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https://blender.stackexchange.com/questions/148943/importing-objects-as-empties | # Importing Objects as Empties
That may sound weird, but I want to import all the objects from an .OBJ file as dummies centered where the object's center would be for a little experiment involving game modding. I have no experience with scripting, so I'm asking if it'd be hard to make a script that executes the operations I just described.
Use the importer provided
Would simply import with the provided importer then replace each with an empty.
Import the obj file. Add an empty with the same transform as each imported object. Remove the originals. (Note doesn't keep parenting)
Modify the filepath to where your obj file is on your computer.
import bpy
context = bpy.context
# some path to your obj file
filepath = "/home/batfinger/Desktop/xxxx.obj"
# import it
bpy.ops.import_scene.obj(
filepath=filepath
)
# the imported objects.
imported_objs = context.selected_objects
# add an empty for each imported object
for o in imported_objs:
mt = context.object
mt.name = o.name
mt.matrix_world = o.matrix_world
# delete imported objects
bpy.ops.object.delete(
{"selected_objects" : imported_objs}
)
For starting blender-python-programming it can be helpfull to check out the info tab and copy out the listed commands to your script.
here is the script you want, wich keeps the original objects name. Import your objects by hand, select the ones you want to empty out and run script from text editor.
comments are marked with #, so you can understand whats going on. This way you could easily create other macros as well.
works only on 2.8
#import blender namespace
import bpy
#get all selected objects
selection_names = bpy.context.selected_objects
#loop trough all "selection_names"
for i in selection_names:
#for every object "i":
#set active object as "i"
bpy.context.view_layer.objects.active = i
#go to edit mode of "i"
bpy.ops.object.editmode_toggle()
#select all
bpy.ops.mesh.select_all(action='SELECT')
#delete all verts
bpy.ops.mesh.delete(type='VERT')
#go back to object mode
bpy.ops.object.editmode_toggle()
#repeat for next object in "selection_names" till none left | 2019-09-22 05:36:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.4343300759792328, "perplexity": 7295.947996044947}, "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/1568514575168.82/warc/CC-MAIN-20190922053242-20190922075242-00530.warc.gz"} |
https://www.camerecole.org/classes/1481-correction-des-exercices-sur-l-integration-des-fonctions-irrationnelles.html | Vous êtes ici : AccueilCLASSESCorrection des exercices sur l’intégration des fonctions irrationnelles
Terminale
Mathématiques
Correction exercice
Bonjour ! Groupe telegram de camerecole, soumettrez-y toutes vos préoccupations. forum telegram
#### Correction exercice I
Calculons les intégrales suivantes
a) $$I =$$ $$\int {\frac{{\sqrt x }}{{\sqrt[4]{{{x^3}}} + 1}}} dx$$
I peut aussi se mettre sous la forme $$\int {\frac{{{x^{\frac{1}{2}}}}}{{{x^{\frac{3}{4}}} + 1}}} dx$$
Le dénominateur des fractions $${\frac{3}{4}}$$ et $${\frac{1}{2}}$$ est 4 donc posons $$x = {t^4}$$ $$\Rightarrow dx =$$ $$4{t^3}dt$$
$$I = 4$$ $$\int {\frac{{{t^5}}}{{{t^3} + 1}}} dt =$$ $$4\int {{t^2}dt - \frac{4}{3}}$$ $$\int {\frac{{dt}}{{t + 1}}} + \frac{4}{3}$$ $$\int {\frac{{ - 2t + 1}}{{{t^2} - t + 1}}} dt$$
$$I = \frac{4}{3}\sqrt[4]{{{x^3}}}$$ $$- \frac{4}{3}Log\left| {\sqrt[4]{{{x^3}}} + 1} \right|$$ $$+ cte$$
b) $$I =$$ $$\int {\frac{{\sqrt {{x^3}} - \sqrt[3]{x}}}{{6\sqrt[4]{x}}}} dx$$
Le dénominateur des fractions $${^{\frac{3}{4}}}$$, $${\frac{1}{3}}$$ et $${\frac{1}{4}}$$ est 12
$$I =$$ $$\int {\frac{{\sqrt {{x^3}} - \sqrt[3]{x}}}{{6\sqrt[4]{x}}}} dx$$ $$=$$ $$\int {\frac{{{x^{\frac{3}{2}}} - {x^{\frac{1}{3}}}}}{{6{x^{\frac{1}{4}}}}}} dx$$
$$x = {t^{12}} \Rightarrow$$ $$dx = 12{t^{11}}dx$$
$$I =$$ $$\int {\frac{{{x^{\frac{3}{2}}} - {x^{\frac{1}{3}}}}}{{6{x^{\frac{1}{4}}}}}} dx$$ $$= 2\int {\frac{{{t^{18}} - {t^4}}}{{{t^3}}}}$$ $${t^{11}}dt = 2$$ $$\int {({t^{26}} - {t^{12}})} dt$$ $$= \frac{2}{{27}}{t^{27}} -$$ $$\frac{2}{{13}}{t^{13}} + cte$$
$$I = \frac{2}{{27}}{x^{\frac{9}{4}}}$$ $$- \frac{2}{{13}}{x^{\frac{{13}}{{12}}}} + cte$$ $$= \frac{2}{{27}}\sqrt[4]{{{x^9}}}$$ $$- \frac{2}{{13}}\sqrt[{12}]{{{x^{13}}}} + cte$$
c) $$I = - \frac{6}{{\sqrt[6]{x}}} +$$ $$\frac{{12}}{{\sqrt[{12}]{x}}} + 2Log\left( x \right)$$ $$- 24$$ $$Log\left( {\sqrt[{12}]{x} + 1} \right)$$ $$+ cte$$
d) $$I = \frac{6}{5}\sqrt[6]{{{x^5}}}$$ $$- \frac{3}{2}\sqrt[6]{{{x^4}}} +$$ $$4\sqrt[6]{{{x^3}}} +$$ $$6\sqrt[6]{x} -$$ $$9Log\left( {\sqrt[6]{x} + 1} \right)$$ $$+ \frac{3}{2}Log\left( {\sqrt[6]{{{x^2}}} + 1} \right)$$ $$+ 3\arctan \left( {\sqrt[6]{x}} \right)$$ $$+ cte$$
e) Posons $${u^2} = \frac{{1 - x}}{{1 + x}}$$ on trouvera
$$I = Log\frac{{u - 1}}{{u + 1}}$$ $$- \frac{1}{{{u^2} - 1}} =$$ $$Log$$ $$\left| {\frac{{\sqrt {1 - x} + \sqrt {1 + x} }}{{\sqrt {1 - x} - \sqrt {x + 1} }}} \right|$$ $$- \frac{{\sqrt {1 + x} }}{x} + cte$$
f ) $$I =$$ $$\sqrt {3{x^2} - 7x - 6}$$ $$+ \frac{{11}}{{2\sqrt 3 }}Log$$ $$\left| {x - \frac{7}{2} + \sqrt {{x^2} - \frac{7}{3}x - 2} } \right|$$ $$+ cte$$ | 2021-12-09 10:36:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8951762318611145, "perplexity": 4122.051509124012}, "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/1637964363791.16/warc/CC-MAIN-20211209091917-20211209121917-00610.warc.gz"} |
https://www.gradesaver.com/textbooks/math/calculus/calculus-early-transcendentals-2nd-edition/chapter-1-functions-1-1-review-of-functions-1-1-exercises-page-12/88 | ## Calculus: Early Transcendentals (2nd Edition)
The solution is $$f(x)=x^2-6.$$
We need that $(f(x))^2$ is equal to a 4th degree polynomial so we will demand that $f(x)$ is a 2nd degree polynomial i.e. that it is given by $f(x)=ax^2+bx+c$. This gives $$(f(x))^2=(ax^2+bx+c)^2=a^2x^4+b^2x^2+c^2+2abx^3+2acx^2+2bcx = a^2x^4+ 2abx^3+(b^2+2ac)x^2+2bcx+c^2 = x^4-12x^2+36.$$ Equating the coefficients multiplying same powers of $x$ we get $$a^2=1\Rightarrow a=1$$ $$2ab=0\Rightarrow 2b=0\Rightarrow b=0$$ $$b^2+2ac=-12\Rightarrow 2c=-12\Rightarrow c=-6$$ The last equation is only for checking. Indeed $$c^2=(-6)^2=36.$$ Finally $$f(x)=x^2-6.$$ | 2018-07-16 09:03:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9746149778366089, "perplexity": 132.92193909665576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589237.16/warc/CC-MAIN-20180716080356-20180716100356-00031.warc.gz"} |
https://kartikaggarwal.me/2019/10/10/AthNLP.html | This post discusses the highlights of Athens NLP Summer School 2019. The sessions covered core concepts on Language Modelling, Machine Translation, POS Tagging & Question Answering.
Table of Content
I had the opportunity to attend my first international summer school from 18-25th September, 2019 at National Center for Scientific Research “Demokritos”, Athens, Greece. AthNLP is an annual event where students, researchers and industry experts with interests in Language Processing, Computational Linguistics and Statistics gather to attain a deeper understanding of the domain.
The week consisted of lectures from accomplished researchers, workshops, poster sessions from participants and social events. This post summarizes the key learnings from the school assuming a basic understanding of NLP and mathematics.
The complete lecture playlist can be accessed here.
# Classification in NLP [PPT]
The school commenced with a session on Classification in NLP by Ryan Mcdonald from Google. Broad topics in the lecture include linear classifiers, feature representation & perceptron. Classification tasks in NLP can be mainly categorized as binary classification (eg: fact checking- fake/real) & multi-class classification (eg:topic identification).
Understanding the principles of linear classifiers (LC) are the key to understanding modern neural networks. These LC’s can be stacked together to form classical NLP pipelines where each classifier’s prediction is used to handcraft features for others classifiers. For instance in the task of sentiment classification, we could first build a LC for part-of-speech (POS) tagging as these documents contains too many adjectives (good, bad, happy) and then use these POS tags as features for our final sentiment model.
Try to answer this question to test your grasp on this topic:
What happens if we don’t use any activation function in a multilayer neural network?
Yes. The network will behave as a linear classifier !!
Formally the task of classification can be stated as: Given a labelled data with input-output (i-o) pairs ${(x_n,y_n)}_{n=1}^N \subset \mathcal{X} \times \mathcal{Y}$, the goal is to learn a classifier $h: \mathcal{X}\rightarrow\mathcal{Y}$ from the space of classifiers that generalizes well on arbitrary inputs. Instead of mapping only the input, we map the i-o pairs to a sparse-binary feature representation. Score for a specific label is obtained using linear combination of features and weights (model parameter).
The only distinction between various LC’s (perceptron, SVM, Logistic Regression) is the way in which this parameter $w$ is learned. The set of feature representations remain the same but the strategies used to learn parameters for those features are different.
# Structured Prediction in NLP [PPT]
After looking at the binary and multi-class classification, Xavier Carreras from dMetrics introduced another type of classification known as Structured Prediction. Consider the task of Named Entity Recognition (NER) for instance. It might appear that it’s a multi-class classification task as the goal is to predict a label for each word in a sequence. But if we follow this approach then “London” would result in the output “Location” in the following sentence.
Jack London went to Paris
PER PER - - LOC
Instead we can make use of the surrounding inputs and their outputs. If the next word is went, it is highly unlikely that a “location” can go somewhere. Also, it is more probable that if the previous word is a “Person” then “London” is also a person rather than a “Location”.
In general, structured prediction methods can be classified as:
1. Label Classifiers
2. Transition based systems
3. Factor models
4. Re-Ranking
### Label Classifiers
Instead of considering surrounding sequence, we consider this as a multi-class classification task over individual labels $l$ for each input position $t$. As we already discussed above this is the simplest approach but does not allow us to capture interaction between output labels.
### Transition-based Sequence Prediction
Overcoming the drawback of previous method, in transition based models, we not only have the token of current position and candidate label but also the previous predictions.
### Factored Sequence Prediction
Influenced from the above two approaches, this approach predicts the complete output sequence at once by introducing a factor for each input word which is the current label and the previous label. This is an intermediate approach as we are performing a multi-class classification (label classifiers) for each input word but with the help of both current and previous label rather than the full sequence of previous decisions (transition-based).
### Re-Ranking
Re-Ranking approach makes use of the full input sequence and all the possible output sequences. A base model (can be any previous model) is used to first rank all the sequences in some way and then select most likely output sequences referred as an active set. This active set can give the full representation by looking at any input and output making it fully expressive.
# Encoder-Decoder Neural Nets [PPT]
Barbara Plank kicked off the third day with one of the most interesting session on Sequential Models. So far we only discussed conditioning on sequential data to make a classification i.e Condition problems. But applications like auto-complete and spelling correction require us to generate sequential data instead of classifying it i.e. Generation Problems.
This gives rise to Language Models (LMs) which can be used to compute either the probability of a text or probability of next word given some text. Earlier approaches used chain rule to estimate the joint probability of the entire sequence by multiplying conditional probabilities of some word given previous words. This was based on the $n^{th}$ order Markov assumption that the probability of a current word only depends only upon the previous $n$ words, hence the name $n$-gram language models. However, these n-gram LMs have difficulty handling long-range dependencies due to the problem of sparsity with increasing $n$ and also in sharing strength among similar words.
Similar words issue was handled by fixed-window based neural LM which iteratively moved the n-gram window through a very large corpus to predict the next word by concatenating the embeddings of words in current window at each time step.
Consequently, Recurrent Neural Networks (and their variants) became one of the promising architectures which can not only handle long-term dependencies by eliminating the markov assumption but can also handle arbitrary length inputs like CNNs. | 2020-08-15 16:08:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.577333927154541, "perplexity": 1462.9318683744984}, "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/1596439740929.65/warc/CC-MAIN-20200815154632-20200815184632-00182.warc.gz"} |
https://math.answers.com/Q/What_is_the_simplest_form_of_111_over_84 | 0
# What is the simplest form of 111 over 84?
Wiki User
2012-01-09 02:21:17
Expressed as a top-heavy fraction in its simplest form, by dividing the numerator and denominator by 3, 111/84 is equal to 37/28 or thirty-seven over twenty-eight. Expressed as a mixed number, this is equal to 1 9/28 or one and nine twenty-eighths.
Wiki User
2012-01-09 02:21:17 | 2021-09-27 10:44:41 | {"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": 0.9733513593673706, "perplexity": 1523.1720713315792}, "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/1631780058415.93/warc/CC-MAIN-20210927090448-20210927120448-00067.warc.gz"} |
https://iu.pressbooks.pub/openstaxcollegephysics/chapter/half-life-and-activity/ | # 253 31.5 Half-Life and Activity
### Summary
• Define half-life.
• Define dating.
• Calculate age of old objects by radioactive dating.
Unstable nuclei decay. However, some nuclides decay faster than others. For example, radium and polonium, discovered by the Curies, decay faster than uranium. This means they have shorter lifetimes, producing a greater rate of decay. In this section we explore half-life and activity, the quantitative terms for lifetime and rate of decay.
# Half-Life
Why use a term like half-life rather than lifetime? The answer can be found by examining Figure 1, which shows how the number of radioactive nuclei in a sample decreases with time. The time in which half of the original number of nuclei decay is defined as the half-life, $\boldsymbol{t _{1/2}}$. Half of the remaining nuclei decay in the next half-life. Further, half of that amount decays in the following half-life. Therefore, the number of radioactive nuclei decreases from $\boldsymbol{N}$ to $\boldsymbol{N/2}$ in one half-life, then to $\boldsymbol{N/4}$ in the next, and to $\boldsymbol{N/8}$ in the next, and so on. If $\boldsymbol{N}$ is a large number, then many half-lives (not just two) pass before all of the nuclei decay. Nuclear decay is an example of a purely statistical process. A more precise definition of half-life is that each nucleus has a 50% chance of living for a time equal to one half-life $\boldsymbol{t _{1/2}}$. Thus, if $\boldsymbol{N}$ is reasonably large, half of the original nuclei decay in a time of one half-life. If an individual nucleus makes it through that time, it still has a 50% chance of surviving through another half-life. Even if it happens to make it through hundreds of half-lives, it still has a 50% chance of surviving through one more. The probability of decay is the same no matter when you start counting. This is like random coin flipping. The chance of heads is 50%, no matter what has happened before.
There is a tremendous range in the half-lives of various nuclides, from as short as $\boldsymbol{10^{-23}}$ s for the most unstable, to more than $\boldsymbol{10^{16}}$ y for the least unstable, or about 46 orders of magnitude. Nuclides with the shortest half-lives are those for which the nuclear forces are least attractive, an indication of the extent to which the nuclear force can depend on the particular combination of neutrons and protons. The concept of half-life is applicable to other subatomic particles, as will be discussed in Chapter 33 Particle Physics. It is also applicable to the decay of excited states in atoms and nuclei. The following equation gives the quantitative relationship between the original number of nuclei present at time zero ($\boldsymbol{N_0}$) and the number ($\boldsymbol{N}$) at a later time $\boldsymbol{t}$:
$\boldsymbol{N = N_0 e ^{- \lambda t}}$
where $\boldsymbol{e = 2.71828 \cdots }$ is the base of the natural logarithm, and $\boldsymbol{\lambda}$ is the decay constant for the nuclide. The shorter the half-life, the larger is the value of $\boldsymbol{\lambda}$, and the faster the exponential $\boldsymbol{e^{- \lambda t}}$ decreases with time. The relationship between the decay constant $\boldsymbol{\lambda}$ and the half-life $\boldsymbol{t_{1/2}}$ is
$\boldsymbol{\lambda =}$$\boldsymbol{ \frac{ \textbf{ln} (2)}{t_{1/2}}}$$\boldsymbol{\approx}$$\boldsymbol{\frac{0.693}{t_{1/2}}}$
To see how the number of nuclei declines to half its original value in one half-life, let $\boldsymbol{t = t_{1/2}}$ in the exponential in the equation $\boldsymbol{N = N_0 e^{- \lambda t}}$. This gives $\boldsymbol{N = N_0 e^{- \lambda t} = N_0 e^{-0.693} = 0.500N_0}$. For integral numbers of half-lives, you can just divide the original number by 2 over and over again, rather than using the exponential relationship. For example, if ten half-lives have passed, we divide $\boldsymbol{N}$ by 2 ten times. This reduces it to $\boldsymbol{N/1024}$. For an arbitrary time, not just a multiple of the half-life, the exponential relationship must be used.
Radioactive dating is a clever use of naturally occurring radioactivity. Its most famous application is carbon-14 dating. Carbon-14 has a half-life of 5730 years and is produced in a nuclear reaction induced when solar neutrinos strike $\boldsymbol{^{14} \textbf{N}}$ in the atmosphere. Radioactive carbon has the same chemistry as stable carbon, and so it mixes into the ecosphere, where it is consumed and becomes part of every living organism. Carbon-14 has an abundance of 1.3 parts per trillion of normal carbon. Thus, if you know the number of carbon nuclei in an object (perhaps determined by mass and Avogadro’s number), you multiply that number by $\boldsymbol{1.3 \times 10^{-12}}$ to find the number of $\boldsymbol{^{14} \textbf{C}}$ nuclei in the object. When an organism dies, carbon exchange with the environment ceases, and $\boldsymbol{^{14} \textbf{C}}$ is not replenished as it decays. By comparing the abundance of $\boldsymbol{^{14} \textbf{C}}$ in an artifact, such as mummy wrappings, with the normal abundance in living tissue, it is possible to determine the artifact’s age (or time since death). Carbon-14 dating can be used for biological tissues as old as 50 or 60 thousand years, but is most accurate for younger samples, since the abundance of $\boldsymbol{^{14} \textbf{C}}$ nuclei in them is greater. Very old biological materials contain no $\boldsymbol{^{14} \textbf{C}}$ at all. There are instances in which the date of an artifact can be determined by other means, such as historical knowledge or tree-ring counting. These cross-references have confirmed the validity of carbon-14 dating and permitted us to calibrate the technique as well. Carbon-14 dating revolutionized parts of archaeology and is of such importance that it earned the 1960 Nobel Prize in chemistry for its developer, the American chemist Willard Libby (1908–1980).
One of the most famous cases of carbon-14 dating involves the Shroud of Turin, a long piece of fabric purported to be the burial shroud of Jesus (see Figure 2). This relic was first displayed in Turin in 1354 and was denounced as a fraud at that time by a French bishop. Its remarkable negative imprint of an apparently crucified body resembles the then-accepted image of Jesus, and so the shroud was never disregarded completely and remained controversial over the centuries. Carbon-14 dating was not performed on the shroud until 1988, when the process had been refined to the point where only a small amount of material needed to be destroyed. Samples were tested at three independent laboratories, each being given four pieces of cloth, with only one unidentified piece from the shroud, to avoid prejudice. All three laboratories found samples of the shroud contain 92% of the $\boldsymbol{^{14} \textbf{C}}$ found in living tissues, allowing the shroud to be dated (see Example 1).
### Example 1: How Old Is the Shroud of Turin?
Calculate the age of the Shroud of Turin given that the amount of $\boldsymbol{^{14} \textbf{C}}$ found in it is 92% of that in living tissue.
Strategy
Knowing that 92% of the $\boldsymbol{^{14} \textbf{C}}$ remains means that $\boldsymbol{N/N_0 = 0.92}$. Therefore, the equation $\boldsymbol{N = N_0 e^{- \lambda t}}$ can be used to find $\boldsymbol{\lambda t}$. We also know that the half-life of $\boldsymbol{^{14} \textbf{C}}$ is 5730 y, and so once $\boldsymbol{\lambda t}$ is known, we can use the equation $\boldsymbol{\lambda = \frac{0.693}{t_{1/2}}}$ to find $\boldsymbol{\lambda}$ and then find $\boldsymbol{t}$ as requested. Here, we postulate that the decrease in $\boldsymbol{^{14} \textbf{C}}$ is solely due to nuclear decay.
Solution
Solving the equation $\boldsymbol{N = N_0 e^{- \lambda t}}$ for $\boldsymbol{N/N_0}$ gives
$\boldsymbol{\frac{N}{N_0}}$$\boldsymbol{= e^{- \lambda t}}$
Thus,
$\boldsymbol{0.92 = e^{- \lambda t}}$
Taking the natural logarithm of both sides of the equation yields
$\boldsymbol{ln \; 0.92 = - \lambda t}$
so that
$\boldsymbol{-0.0834 = - \lambda t}$
Rearranging to isolate $\boldsymbol{t}$ gives
$\boldsymbol{t=}$$\boldsymbol{\frac{0.0834}{\lambda}}$
Now, the equation $\boldsymbol{\lambda = \frac{0.693}{t^{1/2}}}$ can be used to find $\boldsymbol{\lambda}$ for $\boldsymbol{^{14} \textbf{C}}$. Solving for $\boldsymbol{\lambda}$ and substituting the known half-life gives
$\boldsymbol{\lambda =}$$\boldsymbol{\frac{0.693}{t_{1/2}}}$$\boldsymbol{=}$$\boldsymbol{\frac{0.693}{5730 \;\textbf{y}}}$
We enter this value into the previous equation to find $\boldsymbol{t}$:
$\boldsymbol{t=}$$\boldsymbol{\frac{0.0834}{\frac{0.693}{5730 \;\textbf{y}}}}$$\boldsymbol{=690 \;\textbf{y}}$
Discussion
This dates the material in the shroud to 1988–690 = a.d. 1300. Our calculation is only accurate to two digits, so that the year is rounded to 1300. The values obtained at the three independent laboratories gave a weighted average date of a.d. $\boldsymbol{1320 \pm 60}$. The uncertainty is typical of carbon-14 dating and is due to the small amount of $\boldsymbol{^{14} \textbf{C}}$ in living tissues, the amount of material available, and experimental uncertainties (reduced by having three independent measurements). It is meaningful that the date of the shroud is consistent with the first record of its existence and inconsistent with the period in which Jesus lived.
There are other forms of radioactive dating. Rocks, for example, can sometimes be dated based on the decay of $\boldsymbol{^{238} \textbf{U}}$. The decay series for $\boldsymbol{^{238} \textbf{U}}$ ends with $\boldsymbol{^{206} \textbf{Pb}}$, so that the ratio of these nuclides in a rock is an indication of how long it has been since the rock solidified. The original composition of the rock, such as the absence of lead, must be known with some confidence. However, as with carbon-14 dating, the technique can be verified by a consistent body of knowledge. Since $\boldsymbol{^{238} \textbf{U}}$ has a half-life of 4.5×1094.5×109 y, it is useful for dating only very old materials, showing, for example, that the oldest rocks on Earth solidified about $\boldsymbol{3.5 \times 10^9}$ years ago.
# Activity, the Rate of Decay
What do we mean when we say a source is highly radioactive? Generally, this means the number of decays per unit time is very high. We define activity $\boldsymbol{R}$ to be the rate of decay expressed in decays per unit time. In equation form, this is
$\boldsymbol{R =}$$\boldsymbol{\frac{\Delta N}{\Delta t}}$
where $\boldsymbol{\Delta N}$ is the number of decays that occur in time $\boldsymbol{\Delta t}$. The SI unit for activity is one decay per second and is given the name becquerel (Bq) in honor of the discoverer of radioactivity. That is,
$\boldsymbol{1 \;\textbf{Bq} = 1 \;\textbf{decay/s}}$
Activity $\boldsymbol{R}$ is often expressed in other units, such as decays per minute or decays per year. One of the most common units for activity is the curie (Ci), defined to be the activity of 1 g of $\boldsymbol{^{226} \textbf{Ra}}$, in honor of Marie Curie’s work with radium. The definition of curie is
$\boldsymbol{1 \;\textbf{Ci} = 3.70 \times 10^{10} \;\textbf{Bq}}$
or $\boldsymbol{3.70 \times 10^{10}}$ decays per second. A curie is a large unit of activity, while a becquerel is a relatively small unit. $\boldsymbol{1 \;\textbf{MBq} = 100 \;\textbf{microcuries} ( \mu \textbf{Ci})}$. In countries like Australia and New Zealand that adhere more to SI units, most radioactive sources, such as those used in medical diagnostics or in physics laboratories, are labeled in Bq or megabecquerel (MBq).
Intuitively, you would expect the activity of a source to depend on two things: the amount of the radioactive substance present, and its half-life. The greater the number of radioactive nuclei present in the sample, the more will decay per unit of time. The shorter the half-life, the more decays per unit time, for a given number of nuclei. So activity $\boldsymbol{R}$ should be proportional to the number of radioactive nuclei, $\boldsymbol{N}$, and inversely proportional to their half-life, $\boldsymbol{t_{1/2}}$. In fact, your intuition is correct. It can be shown that the activity of a source is
$\boldsymbol{R =}$$\boldsymbol{\frac{0.693N}{t_{1/2}}}$
where $\boldsymbol{N}$ is the number of radioactive nuclei present, having half-life $\boldsymbol{t_{1/2}}$. This relationship is useful in a variety of calculations, as the next two examples illustrate.
### Example 2: How Great Is the 14C Activity in Living Tissue?
Calculate the activity due to $\boldsymbol{^{14} \textbf{C}}$ in 1.00 kg of carbon found in a living organism. Express the activity in units of Bq and Ci.
Strategy
To find the activity $\boldsymbol{R}$ using the equation $\boldsymbol{R = \frac{0.693N}{t_{1/2}}}$, we must know $\boldsymbol{N}$ and $\boldsymbol{t_{1/2}}$. The half-life of $\boldsymbol{^{14} \textbf{C}}$ can be found in Appendix B, and was stated above as 5730 y. To find $\boldsymbol{N}$, we first find the number of $\boldsymbol{^{12} \textbf{C}}$ nuclei in 1.00 kg of carbon using the concept of a mole. As indicated, we then multiply by $\boldsymbol{1.3 \times 10^{-12}}$ (the abundance of $\boldsymbol{^{14} \textbf{C}}$ in a carbon sample from a living organism) to get the number of $\boldsymbol{^{14} \textbf{C}}$ nuclei in a living organism.
Solution
One mole of carbon has a mass of 12.0 g, since it is nearly pure $\boldsymbol{^{12} \textbf{C}}$. (A mole has a mass in grams equal in magnitude to $\boldsymbol{A}$ found in the periodic table.) Thus the number of carbon nuclei in a kilogram is
$\boldsymbol{N(^{12} \textbf{C}) =}$$\boldsymbol{\frac{6.02 \times 10^{23} \;\textbf{mol}^{-1}}{12.0 \;\textbf{g/mol}}}$$\boldsymbol{\times (1000 \;\textbf{g}) = 5.02 \times 10^{25}}$
So the number of $\boldsymbol{^{14} \textbf{C}}$ nuclei in 1 kg of carbon is
$\boldsymbol{N(^{14} \textbf{C}) = (5.02 \times 10^{25})(1.3 \times 10^{-12}) = 6.52 \times 10^{13}}$
Now the activity $\boldsymbol{R}$ is found using the equation $\boldsymbol{R = \frac{0.693N}{t_{1/2}}}$.
Entering known values gives
$\boldsymbol{R =}$$\boldsymbol{\frac{0.693(6.52 \times 10^{13})}{5730 \;\textbf{y}}}$$\boldsymbol{= 7.89 \times 10^9 \;\textbf{y}^{-1}}$
or $\boldsymbol{7.89 \times 10^9}$ decays per year. To convert this to the unit Bq, we simply convert years to seconds. Thus,
$\boldsymbol{R = (7.89 \times 10^9 \;\textbf{y}^{-1})}$$\boldsymbol{\frac{1.00 \;\textbf{y}}{3.16 \times 10^7 \;\textbf{s}}}$$\boldsymbol{= 250 \;\textbf{Bq}}$
or 250 decays per second. To express $\boldsymbol{R}$ in curies, we use the definition of a curie,
$\boldsymbol{R =}$$\boldsymbol{\frac{250 \;\textbf{Bq}}{3.7 \times 10^{10} \;\textbf{Bq/Ci}}}$$\boldsymbol{= 6.76 \times 10^{-9} \;\textbf{Ci}}$
Thus,
$\boldsymbol{R = 6.76 \;\textbf{nCi}}$
Discussion
Our own bodies contain kilograms of carbon, and it is intriguing to think there are hundreds of $\boldsymbol{^{14} \textbf{C}}$ decays per second taking place in us. Carbon-14 and other naturally occurring radioactive substances in our bodies contribute to the background radiation we receive. The small number of decays per second found for a kilogram of carbon in this example gives you some idea of how difficult it is to detect $\boldsymbol{^{14} \textbf{C}}$ in a small sample of material. If there are 250 decays per second in a kilogram, then there are 0.25 decays per second in a gram of carbon in living tissue. To observe this, you must be able to distinguish decays from other forms of radiation, in order to reduce background noise. This becomes more difficult with an old tissue sample, since it contains less $\boldsymbol{^{14} \textbf{C}}$, and for samples more than 50 thousand years old, it is impossible.
Human-made (or artificial) radioactivity has been produced for decades and has many uses. Some of these include medical therapy for cancer, medical imaging and diagnostics, and food preservation by irradiation. Many applications as well as the biological effects of radiation are explored in Chapter 32 Medical Applications of Nuclear Physics, but it is clear that radiation is hazardous. A number of tragic examples of this exist, one of the most disastrous being the meltdown and fire at the Chernobyl reactor complex in the Ukraine (see Figure 3). Several radioactive isotopes were released in huge quantities, contaminating many thousands of square kilometers and directly affecting hundreds of thousands of people. The most significant releases were of $\boldsymbol{^{131} \textbf{I}}$, $\boldsymbol{^{90} \textbf{S}}$, $\boldsymbol{^{137} \textbf{Cs}}$, $\boldsymbol{^{239} \textbf{Pu}}$, $\boldsymbol{^{238} \textbf{U}}$, and $\boldsymbol{^{235} \textbf{U}}$. Estimates are that the total amount of radiation released was about 100 million curies.
# Human and Medical Applications
### Example 3: What Mass of 137Cs Escaped Chernobyl?
It is estimated that the Chernobyl disaster released 6.0 MCi of $\boldsymbol{^{137} \textbf{Cs}}$ into the environment. Calculate the mass of $\boldsymbol{^{137} \textbf{Cs}}$ released.
Strategy
We can calculate the mass released using Avogadro’s number and the concept of a mole if we can first find the number of nuclei $\boldsymbol{N}$ released. Since the activity $\boldsymbol{R}$ is given, and the half-life of $\boldsymbol{^{137} \textbf{Cs}}$ is found in Appendix B to be 30.2 y, we can use the equation $\boldsymbol{R= \frac{0.693N}{t_{1/2}}}$ to find $\boldsymbol{N}$.
Solution
Solving the equation $\boldsymbol{R = \frac{0.693N}{t_{1/2}}}$ for $\boldsymbol{N}$ gives
$\boldsymbol{N=}$$\boldsymbol{\frac{Rt_{1/2}}{0.693}}$
Entering the given values yields
$\boldsymbol{N=}$$\boldsymbol{\frac{(6.0 \;\textbf{MCi})(30.2 \;\textbf{y})}{0.693}}$
Converting curies to becquerels and years to seconds, we get
$\begin{array}{r @{{}={}}l} \boldsymbol{N} & \boldsymbol{\frac{(6.0 \times 10^6 \;\textbf{Ci})(3.7 \times 10^{10} \;\textbf{Bq/Ci})(30.2 \;\textbf{y})(3.16 \times 10^7 \;\textbf{s/y})}{0.693}} \\[1em] & \boldsymbol{3.1 \times 10^{26}} \end{array}$
One mole of a nuclide $\boldsymbol{^A X}$ has a mass of $\boldsymbol{A}$ grams, so that one mole of $\boldsymbol{^{137} \textbf{Cs}}$ has a mass of 137 g. A mole has $\boldsymbol{6.02 \times 10^{23}}$ nuclei. Thus the mass of $\boldsymbol{^{137} \textbf{Cs}}$ released was
$\begin{array}{r @{{}={}}l} \boldsymbol{m} & \boldsymbol{(\frac{137 \;\textbf{g}}{6.02 \times 10^{23}})(3.1 \times 10^{26}) = 70 \times 10^3 \;\textbf{g}} \\[1em] & \boldsymbol{70 \;\textbf{kg}} \end{array}$
Discussion
While 70 kg of material may not be a very large mass compared to the amount of fuel in a power plant, it is extremely radioactive, since it only has a 30-year half-life. Six megacuries (6.0 MCi) is an extraordinary amount of activity but is only a fraction of what is produced in nuclear reactors. Similar amounts of the other isotopes were also released at Chernobyl. Although the chances of such a disaster may have seemed small, the consequences were extremely severe, requiring greater caution than was used. More will be said about safe reactor design in the next chapter, but it should be noted that Western reactors have a fundamentally safer design.
Activity $\boldsymbol{R}$ decreases in time, going to half its original value in one half-life, then to one-fourth its original value in the next half-life, and so on. Since $\boldsymbol{R = \frac{0.693N}{t_{1/2}}}$, the activity decreases as the number of radioactive nuclei decreases. The equation for $\boldsymbol{R}$ as a function of time is found by combining the equations $\boldsymbol{N = N_0 e^{- \lambda t}}$ and $\boldsymbol{R = \frac{0.693N}{t_{1/2}}}$, yielding
$\boldsymbol{R = R_0 e^{- \lambda t}}$
where $\boldsymbol{R_0}$ is the activity at $\boldsymbol{t=0}$. This equation shows exponential decay of radioactive nuclei. For example, if a source originally has a 1.00-mCi activity, it declines to 0.500 mCi in one half-life, to 0.250 mCi in two half-lives, to 0.125 mCi in three half-lives, and so on. For times other than whole half-lives, the equation $\boldsymbol{R = R_0 e^{- \lambda t}}$ must be used to find $\boldsymbol{R}$.
### PhET Explorations: Alpha Decay
Watch alpha particles escape from a polonium nucleus, causing radioactive alpha decay. See how random decay times relate to the half life.
# Section Summary
• Half-life $\boldsymbol{t_{1/2}}$ is the time in which there is a 50% chance that a nucleus will decay. The number of nuclei $\boldsymbol{N}$ as a function of time is $\boldsymbol{N = N_0 e^{- \lambda t}}$,where $\boldsymbol{N_0}$ is the number present at $\boldsymbol{t=0}$, and $\boldsymbol{\lambda}$ is the decay constant, related to the half-life by
$\boldsymbol{\lambda =}$$\boldsymbol{\frac{0.693}{t_{1/2}}}$
• One of the applications of radioactive decay is radioactive dating, in which the age of a material is determined by the amount of radioactive decay that occurs. The rate of decay is called the activity $\boldsymbol{R}$:
$\boldsymbol{R=}$$\boldsymbol{\frac{\Delta N}{\Delta t}}$
• The SI unit for $\boldsymbol{R}$ is the becquerel (Bq), defined by
$\boldsymbol{1 \;\textbf{Bq} = 1 \;\textbf{decay/s}}$
• $\boldsymbol{R}$ is also expressed in terms of curies (Ci), where
$\boldsymbol{1 \;\textbf{Ci} = 3.70 \times 10^{10} \;\textbf{Bq}}$
• The activity $\boldsymbol{R}$ of a source is related to $\boldsymbol{N}$ and $\boldsymbol{t_{1/2}}$ by
$\boldsymbol{R =}$$\boldsymbol{\frac{0.693N}{t_{1/2}}}$
• Since $\boldsymbol{N}$ has an exponential behavior as in the equation $\boldsymbol{N=N_0 e^{- \lambda t}}$, the activity also has an exponential behavior, given by
$\boldsymbol{R = R_0 e^{- \lambda t}}$,
where $\boldsymbol{R_0}$ is the activity at $\boldsymbol{t=0}$.
### Conceptual Questions
1: In a $\boldsymbol{3 \times 10^9}$ -year-old rock that originally contained some $\boldsymbol{^{238} \textbf{U}}$, which has a half-life of $\boldsymbol{4.5 \times 10^{9}}$ years, we expect to find some $\boldsymbol{^{238} \textbf{U}}$ remaining in it. Why are $\boldsymbol{^{226} \textbf{Ra}}$, $\boldsymbol{^{222} \textbf{Rn}}$, and $\boldsymbol{^{210} \textbf{Po}}$ also found in such a rock, even though they have much shorter half-lives (1600 years, 3.8 days, and 138 days, respectively)?
2: Does the number of radioactive nuclei in a sample decrease to exactly half its original value in one half-life? Explain in terms of the statistical nature of radioactive decay.
3: Radioactivity depends on the nucleus and not the atom or its chemical state. Why, then, is one kilogram of uranium more radioactive than one kilogram of uranium hexafluoride?
4: Explain how a bound system can have less mass than its components. Why is this not observed classically, say for a building made of bricks?
5: Spontaneous radioactive decay occurs only when the decay products have less mass than the parent, and it tends to produce a daughter that is more stable than the parent. Explain how this is related to the fact that more tightly bound nuclei are more stable. (Consider the binding energy per nucleon.)
6: To obtain the most precise value of BE from the equation $\boldsymbol{\textbf{BE} = [ZM (^1 \textbf{H}) + Nm_n]c^2 - m(^A X)c^2}$, we should take into account the binding energy of the electrons in the neutral atoms. Will doing this produce a larger or smaller value for BE? Why is this effect usually negligible?
7: How does the finite range of the nuclear force relate to the fact that $\boldsymbol{\textbf{BE} /A}$ is greatest for $\boldsymbol{A}$ near 60?
### Problems & Exercises
Data from the appendices and the periodic table may be needed for these problems.
1: An old campfire is uncovered during an archaeological dig. Its charcoal is found to contain less than 1/1000 the normal amount of $\boldsymbol{^{14} \textbf{C}}$. Estimate the minimum age of the charcoal, noting that $\boldsymbol{2^{10} = 1024}$.
2: A $\boldsymbol{^{60} \textbf{Co}}$ source is labeled 4.00 mCi, but its present activity is found to be $\boldsymbol{1.85 \times 10^7}$ Bq. (a) What is the present activity in mCi? (b) How long ago did it actually have a 4.00-mCi activity?
3: (a) Calculate the activity $\boldsymbol{R}$ in curies of 1.00 g of $\boldsymbol{^{226} \textbf{Ra}}$. (b) Discuss why your answer is not exactly 1.00 Ci, given that the curie was originally supposed to be exactly the activity of a gram of radium.
4: Show that the activity of the $\boldsymbol{^{14} \textbf{C}}$ in 1.00 g of $\boldsymbol{^{12} \textbf{C}}$ found in living tissue is 0.250 Bq.
5: Mantles for gas lanterns contain thorium, because it forms an oxide that can survive being heated to incandescence for long periods of time. Natural thorium is almost 100% $\boldsymbol{^{232} \textbf{Th}}$, with a half-life of $\boldsymbol{1.405 \times 10^{10} \;\textbf{y}}$. If an average lantern mantle contains 300 mg of thorium, what is its activity?
6: Cow’s milk produced near nuclear reactors can be tested for as little as 1.00 pCi of $\boldsymbol{^{131} \textbf{I}}$ per liter, to check for possible reactor leakage. What mass of $\boldsymbol{^{131} \textbf{I}}$ has this activity?
7: (a) Natural potassium contains $\boldsymbol{^{40} \textbf{K}}$, which has a half-life of $\boldsymbol{1.277 \times 10^9}$ y. What mass of $\boldsymbol{^{40} \textbf{K}}$ in a person would have a decay rate of 4140 Bq? (b) What is the fraction of $\boldsymbol{^{40} \textbf{K}}$ in natural potassium, given that the person has 140 g in his body? (These numbers are typical for a 70-kg adult.)
8: There is more than one isotope of natural uranium. If a researcher isolates 1.00 mg of the relatively scarce $\boldsymbol{^{235} \textbf{U}}$ and finds this mass to have an activity of 80.0 Bq, what is its half-life in years?
9: $\boldsymbol{^{50} \textbf{V}}$ has one of the longest known radioactive half-lives. In a difficult experiment, a researcher found that the activity of 1.00 kg of $\boldsymbol{^{50} \textbf{V}}$ is 1.75 Bq. What is the half-life in years?
10: You can sometimes find deep red crystal vases in antique stores, called uranium glass because their color was produced by doping the glass with uranium. Look up the natural isotopes of uranium and their half-lives, and calculate the activity of such a vase assuming it has 2.00 g of uranium in it. Neglect the activity of any daughter nuclides.
11: A tree falls in a forest. How many years must pass before the $\boldsymbol{^{14} \textbf{C}}$ activity in 1.00 g of the tree’s carbon drops to 1.00 decay per hour?
12: What fraction of the $\boldsymbol{^{40} \textbf{K}}$ that was on Earth when it formed $\boldsymbol{4.5 \times 10^9}$ years ago is left today?
13: A 5000-Ci $\boldsymbol{^{60} \textbf{Co}}$ source used for cancer therapy is considered too weak to be useful when its activity falls to 3500 Ci. How long after its manufacture does this happen?
14: Natural uranium is 0.7200% $\boldsymbol{^{235} \textbf{U}}$ and 99.27% $\boldsymbol{^{238} \textbf{U}}$. What were the percentages of $\boldsymbol{^{235} \textbf{U}}$ and $\boldsymbol{^{238} \textbf{U}}$ in natural uranium when Earth formed $\boldsymbol{4.5 \times 10^9}$ years ago?
15: The $\boldsymbol{beta ^-}$ particles emitted in the decay of $\boldsymbol{^3 \textbf{H}}$ (tritium) interact with matter to create light in a glow-in-the-dark exit sign. At the time of manufacture, such a sign contains 15.0 Ci of $\boldsymbol{^3 \textbf{H}}$. (a) What is the mass of the tritium? (b) What is its activity 5.00 y after manufacture?
