings, are all nominal and kept constant in simulations, for which new wheels, new rails, and many designed values are considered. In reality, however, the wheels and rails are constantly worn and regularly re-profiled or ground, and track characteristics vary from section to section and with time. These factors should be born in mind when interpreting the numerical results. For example, the above-shown results suggest that the corrugation wavelength on the monitored Chinese high-speed line should be around 80 mm, while according to the CAT measurement it is around 65 mm (Section 4.1). In other words, the nominal values of the parameters may represent the track section shown in Fig. 1, but not every section of the monitored track, which should be studied further in the future.
Results in Section 3 show that the calculated material response would be regular along the rail if no corrugation was applied. This means that the initiation mechanism of corrugation (growing up from smooth rails) is not included in the FE model. The irregular material response corresponding to the results in Section 4 is the consequence of the existing corrugation or caused by the geometric variation at the corrugation. So, what is the relationship between the initiation mechanism (the wavelength-fixing mechanism) and the existing corrugation’s consequence? The authors here propose a theory, explained in the following paragraph, to answer this question.
The irregular material response caused by the initiation mechanism, hereinafter referred to as the irregular response I, should be larger at corrugation valleys and smaller at crests, i.e., leading to the occurrence of a corrugation. This is the dominant mechanism in the relatively early stages of a corrugation. Once the corrugation comes into being, another irregular material response also starts to act due to the geometric variation at the corrugation, referred hereinafter to as the irregular response II. The irregular response II is larger at crests than at valleys (Fig. 19), being very different from the irregular response I, especially at high speeds (smaller longitudinal shifts at higher speeds, Fig. 26a). Obviously, the irregular response II will alleviate the irregular response I due to their different patterns. Further considering that the irregular response II becomes more irregular with the corrugation depth (Fig. 23), at a certain depth the combined material response (irregular response I+II) starts to become regular, or the initiation mechanism and the consequence of the existing corrugation become balanced, i.e., the corrugation stabilizes. The phenomenon of corrugation stabilization has been observed in the field, e.g., the high-speed corrugations shown in Fig. 14b. Fig. 29 shows a schematic diagram to help understand the mechanism explained above. Note that other factors such as material work hardening and residual stresses could also play certain roles in the stabilization of corrugation.
Fig.29
A schematic diagram of corrugation occurrence
By changing the driving torque definition, braking cases can also be simulated by the 3D transient FE model, although only traction cases are presented in this paper due to space limitations. To authors’ experience, for the same friction exploitation level, the results of the braking and the traction cases indeed show some differences in contact stresses. In the future work, the 3D transient FE model can be employed to further study the material damage mechanisms on corrugated rails. Finally, it might be worthwhile to note that all discussions are mostly based on the numerical results shown above, in which influences of the lateral movement of the wheelset and the initiation mechanism of corrugation are not included.
## 6. Summary and conclusions
On the basis of field measurements and observations of (short-pitch) corrugations occurred on a recently opened Chinese high-speed line, a 3D transient rolling contact FE model is developed by an explicit FE approach to analyze the high-speed vehicle-track interaction in the presence of rail corrugations. The vehicle and the track subsystems are considered to ensure that the vehicle-track interaction is solved accurately in both the vertical and longitudinal directions. Detailed contact solutions on corrugated rails, including both the normal and the tangential solutions, are examined at different traction levels and at rolling speeds of up to 500 km/h. A summary of the results shown above and some conclusions are as follows.
1. The vertical and longitudinal (contact) forces, the pressure, and the surface shear stress all vary following the pattern of the corrugation, but with slightly different longitudinal phases. The patterns of the V-M stress and the frictional work are closer to that of the surface shear stress than to that of the pressure.
2. The discrete supports of the rail have considerable influences on the vehicle-track interaction on the corrugated rail at certain rolling speeds.
3. At certain friction exploitation levels, the state of rolling contact may oscillate between rolling-sliding and full sliding at the passing frequency of corrugation, leading to a significantly higher creepage than on smooth rails.
4. Fluctuation ranges of the V-M stress and the frictional work caused by a corrugation increase with the traction coefficient. This may explain why corrugation is more often observed on curves where the transmitted friction force is relatively high.
5. According to simulations by the nominal parameters, the wavelength of the corrugation occurring on the monitored Chinese high-speed line is most probably around 80 mm and a speed between 250 and 300 km/h is most detrimental, corresponding well to the corrugation shown in Fig. 1. Inevitable variations of the parameters along the track and their changes from the nominal values probably explain why the dominant corrugation wavelength found by CAT measurements is about 65 mm on sections where the running speed is about 300 km/h.
6. The traditional multi-body approach overestimates the dynamic wheel-rail interaction on corrugated rails, whereas results from the transient FE model should be more reasonable.
7. A theory is proposed to explain the observed phenomenon that the corrugation gradually stabilizes.
* Project supported by the National Natural Science Foundation of China (Nos. U1134202, 51275430, and 51305361), the National Basic Research Program (973) of China (No. 2011CB711103), and the Program for Changjiang Scholars and Innovative Research Team in University (Nos. IRT1178 and SWJTU12ZT01), China
## References
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[35] Zhai, W.M., Xia, H., Cai, C.B., 2013. High-speed train-track-bridge dynamic interactions-Part I: theoretical model and numerical simulation. International Journal of Rail Transportation, 1(1-2):3-24.
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[37] Zhou, L., Shen, Z.Y., 2013. Dynamic analysis of a high-speed train operating on a curved track with failed fasteners. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 14(6):447-458. | 2022-12-05 13:41:03 | {"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": 7, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5407381057739258, "perplexity": 1469.5336938130592}, "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/1669446711017.45/warc/CC-MAIN-20221205132617-20221205162617-00455.warc.gz"} |
https://www.gamedev.net/forums/topic/691616-rotating-source-vector-to-destination-vector-2d/ | # Rotating source vector to destination vector (2D)
This topic is 791 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I am trying to rotate a vector to a goal vector.
My main problem now is that I can't wrap my head on is which way to rotate the vector, so finding the "shortest" way to rotate (clockwise or counter clockwise).
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The dot product of the target vector and the normal of the current vector should result in a positive or negative value. This result will determine whether a counter–clockwise or clockwise rotation should be made.
See attached.
Current direction vector A and its normal, A'
And sample target directions B and C:
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Edit, its 2d, what I wrote was 3d.
Edited by ErnieDingo
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If you use complex numbers to represent points and vectors, so you think of (x,y) as the number x+y*i, you can think of rotations as unit-length complex numbers, like cos(alpha)+sin(alpha)*i. Applying a rotation is now just complex-number multiplication. In that language, the rotation that maps a current vector to a target vector is simply z = target / current.
This might not be very satisfying because there is no notion of "shortest". The way mathematicians think of rotations, they are not things that happen progressively over time or anything like that, but just mappings that take an input vector and produce an output vector.
If you want to recover a notion of "shortest", perhaps you are interested in limiting how much the current vector can change in one step. This code would do that:
Complex limited_rotation(Complex current, Complex target, Complex max_rotation) {
Complex z = target / current;
// You may want to normalize z here if you can't guarantee that current and target have the same length
if (z.real() > max_rotation.real())
return z;
if (z.imag() > 0)
return max_rotation;
return conj(max_rotation);
}
Edited by alvaro
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• 11 | 2019-10-21 02:09:03 | {"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": 0.3652454912662506, "perplexity": 1810.8696688629084}, "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-2019-43/segments/1570987751039.81/warc/CC-MAIN-20191021020335-20191021043835-00286.warc.gz"} |
https://www.researchgate.net/publication/200796795_Biomechanical_Properties_of_Fascial_Tissues_and_Their_Role_as_Pain_Generators | ArticlePDF Available
# Biomechanical Properties of Fascial Tissues and Their Role as Pain Generators
Authors:
## Abstract
While fasciae have virtually been treated as the “Cinderella tissue of orthopedic research” during recent decades, new methodological findings and hypotheses suggest that the body-wide fascial network may play a more important role in musculoskeletal medicine than assumed ordinarily. However, there is a great diversity in literature, as to which tissues are included under the term “fascia,” be it the superficial fascia, the endomysium, perineurium, visceral membranes, aponeuroses, retinaculae, or joint/organ capsules. Following the proposed comprehensive terminology of the 1st Fascia Research Congress, this brief review considers all collagenous connective tissues as “fascial tissues” whose morphology is dominantly shaped by tensional loading and can be seen to be a part of an interconnected tensional network throughout the whole body (1). While morphological differences between aponeuroses and lattice-like or irregular fasciae can still be properly described with this terminology, it allows one to see tissue specifications, such as septae, capsules, or ligaments, as local adaptations of this ubiquitous network based on specific loading histories. What are the biomechanical functions of this fascial network, and what role do they play in musculoskeletal dysfunctions? This brief review will highlight the load-bearing function of different fascial tissues and also their proneness to microtearing during physiological or excessive loading. It will review histological studies, indicating a proprioceptive as well as nociceptive innervation of fascia. Finally, the potential role of injury, inflammation, and/or neural sensitization of the posterior layer of the human lumbar fascia in nonspecific low back pain will be explored.