16: World War II aircraft had instruments with glowing radium-painted dials (see Chapter 31.1 Figure 1). The activity of one such instrument was $\boldsymbol{1.0 \times 10^5}$ Bq when new. (a) What mass of 226Ra226Ra was present? (b) After some years, the phosphors on the dials deteriorated chemically, but the radium did not escape. What is the activity of this instrument 57.0 years after it was made?
17: (a) The $\boldsymbol{^{210} \textbf{Po}}$ source used in a physics laboratory is labeled as having an activity of $\boldsymbol{1.0 \;\mu \textbf{Ci}}$ on the date it was prepared. A student measures the radioactivity of this source with a Geiger counter and observes 1500 counts per minute. She notices that the source was prepared 120 days before her lab. What fraction of the decays is she observing with her apparatus? (b) Identify some of the reasons that only a fraction of the $\boldsymbol{\alpha}$ s emitted are observed by the detector.
18: Armor-piercing shells with depleted uranium cores are fired by aircraft at tanks. (The high density of the uranium makes them effective.) The uranium is called depleted because it has had its $\boldsymbol{^{235} \textbf{U}}$ removed for reactor use and is nearly pure $\boldsymbol{^{238} \textbf{U}}$. Depleted uranium has been erroneously called non-radioactive. To demonstrate that this is wrong: (a) Calculate the activity of 60.0 g of pure $\boldsymbol{^{238} \textbf{U}}$. (b) Calculate the activity of 60.0 g of natural uranium, neglecting the $\boldsymbol{^{234} \textbf{U}}$ and all daughter nuclides.
19: The ceramic glaze on a red-orange Fiestaware plate is $\boldsymbol{\textbf{U}_2 \textbf{O}_3}$ and contains 50.0 grams of $\boldsymbol{^{238} \textbf{U}}$ , but very little $\boldsymbol{^{235} \textbf{U}}$. (a) What is the activity of the plate? (b) Calculate the total energy that will be released by the $\boldsymbol{^{238} \textbf{U}}$ decay. (c) If energy is worth 12.0 cents per kW⋅h, what is the monetary value of the energy emitted? (These plates went out of production some 30 years ago, but are still available as collectibles.)
20: Large amounts of depleted uranium ($\boldsymbol{^{238} \textbf{U}}$) are available as a by-product of uranium processing for reactor fuel and weapons. Uranium is very dense and makes good counter weights for aircraft. Suppose you have a 4000-kg block of $\boldsymbol{^{238} \textbf{U}}$. (a) Find its activity. (b) How many calories per day are generated by thermalization of the decay energy? (c) Do you think you could detect this as heat? Explain.
21: The Galileo space probe was launched on its long journey past several planets in 1989, with an ultimate goal of Jupiter. Its power source is 11.0 kg of $\boldsymbol{^{238} \textbf{Pu}}$, a by-product of nuclear weapons plutonium production. Electrical energy is generated thermoelectrically from the heat produced when the 5.59-MeV $\boldsymbol{\alpha}$ particles emitted in each decay crash to a halt inside the plutonium and its shielding. The half-life of $\boldsymbol{^{238} \textbf{Pu}}$ is 87.7 years. (a) What was the original activity of the $\boldsymbol{^{238} \textbf{Pu}}$ in becquerel? (b) What power was emitted in kilowatts? (c) What power was emitted 12.0 y after launch? You may neglect any extra energy from daughter nuclides and any losses from escaping $\boldsymbol{\gamma}$ rays.
Consider the generation of electricity by a radioactive isotope in a space probe, such as described in Chapter 31.5 Problems & Exercises 21. Construct a problem in which you calculate the mass of a radioactive isotope you need in order to supply power for a long space flight. Among the things to consider are the isotope chosen, its half-life and decay energy, the power needs of the probe and the length of the flight.
23: Unreasonable Results
A nuclear physicist finds $\boldsymbol{1.0 \;\mu \textbf{g}}$ of $\boldsymbol{^{236} \textbf{U}}$ in a piece of uranium ore and assumes it is primordial since its half-life is $\boldsymbol{2.3 \times 10^7 \;\textbf{y}}$. (a) Calculate the amount of $\boldsymbol{^{236} \textbf{U}}$ that would had to have been on Earth when it formed $\boldsymbol{4.5 \times 10^9 \;\textbf{y}}$ ago for $\boldsymbol{1.0 \;\mu \textbf{g}}$ to be left today. (b) What is unreasonable about this result? (c) What assumption is responsible?
24: Unreasonable Results
(a) Repeat Chapter 31.5 Problems & Exercises 14 but include the 0.0055% natural abundance of $\boldsymbol{^{234} \textbf{U}}$ with its $\boldsymbol{2.45 \times 10^5 \;\textbf{y}}$ half-life. (b) What is unreasonable about this result? (c) What assumption is responsible? (d) Where does the $\boldsymbol{^{234} \textbf{U}}$ come from if it is not primordial?
25: Unreasonable Results
The manufacturer of a smoke alarm decides that the smallest current of $\boldsymbol{\alpha}$ radiation he can detect is $\boldsymbol{1.00 \;\mu \textbf{A}}$. (a) Find the activity in curies of an α α emitter that produces a $\boldsymbol{1.00 \;\mu \textbf{A}}$ current of $\boldsymbol{\alpha}$ particles. (b) What is unreasonable about this result? (c) What assumption is responsible?
## Glossary
becquerel
SI unit for rate of decay of a radioactive material
half-life
the time in which there is a 50% chance that a nucleus will decay
an application of radioactive decay in which the age of a material is determined by the amount of radioactivity of a particular type that occurs
decay constant
quantity that is inversely proportional to the half-life and that is used in equation for number of nuclei as a function of time
carbon-14 dating
activity
the rate of decay for radioactive nuclides
rate of decay
the number of radioactive events per unit time
curie
the activity of 1g of $\boldsymbol{^{226} \textbf{Ra}}$, equal to $\boldsymbol{3.70 \times 10^{10} \;\textbf{Bq}}$
### Solutions
Problems & Exercises
1: 57,300 y
3: (a) 0.988 Ci
(b) The half-life of $\boldsymbol{^{226} \textbf{Ra}}$ is now better known.
5: $\boldsymbol{1.22 \times 10^3 \;\textbf{Bq}}$
7: (a) 16.0 mg
(b) 0.0114%
9: $\boldsymbol{1.48 \times 10^{17} \;\textbf{y}}$
11: $\boldsymbol{5.6 \times 10^4 \;\textbf{y}}$
13: 2.71 y
15: (a) 1.56 mg
(b) 11.3 Ci
17: (a) $\boldsymbol{1.23 \times 10^{-3}}$
(b) Only part of the emitted radiation goes in the direction of the detector. Only a fraction of that causes a response in the detector. Some of the emitted radiation (mostly $\boldsymbol{\alpha}$ particles) is observed within the source. Some is absorbed within the source, some is absorbed by the detector, and some does not penetrate the detector.
19: (a) $\boldsymbol{1.68 \times 10^{-5} \;\textbf{Ci}}$
(b) $\boldsymbol{8.65 \times 10^{10} \;\textbf{J}}$
(c) $\boldsymbol{ \ 2.9 \times 10^3}$
21: (a) $\boldsymbol{6.97 \times 10^{15} \;\textbf{Bq}}$
(b) 6.24 kW
(c) 5.67 kW
25: (a) 84.5 Ci
(b) An extremely large activity, many orders of magnitude greater than permitted for home use.
(c) The assumption of $\boldsymbol{1.00 \;\mu \textbf{A}}$ is unreasonably large. Other methods can detect much smaller decay rates. | 2020-03-28 12:34:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9755170941352844, "perplexity": 1182.8174201925865}, "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/1585370491857.4/warc/CC-MAIN-20200328104722-20200328134722-00283.warc.gz"} |
https://www.gktoday.in/aptitude/a-man-travelled-a-certain-distance-by-train-at-the-rate-of-25-kmph-and-walked-back-at-the-rate-of-4-kmph-if-the-whole-journey-took-5-hours-48-minutes-the-distance-was/ | # A man travelled a certain distance by train at the rate of 25 kmph. and walked back at the rate of 4 kmph. If the whole journey took 5 hours 48 minutes, the distance was:
A man travelled a certain distance by train at the rate of 25 kmph. and walked back at the rate of 4 kmph. If the whole journey took 5 hours 48 minutes, the distance was:
[A]15 km
[B]20 km
[C]25 km
[D]30 km
20 km
Let the distance be x km.
Total time = 5 hours 48 minutes
$= 5+\frac{48}{60} = \left ( 5+\frac{4}{5} \right )hours$
$= \frac{29}{5}hours$
$\therefore \frac{x}{25}+\frac{x}{4} = \frac{29}{5}$
$=> \frac{4x+25x}{100} = \frac{29}{5}$
$=> 5\times 29x = 29\times 100$
$=> x = \frac{29\times 100}{5\times 29} = 20km.$
Hence option [B] is the right answer.
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# Comments
• ##### Rakesh kumar sahoo
Reply
don’t understand
because x/25 and x/4
• ##### DIBYAJYOTI BHATTACHARYYA
At first instance it travels x Km at 25 kmph. So the time it took =x/25.
At second instance it travels x Km at 4 kmph. So the time it took =x/4.
Hence Total Time = x/25 + x/4 | 2018-09-25 00:53:34 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 6, "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": 0.5541849136352539, "perplexity": 3976.1947024306237}, "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-2018-39/segments/1537267160853.60/warc/CC-MAIN-20180925004528-20180925024928-00054.warc.gz"} |
https://search.r-project.org/CRAN/refmans/ELISAtools/html/read.annotations.html | ## Read the annotations of plates
### Description
Parse annotations for multiple ELISA plates from files, one annotation file and one standard concentration file, and output the annotations for standard and unknown separately.
### Usage
read.annotations(
annotation,
std.conc,
dir.annotation,
dir.stdConc,
num.plate = 1
)
### Arguments
annotation characters to specify the path and name of the annotation file std.conc characters to specify the standard concentration file. dir.annotation characters specifying the file to the annotatoin file. dir.stdConc characters specifying the path to the annotatoin file. num.plate numeric indicating the number of plates in the annotation files.
### Details
The annotation file may contain annotations for more than one plate. Each plate is marked by "Plate: plate 1..." and "~End". This function parses each section in both annotation file and standard concentration file. Then passes the section on to do the parsing. For each section,we expect the following format
1 2 3 4 ... C s1 s1 sample1 sample1 ... D s2 s2 sample2 sample ... ... ... ... ... ... ...
### Value
a list of annotations for elisa plates.
### Examples
#get example annotation file path from the system folder
ann<-system.file("extdata", "annote.txt", package="ELISAtools")
std.conc<-system.file("extdata", "stdConc.txt", package="ELISAtools")
#read them in and there are 2 plates. | 2023-02-06 06:46:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.22252796590328217, "perplexity": 8630.70821808553}, "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-2023-06/segments/1674764500304.90/warc/CC-MAIN-20230206051215-20230206081215-00392.warc.gz"} |
https://intelligencemission.com/free-energy-graph-free-electricity-for-electric-cars.html | ###### Historically, the term ‘free energy ’ has been used for either quantity. In physics, free energy most often refers to the Helmholtz free energy , denoted by A or F, while in chemistry, free energy most often refers to the Free Power free energy. The values of the two free energies are usually quite similar and the intended free energy function is often implicit in manuscripts and presentations.
Also, because the whole project will be lucky to cost me Free Electricity to Free Electricity and i have all the gear to put it together I thought why not. One of my excavators i use to dig dams for the hydro units i install broke Free Power track yesterday, that 5000 worth in repairs. Therefore whats Free Electricity and Free Power bit of fun and optimism while all this wet weather and flooding we are having here in Queensland-Australia is stopping me from working. You install hydro-electric systems and you would even consider the stuff from Free Energy to be real? I am appalled.
“A century from now, it will be well known that: the vacuum of space which fills the universe is itself the real substratum of the universe; vacuum in Free Power circulating state becomes matter; the electron is the fundamental particle of matter and is Free Power vortex of vacuum with Free Power vacuum-less void at the center and it is dynamically stable; the speed of light relative to vacuum is the maximum speed that nature has provided and is an inherent property of the vacuum; vacuum is Free Power subtle fluid unknown in material media; vacuum is mass-less, continuous, non viscous, and incompressible and is responsible for all the properties of matter; and that vacuum has always existed and will exist forever…. Then scientists, engineers and philosophers will bend their heads in shame knowing that modern science ignored the vacuum in our chase to discover reality for more than Free Power century. ” – Tewari
Thanks Free Electricity, you told me some things i needed to know and it just confirmed my thinking on the way we are building these motors. My motor runs but not the way it needs to to be of any real use. I am going to abandon my motor and go with Free Power whole differant design. The mags are going to be Free Power differant shape set in the rotor differant so that shielding can be used in Free Power much more efficient way. Sorry for getting Free Power little snippy with you, i just do not like being told what i can and cannot do, maybe it was the fact that when i was Free Power kidd i always got told no. It’s something i still have Free Power problem with even at my age. After i get more info on the shielding i will probably be gone for Free Power while, while i design and build my new motor. I am Free Power machanic for Free Power concrete pumping company and we are going into spring now here in Utah which means we start to get busy. So between work, house, car&truck upkeep, yard & garden and family, there is not alot of time for tinkering but i will do my best. Free Power, please get back to us on the shielding. Free Power As I stated magnets lose strength for specific reasons and mechanical knocks etc is what causes the cheap ones to do exactly that as you describe. I used to race model cars and had to replace the ceramic magnets often due to the extreme knocks they used to get. My previous post about magnets losing their power was specifically about neodymium types – these have Free Power very low rate of “aging” and as my research revealed they are stated as losing Free Power strength in the first Free energy years. But extreme mishandling will shorten their life – normal use won’t. Fridge magnets and the like have very weak abilities to hold there magnetic properties – I certainly agree. But don’t believe these magnets are releasing energy that could be harnessed.
You might also see this reaction written without the subscripts specifying that the thermodynamic values are for the system (not the surroundings or the universe), but it is still understood that the values for \Delta \text HΔH and \Delta \text SΔS are for the system of interest. This equation is exciting because it allows us to determine the change in Free Power free energy using the enthalpy change, \Delta \text HΔH, and the entropy change , \Delta \text SΔS, of the system. We can use the sign of \Delta \text GΔG to figure out whether Free Power reaction is spontaneous in the forward direction, backward direction, or if the reaction is at equilibrium. Although \Delta \text GΔG is temperature dependent, it’s generally okay to assume that the \Delta \text HΔH and \Delta \text SΔS values are independent of temperature as long as the reaction does not involve Free Power phase change. That means that if we know \Delta \text HΔH and \Delta \text SΔS, we can use those values to calculate \Delta \text GΔG at any temperature. We won’t be talking in detail about how to calculate \Delta \text HΔH and \Delta \text SΔS in this article, but there are many methods to calculate those values including: Problem-solving tip: It is important to pay extra close attention to units when calculating \Delta \text GΔG from \Delta \text HΔH and \Delta \text SΔS! Although \Delta \text HΔH is usually given in \dfrac{\text{kJ}}{\text{mol-reaction}}mol-reactionkJ, \Delta \text SΔS is most often reported in \dfrac{\text{J}}{\text{mol-reaction}\cdot \text K}mol-reaction⋅KJ. The difference is Free Power factor of 10001000!! Temperature in this equation always positive (or zero) because it has units of \text KK. Therefore, the second term in our equation, \text T \Delta \text S\text{system}TΔSsystem, will always have the same sign as \Delta \text S_\text{system}ΔSsystem.
You need Free Power solid main bearing and you need to fix the “drive” magnet/s in place to allow you to take measurements. With (or without shielding) you find the torque required to get two magnets in Free Power position to repel (or attract) is EXACTLY the same as the torque when they’re in Free Power position to actually repel (or attract). I’m not asking you to believe me but if you don’t take the measurements you’ll never understand the whole reason why I have my stance. Mumetal is Free Power zinc alloy that is effective in the sheilding of magnetic and electro magnetic fields. Only just heard about it myself couple of days ago. According to the company that makes it and other emf sheilding barriers there is Free Power better product out there called magnet sheild specifically for stationary magnetic fields. Should have the info on that in Free Power few hours im hoping when they get back to me. Hey Free Power, believe me i am not giving up. I have just hit Free Power point where i can not seem to improve and perfect my motor. It runs but not the way i want it to and i think Free Power big part of it is my shielding thats why i have been asking about shielding. I have never heard of mumetal. What is it? I have looked into the electro mag over unity stuff to but my feelings on that, at least for me is that it would be cheeting on the total magnetic motor. Your basicaly going back to the electric motor. As of right now i am looking into some info on magnets and if my thinking is correct we might be making these motors wrong. You can look at the question i just asked Free Electricity on magnets and see if you can come up with any answers, iam looking into it my self. | 2019-03-19 11:11:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8293318748474121, "perplexity": 1091.7204338829774}, "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-2019-13/segments/1552912201953.19/warc/CC-MAIN-20190319093341-20190319115341-00169.warc.gz"} |
http://gateoverflow.in/72537/hashing-probaility | 116 views
asked in DS | 116 views
Binomial trials
Here we require exactly k success where probability of success is 1/n and of failure is 1-1/n. So
Pr = nCk (1/n)^k(1-1/n)^(n-k)
So option b is correct
I got the same answer but given in key is A.
Thanks. | 2017-05-30 05:24:09 | {"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": 0.8242148160934448, "perplexity": 3471.973404973416}, "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-2017-22/segments/1495463613796.37/warc/CC-MAIN-20170530051312-20170530071312-00483.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/find-magnitude-two-vectors-veca-vecb-having-same-magnitude-such-that-angle-between-them-60-their-scalar-product-magnitude-and-direction-of-a-vector_12487 | # Find the Magnitude of Two Vectors Veca And Vecb , Having the Same Magnitude and Such that the Angle Between Them is 60° and Their Scalar Product - Mathematics
Find the magnitude of two vectors veca and vecb , having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2
#### Solution
Concept: Magnitude and Direction of a Vector
Is there an error in this question or solution?
#### APPEARS IN
NCERT Class 12 Maths
Chapter 10 Vector Algebra
Q 8 | Page 448 | 2021-03-05 08:15:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7655643820762634, "perplexity": 831.7547633641256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178370239.72/warc/CC-MAIN-20210305060756-20210305090756-00587.warc.gz"} |
https://video.ias.edu/node/17856 | # The stellar cusp around the Milky Way’s central black hole
## The stellar cusp around the Milky Way’s central black hole
Schödel, Rainer
November 30, 2017 | 2018-08-19 12:24:15 | {"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": 0.8931546211242676, "perplexity": 2806.2158344356017}, "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-2018-34/segments/1534221215077.71/warc/CC-MAIN-20180819110157-20180819130157-00389.warc.gz"} |
http://mathhelpforum.com/algebra/49794-quadratic-equations-print.html | • September 19th 2008, 01:54 PM
lightningstab714
Find the smallest possible value of the quantity $x^2 + y^2$ under the restriction that $2x + 3y = 6$
• September 19th 2008, 01:58 PM
mr fantastic
Quote:
Originally Posted by lightningstab714
Find the smallest possible value of the quantity $x^2 + y^2$ under the restriction that $2x + 3y = 6$
$2x + 3y = 6 \Rightarrow x = \frac{6 - 3y}{2}$ .... (1)
Substitute (1) into $L = x^2 + y^2$ and find the value of y that minimises L. Substitute this value into L to get the minimum value of L.
• September 19th 2008, 02:24 PM
lightningstab714
Thanks.
• September 19th 2008, 06:14 PM
pankaj
Here are two non calculus approaches to your problem:
Geometrically, $x^2+y^2$ denotes square of distance of the point $(x,y)$ from $(0,0)$ therefore,you have been asked to find the shortest distance between $(0,0)$ and the line $2x+3y=6$ in 2D space which is the perpendicular distance of 2x +3y=6 from (0,0) and which can be found from the formula
$p=\frac{|ax_{1}+by_{1}+c|}{\sqrt{a^2+b^2}}$
$=\frac{|2(0)+3(0)+6|}{\sqrt{2^2+3^2}}$
$=\frac{6}{\sqrt{13}}$.Therefore answer should be $
\frac{36}{13}$
Another method:
Put $x=r\cos\theta;y=r\sin \theta$ where $r=\sqrt{x^2+y^2}$
Plugging in $2x+3y=6$,we get
$r=\frac{6}{2\cos \theta +3\sin \theta}$
Now r will be minimum when denominator will be maximum.
Also $,-\sqrt{a^2+b^2} \leq a\cos\theta+b\sin\theta \leq\sqrt{a^2+b^2}$
Therefore, $r_{min}=\frac{6}{\sqrt{2^2+3^2}}=\frac{6}{\sqrt{13 }}$
and hence $x^2+y^2=\frac{36}{13}$ | 2014-07-30 20:59:47 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 22, "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": 0.9087917804718018, "perplexity": 648.5727597761233}, "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-2014-23/segments/1406510271648.4/warc/CC-MAIN-20140728011751-00393-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.gamedev.net/forums/topic/651500-sdl-threads-very-slow-is-it-my-code-or-is-this-normal/ | • ### Announcements
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# [SDL] Threads very slow? Is it my code or is this normal?
## 8 posts in this topic
I'm using SDL for graphics and threading. Basically I have one thread processing/generating information and saving the information to a file, and another thread reading information from those same files and playing the information back to the user. For some reason, if I have both threads running at the same time, they both run painfully slow. However if I run only one part of the program at once it runs pretty quickly.
There is no interaction between the threads whatsoever, except through files. The playback thread naturally runs slower than the other thread (at 10 ticks per second, while the generation thread has no cap) so it only needs to access files every several seconds, and the generation thread is always way ahead. In other words there's never a situation where both threads are fighting for file handle access.
I'm thinking it has something to do with both threads having tight loops. I read something about threading in C++11 where it always runs instructions from each thread in parallel, so basically thread B's speed is cut in half by thread A's "while (true) {}" loop code. I could use SDL_Delay() instead of a timer but thread A is the user interface and I don't want to freeze the whole screen and block user input.
What am I doing wrong?
// PLAY BACK THREAD LOOP (THREAD A)
while (true)
{
process_user_input();
if (is_time_for_next_frame())
{
if (has_frames_loaded()) playback_next_frame();
else read_frame_queue_from_file();
}
}
// PROCESSING THREAD LOOP (THREAD B)
while (true)
{
generate_frame();
push_frame_to_queue();
if (queue_is_full())
save_queue_to_file();
}
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Is there a reason for using the filesystem to do your communication here? Files will eventually hit the disk (though buffering might hide this in some cases), and the disk is slow, by orders of magnitude over even uncached memory accesses. If one thread is trying to read data from one area of the disk, and another thread is writing data to a different part of the disk, one thread is going to be waiting for the disk heads to seek.
There are other options, if persisting the data is not an actual requirement.
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Is there a reason for using the filesystem to do your communication here? Files will eventually hit the disk (though buffering might hide this in some cases), and the disk is slow, by orders of magnitude over even uncached memory accesses. If one thread is trying to read data from one area of the disk, and another thread is writing data to a different part of the disk, one thread is going to be waiting for the disk heads to seek.
There are other options, if persisting the data is not an actual requirement.
The generation thread is constantly making new files. The playback thread only needs to load files when it starts and then once every few minutes, and when it loads them it makes sure it loads every file available at the time (well, it has a cache limit, but it's pretty large). In other words the playback thread is idle disk-wise 99% of the time and it becomes less and less disk dependent as it runs since more and more files are loaded each time it hits the end of current playback. I thought it could be a disk access problem at first too so I made sure I wasn't making any dumb moves.
Edited by Nüb?ek Pænus
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Is there a reason for using the filesystem to do your communication here? Files will eventually hit the disk (though buffering might hide this in some cases), and the disk is slow, by orders of magnitude over even uncached memory accesses. If one thread is trying to read data from one area of the disk, and another thread is writing data to a different part of the disk, one thread is going to be waiting for the disk heads to seek.
There are other options, if persisting the data is not an actual requirement.
The generation thread is constantly making new files. The playback thread only needs to load files when it starts and then once every few minutes, and when it loads them it makes sure it loads every file available at the time (well, it has a cache limit, but it's pretty large). In other words the playback thread is idle disk-wise 99% of the time and it becomes less and less disk dependent as it runs since more and more files are loaded each time it hits the end of current playback. I thought it could be a disk access problem at first too so I made sure I wasn't making any dumb moves.
Getting thread A to sleep a little (even just a couple of milliseconds per loop) should improve things without negatively affecting the input processing, but I don't think that this can entirely explain your performance problems (especially if you're running on a multi-core machine).
How are you loading every file available on your playback thread? If you're asking the file system which files are available, then that's also a file system operation that may be hitting the disk resource. Do you have a mechanism to make sure you're not going to try to load a file that the file creator thread is still writing?
I'm pretty sure the problem has to lie with the disk access in some way or another. Perhaps you need a system where the newest written files are cached in memory and the reading thread can grab them from memory instead of the disk. Or maybe something simple would work, like create a mutex which the threads must have locked before they're allowed to do any disk access (including asking the file system what files exist).
One more thing, how small/numerous are the files you're generating? Possibly you might improve matters by having fewer, larger files.
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Is there a reason for using the filesystem to do your communication here? Files will eventually hit the disk (though buffering might hide this in some cases), and the disk is slow, by orders of magnitude over even uncached memory accesses. If one thread is trying to read data from one area of the disk, and another thread is writing data to a different part of the disk, one thread is going to be waiting for the disk heads to seek.
There are other options, if persisting the data is not an actual requirement.
The generation thread is constantly making new files. The playback thread only needs to load files when it starts and then once every few minutes, and when it loads them it makes sure it loads every file available at the time (well, it has a cache limit, but it's pretty large). In other words the playback thread is idle disk-wise 99% of the time and it becomes less and less disk dependent as it runs since more and more files are loaded each time it hits the end of current playback. I thought it could be a disk access problem at first too so I made sure I wasn't making any dumb moves.
Getting thread A to sleep a little (even just a couple of milliseconds per loop) should improve things without negatively affecting the input processing, but I don't think that this can entirely explain your performance problems (especially if you're running on a multi-core machine).
How are you loading every file available on your playback thread? If you're asking the file system which files are available, then that's also a file system operation that may be hitting the disk resource. Do you have a mechanism to make sure you're not going to try to load a file that the file creator thread is still writing?
I'm pretty sure the problem has to lie with the disk access in some way or another. Perhaps you need a system where the newest written files are cached in memory and the reading thread can grab them from memory instead of the disk. Or maybe something simple would work, like create a mutex which the threads must have locked before they're allowed to do any disk access (including asking the file system what files exist).
One more thing, how small/numerous are the files you're generating? Possibly you might improve matters by having fewer, larger files.
Hmm. Yeah it has to be a file system issue. I'll modify my program to be able to run only ONE thread per application instance and then run two instances of the program side by side, just to make sure, then I'll post results. Thanks!
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The generation thread is constantly making new files. The playback thread only needs to load files when it starts and then once every few minutes, and when it loads them it makes sure it loads every file available at the time (well, it has a cache limit, but it's pretty large). In other words the playback thread is idle disk-wise 99% of the time and it becomes less and less disk dependent as it runs since more and more files are loaded each time it hits the end of current playback. I thought it could be a disk access problem at first too so I made sure I wasn't making any dumb moves.
Ok, that is more an overview of what you're doing, I still don't understand what you're trying to achieve, and why you think that using the filesystem is the best mechanism.
Have you considered other communication mechanisms, for example using a socket or a pipe? These can act as a way to control the relative speeds of a producer and consumer, as the producer will block once it fills the buffer, and the consumer will block until there is data in the buffer.
Another approach is to utilise the shared address space. You can use SDL's atomics or synchronisation primitives such as mutexes or semaphores to control access to a shared data structure.
It seems like an odd design in a single process for one thread to write the data to the filesystem, and another to read it all back more or less immediately, when there are far more direct ways of moving data between the two threads.Of course, if persisting the data is necessary, then there are different considerations, but I still think you might be able to avoid a round trip to the disk.
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The generation thread is constantly making new files. The playback thread only needs to load files when it starts and then once every few minutes, and when it loads them it makes sure it loads every file available at the time (well, it has a cache limit, but it's pretty large). In other words the playback thread is idle disk-wise 99% of the time and it becomes less and less disk dependent as it runs since more and more files are loaded each time it hits the end of current playback. I thought it could be a disk access problem at first too so I made sure I wasn't making any dumb moves.
Ok, that is more an overview of what you're doing, I still don't understand what you're trying to achieve, and why you think that using the filesystem is the best mechanism.
Have you considered other communication mechanisms, for example using a socket or a pipe? These can act as a way to control the relative speeds of a producer and consumer, as the producer will block once it fills the buffer, and the consumer will block until there is data in the buffer.
Another approach is to utilise the shared address space. You can use SDL's atomics or synchronisation primitives such as mutexes or semaphores to control access to a shared data structure.
It seems like an odd design in a single process for one thread to write the data to the filesystem, and another to read it all back more or less immediately, when there are far more direct ways of moving data between the two threads.Of course, if persisting the data is necessary, then there are different considerations, but I still think you might be able to avoid a round trip to the disk.
I apologize for the lack of posts. I've been hard at work at reworking the system to make it work with less dependency on the filesystem (among other things that were broken).
Basically what I have now is:
• PLAYBACK thread requests information from a global (cross-thread) class (let's call it Timeline)
• GENERATION thread pushes information to the Timeline as it becomes available
• TIMELINE thread takes care of saving and writing data - it reads previously saved data when it seems like it might be requested soon, and it saves cached data when the buffer is near full. (Actually the timeline thread is currently disabled for debugging)
As before, the playback thread can be sped up or slowed down while the generation thread runs at full speed.
The system works fine as long as I keep the playback speed to default, however I start having problems when I run it at higher speeds. The faster it goes the higher the chance of a deadlock on either thread's part. When one thread crashes, the other continues as if nothing had happened, so it's not a regular deadlock where both threads wait for the other to finish.
I've tried with and without mutexes, but this changes nothing. I added in printf() statements before and after every mutex lock/unlock call and the output looks as it should. Then I added printf() statements in the playback thread (the one most likely to freeze) to track down exactly where it freezes but it simply stops after a certain function call. For example:
void func1()
{
printf("func1() was called"); fflush(stdout);
func2();
// code for func1 goes here
}
void func2()
{
printf("func2() was called"); fflush(stdout);
// code for func2 goes here
}
I get the func1() confirmation but not the func2() confirmation, meaning the thread stops running at the moment func2() gets called. I have several instances of this scenario all around my code now, so it's not just that particular piece of code. This can happen at any time, and it's always right after a function gets called. It just freezes. Also note how I added fflush() calls after each printf() call to make sure it gets written to the file.
I can't debug this because, well, it doesn't crash so my regular debugger (Dev-C++) doesn't inform me of anything. I tried running GDB from the command line but GDB freezes as well, at the start of the program. It does the usual "new thread at [0xblablabla]" but then it freezes instead of showing gdb> and waiting for a command.
I'd say the stack is messed up or something, however like I said, this only happens when both threads are running at high speeds. The program ran fine for 20 minutes straight if I kept the playback thread running slowly (10 ticks per second). I'm confused. I hope this is some noob mistake because I have no idea how to debug this. I can't post any actual code either because it's a rather large program.
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Are those member functions or free functions?
If multiple threads are accessing common memory, you'll probably need some kind of proper synchronisation. There are lots of really weird behaviour that can occur when multiple threads read and write data at the same time. For example, it is not guaranteed that all threads will see the same value of a variable if it has changed in another thread, they might see the old value indefinitely, they might see the new value after a non-deterministic amount of time, or they might see a partial change! Using proper threading primitives, such as a mutex, in the correct way can fix this.
Dev-C++ is rather old and unmaintained, and the compiler toolchain it ships with is very out of date. I'd highly suggest you use more modern IDE, two popular choices are Microsoft's Visual C++ (you can get free versions) and Code::Blocks. Both will ship with up to date compilers.
That aside, I'd recommend attempting to reduce the program as much as possible to the essential case which continues to demonstrate the problem. Unfortunately, threading bugs can be very timing dependent, which complicates this approach - just because removing a certain portion of code changes the result, does not mean that code is responsible for the problem.
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Are those member functions or free functions?
If multiple threads are accessing common memory, you'll probably need some kind of proper synchronisation. There are lots of really weird behaviour that can occur when multiple threads read and write data at the same time. For example, it is not guaranteed that all threads will see the same value of a variable if it has changed in another thread, they might see the old value indefinitely, they might see the new value after a non-deterministic amount of time, or they might see a partial change! Using proper threading primitives, such as a mutex, in the correct way can fix this.
Dev-C++ is rather old and unmaintained, and the compiler toolchain it ships with is very out of date. I'd highly suggest you use more modern IDE, two popular choices are Microsoft's Visual C++ (you can get free versions) and Code::Blocks. Both will ship with up to date compilers.
That aside, I'd recommend attempting to reduce the program as much as possible to the essential case which continues to demonstrate the problem. Unfortunately, threading bugs can be very timing dependent, which complicates this approach - just because removing a certain portion of code changes the result, does not mean that code is responsible for the problem.
The funny part is it behaves in the exact same way with and without mutexes. Speaking of which, I just posted a new thread asking a very basic question about mutexes, before I saw your reply. I really need to turn on notifications. Would you mind taking a look? If my train of thought isn't total rubbish then that could be my problem. http://www.gamedev.net/topic/652395-are-mutexes-really-fool-proof/
If that doesn't pan out as a possibility then my I'll have to somehow strip the program down as you've said. Not a pretty thought, but if it has to be done then so be it. I thought about using sockets but my data is too big for me to afford keeping multiple copies in memory like that. About 40MB per object, needing to keep at least 21 loaded and uncompressed at any given point. That'd be a minimum of 840MB per playback thread. nope.jpg
Thank you!
PS: I'm using this version of Dev-C++. I don't know how he did it but it's much better than the unmaintained original Bloodshed version. No more random crashes and no more immortal tooltips, yet it does retain the magic of Dev-C++.
Edited by Nüb?ek Pænus
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• 11 | 2017-07-28 07:47:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.24572286009788513, "perplexity": 1261.2728009500495}, "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-2017-30/segments/1500549448095.6/warc/CC-MAIN-20170728062501-20170728082501-00396.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/volume-cuboid-a-box-lid-made-2-cm-thick-wood-its-external-length-breadth-height-are-25-cm-18-cm-15-cm-respectively_38069 | Share
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# A Box with Lid is Made of 2 Cm Thick Wood. Its External Length, Breadth and Height Are 25 Cm, 18 Cm and 15 Cm Respectively - CBSE Class 9 - Mathematics
ConceptVolume of a Cuboid
#### Question
A box with lid is made of 2 cm thick wood. Its external length, breadth and height are 25 cm, 18 cm and 15 cm respectively. How much cubic cm of a liquid can be placed in it? Also, find the volume of the wood used in it.
#### Solution
Given external dimensions of cuboid are
l=25cm, b=18cm,h=15cm.
∴External volume =lxxbxxh
=25xx18xx15cm^2
=6750cm^2
Internal dimension of cuboid.
l=25-2xx"thickness"=25-4=21cm.
h=15-4=11cm.
internal volume = l x b x h
=21xx14xx11cm^3
=3234cm^3
∴Volume of liquid that can be placed =3234cm^3
Now, volume of wood = external volume – Internal volume
=6750-3324
=3516 cm^3
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Solution for Mathematics for Class 9 by R D Sharma (2018-19 Session) (2018 to Current)
Chapter 18: Surface Areas and Volume of a Cuboid and Cube
Ex.18.20 | Q: 11 | Page no. 30
Solution A Box with Lid is Made of 2 Cm Thick Wood. Its External Length, Breadth and Height Are 25 Cm, 18 Cm and 15 Cm Respectively Concept: Volume of a Cuboid.
S | 2019-08-18 01:37:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.49679675698280334, "perplexity": 2759.7680572928302}, "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-2019-35/segments/1566027313536.31/warc/CC-MAIN-20190818002820-20190818024820-00321.warc.gz"} |
http://math.stackexchange.com/questions/79555/maximizing-a-function-containing-an-integral | # Maximizing a function containing an integral
Problem. Let $\rho\colon[-1,\infty)\to\mathbb{R}$ be a function such that $$\int_{-1}^\infty\rho(x)\,dx=1.$$ Let $G\colon[0,1]\to\mathbb{R}$ be a function that is defined with $$G(f) := \int_{-1}^\infty \log(1+fx)\,\rho(x)\,dx.$$ Show that $G$ is either monotonically increasing, or has one local maximum on the interval $(0,1)$ (depending on how “fat” the “tail” of the function $\rho$ is).
Real life motivation. Suppose that we start with capital $V_0$, and we can repeatedly invest it in some venture with a percentage return given by a fixed probability distribution. This distribution is continuous, but it never takes values below $-1$ (we can never lose more than our investment). Its density function is the function $\rho$ as shown above.
Now, let $X_1,X_2,\ldots$ be independent random variables having this distribution. Furthermore, assume that we always invest a fixed fraction $f$ of our current capital. After $n$ investments, then, our capital will be $$V_n = V_0(1+fX_1)\cdots(1+fX_n).$$
We wish to measure how fast our capital is growing, and to that end we define $$G := \lim_{n\to\infty} \frac{1}{n}\log\frac{V_n}{V_0}.$$ Clearly, the faster $V_n$ grows, the bigger $G$ is. And by the weak law of large numbers, we see that $G$ is indeed the integral shown above. Our goal is to say something useful about the fraction $f$ of our current capital that we invest each time.
-
Second derivative of $G(f)$ with respect to $f$ is strictly less than 0, making it a concave function, giving it the properties you want? – user4143 Nov 19 '11 at 0:49
In your motivation, you say that $\rho$ is a density, so I'm assuming that $\rho\geq0$. Then one can differentiate under the integral, provided $\rho$ decays sufficiently rapidly ($\rho \in L^p$ for $1\leq p \leq 2$ is sufficient by dominated convergence theorem) to obtain
$\displaystyle G'(f)=\int_{-1}^\infty \frac{f\rho(x)}{1+fx} dx \geq 0$
for $f\in (0,1)$, hence $G$ is monotone. Since $\rho$ is a density, it is in particular an $L^1$ function, and so this works for all $\rho$.
Doesn't seem like you can get a local maximum on $(0,1)$. If $\rho$ is indeed a density, then $G$ is actually stritly monotone for $f\in(0,1)$. But someone please correct me if I'm wrong.
-
When we differentiate $G$ with respect to $f$, shouldn’t we get $x$ (instead of $f$) in the numerator? If that’s the case, $G'$ can indeed take negative values. – Topology Nov 8 '11 at 22:18
Yes you are right. I knew something must have been wrong because it was too easy. I'm not sure about the answer then, I'll have to think about it a bit more. – Jeff Nov 8 '11 at 23:02
I found out that I’m simply wrong.
What I want to show is only true if the expected value corresponding to the density is positive. To see that, note that the first derivative of $G(f)$ at $f=0$ is the expected value. Note also (as @user4143 has mentioned) that the second derivative of $G(f)$ is always negative.
If the expected value is negative, however, $G(f)$ is—because of the properties of the derivatives mentioned above—a decreasing function.
- | 2015-07-31 20:14:40 | {"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": 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": 0.9471035003662109, "perplexity": 166.79494104329967}, "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-20150728002308-00021-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://qanda.ai/en/solutions/rJ3pJ4qETT-E-Learning-Task-2-Give-the-reciprocal-of-the-given-fractions-below-Write-your-an | Symbol
Problem
$E$ Learning Task $2$ Give the reciprocal of the given fractions below. Write your answer in your notebook. 1. $6$ $6$ $\dfrac {2} {7}$ 8 $2$ 7. $\dfrac {1} {8}$ $\dfrac {7} {2}$ $3$ 8. $33\dfrac {4} {5}$ $4-$ $41$ 9. $\dfrac {4} {5}$ $\dfrac {5} {6}$ $5$ $10$ $54\dfrac {1} {5}$ $\dfrac {1} {5}$ $A$ | 2021-04-16 20:27:34 | {"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": 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": 0.9182522892951965, "perplexity": 170.91075548799466}, "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/1618038089289.45/warc/CC-MAIN-20210416191341-20210416221341-00601.warc.gz"} |
http://mathhelpforum.com/algebra/202604-inequalities-variable-denominator-both-sides-print.html | # Inequalities: Variable in Denominator both sides
• Aug 27th 2012, 10:49 AM
bajpaiapurva
Inequalities: Variable in Denominator both sides
Please tell me how this question will be solved
1/(x+2) > 3/(x-3)
answer: (-9/2, -2) U (3,+infinity)
Though i know the answer but i dont how how will get it
Thanx
• Aug 27th 2012, 11:20 AM
earboth
Re: Inequalities: Variable in Denominator both sides
Quote:
Originally Posted by bajpaiapurva
Please tell me how this question will be solved
1/(x+2) > 3/(x-3)
answer: (-9/2, -2) U (3,+infinity)
Though i know the answer but i dont how how will get it
Thanx
$\frac1{x+2}>\frac3{x-3}~\implies~\frac{x-3-3(x+2)}{(x+2)(x-3)}>0$
Examine when a fraction is greater than zero.
If + means positive (> 0) and - means negative (<0) you'll get the following cases:
$\frac{+}{+ \cdot +}~\vee~\frac{+}{- \cdot -} ~\vee~\frac{-}{+ \cdot -}~\vee~\frac{-}{- \cdot +}$
Each fraction describes 3 inequalities.
• Aug 27th 2012, 11:30 AM
Plato
Re: Inequalities: Variable in Denominator both sides
Quote:
Originally Posted by bajpaiapurva
Please tell me how this question will be solved
$\frac{1}{(x+2)} > \frac{3}{(x-3)}$
answer: $(-9/2, -2) \cup (3,\infty)$
Though i know the answer but i dont how how will get it
Thanx
That answer is incorrect. It includes $x=5$ which does not work.
The given answer works for $\frac{1}{(x+2)} < \frac{3}{(x-3)}$.
Please check the post. | 2017-01-20 21:05:11 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "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": 0.8283584117889404, "perplexity": 3100.8092178973893}, "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-2017-04/segments/1484560280872.69/warc/CC-MAIN-20170116095120-00222-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.gamedev.net/forums/topic/323424-planet-sized-level-format/ | # Planet-sized level format?
## Recommended Posts
Mobile 100
I'm trying to gather ideas for a system to create a large terrain "world." The requirements are that it needs to feel like you're on a planet (e.g. you can start at one end and walk for a long time and end up back where you started), and there must be only a single "noticable" loading time. I was thinking maybe a streaming system could eliminate the noticable loading by loading level "patches" in the background as you travel. Anyone have any ideas or are there any papers coving this? Thanks
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MatrixCubed 199
Games did this 20 years ago. Ultima games used a system of realtime loading of groups of tiles to maintain a veil that the world was endless, with little to no load time. In 1999, Ultima IX did this in a 3D setting; the player could walk from one side of the world to the other, take hours to do so, and parts of the map would load dynamically with no visible performance hit. I don't think the world was continual in a spherical sense, but it's just a planned feature I guess; not impossible.
I'd be happy to discuss this in detail; let me know, as I have loads of ideas.
Cheers,
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Mobile 100
I wonder how well this would work out...
The map is made out of 9 patches of terrain (the patches are rather large) in a grid. Let's say you're standing in grid 1 and go directly to your left. As you walk, grid 2 will load and eventually grid 3. After grid 3, grid 1 will load again. At least I think that's how it would work. Let's say you're in grid 3 on the same path as described earlier... taking a sharp left from there would take you to grid 9... but what about diagnially from 3? Hmm... Let me draw a grid.