OTHER SOFT TISSUE DISORDERS
Biomechanical Properties of Fascial Tissues and Their Role
as Pain Generators
Robert Schleip
Werner Klingler
ABSTRACT. Objectives: While fasciae have virtually been treated as the “Cinderella tissue of
orthopedic research” during recent decades, new methodological findings and hypotheses suggest
that the body-wide fascial network may play a more important role in musculoskeletal medicine
than assumed ordinarily. However, there is a great diversity in literature, as to which tissues
are included under the term “fascia,” be it the superficial fascia, the endomysium, perineurium,
visceral membranes, aponeuroses, retinaculae, or joint/organ capsules. Following the proposed
comprehensive terminology of the 1st Fascia Research Congress, this brief review considers all
collagenous connective tissues as “fascial tissues” whose morphology is dominantly shaped by
tensional loading and can be seen to be a part of an interconnected tensional network throughout
the whole body (1). While morphological differences between aponeuroses and lattice-like or
irregular fasciae can still be properly described with this terminology, it allows one to see tissue
specifications, such as septae, capsules, or ligaments, as local adaptations of this ubiquitous network
What are the biomechanical functions of this fascial network, and what role do they play
in musculoskeletal dysfunctions? This brief review will highlight the load-bearing function of
different fascial tissues and also their proneness to microtearing during physiological or excessive
loading. It will review histological studies, indicating a proprioceptive as well as nociceptive
innervation of fascia. Finally, the potential role of injury, inflammation, and/or neural sensitization
of the posterior layer of the human lumbar fascia in nonspecific low back pain will be explored.
Findings: While the tensional load-bearing function of tendons and ligaments has never been
disputed, recent publication o Huijing (2) has revealed that muscles also transmit a significant
portion of their force via their epimysia to laterally positioned tissues, such as synergistic or
antagonistic muscles. The reported contribution of M. transversus abdominis to dynamic lumbar
spinal stability has been associated with the load- bearing function of lumbar fasciae’s middle
layer in humans (3). Similarly, electromyography-based measurements of the “flexion–relaxation
phenomenon” suggest a strong tensional load-bearing function of dorsal fascial tissues during
Robert Schleip and Adjo Zorn, Fascia Research Project, Institute of Applied Physiology, Ulm University, Ulm,
Germany.
Werner Klingler, Department of Anesthesiology, Ulm University, Germany.
Journal of Musculoskeletal Pain, Vol. 18(4), 2010
Available online at www.informaworld.com/MUP
doi: 10.3109/10582452.2010.502628 393
394 JOURNAL OF MUSCULOSKELETAL PAIN
healthy forward bending of the human trunk [with a reported absence of such load shifting in low
back pain patients] (4).
Recent ultrasound-based measurements indicate that fascial tissues are commonly used as elastic
springs [catapult action] during oscillatory movements, such as walking, hopping, or running, in
which the supporting skeletal muscles contract rather isometrically (5).
Fascial tissues are prone to viscoelastic deformations, such as creep, hysteresis, and relaxation.
Such temporary deformations alter fascial stiffness, which may take several hours for complete
recovery. Load-bearing tests also reveal the existence of a gradual transition zone between re-
versible viscoelastic deformation and complete tissue tearing. Various degrees of microtearing of
collagenous fibers and their interconnections have been documented to occur within this zone (6).
Fascia is densely innervated by myelinated nerve endings that are assumed to serve a propri-
oceptive function. These include Pacini’s [and paciniform] corpuscles, Golgi tendon organs, and
Ruffini endings (7). In addition, they are innervated by free endings. Newer histological examina-
tions have shown that at least some of these free nerve endings are substance P-containing receptors
that are commonly assumed to be nociceptive (8). Delayed onset muscle soreness can be induced
by repetitive eccentric contraction. A recent experimental study suggests that the epimysial fascia
plays a major role in the generation of related pain symptoms (9).
Panjabi’s (10) new explanatory model of low back pain injuries suggests that single trauma or
cumulative microtrauma causes subfailure injuries of dorsal fascial tissues and their embedded
mechanoreceptors, thereby leading to corrupted mechanoreceptor feedback and further resulting
in connective tissue alterations and neural adaptations. Langevin (11) reports that the posterior
layer of lumbar fascia tends to be thicker in chronic low back pain patients and also expresses
less shear motion during passive trunk flexion. In addition, our group has shown a high density
of myofibroblasts, whose existence is usually associated with excessive loading or injury repair in
the same fascial layer (12). Surgical examinations by Bednar et al. (13) and Dittrich (14) report
frequent signs of injury and inflammation of the lumbar fascia in low back pain patients. Finally,
injection of an inflammatory agent into the rat’s lumbar back muscles resulted in a dramatic
increase of the proportion of dorsal horn neurons with input from the superficial lumbar fascia
(15).
can induce temporary viscoelastic deformation and even microtearing. The innervation of fascia
indicates a potential nociceptive function. Microtearing and/or inflammation of fascia can be a
direct source of musculoskeletal pain. In addition, fascia may be an indirect source of back pain due
to sensitization of fascial nerve endings associated with inflammatory processes in other tissues
within the same segment.
KEYWORDS. Myofibroblasts, fascial tonicity, delayed onset muscle soreness [DOMS], fascial
innervation, microtearing
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Submitted: April 10, 2010
Revision Accepted: April 13, 2010
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... The present results support the hypothesis that the biomechanics may be reflected in the fascia morphology, as previously also referred to by Schleip et al. (2010). All of the previous references (Alexander & Bennet-Clark, 1977;Alexander, 2002;Kielty et al. 2002;Schleip et al. 2010) lend weight to our suggestion that criss-crossing of the collagen fibers in the third layer of the dermis and SF enables a mechanism of elastic recoil with superior capabilities. This mechanism transforms, stores and reduces the energy consumption during locomotion as also described by Clayton (2016), and is beneficial for endurance in terms of long and repeated periods of physical activity in horses. ...
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... Fig. 2 depicts some of these direct physiological pathways, knowledge of which is expanding with ongoing research. For example, Barsotti et al. (2021) detailed how the stress response affects "the function of the fibroblasts and myofibroblasts that reside throughout the body and more specifically in the fascia, a ubiquitous and multi-functional connective tissue", which affects various aspects of musculoskeletal functioning and related pain (Ajimsha et al., 2020;Schleip et al., 2010). A recent EU-OSHA review of evidence-based conceptual models of the relationship between psychosocial hazards and MSD risk concluded that an ideal model should depict "the important double pathway between psychosocial factors and MSDs (both directly and through affecting physical strain)" (pp. ...
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... Fascia as a part of the musculoskeletal (MSK) system has the potential to cause pain, dysfunction and disorders [29][30][31][32][33][34] . A multitude of techniques have been created & utilized by clinicians over the last century targeting this structure [35][36][37][38][39][40] . ...
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... The increased viscosity of hyaluronan leads to the formation of adhesions and the generation of tensional forces. The adhesions alter the activation of mechanoreceptors, leads to the non-physiologic movement of the joints, resulting in pain and dysfunctions [28][29][30][31]. ...
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At a certain position of trunk flexion, there is a sudden onset of electrical silence in back muscles. This is called "flexion-relaxation (F-R) phenomenon." The goals of this study were (1) to evaluate the relationship between flexion angle and activity of back muscles during flexion movement and (2) to determine what the difference is between healthy subjects and patients with chronic low back pain (CLBP). Twenty-five healthy subjects (13 males and 12 females; average age, 28.3 yr) and 20 patients with CLBP (12 males and 8 females; average age, 34.1 yr) volunteered for this study. The subjects were asked to flex forward maximally from the erect position and to maintain full flexion, followed by returning to the initial upright position. Flexion angle of trunk and hip was measured during the examination. Electromyographic activity of erector spinae was also monitored simultaneously. F-R phenomenon was observed in all healthy subjects before reaching the maximum flexion. Electrical silence continued even after extending the trunk began. In contrast, no patients with CLBP demonstrated F-R phenomenon. A significant difference in muscular activities of erector spinae between the groups was obtained when returning to the erect position from the maximum flexion. Moreover, time lag between trunk and hip movement was much greater in patients than in healthy subjects. This study demonstrated that neuromuscular coordination between trunk and hip could be abnormal in patients with CLBP.
Article
FUKUNAGA, T., Y. KAWAKAMI, K. KUBO, and H. KANEHISA. Muscle and tendon interactions during human movements. Exerc. Sport Sci. Rev., Vol. 30, No. 3, pp. 106–110, 2002. Muscle and tendon interaction was estimated in vivo by real-time ultrasonography. Differences between muscles in internal muscle-fiber shortening during isometric actions are due to the elastic properties of tendon. Compliant human tendons allow muscles to contract isometrically during many human movements for efficient force generation. | 2022-09-30 09:04:05 | {"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": 0.3548690378665924, "perplexity": 12436.830351511826}, "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-2022-40/segments/1664030335448.34/warc/CC-MAIN-20220930082656-20220930112656-00265.warc.gz"} |
https://motls.blogspot.com/2005/10/powell-for-harvard-corporation.html?m=0?m=1 | ## Sunday, October 09, 2005 ... //
### Powell for Harvard Corporation
There are some signs that my hottest candidate to replace Conrad Harper in the Harvard Corporation has a nonzero chance to be elected. During the last FAS faculty meeting, some people argued that the new member of the corporation should be tightly connected with the academic world - which may have been an indirect criticism against Colin Powell, proving that the discussion about him is serious.
Harvard should fight almost as much as possible to get him for this position. Is there some evidence that Colin Powell is suited for the administrative positions in the academic world? Sure. He is also being considered to become the twelfth president of Cornell University. How much is the Kremlin on the Charles able and willing to fight back?
#### snail feedback (0) :
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http://onsnetwork.org/chartgerink/2017/02/08/interview-danish-psychology-association-responses/ | # Interview Danish Psychology Association responses
Below, I copy my responses to an interview for the Danish Psychology Association. My responses are in italic. I don’t know when the article will be shared, but I am posting my responses here, licensed CC0. This is also my way of sharing the full responses, which won’t be copied verbatim into an article because they are simply too lengthy.