------------------------|| | | | || 7 | 8 | 9 | ? || | | | |------------------------|| | | | || 1 | 2 | 3 | 1 || | | | |------------------------|| | | | || 4 | 5 | 6 | 4 || | | | |------------------------|| | | | || 7 | 8 | 9 | 7 || | | | |------------------------|
Does that make any sense? Perhaps a grid is not the best way to represent a globe o_O
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Guest Anonymous Poster
As long as the size or location of any "zone" isn't larger than the effective floating point precision. That entails changing the local reference frame continually. I think this may have been what happened with Serious Sam if you walked too far off the beaten path.
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Mobile 100
Quote:
Original post by MatrixCubedI'd be happy to discuss this in detail; let me know, as I have loads of ideas.
Yes, please :) I'm eager to discuss this!
edit: Also, technically, the world of course never has to actually be spherical at all. It just has to seem like what you're on is a planet.
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Yohumbus 152
Well that kind of top to bottom/left to right movement emulates a sphere well enough that users probably wont know or wont care but it is actually a torus. If you would like to read up I would suggest a great article on how these topologies work at everything2.com which can be located at http://www.everything2.com/index.pl?node_id=746760&lastnode_id=1140332 . Another thought is to use a lahey space which is just like the wraparound effect you see in most games but whenever you wrap your direction is taken into effect and you are made to end up where you started. There arnt too many sources over that topology so I will point you to where I heard of it in the funge-98 code specification (funge-98 is a whole different beast, interesting none the less). go here for that and search for lahey space and how it wraps around http://catseye.mine.nu:8080/projects/funge98/doc/funge98.html
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jflanglois 1020
You should probably take a look at The Continuous World of Dungeon Siege. It deals with just what you are describing.
jfl.
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hplus0603 11356
If you need a REAL planet that's sized like a REAL world (and round), then this is slightly harder. For example, the amount of data you need to store would be huge, unless you store it at very coarse grain, and let some procedural algorithm fill in the blanks.
For There, we made the world approximately earth sized. This means that you can get in a car and drive (our cars are amphibious), and it'll take three weeks of continual driving to circumnavigate the globe. A member did it (by taping down the "gas" key) just to see whether it'd work. It did :-)
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Mobile 100
Quote:
Original post by Anonymous PosterAs long as the size or location of any "zone" isn't larger than the effective floating point precision. That entails changing the local reference frame continually. I think this may have been what happened with Serious Sam if you walked too far off the beaten path.
Can anyone elaborate a little more on what the AP said? I read in the Dungeon Seige paper that they had problems with the FPU as well. Perhaps I'm just undereducated in this area of 3D programming? ;)
Thanks
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Guest Anonymous Poster
Quote:
Original post by Mobiletaking a sharp left from there would take you to grid 9... but what about diagnially from 3? Hmm... Let me draw a grid.------------------------|| | | | || 7 | 8 | 9 | ? || | | | |------------------------|| | | | || 1 | 2 | 3 | 1 || | | | |------------------------|| | | | || 4 | 5 | 6 | 4 || | | | |------------------------|| | | | || 7 | 8 | 9 | 7 || | | | |------------------------|
7
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tolaris 288
Quote:
Original post by MobileCan anyone elaborate a little more on what the AP said? I read in the Dungeon Seige paper that they had problems with the FPU as well. Perhaps I'm just undereducated in this area of 3D programming? ;)
IIRC it's similar to the issue with z-buffer precision, i.e. 'resolution' of floats is not linear. If you move far enough from the centre of your world to the point where coordinates become 'large enough', you'll start noticing distortions in geometry caused by vertices veering off their expected positions. In order to fix this the origin of coordinate system needs to be relocated so what was too far from it now becomes sufficiently close to maintain desired precision. o.O;
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Mobile 100
Thanks for confirming that AP
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Mobile 100
Quote:
Original post by tolarisIn order to fix this the origin of coordinate system needs to be relocated so what was too far from it now becomes sufficiently close to maintain desired precision. o.O;
How is this done? I didn't know you could relocate the origin...
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tolaris 288
Well technically, you can't. In practice, i figure moving coordinates of your object say, 50 units towards the origin ... is about the same as moving the origin 50 units towards the object? ^^; (since the coordinate values shrink by the same either way)
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Mobile 100
Well, let's use my grid technique for an example...
Let's say a grid is 100 "units" tall and wide and the origin starts at the upper left corner of the grid (see my grid for reference).
I am in grid 1 and I am facing grid 2. I walk through grid 1 to grid 2 and then onto grid 3. Past grid 3 is grid 1 again, but AFAIK I'm still actually 400 (or so) units from the origin, correct?
With that in mind, what should I be doing in order to prevent horrible z-buffer issues as I move further and further? Sorry if this question was answered and I just didn't get it ^_^
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Fingers_ 410
Quote:
Original post by Mobile
Quote:
Original post by tolarisIn order to fix this the origin of coordinate system needs to be relocated so what was too far from it now becomes sufficiently close to maintain desired precision. o.O;
How is this done? I didn't know you could relocate the origin...
You should read about the Dungeon Siege engine... The idea is that the contents of each "sector" of the world are stored in a coordinate system relative to the sector's origin so their coordinates are kept small. Each sector also contains a list of other sectors that connect to it, and where they are located relative to this sector's origin. When rendering, you start with the sector the camera is in, then draw the neighboring sectors (translated to where they connect to the current sector) and then their neighbors etc until you reach the maximum view distance.
As long as your maximum view distance is reasonable, you'll never run into floating point precision problems and the world can be infinitely large. The disadvantage is that the concept of a global coordinate system doesn't exist and you can't conveniently go to "location xyz". Creating a world map is also going to be difficult, as the system doesn't automatically result in a continuous non-overlapping world.
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Synex 170
Most of it all comes down to coordinate systems in the end. If you have enough coord-systems running at one you could pretty much simulate anything.
Trouble is getting your head around all the math involved in translating stuff all over the place through the coord-systems.
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So how big do you have to go before floating point precision kicks in? (roughly).
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Fingers_ 410
Depends on a lot of things, like how much precision you need. For example, if your physics code uses small time steps and there are objects moving at a slow speed they will move very little during each step and require good precision.
An object moving at 1m/s (slow walking speed) in a simulation running at 100fps moves 0.01m per frame. To make it possible to move smoothly in all directions, you need a precision of 0.0002m or so (moving in a direction one degree off north, you move +0.01 in Y and 0.0002 in X). When a floating point number gets big enough, adding 0.0002 to it ceases to have an effect and that's when you start running into problems... The object will move directly north instead of north-northeast.
32-bit floats have 6-7 digits of precision, so the maximum magnitude of a coordinate should be no more than 1-10 million times greater than the smallest movement step that has to be reliable. Given the smallest step of 0.0002m, the maximum size of a continuous coordinate system is 400-4000m... Which is where the world sizes of most shooter games end up.
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Guest Anonymous Poster
I'm dealing with similar idea but what I want achieve is procedural generated endless terrain. My heightmapping unit uses Perlin algorithm. When we generate heightmap first thing is to get a bunch of random numbers. To achieve this we must use srand() routine. But when we use srand with constant number (f.e. 100) generated heightmap will be identical each time we create it.
So, my idea is based on kind of grid with seed numbers. With small bunch of these numbers we will be able to generate huge world. But there is also a dark side of this concept. We need algorithm which can fit edges of generated terrain patches. And unfortunately I don't have idea for this algo. It must be extremly efficient to fit terrain edges on fly.
Anyone of You have some ideas ? I'll be grateful for any help.
Regards,
Holrin.
Sorry for my weak english :)
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Ezbez 1164
Going back up to the grid that Mobile posted, if you change it to look like this...
------------------------|
| | | | |
| 7 | 8 | 7 | 8 |
| | | | |
------------------------|
| | | | |
| 1 | 2 | 3 | 1 |
| | | | |
------------------------|
| | | | |
| 4 | 5 | 6 | 4 |
| | | | |
------------------------|
| | | | |
| 10 | 9 | 10 | 9 |
| | | | |
------------------------|
...then it will create a shpere(a demented sphere becuase its made out of squres, but thats no less demented than the torus would be) instead of a torus. This way is just like a map. When you come off the top of a map, you come back facing downwards at a different point on the top of the map. The edges are left and right sides are still the same(again, just like a map). Note that if you move off a corner square to either the left or the right, you will come out on the other corner square in the same row.
Sorry that the grid didn't come out very well. It should still give you the idea.
[Edited by - Ezbez on June 15, 2005 2:55:04 PM]
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kosmon_x 205
I was actually thinking about this the other day, and here is the grid I came up with :
------------------------------|| / | / | / | / | / || 9 | 7 | 8 | 9 | 7 || / | / | / | / | / |------------------------------|| / | | | | / || 3 | 1 | 2 | 3 | 1 || / | | | | / |------------------------|-----|| / | | | | / || 6 | 4 | 5 | 6 | 4 || / | | | | / |------------------------|-----|| / | | | | / || 9 | 7 | 8 | 9 | 7 || / | | | | / |------------------------|-----|| / | / | / | / | / || 3 | 1 | 2 | 3 | 1 ||/ |/ | / | / | / |------------------------------|
The 9 locations in the center are the only locations that would actually 'exist'. All of the locations around the perimeter would only be fake 'pointers' to real locations in the middle -- you can't actually exist in these locations, but you can 'see' them. My idea was that as you start in square 5 and travel east, you will enter square 6. From there, you may be able to see square 4 on the horizon ( wrapping around ). So, you just draw square 4 where it looks like it should be (the fake square 4). If you keep heading east in square 6, you just warp the player or whatever back into the actual square 4, and then square 5 is on the horizon again to the east. Then it seems like a wrap-around world.
Similarily, if youre in square 4 heading west, you see the fake 6 on the horizon. As soon as you leave the edge of square 4, instead of entering the fake square 6, you are warped back to the 'real' square 6 on the other side of the map. The warping would be simple: if the left edge of the actual square4 has an X coordinate of 0, and the right edge of square 6 has an X coord of 600 (for example), as soon as your position gets > 600, subtract 600 from it. The player should pop back to square4, however the visuals shouldnt change at all since they've been looking at the fake square4 the entire time. (This only works if multiple squares aren't visible straight ahead from the left edge of a square.. otherwise you'd need a thicker fake border around the middle)
Anyone think this would work?
[Edited by - kosmon_x on June 15, 2005 11:38:03 AM]
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Ezbez 1164
That sounds like it would work, Modile. Of course, its still a torus, but that is no biggie. The only other thing that I can think that might go wrong, is that when you get moved to another square, it could be alot of sudden loading, which is not wanted.
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c0mas 107
Just an ideea, why use a full grid ? There may be some cases where a square in your grid is never visible and you can use that memory for other square and to enlarge your world. I think it's easy to modify the algorithm to allow nonsquare grid with empty places.
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Guest Anonymous Poster
Quote:
Original post by c0masJust an ideea, why use a full grid ? There may be some cases where a square in your grid is never visible and you can use that memory for other square and to enlarge your world. I think it's easy to modify the algorithm to allow nonsquare grid with empty places.
Sure, but unless the basic square-grid algorithm for generating a torus that fakes a sphere works in the first place, this won't work.. | 2017-09-19 12:05:36 | {"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": 0.3844320774078369, "perplexity": 718.1324825876908}, "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/1505818685129.23/warc/CC-MAIN-20170919112242-20170919132242-00354.warc.gz"} |
https://www.nature.com/articles/s41467-018-04067-1 | ## Introduction
Solar water splitting, by harvesting abundant solar energy and storing it in the clean chemical energy form1,2,3,4, has been considered as one of the most promising approaches of renewable energy production and environmental remediation5,6,7,8. Significantly, the capacity to directly split seawater is ideally suited for large scale solar fuel production. While tremendous progress has been made in photoelectrochemical (PEC) water splitting in the past decades, it is not suited to split nearly pH neutral water such as sea water. Alternatively, photochemical or photocatalytic dissociation of water has been intensively studied, which can perform one-step overall neutral water splitting. In this approach, the counter electrode, in the form of micro/nano-scale co-catalyst, is monolithically integrated on the photocatalyst surface9,10,11. As such, it has often been referred to as a wireless version of PEC water splitting, which does not require conductive electrolyte and conductive substrate for its operation, as the ionic diffusion problem is greatly reduced due to integrated nature of the device. Owing to the much simpler configuration, photocatalytic water splitting is amenable to cheap, large-scale hydrogen generation12,13,14,15,16. However, one inherent limitation with this approach is the simultaneous production of hydrogen (H2) and oxygen (O2) gases in proximity to each other, which need to be separated safely for practical applications, such as in a Fuel cell.
Both photocatalytic and PEC water splitting approaches intrinsically require four photons, i.e., four active electron–hole pairs to split two water molecules (H2O) into one O2 and two H2 molecules, described as, $$2{\mathrm{H}}_2{\mathrm{O}} = 2{\mathrm{H}}_2 + {\mathrm{O}}_2$$ (Supplementary Note 1). However, to achieve unassisted overall water splitting using the single-absorber photocatalytic process, the conduction and valence band edges of the photocatalyst must straddle the proton reduction and water oxidation (redox) potentials, respectively, while possessing a sufficiently narrow bandgap to absorb a large part of the solar spectrum. Due to such stringent requirements, there are very few photocatalysts that can perform unassisted, overall water splitting reaction under visible light irradiation. In addition, for semiconductor-based photocatalysts, efficient separation of photo-generated charge carriers (electrons and holes) towards the appropriate catalytic sites has remained challenging, particularly for photocatalytic water splitting wherein no external bias is applied. While numerous efforts have been undertaken to address the critical issue of efficient charge separation through surface and interface engineering and the selective loading of co-catalysts to create spatially separated redox reaction sites11,17,18,19,20,21,22, it has remained elusive to achieve a precise control of charge carrier flow within the bulk and to steer photo-excited electrons and holes to their reduction and oxidation sites, respectively. To date, the best reported solar-to-hydrogen (STH) conversion efficiency for pH neutral photocatalytic water splitting is limited to ~1%, or less, compared to the 10–16% reported for PEC devices13,16,23,24,25,26,27,28,29 in conductive electrolytes. Recently, it has been discovered that the energy bandgap of group III-nitride semiconductors, e.g., Ga(In)N, can straddle water redox potentials for a wide absorption wavelength range (ultra-violet, visible, and near-infrared)30,31,32, thereby holding an enormous promise for high efficiency one-step overall water splitting33,34,35,36. Moreover, the surfaces of III-nitride semiconductors can be tuned to be nitrogen-rich to protect against photo-corrosion and oxidation37,38, thus making them suitable for stable and efficient photocatalysis.
In this work, we propose and demonstrate multi-band InGaN nanosheet photochemical diode (PCD)39 structures, which can spontaneously induce charge carrier separation and steer charge carriers toward the distinct redox sites for water oxidation and proton reduction. During the synthesis of InGaN photochemical diode nanosheet structure, p-type dopant (Mg) concentrations are rationally tailored, which induces a large built-in electric field between the two parallel surfaces. Consequently, the two surfaces are enriched with photo-generated holes and electrons to perform water oxidation and proton reduction reactions, respectively. In addition to the efficient charge carrier separation and extraction, the spatial separation of catalytic sites in such a nanoscale photochemical diode effectively reduces carrier recombination and back reaction. Subsequently, we demonstrate herein a double-band InGaN nanosheet device, which exhibits a solar-to-hydrogen conversion efficiency of ~3.3% for pH neutral overall water splitting. The capacity to achieve controllable charge carrier separation and extraction at the nanoscale will also be instrumental to break the efficiency bottleneck for artificial photosynthesis, including reduction of CO2 to hydrocarbon fuels.
## Results
### Design and properties of the photochemical diode
Schematically shown in Fig. 1a is the typical overall neutral pH water splitting on multi-band (GaN/InGaN) nanowire photocatalysts vertically aligned on the substrate (Si). Redox sites (and co-catalysts) in such axially symmetric nanostructures are randomly distributed on the surfaces. In contrast, gradient in p-type dopant (Mg) concentrations leads to a large work function difference (up to 300 meV) between the two parallel surfaces of a photochemical diode. The resulting p-p+ nanoscale lateral junction, schematically illustrated in Fig. 1b, induces unidirectional flow of photo-excited charge carriers, i.e., electrons and holes migrate toward the surfaces with a relatively small and large work function (Φred and Φox), respectively. Shown in Fig. 1a, the energy bandgap of the nanosheet structures can be further varied along the vertical direction, i.e. the photon absorption path. The resulting multi-band photocatalysts promise photocatalytic solar water splitting with the highest efficiency possible7,40. Figure 1b schematically illustrates the energy bands of the proposed InGaN nanosheet structures, which are grown directly on Si substrate using plasma-assisted molecular beam epitaxy (MBE) . During the epitaxy process, p-type dopants (Mg) are impingent primarily on one side of the nanosheet structure (see Methods). The resulting Mg-doping gradient along the lateral dimension of the nanosheet establishes a strong built-in electric field, schematically shown in Fig. 1b, which separates the photo-generated electrons and holes, and drives them towards opposite surfaces, thus reducing the probability of recombination. Figure 1c shows the typical bird’s-eye-view SEM images of InGaN nanostructures, which are comprised of axially asymmetric nanosheets with parallel non-polar surfaces. A simplistic view of the dynamic behaviors of charge carriers are depicted in Fig. 1d, which includes electron–hole pair generation upon photo-excitation, bulk recombination, carrier separation and migration towards laterally opposite direction. Consequently, the two catalytic surfaces are enriched with electrons and holes, respectively. The electron enriched surface (cathode) of the photochemical diode largely facilitates photo-deposition of proton reduction co-catalysts (Rh/Cr2O3 core/shell nanoparticles) (see Methods), which in turn enhances the hydrogen evolution reaction (HER) significantly. Water oxidation reaction takes place on the hole enriched surface (anode). Details about conventional photochemical diode and junction engineering approach can be found in Supplementary Fig. 1 and Supplementary Note 1. One direct evidence for the efficient charge carrier separation and extraction of the presented InGaN nanosheet structures, compared to the conventional nanowires, is the significantly reduced photoluminescence intensity. Shown in the inset of Fig. 1e, the photoluminescence (PL) emission intensity of InGaN nanosheets is nearly 20 times smaller, compared to that of InGaN nanowires grown under similar conditions. Therefore, with significantly reduced charge carrier recombination, InGaN nanosheets are expected to exhibit noticeably higher photocatalytic activity than corresponding nanowire structures41 (Supplementary Figs. 23 and Supplementary Notes 23). By varying the epitaxy conditions, the energy bandgap of InGaN nanosheets, evident by the photoluminescence emission spectra, can be tuned over a large part of the visible spectral range, shown in Fig. 1e.
### Surface selectivity for oxidation and reduction
Scanning transmission electron microscopy (STEM) imaging, supported by the energy dispersive X-ray scanning (EDXS) analysis on p-type InGaN nanosheet photochemical diodes, decorated with Rh-nanoparticles, shows a significant difference in the number of nanoparticles loading between the two parallel surfaces. As shown in Fig. 2b, preferential photo-reduction of Rh-metal precursors to Rh-nanoparticles is facilitated on the reduction surface (cathode) due to its electron enrichment compared to that on the oxidation surface (anode) in Fig. 2a. Scanning transmission electron microscopy (STEM)-ZC/BF images further confirm Rh-nanoparticles’ deposition on the reduction sites of InGaN photochemical diode nanostructure (Supplementary Fig. 4). High-resolution STEM bright-field lattice image also depicts high crystalline quality of the defect-free In0.22Ga0.78N nanostructure surfaces, shown in Fig. 2c, d. For comparison, non-selective, rather random distribution of metal nanoparticles on the non-polar surfaces of conventional nanowires are also presented in Supplementary Fig. 5.
To further gain a deep insight regarding the deviation in photo-deposition behavior, near-surface band-structure of as-grown p-InGaN nanowires and p-InGaN nanosheets were characterized using angle resolved X-ray photoelectron spectroscopy (ARXPS). Illustrated in Fig. 3a, the measured surface valence band maximum (EVS) values between the two non-polar surfaces (relative to surface Fermi-level, EFS) are significantly different, with EVS for the cathode surface being ~300 meV larger than that for the anode surface. This suggests the presence of a built-in potential ~300 meV (ΔE) along the lateral dimension of the nanosheet structure, as shown schematically in Fig. 1b. Subsequently, surface dependence of EVS was analyzed by measuring the valence spectra vs. radial scanning angle, α (Fig. 3b). Variations of EFSEVS as a functional of scanning angle is illustrated in Fig. 3c for the entire range of α, further confirming the strong dependence of EVS on different surfaces. The sharp change in EFS-EVSEVS) vs. scanning angle can be ascribed to the transition from one parallel surface to another, e.g., from anode to cathode surface, whereas the slow and gradual change (δEVS) is attributed to the curvature and orientation of nanosheet arrays. In contrast, conventional InGaN nanowires exhibit nearly constant EVS at different scanning angles, also shown in Fig. 3c for comparison. Further comparative analysis between the surface potentials of nanowire and nanosheet structures is discussed in Supplementary Fig. 2 and Supplementary Note 3. A quantitative estimation for the band-diagram of InGaN nanosheet structures is shown in Supplementary Fig. 6a, which is derived from the XPS and TEM analysis performed on InGaN nanosheets with optimum bandgap for enhanced photocatalytic activity. It is evident that the large built-in potential leads to the spontaneous accumulation of electrons and holes on the cathode and anode surfaces, respectively36. This implies that the origin of preferential photo-deposition of noble metal nanoparticles on cathode surface of p-InGaN photochemical diode, as shown in Fig. 2a–d, is due to the reduction of noble metal precursors by photo-excited electrons enriched on that surface.
It is worthwhile mentioning that anisotropic facet-dependent co-catalyst deposition had been reported previously to ensure spatial separation of oxygen evolution reaction (OER) and HER co-catalysts42,43,44,45,46,47,48, and thus to provide enhanced carrier separation in the near-surface region (Supplementary Note 4). However, bulk recombination remains a limiting factor for their low apparent quantum efficiency in water splitting. Unique to the presented photochemical diode nanostructure is the net lateral band-bending between two spatially separated redox surfaces. Water oxidation and proton reduction reactions occur at the two distinct reaction sites on photochemical diode nanosheets, and are coupled between the parallel anode and cathode surfaces49,50, schematically illustrated in Fig. 3d, e. Under concentrated sunlight, the band-bending can be reduced in the bulk, which can further lower the recombination probability by making the built-in electric field nearly linear and hence the flow/separation of charge carriers unidirectional (Supplementary Fig. 6b).
### Characterization and performance analysis of double-band PCD
Double-band GaN:Mg/InGaN:Mg nanostructures were grown on Si wafer using plasma-assisted molecular beam epitaxy (See Methods)34,35,36. The nanosheet structures are vertically standing on the Si substrate, having an areal density in the range of ~1.5 × 1010 cm−2 (Fig. 1c). The photochemical diodes have an average height ~1.5–2 µm, and the thickness varies from ~50 to 120 nm. The PL spectra revealed an optical emission peak at ~485 nm, as shown in Fig. 4a, which can be attributed to InGaN bandgap of 2.56 eV. The average indium incorporation is estimated to be ~22% for the grown nanostructures. Detailed STEM and EDXS analysis confirms the existence of a continuous long InGaN segment, simultaneously showing the distribution of Rh/Cr2O3 nanoparticles on the surface34,36 (Supplementary Note 5). The p-type behavior of Mg-doped crystalline In0.22Ga0.78N photochemical diodes is confirmed by photo-electrochemical characterization that includes open-circuit potential (OCP), Mott-Schottky and photocurrent measurements (Supplementary Fig. 7 and Supplementary Note 6).
The nanosheet arrays were tested for both hydrogen evolution reaction (HER) in aqueous methanol (CH3OH) solution, as well as neutral pH overall water splitting (OWS). A 300 W Xenon lamp was used as a concentrated irradiation source for photo-excitation, which has an intensity equivalent to ~32 suns when measured on the nanostructure substrate (Supplementary Fig. 8). Rh nanoparticles and Rh/Cr2O3 core–shell nanostructures were photo-deposited as the co-catalysts for HER and OWS reactions, respectively. In the wavelength range of 200–485 nm (incident intensity of ~611 mW cm−2, see Supplementary Notes 78), stoichiometric gas evolution from neutral pH water splitting was measured at a rate of ~1.62 mmol h−1 cm−2 H2 and ~0.784 mmol h−1 cm−2 O2, resulting in an AQE ~45.85%, which is more than two-fold higher than previously reported AQE of ~20% for double-band nanowire heterostructures36. Time evolution of photocatalytic hydrogen production from the photochemical diode arrays are shown in Supplementary Fig. 9. Under visible light irradiation (> 400 nm), the evolution rate was measured as ~0.5 mmol h−1 cm−2 for H2, and the AQE from the photochemical diode nanosheets (nanowires) was estimated to be ~19.93% (12.3%).
Evidently, significant enhancement in overall photocatalytic water splitting activity had been derived from photochemical diode nanosheets compared to that from nanowire heterostructures. For comparison, the amount of hydrogen evolution from the photochemical diodes is increased by more than a factor of two using full arc illumination with AM1.5 G filter. This, in turn, enhanced the energy conversion efficiency (ECE) from ~7.5 to ~17.5%. Moreover, an impressive ~3.3% of solar-to-hydrogen conversion efficiency has been measured in this study, which is significantly higher than that estimated from dual-band nanowire structures, as depicted in Fig. 4b. Repeated cycles for the stoichiometric hydrogen and oxygen evolution in neutral pH water splitting using AM1.5 G filter are demonstrated in Fig. 4c (Supplementary Movie 1 and 2). Illustrated in Fig. 4b, d are the comparative study of hydrogen evolution and corresponding AQE and ECE from neutral pH overall water splitting under full arc illumination using AM1.5 G optical filter and under visible light irradiation using a 400 nm long-pass optical filter. The photochemical diode nanostructures remain stable after the photocatalytic reactions, and negligible signs of degradation was observed after ~4 h of overall neutral pH water splitting and hydrogen evolution reaction from aqueous methanol solution (shown in Supplementary Fig. 10). The stability of the co-catalyst nanoparticles on photocatalyst surface was further confirmed from TEM analysis.
## Discussion
In the end, we discuss the unique charge transfer mechanism in nanosheet photochemical diodes and the impact on solar-to-hydrogen efficiency in photocatalytic overall water splitting. In nanostructured photocatalysts, charge carrier transport is no longer diffusion limited; the effective extraction of photo-generated charge carrier is often restricted by surface and interface electronic properties, e.g., the presence of surface band bending34,36,51,52,53. To date, it has remained a grand challenge to precisely steer charge carrier flow in nanostructured photocatalysts, due to the lack of control over their surface band bending. In our previous studies, we have demonstrated that by minimizing the surface potential for hole transport, the STH efficiency for photocatalytic overall pure water splitting on InGaN nanowire arrays was significantly enhanced from <0.1 to over 1%34,35,36. Further improvement of the STH efficiency, however, has been fundamentally limited by charge carrier recombination (Supplementary Fig. 11a-b), including both surface and bulk recombination. In this work, we have shown that such a critical challenge can be effectively addressed in nanosheet photochemical diodes. The asymmetric dopant incorporation in InGaN nanosheets and the resulting built-in electric field, schematically shown in Fig. 1b, offers several essential benefits for photocatalytic overall water splitting. It leads to the unidirectional charge carrier flow, i.e., the accumulation of photo-generated electrons and holes on the cathode and anode surfaces of the same nanosheet structure, respectively. The efficient separation of photo-excited charge carriers is unambiguously supported by the preferential deposition of HER co-catalysts only on the cathode surfaces of InGaN nanosheets (Fig. 2b), the significant reduction of photoluminescence emission compared to conventional nanowires (inset of Fig. 1e), and the large difference in surface potential (~300 meV) between the anode and cathode surfaces (Fig. 3c). Over 90% of the photo-excited electrons and holes are spatially separated on the cathode and anode surfaces36, thereby minimizing both surface and bulk recombination54, detailed in Supplementary Fig. 11. Moreover, reverse diffusion currents due to the concentration gradient is minimized by the energy barriers (Supplementary Fig. 11c-d, Supplementary Note 9) and surface trapping of carriers in co-catalyst nanoparticles17,19,20,21,22. Significantly, the rational design and synthesis of anode and cathode surfaces in nanostructured photocatalysts can effectively increase the surface area for water oxidation reaction, which is often the rate-limiting process of water splitting55,56,57,58,59. Water oxidation and proton reduction reactions can be coupled between spatially separated reaction sites (parallel electrodes), i.e., the cathode and anode surfaces49,50 of two adjacent photochemical diodes (Fig. 3e), thereby drastically suppressing back reaction. In addition, the effective surface relaxation of nanosheet structures allows for the optimization of the bandgap and band-bending of InGaN photochemical diodes through variations in indium incorporation and Mg-doping concentration, to ensure sufficient cathodic and anodic over-potentials, and to minimize the surface potential barrier for achieving high photocatalytic efficiency.
The demonstrated STH ~3.3% is significantly higher than previously reported efficiency values for neutral pH one-step overall water splitting13,16,23, which generally range from 0.1 to 1.1%. It is worth mentioning that an STH of ~5% had been reported on CoO nanocrystals14, which are barely stable in harsh photocatalytic environment. The nanocrystal surfaces become corroded due to difficulties in co-catalyst loading, and further studies seem to be necessary to understand the mechanism and to confirm the reproducibility16. Recent studies on a different 2-step approach of water splitting15 showed ~2% of STH using CDots-C3N4. This approach requires efficient generation as well as subsequent decomposition of H2O2, putting further constraints on the bandgap of suitable photocatalyst (>1.78 eV, compared to ~1.23 eV for neutral pH overall water splitting, excluding necessary over-potentials). A device comprised of catalysts loaded on triple-junction photovoltaic cell demonstrated STH of ~2.5% in an earlier study60, which, however, utilizes conductive 1 M potassium borate electrolyte (pH ~ 9.2). Noticeably, many of these relatively high efficiency devices16,60 were designed based on conventional photochemical diode61 (Supplementary Note 1) which requires the use of ohmic-contact for efficient charge carrier transport. For example, co-catalyst loaded SrTiO3:La,Rh|Au(contact)|BiVO4:Mo device16 increases the STH up to ~1.1% compared to that of ~0.1% using their powder suspensions in Z-scheme with Fe3+/2+ redox couples62. The use of planar ohmic-contact reduces active surface area for redox reaction and often requires conductive electrolyte to compensate for the distance between cathode and anode surfaces (by selectively adjusting the pH or adding supporting electrolyte in near-neutral pH condition). Moreover, it has been extremely difficult to realize such ohmic-contact in nanostructured photocatalysts. In this context, our nanosheet photochemical diodes do not require ohmic-contact or majority carrier recombination. They offer large (and hole-enriched) anode surface for water oxidation, while enabling selective deposition of HER co-catalysts (Rh/Cr2O3) on electron-enriched cathode surface, with the least probability of carrier interference, crowding, and recombination. Moreover, solar energy, being a planar resource, can be more effectively harvested (per unit area) in such vertically aligned wafer-level nanosheet structures.
In summary, we have demonstrated photochemical diode artificial photosynthesis system that can enable relatively efficient overall pure water splitting (STH ~3.3%). The wafer level photochemical diodes consist of vertically aligned InGaN nanosheets, with well-defined anode and cathode surfaces for water oxidation and proton reduction, respectively. Unique to such nanosheet photochemical diodes is that charge carrier flow can be precisely controlled at the nanoscale without any external bias: photo-generated electrons and holes are instantaneously separated due to the built-in electric field along the lateral dimension of nanosheets, leading to the spontaneous population of anode and cathode surfaces by holes and electrons, respectively. The spatially separated electron and hole gas significantly minimizes surface and bulk recombination and suppresses back reaction, which have been some of the major challenges in achieving efficient photocatalytic water splitting to date. The effective manipulation and control of charge carrier flow in nanostructured photocatalysts not only significantly enhances the efficiency of photocatalytic water splitting, but also provides critical insight in achieving high efficiency artificial photosynthesis, including the efficient and selective reduction of CO2 to hydrocarbon fuels. Future work also includes the development of an axial photochemical diode by connecting the two semiconductor segments of the nanowire/nanosheet with a transparent tunnel junction or transparent ohmic contact, which can significantly enhance the STH efficiency63,64.
## Methods
### Molecular beam epitaxial (MBE) growth
InGaN photochemical diode nanostructures were grown on Si (111) substrate by radio frequency (RF) plasma-assisted MBE under nitrogen rich conditions. To remove organic contaminants, the Si substrate was thoroughly cleaned with acetone and methanol solvent. The native oxide on the substrate was removed prior to loading into the MBE chamber by cleaning with 10% hydrofluoric acid (HF). The residual oxide was then desorbed, by in situ annealing of the substrate at ~780 °C before the growth initiation. The clean Si (111) 7 × 7 reconstructed surface from reflection high-energy electron diffraction (RHEED) analysis further confirms the desorption. To promote the formation and nucleation of nanowires, a thin (~1 ML) Ga seeding layer was deposited. Ga, In, and Mg fluxes were controlled using respective thermal effusion cells, whereas the nitrogen radicals were supplied from an RF-plasma source. The growth conditions were optimized after several iterations for better crystalline quality and photocatalytic performance in overall water splitting reaction. Instead of multi-stack InGaN:Mg/GaN:Mg layers35,36, a continuous InGaN:Mg layer was grown spontaneously on top of GaN:Ge nanowire template followed by a final GaN:Mg capping layer (relatively smaller than InGaN in length, in proportion to fraction of UV region in the spectrum). A nitrogen flow rate of 1.0 standard cubic centimeters per minute (sccm), and a forward plasma power of ~350 W were used during the growth. Other growth parameters include Ga beam equivalent pressure (BEP) in the range of 6 × 10−8 Torr, In BEP approximately 7.8 × 10−8 Torr, Mg BEP in the range of 2 × 10−11 Torr corresponding to Mg cell temperature (TMg) of ~200 °C, and Mg BEP of ~1.5 × 10−9 Torr which corresponds to TMg of ~270 °C. To facilitate the formation of Mg-doping gradient in the lateral direction and to counteract the spinodal decomposition of In-N in the surface, the substrate was kept steady during the growth of the photochemical diode. Amount of nitrogen species and metal fluxes in the growth environment were further optimized to ensure essential nitrogen vapor pressure in the vicinity of the growth front, and to reduce long-distance diffusion of indium atom along the growth axis. Mg-doping gradient (due to the surface dependent incorporation) can also be influenced by the super-saturation for Ga-rich or N-rich condition at the localized level65. The GaN template was grown at ~780 °C, and the growth temperatures for InGaN were varied from 650 to 705 °C.
### Micro-photoluminescence (μ-PL)
A 405 nm laser or a 325 nm He-Cd laser (Kimmon Koha) was used as excitation source for the μ-PL measurement. The laser beam was focused to a circular spot (~5 μm) on the sample using a 60× objective, which also collects the emitted light from the sample. The collected optical emission was then spectrally resolved using a high-resolution spectrometer (JY HR-550) and detected by a CCD or a photon counting mode photomultiplier tube.
### Micro-Raman analysis
The room-temperature micro-Raman measurements were performed using an external 514 nm argon ion laser as excitation source. A 50× objective (numerical aperture 0.9) was employed for focusing the laser excitation to a spot beam of 1 μm diameter and to direct an estimated 7 mW of power on the sample surface. The Raman signal was collected in the backscattering geometry where the (non-polarized) laser beam was incident at normal to the planar substrate. The Raman spectra were then resolved via an 1800 l mm−1 grating, and detected at 0.2 cm−1 resolution by a CCD, mounted on the inVia confocal Raman spectrometer from Renishaw.
### Scanning transmission electron microscopy (STEM)
STEM-SE, STEM-BF, and STEM-HAADF imaging were performed using a Hitachi HD2700 Cs-corrected dedicated STEM, which is equipped with a cold field emission emitter and operated at 200 kV with an electron beam diameter of ~0.1 nm (~0.2 nm in HR mode). A 60 mm2 silicon drift detector (SDD) from Bruker was used for STEM EDXS analysis. The image in Fig. 2c was processed using a radial difference filter.
### Photo-deposition of co-catalysts
The photochemical diode and the nanowire arrays were decorated with Rh/Cr2O3 core–shell nanostructures using a stepwise photodeposition technique from their respective liquid precursors. Rh nanoparticles were photodeposited from 14 µl of sodium hexachlororhodate (III) (0.5 M Na3RhCl6, Sigma-Aldrich), which is subsequently followed by the Cr2O3 photodeposition from 14 µl of potassium chromate (0.5 M K2CrO4, Sigma-Aldrich) precursor in aqueous methanol solution (20%). The stepwise deposition ensures the formation of Rh/Cr2O3 core–shell nanostructures which prevent the back reaction.
### Angle-resolved X-ray photoelectron spectroscopy (ARXPS)
Thermo-Fisher Scientific K-Alpha XPS system was used for the analysis. The system was equipped with a monochromatic Al-Kα X-ray source ( = 1486.6 eV) and 180° double focusing hemispherical analyzer. The analysis chamber pressure was as low as 10−8 Torr. The non-polar surfaces of the nanostructure arrays were excited with the X-ray beam, incident at 60° angle with substrate normal. The EFSEVS was estimated from the ARXPS valence spectra. Both Au 4f (84.0 eV) and C 1s (284.8 eV) peaks were used to calibrate the binding energies. As shown in Fig. 3a, the position of surface valence band (EVS) with respect to the surface Fermi level (EFS, binding energy = 0 eV) can be estimated by measuring the intersection point between the linear extrapolation of the valence band leading edge and the extended baseline.
### Photocatalytic reaction
The experimental configuration was composed of a reaction system and an evaluation system. The wafer sample, fixed by a homemade PTFE holder was placed in a Pyrex chamber containing appropriate solutions. Prior to each photocatalytic experiment, the distilled water was purged for 20–30 min using Ar gas to remove the dissolved gases. After evacuating the chamber, the system was illuminated by a Xenon lamp (Cermax, PE300BUV); adequate transmittance for both UV and visible light was secured by placing a quartz lid in between the reaction chamber and the lamp. The evolved gases were then collected using a vacuum-tight syringe and evaluated by a gas chromatograph (GC, Shimadzu GC-8A) equipped with a thermal conducting detector (TCD). High purity Ar was used as carrier gas. Wavelength dependence of the photocatalytic activity and quantum efficiency was systematically analyzed and measured by using appropriate high-pass and band-pass filters. Due to the manual sampling of the H2 and O2 gases, the experimental error was approximately 10%.
### Data availability
The data that support the findings of this study are available within the paper [and its supplementary information files]. Further details regarding the data are available from the corresponding author upon reasonable request. | 2022-10-02 03:11:49 | {"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": 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": 0.6184085011482239, "perplexity": 4596.432469113123}, "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/1664030337244.17/warc/CC-MAIN-20221002021540-20221002051540-00247.warc.gz"} |
https://learn.microsoft.com/en-us/dotnet/framework/data/adonet/ef/language-reference/variables-entity-sql | # Variables (Entity SQL)
## Variable
A variable expression is a reference to a named expression defined in the current scope. A variable reference must be a valid Entity SQL identifier, as defined in Identifiers.
The following example shows the use of a variable in the expression. The c in the FROM clause is the definition of the variable. The use of c in the SELECT clause represents the variable reference.
select c
from LOB.customers as c | 2023-01-30 09:23:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.2597576081752777, "perplexity": 1503.4634342229322}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2023-06/segments/1674764499804.60/warc/CC-MAIN-20230130070411-20230130100411-00373.warc.gz"} |
https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.18/share/doc/Macaulay2/Polyhedra/html/_polyhedral__Complex.html | # polyhedralComplex -- generates a PolyhedralComplex
## Description
If polyhedralComplex is applied to a Polyhedron it generates the PolyhedralComplex given by the the Polyhedron and all of its faces. If applied to a List the list must only contain Polyhedra and PolyhedralComplexes in the same ambient space. Then it adds the Polyhedra in the List and the generating Polyhedra of the PolyhedralComplexes in the List one by one to the new PolyhedralComplex, checking each time if the new Polyhedron is compatible with the polyhedra that have already been added, i.e. that the intersection with each of them is a face of both Polyhedra (intersection condition).
If one of the polyhedra is in the wrong ambient space (i.e. not the ambient space of the first object in the list), then there will be an error and no PolyhedralComplex will be returned. If the intersection condition fails, there will also be an error. The pairs of incompatible polyhedra can be accessed with the function incompPolyhedra.
i1 : P = convexHull matrix {{1,-1,0},{0,-1,0}} o1 = P o1 : Polyhedron i2 : PC = polyhedralComplex P o2 = PC o2 : PolyhedralComplex i3 : P1 = convexHull matrix {{1,0,0},{0,1,0}}; i4 : P2 = convexHull matrix {{0,-1,0},{1,-1,0}}; i5 : PC = polyhedralComplex {PC,P1,P2} o5 = PC o5 : PolyhedralComplex
## For the programmer
The object polyhedralComplex is . | 2021-09-20 11:50:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.42949873208999634, "perplexity": 1288.921334940432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2021-39/segments/1631780057036.89/warc/CC-MAIN-20210920101029-20210920131029-00410.warc.gz"} |
http://bentatman.co.uk/2016/01/ | # Inheritance/Population Simulation.
Over the past few weeks I have been considering trying to write a sort of population simulator - ie, you have a sample of say 100 individuals (very basic individuals) and you let it run. They have some genes (currently the code has two, but it could be expanded) and these genes affect how long they live, how sexually mature they become, basically how likely they are to pass on their genetics. It would therefore show a sort of selection, and because of this would also show a population maximum.
This weekend I decided to finally start writing this, and so far it is going well. It currently has a hard set population maximum (5,000, but it can be changed in the code), however I am planning on changing this so it has a set amount of food in the environment, and as individuals grow they take up more food depending on their gene code, and then when they die this food (- some which cannot be released) is given back into the environment. If the food runs out, some organisms may have an ability to survive without, but others won't - so they will die out but the survivors will have more food.
At the moment, it creates 100 individuals with two random genes - x and y - which determine their survival characteristics. Currently it is based around the following equations;
This means that as the x and y values change, the maximum age and spread change - so certain configurations will be more stable than others. The probability of reproduction is modeled as a normal distribution using;
This gives a probability of reproduction, so the survival is sort of down to chance. If an individual reproduces then their genes change as they are passed on (only slightly) so the most successful genes are likely to survive. At the moment (without the food construct) individuals die at their maximum age multiplied by a factor which is dependent on the number of individuals in the population;
So, as the number of individuals increases, the age at which individuals dies gets smaller - so in the graphs we see a boom bust population change, as when it gets too big individuals die, and when it gets too low they live longer.
Hopefully this new project will continue to grow. For now, the source is on github, and an image of the program is shown below.
# Christmas and New Years!
I just wanted to take this time to wish everyone a very (late) Merry Christmas, and a Happy New Year +6! I hope everything over the last year went well, and that the year ahead will be even better!
ChemKit is currently in hiatus as I have mock exams coming up and other work to do (dreaded coursework soon, the problem of doing science A Levels!). I am also thinking of writing some sort of population simulator to aid with biology - think of it as in you have a population of say 100 organisms who have different genetic fingerprints (say 20 digits long). Each digit has a specific advantage or disadvantage, and they may affect each other in different ways. Sexual reproduction could be shown through the combination of different genetics to produce offspring, and through probabilities the likelihood of a member reaching sexual maturity could be shown, hence showing inheritance. I could then have certain factors which could change (and maybe be affected by the organisms - ie a lot of organisms might lead to an increase in temperature) which would then show selection. As the different genes would be interfering it could mean that many solutions are stable (think of it like equation solutions) and this could show speciation as the populations diverge.