***What do you envision that this kind of technology could do in a foreseable future?What do you mean by “this” kind of technology? If you mean computerized tools assisting scholars, I think there is massive potential in both development of new tools to extract information (for example what ContentMine is doing) and in application. Some formidable means are already here. For example, how much time do you spend as a scholar to produce your manuscript when you want to submit it? This does not need to cost half a day when there are highly advanced, modern submission managers. Same when submitting revisions. Additionally, annotating documents colloboratively on the Internet with hypothes.is is great fun, highly educational, and productive. I could go on and on about the potential of computerized tools for scholars.
Why do you think this kind of computerized statistical policing is necessary in the field of psychology and in science in general?
Again, what is “this kind of computerized statistical policing”? I assume you’re talking about statcheck only for the rest of my answer. Moreover, it is not policing — a spell-checker does not police your grammar, it helps you improve your grammar. statcheck does not police your reporting, it helps you improve your reporting. Additionaly, I would like to reverse the question: should science not care about the precision of scientific results? With all the rhetoric going on in the USA about ‘alternative facts’, I think it highlights how dangerous it is to let go of our desire to be precise in what we do. Science’s inprecision has trickle down effects in the policies that are subsequently put in place, for example. We put in all kinds of creative and financial effort to progress our society, why should we let it be diminished by simple mistakes that can be prevented so easily? If we agree that science has to be precise in the evidence it presents, we need to take steps to make sure it is. Making a mistake is not a problem, it is all about how you deal with it.
So far the Statcheck tool is only checking if the math behind the statistical calculations in the published articles are wrong when the null-hypothesis significance testing has been used. What you refer to as reporting errors in your article from December last year published in Behaviour Research Methods. But these findings aren’t problematic as long as the conclusions in the articles aren’t affected by the reporting errors?
They aren’t problematic?—who is the judge of whether errors aren’t problematic? If you consider just statistical significance, there are still 1/8 papers that contain such a problem. Moreover, all errors in reported results affect meta-analyses — is that not also problematic down-the-line? I find it showing of hubris for any individual to say that they can determine whether something is problematic or not, when there can be many things that that person doesn’t realize even can be affected. It should be open to discussion, so information about problems need to be shared and discussed. This is exactly what I aimed to do with the statcheck reports on PubPeer for a very specific problem.
In the article in Behaviour Research Methods you find that half of all published psychology papers that use NHST contained at least one p-value that was inconsistent with its test statistic and degrees of freedom. And that One in eight papers contained a grossly inconsistent p-value that may have affected the statistical conclusion. What does this mean? I’m not a mathematician.
You don’t need to be a mathematician to understand this. Say we have a set of eight research articles presenting statistical results with certain conclusions. Four of those eight will contain a result that does not match up to the results presented (i.e., inconsistent), but does not affect the broad strokes of the conclusion. One of those eight contains a result that does not match up to the conclusion and potentially nullifies the conclusions. For example, if a study contains a result that does not match up with the conclusion, but concluded that a new behavioral therapy is effective at treating depression. That means the evidence for the therapy effectiveness is undermined — affecting direct clinical benefits as a result.
Why are these findings important?
Science is vital to our society. Science is based on empirical evidence. Hence, it is vital to our society that empirical evidence is precise and not distorted by preventable or remediable mistakes. Researchers make mistakes, no big deal. People like to believe scientists are more objective and more precise than other humans — but we’re not. The way we build checks- and balances to prevent mistakes from proliferating and propagating into (for example) policy is crucial. statcheck contributes to understanding and correcting one specific aspect of such mistakes we can all make.
Why did you decide to run the statcheck on psychology papers specifically?
statcheck was designed to extract statistical results reported as prescribed by the American Psychological Association. It is one of the most standardized ways of reporting statistical results. It makes sense to apply software developed on standards in psychology to psychology.
Why do you find so many statistical errors in psychology papers specifically?
I don’t think this is a problem to psychology specifically, but more a problem of how empirical evidence is reported and how manuscripts are written.
Are psychologists not as skilled at doing statistical calculations as other scholars?
I don’t think psychologists are worse at doing statistical calculations. I think point-and-click software has made it easy for scholars to compute statistical results, but not to insert them into manuscripts reliably. Typing in those results is error prone. I make mistakes when I’m doing my finances at home, because I have to copy the numbers. I wish I had something like statcheck for my finances. But I don’t. For scientific results, I promote writing manuscripts dynamically. This means that you no longer type in the results manually, but inject the code that contains the result. This is already possible with tools such as Rmarkdown and can greatly increase the productivity of the researcher. It has saved my skin multiple times, although you still have to be vigilant for mistakes (wrong code produces wrong results).
Have you run the Statcheck tool on your own statistical NHST-testing in the mentioned article?
Yes! This was the first thing I did, way before I was running it on other papers. Moreover, I was non-selective when I started scanning other people’s papers — I apparently even made a statcheck report that got posted on PubPeer for my supervisor (see here). He laughed, because the paper was on reporting inconsistencies and the gross inconsistency was simply an example of one in the running text. A false positive, highlighting that statcheck‘s results always need to be checked by a human before concluding anything definitive.
Critics call Statcheck “a new form of harassment” and accuse you of being “a self appointed data police”. Can you understand these reactions?
Proponents of statcheck praise it as a good service. Researchers who study how researchers conduct research are called methodological terrorists. Any change comes with proponents and critics. Am I a self-appointed data policer? To some, maybe. To others, I am simply providing a service. I don’t chase individuals and I am not interested in that at all — I do not see myself as part of a “data police”. That people think these reports is like getting reprimanded highlights to me that there still rests a taboo on skepticism within science. Skepticism is one of the ideals of science, so let’s aim for that.
Why do you find it necessary to send out thousands of emails to scholars around the world informing them that their work has been reviewed and point out to them if they have miscalculated?
It was not necessary — I thought it was worthwhile. Why do some scholars find it necessary to e-mail a colleague about their thoughts on a paper? Because they think it is worthwhile and can help them or the original authors. Exactly my intentions by teaming up with PubPeer and posting those 50,000 statcheck reports.
Isn’t it necessary and important for ethical reasons to be able to make a distinction between deliberate miscalculations and miscalculations by mistake when you do this kind of statcheck?
If I was making accusations about gross incompetence towards the original authors, such a distinction would clearly be needed. But I did not make accusations at all. I simply stated the information available, without any normative or judging statements. Mass-scale post-publication peer review of course brings with it ethical problems, which I carefully weighed before I started posting statcheck reports with the PubPeer team. The formulation of these reports was discussed within our group and we all agreed this was worthwhile to do.
As a journalist I can write and publish an article with one or two factual errors. This doesn’t mean the article isn’t of a general high journalistic standard or that the content of the article isn’t of great relevance for the public- couldn’t you make the same argument about a scientific article? And when you catalogue these errors online you are at the risk of blowing up a storm in a tea cup and turn everybody’s eyes away from the actual scientific findings?
Journalists and scholars are playing different games. An offside in football is not a problem in tennis and the comparison between journalists and scholars seems similar to me. I am not saying that an article is worthless if it contains an inconsistency, I just say that it is worth looking at before building new research lines on it. Psychology has wasted millions and millions of euros/dollars/pounds/etc on chasing ephemeral effects that are totally unreasonable, as several replication projects have highlighted in the last years. Moreover, I think the general opinion of science will only improve if we are more skeptical and critical of each other instead of trusting findings based on reputation, historical precedent, or ease with which we can assimilate the findings. | 2019-02-18 23:29:22 | {"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": 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": 0.3649708926677704, "perplexity": 1229.1374806649444}, "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-2019-09/segments/1550247488490.40/warc/CC-MAIN-20190218220415-20190219002415-00401.warc.gz"} |
https://zbmath.org/?q=an%3A1329.76266 | zbMATH — the first resource for mathematics
On velocity and reactive scalar spectra in turbulent premixed flames. (English) Zbl 1329.76266
Summary: Kinetic energy and reactive scalar spectra in turbulent premixed flames are studied from compressible three-dimensional direct numerical simulations (DNS) of a temporally evolving rectangular slot-jet premixed flame, a statistically one-dimensional configuration. The flames correspond to a lean premixed hydrogen-air mixture at an equivalence ratio of 0.7, preheated to 700 K and at 1 atm, and three DNS are considered with a fixed jet Reynolds number of 10000 and a jet Damköhler number varying between 0.13 and 0.54. For the study of spectra, motivated by the need to account for density change, which can be locally strong in premixed flames, a new density-weighted definition for two-point velocity/scalar correlations is proposed. The density-weighted two-point correlation tensor retains the essential properties of its constant-density (incompressible) counterpart and recovers the density-weighted Reynolds stress tensor in the limit of zero separation. The density weighting also allows the derivation of balance equations for velocity and scalar spectrum functions in the wavenumber space that illuminate physics unique to combusting flows. Pressure-dilatation correlation is a source of kinetic energy at high wavenumbers and, analogously, reaction rate-scalar fluctuation correlation is a high-wavenumber source of scalar energy. These results are verified by the spectra constructed from the DNS data. The kinetic energy spectra show a distinct inertial range with a $$-5/3$$ scaling followed by a ’diffusive-reactive’ range at higher wavenumbers. The exponential drop-off in this range shows a distinct inflection in the vicinity of the wavenumber corresponding to a laminar flame thickness, $$\delta_L$$, and this is attributed to the contribution from the pressure-dilatation term in the energy balance in wavenumber space. Likewise, a clear spike in spectra of major reactant species (hydrogen) arising from the reaction-rate term is observed at wavenumbers close to $$\delta_L$$. It appears that in the inertial range classical scaling laws for the spectra involving the Kolmogorov scale are applicable, but in the high-wavenumber range where chemical reactions have a strong signature the laminar flame thickness produces a better collapse. It is suggested that a full scaling should perhaps involve the Kolmogorov scale, laminar flame thickness, Damköhler number and Karlovitz number.
MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010) 76Fxx Turbulence 76V05 Reaction effects in flows 80A25 Combustion
Full Text:
References:
[1] DOI: 10.1016/0016-0032(52)90903-4 [2] DOI: 10.1017/S0022112093000230 [3] DOI: 10.1016/0010-2180(95)00036-6 [4] DOI: 10.1063/1.857971 [5] DOI: 10.1063/1.4813811 · Zbl 06480072 [6] DOI: 10.1080/13647830600898995 · Zbl 1121.80342 [7] DOI: 10.1017/S0022112089002697 [8] DOI: 10.1063/1.3371715 · Zbl 1190.76130 [9] DOI: 10.1146/annurev.fluid.32.1.203 · Zbl 0988.76042 [10] DOI: 10.1016/j.combustflame.2006.09.005 [11] DOI: 10.1017/S0022112007009330 · Zbl 1131.76057 [12] DOI: 10.1017/S0022112008005636 · Zbl 1171.76458 [13] DOI: 10.1016/S0168-9274(99)00141-5 · Zbl 0986.76060 [14] DOI: 10.1016/0168-9274(94)00004-2 · Zbl 0804.76062 [15] DOI: 10.1017/S0022112002008650 · Zbl 1009.76503 [16] DOI: 10.1146/annurev.fluid.36.050802.122015 · Zbl 1117.76029 [17] DOI: 10.1017/S0022112061000615 · Zbl 0111.39205 [18] DOI: 10.1016/j.combustflame.2012.01.005 [19] DOI: 10.1088/1749-4699/2/1/015001 [20] Jones, Turbulent Reacting Flows pp 309– (1993) [21] Hinze, Turbulence (1975) [22] Chakraborty, Phys. Fluids 19 (2007) [23] DOI: 10.1016/j.combustflame.2011.11.020 [24] DOI: 10.1017/S0022112091000460 [25] DOI: 10.1016/S0010-2180(03)00184-6 [26] DOI: 10.1023/B:APPL.0000044404.24369.f1 · Zbl 1081.76550 [27] DOI: 10.1017/S0022112059000106 · Zbl 0085.39702 [28] DOI: 10.1017/S002211205900009X · Zbl 0085.39701 [29] DOI: 10.1063/1.869452 · Zbl 1185.76785 [30] DOI: 10.1063/1.2186590 · Zbl 1185.80011 [31] DOI: 10.1017/S0022112091000204 · Zbl 0721.76037 [32] DOI: 10.1002/kin.20026 [33] DOI: 10.1017/CBO9780511840531 · Zbl 0966.76002 [34] DOI: 10.1016/0021-9991(92)90046-2 · Zbl 0766.76084 [35] DOI: 10.1017/S0022112087002167 · Zbl 0633.76054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-08-01 05:24:43 | {"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": 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": 0.3436610996723175, "perplexity": 7748.412961111229}, "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-2021-31/segments/1627046154158.4/warc/CC-MAIN-20210801030158-20210801060158-00482.warc.gz"} |
https://pypi.org/project/fancycompleter/ | colorful TAB completion for Python prompt
# fancycompleter: colorful Python TAB completion
## What is is?
fancycompleter is a module to improve your experience in Python by adding TAB completion to the interactive prompt. It is an extension of the stdlib's rlcompleter module.
Its best feature is that the completions are displayed in different colors, depending on their type:
In the image above, strings are shown in green, functions in blue, integers and boolean in yellows, None in gray, types and classes in fuchsia. Everything else is plain white.
fancycompleter is compatible with Python 3. However, by default colors don't work on Python 3, see the section How do I get colors? for details.
## Other features
• To save space on screen, fancycompleter only shows the characters "after the dot". By contrast, in the example above rlcompleter shows everything prepended by "sys.".
• If we press <TAB> at the beginning of the line, a real tab character is inserted, instead of trying to complete. This is useful when typing function bodies or multi-line statements at the prompt.
• Unlike rlcompleter, fancycompleter does complete expressions containing dictionary or list indexing. For example, mydict['foo'].<TAB> works (assuming that mydict is a dictionary and that it contains the key 'foo', of course :-)).
• Starting from Python 2.6, if the completed name is a callable, rlcompleter automatically adds an open parenthesis (. This is annoying in case we do not want to really call it, so fancycompleter disable this behaviour.
## Installation
First, install the module with pip or easy_install:
$pip install fancycompleter Then, at the Python interactive prompt: >>> import fancycompleter >>> fancycompleter.interact(persist_history=True) >>> If you want to enable fancycompleter automatically at startup, you can add those two lines at the end of your PYTHONSTARTUP script. If you do not have a PYTHONSTARTUP script, the following command will create one for you in ~/python_startup.py: $ python -m fancycompleter install
On Windows, install automatically sets the PYTHONSTARTUP environment variable. On other systems, you need to add the proper command in ~/.bashrc or equivalent.
Note: depending on your particular system, interact might need to play dirty tricks in order to display colors, although everything should "just work". In particular, the call to interact should be the last line in the startup file, else the next lines might not be executed. See section What is really going on? for details.
## How do I get colors?
If you are using PyPy, you can stop reading now, as fancycompleter will work out of the box.
If you are using CPython on Linux/OSX and you installed fancycompleter with pip or easy_install, they automatically installed pyrepl as a requirement, and you should also get colors out of the box. If for some reason you don't want to use pyrepl, you should keep on reading.
By default, in CPython line input and TAB completion are handled by GNU readline (at least on Linux). However, readline explicitly strips escape sequences from the completions, so completions with colors are not displayed correctly.
There are two ways to solve it:
• (suggested) don't use readline at all and rely on pyrepl
• use a patched version of readline to allow colors
By default, fancycompleter tries to use pyrepl if it finds it. To get colors you need a recent version, >= 0.8.2.
Starting from version 0.6.1, fancycompleter works also on Windows, relying on pyreadline. At the moment of writing, the latest version of pyreadline is 2.1, which does not support colored completions; here is the pull request which adds support for them. To enable colors, you can install pyreadline from this fork using the following command:
pip install --upgrade https://github.com/antocuni/pyreadline/tarball/master
If you are using Python 3, pyrepl does not work, and thus is not installed. Your only option to get colors is to use a patched readline, as explained below.
## I really want to use readline
This method is not really recommended, but if you really want, you can use use a patched readline: you can find the patches in the misc/ directory:
You can also try one of the following precompiled versions, which has been tested on Ubuntu 10.10: remember to put them in a place where the linker can find them, e.g. by setting LD_LIBRARY_PATH:
Once it is installed, you should double-check that you can find it, e.g. by running ldd on Python's readline.so module:
\$ ldd /usr/lib/python2.6/lib-dynload/readline.so | grep readline
Finally, you need to force fancycompleter to use colors, since by default, it uses colors only with pyrepl: you can do it by placing a custom config file in ~/.fancycompleterrc.py. An example config file is here (remind that you need to put a dot in front of the filename!).
## Customization
To customize the configuration of fancycompleter, you need to put a file named .fancycompleterrc.py in your home directory. The file must contain a class named Config inheriting from DefaultConfig and overridding the desired values.
## What is really going on?
The default and preferred way to get colors is to use pyrepl. However, there is no way to tell CPython to use pyrepl instead of the built-in readline at the interactive prompt: this means that even if we install our completer inside pyrepl's readline library, the interactive prompt won't see it.
The issue is simply solved by avoiding to use the built-in prompt: instead, we use a pure Python replacement based on code.InteractiveConsole. This brings us also some niceties, such as the ability to do multi-line editing of the history.
The console is automatically run by fancycompleter.interact(), followed by sys.exit(): this way, if we execute it from the script in PYTHONSTARTUP, the interpreter exits as soon as we finish the use the prompt (e.g. by pressing CTRL-D, or by calling quit()). This way, we avoid to enter the built-in prompt and we get a behaviour which closely resembles the default one. This is why in this configuration lines after fancycompleter.interact() might not be run.
Note that if we are using readline instead of pyrepl, the trick is not needed and thus interact() will simply returns, letting the built-in prompt to show up. The same is true if we are running PyPy, as its built-in prompt is based on pyrepl anyway.
## Project details
Uploaded source
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http://blogs.mathworks.com/loren/2013/07/23/deconstructing-destructors/ | # Loren on the Art of MATLAB
## Deconstructing Destructors
I'm pleased to introduce guest blogger Jennifer Black, manager of the MATLAB object system team. Today Jennifer will be sharing some thoughts on what happens when objects are destroyed in MATLAB and how you can control aspects of this process.
### Contents
#### Why Define a Destructor
When an instance of a class is cleared, MATLAB automatically cleans up the values of the properties owned by that object. For example, if I have a Signal class with a property that stores a matrix of double values, that matrix is destroyed when a Signal is destroyed. Sometimes, however, additional actions might be needed to handle external resources.
Let’s consider the following class:
type FileReader1
classdef FileReader1 < handle
properties
FileID
end
end
In MATLAB, a file identifier is an integer returned by the fopen function. It is used by file I/O functions such as fread and fwrite to access the file. If fopen cannot open the file, it returns -1.
If I have an open FileReader named reader in my workspace, clearing reader will cause MATLAB to clear the value stored in my FileID property. Now the number is gone, but the file is still open. Clearing FileID without first closing the file means the file is left open even though it is no longer being used. If this happens enough times I might run out of system file handles.
Let's look at defining a destructor to close the file handle. A destructor is a method of a class responsible for cleaning up resources owned by objects. In MATLAB, the handle superclass is used for all kinds of objects that have a unique identity independent of their current state. Unlike numbers, matrices, etc., handle objects represent unique things that have a beginning and an end and may change internal state along the way. Any subclass of handle may define a special method named delete, often referred to as a destructor. In the case of my FileReader class, I can correct the problem of leaking file handles by implementing my own destructor:
type FileReader2
classdef FileReader2 < handle
properties
FileID
end
methods
function delete(readerObj)
fclose(readerObj.FileID);
end
end
end
#### Defining a Destructor
A MATLAB destructor takes a single input argument - the object being destroyed - and returns no outputs. The input object is always scalar even if an array of MATLAB objects goes out of scope all at once.