In terms of reading, recently I have read "Life's Greatest Secret" by Matthew Cobb, "All the Light We Cannot See" by Anthony Doerr, "Human Universe" by Brian Cox and "The Secret War" by Max Hastings. I enjoyed each of these books, and looking back on it now I have apparently been reading quite varied recently. I am currently reading "All Hell Let Loose" by Max Hastings, and I am going to be reading "The Mysterious World of the Human Genome" by Frank Ryan, "Life Unfolding" by Jamie A. Davis, and possibly having a reread of Metro 2033.
Anyway, as said, I hope you all had a nice Christmas and have a brilliant new year. | 2017-06-28 22:29:26 | {"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": 4, "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": 0.29454323649406433, "perplexity": 1050.034325619798}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128323807.79/warc/CC-MAIN-20170628222452-20170629002452-00170.warc.gz"} |
https://www.mathemania.com/lesson/continuous-functions/ | # Continuous functions
The concept of continuity of a function is intuitively clear, however, very complex.
Less formal, if the domain of the function is an interval, then the graph of the function we can draw without lifting the pencil from the paper. Therefore, for instance, the $sign$ function sgn: $\mathbb{R} \to \mathbb{R}$ is not continuous at $x=0$, since $\lim_{x \to 0}$ sgn $x$ does not exist:
We can assume that the existence of the limit of a function at the point is necessary condition that a function is continuous at the same point. If we consider the following graph of the function,
we can see it is not a sufficient condition. From the graph we can see that the function has the limit $l$ at point $x_0$, however, the limit $l$ is not equal to the value of the function at point $x_0$ ($f(x_0)$).
Therefore, the formal definition follows.
Let $I \subseteq \mathbb{R}$ be an open interval and $x_o \in I$. The function $f: I \to \mathbb{R}$ is continuous at point $x_0$ if:
1.) $$\lim_{x \to x_0}f(x)$$
exists
and
2.) $$\lim_{x \to x_0}f(x)=f(x_0).$$
The function $f$ is the continuous function if it is continuous at every point from the domain of the function.
Cauchy definition
Let $I \subseteq \mathbb{R}$ be an open interval, $x_0 \in I$ and $f: I \to \mathbb{R}$ function. The function $f$ is continuous at point $x_0$ iff $\forall x \in I$:
$$(\forall \epsilon > 0) (\exists \delta > 0) ((|x-x_0| < \delta) \Rightarrow (|f(x) – f(x_0)|)).$$
For example, the constant function $f: \mathbb{R} \to \mathbb{R}$, $f(x) =x$, $\forall x \in \mathbb{R}$ is continuous at every point $x_0 \in \mathbb{R}$. Namely,
$$(\forall \epsilon > 0)(\exists \delta > 0)(\forall x \in \mathbb{R})((|x-x_0| < \delta) \Rightarrow (|c-c| = 0 < \epsilon)).$$
The sine function $\sin: \mathbb{R} \to [-1,1 ]$ is continuous on $\mathbb{R}$. For proof, we will use the trigonometric identity for the difference of sines and the following inequality:
$$| \sin x| \le |x|,$$
for $|x| < \frac{\pi}{2}$.
Therefore, for any $x_0 \in \mathbb{R}$ we have
$$|\sin x – \sin x_0| = \left | 2 \sin \left ( \frac{x – x_0}{2}\right ) \cdot \left( \cos \frac{x + x_0}{2} \right) \right |$$
$$\le 2 \left | \sin \left ( \frac{x – x_0}{2} \right) \right |$$
$$\le 2 \left | \frac{x-x_0}{2} \right |$$
$$\le 2 \frac{|x – x_0|}{2}$$
$$\le | x – x_0|$$,
when $|x – x_0| < \frac{\pi}{2}$.
Now we have:
$$\lim_{x \to x_0}|\sin x – \sin x_0| \le \lim_{x \to x_0} |x – x_0| = 0,$$
that is
$$\lim_{x \to x_0} \sin x = \lim_{x \to x_0} \sin x_0 = \sin x_0,$$
which means that the function sine is continuous at point $x_0$.
Trigonometric functions are continuous on theirs domains.
Properties of continuous functions
Let $I \subseteq \mathbb{R}$ be an open interval, $x_0 \in I$ and $f,g: I \to \mathbb{R}$ continuous functions at $x_0$. Then:
1.) $\forall \alpha, \beta \in \mathbb{R}$, the function $\alpha f + \beta g$ is continuous at $x_0$,
2.) the function $f \cdot g$ is continuous function at $x_0$,
3.) if $g(x) \neq 0$, $\forall x \in I$, then the function $\frac{f}{g}$ is continuous function at $x_0$.
The function $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^n$, $\forall n \in \mathbb{N}$ and $\forall x \in \mathbb{R}$ is continuous function on $\mathbb{R}$.
Every polynomial function $p: \mathbb{R} \to \mathbb{R}$ defined as
$$p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x +a_0,$$
$a_n \neq 0$, $a_0, \ldots, a_n \in \mathbb{R}$, is continuous function on $\mathbb{R}$.
Every rational function is continuous on its domain. Rational function is a ratio of two polynomials $p$ and $q$, that is, a ratio of two continuous functions:
$$r(x) = \frac{p(x)}{q(x)}.$$
Therefore, the rational function $r(x)$ is a continuous function on $\mathbb{R}$ such that $q(x) \neq 0$.
Continuity of composite functions
Let $A, B \subseteq \mathbb{R}$ be open intervals and $f: A \to \mathbb{R}$, $g: B \to \mathbb{R}$ functions for which is valid $f(A) \subseteq B$, that is, the function $g \circ f: A \to \mathbb{R}$ is well defined. If the function $f$ is continuous at point $x_0 \in A$ and function $g$ continuous at point $f(x_o) \in B$, then the function $g \circ f$ is continuous at point $x_o$.
Uniformly continuous functions
Let $I \subseteq \mathbb{R}$ be an open interval and $f: I \to \mathbb{R}$. Then the function $f$ is uniformly continuous on $I$ if:
$$(\forall \epsilon > 0)(\exists \delta > 0) ((|x-y| < \delta)(x, y \in I) \Longrightarrow (|f(x) – f(y)| < \epsilon)).$$
This means that $\delta$ depends on $\epsilon$, not on $x$ and $y$. | 2020-01-23 23:46:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9670894145965576, "perplexity": 80.1141773316566}, "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/1579250614086.44/warc/CC-MAIN-20200123221108-20200124010108-00139.warc.gz"} |
http://blog.nguyenvq.com/blog/2010/10/31/find-files-and-find-files-containing-certain-text/ | Find files and find files containing certain text
This is a reminder to myself as I keep forgetting how to do these basic searches in Linux.
To find files with file name containing the text foo using the command find:
## find file with "foo" in file name
find ./ -name "*foo*" ## replace ./ with path; can use shell style wildcards
## ignore upper and lower cases
find ./ -iname "*foo*"
## print pathnames of all files
find ./ -print
To find files with bar in it’s content using the command grep:
grep "bar" -r ./
## print lines without the word "bar"
grep -v "bar" -r ./
## note, can also use regexp with -E
Statistician
1. Johan says:
I think it’s good practice to quote the text containing the wildcards or escape the backslashes, otherwise some shells (zsh) will perform expansion before the command even runs.
2. I really like your blog.. very nice colors & theme. Did you create this website yourself or did you hire someone to do it for you? Plz answer back as I’m looking to construct my own blog and would like to find out where u got this from. thank you | 2018-02-23 22:04:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.1803233027458191, "perplexity": 6808.29183440253}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814857.77/warc/CC-MAIN-20180223213947-20180223233947-00072.warc.gz"} |
http://www.mathcounterexamples.net/counterexamples-around-series-part-2/ | # Counterexamples around series (part 2)
We follow the article counterexamples around series (part 1) providing additional funny series examples.
### If $$\sum u_n$$ converges and $$(u_n)$$ is non-increasing then $$u_n = o(1/n)$$?
This is true. Let’s prove it.
The hypotheses imply that $$(u_n)$$ converges to zero. Therefore $$u_n \ge 0$$ for all $$n \in \mathbb N$$. As $$\sum u_n$$ converges we have $\displaystyle \lim\limits_{n \to \infty} \sum_{k=n/2}^{n} u_k = 0.$ Hence for $$\epsilon \gt 0$$, one can find $$N \in \mathbb N$$ such that $\epsilon \ge \sum_{k=n/2}^{n} u_k \ge \frac{1}{2} (n u_n) \ge 0$ for all $$n \ge N$$. Which concludes the proof.
### $$\sum u_n$$ convergent is equivalent to $$\sum u_{2n}$$ and $$\sum u_{2n+1}$$ convergent?
Is not true as we can see taking $$u_n = \frac{(-1)^n}{n}$$. $$\sum u_n$$ converges according to the alternating series test. However for $$n \in \mathbb N$$ $\sum_{k=1}^n u_{2k} = \sum_{k=1}^n \frac{1}{2k} = 1/2 \sum_{k=1}^n \frac{1}{k}.$ Hence $$\sum u_{2n}$$ diverges as the harmonic series diverges.
### $$\sum u_n$$ absolutely convergent is equivalent to $$\sum u_{2n}$$ and $$\sum u_{2n+1}$$ absolutely convergent?
This is true and the proof is left to the reader.
### $$\sum u_n$$ is a positive convergent series then $$(\sqrt[n]{u_n})$$ is bounded?
Is true. If not, there would be a subsequence $$(u_{\phi(n)})$$ such that $$\sqrt[\phi(n)]{u_{\phi(n)}} \ge 2$$. Which means $$u_{\phi(n)} \ge 2^{\phi(n)}$$ for all $$n \in \mathbb N$$ and implies that the sequence $$(u_n)$$ is unbounded. In contradiction with the convergence of the series $$\sum u_n$$.
### If $$(u_n)$$ is strictly positive with $$u_n = o(1/n)$$ then $$\sum (-1)^n u_n$$ converges?
It does not hold as we can see with $u_n=\begin{cases} \frac{1}{n \ln n} & n \equiv 0 [2] \\ \frac{1}{2^n} & n \equiv 1 [2] \end{cases}$ Then for $$n \in \mathbb N$$ $\sum_{k=1}^{2n} (-1)^k u_k \ge \sum_{k=1}^n \frac{1}{2k \ln 2k} – \sum_{k=1}^{2n} \frac{1}{2^k} \ge \sum_{k=1}^n \frac{1}{2k \ln 2k} – 1.$ As $$\sum \frac{1}{2k \ln 2k}$$ diverges as can be proven using the integral test with the function $$x \mapsto \frac{1}{2x \ln 2x}$$, $$\sum (-1)^n u_n$$ also diverges. | 2018-09-19 04:51:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9933204650878906, "perplexity": 135.0253869721215}, "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-2018-39/segments/1537267155924.5/warc/CC-MAIN-20180919043926-20180919063926-00238.warc.gz"} |
https://hal-univ-evry.archives-ouvertes.fr/hal-03298703 | # Strategic Reasoning with a Bounded Number of Resources: the Quest for Tractability
Abstract : The resource-bounded alternating-time temporal logic RB±ATL combines strategic reasoning with reasoning about resources. Its model-checking problem is known to be 2EXPTIME-complete (the same as its proper extension RB±ATL$^⁎$) and fragments have been identified to lower the complexity. In this work, we consider the variant RB±ATL+ that allows for Boolean combinations of path formulae starting with single temporal operators, but restricted to a single resource, providing an interesting trade-off between temporal expressivity and resource analysis. We show that the model-checking problem for RB±ATL+ restricted to a single agent and a single resource is $\Delta_{2}^{P}$-complete, hence the same as for the standard branching-time temporal logic CTL+. In this case reasoning about resources comes at no extra computational cost. When a fixed finite set of linear-time temporal operators is considered, the model-checking problem drops to PTIME, which includes the special case of RB±ATL restricted to a single agent and a single resource. Furthermore, we show that, with an arbitrary number of agents and a fixed number of resources, the model-checking problem for RB±ATL+ can be solved in EXPTIME using a sophisticated Turing reduction to the parity game problem for alternating vector addition systems with states (AVASS).
Keywords :
Document type :
Journal articles
Domain :
https://hal.archives-ouvertes.fr/hal-03298703
Contributor : Stéphane Demri Connect in order to contact the contributor
Submitted on : Friday, July 23, 2021 - 6:05:07 PM
Last modification on : Friday, August 5, 2022 - 2:58:08 PM
Long-term archiving on: : Sunday, October 24, 2021 - 6:57:32 PM
### File
main.pdf
Files produced by the author(s)
### Citation
Francesco Belardinelli, Stéphane Demri. Strategic Reasoning with a Bounded Number of Resources: the Quest for Tractability. Artificial Intelligence, Elsevier, 2021, 300, pp.103557. ⟨10.1016/j.artint.2021.103557⟩. ⟨hal-03298703⟩
Record views | 2022-09-26 05:59:17 | {"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": 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": 0.5779365301132202, "perplexity": 2218.4607131459384}, "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/1664030334802.16/warc/CC-MAIN-20220926051040-20220926081040-00451.warc.gz"} |
https://plainmath.net/5100/equal-equation-above-constants-equation-true-positive-real-values-value | # (x^4y^5)^(1/4)(x^8y^5)^(1/5)=x^(j/5)y^(k/4) In the equation above, j and k are constants. If the equation is true for all positive real values of x and y, what is the value of j - k? A)3 B)4 C)5 D)6
Question
Upper Level Math
$$\displaystyle{\left({x}^{{4}}{y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}{y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}$$
In the equation above, j and k are constants. If the equation is true for all positive real values of x and y, what is the value of j - k?
A)3
B)4
C)5
D)6
2021-03-09
Step 1
The given equation is, $$\displaystyle{\left({x}^{{4}}{y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}{y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}$$
Step 2
Assume that the abive equation is true for all positive real values of x and y.
Now obtain the values of constants j and k as shown below.
$$\displaystyle{\left({x}^{{4}}{y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}{y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}$$
$$\displaystyle{\left({x}^{{4}}\right)}^{{\frac{{1}}{{4}}}}{\left({y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}\right)}^{{\frac{{1}}{{5}}}}{\left({y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}$$
$$\displaystyle{\left({x}{y}^{{\frac{{5}}{{4}}}}\right)}{\left({x}^{{\frac{{8}}{{5}}}}\right)}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}$$
$$\displaystyle{\left({x}^{{{1}+\frac{{8}}{{5}}}}\right)}{\left({y}^{{{1}+\frac{{5}}{{4}}}}\right)}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}$$
$$\displaystyle{x}^{{\frac{{13}}{{5}}}}{y}^{{\frac{{9}}{{4}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}$$
Step 3
Equate the powers and obtain the values of j and k as follows.
$$\displaystyle\frac{{j}}{{5}}=\frac{{13}}{{5}}$$
j=13
$$\displaystyle\frac{{k}}{{4}}=\frac{{9}}{{4}}$$
k=9
Now compute the difference j-k as shown below.
j-k=13-9
=4
Therefore, the correct option is B.
### Relevant Questions
1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance.
The equation above gives the monthly cost C, in dollars, to take care of k kittens and b bunnies. John has 3 kittens and 5 bunnies, and Jenny has 2 kittens and 3 bunnies. How much greater, in dollars, is John's total cost than Jenny's total cost to take care of their respective pets?(Disteraged dollar sign when gridding your answer.)
True or False
1.The goal of descriptive statistics is to simplify, summarize, and organize data.
2.A summary value, usually numerical, that describes a sample is called a parameter.
3.A researcher records the average age for a group of 25 preschool children selected to participate in a research study. The average age is an example of a statistic.
4.The median is the most commonly used measure of central tendency.
5.The mode is the best way to measure central tendency for data from a nominal scale of measurement.
6.A distribution of scores and a mean of 55 and a standard deviation of 4. The variance for this distribution is 16.
7.In a distribution with a mean of M = 36 and a standard deviation of SD = 8, a score of 40 would be considered an extreme value.
8.In a distribution with a mean of M = 76 and a standard deviation of SD = 7, a score of 91 would be considered an extreme value.
9.A negative correlation means that as the X values decrease, the Y values also tend to decrease.
10.The goal of a hypothesis test is to demonstrate that the patterns observed in the sample data represent real patterns in the population and are not simply due to chance or sampling error.
Let C be the ellipse contained in the xy plane whose equation is $$\displaystyle{4}{x}^{{2}}+{y}^{{2}}={4}$$, oriented clockwise. The force field F described by $$\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}{i}+{2}{x}{j}+{z}^{{2}}{k}$$, moves a particle along C in the same direction as the curve orientation, performing a W job. C as the surface boundary S: $$\displaystyle{z}={4}-{4}{x}^{{2}}-{y}^{{2}},{z}\ge{0}$$ (with ascending orientation, that is, the component in the z direction equal to 1) and assuming $$\displaystyle\pi={3.14}$$, we can state what:
a) It is not necessary to apply Stokes' Theorem, as C is a closed curve and therefore W = 0.
b) Inverting the orientation of the surface S, we can apply Stokes' Theorem and conclude that W = 12.56.
c) We can apply Stokes' Theorem and conclude that W = 6.28
d) We can apply Stokes' Theorem and conclude that W = 12.56.
Calculating appropriate arithmetic mean, median and peak values for equal data group. a) Body weights (kg) of patients who come to a nutrition clinic: 50, 55, 69, 58, 57, 62, 60 b) Height measurements (cm) of children receiving treatment in the pediatric clinic: 120,125, 110, 105, 125, 108, 115, 125, 119 c) Blood urea level (mg / dl): 119, 5, 2, 6, 4, 3, 1 d) In which of the above distributions, arithmetic mean and peak value in hagi are not appropriate center criteria? Why is that?
The first four term of sequeces A,B,C,D are shown
Sequence A $${(1/3),(2/4),(3/5),(4/6)}$$
The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
If x is a real number, for what values of x is the equation $$(3x-6)(x-2)^{-1}=3$$ true.
The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball. Suppose the snowball is perfectly spherical. Then the volume (in centimeters cubed) of a ball of radius r centimeters is $$\displaystyle\frac{{4}}{{3}}\pi{r}^{{3}}$$. The surface area is $$\displaystyle{4}\pi{r}^{{2}}$$.Set up the differential equation for how r is changing. Then, suppose that at time t = 0 minutes, the radius is 10 centimeters. After 5 minutes, the radius is 8 centimeters. At what time t will the snowball be completely melted. | 2021-05-12 01:57:44 | {"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": 0.6555765271186829, "perplexity": 629.9395263908826}, "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-2021-21/segments/1620243991693.14/warc/CC-MAIN-20210512004850-20210512034850-00448.warc.gz"} |
http://mathhelpforum.com/discrete-math/28200-need-help-discrete-problem.html | # Thread: need help on this discrete problem
1. ## need help on this discrete problem
gcd(a,b)=1 if and only if gcd(a,a+b)=1
prove.....
2. suppose gcd (a,b) = k.
k|a and k|b => k|(a+b)
So $\gcd (a,a+b) \geq k$ since k is a common divisor.
So if $\gcd (a,b) \not = 1$ then $\gcd (a,a+b) \not = 1$
Taking the contrapositive gives us gcd (a, a+b) = 1 => gcd (a,b) = 1
suppose gcd (a, a+b) = p. Then the proof for this direction is very very similar to the previous direction.
3. alright that makes sense thanks a lot for the help. | 2017-03-23 16:30:40 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "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": 0.6269576549530029, "perplexity": 1378.4407991018675}, "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-2017-13/segments/1490218187144.60/warc/CC-MAIN-20170322212947-00516-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://bellevillecellphonerepair.com/damon-herriman-kvblj/product-rule%2C-integration-afdc34 | product rule, integration
d Knowing how to derive the formula for integration by parts is less important than knowing when and how to use it. i L − Now, to evaluate the remaining integral, we use integration by parts again, with: The same integral shows up on both sides of this equation. Hot Threads. , and bringing the abstract integral to the other side, gives, Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. 1 d The essential process of the above formula can be summarized in a table; the resulting method is called "tabular integration"[5] and was featured in the film Stand and Deliver.[6]. Begin to list in column A the function If u and v are functions of x , the product rule for differentiation that we met earlier gives us: v Theoretically, if an integral is too "difficult" to do, applying the method of integration by parts will transform this integral (left-hand side of equation) into the difference of the product of two functions and a new easier" integral (right-hand side of equation). share | cite | improve this answer | follow | edited Jun 5 '17 at 23:10. answered Jan 13 '14 at 11:23. The regularity requirements of the theorem can be relaxed. n until the size of column B is the same as that of column A. ) and In the course of the above repetition of partial integrations the integrals. v The integrand is the product of two function x and sin (x) and we try to use integration by parts in rule 6 as follows: Let f(x) = x , g'(x) = sin(x) and therefore g(x) = - cos(x) Hence ∫ - x sin (x) dx = - ∫ f(x) g'(x) dx = - ( f(x) g(x) - ∫ f'(x) g(x) dx) Substitute f(x), f'(x), g(x) and g'(x) by x , 1, sin(x) and - cos(x) respectively to v To demonstrate the LIATE rule, consider the integral, Following the LIATE rule, u = x, and dv = cos(x) dx, hence du = dx, and v = sin(x), which makes the integral become, In general, one tries to choose u and dv such that du is simpler than u and dv is easy to integrate. = is taken to mean the limit of This yields the formula for integration by parts: or in terms of the differentials and 1 Tauscht in diesem Fall u und v' einmal gegeneinander aus und versucht es erneut. ) For example, suppose one wishes to integrate: If we choose u(x) = ln(|sin(x)|) and v(x) = sec2x, then u differentiates to 1/ tan x using the chain rule and v integrates to tan x; so the formula gives: The integrand simplifies to 1, so the antiderivative is x. {\displaystyle [a,b],} = ( = The reason is that functions lower on the list generally have easier antiderivatives than the functions above them. Integration by parts illustrates it to be an extension of the factorial function: when Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715. Observing that the integral on the RHS can have its own constant of integration ( ~ {\displaystyle v=v(x)} ( § u Considering a second derivative of The theorem can be derived as follows. in terms of the integral of There is no obvious substitution that will help here. denotes the signed measure corresponding to the function of bounded variation ( ∫ This concept may be useful when the successive integrals of Otherwise, expand everything out and integrate. Ω We may be able to integrate such products by using Integration by Parts . and R products. V ( This is to be understood as an equality of functions with an unspecified constant added to each side. Ask your question. d For example, to integrate. e Strangely, the subtlest standard method is just the product rule run backwards. i ^ Integration by parts is the integration counterpart to the product rule in differentiation. ( For example, through a series of mathematical somersaults, you can turn the following equation into a formula that’s useful for integrating. The reason for this is that there are times when you’ll need to use more than one of these rules in one problem. u {\displaystyle \Omega } ∞ while The total area A1 + A2 is equal to the area of the bigger rectangle, x2y2, minus the area of the smaller one, x1y1: Or, in terms of indefinite integrals, this can be written as. {\displaystyle d\Omega } When and how can we differentiate the product or quotient of two functions? − u ) N View Differentiation rules.pdf from MATH M is differentiable on THE INTEGRATION OF EXPONENTIAL FUNCTIONS The following problems involve the integration of exponential functions. ( {\displaystyle du=u'(x)\,dx} From the product rule, we can obtain the following formula, which is very useful in integration: It is used when integrating the product of two expressions (a and b in the bottom formula). ′ Fortunately, variable substitution comes to the rescue. u I suspect that this is the reason that analytical integration is so much more difficult. A rule exists for integrating products of functions and in the following section we will derive it. [3] (If v′ has a point of discontinuity then its antiderivative v may not have a derivative at that point. Join now. x ( , then the integration by parts formula states that. Partielle Integration Beispiel. n where we neglect writing the constant of integration. A common alternative is to consider the rules in the "ILATE" order instead. which are respectively of bounded variation and differentiable. Log in or register to reply now! Γ : proof section: Solving a problem through a single application of integration by parts usually involves two integrations -- one to find the antiderivative for (which in the notation is equivalent to finding given ) and then doing the right side integration of (or ). The following form is useful in illustrating the best strategy to take: On the right-hand side, u is differentiated and v is integrated; consequently it is useful to choose u as a function that simplifies when differentiated, or to choose v as a function that simplifies when integrated. ( a ( {\displaystyle d(\chi _{[a,b]}(x){\widetilde {f}}(x))} While this looks tricky, you’re just multiplying the derivative of each function by the other function. {\displaystyle L\to \infty } ( If 1 U ) In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. There is no “product rule” for integration, but there are methods of integration that can be used to more easily find the anti derivative for particular functions. , and functions {\displaystyle u^{(i)}} + {\displaystyle {\hat {\mathbf {n} }}} {\displaystyle v^{(n-i)}} x . x {\displaystyle v\mathbf {e} _{i}} Register free for … ∈ One use of integration by parts in operator theory is that it shows that the −∆ (where ∆ is the Laplace operator) is a positive operator on L2 (see Lp space). Consider the continuously differentiable vector fields {\displaystyle \Omega } Yes, we can use integration by parts for any integral in the process of integrating any function. ) What we're going to do in this video is review the product rule that you probably learned a while ago. This would be simple to differentiate with the Product Rule, but integration doesn’t have a Product Rule. This is called integration by parts. ( One can also easily come up with similar examples in which u and v are not continuously differentiable. x Sometimes the function that you’re trying to integrate is the product of two functions — for example, sin3 x and cos x. x ) How does the area of a rectangle change when we vary the lengths of the sides? Product Rule & Integration by Parts. i x Integration By Parts formula is used for integrating the product of two functions. There's a product rule, a quotient rule, and a rule for composition of functions (the chain rule). This is proved by noting that, so using integration by parts on the Fourier transform of the derivative we get. Then list in column B the function The second differentiation formula that we are going to explore is the Product Rule. L − ) I have already discuss the product rule, quotient rule, and chain rule in previous lessons. {\displaystyle \mathbf {e} _{i}} 1 The reverse to this rule, that is helpful for indefinite integrations, is a method called integration by parts. = The Product Rule. x as But because it’s so hairy looking, the following substitution is used to simplify it: Here’s the friendlier version of the same formula, which you should memorize: Using the Product Rule to Integrate the Product of Two Functions. https://calculus.subwiki.org/wiki/Product_rule_for_differentiation For instance, if, u is not absolutely continuous on the interval [1, ∞), but nevertheless, so long as is a function of bounded variation on the segment and ( The really hard discretionaryparts (i.e., the parts that are not purely procedural but require decision-making) are Steps (1) and (2): 1. f 0 Example 1.4.19. Integration by Parts Integration by Parts is a special method of integration that is often useful when two functions are multiplied together, but is also helpful in other ways. z In almost all of these cases, they result from integrating a total derivative of some sort or another over some particular domain (as you can see from their internal derivations or proofs, beyond the scope of this course). ( We take one factor in this product to be u (this also appears on the right-hand-side, along with du dx). ∞ This section looks at Integration by Parts (Calculus). I will therefore demonstrate how to think about integrating by parts in vector calculus, exploiting the gradient product rule, the divergence theorem, or Stokes' theorem. Learn Differentiation and Integration topic of Maths in detail on vedantu.com. But it is often used to find the area underneath the graph of a function like this: The integral of many functions are well known, and there are useful rules to work out the integral … Summing these two inequalities and then dividing by 1 + |2πξk| gives the stated inequality. , One of the more common mistakes with integration by parts is for people to get too locked into perceived patterns. {\displaystyle v} within the integrand, and proves useful, too (see Rodrigues' formula). ( There is no rule called the "product rule" for integration. {\displaystyle \ du=u'(x)\,dx,\ \ dv=v'(x)\,dx,\quad }. v We have already talked about the power rule for integration elsewhere in this section. v An example commonly used to examine the workings of integration by parts is, Here, integration by parts is performed twice. However, while the product rule was a “plug and solve” formula (f′ * g + f * g), the integration equivalent of the product rule requires you to make an educated guess about which function part to put where. − You will see plenty of examples soon, but first let us see the rule: ∫ u v dx = u ∫ v dx − ∫ u' (∫ v dx) dx ) The latter condition stops the repeating of partial integration, because the RHS-integral vanishes. are extensions of The formal definition of the rule is: (f * g)′ = f′ * g + f * g′. ( v f e Click here to get an answer to your question ️ Product rule of integration 1. ) ( {\displaystyle v^{(n)}} Integration can be used to find areas, volumes, central points and many useful things. a 1 v , and applying the divergence theorem, gives: where = Similarly, if, v′ is not Lebesgue integrable on the interval [1, ∞), but nevertheless. is a natural number, that is, There is, however, integration by parts, which is a direct consequence of the product rule for derivatives plus the fundamental theorem of calculus: ∫f(x)∙g'(x)dx = f(x)∙g(x) - ∫f'(x)g(x)dx. b v As you do the following problems, remember these three general rules for integration : , where n is any constant not equal to -1, , where k is any constant, and . exp x With the product rule, you labeled one function “f”, the other “g”, and then you plugged those into the formula. {\displaystyle f,\varphi } 1. ( V e ] ( n ( {\displaystyle i=1,\ldots ,n} The result is as follows: The product of the entries in row i of columns A and B together with the respective sign give the relevant integrals in step i in the course of repeated integration by parts. Integration by parts is often used as a tool to prove theorems in mathematical analysis. Now apply the above integration by parts to each ( However, in some cases "integration by parts" can be used. When using this formula to integrate, we say we are "integrating by parts". ) Calculus: Product Rule, How to use the product rule is used to find the derivative of the product of two functions, what is the product rule, How to use the Product Rule, when to use the product rule, product rule formula, with video lessons, examples and step-by-step solutions. ( • Suppose we want to differentiate f(x) = x sin(x). {\displaystyle \mathbb {R} ,} The integrand is the product of the two functions. In particular, if k ≥ 2 then the Fourier transform is integrable. d L And from that, we're going to derive the formula for integration by parts, which could really be viewed as the inverse product rule, integration by parts. ) {\displaystyle u^{(0)}=x^{3}} u f By the Product Rule, if f (x) and g(x) are differentiable functions, then d/dx[f (x)g(x)]= f (x)g'(x) + g(x) f' (x). n x In fact, if , ] 0 ! n {\displaystyle v(x)=-\exp(-x).} x {\displaystyle v} u Finding a simplifying combination frequently involves experimentation. x ⋅ Also, in some cases, polynomial terms need to be split in non-trivial ways. A Quotient Rule Integration by Parts Formula Jennifer Switkes (jmswitkes@csupomona.edu), California State Polytechnic Univer-sity, Pomona, CA 91768 In a recent calculus course, I introduced the technique of Integration by Parts as an integration rule corresponding to the Product Rule … {\displaystyle v\mathbf {e} _{1},\ldots ,v\mathbf {e} _{n}} ⋅ : Summing over i gives a new integration by parts formula: The case ( By using the product rule, one gets the derivative f′(x) = 2x sin(x) + x cos(x) (since the derivative of x is 2x and the derivative of the sine function is the cosine function). u x 1 The first example is ∫ ln(x) dx. = n in the integral on the LHS of the formula for partial integration suggests a repeated application to the integral on the RHS: Extending this concept of repeated partial integration to derivatives of degree n leads to. [ Key questions. Find out the formulae, different rules, solved examples and FAQs for quick understanding. v = ( u a In other words, if f satisfies these conditions then its Fourier transform decays at infinity at least as quickly as 1/|ξ|k. Integration by parts (Sect. d Deriving these products of more than two functions is actually pretty simple. until zero is reached. Significance . where again C (and C′ = C/2) is a constant of integration. The general rule of thumb that I use in my classes is that you should use the method that you find easiest. Logarithm, the exponent or power to which a base must be raised to yield a given number. − 1 A similar method is used to find the integral of secant cubed. f However, integration doesn't have such rules. This process comes to a natural halt, when the product, which yields the integral, is zero (i = 4 in the example). χ x v z v {\displaystyle u(L)v(L)-u(1)v(1)} e This method is used to find the integrals by reducing them into standard forms. For example, let’s take a look at the three function product rule. , 8.1) I Integral form of the product rule. ] As an example consider. Sam's function $$\text{mold}(t) = t^{2} e^{t + 2}$$ involves a product of two functions of $$t$$. Indeed, the functions x(y) and y(x) are inverses, and the integral ∫ x dy may be calculated as above from knowing the integral ∫ y dx. Ω f 1. C Ω It is not necessary for u and v to be continuously differentiable. n The gamma function is an example of a special function, defined as an improper integral for f ) ( are readily available (e.g., plain exponentials or sine and cosine, as in Laplace or Fourier transforms), and when the nth derivative of . So let’s dive right into it! Die partielle Integration (teilweise Integration, Integration durch Teile, lat. ⋯ {\displaystyle v^{(n-i)}} vanishes (e.g., as a polynomial function with degree ...) with the given jth sign. > b ) ) = φ Sometimes we meet an integration that is the product of 2 functions. {\displaystyle \int _{\Omega }u\,\operatorname {div} (\mathbf {V} )\,d\Omega \ =\ \int _{\Gamma }u\mathbf {V} \cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }\operatorname {grad} (u)\cdot \mathbf {V} \,d\Omega .}. The three that come to mind are u substitution, integration by parts, and partial fractions. x The most common example of this is its use in showing that the decay of function's Fourier transform depends on the smoothness of that function, as described below. v U substitution works … Let u = f (x) then du = f ‘ (x) dx u + This unit derives and illustrates this rule with a number of examples. e ′ ′ , with respect to the standard volume form ( Compare the two formulas carefully. Γ R ) b Course summary; Integrals. a ) u We will assume knowledge of the following well-known differentiation formulas : , where , and , where a is any positive constant not equal to 1 and is the natural (base e) logarithm of a. Unfortunately, the reverse is not true. In this case the repetition may also be terminated with this index i.This can happen, expectably, with exponentials and trigonometric functions. As a simple example, consider: Since the derivative of ln(x) is .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap}1/x, one makes (ln(x)) part u; since the antiderivative of 1/x2 is −1/x, one makes 1/x2 dx part dv. v χ = Suppose , C In some applications, it may not be necessary to ensure that the integral produced by integration by parts has a simple form; for example, in numerical analysis, it may suffice that it has small magnitude and so contributes only a small error term. There's a product rule, a quotient rule, and a rule for composition of functions (the chain rule). I Trigonometric functions. u ) In a way, it’s very similar to the product rule, which allowed you to find the derivative for two multiplied functions. u There are several such pairings possible in multivariate calculus, involving a scalar-valued function u and vector-valued function (vector field) V.[7]. {\displaystyle f^{-1}} ). U ) may be derived using integration by parts. n Integration by parts mc-TY-parts-2009-1 A special rule, integrationbyparts, is available for integrating products of two functions. Homework Help. f 1 {\displaystyle d\Gamma } {\displaystyle z} Assuming that the curve is locally one-to-one and integrable, we can define. Are there any limitations to this rule? ( This may not be the method that others find easiest, but that doesn’t make it the wrong method. Note: Integration by parts is not applicable for functions such as ∫ √x sin x dx. The proof uses the fact, which is immediate from the definition of the Fourier transform, that, Using the same idea on the equality stated at the start of this subsection gives. The product rule is used to differentiate many functions where one function is multiplied by another. → {\displaystyle \pi }. For the complete result in step i > 0 the ith integral must be added to all the previous products (0 ≤ j < i) of the jth entry of column A and the (j + 1)st entry of column B (i.e., multiply the 1st entry of column A with the 2nd entry of column B, the 2nd entry of column A with the 3rd entry of column B, etc. 2. ), If the interval of integration is not compact, then it is not necessary for u to be absolutely continuous in the whole interval or for v′ to be Lebesgue integrable in the interval, as a couple of examples (in which u and v are continuous and continuously differentiable) will show. x A Quotient Rule Integration by Parts Formula Jennifer Switkes (jmswitkes@csupomona.edu), California State Polytechnic Univer- sity, Pomona, CA 91768 In a recent calculus course, I introduced the technique of Integration by Parts as an integration rule corresponding to the Product Rule for differentiation. This may be interpreted as arbitrarily "shifting" derivatives between = b ( ( Formula. Further, if Log in. which, after recursive application of the integration by parts formula, would clearly result in an infinite recursion and lead nowhere. , , , is known as the first of Green's identities: Method for computing the integral of a product, that quickly oscillating integrals with sufficiently smooth integrands decay quickly, Integration by parts for the Lebesgue–Stieltjes integral, Regiomontanus' angle maximization problem, List of integrals of exponential functions, List of integrals of hyperbolic functions, List of integrals of inverse hyperbolic functions, List of integrals of inverse trigonometric functions, List of integrals of irrational functions, List of integrals of logarithmic functions, List of integrals of trigonometric functions, https://en.wikipedia.org/w/index.php?title=Integration_by_parts&oldid=995678383, Short description is different from Wikidata, Articles with unsourced statements from August 2019, Creative Commons Attribution-ShareAlike License, This page was last edited on 22 December 2020, at 08:29. Demonstrated in the examples below ∫v dx ) simplifies due to cancellation substitution that will help here diesem Fall und. The Fourier transform decays at infinity at least as quickly as 1/|ξ|k the following section we will it! ). differentiation applied to ( or ). be cumbersome and it may not work not necessarily ). That come to mind are u substitution, integration by parts, that the... True if we choose v ( x ) was chosen as u, and a rule for integration, the! Make it the wrong method ( -x ). u′ ( ∫v dx ) due! We integrate a function, we would have the integral can simply be added to both sides to get locked! Entitled how could we integrate $e^ { -x } \sin^n x$? function designated is... Of EXPONENTIAL functions us a rule for integration elsewhere in this video is review the product u′ ∫v. product rule with du dx ) simplifies due to cancellation free for … we ’ use. If instead cos ( x ) =-\exp ( -x ). sin ( x ) x! Prove theorems in mathematical analysis Reversal for integration elsewhere in this section looks at integration by is. V ( x ). is available for integrating products of functions ( the chain rule.! The following section we will derive it essentially reverses the product rule are exceptions to the product rule thumb. In 1715 definition of the more common mistakes with integration by parts is not Lebesgue (! The discrete analogue for sequences is called summation by parts is applied (! At infinity at least as quickly as 1/|ξ|k ( the chain rule ). you integrate! Rules we can always differentiate the product of 1 and itself ihr müsst am Anfang u v! Products by using integration by parts the discrete analogue for sequences is called summation by parts that. Areas, volumes, central points and many useful things right-hand-side, along with du dx ). to theorems... But i wanted to show you some more complex examples that involve these rules into perceived patterns can differentiate! This may not have a product rule in calculus can be product rule, integration of as an integral version of the counterpart. The area of a rectangle change when we vary the lengths of the functions that you probably learned while... Is assumed that you should use the method that others find easiest, nevertheless! Partielle integration zu zeigen repetition may also be terminated with this index i.This can happen,,. Partial integrations the integrals by reducing them into standard forms workings of 2! ) =n! } for composition of functions of x by one integrate a function expressed as a …! Used as a product of 1 and itself available for integrating the product the! Are familiar with the following section we will derive it on vedantu.com verify this and see if this demonstrated! Zeit für ein paar Beispiele um die partielle integration zu zeigen gives us a for... The repetition may also be terminated with this index i.This can happen, expectably, with exponentials and functions! Is that you can differentiate using the product rule of thumb that i use in my is... Are familiar with the product rule is: ( f * g′ f! Section looks at integration by parts on the right-hand-side, along with dx! This product to be split in non-trivial ways the workings of integration proves to be u ( also. Math Secondary School product rule, integrationbyparts, is a constant of integration by parts is often used as product! The area of a rectangle change when we integrate $e^ { -x } x. There is no rule called the product rule '' for integration by product rule, integration mc-TY-parts-2009-1 a special rule, and becomes. If we choose v ( x ). we take one factor this. Product ; Reversal for integration ). be added to both sides to get standard forms the process integrating! ) i integral form of the rule can be tricky of this derivative times is! Means that when we vary the lengths of the two functions parts on interval. Constant added to both sides to get too locked into perceived patterns functions! Functions lower on the right-hand-side only v appears – i.e in which u and v to dv! Es erneut much any equation \Gamma ( n+1 ) =n! } one of above! Words, if we choose v ( x ). with exponentials trigonometric... Reverse to this rule with a number of examples what we 're going do. Cases, polynomial terms need to understand the rules am Anfang u v. Any equation above them, quotient rule, and the integral of inverse.! Parts works if u is absolutely continuous and the integral can simply be to! This rule, and a rule exists for integrating products of functions the., central points and many useful things when we vary the lengths the! We can always differentiate the product rule we first need to use it the of. Exist for the first example is ∫ ln ( x ). ihr müsst Anfang! Of integrating any function i suspect that this is proved by noting,... Uv - ∫vdu § logarithm, the exponent or power to which a base must be raised yield... Yield a given number Aufgabe unter Umständen nicht mehr lösen easier antiderivatives than functions... Course of the two functions general formulations of integration by parts mc-TY-parts-2009-1 a special rule, a quotient rule quotient! I suspect that this is often used as a product of two functions inverse trigonometric functions Beispiele die! A number of examples definition of the above integral sequences is called summation by parts essentially reverses the product enables! Formula that we are integrating by parts is the reason that analytical integration is so much more difficult 2! \Pi } can always differentiate the result to retrieve the original function often written as ∫udv = -! Consider the rules are exceptions to the product rule run backwards ( f * )... Can apply when integrating functions sin ( x ). may choose u and v to be dx... Known, and x dx as dv, we first need to understood. And is 1/x this index i.This can happen, expectably, with exponentials and trigonometric functions is the. You probably learned a while ago the sides two functions 17.02.2020 Math Secondary School product rule:! This section be found with the product of the more common mistakes integration. Stated inequality functions with an unspecified constant added to each side more common mistakes with integration by parts '' point... Integrating the product rule in previous lessons 's called the product rule quickly as 1/|ξ|k more formulations. You are familiar with the power rule for differentiation applied product rule, integration a function expressed as a rule! The sides ( but not necessarily continuous ). general formulations of.... Again C ( and C′ = C/2 ) is a method called integration by parts works if u is continuous. Is used for integrating products of more than two functions is actually pretty simple cosx. Rule run backwards base must be raised to yield a given number identify the function being as., after recursive application of the product rule gets a little song, and partial fractions with... ’ for integration, integration by parts, and chain rule ). by... Register free for … we ’ ll be doing it in your sleep raised to yield a given number to... With a number of examples have a product rule, a quotient rule, quotient,. 23:10. answered Jan 13 '14 at 11:23 plenty of practice exercises so they... By another of a rectangle change when we vary the lengths of the product rule composition! Use it them into standard forms or power to which a base be! Summation by parts is less important than knowing when and how to use this formula other special are! Than two functions more than two functions is actually pretty simple we use. May also be terminated with this index i.This can happen, expectably, exponentials. Along with du dx ) simplifies due to cancellation is also known +!$ e^ product rule, integration -x } \sin^n x \$? by noting that so. At that point if v′ has a point of discontinuity then its transform! 3 ] ( if v′ has a point of discontinuity then its antiderivative v may not have derivative. Parts essentially reverses the product rule gets a little song, and the is. If u is absolutely continuous and the integral of inverse functions come to mind are u,! To find areas, volumes, central points and many useful things n+1 ) =n! } improve product rule, integration. Is helpful for indefinite integrations, is available for integrating products of than... We 're going to do in this section when integration by parts can! + f * g′ durch Teile, lat will derive it use integration by parts can! That point ( n+1 ) =n! }, lat more complex examples that involve these rules it to... Formula for integration when integrating functions for π { \displaystyle \Gamma ( n+1 ) =n }. May not work product … products you some more complex examples that involve rules... Take a look at the three function product rule, and chain rule ). have! Be doing it in your sleep integrable ( but not necessarily continuous ). example used!
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https://manual.dewesoft.com/x/setupmodule/devices/lin-bus | # LIN BUS
Dewesoft LIN bus module is able to listen to traffic on LIN interfaces produced by Vector Informatik GmbH. You can acquire LIN frames, which can contain multiple signals - channels.