In a MATLAB class you can define a method named delete that is not a destructor. A delete method with a different signature is not a destructor, nor is a static delete method. For example, defining the method to take more than one input argument or to return any output arguments means the method is not treated as an object destructor.
#### Calling Destructors
An object destructor can be called either explicitly by calling the delete method, or implicitly by MATLAB, for example when a variable is cleared or goes out of scope. Let's consider the difference between these two actions. To do so I will add a constructor and a readData method to my class:
type FileReader3
classdef FileReader3 < handle
properties(SetAccess = protected)
FileID = -1;
FileName
end
methods
function reader = FileReader3(fname)
if ischar(fname)
reader.FileName = fname;
reader.FileID = fopen(fname,'r');
end
end
function colorData = readData(reader)
if reader.FileID == -1
error('No file name has been specified for this FileReader. No data will be read.');
else
colorData = fscanf(reader.FileID,'%f',[3,inf]);
colorData = colorData';
frewind(reader.FileID);
end
end
function delete(reader)
if reader.FileID ~= -1
s = sprintf('Closing %s', reader.FileName);
disp(s);
fclose(reader.FileID);
end
end
end
end
First, let's consider the case where I have one variable holding a FileReader:
myReader = FileReader3('colorData.txt');
Explicitly calling delete on myReader invokes the destructor and then destroys the FileReader. The myReader variable remains in my workspace, but its handle is no longer valid:
delete(myReader);
Closing colorData.txt
isvalid(myReader)
ans =
0
An implicit call to a destructor will occur if I clear myReader from my workspace:
myReader = FileReader3('colorData.txt');
clear myReader
Closing colorData.txt
The destructor will also be implicitly called if my variable goes out of scope, for example because the end of a function has been reached. To illustrate, let's add a helper function in which I create a FileReader:
type readDataFromFile
function colorData = readDataFromFile(filename)
myReader = FileReader3(filename);
colorData = readData(myReader);
end
When I call my new helper function, myReader is created in the function and will go out of scope when the function ends. This causes MATLAB to clear the variable, which results in an implicit call to the delete method:
colordata = readDataFromFile('colorData.txt');
Closing colorData.txt
Now, let's contrast what we have just discussed about having a single handle with the case where I have multiple handles to the same FileReader. This happens, for example, when I create a handle object and then assign that handle to another workspace variable:
reader1 = FileReader3('colorData.txt');
reader2 = reader1;
With these two handles to the same FileReader now in my workspace, I will explicitly call delete, and then use the isvalid method to see what happened to the FileReader object:
delete(reader1);
Closing colorData.txt
isvalid(reader1)
ans =
0
isvalid(reader2)
ans =
0
As expected, the reader1 handle is no longer valid, but note that reader2 is also no longer valid. Why did this happen? Because when I explicitly call delete, the destructor is called and the object is destroyed no matter how many handles there are referencing that object.
Now let's see what happens when a destructor is called implicitly. Once again let's create two handles to the same FileReader:
reader1 = FileReader3('colorData.txt');
reader2 = reader1;
But this time, rather than call the destructor explicitly, I will just clear one of the handles. As we saw earlier, this can result in an implicit call to the delete method:
clear reader1;
isvalid(reader2)
ans =
1
Why is reader2 still valid, and why was my destructor not called? Because MATLAB will only implicitly call a destructor when the last reference to an object is cleared. As reader2 still exists in my workspace, the underlying FileReader is not destroyed.
Most often a destructor is defined as a public method, but it can also be declared as a private or protected method. Making it private will prevent code outside the class from explicitly calling the destructor. Similarly, a protected destructor can only be explictly called from methods of the same class or from subclass methods. MATLAB will always be able to implicitly call a destructor, even one declared private or protected. You might choose to restrict access to your destructor in a situation where you want to prevent the accidental deletion of an object, such as when you have a singleton object.
#### Handles Contained within Other Structures
What happens when a handle is stored as a field of a struct or a cell of a cell array, or as a property of another object? When that top-level container variable is destroyed, the handle object will also be destroyed and its delete method implicitly called if and only if no other references to the object exist. Let's look at an example with structs:
s.myReader = FileReader3('colorData.txt');
clear s
Closing colorData.txt
Note that the FileReader stored in the myReader field of s has been destroyed. However, as we saw previously, another handle to the same FileReader will prevent the destruction of the object:
reader4 = FileReader3('colorData.txt');
s.myReader = reader4;
clear s
isvalid(reader4)
ans =
1
Even after clearing s, we see that reader4 is still valid. The second handle prevented the implicit destruction of the object.
#### Classes in Hierarchies
So far we have been looking at examples of stand-alone classes, but what about classes that are part of a hierarchy? In MATLAB, every class has the ability to control its own destruction behavior. In a hierarchy of classes, every class can define its own destructor. As a result, a destructor cannot be defined as a Sealed method.
When an object is destroyed, MATLAB will call the destructor of the class of the object first, if it has been defined. Next, the destructor of each superclass is called, starting with the most direct superclass.
A destructor can reference the properties of the class itself, including properties inherited from superclasses, but it generally should not reference properties of a subclass. Why? Because when a superclass destructor is executing, the destructor of its subclass has already executed, and could have invalidated the values in its properties.
#### How Do You Use Destructors?
Now that we've discussed the basics of working with destructor methods in MATLAB, I'd like to hear of your experiences. Are you already using destructors? If you have any interesting applications or questions, I'd be very happy to hear about them here.
Get the MATLAB code
Published with MATLAB® R2013a
### 5 Responses to “Deconstructing Destructors”
1. Sven replied on :
Loren, see the 4 lines of code above the text “Even after clearing s, we see that reader4 is still valid. The second handle prevented the implicit destruction of the object.”
In these lines, you make a reader, assign a (handle) copy of it, and then clear that handle copy. As you write above, “the second handle prevented the implicit destruction of the object”. HOWEVER, the output from that code chunk contains “Closing colorData.txt”, which indicates that the destruction method _was_ called. I’m confused.
Similarly, you have the two lines:
reader1 = FileReader3(‘colorData.txt’);
reader2 = reader1;
which somehow ran the destructor method because it produced the output “Closing colorData.txt”. I don’t see how making an object and then making a copy of it calls any destructors at all… is this just a garbling of the wrong MATLAB output under each chunk of code?
2. Nick replied on :
I use destructors quite a lot. As clean up actions for temporary files. However, in the rare case that Matlab crashes (sic) the destructors are not executed. This way unwanted files are kept on the system. Do you see a solution for this?
3. Jennifer replied on :
Hi Sven,
Thank you for catching the problem of incorrect output in the original posting. It turns out that we made a mistake during the publishing process. There were extra variables in the workspace, which affected the output being generated. The posting has since been updated and the output is now consistent with the descriptions.
4. Jennifer replied on :
Hi Nick,
Thanks for sharing the ways that you use destructors. You raise a good point about what happens when MATLAB crashes. Unfortunately, when MATLAB crashes we don’t today have the ability to run destructors as MATLAB can’t reliably execute the destructor code after a crash.
5. Andrew replied on :
I like to keep in mind this sentence from the delete method documentation:
”A delete method should not generate errors or create new handles to the object being destroyed.”
@Nick: Do you mean the entire Matlab application or just your running code? if it’s the latter you might look into the always useful onCleanup.
Loren Shure works on design of the MATLAB language at MathWorks. She writes here about once a week on MATLAB programming and related topics.
These postings are the author's and don't necessarily represent the opinions of MathWorks. | 2013-12-08 12:46:43 | {"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": 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": 0.3159208297729492, "perplexity": 1965.638891702904}, "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-2013-48/segments/1386163065688/warc/CC-MAIN-20131204131745-00080-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/106481/a-remark-in-jantzens-lectures-on-quantum-groups | # A remark in Jantzen's 'Lectures on Quantum Groups'
In Jantzen's AMS text 'Lectures on Quantum Groups' he makes the following remark (p.187, preface to Chapter 9):
"For general (complex semisimple f.d. Lie algebra) $\frak{g}$ we can consider for each simple root $\alpha$... a Lie subalgebra (of $\frak{g}$) isomorphic to $\frak{sl}_{2}$ (ie, the Lie subalgebra $\frak{s}_{\alpha}$ generated by suitable $X_{\alpha}\in \frak{g}_{\alpha}$, $Y_{\alpha}\in\frak{g}_{-\alpha}$). So, if $M$ is a f.d. $\frak{g}$-module, then one can find (for fixed $\alpha$) a basis $v_{1},\ldots,v_{n}$ such that $Y_{\alpha}v_{i}$ is either $0$ or a nonzero multiple of another $v_{j}$ and such that also each $X_{\alpha}v_{h}$ is either $0$ or a nonzero multiple of another $v_{l}$. However, in general, there does not exist a basis that works simultaneously for all simple $\alpha$. (There are exceptions, such as the adjoint representations...)..."
The first part of this statement is standard (we are applying complete reducibility of $M$ as an $\frak{s}_{\alpha}$-module). However, I hope that someone can illuminate the last sentence on 'exceptions' - is this a typo/mis-statement? Or am I missing something here?
It is not possible to obtain a simultaneous basis for the adjoint representation of $\frak{sl}_{3}$: indeed, if $\alpha_{1},\alpha_{2}$ are the simple roots and $\frak{s}_{1},\frak{s}_{2}$ the corresponding $\frak{sl}_{2}$-triples then we can decompose $\frak{sl}_{3}$ as
$\frak{sl}_{3}$$\cong L(1)\oplus L(2)\oplus L(0)\oplus L(1)$
when we consider $\frak{sl}_{3}$ as either a $\frak{s}_{1}$-or $\frak{s}_{2}$-module. Here, $L(n)$ is the irreducible $\frak{sl}_{2}$-module of dimension $n+1$. Also, in both decompositions we have $L(0)$ appears as a subspace of $\frak{h}$ (the $0$-weight space).