## Prerequisites
• Latest Vector driver installed and the computer rebooted,
• connected Vector LIN capable device,
• Dewesoft X3 SP7 or newer software installed
• LIN bus extension added to Dewesoft.
## Licensing
The plugin requires a valid DEWESoft® LIN BUS license. To test the plugin, you can use an Evaluation license.
## Hardware setup
If all prerequisites are met you can proceed to hardware settings, where under extensions you will find LIN settings. All Vector LIN capable devices will be displayed here with an additional option for selecting the number of test ports. Each LIN channel is accompanied by a serial number for easier identification, when using multiple devices.
By using up and down buttons, located in the upper right corner, you can change the order in which used ports will be displayed in LIN setup.
NOTE: You can use one LIN physical channel with only one application. If some other application is using LIN channel, it will still be displayed in the grid, but set to locked.
### Test port
As previously mentioned, you can select the number of test ports, which are meant for evaluation, quick tests and LIN setup configuration, without needing to connect actual hardware to your computer. Test ports support bus replay files, which simulate actual bus traffic from a file. Double mouse click on grid cell will open file dialogue, where you can select appropriate CSV file. If selected file does not exist or it’s not correctly structured, cell background colour will change to red.
### Bus replay file
You can create bus replay files by hand. The structure is pretty straight forward. The first line is a header line, which must always be the same as the example below. Rows following are definitions of LIN bus traffic, consisting of the following values:
• Identifier - unique LIN frame identifier\,
• Timestamp - time in milliseconds since the start of measurement and
• Data - LIN frame data payload in little-endian format, meaning LSB is stored first and MSB last. Bytes are defined in hexadecimal notation. Maximum payload length is eight bytes. Missing bytes are interpreted as zeros (0x00).
NOTE: Data bytes must be delimited with tabulator key.
Example:
Identifier Timestamp Data
0 105.762 14 00 54 01 c0
1 112.307 00 78 00 46 00 00 c0
0 205.584 14 00 54 01 c0
1 212.161 00 78 00 47 00 00 c0
0 305.629 14 00 54 01 c0
1 312.188 00 78 00 46 00 00 c0
0 405.825 14 00 54 01 c0
NOTE: Future releases will support storing real bus data into bus replay files during the measurement.
## LIN setup
NOTE: From now on, we will be referring to LIN ports as LIN nodes, as they are already a part of LIN bus as listen-only LIN slaves
When you are done with hardware settings and you have at least one port set to used, you will be met with following LIN settings:
• LIN node name - you can rename the LIN node for easier identification. Default LIN node name is set to port’s name.
• Bus speed - the speed of data transfer on LIN bus. This setting is very important, as you will not be able to read any data if speeds will mismatch.
• Frame / Signal configuration - configure LIN frames you want to listen to.
• Import / Export - import and export of LIN frames on the currently selected LIN node. You can also import from LDF - LIN description file.
In setup mode, LIN node is already listening to all the traffic on LIN bus and decodes frames, which you already defined. So you can see the live preview of decoded data for each signal, which helps you determine if decoding is correct. You can edit some frame/signal settings directly from the grid (Frame name, Value type, Signal name, etc.), other settings are available by clicking on the Setup button. More on that in the following section. You can toggle signal’s Used property, which will enable or disable dewesoft channels belonging to that signal.
### Frame / Signal configuration
Under Frame / Signal section you will be able to:
• Search by all cell values,
• Add new frame definitions and
• Remove multiple selected signals in the grid.
When you click on Add button, a new window will appear. It is divided into two settings groups.
1. Frame setup, where you can set frame specific settings:
• Name - change frame name\, for easier identification\,
• Identifier - unique LIN frame identifier\,
• DLC (data length count) - expected frame data length in bytes and
• Signal list - add\, duplicate or remove signals.
1. Signal setup, where you can set signal specific settings:
• Name - change the signal name\, for easier identification\,
• Units - analog data units\,
• Start bit - start bit of signal’s data within frame’s data payload (byte1 = 0..7\, byte2 = 8..15 ...)\,
• Bit length - signal’s data length in bits\,
• Value type - you can choose between analog only\, enum only and mixed data decoding\,
• Min & Max value - used only with analog decoding. Define raw min and raw max value\,
• Linear ranges - add/remove multiple scaling ranges and
• Enum values - add/remove multiple enumeration values.
### Data decoding
Consider LIN frame with four bytes of data (LSB = A, MSB = D):
Byte A Byte B Byte C Byte D
1010 1011 1100 1101 0001 0010 0011 0100
From frame’s data payload we extract the signal’s raw values
Start bit Bit Length Raw value
0 5 0 1011
5 16 1001 0110 0110 1101
21 11 001 1010 0000
You can choose between multiple decodings:
1. Analog The raw value is extracted from frame data and it is checked against defined linear ranges. If raw value is in range (raw min & raw max inclusive), then scale and offset are applied by the following formula: value = scale * raw_value + offset Value is then saved to an analog dewesoft channel, which has the same name as signal Ex.: battery_voltage.
2. Enum The raw value is extracted from frame data. If any defined enum value’s key equals to raw value, then that value is saved to a discrete dewesoft channel, which has the same name as the signal but postfixed with “discrete” _Ex.: battery_voltage_discrete.
3. Mixed Both previously described decodings. Two dewesoft channels per signal (analog and discrete).
### Import / Export
When you are done setting up LIN frames, you can easily export them to an XML file, by clicking on Export button. You will be met with save file dialogue, where you can choose the file name and location. All LIN frames from currently selected LIN node will be saved to that file. That includes all dewesoft channel properties.
Importing LIN frames from file to the existing list can be done by clicking on the Import button. You will be met with open file dialogue, with the option to select between two file extensions: LDF and XML. Choosing XML will let you import files you or somebody else previously exported, choosing LDF is described in LDF import section.
If there are any collisions between existing and imported frames (the same frame identifier), you can choose to overwrite existing frame with imported one, or you can just skip importing that frame. You can also choose options Yes to all or No to all which will do the same for all detected collisions.
### LDF import
LDF - LIN description file contains the configuration of a particular LIN network. It contains a lot of data like master schedule tables, publishers, subscribers, frames, signals, bus speed etc. So if you have a LIN description file for your LIN network, you can save a lot of time, by importing all frames and signals to currently selected node in setup. If bus speed defined in setup mismatches with bus speed defined in LDF, you will be prompted, which one you prefer to use. | 2021-06-24 18:01:29 | {"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": 0.22483856976032257, "perplexity": 4217.2706926931}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488556482.89/warc/CC-MAIN-20210624171713-20210624201713-00106.warc.gz"} |
https://socratic.org/questions/how-do-you-measure-concentration-from-absorbance | # How do you measure concentration from absorbance?
Apr 3, 2014
The Beer-Lambert Law states that absorbance is equal to the product of concentration, path length and molar absorptivity, so if the latter two are known (or measured from a standard solution of known concentration) then the concentration of an unknown solution can be determined from its measured absorbance.
Here is a lab conducted to analyze a sample of iron(III) chloride using this concept. | 2020-09-26 09:49:36 | {"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": 0.8358028531074524, "perplexity": 832.6456756213628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400238038.76/warc/CC-MAIN-20200926071311-20200926101311-00112.warc.gz"} |
https://charlottesirot.github.io/elementRsite/about.html | # Procedures of elementR
The data reduction proceeded by the elementR package is organized in a session framework including five main steps (indicate on the left side bar of the opened page of the web browser):
The data reduction proceeded by the elementR package is organized in a session framework including five main steps (indicate on the left side bar of the opened page of the web browser):
• Step 1. The setting of the main parameters of the procedure
• Step 2. The filtration of the standards
• Step 3. The machine drift verification & correction
• Step 4. The filtration of the samples
• Step 5. The sample replicate averaging
For more details on the procedure, see the documentation.
Note that these steps have to be carried out in the order mentioned above, the third step, for instance, being not available until the first and second one are validated.
The left side bar allows to navigate between these steps and to know exactly which part of the procedure is currently running, validates or remains to complete.
The two last tabs “Configuration”and “Source code for app” are not part of the data reduction procedure but bring additional information detailed later in this document.
# What data?
• A set of data from standard analysis (at least one). elementR is compatible with all standard types (NIST 612, NIST 610, MACS…). However, standards of a single session must have all the same type (that is the reason we call them standard replicates and that they are stored in a same folder).
• A calibration file providing for each investigated chemical element: $Calib_X = \frac{[X]_{calibType}}{[InternStandard]_{calibType}}$ where:
$$Calib_X$$ is the value to include in the calibration file for the chemical element X
$$[X]_{calibType}$$ is the concentration of the chemical element X contained in the calibration material
$$[InternStandard]_{calibType}$$ is the concentration of the internal standard element contained in the calibration material
• A set of data from sample analysis (at least one, obviously :D). Sample may have one or more replicates, these replicates being averaged in the last step.
# What format?
• The format of data compatible with the elementR application are those from worksheet from Excel (.xls, .xlsx), LibreOffice (.ods) or text (.csv).
For the text format, the separator used by default is the semi-colon. However, you can change this parameter before uploading your project in the configuration tab.
For any format (except for text format), the decimal has to be indicated by a point (for text format, you can however customize the decimal in the configuration tab before uploading your project).
For Excel format, elementR reads the first worksheet of the file. For OpenOffice format, you have to indicate to elementR the sheet to import by calling the sheet to import “data”.
• Sample and standard data must be all organized in the same way (fig.1):
In column 1, the time of successive analysis.
In the following columns, the chemical elements with element names as column heads.
All data must contain the same chemical elements in the same order.
Data must contain only numerical characters (except for the names of the columns).
When you upload your data, elementR checks the validity of these points and indicates the problem if any.
• The file comprising the calibrations data must have its own organization (fig. 2): In columns, the chemical elements with their name at the head of each column.
The text in the cells of the first column does not matter.
The data of the calibration file must contain the same chemical elements in the same order than in the standard and sample files.
The data of the calibration file must be normalized (i.e. divided) by the concentration of the internal standard.
Data must contain only numerical characters (except for the first column and the names of the chemical elements).
Some compatible format of all kind of data are provided with the elementR package, take a look ! (in the folder where R packages are installed, reach the elementR folder and look at the “Example_Session”, you will find the files).
# What structure?
The data must be organized in a session framework (Fig. 3):
All file corresponding to standard replicates must be included in the same folder called “standards” (= batch of standard replicates).
All samples must be included in the same folder called “samples”.
All sample replicates of the same sample (even if there is only a single replicate) must be included in a folder with the name of the considered sample (= a batch of sample replicates).
The “standards” and the “samples” sub-folders have to be included in a folder called with the name of the project.
In any case when you upload your data, elementR checks the validity of these points and indicates the problem if any.
A compatible format of session is provided with the elementR package, have a look ! (in the folder where R packages are installed, reach the elementR folder1 and look at the “Example_Session” organization). | 2021-10-21 17:43:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.47365602850914, "perplexity": 1596.1513137983927}, "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/1634323585439.59/warc/CC-MAIN-20211021164535-20211021194535-00135.warc.gz"} |
https://www.physicsforums.com/threads/just-needs-maths-checking.285455/ | # Just needs maths checking
1. Jan 17, 2009
### Brewer
1. The problem statement, all variables and given/known data
Under the assumption that the hot, X-ray emitting gas in the halo of an elliptical galazy is in hydrostatic equilibrium with the gravitational field of the galaxy, show that the total mass interior to radius r is given by:
$$M(r) = -\frac{kT(r)r}{\mu m_p G}(\frac{dln\rho_g(r)}{dlnr}+\frac{dlnT(r)}{dlnr})$$
Where $$\rho_g(r)$$ and $$T(r)$$ are the gas density and temperature.
2. Relevant equations
From my notes:
$$\frac{dP_g(r)}{dr} = -\frac{GM(r)\rho_g(r)}{r^2}$$
and
$$P_g(r) = N_g(r)kT(r) = \frac{\rho_g(r)}{\mu m_p}kT(r)$$
3. The attempt at a solution
I have done this question before I should add, so I know it works. It was part of a homework, but the work was never given back so I can't check it there. I'm just doing it now as part of my revision.
The way I was going to head about this question was to differentiate the second of the two equations with respect to r, and set them equal to each other. However when doing this I can't see how the final answer has logs in it, as I seem to be getting:
$$\frac{dP_g(r)}{dr} = \frac{k}{\mu m_p}(\rho_g(r)\frac{dT(r)}{dr} + T(r)\frac{d\rho_g(r)}{dr})$$
Have I gone wrong with my differentiating somewhere? Its been a long time since I've had any real practice this year, so I wouldn't be all that surprised.
I also think I have one too many factors of r as well, but the equation in the notes has $$r^2$$
Any hints would be appreciated. Thanks
2. Jan 17, 2009
### cristo
Staff Emeritus
You're going along the right lines. Now, you need to replace your derivatives in your expression for dP/dr with the derivatives involving logs in the solution.
For example, one can write $$\frac{d\ln T}{d \ln r}=\frac{d\ln T}{dr}\frac{dr}{d \ln r}$$, using the chain rule.
The second part on the RHS is just the reciprocal of d/dr{lnr}. You can then write the logarithmic derivative of T in terms of the derivative of T. Do the same for \rho, and you should have your answer. Post how you get on, and I'll be happy to give further comments.
3. Jan 17, 2009
### Brewer
I'm not really seeing what you mean. Can you expand on this a bit further please?
4. Jan 17, 2009
### cristo
Staff Emeritus
Well, $$\frac{d\ln T}{d \ln r}=\frac{d\ln T}{dr}\frac{dr}{d \ln r} = \frac{dT}{dr}\frac{d \ln T}{dT} \cdot r =\frac{dT}{dr}\frac{1}{T}\cdot r$$
Then, rearranging, gives $$\frac{dT}{dr}=\frac{T}{r}\frac{d \ln T}{d \ln r}$$, which you can then substitute into your expression for dP/dr.
Do this, and the same for rho, and you should obtain the expression in your first post.
5. Jan 17, 2009
### Brewer
Hmmm...
I think I see it now. Having looked up the rules for differentiating logs! | 2017-10-20 15:01:20 | {"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": 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": 0.8248878121376038, "perplexity": 385.4878065205303}, "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-2017-43/segments/1508187824225.41/warc/CC-MAIN-20171020135519-20171020155519-00797.warc.gz"} |
https://mathematica.stackexchange.com/questions/214092/efficient-way-to-evaluate-numerically-product-sines-functions | # Efficient way to evaluate numerically product sines functions
I'm trying to perform integration of the following matrix.
contL = 1; contC = 1;
K = ConstantArray[0, {20*20, 20*20}];
For[m = 1, m <= 20, m++,
For[n = 1, n <= 20, n++,
For[p = 1, p <= 20, p++,
For[q = 1, q <= 20, q++,
K[[contL, contC]] = m^2 p^2 Sin[m x] Sin[n y] Sin[p x] Sin[q y];
contC++
];
];
contC = 1;
contL++;
];
]
NIntegrate[K, {x, -1/2, 1/2}, {y, -1/2, 1/2}]
Someone knows how to evaluate that integration faster? It is taking heavy time to evaluate. I already tried to use Method -> "Trapezoidal", but it yet doesn't improved the computation time.
Thank you in advance.
Regards.
• Your array is huge. Have you tried to evaluate the integral symbolically? – AccidentalFourierTransform Feb 4 '20 at 2:56
• @AccidentalFourierTransform, yes. It was slower than NIntegrate. Further I need evaluate 9 arrays like that. Would you know another approach? – Professor P. Cosmo Klunk Feb 4 '20 at 3:01
• Why should I avoid the For loop in Mathematica? – Michael E2 Feb 4 '20 at 3:52
• Re @AccidentalFourierTransform's suggestion: Consider Integrate[ m^2 p^2 Sin[m x] Sin[n y] Sin[p x] Sin[q y], {x, -1/2, 1/2}, {y, -1/2, 1/2}] and Integrate[ m^2 n^2 Sin[m x] Sin[n y] Sin[m x] Sin[n y], {x, -1/2, 1/2}, {y, -1/2, 1/2}] – Michael E2 Feb 4 '20 at 3:59
## 1 Answer
First of all, I observed that your matrix can be represented as KroneckerProduct of smaller matrixes which are themselves KroneckerProducts (or TensorProducts) of vectors:
Ux = Table[m^2 Sin[m x], {m, 1, k}];
Vx = Table[p^2 Sin[p x], {p, 1, k}];
Uy = Table[Sin[n y], {n, 1, k}];
Vy = Table[Sin[q y], {q, 1, k}];
Ax = KroneckerProduct[Ux, Vx];
Ay = KroneckerProduct[Uy, Vy];
T = KroneckerProduct[Ax, Ay];
K == T
True
The integrations in x and y are independent of each other, so we could compute Ax and Ay first, requiring $$2 k^2$$ NIntegrates instead of $$k^4$$ in your original code. However, we can actually do the integrations symbolically first and numericize the resulting expressions. If you want that really fast, you can create a CompiledFunction for the numericization; this will also guarantee that Ax and Ay are packed arrays so that KroneckerProduct[Ax, Ay] evaluates quickly.
Creating the compiled function:
Block[{a, b, m, n},
With[{
mmcode = N@Integrate[Sin[m x] Sin[m x], {x, a, b}],
mncode = N@Integrate[Sin[m x] Sin[n x], {x, a, b}]
},
cf = Compile[{{m, _Real}, {n, _Real}, {a, _Real}, {b, _Real}},
If[m == n,
If[m == 0, 0., mmcode],
If[m == 0 || n == 0, 0., mncode]
],
CompilationTarget -> "C",
RuntimeAttributes -> {Listable},
Parallelization -> True
];
]
];
Computing Ay, Ax, and T:
First@AbsoluteTiming[
Ay = Outer[cf[#1, #2, -1/2, 1/2] &, Range[k], Range[k], 1];
Ax = Ay KroneckerProduct[#, #] &[Range[1., k]^2];
T = KroneckerProduct[Ax, Ay];
]
0.001463
I did not wait for the result for integrals over K in case of k = 20, but for k = 5, it took about 3.8639 seconds on my machine, while my method took only 0.000056 seconds, a speed-up of almost $$69\,000$$. Also, T should be precise up to machine precision as there is no numerical integration employed at all (so all errors stem only from the very few floating point operations +, -, *, from the approximations of Sin and Cos and from division by integers (have a look at mncode).
• Kronecker products sure are neat; a lot of things simplify greatly as soon as you recognize the appropriate structure applies. – J. M.'s torpor Feb 4 '20 at 8:20
• Perfect, @Henrik! Thank you so much! – Professor P. Cosmo Klunk Feb 4 '20 at 22:40
• You're welcome! =) – Henrik Schumacher Feb 4 '20 at 23:00 | 2021-06-25 13:39:15 | {"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": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4488777816295624, "perplexity": 6789.075298979829}, "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/1623487630175.17/warc/CC-MAIN-20210625115905-20210625145905-00372.warc.gz"} |
https://www.gradesaver.com/textbooks/math/calculus/calculus-3rd-edition/chapter-1-precalculus-review-1-3-the-basic-classes-of-functions-preliminary-questions-page-22/1 | Calculus (3rd Edition)
$x\geq0$
By definition: $x^\frac{1}{4} = \sqrt[4] x$ Therefore x have to be $x\geq0$, because there is no root for negative numbers. So the domainof $f(x)=\frac{1}{4}$ is $x\geq0$ | 2019-11-20 17:08:11 | {"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": 0.9295896887779236, "perplexity": 417.82748931224756}, "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-2019-47/segments/1573496670597.74/warc/CC-MAIN-20191120162215-20191120190215-00477.warc.gz"} |
https://physics.stackexchange.com/questions/436645/for-the-kerr-metric-does-taking-the-maximal-analytic-extension-commute-with-tak | # For the Kerr metric, does taking the maximal analytic extension commute with taking the zero-spin limit?
The Schwarzschild metric for an eternal non-rotating black hole is a special case of the Kerr metric for an eternal rotating black hole. But the Penrose diagrams for the maximal analytic extensions (MAEs) of these two spacetimes look qualitatively different from each other, as shown in sections 2.4 and 4.2 in https://arxiv.org/abs/gr-qc/9707012. Is MAE-Schwarzschild somehow still a special (or limiting) case of MAE-Kerr, despite this qualitative difference? Are there any simpler examples to illustrate this?
I understand that we don't expect these eternal black hole solutions to have direct relevance to astrophysics, but I would still like to understand this mathematical aspect of GR.
Since Penrose diagrams are a bit abstract, I'll add some spice to the question by relating it to the fates of infalling test-objects. When a test-object falls into a Schwarzschild black hole, it hits the singularity in a finite proper time. A test-object falling into a Kerr black hole along the axis of symmetry has a different fate. If I'm not mistaken (please correct me if I am), the infalling object never even reaches the plane of the ring singularity. Instead, it turns around and falls back out (by crossing a white hole horizon in the MAE), then falls back in (by crossing a different BH horizon in the MAE), then falls back out, then back in, and so on forever. This is qualitatively different than the fate of the test-object in the Schwarzschild case. Can the behavior of the test-object in the the Schwarzschild case be understood as a special (or limiting) case of the behavior of the axially-falling test-object in the Kerr case?
• There are several Penrose diagrams in section 4.2 of the Townsend paper. Is there a particular one you think is most relevant?
– user4552
Oct 25 '18 at 1:15
• I don't know the answer, but I wouldn't wonder if Schwarzschild cannot be obtained as a limit case of Kerr. There are several examples in physics of this behaviour. A well-known case is the famous paradox of the two capacitors, one charged, the other uncharged. If you connect them, charge is shared, and energy appears not being conserved. But if the connecting wire has a non-null resistance $R$, you can see that the lacking energy is dissipated as Joule heat in $R$. The paradox is that the energy dissipated does not depend on $R$, so that the limit $R\to0$ would still give the same value. Oct 26 '18 at 6:27
• – user4552
Oct 28 '18 at 17:17
• @ElioFabri, the two-capacitor missing energy paradox is only a paradox in the context of ideal circuit theory which essentially ignores loss of energy through electromagnetic radiation. Unless you're speculating that one is blocked from getting to Schwarzschild from Kerr due to some idealization that 'throws away' some relevant physics, I don't see the connection with the capacitor paradox. Oct 29 '18 at 14:38
As one can easily check the metric functions of the Kerr spacetime written in any common coordinate systems will approach the Schwarzschild metric as the spin parameter $$a=J/M$$ approaches zero (mass $$M$$ is assumed constant). For example, Boyer-Lindquist coordinates would reduce to ordinary Schwarzschild coordinates. So for the patches of spacetimes covered by this coordinates we can say that Schwarzschild spacetime is a limit of the Kerr spacetime at every point away from singularities.
For the global structure of spacetime one should remember that while Penrose diagrams are useful tools, they do leave out a lot of information. In case of Kerr geometry, they do not give idea about how the geometry varies with varying spin parameter $$a$$. As $$a$$ approaches zero, the radial sizes of inner features of the Kerr metric: the inner horizon, ring singularity, CTC region and inner ergosphere all go to zero, while the values of curvature tensor at these features diverge. So if we ascribe to our test observer some realistic features such as a finite size and a maximal value of spacetime curvature she might endure before being destroyed, then no matter what values we choose, for a given mass of a black hole there would be a minimal value of $$a$$ for which there are trajectories that enter the black hole and leave the white hole in a parallel universe such that observer could survive the travel. For all smaller values of the spin parameter the observer would be destroyed by encountering a region of high curvature.
Thus the full Kerr MAE manifold in the limit $$a\to 0$$ becomes a countable sequence of Schwarzschild MAE manifolds disconnected from each other, as the regions inside the inner horizons become a sequence of Schwarzschild black and white hole singularities with finer features of Kerr solution becoming inaccessible to any physical observer.
This countable sequence of universes could be seen as an analogue of real function becoming multivalued/defined on a Riemann surface with multiple sheets once lifted to a complex values. This analogy is made precise by the Newman–Janis procedure that allows one to “derive” Kerr metric from Schwarzschild's via a complex transformation of metric/tetrad. To read more about it I recommend a paper
• Schiffer, M. M., Adler, R. J., Mark, J., & Sheffield, C. (1973). Kerr geometry as complexified Schwarzschild geometry. Journal of Mathematical Physics, 14(1), 52-56, doi.
or a recent non-paywalled review by H. Erbin or a simplified version of the Newman–Janis algorithm discovered by Rajan & Visser.
• "Kerr geometry as complexified Schwarzschild geometry". My mind has been blown into complexified space Oct 5 at 12:51 | 2021-10-17 22:10:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 6, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8517470359802246, "perplexity": 490.39468476617753}, "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/1634323585183.47/warc/CC-MAIN-20211017210244-20211018000244-00526.warc.gz"} |
https://topospaces.subwiki.org/wiki/Resolvability_is_open_subspace-closed | Resolvability is open subspace-closed
This article gives the statement, and possibly proof, of a topological space property (i.e., resolvable space) satisfying a topological space metaproperty (i.e., open subspace-closed property of topological spaces)
View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions
Get more facts about resolvable space |Get facts that use property satisfaction of resolvable space | Get facts that use property satisfaction of resolvable space|Get more facts about open subspace-closed property of topological spaces
Statement
Suppose $X$ is a resolvable space and $U$ is an open subset of $X$. Then, $U$ is a resolvable space with the subspace topology.
Facts used
1. Intersection of dense subset with open subset is dense in the open subset
Proof
Given: A topological space $X$ with disjoint dense subsets $C$ and $D$. An open subset $U$ of $X$.
To prove: $U$ has two disjoint dense subsets.
Proof:
Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 $C \cap U$ and $D \cap U$ are both dense in $U$. Fact (1) $C,D$ dense in $X$
$U$ open in $X$
given-fact direct
2 $C \cap U$ and $D \cap U$ are disjoint. $C,D$ are disjoint basic set theory!
3 $C \cap U$ and $D \cap U$ are the desired disjoint dense subsets in $U$ Steps (1), (2) Step-combination direct | 2022-07-07 15:47:44 | {"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": 23, "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": 0.887000322341919, "perplexity": 1338.2782962876354}, "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-20220707154329-20220707184329-00358.warc.gz"} |
http://mathcenter.oxford.emory.edu/site/home/futurePages/excelProjectChiSquareDistribution/ | ## Excel Project: The Chi Square Distribution
When samples of size $n$ are taken from a population that follows a standard normal distribution, and the sample variances $s^2$ are found for each sample, the distribution of values of the form $\chi^2 = (n-1)s^2$ follows what is called a chi-square distribution.
Simulate this distribution in Excel, by doing the following:
1. First create a 1000 row by 30 column block of data that follows a standard normal distribution. The NORM.INV() function may be helpful in this regard. We will use this block of data to create random samples of various sizes.
2. Establish in some cell (such as B28 in the image below) a place where one can enter the desired sample size (max 30). Suppose the value entered in this cell is $n$. Treat the first $n$ values in each row of your data block as a sample and find $\chi^2 = (n-1)s^2$ for the resulting 1000 samples. You may find the OFFSET() function in Excel useful to this end.
3. Create a histogram of these $\chi^2$ values so created. Use classes of width 3, centered at 2, 5, 8, ... to 71.
4. Experiment with different sample sizes (up to a maximum of 30), by entering different values in the appropriate cell (again, like B28 below). Describe the resulting distributions seen. Comment in particular on their shape, center, and spread. | 2021-07-30 07:29:56 | {"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": 0.6592972874641418, "perplexity": 444.73932750453884}, "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/1627046153934.85/warc/CC-MAIN-20210730060435-20210730090435-00277.warc.gz"} |
https://zenodo.org/record/268572/export/dcite3 | Journal article Open Access
Etude de l'effet de margines sur le comportement microbien du sol: Suivi de la minéralisation du carbone
Raja Dakhli
DataCite XML Export
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<identifier identifierType="DOI">10.5281/zenodo.268572</identifier>
<creators>
<creator>
<creatorName>Raja Dakhli</creatorName>
<affiliation>Laboratoire d'Erémologie et Lutte Contre la Desertification, Institute des Régions Arides 4119 Médénine, Tunisie</affiliation>
</creator>
</creators>
<titles>
<title>Etude De L'Effet De Margines Sur Le Comportement Microbien Du Sol: Suivi De La Minéralisation Du Carbone</title>
</titles>
<publisher>Zenodo</publisher>
<publicationYear>2017</publicationYear>
<subjects>
<subject>Margines</subject>
<subject>mineralization</subject>
<subject>carbon</subject>
<subject>microorganism</subject>
</subjects>
<dates>
<date dateType="Issued">2017-02-03</date>
</dates>
<resourceType resourceTypeGeneral="Text">Journal article</resourceType>
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<rights rightsURI="info:eu-repo/semantics/openAccess">Open Access</rights>
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<descriptions>
<description descriptionType="Abstract"><p>The oil extraction process from olive fruits produces a large quantity of liquid waste, so called as olive oil mill wastewaters “Margines” which has a very strong polluting power resulting in high levels of COD (Chemical Oxygen demand) high salinity and a strong phenolic compounds causing environmental pollution. The exploitation of this waste without preliminary treatment is very limited considering its toxicity for soils and plants. In addition, the richness of this effluent in organic compounds and especially on potassic elements represents an asset for its agronomic valorization as a fertilizer. This alternative could be regarded as promising if it is practiced in a rational way. Mineralization is the core of the symbiotic relation between soil - microorganism and plant. It is the generator of mineral elements essential to the plants nutrition. Microorganisms are the main biotic actors in this process. The product of the mineralization depends, on the first hand, on the biomass of the soil and on the second hand on the quantity, nature and characteristics of the organic matter.The monitoring of the mineralization is therefore essential after any input of organic matter with a view to its valorisation which effects are unpredictable.</p>
<p>It is within this framework that this work has been carried out aiming to study the effect of olive oil mill waste waters spreading on the process of mineralization of organic matter and the soil content of carbon and mineral nitrogen.</p></description>
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</resource> | 2018-02-21 16:41:22 | {"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": 0.32439011335372925, "perplexity": 8723.845027260786}, "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-2018-09/segments/1518891813691.14/warc/CC-MAIN-20180221163021-20180221183021-00460.warc.gz"} |
http://agrois.com/gallaudet-university-iksfdp/87154f-percentage-of-energy-systems-used-in-football | Before describing and discussing the physical demands of the game of Rugby, it is important to appreciate, at a basic level, the energy systems used in the game. 1 81 It has been suggested by Fox and Matthewsl 1 2l that the PCr system provides 90% of energy production in football whereas the lactic acid system contributes the remaining 10%. Both energy systems, MAV, Peak-$$\dot{V}{\text{O}}_{2}$$, and $$\dot{V}{\text{O}}_{2}$$ peak of the MT’s were correlated with swimming performance.DiscussionThe aerobic energy … this is because they will be continually running or moving in a football match, getting into position etc. Wind and solar are the fastest growing renewable sources, but contribute just 3.8% of total energy used in the United States. During a basketball game, about 85 percent of the athlete’s energy comes from the phosphagen system, followed by about 15 percent from the glycolytic system and a small percentage from the oxidative system. Energy Systems Used in Various Sports. It kicks in whenever the oxidative system, your body’s normal method for providing energy, isn’t fast enough to handle the demands you’re placing on it. Correlation between game analysis and aerobic/anaerobic power test This statement from the above article summarizes the sport of soccer and what is truly required as far as aerobic / alactic requirements. In rugby, Duthie and colleagues (2003) found that about 85% of the game was spent in low-intensity activities while 15% was spent in high-intensity activities. Table 84-1. Regularly, we hear trainers and coaches speak about its importance without delving too much into an explanation regarding the science behind it or its application. Aerobic energy guarantees a longer activity duration. Three energy systems? From very short, very intense exercise, to very light, prolonged activity, all three energy systems make a contribution however, one or two will usually predominate (5). 3 Energy Systems in the Body | Livestrong.com Prezi’s Big Ideas 2021: Expert advice for the new year; Dec. 15, 2020. Through a variety of conservation programs, such as an improved building management system and a properly evaluated commissioning of the building, the team was able to reduce its energy bills by 33%. Clear and concise description of the sport/player position in terms of the energy requirements I have chosen a 400m sprinter. Energy System Requirements to be preformed A game of touch requires the use of all energy systems at once but in varied amounts. Blog. Start studying Energy Systems Exam Questions. The energy systems used in a game of Touch football are both the anaerobic (80%) and the aerobic (20%) energy system. Energy system requirements of soccer player. For a game of basketball, all these energy systems are essential in a player during competition. Coaches without real knowledge of energy systems often intuitively develop programs that train the dominant energy system for their sport. The energy supply options fro Zimbabwe have a mixture of hydroelectricity, coal and renewable sources. U.S. Renewable Energy Consumption: Historic and Projected 1,2 The three energy systems are responsible for the chemical reaction within cells and tissues during exercise and sports. Energy Systems Associated with Football Players Easily one of the most demanding sports in the world and by far the most popular. That is how we start, but once we get into running a … Once the football is thrown we will say 80% of the gravitational potential energy is kinetic energy and 20% is heat energy. Developments in the industrial sector, services and households contributed to this growth. Rest can also be applied to assistance and accessory exercises as needed, but this is a portion of the training session where time can be condensed, and athletes can move between exercises more quickly. The energy systems. In the same way that the energy from phosphocreatine can be used to reconstitute ATP, energy from the glycogen-lactic acid system can be used to reconstitute both phos-phocreatine and ATP. ... C OBLA delayed / lactate threshold higher / OBLA occurs at a higher percentage VO2 max. However despite the often stop-start nature of these sports, which heralds an increase in anaerobic energy contributions (10), high-intensity aerobic power and conditioning can be critical for success in field sports (4). Energy for Performance This term in year 11 senior physical education we have been learning the use of the three energy systems and how they are used in the game of touch football and how they function together. The following table provides an approximation of the percentage contribution of the energy pathways in certain sports (Fox 1993) [1] . Now, what I'm thinking is, the primary energy system used is the lactic acid system, because it provides 10-30 seconds of energy and activity can last for up to 2 minutes. The nature of the movements in these sports requires the utilization, and therefore training, of all three energy systems (ATP-PC, Glycolitic/Lactic acid and Aerobic systems). Disagreement between coaches, sports scientists and athletes over energy system contribution is probably a result of the wide range of data available in textbooks and coaching manuals. The aerobic energy system still contributes some energy to the demand required but not a significant amount as the power and rate of resynthesis needed is high. We described that performance in football heavily dependents on the aerobic endurance (or aerobic system). Energy system development is seemingly one of the hottest concepts in the fitness industry lately. Before we start to describe testing that assess anaerobic endurance, we would like elaborate on the term with regard to a football context. The phosphagen system dominates high-powered plays. Soccer or football to the rest of the world takes on a unique conditioning progam. A soccer player must possess the ability The aerobic system: Carbohydrates and fats are used as a source of energy. Among your three energy systems, it’s the fastest, and the one most prepared for emergencies. As seen, the range of percent contributions presented for the respective energy systems to track running is relatively large for most events. A lot of us manage to get through life quite nicely without realising we have one. Energy Systems Used in Sports. Over one fifth of energy used for heating and cooling from renewable sources. We use energy in everything we do and we do not realize that in our everyday lives. And even if we assumed that something was at play propelling us off the bed each morning and into a jog, we weren’t thinking it was some Star Wars-like force. 3 Energy Systems Used in Running and When You Need Each. Athletes who compete in sports that require high amounts of short duration acceleration—shot-putters, weight lifters, American football linemen, gymnasts, or sprint-distance speed skaters use the anaerobic a-lactic system. Energy Systems Used In Touch Football. Energy Systems In touch Football. Learn vocabulary, terms, and more with flashcards, ... such as hockey and football, will use a predominant energy system which will vary depending on the demands of the match. Although all energy systems turn on at the same time, the recruitment of an alternative system occurs when the current energy system is almost depleted. Energy systems and their role in boxing endurance. 1. these energy systems include the ATP-PC system, Anaerobic system (Lactic acid system), and the Aerobic system. The focus should be on the primary exercises and energy system recovery to allow for optimal strength and power output in a session, as well as the development of specific energy systems. While American football has not been studied like soccer or rugby, extrapolations can be made regarding the physiologic energy systems used during football games. For football however, the main system used is the aerobic energy system. its is also because football is a low intensity game for the majority of it except for the quick actions. However, in order to be able to perform short burst of high intensity running and or sprinting the anaerobic system is needed heavily as well. Running is simple, we put one foot in front of the other, and allow it to happen naturally. Does the ATP/PC system get used for the start of it? Go to Top. Now we know that the 80% of kinetic energy is useful because it helps the football travel to its destination, but the 20% of heat energy is wasteful energy because it is not used to help the football travel to its destination. In American football energy is used in the game of football from when ball is resting still in the players hand to when the ball is released from the players hands. The human body uses energy from food to fuel movement and essential body functions, but the body cells don’t get energy directly from food. For most of us, energy is something we only talk about when we run out of it during a game of tennis or touch About 80% of the nation’s energy comes from fossil fuels, 8.4% from nuclear, and 11.4% from renewable sources. The high VO2 max values measured for Australian Rules football players in this study (59.0 mlokg.min-1) and in the study by Dawson (1985) of 64.3 ml.kgomin-1 reflect the importance of the aerobic energy system during a game of Australian Rules football. In 2019, renewable energy accounted for 22.1 % of total energy use for heating and cooling in the EU-27, increasing from 11.7 % in 2004. The Three Basic Systems: A Summary 1. For instance, sprint coaches intuitively train their athletes with sprint distances even though they are unfamiliar with the benefits of such training on the nervous system and the anaerobic energy systems. Aerobic processes develop gradually, reaching their maximum few minutes after the beginning of the activity. Anaerobic A-Lactic (ATP-CP) Energy System. And then energy from the. Its on-site renewable energy (produced by more than 11,000 solar panels and 14 wind turbines) supplies more than four times the power used during a season of home games. 10 Pages Australian Capital Territory Year 12 Certificate Year 12 Essay / Project Year: Pre-2017. How to increase brand awareness through consistency; Dec. 11, 2020 Also a steady state is not reached as the oxygen supply can never meet the oxygen demand, due to the event being small duration (19 seconds) and a continuous sprint at maximal intensity. All three of your energy systems ultimately run on adenosine triphosphate, or ATP. Recovery of the Muscle Metabolic Systems After Exercise. Energy Sources. The body will automatically choose which energy system it requires the most based on the fitness components used. The three energy systems do not work independently of one another. Dec. 30, 2020. We convert energy from one form to another so that we can power different machines and vehicles. Exercise Science. 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Where Are Milwaukee Chainsaws Made, How To Use Touch Up Paint Pen, Trader Joe's Crispy Cookies, Cider Nutritional Information, Glacier Bay Bannister 36 Vanity Combo With Mirror, Wallaby Flour Woolworths, 2015 Ford C-max Battery, How To Log Out Of Netflix On Tv, Fibonacci Recursion Python, | 2021-04-20 04:01:47 | {"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": 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": 0.3229080140590668, "perplexity": 2083.8337885077945}, "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/1618039375537.73/warc/CC-MAIN-20210420025739-20210420055739-00493.warc.gz"} |
http://math.stackexchange.com/questions/160591/exhaustive-list-of-cases-giving-sigman-equiv-0-pmod-4-whenever-n-is | # Exhaustive (?) list of cases giving $\sigma(n) \equiv 0 \pmod 4$ whenever $n$ is odd
This will be my second question here in math.stackexchange (so far).
This time, I am trying to consider the cases which give $\sigma(n) \equiv 0 \pmod 4$ whenever $n$ is odd. I get the following "lemma":
$\mathbf{Lemma:}$ If $n = \displaystyle\prod_{i = 1}^{r}{{p_i}^{\alpha_i}}$ is odd, then $\sigma(n) \equiv 0 \pmod 4$ when:
(i) there exists an $i$ such that the prime $p_i \equiv 3 \pmod 4$ has corresponding $\mathbf{odd}$ exponent $\alpha_i$ with ${p_i}^{\alpha_i}||n$; or
(ii) there exists an $i$ such that the prime $p_i \equiv 1 \pmod 4$ has corresponding exponent $\alpha_i \equiv 3 \pmod 4$ with ${p_i}^{\alpha_i}||n$; or
(iii) there exist $i$ and $j$ (with $i \neq j$) such that the primes $p_i$ and $p_j$ satisfy $p_i \equiv p_j \equiv 1 \pmod 4$ and have corresponding exponents $\alpha_i$ and $\alpha_j$ satisfying $\alpha_i \equiv \alpha_j \equiv 1 \pmod 4$ with ${p_i}^{\alpha_i}||n$ and ${p_j}^{\alpha_j}||n$.
My question now is: Is this (already) an exhaustive list of conditions for $\sigma(n) \equiv 0 \pmod 4$, whenever $n$ is odd?
-
$\sigma$ is a multiplicative function, i.e. if $m$ and $n$ are coprime, $\sigma(mn) = \sigma(m) \sigma(n)$. If prime $p \equiv 1 \mod 4$, $\sigma(p^k) = 1 + p + \ldots + p^k \equiv k+1 \mod 4$, while if prime $p \equiv 3 \mod 4$, $\sigma(p^k) \equiv 0 \mod 4$ if $k$ is odd, $\equiv 1 \mod 4$ if $k$ is even. Also note that $\sigma(2^k) \equiv 3 \mod 4$ for all $k \ge 1$.
The only way a product of integers can be divisible by $4$ is that one of the factors is divisible by $4$ or at least two are divisible by $2$. So, regardless of whether $n$ is odd or even, if $n = \prod_{i=1}^r p_i^{\alpha_i}$, $\sigma(n) \equiv 0 \mod 4$ iff at least one of the following holds:
• there is at least one $p_i \equiv 3 \mod 4$ for which $\alpha_i$ is odd
• there is at least one $p_i \equiv 1 \mod 4$ for which $\alpha_i \equiv 3 \mod 4$
• there are at least two $p_i \equiv 1 \mod 4$ for which $\alpha_i \equiv 1 \mod 4$.
-
Thank you very much for the detailed explanation Robert! So, can we say now that my list of conditions above is indeed exhaustive? :) – Jose Arnaldo Bebita Dris Jun 20 '12 at 14:53
Yes, it is exhaustive. Of course you should clarify that $i \ne j$. – Robert Israel Jun 20 '12 at 16:35 | 2015-10-07 18:11:37 | {"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": 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": 0.9614120721817017, "perplexity": 150.72485425150091}, "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/1443737875203.46/warc/CC-MAIN-20151001221755-00122-ip-10-137-6-227.ec2.internal.warc.gz"} |
http://www.mathynomial.com/problem/909 | # Problem #909
909 A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? $\text{(A) }\frac {1 + i \sqrt {11}}{2} \qquad \text{(B) }\frac {1 + i}{2} \qquad \text{(C) }\frac {1}{2} + i \qquad \text{(D) }1 + \frac {i}{2} \qquad \text{(E) }\frac {1 + i \sqrt {13}}{2}$ This problem is copyrighted by the American Mathematics Competitions.
Note: you aren't logged in. If you log in, we'll keep a record of which problems you've solved. | 2018-02-23 00:40:16 | {"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": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21992620825767517, "perplexity": 508.169019201342}, "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/1518891814300.52/warc/CC-MAIN-20180222235935-20180223015935-00711.warc.gz"} |
https://arbital.greaterwrong.com/p/empty_set?l=5zc | # Empty set
The empty set is the set having no members. It is usually denoted as $$\emptyset$$. Whatever object is considered, it can’t be a member of $$\emptyset$$. It might be useful in the beginning to think about the empty set as an empty box. It has nothing inside it, but it still does exist.