If we were to have a simultaneous basis as described above we would need a basis vector $u\in\frak{h}\subset\frak{sl}_{3}$ corresponding to the copy of $L(0)$ appearing in the $\frak{s}_{1}$- and $\frak{s}_{2}$-decompositions of $\frak{sl}_{3}$. This would imply that $\ker ad \;X_{\alpha_{1}}\cap \ker ad \; X_{\alpha_{2}}\cap \frak{h}$ is nonzero, which is impossible (as can be seen by a basic calculation) since we are in characteristic $0$.
-
To me it is not clear what the exceptions are meant to be exceptions to. So it seems best to ignore the sentence entirely. You understand perfectly what is happening. – Wilberd van der Kallen Sep 6 '12 at 9:07
Your argument in the second to last paragraph is wrong. The basis of the adjoint representation given by any basis vector in each root space, and the basis of $\mathfrak{h}$ given by the simple coroots $H_i=[E_i,F_i]$ has this property for any semi-simple Lie algebra.
The mistake you made was thinking that the basis had to be compatible with each $\mathfrak{sl}_2$ decomposition, and in particular that all but one basis vector in $\mathfrak{h}$ must be sent to zero by bracket with $E_i$ or $F_i$. This is not what Jantzen said (you're right that no basis with that property can exist except in products of $\mathfrak{sl}_2$'s); he only said that the bracket of one basis vector with $E_i$ or $F_i$ must be a multiple of another basis vector, and any basis of $\mathfrak{h}$ has that property. In fact, the only place where a basis vector from each root space and a completely arbitrary basis of $\mathfrak{h}$ will fail is that $[E_i,F_i]$ must be a multiple of a basis vector, which forces us to take the coroots.
-
Ahh, thanks for your comment. I was certain that I must have been misreading this and glad for a fresh set of eyes to confirm. Cheers – George Melvin Sep 6 '12 at 16:46
My reading of the passage is different. Take for example the adjoint representation. Here one can choose a Chevalley basis, whose properties meet Jantzen's requirements relative to each fixed simple root. In your set-up you need to keep in mind that there is no canonical direct sum decomposition of the type you indicate, but a Chevalley basis does give (up to sign choices) something unique.
One thing that makes this example (or some "natural" modules) work in Jantzen's discussion is the fact that weight spaces are 1-dimensional, apart from the zero weight space. In general, it's much trickier to fix an all-purpose basis in each separate weight space. This is what makes Lusztig's "canonical basis" ideas so striking, along with Kashiwara's development of "crystal bases" (the subject of Jantzen's Chapters 9-11.
Anyway, as Wilberd points out, the chapter introduction includes some informal language which by itself isn't so important compared to the rigorous material following.
It's usually a good idea to address a question like this to the author (when possible) and also check to see if errata are posted anywhere. For instance, there are some lists (mostly involving very minor corrections) posted at Jantzen's homepage in Aarhus: http://home.imf.au.dk/jantzen/
-
Thanks for the comment and helpful advice. Cheers – George Melvin Sep 6 '12 at 16:48 | 2016-06-29 09:28:29 | {"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": 0.8928812146186829, "perplexity": 622.3640912851313}, "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-2016-26/segments/1466783397695.90/warc/CC-MAIN-20160624154957-00011-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/199699 | ## Low-temperature Collisions between Neutral Molecules in Merged Molecular Beams
We have developed an experiment for the investigation of neutral molecular collisions in the gas phase at temperatures as low as 100 mK. These low temperatures are obtained by merging two supersonic expansions, using an electric and a magnetic guide, and by matching the velocities of the beams. Since the energy available for the collisions, or the temperature, is determined only by the relative velocity of the reaction partners this enables the study of chemical processes at very low temperatures without the need to prepare slow molecules in the laboratory frame of reference. This paper describes the method and presents results on the Ne(P-3(2))+NH3 Penning ionization.
Published in:
Chimia, 68, 4, 256-259
Year:
2014
Publisher:
Bern, Schweizerische Chemische Gesellschaft
ISSN:
0009-4293
Keywords:
Laboratories:
Note: The status of this file is: EPFL only | 2018-04-22 05:11:32 | {"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": 0.8611959218978882, "perplexity": 804.446043564641}, "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-2018-17/segments/1524125945493.69/warc/CC-MAIN-20180422041610-20180422061610-00078.warc.gz"} |
https://physics.stackexchange.com/questions/600347/why-is-g-mu-nu-fracdx-mud-lambda-fracdx-nud-lambda-a-constant-for | # Why is $g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}$ a constant for geodesics in GR?
In Sean Carroll's spacetime and geometry chapter 5 Carroll states the following
In addition we always have another constant of the motion for geodesics: the geodesic equation (together with metric compatibility) implies that the quantity$$\epsilon=-g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}\tag{5.55}$$ is constant along the path. (For any trajectory we can choose the parameter $$\lambda$$ such that $$\epsilon$$ is a constnat; we are simply noting that this is compatible with affine parameterization along a geodesic.)
I feel like I understand this now but my understanding of formal tensors is still a bit shaky so could you tell if my reasoning is valid and correct any misunderstandings?
1. From the metric components $$g_{\mu\nu}$$ we can construct the metric tensor given by $$g(x)=g_{\mu\nu}(x)dx^\mu\otimes dx^\nu$$. Since it is a tensor it is reparametrization invariant but can still vary over space.
2. Metric compatibility tells us that actually it doesn't vary over space. Since $$\nabla_\alpha g_{\mu\nu}=0$$ the metric can be parallel transported to any point in space so it is constant.
3. If $$x(\lambda)$$ is a geodesic then from the geodesic equation we have $$\nabla_{\dot x}\dot x=0$$. This means the direction of the tangent vector $$\frac{d}{d\lambda}$$ is conserved. By a suitable (reparametrization) of $$\lambda$$ we get that $$\frac{d}{d\lambda}$$ is conserved.
4. Since $$g_{\mu\nu}dx^\mu\otimes dx^\nu$$ and $$\frac{d}{d\lambda}$$ are both conserved we have that $$g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}=g\left(\frac{d}{d\lambda}\otimes \frac{d}{d\lambda}\right)=\left(g_{\mu\nu}dx^\mu\otimes dx^\nu\right)\left(\frac{d}{d\lambda}\otimes \frac{d}{d\lambda}\right)$$ is also conserved.
• I think you're a bit confused with what a tensor is. The metric g takes two vectors as arguments, not a (2,0)-tensor. Dec 13, 2020 at 14:42
• @Krup'a So would you write 4. as $g\left(\frac{d}{d\lambda},\frac{d}{d\lambda}\right)=\left(g_{\mu\nu}dx^\mu\otimes dx^\nu\right)\left(\frac{d}{d\lambda},\frac{d}{d\lambda}\right)$? So still with a tensor product between the differentials? Dec 13, 2020 at 14:46
• As a note, the metric tensor $\textbf{g}$ should be written as $\textbf{g}=g_{\mu\nu}(\textbf{e}^\mu \otimes \textbf{e}^\nu)$. Dec 13, 2020 at 15:51
You don't need any knowledge of tensor calculus to understand this.
What Does $$g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^ \nu}{d\lambda}$$ Represent?
For a curve $$x^\mu(\lambda)$$ in a manifold, the quantity $$g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^ \nu}{d\lambda}$$ is the square of the length of the tangent vector $$\textbf{t}$$ to the curve $$x^\mu(\lambda)$$ at any point $$P$$.
To see this, note that at any point $$P$$ on the curve, the tangent vector $$\textbf{t}$$ is defined as $$\textbf{t}=\frac{d\textbf{s}}{d\lambda},$$ where $$d\textbf{s}$$ is the infinitesimal separation vector between point $$P$$ and a nearby point $$Q$$ on the curve corresponding to the parameter value $$\lambda+d\lambda$$.
In a given coordinate system with basis vectors $$\textbf{e}_\mu$$, we can write $$d\textbf{s}=\textbf{e}_\mu dx^\mu$$ so that the tangent vector is now $$\textbf{t}=\frac{dx^\mu}{d\lambda}\textbf{e}_\mu.$$ (Note: More explicitly, $$dx^\mu \equiv dx^\mu(\lambda)$$.)
The square of the length of the tangent vector $$\textbf{t}$$ is then $$|\textbf{t}|^2=g_{\mu\nu}t^\mu t^\nu=g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}.$$
So the quantity $$g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^ \nu}{d\lambda}$$ being constant throughout the curve $$x^\mu(\lambda)$$ means that the tangent vector $$\textbf{t}$$ has a constant length throughout the curve.
Parameterising A Curve
For any curve $$x^\mu(\lambda)$$, we can paramaterise the curve such that the length of the tangent vector is constant. Note that if we choose the parameter $$\lambda$$ to be $$\lambda=as+b,$$ where $$s$$ is the distance measured along the curve and $$a$$, $$b$$ are constants, the length of the tangent vector will be constant. This can be shown through $$|\textbf{t}|=\frac{d|\textbf{s}|}{d\lambda}=\frac{ds}{d\lambda}={1\over a},$$ where $$1\over a$$ is a constant.
Summary: For any curve $$x^\mu(\lambda)$$, we can always parameterise it such that the length of the tangent vector is constant throughout the curve. The length of the tangent vector is given by $$|\textbf{t}|=(g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^ \nu}{d\lambda})^{1\over2}$$.
References:
1. Hobson, Efstathiou & Lasenby General Relativity: An Introduction for Physicists pg. 75
This is simply a choice of parametrization of the geodesic $$x^\mu(\lambda)$$. If we were in a Euclidean-signature manifold, $$\lambda$$ would be proportional to the arc length along the curve. Here, for a timelike geodesic, it would be proportional to the proper time along the curve.
• So would you say 3. is correct? Is the direction of $d/d\lambda$ conservered for a general geodesic? Dec 13, 2020 at 14:06
• It depends on what you mean by $d/d\lambda$. Do you mean $(d x^\mu/d \lambda) \partial_\mu$ with $\partial_\mu$ considered as a tangent space basis vector? Certainly geodesics are autoparallels. I don't like saying that "direction is conserved" as directions at different points are not comparable. Dec 13, 2020 at 14:14
• To me $d/d\lambda$ and $(dx^\mu/\lambda)\partial_\mu$ are two expressions of the same thing. I don't know what autoparallels are. Dec 13, 2020 at 14:17
• But I have to agree that saying directions are conserved is a bit of a vague statement. Dec 13, 2020 at 14:19
• I mean that parallel trasnporting the tangent along the geodesic takes you to the tangent at the new point. If you parallel transport to the same point along a different curve you will get a different vector. So directions at different points are not comparable. The attempted comparison depends on how you try to do it. Dec 13, 2020 at 14:19 | 2022-08-15 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": 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": 48, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8837172985076904, "perplexity": 180.81858063085946}, "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/1659882572212.96/warc/CC-MAIN-20220815205848-20220815235848-00161.warc.gz"} |
https://mathematica.stackexchange.com/questions/88044/equality-of-expressions-containing-complex-numbers/88047 | # Equality of expressions containing complex numbers [closed]
I am using Mathematica to test the equivalence of symbolic expressions. I run into what appears to be a bug when I test the equivalence of expressions involving an imaginary number taken to a complex power. When I test the equivalence of such expressions that are clearly equal, mathematica does not recognize their equivalence. Here is an example:
Assuming[k > 0 && m > 0,
{TrueQ[(2*I*k)^(1 + 0.1*I) (2 I k)^(I*m) == (2*I*k)^(1 + 0.1*I + I*m)]}]
False
Interestingly, when I repeat the above test, except changing the 2 I k to I k, Mathematica recognizes the equivalence:
Assuming[k > 0 && m > 0,
{TrueQ[(I*k)^(1 + 0.1*I)* (I k)^(I m) == (I*k)^(1 + 0.1*I + I*m)]}]
True
Another interesting example I encountered was when taking an imaginary number to the power Pi vs. N[Pi]:
Assuming[k > 0 && m > 0, {TrueQ[(I k)^(Pi) (I k)^(I m) == (I k)^(Pi + I m)]}]
True
Assuming[k > 0 && m > 0, {TrueQ[(I k)^(N[Pi]) (I k)^(I m) == (I k)^(N[Pi] + I m)]}]
False
I suspect these to be a bug in Mathematica. Has anyone encountered similar problems with equivalence tests in Mathematica, or does anyone have an idea of what might be the cause if this apparent problem?
• You will want to avoid inexact numbers in this application. Try replacing 0.1 with 1/10 in your first example, for instance. – J. M. is in limbo Jul 12 '15 at 11:19
• Please, try to format your questions properly. If you visit the help centre you can read more about it. – Sektor Jul 12 '15 at 11:23
This is not a bug. It's a misunderstanding about what TrueQ does.
From the documentation,
TrueQ will return True only if the input is explicitly True
To put it more explicitly, it's equivalent to trueQ[expr_] := If[expr === True, True, False].
The expression (2*I*k)^(1 + 0.1*I) (2 I k)^(I*m) == (2*I*k)^(1 + 0.1*I + I*m) is not the symbol True (regardless of the fact whether it's mathematically true or not), so TrueQ returns False.
(I k)^(Pi) (I k)^(I m) == (I k)^(Pi + I m) evaluates to True immediately due to canonicalization of the power expression, so TrueQ returns True.
TrueQ does not deal with symbolic mathematics at all. It simply checks if an expression is True, literally, or something else. It does not even care about the meaning of the symbol True.
The correct way to handle this problem is to use Simplify instead of TrueQ and to only use exact numbers. 0.1 is not exact.
• Assuming[k > 0 && m > 0, Simplify[(2*I*k)^(1 + 0.1*I) (2 I k)^(I*m) == (2*I*k)^(1 + 0.1*I + I*m)]] ( i.e., with inexact numbers) evaluates to True on my system (Mma v10.1 with Mac OS 10.10.4). Although I agree that generally it is best when checking for equality to use exact numbers to avoid potential problems. – Bob Hanlon Jul 12 '15 at 12:40
What you are encountering is something all Mathematica users encounter because it is the way Mathematica works. Szabolcs has explained this well. However, I would like to add that you can fix the "problem" by using Simplify
Simplify[(2*I*k)^(1 + 0.1*I) (2*I*k)^(I*m) == (2*I*k)^(1 + 0.1*I + I*m)]
True
Simplify[(I k)^(N[Pi]) (I k)^(I m) == (I k)^(N[Pi] + I m)]
True | 2020-02-16 21:28:36 | {"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": 0.2017345279455185, "perplexity": 2305.851564868392}, "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-2020-10/segments/1581875141430.58/warc/CC-MAIN-20200216211424-20200217001424-00307.warc.gz"} |
https://electronics.stackexchange.com/questions/223959/pspice-common-emitter-schematic-missing-values/223967 | # Pspice Common Emitter Schematic Missing Values
I am doing some homework for universiy and PSpice keeps saying that there is some missing values.I cant seem to find a solution in my books so I though that i might share my issue with you and understand why that happens.Here is the schematic
I'm no expert in PSpice, but the following lines seem to spell it out:
R_R2 0 2 4,9k
------------------------\$
ERROR -- Missing value
I take it you are from a country where the comma (,) is used as a decimal mark. PSpice is likely expecting a period (.) as the decimal mark. Change the value of R_R2 to 4.9k, that should sort it out.
As I understand it, because SPICE uses a comma to separate parameters or other tokens in many contexts, a period/dot must be used as a decimal mark to avoid ambiguity.
• Yes - uint128_t is correct. And - more than that, are you aware that there is no input signal (Vin=0)? – LvW Mar 22 '16 at 8:16
• Yeah pretty dumb mistake made by me.Thank you very much.As for the input signal thats the value i recieved from my teacher and its working fine now even without it. – Yani Stoyanov Mar 23 '16 at 22:00
• @LvW I suspect that OP is doing AC analysis or something similar, and therefore the simulator ignores that particular parameter. – uint128_t Mar 24 '16 at 0:53
• AC analysis without any ac source? Not very probable. – LvW Mar 24 '16 at 9:31
• @LvW I think Vin is the AC source, and that the AC amplitude of that source is hidden on the schematic. I don't know PSpice, but that's what LTSpice does. – uint128_t Mar 24 '16 at 13:59 | 2019-09-15 22:38:37 | {"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": 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": 0.5911592841148376, "perplexity": 1756.883745461285}, "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-2019-39/segments/1568514572436.52/warc/CC-MAIN-20190915215643-20190916001643-00480.warc.gz"} |
https://www.coursehero.com/sg/college-algebra/quadratic-modeling/ | ### Using a Parabola to Model Data
Scatterplot Parabola of Best Fit
The scatterplot shows a trend that is modeled by a parabola, which can be sketched by hand or generated using technology to approximate the data. A parabola can be used to approximate the trend. Note that a curve of best fit may not touch all or even any of the points on the scatterplot.
Step-By-Step Example
The table shows data collected in an experiment measuring the height of an object at certain points in time.
Time (s) Height (m)
0 1
2 159
4 269
5 275
7 342
9 330
10 246
12 220
14 86
Estimate the time at which the object landed on the ground.
Step 1
Make a scatterplot of the data set. The scatterplot shows that a quadratic model is probably a good fit for the data.
Step 2
Sketch a parabola that approximates the pattern shown by the data set. The parabola shown models the height of the object over time.
Solution
The parabola intersects the $x$-axis at approximately $x=15$. So the object hit the ground after approximately 15 seconds.
### Quadratic Functions of Best Fit
Technology can be used to generate a quadratic function that best fits the quadratic relationship between two variables.
There are many computer programs and calculators with the capability to find the equation of the parabola that best fits a set of data. Spreadsheets are commonly used to create scatterplots, fitted curves, and equations of the curves of best fit. A quadratic model generated by technology can be used to make predictions.
Step-By-Step Example
Modeling Data with a Parabola
A corporation uses the data in the table to track its profits for 10 years and wants to do an analysis for a presentation.
Year Profit (millions of ) 1 2.5 2 1.9 3 1.2 4 1.1 5 1.4 6 1.7 7 2.4 8 3.1 9 3.9 10 4.6 Use a spreadsheet to create a scatterplot of the data table, and estimate the corporation's profit in 3 more years. Step 1 Enter the data into a spreadsheet. Step 2 Select the simple scatterplot from the chart options. In most spreadsheets, the data should be highlighted before selecting the chart. If a chart with no points appears, the data set needs to be highlighted. Create the graph, and label the axes. Step 3 Using the available trend line or regression function, fit the parabola to the data, and calculate the quadratic equation. Solution Use the equation to estimate the company's profit in 3 more years. Substitute 13 into the equation for $x$, and evaluate to find $y$. \begin{aligned}y &= 0.1023x^2 - 0.8426x + 3.0767\\y &= 0.1023\cdot 13^2 - 0.8426\cdot 13 + 3.0767\\y &= 9.4116\end{aligned} If the trend continues, in 3 years, the company will make about9.4 million in profit. | 2022-05-19 12:46:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "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": 0.6585255265235901, "perplexity": 592.96006282236}, "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/1652662527626.15/warc/CC-MAIN-20220519105247-20220519135247-00521.warc.gz"} |
https://solvedlib.com/a-chemist-mixes-a-20-solution-and-a-50-solution,3576 | # A chemist mixes a 20% solution and a 50% solution to obtain a 40% solution
###### Question:
A chemist mixes a 20% solution and a 50% solution to obtain a 40% solution. If 300g of solution was obtained, how much of each type of the original solutions were used?