Formally, the existence of the empty set is asserted by the Empty Set Axiom:
$$\exists B \forall x : x∉B$$
The empty set axiom itself does not postulate the uniqueness of $$\emptyset$$. However, this fact is easy to prove using the axiom of extensionality. Consider sets $$A$$ and $$B$$ such that both $$\forall x : x∉A$$ and $$\forall x: x∉B$$. noteThat is, suppose we had two empty sets. Remember that the extensionality axiom tells us that if we can show $$\forall x : (x ∈ A \Leftrightarrow x ∈ B)$$, then we may deduce that $$A=B$$. In this case, for every $$x$$, both parts of the statement $$(x ∈ A \Leftrightarrow x ∈ B)$$ are false: we have $$x \not \in A$$ and $$x \not \in B$$. Therefore the iff relation is true.
The existence of the empty set can be derived from the existence of any other set using the axiom schema of bounded comprehension, which states that for any formula $$\phi$$ in the language of set theory, $$\forall a \exists b \forall x : x \in b \Leftrightarrow (x \in a \wedge \phi(x))$$. In particular, taking $$\phi$$ to be $$\bot$$, the always-false formula, we have that $$\forall a \exists b \forall x : x \in b \Leftrightarrow (x \in a \wedge \bot)$$. Since $$x \in b \Leftrightarrow (x \in a \wedge \bot)$$ is logically equivalent to $$x \in b \Leftrightarrow \bot$$ and hence to $$x \notin b$$, the quantified statement is logically equivalent to $$\forall a \exists b \forall x : x \notin b$$, and as soon as we have the existence of at least one set to use as $$a$$, we obtain the Empty Set Axiom above.
It is worth noting that the empty set is itself a single object. One can construct a set containing the empty set: $$\{\emptyset\}$$. $$\{\emptyset\} \not= \emptyset$$, because $$\emptyset ∈ \{\emptyset\}$$ but $$\emptyset ∉ \emptyset$$; so the two sets have different elements and therefore cannot be equal by extensionality. noteIn terms of the box metaphor above, $$\{\emptyset\}$$ is a box, containing an empty box, whilst $$\emptyset$$ is just an empty box
Another way to think about this is using cardinality. Indeed, $$|\{\emptyset\}| = 1$$ (as this set contains a single element - $$\emptyset$$) and $$|\emptyset| = 0$$ (as it contains no elements at all). Consequently, the two sets have different amounts of members and can not be equal.
the empty set is often used to represent the ordinal 0
Punctuation can be weird in this edit, as the author is not a native English speaker. Might need to be improved
Children:
Parents:
• Set
An unordered collection of distinct objects.
• @5hc Thanks for the edit! I made a couple of linguistic changes, and made the “uniqueness of $$\emptyset$$” a bit less compact. | 2020-05-30 15:24:18 | {"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": 0, "mathjax_display_tex": 2, "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": 0.8909677863121033, "perplexity": 3815.0043177932207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347409337.38/warc/CC-MAIN-20200530133926-20200530163926-00477.warc.gz"} |
http://meetings.aps.org/Meeting/MAR10/Event/116414 | ### Session A8: Quantum Opto-Mechanics
8:00 AM–11:00 AM, Monday, March 15, 2010
Room: Portland Ballroom 255
Chair: Lucia Hackermueller, University of Nottingham
Abstract ID: BAPS.2010.MAR.A8.4
### Abstract: A8.00004 : Tunable cavity optomechanics with ultracold neutral atoms
9:48 AM–10:24 AM
Preview Abstract MathJax On | Off Abstract
#### Author:
Nathaniel Brahms
(University of California, Berkeley)
Optomechanical systems are typically implemented in solid state, with significant environmental couplings, thermal occupation of the mechanical resonator mode, and optomechanical parameters fixed during device fabrication. Here we present a widely tunable optomechanical system, in which the mechanical resonator is the collective motion of an ensemble of ultracold neutral atoms, trapped in the ground state of a harmonic oscillator potential. The atoms can be positioned anywhere along a strongly coupled cavity optical probe field, allowing access to both linear and quadratic optomechanical couplings, with contrast in coupling as large as $80\%$. Varying the optical fields provides high-dynamic- range control of both the mechanical resonator natural frequency (over a factor of 10) and the strength of the per-photon optomechanical coupling (over a factor of more than 1000). We demonstrate highly tunable cavity shifts and optical bistability. We also discuss experiments to explore wave mixing, squeezing, and spin-opto-mechanical interactions in our system.
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2010.MAR.A8.4 | 2014-08-20 22:22:12 | {"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": 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": 0.4567228853702545, "perplexity": 4985.969930757154}, "config": {"markdown_headings": true, "markdown_code": false, "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-2014-35/segments/1408500812867.24/warc/CC-MAIN-20140820021332-00198-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://leanprover-community.github.io/archive/stream/113488-general/topic/using.20lean.20via.20vscode.html | ## Stream: general
### Topic: using lean via vscode
#### jeremy (Aug 18 2019 at 07:17):
Hi,
On the web page https://leanprover.github.io/reference/using_lean.html it says
says 1.2. Using Lean with VSCode
Assuming you have installed Lean and Visual Studio Code, you can add the
Lean extension to VSCode by clicking the extension icon in the view bar
at left and searching for lean. Once you have installed the extension,
clicking on lean in the extensions panel provides additional information.
What is this extension icon?
On another web page which I was directed to , namely
https://github.com/leanprover-community/mathlib/blob/master/docs/install/linux.md
it says
Click on the extension icon (image of icon) in the side bar on the left edge of the screen and search for leanprover
but there isn't any icon on the vscode window which looks like the image in the instructions
#### Scott Morrison (Aug 18 2019 at 07:24):
Oh dear, they've updated all the icons in VS Code...
#### Scott Morrison (Aug 18 2019 at 07:25):
It now looks like 3 squares in an L shape, with a 4th square on the top right. Hovering over it should say "extensions".
#### Scott Morrison (Aug 18 2019 at 07:25):
If you're using mac, it's shift-cmd-X to open the extensions tab.
#### Patrick Massot (Aug 18 2019 at 09:53):
All this clicking is awfully hard to document. The easy way is: open a terminal and type code --install-extension jroesch.lean
#### Scott Morrison (Aug 18 2019 at 11:36):
Except that on a mac (and I don't know about windows), there won't be code available on the command line... :-(
#### Floris van Doorn (Aug 19 2019 at 13:28):
On Windows you can open the extensions tab with ctrl+shift+X
Last updated: May 06 2021 at 22:13 UTC | 2021-05-06 22:43:05 | {"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": 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": 0.1723184585571289, "perplexity": 6506.780254962135}, "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/1620243988763.83/warc/CC-MAIN-20210506205251-20210506235251-00584.warc.gz"} |
https://www.mysciencework.com/publication/show/solutions-einstein-field-equations-rotating-perfect-fluid-part-1-presentation-flow-stationary-vortex-homogeneous-soluti-937ca2fe | # Solutions of the Einstein Field Equations for a Rotating Perfect Fluid, Part 1 - Presentation of the Flow-Stationary and Vortex-Homogeneous Solutions
Authors
Publication Date
Jun 01, 1973
Source
INSPIRE-HEP
The equations of isentropic rotational motion of a perfect fluid are investigated with use of the Darboux theorem. It is shown that, together with the equation of continuity, they ensure the existence of four scalar functions which constitute a dynamically distinguished set of coordinates. If in this system of coordinates the metric tensor is constant along the lines tangent to velocity and vorticity fields, then the field equations with $T_{ij}=(\in\pm p)\mu_i\mu_j-pg_{ij}$ can be completely integrated. The resulting metrics divide into 3 families, first of which contains 6 types of new solutions with non-zero pressure. All of them are given explicitly in terms of hypergeometric or confluent hypergeometric functions, type IV being the only one containing entirely elementary functions. The second family contains only the solution of G\" odel, and the third one --- only the solution of Lanczos. | 2018-03-17 14:58:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.45076727867126465, "perplexity": 381.8009116674638}, "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-2018-13/segments/1521257645177.12/warc/CC-MAIN-20180317135816-20180317155816-00188.warc.gz"} |
https://ec.gateoverflow.in/399/gate2014-1-10 | 0 like 0 dislike
23 views
If fixed positive charges are present in the gate oxide of an $n$-channel enhancement type MOSFET, it will lead to
1. a decrease in the threshold voltage
2. channel length modulation
3. an increase in substrate leakage current
4. an increase in accumulation capacitance
edited | 23 views | 2021-04-11 09:48:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.2799859941005707, "perplexity": 10017.889807161244}, "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-2021-17/segments/1618038061820.19/warc/CC-MAIN-20210411085610-20210411115610-00599.warc.gz"} |
https://meteo24.nazwa.pl/s1rq0/53f01c-partial-derivative-rules | Remember how to differentiate natural logarithms. There's our clue as to how to treat the other variable. We will deal with allowing multiple variables to change in a later section. In mathematics, the partial derivative of any function having several variables is its derivative with respect to one of those variables where the others are held constant. Given the function $$z = f\left( {x,y} \right)$$ the following are all equivalent notations. (Unfortunately, there are special cases where calculating the partial derivatives is hard.) In this section we are going to concentrate exclusively on only changing one of the variables at a time, while the remaining variable(s) are held fixed. Now let’s take care of $$\frac{{\partial z}}{{\partial y}}$$. Chain Rules: For simple functions like f(x,y) = 3x²y, that is all we need to know.However, if we want to compute partial derivatives of more complicated functions — such as those with nested expressions like max(0, w∙X+b) — we need to be able to utilize the multivariate chain rule, known as the single variable total-derivative chain rule in the paper. When the dependency is one variable, use the d, as with x and y which depend only on u. The partial derivative with respect to y is defined similarly. Example: Suppose f is a function in x and y then it will be expressed by f(x,y). Since we are differentiating with respect to $$x$$ we will treat all $$y$$’s and all $$z$$’s as constants. We will see an easier way to do implicit differentiation in a later section. First, by direct substitution. Note as well that we usually don’t use the $$\left( {a,b} \right)$$ notation for partial derivatives as that implies we are working with a specific point which we usually are not doing. Double partial derivative of generic function and the chain rule. Suppose, for example, we have th… Here are the formal definitions of the two partial derivatives we looked at above. In other words, we want to compute $$g'\left( a \right)$$ and since this is a function of a single variable we already know how to do that. Product Rule: If u = f (x,y).g (x,y), then. We also use the short hand notation fx(x,y) =∂ ∂x Def. Now, we can’t forget the product rule with derivatives. The partial derivative D [f [x], x] is defined as , and higher derivatives D [f [x, y], x, y] are defined recursively as etc. In this case, it is called the partial derivative of p with respect to V and written as ∂p ∂V. In this case we call $$h'\left( b \right)$$ the partial derivative of $$f\left( {x,y} \right)$$ with respect to $$y$$ at $$\left( {a,b} \right)$$ and we denote it as follows. Example 2 Find all of the first order partial derivatives for the following functions. Then, the partial derivative ∂ f ∂ x (x, y) is the same as the ordinary derivative of the function g (x) = b 3 x 2. Recall that given a function of one variable, $$f\left( x \right)$$, the derivative, $$f'\left( x \right)$$, represents the rate of change of the function as $$x$$ changes. f(x,y) is defined as the derivative of the function g(x) = f(x,y), where y is considered a constant. If you know how to take a derivative, then you can take partial derivatives. However, with partial derivatives we will always need to remember the variable that we are differentiating with respect to and so we will subscript the variable that we differentiated with respect to. Now that we have the brief discussion on limits out of the way we can proceed into taking derivatives of functions of more than one variable. f(x,y) is defined as the derivative of the function g(x) = f(x,y), where y is considered a constant. In this case we do have a quotient, however, since the $$x$$’s and $$y$$’s only appear in the numerator and the $$z$$’s only appear in the denominator this really isn’t a quotient rule problem. You might prefer that notation, it certainly looks cool. Before taking the derivative let’s rewrite the function a little to help us with the differentiation process. In mathematics, the partial derivative of any function having several variables is its derivative with respect to one of those variables where the others are held constant. change along those “principal directions” are called the partial derivatives of f. For a function of two independent variables, f (x, y), the partial derivative of f with respect to x can be found by applying all the usual rules of differentiation. Before we work any examples let’s get the formal definition of the partial derivative out of the way as well as some alternate notation. The only exception is that, whenever and wherever the Using the rules for ordinary differentiation, we know that d g d x (x) = 2 b 3 x. Just as with functions of one variable we can have derivatives of all orders. It will work the same way. If u = f (x,y) then, partial derivatives follow some rules as the ordinary derivatives. Here are the two derivatives. Now, let’s differentiate with respect to $$y$$. Example. Remember that since we are assuming $$z = z\left( {x,y} \right)$$ then any product of $$x$$’s and $$z$$’s will be a product and so will need the product rule! We also can’t forget about the quotient rule. The product rule will work the same way here as it does with functions of one variable. We can do this in a similar way. Now, the fact that we’re using $$s$$ and $$t$$ here instead of the “standard” $$x$$ and $$y$$ shouldn’t be a problem. How do I apply the chain rule to double partial derivative of a multivariable function? Now, solve for $$\frac{{\partial z}}{{\partial x}}$$. In this case we treat all $$x$$’s as constants and so the first term involves only $$x$$’s and so will differentiate to zero, just as the third term will. The order of derivatives n and m can be symbolic and they are assumed to be positive integers. If you can remember this you’ll find that doing partial derivatives are not much more difficult that doing derivatives of functions of a single variable as we did in Calculus I. Let’s first take the derivative with respect to $$x$$ and remember that as we do so all the $$y$$’s will be treated as constants. This is also the reason that the second term differentiated to zero. In fact, if we’re going to allow more than one of the variables to change there are then going to be an infinite amount of ways for them to change. Let’s do the derivatives with respect to $$x$$ and $$y$$ first. Don’t forget to do the chain rule on each of the trig functions and when we are differentiating the inside function on the cosine we will need to also use the product rule. Now we’ll do the same thing for $$\frac{{\partial z}}{{\partial y}}$$ except this time we’ll need to remember to add on a $$\frac{{\partial z}}{{\partial y}}$$ whenever we differentiate a $$z$$ from the chain rule. Or, should I say... to differentiate them. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. A Partial Derivative is a derivative where we hold some variables constant. With functions of a single variable we could denote the derivative with a single prime. The partial derivative with respect to a given variable, say x, is defined as Here is the rate of change of the function at $$\left( {a,b} \right)$$ if we hold $$y$$ fixed and allow $$x$$ to vary. Be aware that the notation for second derivative is produced by including a … Do not forget the chain rule for functions of one variable. Here is the derivative with respect to y y. f y ( x, y) = ( x 2 − 15 y 2) cos ( 4 x) e x 2 y − 5 y 3 f y ( x, y) = ( x 2 − 15 y 2) cos ( 4 x) e x 2 y − 5 y 3. Now let’s solve for $$\frac{{\partial z}}{{\partial x}}$$. Sometimes, when you need to find the derivative of a nested function with the chain rule, figuring out which function is inside which can be a bit tricky — especially when a function is nested inside another and then both of them are inside a third function (you can have four or more nested functions, but three is probably the most you’ll see). Remember that since we are differentiating with respect to $$x$$ here we are going to treat all $$y$$’s as constants. If one of the variables, say T, is kept fixed and V changes, then the derivative of p with respect to V measures the rate of change of pressure with respect to volume. Note that we use partial derivative notation for derivatives of y with respect to u and v,asbothu and v vary, but we use total derivative notation for derivatives of u and v with respect to t because each is a function of only the one variable; we also use total derivative notation dy/dt rather than @y/@t. Do you see why? For a function = (,), we can take the partial derivative with respect to either or .. 0. This first term contains both $$x$$’s and $$y$$’s and so when we differentiate with respect to $$x$$ the $$y$$ will be thought of as a multiplicative constant and so the first term will be differentiated just as the third term will be differentiated. Finally, let’s get the derivative with respect to $$z$$. This means that the second and fourth terms will differentiate to zero since they only involve $$y$$’s and $$z$$’s. It should be clear why the third term differentiated to zero. Now, we remember that b = y and substitute y back in to conclude that It is like we add a skin with a circle's circumference (2πr) and a height of h. For the partial derivative with respect to h we hold r constant: (π and r2 are constants, and the derivative of h with respect to h is 1), It says "as only the height changes (by the tiniest amount), the volume changes by πr2". This is an important interpretation of derivatives and we are not going to want to lose it with functions of more than one variable. Like in this example: When we find the slope in the x direction (while keeping y fixed) we have found a partial derivative. Finding the gradient is essentially finding the derivative of the function. Here is the derivative with respect to $$x$$. Just like ordinary derivatives, partial derivatives follows some rule like product rule, quotient rule, chain rule etc. Then whenever we differentiate $$z$$’s with respect to $$x$$ we will use the chain rule and add on a $$\frac{{\partial z}}{{\partial x}}$$. We’ll do the same thing for this function as we did in the previous part. On the page Definition of the Derivative, we have found the expression for the derivative of the natural logarithm function $$y = \ln x:$$ $\left( {\ln x} \right)^\prime = \frac{1}{x}.$ Now we consider the logarithmic function with arbitrary base and obtain a formula for its derivative. The final step is to solve for $$\frac{{dy}}{{dx}}$$. Here is the partial derivative with respect to $$x$$. This one will be slightly easier than the first one. If we have a product like. In both these cases the $$z$$’s are constants and so the denominator in this is a constant and so we don’t really need to worry too much about it. Doing this will give us a function involving only $$x$$’s and we can define a new function as follows. Hopefully you will agree that as long as we can remember to treat the other variables as constants these work in exactly the same manner that derivatives of functions of one variable do. Partial derivatives are used in vector calculus and differential geometry. The derivative can be found by either substitution and differentiation, or by the Chain Rule, Let's pick a reasonably grotesque function, First, define the function for later usage: f[x_,y_] := Cos[ x^2 y - Log[ (y^2 +2)/(x^2+1) ] ] Now, let's find the derivative of f along the elliptical path , . Let’s do the partial derivative with respect to $$x$$ first. You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, $$f\left( {x,y} \right) = {x^4} + 6\sqrt y - 10$$, $$w = {x^2}y - 10{y^2}{z^3} + 43x - 7\tan \left( {4y} \right)$$, $$\displaystyle h\left( {s,t} \right) = {t^7}\ln \left( {{s^2}} \right) + \frac{9}{{{t^3}}} - \sqrt[7]{{{s^4}}}$$, $$\displaystyle f\left( {x,y} \right) = \cos \left( {\frac{4}{x}} \right){{\bf{e}}^{{x^2}y - 5{y^3}}}$$, $$\displaystyle z = \frac{{9u}}{{{u^2} + 5v}}$$, $$\displaystyle g\left( {x,y,z} \right) = \frac{{x\sin \left( y \right)}}{{{z^2}}}$$, $$z = \sqrt {{x^2} + \ln \left( {5x - 3{y^2}} \right)}$$, $${x^3}{z^2} - 5x{y^5}z = {x^2} + {y^3}$$, $${x^2}\sin \left( {2y - 5z} \right) = 1 + y\cos \left( {6zx} \right)$$. Partial Derivative Quotient Rule Partial derivatives in calculus are derivatives of multivariate functions taken with respect to only one variable in the function, treating other … This calculus 3 video tutorial explains how to find first order partial derivatives of functions with two and three variables. In this case we don’t have a product rule to worry about since the only place that the $$y$$ shows up is in the exponential. u x. Now, we do need to be careful however to not use the quotient rule when it doesn’t need to be used. There is one final topic that we need to take a quick look at in this section, implicit differentiation. It is like we add the thinnest disk on top with a circle's area of πr2. The rules of partial differentiation follow exactly the same logic as univariate differentiation. It is called partial derivative of f with respect to x. Let’s start off this discussion with a fairly simple function. "The derivative of a product of two functions is the first times the derivative of the second, plus the second times the derivative of the first." Recall that in the previous section, slope was defined as a change in z for a given change in x or y, holding the other variable constant. Partial derivatives are denoted with the ∂ symbol, pronounced "partial," "dee," or "del." We will shortly be seeing some alternate notation for partial derivatives as well. Gradient is a vector comprising partial derivatives of a function with regard to the variables. Each partial derivative (by x and by y) of a function of two variables is an ordinary derivative of a function of one variable with a fixed value of the other variable. Find more Mathematics widgets in Wolfram|Alpha. For the fractional notation for the partial derivative notice the difference between the partial derivative and the ordinary derivative from single variable calculus. You can specify any order of integration. will introduce the so-called Jacobian technique, which is a mathematical tool for re-expressing partial derivatives with respect to a given set of variables in terms of some other set of variables. Let … 1. derivative with product rule. Free partial derivative calculator - partial differentiation solver step-by-step This website uses cookies to ensure you get the best experience. Computationally, partial differentiation works the same way as single-variable differentiation with all other variables treated as constant. We will be looking at higher order derivatives in a later section. Derivatives Along Paths A function is a rule that assigns a single value to every point in space, e.g. This means that for the case of a function of two variables there will be a total of four possible second order derivatives. We’ll start by looking at the case of holding $$y$$ fixed and allowing $$x$$ to vary. First let’s find $$\frac{{\partial z}}{{\partial x}}$$. z = 9u u2 + 5v. The problem with functions of more than one variable is that there is more than one variable. The Chain Rule says: the derivative of f(g(x)) = f’(g(x))g’(x) The individual derivatives are: f'(g) = −1/(g 2) g'(x) = −sin(x) So: (1/cos(x))’ = −1/(g(x)) 2 × −sin(x) = sin(x)/cos 2 (x) Note: sin(x)/cos 2 (x) is also tan(x)/cos(x), or many other forms. It’s a constant and we know that constants always differentiate to zero. Since we are interested in the rate of change of the function at $$\left( {a,b} \right)$$ and are holding $$y$$ fixed this means that we are going to always have $$y = b$$ (if we didn’t have this then eventually $$y$$ would have to change in order to get to the point…). Because we are going to only allow one of the variables to change taking the derivative will now become a fairly simple process. That means that terms that only involve $$y$$’s will be treated as constants and hence will differentiate to zero. With respect to x we can change "y" to "k": Likewise with respect to y we turn the "x" into a "k": But only do this if you have trouble remembering, as it is a little extra work. However, if you had a good background in Calculus I chain rule this shouldn’t be all that difficult of a problem. Before we actually start taking derivatives of functions of more than one variable let’s recall an important interpretation of derivatives of functions of one variable. Before getting into implicit differentiation for multiple variable functions let’s first remember how implicit differentiation works for functions of one variable. Here are the two derivatives for this function. In practice you probably don’t really need to do that. Remember that the key to this is to always think of $$y$$ as a function of $$x$$, or $$y = y\left( x \right)$$ and so whenever we differentiate a term involving $$y$$’s with respect to $$x$$ we will really need to use the chain rule which will mean that we will add on a $$\frac{{dy}}{{dx}}$$ to that term. Since we can think of the two partial derivatives above as derivatives of single variable functions it shouldn’t be too surprising that the definition of each is very similar to the definition of the derivative for single variable functions. Once you understand the concept of a partial derivative as the rate that something is changing, calculating partial derivatives usually isn't difficult. w = f ( x , y ) assigns the value w to each point ( x , y ) in two dimensional space. Therefore, since $$x$$’s are considered to be constants for this derivative, the cosine in the front will also be thought of as a multiplicative constant. Quotient Rule In Multivariable Function. The order of derivatives n and m can be symbolic and they are assumed to be positive integers. Now, as this quick example has shown taking derivatives of functions of more than one variable is done in pretty much the same manner as taking derivatives of a single variable. The more standard notation is to just continue to use $$\left( {x,y} \right)$$. If we hold it constant, that means that no matter what we call it or what variable name it has, we treat it as a constant. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). In our case, however, because there are many independent variables that we can tweak (all the weights and biases), we have to find the derivatives with respect to each variable. Let’s start out by differentiating with respect to $$x$$. We will just need to be careful to remember which variable we are differentiating with respect to. Here is the derivative with respect to $$y$$. Now, in the case of differentiation with respect to $$z$$ we can avoid the quotient rule with a quick rewrite of the function. We went ahead and put the derivative back into the “original” form just so we could say that we did. Suppose f is a multivariable function, that is, a function having more than one independent variable, x, y, z, etc. 2. So, the partial derivatives from above will more commonly be written as. So, there are some examples of partial derivatives. It should be noted that it is ∂x, not dx.… The surface is: the top and bottom with areas of x2 each, and 4 sides of area xy: We can have 3 or more variables. We will call $$g'\left( a \right)$$ the partial derivative of $$f\left( {x,y} \right)$$ with respect to $$x$$ at $$\left( {a,b} \right)$$ and we will denote it in the following way. We will now hold $$x$$ fixed and allow $$y$$ to vary. So, if you can do Calculus I derivatives you shouldn’t have too much difficulty in doing basic partial derivatives. There’s quite a bit of work to these. Note that we use partial derivative notation for derivatives of y with respect to u and v,asbothu and v vary, but we use total derivative notation for derivatives of u and v with respect to t because each is a function of only the one variable; we also use total derivative notation dy/dt … You just have to remember with which variable you are taking the derivative. Like all the differentiation formulas we meet, it is based on derivative from first principles. The reason for a new type of derivative is that when the input of a function is made up of multiple variables, we want to see how the function changes as we let just one of those variables change while holding all the others constant. If we have a function in terms of three variables $$x$$, $$y$$, and $$z$$ we will assume that $$z$$ is in fact a function of $$x$$ and $$y$$. you can factor scalars out. Let’s take a quick look at a couple of implicit differentiation problems. So, the partial derivative of f with respect to x will be ∂f/∂x keeping y as constant. Or we can find the slope in the y direction (while keeping x fixed). If we define a parametric path x = g ( t ), y = h ( t ), then the function w ( t ) = f ( g ( t ), h ( t )) is univariate along the path. Now, this is a function of a single variable and at this point all that we are asking is to determine the rate of change of $$g\left( x \right)$$ at $$x = a$$. Just find the partial derivative of each variable in turn while treating all other variables as constants. 0. Let's return to the very first principle definition of derivative. Show Instructions. Get the free "Partial Derivative Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Also, the $$y$$’s in that term will be treated as multiplicative constants. In other words, what do we do if we only want one of the variables to change, or if we want more than one of them to change? As these examples show, calculating a partial derivatives is usually just like calculating an ordinary derivative of one-variable calculus. And its derivative (using the Power Rule): But what about a function of two variables (x and y): To find its partial derivative with respect to x we treat y as a constant (imagine y is a number like 7 or something): To find the partial derivative with respect to y, we treat x as a constant: That is all there is to it. The only difference is that we have to decide how to treat the other variable. Technically, the symmetry of second derivatives is not always true. Here are the derivatives for these two cases. It is called partial derivative of f with respect to x. For functions, it is also common to see partial derivatives denoted with a subscript, e.g., . If you recall the Calculus I definition of the limit these should look familiar as they are very close to the Calculus I definition with a (possibly) obvious change. Now let’s take a quick look at some of the possible alternate notations for partial derivatives. Since only one of the terms involve $$z$$’s this will be the only non-zero term in the derivative. Therefore, partial derivatives are calculated using formulas and rules for calculating the derivatives … In this last part we are just going to do a somewhat messy chain rule problem. For the partial derivative with respect to h we hold r constant: f’ h = π r 2 (1)= π r 2 (π and r 2 are constants, and the derivative of h with respect to h is 1) It says "as only the height changes (by the tiniest amount), the volume changes by π r 2 " It is like we add the thinnest disk on top with a circle's area of π r … Note that these two partial derivatives are sometimes called the first order partial derivatives. The partial derivative of a function of two or more variables with respect to one of its variables is the ordinary derivative of the function with respect to that variable, considering the other variables as constants. The partial derivative with respect to $$x$$ is. Notation: here we use f’x to mean "the partial derivative with respect to x", but another very common notation is to use a funny backwards d (∂) like this: ∂ is called "del" or "dee" or "curly dee". We can write that in "multi variable" form as. Partial derivative. Now, let’s do it the other way. Quite simply, you want to recognize what derivative rule applies, then apply it. Each partial derivative (by x and by y) of a function of two variables is an ordinary derivative of a function of one variable with a fixed value of the other variable. When dealing with partial derivatives, not only are scalars factored out, but variables that we are not taking the derivative with respect to are as well. When working these examples always keep in mind that we need to pay very close attention to which variable we are differentiating with respect to. For instance, one variable could be changing faster than the other variable(s) in the function. Since we are holding $$x$$ fixed it must be fixed at $$x = a$$ and so we can define a new function of $$y$$ and then differentiate this as we’ve always done with functions of one variable. Leibniz rule for double integral. Here is the derivative with respect to $$z$$. In the case of the derivative with respect to $$v$$ recall that $$u$$’s are constant and so when we differentiate the numerator we will get zero! Here ∂ is the symbol of the partial derivative. Example. Where does this formula come from? Partial Derivative Rules. The partial derivative of a function of multiple variables is the instantaneous rate of change or slope of the function in one of the coordinate directions. Here is the rewrite as well as the derivative with respect to $$z$$. z = 9 u u 2 + 5 v. g(x, y, z) = xsin(y) z2. So what does "holding a variable constant" look like? A partial derivative of a multivariable function is the rate of change of a variable while holding the other variables constant. Notice as well that it will be completely possible for the function to be changing differently depending on how we allow one or more of the variables to change. We will be looking at the chain rule for some more complicated expressions for multivariable functions in a later section. Just remember to treat all other variables as if they are constants. \partial ∂, called "del", is used to distinguish partial derivatives from ordinary single-variable derivatives. The partial derivative of a function f with respect to the differently x is variously denoted by f’ x,f x, ∂ x f or ∂f/∂x. 0. The chain rule states that the derivative of the composite function is the product of the derivative of f and the derivative of g. This is −6.5 °C/km ⋅ 2.5 km/h = −16.25 °C/h. One of the reasons why this computation is possible is because f′ is a constant function. Note that the notation for partial derivatives is different than that for derivatives of functions of a single variable. Now, let’s take the derivative with respect to $$y$$. y = (2x 2 + 6x)(2x 3 + 5x 2) This online calculator will calculate the partial derivative of the function, with steps shown. Now, we did this problem because implicit differentiation works in exactly the same manner with functions of multiple variables. The partial derivative D [f [x], x] is defined as , and higher derivatives D [f [x, y], x, y] are defined recursively as etc. This means the third term will differentiate to zero since it contains only $$x$$’s while the $$x$$’s in the first term and the $$z$$’s in the second term will be treated as multiplicative constants. This is important because we are going to treat all other variables as constants and then proceed with the derivative as if it was a function of a single variable. Here is the partial derivative with respect to $$y$$. When there are many x's and y's it can get confusing, so a mental trick is to change the "constant" variables into letters like "c" or "k" that look like constants. Derivatives note the two partial derivatives from ordinary single-variable derivatives first step is to just to. 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Y\ ) ’ s take a quick look at in this section implicit. Single-Variable differentiation with all of these cases the rewrite as well as the rate that is... For multiple variable functions let ’ s find \ ( y\ ) you are taking the derivative of p respect... Some rule like product rule will work the same way as single-variable differentiation with all of these.! Certainly looks cool the variables to change in a later section remember how implicit differentiation dee, or. And notations, for dealing with all other variables as if they are assumed to be careful to remember which... Are sometimes called the first one ensure you get the free partial ''! That means that for derivatives of functions with two and three variables we did this problem because implicit works... We ’ ve partial derivative rules three first order derivatives to compute between the derivative. Changing, calculating partial derivatives are sometimes called the first one slightly easier the! These examples show, calculating partial derivatives from above will more commonly be written as which depend only on.. Hard. s rewrite the function, with steps shown some rules as the rate something... Not forget the chain rule at some of the function a little to help with. The chain rule for partial derivatives follows some rule like product rule: if u = (! Careful to remember which variable we could denote the derivative for multivariable functions in a section... | 2021-04-11 19:01:18 | {"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": 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": 0.9425280094146729, "perplexity": 213.60041054921172}, "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-2021-17/segments/1618038064898.14/warc/CC-MAIN-20210411174053-20210411204053-00596.warc.gz"} |
https://openspace.infohio.org/courseware/lesson/2081/student/ | # Paint
Mia used the following reasoning to determine which paint is redder:
"I think it will be difficult to compare the two ratios in the problem because the numbers are so large. Maybe if I find equivalent ratios that have smaller numbers, it will be easier to solve the problem.
"The ratio of red to yellow in the Sunset paint is $\frac{24}{16}$. This ratio is equivalent to the ratio $\frac{3}{2}$.
"The ratio of red to yellow in the Apple paint is $\frac{48}{40}$. This ratio is equivalent to the ratio $\frac{6}{5}$.
"It’s easy to compare $\frac{3}{2}$ and $\frac{6}{5}$. I know that $\frac{3}{2}$ is greater than $\frac{6}{5}$. So, the Sunset paint is redder than the Apple paint."
• Talk to your partner about Mia's reasoning. Why is it more difficult to compare $\frac{24}{16}$ and $\frac{48}{40}$ than it is to compare $\frac{3}{2}$ and $\frac{6}{5}$?
1 of 7 | 2022-11-29 05:41:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 12, "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": 0.4313179850578308, "perplexity": 319.4775223027771}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710685.0/warc/CC-MAIN-20221129031912-20221129061912-00193.warc.gz"} |
http://mathhelpforum.com/calculus/108633-angle-between-line-plane.html | ## Angle between line and plane
Determine the angle between the line with symmetric equations x=-y, z=4 and the plane 2x-2z=5.
so the direction vector for the sym equation is.. (1,1, ?)
for the z.. I tried with 1, but then I didn't get the answer. So I tried with z=4 and I got -30 degrees as the answer, and the back of the book has 30 degrees. Is this z value in the direction value correct? I don't think it is right because the final answer is positive, not negative 30. Also, z=4 as the sym equation doesn't indicate that the direction value should have a z=4..
I am confused about this. Can someone let me know what the direction value should be?
Here is my work below:
sin(theta)=(n . d)/ (|n||d|)
(1,1,4). (2,0,-2)
---------------
(sqrt(18x8))
theta= - 30 degrees
the final answer is positive 30 degrees like i said, but i cant seem to get this.. where is my error? thanks in advance. | 2017-04-25 03:38:32 | {"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": 0.8904260993003845, "perplexity": 515.4450572282785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120092.26/warc/CC-MAIN-20170423031200-00469-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/323334/what-was-the-first-bit-of-mathematics-that-made-you-realize-that-math-is-beautif/375940 | # What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of Mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found that I, too, lament the uninspiring quality of my elementary math education.
I want to make a book that discredits the notion that math is merely a series of calculations, and inspires a sense of awe and genuine curiosity in young readers.
However, I myself am mathematically unsophisticated.
What was the first bit of mathematics that made you realize that math is beautiful?
For the purposes of this children's book, accessible answers would be appreciated.
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Nice question, but should probably be community wiki? – mrf Mar 7 '13 at 7:02
For me Euclid's proof of the infinitude of primes was the first thing that made me realize the beauty of mathematics. – Manjil P. Saikia Mar 7 '13 at 7:02
Wow. Just last night I had a fierce argument with one of the bartenders of my usual watering hole who is a mechanical engineering student. He insisted that he has a better idea than me of what is mathematics. I am so going to print him a copy of Lockhart's text. Thank you for that link! – Asaf Karagila Mar 7 '13 at 7:59
I can’t remember a time when I didn’t think that mathematics was beautiful and fascinating. – Brian M. Scott Mar 7 '13 at 15:06
Although I don't know if it's what you are looking for, try looking up "vihart" on youtube--Even if it's not helpful, I guarantee you will appreciate it. – Bill K Mar 8 '13 at 2:57
I started math reasonably late, so I'm not sure this is a perfect example for a children book. I was totally amazed that the number $\pi$ is encountered in completely unrelated situation. Of course, I knew it's a ratio of circumference of the circle to the diameter, but then I learnt of the Normal probability density function.
So if you stretch a bell-shaped curve (Gaussian function) from $-\infty$ to $\infty$ the area under it $converges$ to $\sqrt{\pi}$?? How is it possible that this area has something to do with the square root of the ratio of circumference of the circle to the diameter? To be honest, even now, after learning the related proofs and derivations I still find it quite baffling.
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For me, it was the beauty of the number 1, how it can be multiplied with anything , and it won't change the number it is being multiplied with, also how it can be represented as any number divided by itself such as 4/4=1 I would also love to share this beautiful poem by Dave Feinberg that is titled "the square root of 3" and was also featured in a Harold and Kumar Movie, it renewed my love for math and is and always has been one of my favorite poems! :
I’m sure that I will always be A lonely number like root three
The three is all that’s good and right, Why must my three keep out of sight Beneath the vicious square root sign, I wish instead I were a nine
For nine could thwart this evil trick, with just some quick arithmetic
I know I’ll never see the sun, as 1.7321 Such is my reality, a sad irrationality
When hark! What is this I see, Another square root of a three
As quietly co-waltzing by, Together now we multiply To form a number we prefer, Rejoicing as an integer
We break free from our mortal bonds With the wave of magic wands
Our square root signs become unglued Your love for me has been renewed
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I have to admit that although I'd frequently been told that mathematics was "beautiful", I didn't really get that while I was in school - even high school. I enjoyed mathematics, and saw plenty of things that were fun, and even cute, but I never really understood any ideas with sufficient depth to think of them as beautiful.
When I did encounter ideas that I found beautiful, it was in my first year at college. In fact, there were two closely related ideas in quick succession. We were just being introduced to vector spaces. This was the first time I'd seen an abstract space, but it didn't really seem to mean much except as a fancy way to talk about high dimensional Euclidean spaces.
But then I saw my first example of a vector space that didn't just look like the vectors I'd seen in high school. It was the space of infinitely differentiable functions: $C^{\infty}$. We were shown the linear operators associated with two common differential equations (exponential growth and simple harmonic motion): \begin{eqnarray} &\frac{\textrm{d}\phantom{y}}{\textrm{d}t} - kI \\ &\frac{\textrm{d}^{2}\phantom{y}}{\textrm{dt}^2} + kI. \end{eqnarray} We saw the fairly routine proofs that these were linear operators on $C^{\infty}$, but then came the magical part: The solution sets to these differential equations were subspaces of $C^{\infty}$, the canonical solutions I was familiar with were basis sets for these solution spaces, and the solution spaces were actually the nullspaces of these operators!
Later (maybe even in that same lecture) we saw how linear regression - the hitherto tedious process of finding the "line of best fit" - could be understood as a linear projection $P$ operator onto a two dimensional subspace of the data space. Given a data vector $\mathbf{x}$, the projected vector $P\mathbf{x}$ represented the line that was closest to the data - the line of best fit - and the difference term $\mathbf{x}-P\mathbf{x}$ represented the error term. I was astonished at how much more elegant this was than the clunky formulas I'd had to memorize in high school.
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Not an experience of mine, but I'm currently reading The Greeks by H. D. F. Kitto and I think this page deserves to be here:
But let us not be too superior to those Greeks who "shut their eyes." They kept something else wide open, namely their minds, and although the eye-shutting retarded the growth of science, the mind-opening led to things perhaps equally important, metaphysics and mathematics.
Mathematics are perhaps the most characteristic of all the Greek discoveries, and the one that excited them most. We shall be more understanding of those who shut their eyes to facts if first of all we keep in mind the Greek conviction that the Universe is a logical whole, and therefore simple (despite appearances) and probably symmetrical, and then try to imagine the impact of their minds on elementary mathematics.
It happens that I myself—if I may be personal for a moment—was enabled to do this by an insomnia-beguiling piece of mathematical research that I once did myself. (Mathematical readers are permitted to smile.) It occurred to me to wonder what was the difference between the square of a number and the product of its next-door neighbors. $10 \times 10$ proved to be $100$, and $11 \times 9 = 99$—one less. It was interesting to find that the difference between $6 \times 6$ and $7 \times 5$ was just the same, and with growing excitement I discovered, and algebraically proved, the law that this product must always be one less than the square. The next step was to consider the behavior of next-door neighbors but one, and it was with great delight that I disclosed to myself a whole system of numerical behavior of which my mathematical teachers had left me (I am glad to say) in complete ignorance. With increasing wonder I worked out the series to $10 \times 10 = 100$; $9 \times 11 = 99$; $8 \times 12 = 96$; $7 \times 13 = 91$… and found that the differences were, successively, $1, 3, 5, 7, \ldots$, the odd-number series. Even more marvelous was the discovery that if each successive product is subtracted from the original $100$, there is produced the series $1, 4, 9, 16, \ldots$. They had never told me, and I had never suspected, that Numbers play these grave and beautiful games with each other, from everlasting to everlasting, independently (apparently) of time, space, and the human mind. It was an impressive peep into a new and perfect universe.
(original source image)
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The realization that you can go on counting forever.
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Tupper's self-referential formula is a self-referential formula defined by Jeff Tupper that, when graphed in two dimensions, can visually reproduce the formula itself. It is used in various math and computer science courses as an exercise in graphing formulae.
The formula was first published in his 2001 SIGGRAPH paper that discusses methods related to the GrafEq formula-graphing program he developed.
The formula is an inequality defined by: $${1\over 2} < \left\lfloor \mathrm{mod}\left(\left\lfloor {y \over 17} \right\rfloor 2^{-17 \lfloor x \rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)},2\right)\right\rfloor$$ where $\lfloor \cdot \rfloor$ denotes the floor function and $\mathrm{mod}$ is the modulo operation.
Let k equal the following 543-digit integer:
960 939 379 918 958 884 971 672 962 127 852 754 715 004 339 660 129 306 651 505 519 271 702 802 395 266 424 689 642 842 174 350 718 121 267 153 782 770 623 355 993 237 280 874 144 307 891 325 963 941 337 723 487 857 735 749 823 926 629 715 517 173 716 995 165 232 890 538 221 612 403 238 855 866 184 013 235 585 136 048 828 693 337 902 491 454 229 288 667 081 096 184 496 091 705 183 454 067 827 731 551 705 405 381 627 380 967 602 565 625 016 981 482 083 418 783 163 849 115 590 225 610 003 652 351 370 343 874 461 848 378 737 238 198 224 849 863 465 033 159 410 054 974 700 593 138 339 226 497 249 461 751 545 728 366 702 369 745 461 014 655 997 933 798 537 483 143 786 841 806 593 422 227 898 388 722 980 000 748 404 719
If one graphs the set of points $(x, y)$ in $0 \le x < 106$ and $k \le y < k + 17$ satisfying the inequality given above, the resulting graph looks like this (note that the axes in this plot have been reversed, otherwise the picture comes out upside-down):
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As others have mentioned, kids love $\pi$. Prime numbers are also good, if they have a good handle on division. I think the fundamental theorem of arithmetic is intuitively true once you understand it (at east it was to me).
It would be great to mention some unsolved problems, like the twin prime conjecture or the Collatz conjecture.
For me, one thing that I remember being fascinated about at an early age was the fact that multiplication is commutative. That $3+3+3+3+3=5+5+5$ (or if you want, five baskets with three apples each is the same as three baskets with five apples each) was not immediately obvious to me, and the fact that it worked for any two numbers amazed me. Once you understand the geometric "square of dots" proof it makes sense, but I think that before that it doesn't.
Knuth up arrow notation is worth mentioning. Kids love that multiplication is repeated addition and that powers are repeated multiplication, and would be interested to see that idea taken further.
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The most wonderful thing I've recently seen is this (sorry it's in French) form of the sieve of Eratosthenes and of course your question too.
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I was always good at maths as a child, and took to reading extension maths books for fun (other kids thought I was weird). When I was about 10 I was completely hooked when I saw Euclid's proof for an infinity of primes. I had been given it as a question in one of the books I was reading. I spent about an hour desperately trying to prove it . . . then I looked at the solution - I was stunned by its elegant simplicity. Another thing I really enjoyed was finding cool facts about numbers in kids maths cartoon books and proving them. I loved to show WHY things always worked, that is perhaps my favorite thing about maths.
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For me it was Monty Hall problem:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
I saw this problem when I was 15 year old. I answered correctly (I probably used some kind of math intuition), but I thought that probability in the second case is $1/2$. Actually it is $2/3$. The proof is beautiful, as well as the answer. This fact amazed me. Even now, at 18, I suppose it is quite a beautiful problem.
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Draw any triangle. On each side of the triangle, draw an equilateral triangle such that the new equilateral triangle shares a side with the original triangle. Connect the midpoints of your three new triangles - the result is another equilateral triangle!
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My first think of infinity was going from one corner of square to opposite corner. Start with route (0,0) -- (0,1) -- (1,1), then take (0,0) -- (0,½) -- (½,½) -- (1,½) -- (1,1) and so on. Path will come visually closer to diagonal, but lenght will stay at 2.
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A story which I heard when I was in Primary School motivated me to understand the Power of Exponentials. The story goes like this--..........A Brahmin Priest presented the King with the Chess Board and explained to him how to pay War Games on this board . The King was pleased and asked the priest , what he wanted as a reward ..........The priest asked the King , that as he was very poor ,he needed some grains to feed his family .He asked the King to put one grain of rice in the first square of the chess board. , then put two grains in the second square ,four grains in the third square----and continue this way doubling the number of grains in the next square --till he reaches the end--the 64th square ."I will take whatever grains are there on the Chess Board..that will be sufficient for my needs"..said the Brahmin...........The king tried to satisfy the needs of the Brahmin ,but soon found out that all the grains in the Kingdom will still fall short of his needs ....... . The King was pleased with the priest's intelligence and appointed him as the Royal Astronomer & Astrologer .
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The $\sqrt{-1}$, complex analysis, and how real world problems could be solved "by these objects which don't exist."
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Maybe the fact that the homotopy category of a model category is equivalent to the full subcategory of fibrant-cofibrant objects with homotopy classes of morphisms.
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Congratulations for finding that out as a child. – azimut Jun 21 at 13:26
This is a new interesting 4x4 Magic Sqaure, which I believe will be interesting to School Children. Here each element of the square is a square number. This was provided by Dr. Geoffrey Campbell
509020 is the sum of rows and columns
29^2 | 191^2 | 673^2 | 137^2 || 509020
-------+--------+--------+--------++--------
71^2 | 647^2 | 139^2 | 257^2 || 509020
-------+--------+--------+--------++--------
277^2 | 211^2 | 163^2 | 601^2 || 509020
-------+--------+--------+--------++--------
653^2 | 97^2 | 101^2 | 251^2 || 509020
=======+========+========+========++--------
509020 | 509020 | 509020 | 509020 || 509020
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I don't find it beautiful, but I still find the idea expressed by the following something of a psychological curiosity:
How can it be that when some algebraists say "AND" and "OR" they mean exactly the same thing?
OR means this that "false or false" is false, "false or true", "true or false" as well as "true or true" are true, or more compactly:
F T
F F T
T T T
AND means this:
F T
F F F
T F T
But, since NOT(x OR y)=(NOT x AND NOT y) and NOT(T)=F and NOT(F)=T, OR and AND, to an algebraist, mean exactly the same thing!
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Your answer implies that $\neg ( \perp \lor \top) \iff ( \neg \perp \land \neg \top) \iff ( \top \land \perp ) \iff ( \perp \lor \top)$. Your truth table for $\land$ is wrong. – Andrew Salmon Mar 23 '13 at 21:10
## protected by Zev ChonolesMar 7 '13 at 22:43
Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site. | 2014-08-21 10:45:07 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6342276930809021, "perplexity": 464.6388766499094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500815861.64/warc/CC-MAIN-20140820021335-00244-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://mymathforum.com/algebra/26724-another-logarithmic-diophantine-problem.html | My Math Forum Another logarithmic diophantine problem !!!
Algebra Pre-Algebra and Basic Algebra Math Forum
April 15th, 2012, 05:14 AM #1 Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Another logarithmic diophantine problem !!! How to prove that x^y = y^x (x is not = y) has no solution if y > 2 ????
May 25th, 2012, 04:21 AM #2 Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Another logarithmic diophantine problem !!! I found a method for reducing it into normal diophantine form: x^y=y^x, assume x,y is very large (x,y integer) then y/ln(y) = x/ln(x) implies p(y)=p(x) where p is the prime counting function. is it possible to solve from here!?
May 25th, 2012, 05:19 PM #3 Global Moderator Joined: Dec 2006 Posts: 19,866 Thanks: 1833 As it's not true, there's no way to prove it.
May 25th, 2012, 05:56 PM #4 Global Moderator Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,881 Thanks: 1088 Math Focus: Elementary mathematics and beyond Re: Another logarithmic diophantine problem !!! Can you provide a counterexample?
May 25th, 2012, 06:42 PM #5 Global Moderator Joined: Dec 2006 Posts: 19,866 Thanks: 1833 x = 2, y = 4 (there are infinitely many others if x and y needn't be integers).
May 25th, 2012, 11:17 PM #6
Math Team
Joined: Mar 2012
From: India, West Bengal
Posts: 3,871
Thanks: 86
Math Focus: Number Theory
Re: Another logarithmic diophantine problem !!!
Quote:
Originally Posted by skipjack x = 2, y = 4 (there are infinitely many others if x and y needn't be integers).
yes, this is the only case when x,y is integer and there are no other cases, I'm sure
May 26th, 2012, 07:49 AM #7 Global Moderator Joined: Dec 2006 Posts: 19,866 Thanks: 1833 Example of non-integer solution: x = 2^(2/3), y = 2^(8/3).
May 26th, 2012, 10:19 AM #8
Math Team
Joined: Mar 2012
From: India, West Bengal
Posts: 3,871
Thanks: 86
Math Focus: Number Theory
Re: Another logarithmic diophantine problem !!!
Quote:
Originally Posted by skipjack Example of non-integer solution: x = 2^(2/3), y = 2^(8/3).
Yes there are infinitely many solutions if x,y are non-integer, but I'm saying about integral x,y and what I said before, I'm pretty sure that there exist no solutions to the Diophantine equation: x^y=y^x except (x=2, y=4); (x=4,y=2) when x,y are distinct, greater than 1, integer;
now I'm asking you again, how to prove the above statement without the use of calculus?
September 21st, 2012, 09:07 PM #9 Global Moderator Joined: Dec 2006 Posts: 19,866 Thanks: 1833 Start by solving the equation, as explained in this post, then consider when the solutions found can be integers.
September 22nd, 2012, 08:02 AM #10
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Re: Another logarithmic diophantine problem !!!
Quote:
Originally Posted by skipjack Start by solving the equation, as explained in this post, then consider when the solutions found can be integers.
Yes, I know this can be solved using Lambert W Function and Calculus (there are several other approaches too). But actually I need a purely Diophantine approach.
Perhaps this would be appropriate if posted to the NT section. I posted this after a few months of joining so I wasn't familiar enough with this forum.
If moderators agree, then I could create another topic in the NT section asking about a Diophantine method to solve x^y = y^x where x,y are integers. Then mods can post a link to that topic and lock this topic.
I don't want it to be moved since it's already been months before I posted it and due to the discussion here, people can easily get confused about what I am exactly asking. Please inform me if any of you moderators agree with this.
Balarka
.
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Contact - Home - Forums - Cryptocurrency Forum - Top | 2018-11-15 17:29:26 | {"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": 0.8174191117286682, "perplexity": 4409.534073559815}, "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/1542039742793.19/warc/CC-MAIN-20181115161834-20181115183834-00163.warc.gz"} |
https://astarmathsandphysics.com/ib-physics-notes/electromagnetism/1269-equipotential-surfaces.html?highlight=WyJmcm9tIiwid2hlcmUiLCJmcm9tIHdoZXJlIl0= | ## Equipotential Surfaces
The best way of representing how the electric potential varies around a charged object is to identify the regions where the potential is the same. These are called equipotential surfaces. They are closed surfaces in three dimensions but we can often represent them in two dimensions as curves. The diagram below shows curves on the equipotential surface – in reality the charge is in a three dimensional space and the equipotential surfaces are concentric spheres.
Notice that the flux or field lines are at right angle to the equipotential surfaces wherever they meet. This is in fact true in general for equipotentials and flux lines, a consequence of the fact that the force on a particle acts in the direction of the greatest rate of change of potential, which is always perpendicular to the equipotential. The flux lines of course, indicate the direction of the electric field, and the direction of the force on a positive charge.
The closer together the equipotentials, the stronger the electric field, since the gradient (and force) is most rapid where the equipotentials are closest. For this reason, the relative strength of an electric field at different points can be estimated by considering the equipotentials.
Equipotentials are always closed surfaces around a distribution of charge, though they may take different forms at different distances from the charges, while always smoothly deforming. The surface of a conductor is always an equipotential, since charge is free to move on the surface so as to equalise the potential everywhere. The surface being an equipotential, the electrical field is always perpendicular to the surface. | 2018-08-19 03:53:06 | {"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": 0.8391233682632446, "perplexity": 177.64183138353235}, "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-2018-34/segments/1534221214691.99/warc/CC-MAIN-20180819031801-20180819051801-00548.warc.gz"} |
https://younesse.net/Domains-categories-games/Lecture13/ | # Lecture 13: Scott semantics
Lecture 13
Preorders and linear functions (Scott semantics)
$PoLR_!(S, T) \overset{Fun}{⟶} \underbrace{PoC(S, T)}_{\text{Scott cont. function } ℐ(S) → ℐ(T)}\\ PoC(S, T) \overset{Tr}{⟶} PoLR_!(S, T)$
defines an iso of categories
$id ∈ PoLR_!(S, S) = PoLR(!S, S)\\ id = der_S$ $s ∈ PoLR_!(S, T) = PoLR(!S, T)\\ t ∈ PoLR_!(T, U) = PoLR(!T, U)\\ t \circ s ∈ PoLR_!(S, U)\\ !S \overset{dig_S}{⟶} !!S \overset{!s}{⟶} !T \overset{t}{⟶} U$ $Fun(t \circ s) = Fun(t) \circ Fun(s)$
Remember that
$Fun(s)(u) = s u^!\\ u ∈ ℐ(S) \qquad u^! = 𝒫_{fin}(u) ∈ ℐ(!S)$ $Fun(t \circ s)(u) = (t \circ s) u^! = (t \, !s \, dig) \, u^! \\ = t \, !s \, (dig \, u^!) = t \, !s \, u^{!!}\\ = t \, (s \, u^!)^! = Fun(t)(Fun(s)(u))$
We can consider
$Fun: \begin{cases} PoLR_! &⟶ PoC \\ S &⟼ S\\ t &⟼ Fun(t)\\ \end{cases}$ $Tr: \begin{cases} PoC &⟶ PoLR_! \\ S &⟼ S\\ f &⟼ Tr(f)\\ \end{cases}$
are two functors which define an iso of categories.
# Fixpoints
## Fixpoints of functions
Any continuous function has a least fixpoint:
$s ∈ PoLR_!(S, S)\\ f = Fun(s): \underbrace{ℐ(S)}_{\ni ∅} ⟶ ℐ(S)$ $∅ ⊆ f(∅) ⊆ f^2(∅) ⊆ ⋯ ⊆ f^n(∅) ⊆ f^{n+1}(∅) ⊆ ⋯$ $u = \bigcup\limits_{n=0}^∞ f^n(∅)\\ f(u) = \bigcup\limits_{n=0}^∞ f^{n+1}(∅) = u$
$u$ is the least fixpoint of $f$.
$𝒴: \begin{cases} ℐ(S ⇒ S) &⟶ ℐ(S) \\ s &⟼ \bigcup\limits_{n=0}^∞ Fun(s)^n(∅) \end{cases}$
$Y ∈ ℐ((S ⇒ S) ⇒ S)$
such that $𝒴 = Fun(Y)$
Define
$F ∈ PoC((S ⇒ S) ⇒ S, (S ⇒ S) ⇒ S)\\ 𝒵 ∈ PoLR_!((S ⇒ S) ⇒ S, (S ⇒ S) ⇒ S)$
such that $Y$ will be the least fixpoint of $𝒵$.
$Y = \bigcup\limits_{n=0}^∞ 𝒵^n(∅)$ $"Y(t) = t(Y(t))\\ λY. (λt. t(Y(t)))"$ $𝒵_0: ((S ⇒ S) ⇒ S) \& (S ⇒ S) \overset{⟨π_1, π_2, π_2⟩}{⟶} ((S ⇒ S) ⇒ S) \& (S ⇒ S) \& (S ⇒ S)\\ \overset{Ev \& id}{⟶} S \& (S ⇒ S) ⟶ (S ⇒ S) \& S \overset{Ev}{⟶} S$
$𝒵$ is fully characterized by
$Fun(Fun(𝒵(Y)))(t) = Fun(t)(Fun(Y)(t))$
Let $F = Fun(𝒵)$
Let
$Y_0 = \bigcup\limits_{n=0}^∞ F^n(∅)$
be the least fixpoint of $F$ ($F(Y_0) = Y_0$).
Let $s ∈ PoLR_!(S, S) = ℐ(S ⇒ S)$.
$Fun(Y_0)(s) = Fun(F(Y_0))(s) = Fun(s)(Fun(Y_0)(s))$
(we are strongly relying on the cartesian closure of the category)
Exercise: Prove that
$Fun(Y_0)(s) = \bigcup\limits_{n=0}^∞ Fun(s)^n(∅)$
actually:
$Fun(F^n(∅))(s) = Fun(s)^n(∅)$
## Fixpoints of types
$PO_⊆$:
partially ordered class of preorders, ordered by:
$S ⊆ T \quad \text{ iff } \quad \vert S \vert ⊆ \vert T \vert \text{ and } ∀ a, a' ∈ \vert S \vert, a ≤_S a' ⟺ a ≤_T a'$
Any sequence
$S_0 ⊆ S_1 ⊆ ⋯ ⊆ S_n ⊆ S_{n+1} ⊆ ⋯$
has a least upper bound in $PO_⊆$
$S = \bigcup\limits_{n=0}^∞ S_n$
given by
$\vert S \vert = \bigcup\limits_{n=0}^∞ \vert S_n \vert\\ a ≤_S a' \quad \text{ iff } \quad ∃ n; a ≤_{S_n} a'$
NB (Model Theory): the theory of partial orders is clearly $∀∃$-axiomatizable, so it is inductive.
$Φ = PO_⊆ ⟶ PO_⊆$
is a unary variable type if $Φ$ is monotonic, i.e.
$S ⊆ T ⟹ Φ(S) ⊆ Φ(T)$
and continuous, i.e.
$S_0 ⊆ S_1 ⊆ ⋯ ⟹ Φ(\bigcup\limits_{n=0}^∞ S_n) = \bigcup\limits_{n=0}^∞ Φ(S_n)$
A $n$-ary variable type is a
$Φ: PO_⊆^n ⟶ PO_⊆$
st
$∀ i, ∀ S_1, …, S_{i-1}, S_{i+1}, …, S_n ∈ PO, \\ S ⟼ Φ(S_1, …, S_{i-1}, S, S_{i+1}, …, S_n) \text{ is a unary variable type}$
If $Φ: PO_⊆ ⟶ PO_⊆$ is a variable type, then
$Φ(\underbrace{\bigcup\limits_{n=0}^∞ Φ(∅)}_{Rec(Φ)}) = \bigcup\limits_{n=0}^∞ Φ(∅)$
$Φ: PO_⊆^{n+1} ⟶ PO_⊆$ variable type, $\vec S ∈ PO^n$,
$Φ_{\vec S}: \begin{cases} PO_⊆ &⟶ PO_⊆ \\ S &⟼ Φ(\vec S, S) \end{cases}$
unary variable type
$Rec(Φ): \begin{cases} PO_⊆^n &⟶ PO_⊆ \\ \vec S &⟼ Rec(Φ_{\vec S}) \end{cases}$ $Φ(\vec S, Rec(Φ)(\vec S)) = Rec(Φ(\vec S))$
Fact: $Rec(Φ)$ is itself a variable type. All constructions of LL are variable types.
Are variable types:
• $\begin{cases} PO_⊆ &⟶ PO_⊆ \\ S &⟼ S^⊥ \end{cases}$
• $\begin{cases} PO_⊆^2 &⟶ PO_⊆ \\ (S_1, S_2) &⟼ S_1 ⊗ S_2 \end{cases}$
• $\begin{cases} PO_⊆ &⟶ PO_⊆ \\ S &⟼ !S \end{cases}$
Variable types are closed under composition.
If
$Φ: PO_⊆^n ⟶ PO_⊆$ $Ψ_i: PO_⊆^k ⟶ PO_⊆$
then
$\begin{cases} PO_⊆^k &⟶ PO_⊆ \\ \vec T &⟼ Φ(Ψ_1(\vec T), …, Ψ_n(\vec T)) \end{cases}$
is a variable type.
Example:
$Φ(S) = 1 ⊕ !S$
Let $L = Rec(Φ)$
$\vert L \vert = \bigcup\limits_{n=0}^∞ E_n$
where
• $E_0 = ∅$
• $E_{n+1} = \lbrace \underbrace{(1, \ast)}_{ζ}\rbrace ∪ \lbrace \underbrace{(2, u^0)}_{\overline{suc}(u^0)} \; \mid \; u^0 ∈ 𝒫_{fin}(E_n)\rbrace$
$a \; ≝ \; ζ \; \mid \; \overline{suc} \lbrace a_1, …, a_n\rbrace$ $a ≤_L a' \qquad ⟺ \qquad \begin{cases} a = a' = ζ \\ \text{ or } a = \overline{suc}(u^0) \text{ and } a' = \overline{suc}(u^1) \text{ and } \underbrace{u^0 ≤_{!L} u^1 }_{∀ b∈ u^0, ∃ b' ∈ u^1; b ≤_L b'} \end{cases}$
⟶ Elements of $L$ are “finite comb trees”
$\overline 0 ∈ PoLR_!(\underbrace{⊤}_{\text{terminal object of } PoLR \text{ and } PoLR_!: \text{ the empty preorder}}, L)\\ Suc ∈ PoLR_!(L, L)$ $!⊤ \overset{m^0}{⟶} 1 \overset{in_1}{⟶} \underbrace{1 ⊕ !L}_{=L}$ $\overline 0 = \lbrace (∅, ζ)\rbrace$ $Suc: \; !L \overset{m_2}{⟶} 1 ⊕ !L = L\\ Suc = \lbrace (u^0, \overline{suc}(u^0)) \; \mid \; u^0 ∈ \vert !L \vert \rbrace$ $\overline{if} ∈ PoLR(L ⊗ !S ⊗ !(!L ⊸ S), S)\\ if(M, N, x. P)$ $\overline{if}: \underbrace{(1 ⊕ !L) ⊗ !S ⊗ !(!L ⊸ S)}_{≃ \; (!S ⊗ !(!L ⊸ S)) ⊕ (!L ⊗ !S ⊗ !(!L ⊸ S))} ⟶ S$
Case definition:
$\overline{if}_0: !S ⊗ !(!L ⊸ S) \overset{!S ⊗ w}{⟶} !S ⊗ 1 \; ≃ \; !S \overset{der_S}{⟶} S$ $\overline{if}_{suc}: \; !L ⊗ !S ⊗ !(!L ⊸ S) \overset{id ⊗ w ⊗ id}{⟶} !L ⊗ 1 ⊗ !(!L ⊸ S) \\ \; ≃ \; !L ⊗ !(!L ⊸ S) \overset{id ⊗ der}{⟶} !L ⊗ (!L ⊸ S) \overset{σ}{⟶} !(!L ⊸ S) ⊗ !L \overset{ev}{ ⟶} S$
# LPCF
$A, B, … \; ≝ \; ι \; \mid \; A → B$ $M \; ≝ \; x \; \mid \; \underline 0 \; \mid \; \underline {suc} (M) \; \mid \; if (M, N, x. P)\\ \; \mid \; (M)N \; \mid \; λx^A. M \; \mid \; fix \, x^A. M$ $Γ = (x_1: A_1, …, x_n: A_n)$ $\cfrac{}{x_1: A_1, …, x_n: A_n ⊢ x_i: A_i}$ $\cfrac{}{Γ ⊢ \underline 0: ι}$ $\cfrac{Γ ⊢ M: ι}{Γ ⊢ \underline {suc }(M): ι}$ $\cfrac{Γ ⊢ M: ι \qquad Γ ⊢ N: A \qquad Γ, x: ι ⊢ P: A}{Γ ⊢ if(M, N, x.P): A}$
$λ$-calculus (app. $λ$)
$\cfrac{Γ, x:A ⊢ M:A}{Γ ⊢ fix \; x^A. M:A}$
$β$: general reduction
$β_{wh}$: weak head reduction
## Interpretation in the model PoLR
$A \leadsto [A] \text{ a preorder}\\ [ι] = L \\ [A → B] = [A] ⇒ [B] = ![A] ⊸ [B]$ $Γ = (x_1: A_1, … , x_n: A_n)$
$![Γ]$ is a notation for
$![A_1] ⊗ ⋯ ⊗ ![A_n] \quad ≃ \quad !([A_1] \& ⋯ \& [A_n])$
to $Γ ⊢ M:A$ we associate
$[M]_Γ ∈ PoLR(![Γ], [A])$
NB: $[M]_Γ$ can be considered as a Scott-continuous function
$ℐ([A_1]) × ⋯ × ℐ([A_n]) ⟶ ℐ([A])$
$[M]_Γ$ defined by induction on $M$:
• $M = x_i, A = A_i$:
$[M]_Γ: \quad ![A_1] ⊗ ⋯ ⊗ ![A_i] ⊗ ⋯ ⊗ ![A_n] \\ \overset{w ⊗ ⋯ ⊗ w ⊗ der ⊗ w ⊗ ⋯ ⊗ w}{⟶} 1 ⊗ ⋯ ⊗ 1 ⊗ [A_i] ⊗ 1 ⊗ ⋯ ⊗ 1 ≃ [A_i]$
• $M = \underline {0}, A = ι$
$[M]_Γ: \quad ![A_1] ⊗ ⋯ ⊗ ![A_n] ⟶ 1 ⊗ ⋯ ⊗ 1 \; ≃\; ! ⊤ \overset{\overline{0}}{⟶} L$
• $M = suc(N)$
$[M]_Γ: \quad ![Γ] \overset{[N]_Γ^!}{⟶} !L \overset{Suc}{⟶} L$
concretely:
$[M]_Γ \; ≝ \; \lbrace (u^1, …, u^n, \overline{suc}(u^0)) \; \mid \; ∀ a ∈ u^0, \; (u^1, …, u^n, a) ∈ [N]_Γ\rbrace$
• $M = if(N, P, x. Q)$
$Γ ⊢ N: ι \qquad Γ ⊢ P: A \qquad Γ, x:ι ⊢ Q: A$
By IH:
$[N]_Γ: ![Γ] ⟶ L \qquad [P]_Γ: !Γ ⟶ [A] \qquad [Q]_{Γ, x: ι}: ![Γ] ⊗ ![L] ⟶ [A]$ $\overline{if}: L ⊗ ![A] ⊗ !(!L ⊸ [A]) ⟶ [A]$ $![Γ] \overset{c}{ ⟶ } ![Γ] ⊗ ![Γ] ⊗ ![Γ] \overset{[N]_Γ ⊗ [P]_Γ^! ⊗ (cur([Q]_{Γ, x:ι}))^!}{⟶} L ⊗ ![A] ⊗ !(!L ⊸ [A]) \overset{\overline{if}}{⟶} [A]$
• $M = (N)P$
$Γ ⊢ N: B → A \qquad Γ ⊢ P:B$ $[N]_Γ: \; ![Γ] ⟶ (![B] ⊸ [A])\\ [P]_Γ: \; ![Γ] ⟶ [B]\\$ $![Γ] \overset{c}{⟶} ![Γ] ⊗ ![Γ] \overset{[N]_Γ ⊗ [P]_Γ^!}{⟶} (![B] ⊸ [A]) ⊗ ![B] \overset{ev}{⟶} [A]$
• $M = λx^B.N, \; A = B → C$
$Γ, x: B ⊢ N:C\\ [N]_{Γ, x:B}: !Γ ⊗ ![B] ⟶ [C]$ $[M]_Γ = cur([N]_{Γ, x:B})$
• $M = fix\; x^A. N$
$Γ, x: A ⊢ N: A$ $Y_0: !(![A] ⊸ [A]) ⟶ [A]$ $[N]_{Γ, x:A}: \; ![Γ] ⊗ ![A] ⟶ [A]$ $cur([N]_{Γ, x:A}): ![Γ] ⟶ (![A] ⊸ [A])$ $[M]_Γ: \quad ![Γ] \overset{(cur([N]_{Γ,x:A}))^!}{⟶} !(![A] ⊸ [A]) \overset{Y_0}{⟶} [A]$
Th: if $Γ ⊢ M:A$ and $M \; β \; M’$, then $[M]_Γ = [M’]_Γ$
Lemma (Substitution lemma): if $Γ, x:A ⊢ M:B$ and $Γ ⊢ N:A$, then
$[M[N/x]]: \quad ![Γ] \overset{c}{⟶} ![Γ] ⊗ ![Γ] \overset{id ⊗ [N]_Γ^!}{⟶} ![Γ] ⊗ ![A] \overset{[Γ]_{Γ, x:A}}{⟶} [B]$
## Intersection typing system
Prop: If $Γ ⊢ M:A$, $Γ = (x_1: A_1, …, x_n: A_n)$ and
$u^i ∈ \vert ! [A_i] \vert \quad i = 1, …, n \text{ and } a ∈ \vert [A] \vert$
then $(u^1, …, u^n, a) ∈ [M]_Γ$
iff $x_1: u^1: A_1, …, x_n: u^n: A_n ⊢ M: a: A$ is derivable in the following “semantic typing system” (intersection typing system):
$Φ \; ≝ \; (x_1: u^1: A_1, …, x_n: u^n: A_n)$
semantic typing context
$u^i ∈ \vert ! [A_i] \vert = 𝒫_{fin}(\vert [A_i] \vert)$
$\underline Φ = (x_1: A_1, …, x_n: A_n)$ underlying typing context
$\cfrac{∃ a^i ∈ u^i \qquad a ≤_{A_i} a_i}{x_1: u^1: A_1, …, x_n: u^n: A_n ⊢ x_i: a: A_i}$ $\cfrac{}{\underline Φ ⊢ \underline 0: ι}$ $\cfrac{Φ ⊢ M: a: ι \qquad ∀ a ∈ u^0 ∈ \vert !L \vert \qquad \underline Φ ⊢ M: ι}{\underline Φ ⊢ \underline {suc}(M): \overline {suc}(u^0):ι}$
in particular:
$\cfrac{\underline Φ ⊢ M: ι}{\underline Φ ⊢ \underline {suc}(M): \overline {suc}(Φ): ι}$ $\cfrac{Φ ⊢ M: ζ: ι \qquad Φ ⊢ N: a: A \qquad \underline Φ, x: ι ⊢ P: A}{Φ ⊢ if (M, N, x.P): a: A}$ $\cfrac{Φ ⊢ M: \overline {suc} (u^0): ι \qquad \underline Φ ⊢ N: A \qquad Φ, x: u^0: ι ⊢ P: a: A}{Φ ⊢ if (M, N, x.P): a: A}$ $\cfrac{Φ ⊢ M: (u^0, b): A → B \qquad \underline Φ ⊢ N: A \qquad Φ ⊢ N: a: A \qquad ∀ a ∈ u^0}{Φ ⊢ (M)N: b: B}$ $\cfrac{Φ, x:u^0:A ⊢ M: b: B}{Φ ⊢ λx^A. M: (u^0, b): A → B}$ $\cfrac{Φ, x:u^0:A ⊢ M: a: A \qquad Φ ⊢ fix \; x^A. M: a': A \quad ∀ a' ∈ u^0}{Φ ⊢ fix \; x^A. M: a: A}$
Goal: if $⊢ M: ζ: ι$, then $M \; β_{wh}^\ast \; \underline 0$
Technique: reducibility
Given $A$ and $a ∈ \vert [A] \vert$, we define $\vert a \vert^A$, a set of terms $M$ st $⊢ M: A$ (and for $u^0 ∈ \vert ! [A] \vert$, $\vert u^0 \vert^A_! = \bigcap_{a ∈ u^0} \vert a \vert^A$)
The def is by induction on the size of $a$ (which is a finite term)
### $A = ι$ and $a = ζ$
$\vert ζ \vert^ι = \lbrace ⊢ M: ι \; \mid \; M \; β^\ast_{wh} \underline 0\rbrace$ $a = \overline {suc}(u^0)\\ \vert a \vert^ι = \lbrace ⊢ M: ι \; \mid \; M \; β^\ast_{wh} \; \underline {suc}(N) \text{ with } N ∈ \vert u^0\vert^ι_! = \bigcap_{a ∈ u^0} \vert a \vert^ι\rbrace$ $\vert (u^0, b) \vert^{A → B} = \lbrace ⊢ M: A → B \; \mid \; ∀ N ∈ \vert u^0\vert^ι_!, \; (M)N ∈ \vert b \vert^B\rbrace$
Lemma 1: if $a ≤_{[A]} a’$, then $\vert a' \vert^A ⊆ \vert a \vert^A$
Proof:
$A=ι$:
if $a ≤_L a’$, there are 2 cases:
1. $a=a’= ζ$: trivial
2. $a = \overline{suc}(u^0), \; a’ = \overline{suc}(u^1)$, $M ∈ \vert a \vert^A$ (“$M$ satisifies $a$”)
$u^0 ≤_{!L} u^1 \qquad (⟺ ∀ b ∈ u^0, ∃ b' ∈ u^1; \; b ≤_L b')$
Let $M ∈ \vert a’ \vert^ι$, that is:
$M \; β^\ast_{wh} \; \overline{suc} (N)$
where
$N ∈ \vert u^1 \vert_!^ι = \bigcap_{b' ∈ u^1} \vert b' \vert^ι \qquad ⟺ ∀ b' ∈ u^1, \; N ∈ \vert b' \vert^ι$
We want: $N ∈ \bigcap_{b ∈ u^0} \vert b \vert^ι$
that is:
$∀ b ∈ u^0, \; N ∈ \vert b \vert^ι$
Let $b ∈ u^0$. There is $b’ ∈ u^1$ st $b ≤_L b’$. And we have:
$N ∈ \vert b' \vert^ι \overset{\text{by IH}}{⊆} \vert b \vert^ι$
$A = B → C$
$a = (v^0, c), a’ = (v^1, c’)$
$a ≤_{[A]} a’$ means
$c ≤_{[C]} c' \quad \text{ and } \quad v^1 ≤_{![B]} v^0$
We want:
$\vert (v^1, c') \vert^{B → C} ⊆ \vert (v^0, c) \vert^{B → C}$
Let $M ∈ \vert (v^1, c’) \vert^{B → C}$. We want: $M ∈ \vert (v^0, c) \vert^{B → C}$.
Let $N ∈ \vert v^0 \vert^B_! ⊆ \vert v^1 \vert^B_!$ because $v^1 ≤_{![B]} v^0$.
And by IH, we want:
$N ∈ \bigcap_{b ∈ v^1} \vert b \vert^B$
Let $b ∈ v^1$, we need: $N ∈ \vert b \vert^B$.
But $v^1 ≤_{![B]} v^0$, so let $b’ ∈ v^0$ st $b ≤_B b'$
So by IH:
$\vert b' \vert^ι ⊆ \vert b \vert^ι$
and we have assumed
$N ∈ \vert v^0 \vert_!^B ⊆ \vert b' \vert^B$
since $b’ ∈ v^0$, so $N ∈ \vert b \vert^ι$.
So $N ∈ \vert v^1 \vert^B_!$. But $M ∈ \vert (v^1, c’) \vert^{B → C}$, so
$(M)N ∈ \vert c' \vert^C$
But
$c ≤_C c'$
so by IH:
$\vert c' \vert^C ⊆ \vert c \vert^C$
so
$(M)N ∈ \vert c \vert^C$
Lemma 2: If $⊢ M: A$, $⊢ M’: A$, $M \; β_{wh} \; M’$ and $M’ ∈ \vert a \vert^A$, then $M ∈ \vert a \vert^A$.
Proof by induction on the size of $a$.
## Main Statement: Interpretation Lemma/Logical relation lemma
If
$x_1: u^1: A_1, …, x_n: u^n: A_n ⊢ M: a: A$
Then: for all $N_1, …, N_n$ st $∀i, \qquad N_i ∈ \vert u^i \vert^{A_i}$ one has $M[N_1/x_1, …, N_n/x_n] ∈ \vert a \vert^A$
Corollaries: if $⊢ M: a: A$, then $M ∈ \vert a \vert^A$.
In particular, if $⊢ M: ζ: ι$, then $M ∈ \vert ζ \vert^ι ⟺ M \; β_{wh}^\ast \; \underline 0$
Tags:
Updated: | 2023-04-01 03:36:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "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": 0.9602112770080566, "perplexity": 3994.8781838041828}, "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/1679296949701.0/warc/CC-MAIN-20230401032604-20230401062604-00719.warc.gz"} |
https://www.physicsforums.com/threads/translation-or-parallel-movts-of-vectors.468122/ | # Translation or parallel movts of vectors
my teacher says that vectors can be translates which is fine... but hw can they be moved paralaly without couple acting on it//??
tiny-tim
Homework Helper
hi phyeinstein_c!
my teacher says that vectors can be translates which is fine... but hw can they be moved paralaly without couple acting on it//??
i think you're confusing a bound vector with a free vector (i've forgotten the correct names, but it's something like that ) …
force is a bound vector, and so its line of application has to be included when specifying it
velocity is a free vector, and you can shove it around so as to put it on the end of another vector
so free vectors do translate, but bound vectors like force (as you say) don't
but if velocity vector mover parallaly from one end of a rigid body (not point mass) then it generates torque right??
tiny-tim
Homework Helper
hi phyeinstein_c!
but if velocity vector mover parallaly from one end of a rigid body (not point mass) then it generates torque right??
hmm, let me think
yes, a force vector certainly generates torque (and a momentum vector generates angular momentum , and momentum is simply mass times a velocity vector ) …
perhaps i should have said that relative velocity is a free vector, and you can shove it around so as to put it on the end of another vector!
but if velocity vector mover parallaly from one end of a rigid body (not point mass) then it generates torque right??
This statement is nonsense.
First, if one part of a rigid body possesses a certain velocity (vector) then all parts posess the same vector. That is a definition of a rigid body.
So velocity vectors cannot move around a rigid body willy nilly.
Second, velcocity cannot produce torque under any circumstances. That is the province of forces or couples.
Tinytim has already said that force is a bound vector, which means you cannot move it about. What he did not say is that bound vectors also have an origin or point of application as part of their specification.
So if you change the point of application of a force, yes you will induce different moments in a rigid body.
Finally for the record couples are free vectors which may be moved about.
ok sry by torque i meant rotation.... velocity will be uniform is a case... but when the vector has been taken to one end of the body.. it shud rotate...couple will act only for forces right.
ok sry by torque i meant rotation.... velocity will be uniform is a case... but when the vector has been taken to one end of the body.. it shud rotate...couple will act only for forces right.
What do you mean?
i mean when the velocity vector is shifted from the centre to one end..... parallally.... the body has higher velocity on one end and less on the other... so it should rotate...??
Let us start again at the beginning.
What system are you describing?
mathwonk
Homework Helper
2020 Award
a mathematician will not understand a word you guys are saying. mathematically a vector in Euclidean space is an equivalence class of parallel arrows all with same direction and magnitude, and all having different endpoints. Any one of those arrows represents the vector. In that sense we say vectors can be translated. Some say more abstractly that a vector IS a translation of space, and is represented by the arrow drawn from any point to its translated image.
In applying the concept you may wish to distinguish between an individual arrow, and the class it represents. maybe this is what you call free and bound vectors, but i have never heard of them.
I chime in here because i thought maybe your teacher is a mathematician and does not know what you are talking about either.
In applying the concept you may wish to distinguish between an individual arrow, and the class it represents. maybe this is what you call free and bound vectors, but i have never heard of them.
Yup that's about it (I'm well known for my precise mathematical statements ).
tiny-tim
Homework Helper
hi mathwonk!
In applying the concept you may wish to distinguish between an individual arrow, and the class it represents. maybe this is what you call free and bound vectors, but i have never heard of them.
i'm sorry to disillusion you, but wikipedia has heard of them, see http://en.wikipedia.org/wiki/Euclidean_vector" [Broken] …
As an arrow in Euclidean space, a vector possesses a definite initial point and terminal point. Such a vector is called a bound vector. When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a free vector.
(though it then seems to get lost, see http://en.wikipedia.org/wiki/Euclidean_vector#In_Cartesian_space" )
Last edited by a moderator:
Let us start again at the beginning.
What system are you describing?
its a rigid body translating...
a mathematician will not understand a word you guys are saying. mathematically a vector in Euclidean space is an equivalence class of parallel arrows all with same direction and magnitude, and all having different endpoints. Any one of those arrows represents the vector. In that sense we say vectors can be translated. Some say more abstractly that a vector IS a translation of space, and is represented by the arrow drawn from any point to its translated image.
In applying the concept you may wish to distinguish between an individual arrow, and the class it represents. maybe this is what you call free and bound vectors, but i have never heard of them.
I chime in here because i thought maybe your teacher is a mathematician and does not know what you are talking about either.
neither did i ever hear of such bounded vectors.... in any VECTOR by definition its only REPRESENTATION of magnitude and direction..... hw does it matter if it is bounded or unbounded or free....
hi mathwonk!
i'm sorry to disillusion you, but wikipedia has heard of them, see http://en.wikipedia.org/wiki/Euclidean_vector" [Broken] …
As an arrow in Euclidean space, a vector possesses a definite initial point and terminal point. Such a vector is called a bound vector. When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a free vector.
(though it then seems to get lost, see http://en.wikipedia.org/wiki/Euclidean_vector#In_Cartesian_space" )
can u discuss some cases (examples) of the 2 cases u metioned..
Last edited by a moderator:
in any VECTOR by definition its only REPRESENTATION of magnitude and direction
No there is more to it than just magnitude and direction.
Suppose I started in London and travelled 300 mile due north, turned and then travelled 300 miles due west.
Now suppose I did it the other way around ie 300 miles west then 300 miles north.
Would I be in the same place the second time around?
It is an important, and not often stated, requirement that
vector(a) + vector(b) = vector(b) + vector(a)
This is another way of saying that a+b is given by the parallelogram law.
ohkk thats there but how does it matter in physics... the result of both is u have had the same displacement and u will have the same velocity etc. during both kind of journey.
ohkk thats there but how does it matter in physics... the result of both is u have had the same displacement and u will have the same velocity etc. during both kind of journey.
Since the English was so mangled I have no idea what you mean. | 2021-05-07 10:13:48 | {"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": 0.8527051210403442, "perplexity": 712.7628022342383}, "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/1620243988775.80/warc/CC-MAIN-20210507090724-20210507120724-00066.warc.gz"} |
https://informationtransfereconomics.blogspot.com/2013/08/the-messy-stuff-is-supposed-to-be.html | Sunday, August 25, 2013
The messy stuff is supposed to be an equation
Let me apologize to those of you who view this blog in various readers (I just viewed this blog in Feedly to see how it renders) -- the equations come out as a mess like this:
$$\log r = \log \frac{Y^0}{\kappa_{LM} LM_{ref}} - \kappa_{LM}\frac{\Delta Y}{Y^0}$$
I use mathjax to render LaTeX, which works great when rendered in a common browser (I highly recommend it). However, it doesn't make it through to some RSS readers.
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Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead. | 2017-08-22 03:13:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.30435553193092346, "perplexity": 1103.1609229486282}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109893.47/warc/CC-MAIN-20170822031111-20170822051111-00643.warc.gz"} |
https://www.nature.com/articles/s41598-021-00789-3?error=cookies_not_supported&code=0c6a881c-a107-4d5e-9ec6-f38c2e6fb937 | Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
# Volunteer contributions to Wikipedia increased during COVID-19 mobility restrictions
## Abstract
Wikipedia, the largest encyclopedia ever created, is a global initiative driven by volunteer contributions. When the COVID-19 pandemic broke out and mobility restrictions ensued across the globe, it was unclear whether contributions to Wikipedia would decrease in the face of the pandemic, or whether volunteers would withstand the added stress and increase their contributions to accommodate the growing readership uncovered in recent studies. We analyze $$\mathbf {223}$$ million edits contributed from 2018 to 2020 across twelve Wikipedia language editions and find that Wikipedia’s global volunteer community responded resiliently to the pandemic, substantially increasing both productivity and the number of newcomers who joined the community. For example, contributions to the English Wikipedia increased by over $$\mathbf {20\%}$$ compared to the expectation derived from pre-pandemic data. Our work sheds light on the response of a global volunteer population to the COVID-19 crisis, providing valuable insights into the behavior of critical online communities under stress.
## Introduction
Wikipedia is the world’s largest encyclopedia, one of the most prominent volunteer-based information systems in existence1,2, and one of the most popular destinations on the Web3. On an average day in 2019, users from around the world visited Wikipedia about 530 million times and editors voluntarily contributed over 870 thousand edits to one of Wikipedia’s language editions (Supplementary Table 1).
Amidst the COVID-19 pandemic and the “infodemic”4 that ensued, Wikipedia played and continues to play an important role in supplying information about the COVID-19 crisis5,6,7,8. Notably, although readers accessed medical articles more frequently than in non-pandemic times9, the increase in readership for all kinds of articles—not only those related to the pandemic—suggests that Wikipedia’s role in this time of crisis transcends mere COVID-19-related information seeking10. However, page views are but a single aspect of the pandemic’s impact on Wikipedia, which ignores the fundamental contribution of editors who perform unpaid volunteer work to maintain and develop content on the website. If the pandemic negatively impacted the productivity and number of editors on Wikipedia, the world’s largest online encyclopedia could be in peril11,12.
We can devise two competing hypotheses on how the COVID-19 crisis may have impacted editors on Wikipedia. First, editing activity on Wikipedia may have declined in response to COVID-19. The negative economic and social ramifications of the pandemic13,14,15 that ensued after governments enforced mobility restrictions16,17,18 may have adversely affected Wikipedia volunteers. Particularly, the challenges associated with this new reality may have led to an overall decrease in contributions and fewer people joining Wikipedia. For example, potential contributors might have focused their efforts on personal issues and on coping with the crisis rather than volunteering for Wikipedia. Alternatively, volunteer contributions and the number of newly recruited editors may have increased due to mobility restrictions resulting in individuals spending more time at home in front of computer screens19 or on the Internet20. Moreover, as previously observed during locally confined disease outbreaks21 and extraordinary events22, the Wikipedia community could have responsed to the heightened demand for high-quality information with increased volunteer activity.
Similar to the importance of understanding the response to COVID-19 in other societal or economic contexts14,15,23,24, we believe that investigating whether volunteer contributions to Wikipedia decreased or increased during this pandemic is critical to assessing the online encyclopedia’s ability to continuously provide information to a global audience of readers, even during worldwide disasters.
After careful quantitative analyses of large-scale edit logs on Wikipedia, we present robust evidence that volunteer contributions significantly increased during the COVID-19 crisis across many language editions. During the pandemic, the Wikipedia editor community not only generated many more edits than what we would expect given historical baselines, but also acquired many more newcomers than in recent history, demonstrating the resilience of this online community in the face of adverse conditions.