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##### St[Thlnk About ItClasslfying: What variations did you identify that distinguish desert plants and rain forest plants?Formulating Hypotheses: How could these variations help desert and rain forest plants survive in their environments?
St[ Thlnk About It Classlfying: What variations did you identify that distinguish desert plants and rain forest plants? Formulating Hypotheses: How could these variations help desert and rain forest plants survive in their environments?... | 2023-01-29 15:20:16 | {"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": 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": 0.9726012349128723, "perplexity": 3488.080230654878}, "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-2023-06/segments/1674764499744.74/warc/CC-MAIN-20230129144110-20230129174110-00000.warc.gz"} |
https://www.tutorialspoint.com/finding-n-th-term-of-series-3-13-42-108-235-in-cplusplus | # Finding n-th term of series 3, 13, 42, 108, 235... in C++
In this problem, we are given a number n. Our task is to find the n-th term of series 3, 13, 42, 108, 235...
Let's take an example to understand the problem,
Input : 5
Output : 235
## Solution Approach
The series can be represented as the sum of cubes of first n natural numbers. The formula for that is (n*(n+1)/2)2. Also if we add 2* to it we will get the required series.
The formula for sum of the series is (n*(n+1)/2)2+2*n.
For n = 5 sum by the formula is
(5 * (5 + 1 ) / 2)) ^ 2 + 2*5
= (5 * 6 / 2) ^ 2 + 10
= (15) ^ 2 + 10
= 225 + 10
= 235
## Example
Program to illustrate the working of our solution
#include <iostream>
using namespace std;
int findNthTerm(int N)
{
return ((N * (N + 1) / 2)*(N * (N + 1) / 2) ) + 2 * N;
}
int main()
{
int N = 5;
cout<<"The Nth term fo the series n is "<<findNthTerm(N);
return 0;
}
## Output
The Nth term fo the series n is 235 | 2023-02-08 08:32:48 | {"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": 0.6092777252197266, "perplexity": 840.8546567593305}, "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/1674764500719.31/warc/CC-MAIN-20230208060523-20230208090523-00431.warc.gz"} |
https://en.khanacademy.org/math/get-ready-for-7th-grade/xa46d6dd638f86863:get-ready-for-negative-number-operations/xa46d6dd638f86863:negative-symbol-as-opposite/v/negative-symbol-as-opposite | If you're seeing this message, it means we're having trouble loading external resources on our website.
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# Negative symbol as opposite
CCSS.Math:
## Video transcript
- [Voiceover] In another video we've already talked about what an opposite of a number is, but we'll review it a little bit. If we start with a positive number, say, the number four. If you have a number and then we want to think about what the opposite of that number is. Opposite. This is just the symbol for number or shorthand for number. So if the number is four, what is its opposite? Well, four is one, two, three, four to the right of zero. Positive four is four to the right of zero. So its opposite is going to be one, two, three, four to the left. So it's going to be negative four. Or another way to think about it, if you have a positive number, its opposite is going to be the negative of that number. So another way of thinking about it is, is that this negative literally means opposite. One way to think about it, this is the number that is the opposite of four. So let me write this down. So if I write negative four. Negative, let me write this, negative four. That literally means... So you can take this negative symbol as meaning opposite. Opposite. Opposite of four. Opposite of four. If you were to say negative, instead of saying a specific number let's just say a letter that could represent a number. So if I said negative A, that means the opposite of the number A. That means the opposite... Opposite of... Let me do that in blue color. Opposite of A. Of A. And if this confuses you a little bit just look, A could be any number right over here. So let me draw a number line just so this can be a little bit, make a little bit more sense hopefully. So this is a number line. Let me put some tick marks here. I don't know much each of these tick marks jump but let's say that A is some number that is right over here. Well negative A is going to be the opposite of A. So if A is three tick marks to the right negative A is going to be three tick marks to the left. One, two, three. And so the opposite of A is going to be this value right over here. And we can write that, we can write opposite of A over here. We could literally write opposite, opposite of A is that number right over there. Or as a shorthand, we can just write-- We can just write negative. We can just write this is negative A right over here. Negative A. So with that in mind, if we literally view this negative symbol as meaning the opposite of whatever this is, let's try something interesting. What would be the negative-- Let me do this in another color. What would be the negative of negative three? And I encourage you to pause the video and think about it. Well we just said, this negative means the opposite. So you can think about this as meaning this means the opposite... Opposite of negative-- The opposite of negative three. So what is the opposite of negative three? Well negative three is three to the left of zero. One, two, three. So it's opposite is going to be three to the right of zero. One, two, three. So it's going to be positive three. So this is equal to positive three. Or we could just write positive three like that. So hopefully this gives you a better appreciation for what opposite means and also how it relates to the actual negative symbol. We could keep going. We could do something like what is the negative of the negative of negative-- Let me do a different number. Of negative two. Well, this part right over here the negative, the opposite of negative two, which is really the opposite of the opposite of two, well that's just going to be two. Every time you say opposite you flip over the number line. So this flips you over the number line two to the left and this flips you back two to the right so all of this is going to be two but then you're going to take the opposite of two so that's just going to be negative two. If you threw another negative in front of this, it would be the opposite of all of this. So it would be the opposite of negative two and then all of a sudden it would become positive. So every time you put that negative in front there you're flipping on the other side of the number line. | 2021-01-21 18:36:12 | {"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": 0.8336523175239563, "perplexity": 323.84556923247357}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2021-04/segments/1610703527224.75/warc/CC-MAIN-20210121163356-20210121193356-00010.warc.gz"} |
https://zzapomni.com/konygina-metodicheskie-ukazaniya-po-1999-1354/18 | # Методические указания по английскому языку для студентов 5-го курса исторического факультета. Коныгина Г.И. - 18 стр.
Составители:
Рубрика:
• ## Иностранный язык
18
2. precedes an intransitive verb.
e.g.
How do you answer the first question?
How should you write it?
How does number 5 go?
How did you get on?
“What”
1. requires a transitive verb where
“what”
is the object;
2. precedes
“to be”
as a complement.
e.g
. What did you write?
What should you say?
What was number 6?
____________________________________________________________________________________________________________________________
Translate this / that / last sentence into English.
Could you put that into Russian for us.
Translate from Russian into English.
Don’t translate word for word.
Think about the meaning of the whole sentence.
Use English.
Try it in English.
Now the same thing in English.
This is supposed to be an English lesson, so let’s speak English.
COMPREHENSION
Let’s see if you’ve understood.
You had the job of preparing five questions each on this unit.
Let’s look at this chapter in more detail.
____________________________________________________________________________________________________________________________
Note:
You can sometimes hear:
Let’s discuss
this.
“Discuss
” is not followed by a preposition. It is also rather inappropriate
in the classroom.
____________________________________________________________________________________________________________________________
2. We’ll have a look at the new words.
I don’t think you’ve had / met this word before.
Let’s read through the vocabulary first.
I think we had this verb last time.
We looked at / dealt with these forms last week.
____________________________________________________________________________________________________________________________
Note: I think you haven’t had this word before.
I think you haven’t got time.
18
2. precedes an intransitive verb.
e.g. How do you answer the first question?
How should you write it?
How does number 5 go?
How did you get on?
“What” 1. requires a transitive verb where “what” is the object;
2. precedes “to be” as a complement.
e.g. What did you write?
What should you say?
What was number 6?
____________________________________________________________________________________________________________________________
Translate this / that / last sentence into English.
Could you put that into Russian for us.
Translate from Russian into English.
Don’t translate word for word.
Think about the meaning of the whole sentence.
Use English.
Try it in English.
Now the same thing in English.
This is supposed to be an English lesson, so let’s speak English.
COMPREHENSION
Let’s see if you’ve understood.
You had the job of preparing five questions each on this unit.
Let’s look at this chapter in more detail.
____________________________________________________________________________________________________________________________
Note: You can sometimes hear:
“Discuss” is not followed by a preposition. It is also rather inappropriate
in the classroom.
____________________________________________________________________________________________________________________________
2. We’ll have a look at the new words.
I don’t think you’ve had / met this word before.
Let’s read through the vocabulary first.
I think we had this verb last time.
We looked at / dealt with these forms last week.
____________________________________________________________________________________________________________________________
Note: • I think you haven’t had this word before.
• I think you haven’t got time. | 2022-12-02 20:04:45 | {"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": 0.5100935101509094, "perplexity": 6181.767446126078}, "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-2022-49/segments/1669446710916.40/warc/CC-MAIN-20221202183117-20221202213117-00544.warc.gz"} |
https://www.physicsforums.com/threads/verifying-trig-identity.349296/ | # Verifying trig identity
## Homework Statement
(sin 3α/sin α) - (cos 3α/cosα) =2
## The Attempt at a Solution
I know for sin 2 α I would put 2 sinαcosα, so for 3α, do I just put 3sinαcosα?
for cos 3α, I'm sorta clueless because there's 3 we can use for cosine,
Then after that step, I know to get both of them on the LHS to have a common denominator, which sinα cosα, please help. Thank yyou in advance! | 2020-07-09 02:34:47 | {"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": 0.8120084404945374, "perplexity": 3774.8192304583845}, "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/1593655897844.44/warc/CC-MAIN-20200709002952-20200709032952-00277.warc.gz"} |