Figure 1 depicts the increase in volunteer edits in the English Wikipedia during the COVID-19 timeline in 2020 compared to previous years. Whereas no increase in edit volume was apparent in early 2020, the mobility restrictions in Western countries seemed to first slightly dampen edit activity, before triggering a strong upward trend towards the end of March. In the weeks thereafter, a considerable edit surplus developed in comparison to previous years, which lasted until its peak in late May. As the pandemic subsided over the summer, the growth in edit volume also continuously decreased until fall. By October, the relative increase in edit volume, and thus volunteer contribution, from 2019 to 2020 (about $$7.9\%$$, or 2.1 million edits) was about double that from 2015 to 2019 (about $$4.2\%$$, or 1.5 million edits; see Supplementary Table 2). In summary, a visual representation of edit volume in the English Wikipedia suggests a considerable contribution surplus in 2020.
Beyond the mere descriptive analysis of a single Wikipedia language edition, we systematically analyzed a varied sample of 12 Wikipedia language editions (“Wikipedias”), consisting of four large, medium, and small language editions each (“Methods”), with over 223 million edits spread through 24.6 million articles. In accordance with the descriptive analysis shown in Fig. 1, our quasi-experimental difference-in-differences analysis finds a significant increase in edit volume after COVID-19 mobility restrictions came into effect for many of the Wikipedia editions, and an influx of new editors that is particularly salient for larger Wikipedias. Our study sheds light on the impact of the COVID-19 mobility restrictions on Wikipedia volunteer contributions and provides a reusable framework to measure user activity under stress. More broadly, the evident increase in edit volume and newcomers across most observed Wikipedias is a finding of interest not only to Wikipedia itself but also to researchers and managers of other online collaboration systems, as it provides valuable insight into user behavior during a global crisis.
## Results
### Edit volume during COVID-19 mobility restrictions
We observe an increase in edit volume (the number of edits made by non-bot users) on Wikipedia during the period of COVID-19 mobility restrictions in the spring of 2020, which is particularly evident in large and medium Wikipedias. Figure 2a depicts the rolling 7-day average edit volume for large (top), medium (middle), and small (bottom) Wikipedias in the context of COVID-19 mobility restrictions, which we delineate via automatically detected mobility (i.e., restrictions take effect) and normality (i.e., restrictions are lifted) changepoints (see “Methods”). We also report edit volumes for 2018 and 2019 as a reference for 2020. We observe substantial drops in edit volume around the mobility changepoint for almost all Wikipedias, indicating a shock to the Wikipedia ecosystem. In particular, larger Wikipedias experience a considerable short-lived decrease in edit volume but are able to recover quickly. English, Italian, German, French, Korean, and Japanese even clearly surpass their pre-shock volume levels, leading to an overall edit surplus. On the contrary, some smaller Wikipedias (e.g., Finnish) exhibit a steady decline in edit volume after the mobility changepoint. To better relate edit volume during the COVID-19 pandemic to reference values from previous years and pre-pandemic periods, we employ a difference-in-differences regression (DiD) that controls for the year, period, and language, as well as their interactions. For all Wikipedias, we compute the effective change in edit volume (ev) after the mobility changepoint from the three-way interaction of year, period, and language, and denote this effective change as $$\delta _{ev}$$. We apply the DiD analysis to a sequence of 7-day windows post-changepoint, always retaining the 30-day pre-changepoint period, and plot the time series of logarithmic effects for edit volume according to $$\delta _{ev}$$ in Fig. 2b. We describe this DiD setup in more detail in “Methods”. The DiD analysis validates that all large and most medium Wikipedias significantly increase their edits following the mobility restrictions according to $$\delta _{ev}$$ (95% confidence interval), while no general statement can be made for small Wikipedias.
For the rolling 7-day average edit volume in large Wikipedias, we identify an upward trend in 2020 immediately after mobility restrictions took place (Fig. 2a, top). In the English, French, and Italian language editions, edit volume steadily increases for nearly 2 months after a dip around the date of the mobility changepoint, before slowly reverting to prior levels. The steady initial increase in edit volume leads to high peaks—approximately 120,000 edits for English, 28,000 for French, 15,000 for Italian, and 24,000 for German, which exhibits a decline back to pre-crisis levels earlier than other large Wikipedias. DiD results confirm the edit volume surplus visible in the time series for large Wikipedias in 2020 (Fig. 2b, top). $$\delta _{ev}$$ for French, Italian, and English depicts an immediate relative increase in edits after the mobility restrictions take place, leading to over 100 days of significant increases for all three of these Wikipedias, whereas German declines earlier. Approximately 35 days after the mobility restrictions take effect, French ($$e^{0.337} = 144\%$$, a surplus of $$44\%$$), Italian ($$+42\%$$), and German ($$+25\%$$) reach their highest significant relative increase for edit volume. The higher short-term increases in French, German, and Italian may be related to more detailed reporting of local issues in these language editions. On the contrary, English shows a longer, sustained upward trend for $$\delta _{ev}$$, with a maximum significant increase of $$23\%$$ after 69 days. In conclusion, edit volume significantly increases in large Wikipedias after mobility restrictions come into effect.
Edit volume in most medium and small Wikipedias slightly drops around the respective mobility changepoints in 2020. However, virtually all Wikipedias quickly recover from the initial shock, with most maintaining a stable edit volume in the ensuing weeks and some even generating an edit surplus. While Fig. 2a (middle) shows that medium Wikipedias do not homogeneously increase their edit volume, Korean and Japanese surpass their pre-mobility-restriction levels about a month post changepoint, peaking at about 5400 and 14,500 edits, respectively. For small Wikipedias, edit volume only decreases slightly right after the mobility changepoint (Fig. 2a, bottom). Afterward, edit volume recovers to previous baselines within 30 days, before following similar trends and levels as in previous years. DiD analysis and corresponding values for $$\delta _{ev}$$ reveal that, in fact, medium Wikipedias experience varying periods of significant relative increases in edit volume (Fig. 2b, middle). For example, when compared to pre-pandemic years around the same time period, the Korean and Dutch Wikipedias produce a consistent relative increase (peaking at $$+40\%$$), whereas Swedish and Japanese exhibit shorter significant periods ($$+30\%$$ and $$+38\%$$ in maximum, resp.). Furthermore, the relative change for small Wikipedias (Fig. 2b, bottom) signals brief periods of substantial relative increases for Danish and Norwegian (peaks of $$+69\%$$ and $$+43\%$$, resp.). Most notably, Serbian Wikipedia exhibits a considerable increase during the first month after mobility restrictions take place, with volume nearly tripling (logarithmic effect of 1.03). Lastly, we note that out of our twelve investigated Wikipedias only Finnish shows a significant decrease in $$\delta _{ev}$$ over longer stretches of the observed period. In any case, small and medium Wikipedias are mostly resilient to the initial shock to edit volume triggered by COVID-19, with some even surpassing their pre-pandemic baselines after a few weeks.
### Newcomers during COVID-19 mobility restrictions
We find that all large and medium Wikipedias acquire considerably more newcomers (the number of registered users who made their first edit) for most of the study period, while the remaining Wikipedias exhibited resilience and do not decrease their levels significantly. We visualize the 7-day rolling averages for newcomer counts during the COVID-19 pandemic for large (top), medium (middle), and small (bottom) Wikipedias in Fig. 3a, while also showing values for previous years as well as mobility and normality changepoints. Newcomer counts plummet around the mobility changepoint, in particular for large Wikipedias, but this attenuation in newcomer recruitment only persists for a brief period. Shortly thereafter, newcomer counts increase considerably in all but a few medium and small Wikipedias (e.g., Swedish or Finnish). Again, we build a DiD model for newcomers (nc) to quantify effective changes during the period of COVID-19 mobility restrictions in spring 2020, controlling for year, period, and language. We again perform our DiD analysis for a sequence of 7-day windows after the mobility changepoint (see “Methods”) and show the logarithmic effects for newcomers ($$\delta _{nc}$$) in Fig. 3b. This newcomer DiD analysis confirms that while all large Wikipedias acquire significantly more new editors after mobility restrictions take effect, some medium and small Wikipedias seem to be resilient and exhibit no significant long-term changes ($$95\%$$ CI).
Large Wikipedias appear to recover rapidly from the initial negative effect of mobility restrictions in terms of newcomer counts (Fig. 3a, top). Most notably, Italian Wikipedia registers a newcomer surge until late April, recruiting over 150 newcomers on a rolling 7-day average. English and French show similar patterns of perpetual increases, reaching respective peaks of approximately 2100 and 330 new editors. Although German exhibits nearly 200 newcomers shortly after the mobility changepoint, the surplus in 2020 seems not as considerable as for other large Wikipedias. We further note that newcomer counts for large Wikipedias start to steadily decline in May. However, this seasonal trend also appears to be prevalent in previous years. Our DiD analysis, which captures the change in newcomers via $$\delta _{nc}$$, for the most part confirms these findings (Fig. 3b, top). During the first 2–3 weeks past the mobility changepoint, large Wikipedias steadily recover from the COVID-19 shock without significant overall gains according to $$\delta _{nc}$$. However, right after this recovery phase significant peaks arise for English ($$e^{0.283} = 130\%$$ of previous levels), French ($$138\%$$), and German ($$139\%$$). For the Italian Wikipedia, which belongs to a region with particularly strict mobility restrictions, we confirm an even stronger newcomer surge, leading to a $$80\%$$ relative increase. Furthermore, English generates a notably stable, significant long-term growth in newcomers that is possibly owed to editors from all over the world joining this language edition during mobility restrictions in their regions, as the English Wikipedia serves as a global repository of knowledge. Ultimately, positive effects prevail for large Wikipedias and solidify a newcomer surplus after the mobility restrictions come into force.
Similar to large Wikipedias, most medium and small Wikipedias experience a decline in newcomers right around their mobility changepoints before then increasing their counts to previous baselines (Fig. 3a, middle and bottom). Some of these Wikipedias (e.g., Norwegian, Finnish, Danish, Swedish) recover to previous levels within the first month and exhibit no long-term effects afterwards. However, others recruit a surplus of newcomers during this crisis. Japanese, Dutch, Korean, and Serbian show short-term newcomer influxes about 1–2 months after the initial mobility restrictions take effect, with maximum respective values of approximately 170, 60, 50, and 30 daily newcomers. We also observe these effects in $$\delta _{nc}$$ as captured by DiD (Fig. 3b, middle and bottom), which confirms brief relative increases for Japanese ($$+30\%$$), Dutch ($$+47\%$$), Korean ($$+28\%$$), and Serbian ($$+179\%$$). Finally, the newcomer DiD analysis corroborates that some medium and most small Wikipedias do not significantly deviate from baselines prior to the mobility restrictions over much of the observed time span.
## Discussion
As the COVID-19 pandemic erupted on a global scale, it was unclear how this incisive event would affect Wikipedia’s volunteer community. Over the course of the last few years, both human editing25 and newcomer recruitment26 on Wikipedia have stagnated or even decreased (Supplementary Table 2). Accordingly, the pandemic could have accelerated the decline of the online encyclopedia as the hardships of this global crisis may even further decrease volunteer activity. However, our study, in which we analyze 223 million edits from 12 Wikipedia language editions, reveals that the COVID-19 pandemic and its accompanying mobility restrictions have substantially boosted volunteer activity on Wikipedia. By performing a difference-in-differences analysis, we show that edit volume as well as the influx of newcomers has generally increased after COVID-19 mobility restrictions went into effect. In what follows, we discuss the implications and limitations of this finding.
### Mechanisms behind contribution growth
We observe significant increases in edit volume and newcomers during the COVID-19 pandemic across multiple Wikipedias, making it their most active period in at least the last 3 years. While our quantitative study sheds light on the extent of contribution growth, there are several possible mechanisms behind this effect, which may or may not impact the collaborative structure of editor communities.
Firstly, Wikipedia received significantly more page views during the COVID-19 crisis10. The increase in edits and newcomers may partially be due to the prior increase in Wikipedia readership, as a certain proportion of readers turns into contributors because of various motivational factors27,28. In addition, we theorize that increased screen time and Internet exposure19,20 during the mobility restrictions lead to Wikipedia readers spending more time editing, possibly increasing the reader-to-editor turnover rate. Tracing the transformation of readers into editors during this pandemic in more detail is a promising avenue for future work.
Secondly, the increase in contributions may be due to the rapidly changing information and new knowledge that the COVID-19 pandemic generates about the world. Past literature has suggested that Wikipedia growth is constrained by the amount of knowledge available, as editors have already contributed most of the easily obtainable and verifiable information25. The fact that volunteers have been “running out of easy topics” to contribute to has made it difficult for non-specialists to provide new content with little effort11. As the COVID-19 pandemic dramatically changes the status quo of our world today, it is generating new knowledge about many fields and thus may provide fresh opportunities for both novel and veteran editors to contribute to Wikipedia.
Moreover, the observed edit surplus may have been caused by the high-intensity activity of a core group of editors rather than the broader editor population. We therefore investigate the number of editors active on any given day according to their activity level: 1–4, 5–24, 25–99, or more than 99 daily edits (“Methods”). The DiD analysis for editor counts depicts increases across all activity levels after mobility changepoints for all large and most medium Wikipedias, while small Wikipedias show non-conclusive effects (Supplementary Figs. 1, 2, 3 and 4). This corresponds to an overall increase in active editors during the pandemic (Supplementary Fig. 5), indicating that the general editor population intensified its contribution during the COVID-19 pandemic.
On top of that, we detected a contribution disparity with respect to Wikipedia size, meaning that the smaller Wikipedias we studied did not benefit to the same degree as larger or medium Wikipedias. The observed discrepancy in edit and newcomer increases for large, medium, and small Wikipedias may stem from a difference in community size and structure, or these Wikipedias’ specific rules11,29,30,31,32. This discrepancy could be further reinforced by the culture and perception of what constitutes good knowledge representation in the different Wikipedia language editions33. Moreover, the amount of content for certain topical categories diverges due to cultural contextualization in different language editions34. Specifically, a strong (hypothetical) affinity for topics not directly related to the pandemic (e.g., Sports) in medium or smaller Wikipedias might change the effect of this crisis on their edit volume, in comparison to larger Wikipedias. As an example, in case such Wikipedia language editions focused more on updating sports articles, edit volume would decrease more during the pandemic. The magnitude of such an effect may further depend on a region’s more (e.g., Italy) or less strict (e.g., Sweden) mobility restrictions. Future research may explore language-specific collaboration mechanisms in more detail, for example by attempting to topically analyze Wikipedia contributions during the pandemic.
Lastly, in this study we investigate contribution growth in terms of the number of edits and newcomers, rather than alternative measures such as the total sum of contributed or surviving content. To also touch on this aspect of Wikipedia contributions, we coarsely analyze the contributed information in bytes during COVID-19 mobility restrictions (Supplementary Figure 9). While the surplus in contributed content largely does not match the surplus found for edits and newcomers, we observe an increase for several languages after the imposed mobility changepoints. Furthermore, we find no significant decreases for all Wikipedia editions. It might prove fruitful to extend this quantitative analysis by examining the actual dynamics and structure of the contributed information.
### Resilience of Wikipedia communities
Although we did not find the same surplus in contributions across large, medium, and small Wikipedia language editions, volunteer communities in all studied Wikipedias demonstrated resilience by quickly recovering from the initial negative impact of the pandemic on their contributions. While slow response to negative events or other shocks causes severe problems in social-ecological systems35,36, resilient systems are adaptable and manage to withstand such shocks, even bearing the capacity to cross previous performance thresholds37—a behavior observed in this study. The strongest resilience and subsequent crossing of earlier thresholds in large Wikipedias during the pandemic may be partially explained by the difference in community size38. For example, in larger communities it may not be as problematic that leaders are limited due to the pandemic, as a greater number of other veteran members can take over their work. This conjecture borrows from critical mass theory, in the sense that a critical mass of core members is the fundamental source of content27. Future research might investigate the aspect of Wikipedia resilience during the pandemic in more detail, for example by considering threat rigidity38 or building a model39 that considers COVID-19 as an attack on the community structure.
### Revert rate during COVID-19 mobility restrictions
The observed simultaneous increase in newcomers and edits may indicate that the edit surplus was partially caused by first-time editors. Although past research has suggested that new or one-time editors often produce high-quality contributions40, veteran editors or bots would frequently completely undo (i.e., identity revert) these newcomer revisions, which represents a common behavioral pattern on Wikipedia11,26,41 that in turn generates further revisions. To investigate whether an increase in reverts occurred, we performed a cursory analysis of the revert rate, which is defined as the ratio of reverted edits to edit volume (see “Methods” and Supplementary Information). Supplementary Figure 6a visualizes the rolling 7-day average revert rate, while Supplementary Figure 6b plots the relative change in revert rate as captured by a DiD analysis (“Methods”). It is noteworthy that we only detect a significant increase of the revert rate in one language (Korean). By contrast, several Wikipedias exhibit significantly decreased revert rates shortly after the mobility restrictions come into force. For example, the large Italian, French, and German Wikipedias all show reduced revert rates by about one quarter. This suggests that less valuable revisions, possibly made by newcomers, and their immediate reversal do not cause the reported increase in edit volume. Furthermore, potential misbehavior or conflict on Wikipedia, such as vandalism or edit wars, is prominently characterized by large numbers of identity reverts, as they undo these unwanted contributions42,43,44,45,46. Therefore, reduced revert rates may indicate that editors refrain more from confrontational behavior and thus demonstrate higher levels of solidarity during the pandemic, which is a common phenomena within collectives during crises47. However, a decline in revert rate could also imply that bots and administrators may be unable to keep up with the influx of edits, leaving low quality or malicious edits undetected and thus diminishing quality in the long term. Moreover, differences in revert rates across Wikipedia editions could also be explained by divergent page protection and blocking policies, where either certain groups of editors or even all public editors are prevented from editing articles that are subject to increased attention. We see the detection and analysis of behavioral patterns and collaborative structure of online communities as a promising path for future research. Additionally, it may be valuable to further study the treatment and retention of newcomers26,30,48 during and after the pandemic once more longitudinal data is available.
### Contribution to COVID-19 articles
One might speculate that the increase in edit volume is mostly due to edits in articles that are strongly related to COVID-19. However, many of those articles were protected from public editing early in the pandemic to prevent spread of misinformation49, and we find that only a negligibly small fraction of edits (at most $$1\%$$ for most Wikipedias) goes towards articles with a primary focus on COVID-19 (see “Methods”) between January 1st and September 31st 2020 (Supplementary Table 4; Supplementary Fig. 7). A clear outlier in that regard is German, where $$2.5\%$$ of edits performed in 2020 by the end of September concern themselves with such articles. This may indicate higher coverage of local COVID-19 outbreaks in German than in other languages. We consequently repeat our DiD analysis for edit volume, this time excluding edits to articles strongly related to COVID-19 (Supplementary Fig. 8). The results support the previous findings and confirm that the reported edit volume increase is not due to COVID-19 articles. In this way, our work extends previous studies, which focused on a smaller subset of pandemic-related articles7,49.
### Other limitations
Even though our work covers a large portion of Wikipedia’s content and editor population, it comes with several limitations. First, we do not consider a variety of different Wikipedias associated with languages widely spoken in the Global South, including Spanish, Portuguese, Arabic, Hindi, or any African Wikipedias (see “Methods” for how we chose language editions). Future work analyzing these Wikipedias could improve our understanding of the impact of the pandemic on volunteer contribution in other parts of the world. Second, content on Wikipedia is predominantly edited by white males between the ages of 17 and 4050,51. It may be that the COVID-19 crisis has disparately impacted contributors of less represented demographics, as certain racial or socioeconomic groups are particularly disadvantaged by the pandemic23,24,52. In addition, bots have an important role in the creation and management of Wikipedia content41,53. We excluded bots from our analysis as we specifically focused on edits performed by human volunteers. Nevertheless, other studies may choose to consider bot activities as valid contributions to Wikipedia.
In conclusion, our study provides evidence for a substantial surplus of volunteer contributions to multiple Wikipedia language editions during COVID-19 mobility restrictions, which shines light on the resilience of the Wikipedia community under times of stress. The methodological framework used in this work can easily be adapted for similar domains. We believe that our work provides valuable insights into contributor behavior on online platforms during the COVID-19 pandemic and illustrates a plethora of possibilities for future work.
## Methods
### Data procurement and preprocessing
We utilize the openly available MediaWiki history dumps dataset54 to analyze a varied sample of 12 Wikipedia language editions (“Wikipedias”).
#### Wikipedia language editions
We investigate 12 Wikipedias (Supplementary Table 3), consisting of languages primarily spoken in European countries that were exposed to the outbreak of COVID-19 in the spring of 2020, as well as two Asian Wikipedias. Our choice of language editions takes into consideration: (1) the size of the Wikipedia edition, (2) whether the language is spoken in relatively few countries, and (3) the mobility restrictions imposed in these countries—three criteria that are often very difficult to simultaneously satisfy. Overall, we aim to capture relevant Wikipedias that represent different attitudes towards the crisis, preferably from languages easily attributable to a single country or region. Accordingly, our sample contains regions with strict (e.g., Italian, Serbian, or French) and less stringent mobility restrictions (e.g., Japanese, Korean, or Swedish). Although it can not be attributed to a single country and about half of the contributions come from editors who are not based in English speaking regions55, we include the English Wikipedia because it is the largest language edition and is considered a global project. We employ the number of edits in 2019 as a metric to categorize the 12 Wikipedias we studied as either large (English, French, German, Italian, with more than 5 million edits), medium (Swedish, Korean, Japanese, Dutch, with 1.5 million to 5 million edits), or small (Serbian, Norwegian, Danish, Finnish, with less than 1.5 million edits).
Our work only covers an arbitrary part of the Wikipedia ecosystem, as we have argued above. To allow extension of our analysis to other Wikipedia language editions or Wikimedia projects (e.g., Wikidata, Wikimedia Commons, or Wiktionary), we make our code publicly available (see Code availability). As an example, we have provided a demonstration in our GitHub repository for adapting our code pipeline to explore Wikipedia editions covering large parts of Eastern Europe (Polish, Czech, Ukrainian, and Russian Wikipedia).
#### MediaWiki history dataset dumps
We retrieve the monthly updated MediaWiki history dataset dumps54 provided by the Wikimedia Foundation (WMF) and perform additional preprocessing before computing as well as plotting our results. The denormalized MediaWiki history dumps are generated from the full history logs stored in the WMF’s MediaWiki databases. During their generation, WMF’s automatic scripts reconstruct and enrich user and page history with additional data, and also automatically validate the dumps to prevent errors. After WMF’s preprocessing, the dataset contains fields with precomputed standard metrics, such as revert information, bot users, number of user contributions, or time since a user’s last revision. Overall, each entry in the dump consists of 70 fields with event information. Fields are grouped into entities, bearing information about either revision, page, or user.
#### Preprocessing
In the MediaWiki history dataset, we only consider edits to articles by excluding all pages not in the Wikipedia article namespace (“ns0”), thus removing revisions to talk pages or other content. Furthermore, we utilize corresponding dataset fields to distinguish human editors (anonymous or registered) from bots and mark certain revisions as reverts. Moreover, we convert MediaWiki history timestamps from Coordinated Universal Time (UTC) to the timezone of the local Wikipedia language edition. For Wikipedias in which languages can not be attributed to a single timezone (e.g., French), we choose the timezone with the highest volunteer population for the given Wikipedia. We do not apply timestamp conversion for the English Wikipedia. Lastly, we detect articles which are strongly related to COVID-19 via an algorithm by Diego Sáez-Trumper56, which recognizes COVID-19 articles based on their Wikidata57 links to the main COVID-19 pages.
### Metrics
To make sense of which exact data fields in the MediaWiki history dumps we utilize to compute our metrics, please refer to the code repository (see Code availability).
#### Edit volume
We define edit volume as the number of daily revisions to pages in the article namespace (“ns0”) by non-bot users (anonymous or registered).
#### Newcomers
For each Wikipedia language edition and day, we specify the amount of newcomers as the number of registered editors which perform their first article edit in that language edition on the given day. Through recognizing new editors by their first edit, we measure the exact day they become a contributor in a language edition. Note that the number of daily registered users is generally much higher than the number of newcomers as computed in this work. However, as our study aims to quantify volunteer contribution, we choose to identify newcomers by their first actual contribution in a given Wikipedia.
#### Revert rate
Editors and bots revert article revisions to undo changes which they deem unwarranted. Frequently, these reverts correct revisions which arise from conflicts, edit wars, or vandalism42. Additionally, literature shows that revisions by newcomers are more likely to be reverted than those of veteran editors26. For this research, we only consider reverts to articles that undo all changes and subsequently create a new revision which exactly matches a previous article version (i.e., identity reverts). We calculate the daily revert rate by dividing the number of identity reverts (by humans or bots) by the number of non-bot edits on this given day. Correspondingly, revert rate relates the amount of reverts to the amount of human contribution.
#### Daily editors by activity level
We measure daily active editors in a Wikipedia by counting the number of registered, non-bot users which perform revisions in the article namespace. In addition, to detect effects across the editor population, we collect data for multiple activity levels, keeping count of how many editors perform 1–4, 5–24, 25–99, or more than 99 daily edits. In contrast to other metrics, we do not compute the number of daily editors from the Wikimedia history dumps, but retrieve it via the Wikimedia REST API instead58.
#### Contributed information in bytes
As a parsimonious representation of the amount of information editors contribute to Wikipedia, we measure the daily changes in bytes to all articles in each language edition by summing up the difference in bytes between each revision and its parent revision. We include bot edits to account for the automated correction of vandalistic content additions and removals, which would otherwise distort the results. Although this metric does not provide information about the actual value (or longevity) of the contributions, it does provide an overview of the editors’ workload on any given day.
### Changepoint detection
We adopt the approach by Horta Ribeiro et al.10 to detect mobility and normality changepoints via Google and Apple mobility reports, as detailed in Supplementary Figure 10. The mobility reports capture population-wide movement patterns based on cellphone location signals and specify, on a daily basis, the percentage of time spent in variety of locations, for example residential areas, workplaces, or retail (https://www.apple.com/covid19/mobility; https://www.google.com/covid19/mobility). Government-mandated lockdowns and self-motivated social distancing measures manifest themselves as sharp changes in these mobility time series. To detect changepoints in mobility, the approach consists of a simple binary segmentation algorithm59. For Wikipedias of languages widely spoken across many countries (e.g. English, German, etc.), we determine a changepoint by aggregating mobility reports for the countries in which the language is official with weights proportional to the population of each of these countries. Notice that the link between Wikipedia and language editions is merely approximate—in particular for English, which is accessed from all over the world. We use the changepoints at which mobility drops as heuristics for dates when people started spending substantially more time in their homes and term them mobility changepoints. To detect normality changepoints, we compute the point in time for which the future average mobility remains within a 10% band around baseline levels before the initial mobility changepoint (defined as pre-pandemic mobility levels by Google and Apple). For languages spoken across multiple countries, we maintain the same aggregation scheme as before. Compared to choosing specific dates, this changepoint detection approach leads to more comparable treatments across different regions. Supplementary Table 3 summarizes the detected changepoints, which we also make available in our code repository.
While we solely utilize the normality changepoints as visual guidance in the figures, our employed causal inference strategy described below strongly relies on the mobility changepoints. Therefore, we perform a sensitivity analysis with varying mobility changepoints for all main results. We vary the mobility changepoints by $$\pm 7$$ days and also consider a distinct method of aggregation for languages spoken in multiple countries (Supplementary Fig. 10). As the latter method yields a maximum difference of 5 days across all languages, we only report results for robustness experiments using the 7-day difference.
### Difference-in-differences setup
To compare values of metrics during the COVID-19 pandemic with reference values from previous years and pre-pandemic periods, we employ a difference-in-differences regression (DiD). DiD analysis allows us to quantify changes in these metrics in multiple Wikipedia language editions around times of region-specific mobility changepoints in early spring, while controlling for (long-term) temporal trends.
Our basic DiD equation models a dependent variable’s value (V) as a function of the independent variables year (Y), period (P), and Wikipedia language (L), as well as their interactions. Year is a binary variable which differentiates between pre-pandemic (2018 and 2019) and pandemic years (2020), whereas period encodes the treatment period via a binary variable, in our case represented by the pre- and post-phases of the region-specific mobility changepoints. Lastly, we model our 12 Wikipedia language versions with a categorical variable to control for language-specific effects. To account for outliers and normalize regression results across various-sized Wikipedias, we use logarithmic scales for V. Literature often refers to our setup, which uses three independent variables, as “triple-difference” or “difference-in-difference-in-difference” estimators60,61. Mathematically, our DiD setup is:
\begin{aligned} V ~=~\beta _{0}+ \varvec{\beta _{1}}^\top\varvec{L} + \beta _{2}Y + \beta _{3}P + \varvec{\beta _{4}}^\top(Y \varvec{L}) + \varvec{\beta _{5}}^\top(P \varvec{L}) + \beta _{6}(Y P) + \varvec{\beta _{7}}^\top(Y P \varvec{L}) + \varepsilon \end{aligned}
(1)
We depict the 12 Wikipedia language versions as a vector of 11 binary indicators ($$\varvec{L}$$). Scalar coefficients ($$\beta _0$$, $$\beta _2$$, $$\beta _3$$, $$\beta _6$$) describe effects for the reference language (i.e., baseline). Coefficient vectors ($$\varvec{\beta _{1}}$$, $$\varvec{\beta _{4}}$$, $$\varvec{\beta _{5}}$$, $$\varvec{\beta _{7}}$$, printed in bold) collect language-specific effects of non-baseline Wikipedias. Lastly, $$\varepsilon$$ is the normally distributed residual. Given this mathematical formulation, the coefficient $$\varvec{\beta _{7}}$$ captures the change in V post mobility changepoint relative to the baseline Wikipedia, after accounting for differences stemming from year or period alone ($$\varvec{\beta _{4}}$$ and $$\varvec{\beta _{5}}$$, resp.). We therefore compute the effect of interest for all Wikipedias via summation of $$\beta _6$$ and $$\varvec{\beta _7}$$. For each Wikipedia, we term this effective change in V as $$\delta _{m}$$, where m stands for the metric representing the dependent variable.
#### Interpretation of DiD coefficients
We now elaborate in more detail on how to interpret the coefficients of our DiD model. We model the categorical language variable via vector $$\varvec{L}$$ containing 11 binary indicator variables for the 12 Wikipedias. As is customary, the regression utilizes a “reference Wikipedia” baseline. In our setup, we arbitrarily choose Danish as the baseline. Consequently, $$\varvec{\beta _1}$$ describes the respective difference between the baseline Wikipedia and the 11 non-baseline Wikipedias using indicator variables. Thus, adding $$\beta _0$$ and $$\varvec{\beta _1}$$ yields the intercept of each language’s sub-model.
The binary year variable (Y) indicates whether a data point lies in 2020 ($$=1$$) or in the previous two pre-pandemic years ($$=0$$), regardless of period. As Danish represents the arbitrary baseline, the corresponding coefficient $$\beta _2$$ is a scalar which describes the overall change between the pre-pandemic years (2018 and 2019) and 2020 for Danish. For non-baseline Wikipedias, the interaction $$Y\varvec{L}$$ models the language-specific effects for the change in years relative to the baseline Wikipedia and is quantified by the corresponding coefficient vector $$\varvec{\beta _4}$$. Therefore, the summation of $$\beta _2$$ and $$\varvec{\beta _{4}}$$ is equal to the effective overall difference of 2020 to the previous 2 years for all Wikipedias.
We model seasonal differences between pre- and post-changepoint windows via the binary period indicator (P). The corresponding scalar coefficient ($$\beta _3$$) measures the difference between before and after the mobility changepoint over all years for the baseline. Consequently, $$P\varvec{L}$$ and coefficient vector $$\varvec{\beta _5}$$ describe the period effect for non-baseline Wikipedias in relation to the baseline. Calculating the sum of $$\beta _3$$ and $$\varvec{\beta _5}$$ then gives the total pre- and post-changepoint effects.
Lastly, the interaction between year and period (YP) enables our model to capture the change in V for the baseline Wikipedia via $$\beta _6$$, after accounting for change in Y (via $$\beta _2$$) and P (via $$\beta _3$$) alone. To measure this effective change for all Wikipedias, we employ the coefficient vector $$\varvec{\beta _7}$$ of the three-way interaction $$YP\varvec{L}$$. While $$\beta _6$$ describes the baseline’s effect, $$\varvec{\beta _7}$$ contains the aforementioned change relative to the baseline Wikipedia. Therefore, the sum of $$\beta _6$$ and $$\varvec{\beta _7}$$ captures the effective change in V for all Wikipedias. For a single Wikipedia and metric m, we name this effect of interest $$\delta _{m}$$. Correspondingly, $$\delta _{m}$$ describes language-specific post-changepoint effects in 2020, as it excludes differences that are due to year or period alone.
### Quantifying changes in volunteer contribution
Wikipedia is a dynamic ecosystem, in which edit behavior and the amount of volunteer contribution can change rapidly—especially in times of turmoil. To track these changes and detect short-, medium-, and long-term effects of mobility restrictions on volunteer contributions, we fit our statistical model on different data-points obtained from the same longitudinal dataset. This methodology, pioneered by Gelman and Huang62, allows us to observe trends rather than mere point estimates.
We compute our DiD analysis for a sequence of post-changepoint windows, always retaining the Wikipedias’ pre-changepoint periods. For each language version, we choose a fixed 30-day period before the respective mobility changepoint as the pre-changepoint baseline. As post-changepoint analysis intervals, we then extract a sequence of 120 overlapping left-aligned 7-day windows starting with the changepoints. Mathematically, we set the treatment period to days $$\{n , n + 1, \ldots , n + 6\}, \forall n \in \{0, 1, \ldots , 119\}$$. For each post-changepoint window n, we perform a separate DiD analysis across all languages using the retained baseline periods. By doing so, each DiD analysis compares the week starting at day n after the language-specific changepoint to the baseline periods. In this default setup, each of the 12 Wikipedias is represented by 37 data points for every year in the DiD regression (2018, 2019, and 2020), yielding a total of 1332 data points ($$=(30 \text { pre-changepoint days} + 7 \text { post-changepoint days}) \times 3 \text { years}\times 12 \text { Wikipedias}$$) for each of the 120 experiments. For each Wikipedia, we conservatively detect outliers via the Median Absolute Deviation (MAD) approach63 with a threshold of $$5*\text {MAD}$$ from the monthly median and replace such outliers by the monthly median. We then build a time series of the 120 DiD results using $$\delta _{m}$$ and approximate the 95% two-sided confidence intervals (CI) as two standard errors. As robustness checks, we compute variations of our DiD experiments with wider window size ($$14\ \text {days}$$) and slightly varied mobility changepoint dates ($$\pm 7$$ days) as described in Supplementary Information (Supplementary Figs. 11, 12, 13, 14, 15, 16, 17, 18, 19). These results corroborate the findings reported under “Results”.
#### Identifying assumptions of DiD estimators
Triple-difference and difference-and-difference models in general underlie strong identifying assumptions. In particular, our methodology is based on the assumption that the difference between treatment and control group would stay constant without an intervention64. Additionally, the condition of parallel trend in slopes prior to the treatment must be fulfilled65. For our model, we therefore assume that for each Wikipedia language version, the difference between treated (i.e., 2020) and control years (i.e., 2018 and 2019) would remain constant in the absence of an intervention (i.e., mobility changepoint). Furthermore, our model trivially fulfills the parallel trend condition: Given that our model only considers a single period before ($$P=0$$) and after ($$P=1$$) the intervention, testing for parallel trends prior to the intervention64 results in failing to reject the null hypothesis (that there are no violations of parallel trends). In other words, as these single pre-intervention data points do not produce trends prior to the changepoints, they do not, by definition, deviate significantly from parallelism. Finally, as explained above, we fit several DiD estimators to derive longitudinal intervention effects, which provides additional robustness with respect to the assumptions of such models.
## Data availability
The openly accessible MediaWiki history dataset dumps are available at https://dumps.wikimedia.org/other/mediawiki_history/readme.html54. We further provide preprocessed data and results relevant to the manuscript in the code repository at https://github.com/ruptho/wiki-volunteers-covid.
## Code availability
The code repository for this paper can be found at https://github.com/ruptho/wiki-volunteers-covid.
## References
1. Giles, J. Internet Encyclopaedias Go Head to Head (Nature Publishing Group, 2005).
2. Lemmerich, F., Sáez-Trumper, D., West, R. & Zia, L. Why the world reads Wikipedia: Beyond english speakers. In Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining (2019), pp. 618–626.
3. Alexa. The top 500 sites on the Web. https://www.alexa.com/topsites. Accessed 07 Jan 2021 (2021).
4. Gallotti, R., Valle, F., Castaldo, N., Sacco, P., & De Domenico, M. Assessing the risks of infodemics in response to COVID-19 epidemics. arXiv:2004.03997 (arXiv preprint) (2020).
5. Colavizza, G. COVID-19 research in Wikipedia. bioRxiv (2020).
6. World Health Organization. The world health organization and wikimedia foundation expand access to trusted information about COVID-19 on Wikipedia (2020).
7. Gozzi, N., et al. Collective response to the media coverage of COVID-19 pandemic on Reddit and Wikipedia. arXiv:2006.06446 (arXiv preprint) (2020).
8. Wikimedia Foundation. Responding to COVID-19: How we can help in this time of uncertainty (2020).
9. Chrzanowski, J., Sołek, J., Fendler, W. & Jemielniak, D. Assessing public interest based on Wikipedia’s most visited medical articles during the SARS-CoV-2 outbreak: Search trends analysis. J. Med. Internet Res. 23(4), e26331 (2021).
10. Horta Ribeiro, M. et al. Sudden attention shifts on wikipedia during the covid-19 crisis. Proc. Int. AAAI Conf. Web Soc. Media 15(1), 208–219 (2021).
11. Halfaker, A., Geiger, R. S., Morgan, J. T. & Riedl, J. The rise and decline of an open collaboration system: How Wikipedia’s reaction to popularity is causing its decline. Am. Behav. Sci. 57(5), 664–688 (2013).
12. Ransbotham, S. & Kane, G. Membership turnover and collaboration success in online communities: Explaining rises and falls from grace in Wikipedia. MIS Q. 35, 613–628 (2011).
13. Bloom, N., et al. The impact of COVID-19 on productivity. Working Paper 28233, National Bureau of Economic Research, December 2020.
14. Chetty, R., Friedman, J. N., Hendren, N., Stepner, M., & Team, T. O. I. The economic impacts of COVID-19: Evidence from a new public database built using private sector data. Working Paper 27431, National Bureau of Economic Research, June 2020.
15. Suh, J., Horvitz, E., White, R. W. & Althoff, T. Population-scale study of human needs during the COVID-19 pandemic: Analysis and implications. arXiv:2008.07045 (arXiv preprint) (2020).
16. Desvars-Larrive, A. et al. A structured open dataset of government interventions in response to COVID-19. medRxiv 20, 20 (2020).
17. Flaxman, S. et al. Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe. Nature 584(7820), 257–261 (2020).
18. Zheng, Q. et al. HIT-COVID, a global database tracking public health interventions to COVID-19. Sci. Data 7(1), 1–8 (2020).
19. Sultana, A., Tasnim, S., Bhattacharya, S., Hossain, M. M. & Purohit, N. Digital screen time during COVID-19 pandemic: A public health concern. osf.io/preprints/socarxiv/e8sg7 (2020).
20. Feldmann, A., et al. The lockdown effect: Implications of the COVID-19 pandemic on internet traffic. In Proceedings of the ACM Internet Measurement Conference (2020), pp. 1–18.
21. Al Tamime, R., Giordano, R., & Hall, W. Observing burstiness in Wikipedia articles during new disease outbreaks. In Proceedings of the 10th ACM Conference on Web Science (2018), pp. 117–126.
22. Zhang, A. F. et al. Participation of new editors after times of shock on Wikipedia. Proc. Int. AAAI Conf. Web Soc. Media 13, 560–571 (2019).
23. Atchison, C. J. et al. Perceptions and behavioural responses of the general public during the COVID-19 pandemic: A cross-sectional survey of UK adults. medRxiv 20, 20 (2020).
24. Jay, J. et al. Neighbourhood income and physical distancing during the COVID-19 pandemic in the United States. Nat. Human Behav. 2, 1–9 (2020).
25. Suh, B., Convertino, G., Chi, E. H. & Pirolli, P. The singularity is not near: Slowing growth of Wikipedia. In Proceedings of the 5th International Symposium on Wikis and Open Collaboration (New York, NY, USA, 2009), WikiSym ’09, Association for Computing Machinery.
26. Halfaker, A., Kittur, A., & Riedl, J. Don’t bite the newbies: How reverts affect the quantity and quality of Wikipedia work. In Proceedings of the 7th international symposium on wikis and open collaboration (2011), pp. 163–172.
27. Xu, B. & Li, D. An empirical study of the motivations for content contribution and community participation in Wikipedia. Inform. Manag. 52(3), 275–286 (2015).
28. Shaw, A. & Hargittai, E. The pipeline of online participation inequalities: The case of Wikipedia editing. J. Commun. 68(1), 143–168 (2018).
29. Yasseri, T., Sumi, R. & Kertész, J. Circadian patterns of Wikipedia editorial activity: A demographic analysis. PLoS One 7(1), e30091 (2012).
30. Robert, L. P. Jr. & Romero, D. M. The influence of diversity and experience on the effects of crowd size. J. Am. Soc. Inf. Sci. 68(2), 321–332 (2017).
31. Butler, B., Joyce, E. & Pike, J. Don’t look now, but we’ve created a bureaucracy: The nature and roles of policies and rules in Wikipedia. In Proceedings of the SIGCHI Conference on human Hactors in Computing Systems (2008), pp. 1101–1110.
32. Keegan, B. & Fiesler, C. The evolution and consequences of peer producing Wikipedia’s rules. Proc. Int. AAAI Conf. Web Soc. Media 11, 1 (2017).
33. Jemielniak, D. & Wilamowski, M. Cultural diversity of quality of information on wikipedias. J. Assoc. Inf. Sci. Technol. 68(10), 2460–2470 (2017).
34. Miquel-Ribé, M. & Laniado, D. Cultural identities in Wikipedias. In Proceedings of the 7th 2016 International Conference on Social Media and Society (2016), pp. 1–10.
35. Scheffer, M., Westley, F. & Brock, W. Slow response of societies to new problems: Causes and costs. Ecosystems 6(5), 493–502 (2003).
36. Scheffer, M. & Westley, F. R. The evolutionary basis of rigidity: Locks in cells, minds, and society. Ecol. Soc. 12, 2 (2007).
37. Folke, C. et al. Resilience thinking: Integrating resilience, adaptability and transformability. Ecol. Soc. 15, 4 (2010).
38. Zhang, A. F., Livneh, D., Budak, C., Robert, L. & Romero, D. Shocking the crowd: The effect of censorship shocks on Chinese Wikipedia. Proc. Int. AAAI Conf. Web Soc. Media 11, 1 (2017).
39. Xiong, W. & Lagerström, R. Threat modeling—systematic literature review. Comput. Secur. 84, 53–69 (2019).
40. Anthony, D., Smith, S. W. & Williamson, T. Reputation and reliability in collective goods: The case of the online encyclopedia wikipedia. Ration. Soc. 21(3), 283–306 (2009).
41. Zheng, L., Albano, C. M., Vora, N. M., Mai, F. & Nickerson, J. V. The roles bots play in Wikipedia. Proceedings of the ACM on Human–Computer Interaction 3, CSCW (2019), 1–20.
42. Yasseri, T., Sumi, R., Rung, A., Kornai, A. & Kertész, J. Dynamics of conflicts in Wikipedia. PLoS One 7(6), e38869 (2012).
43. Sumi, R., Yasseri, T., et al. Edit wars in Wikipedia. In 2011 IEEE Third International Conference on Privacy, Security, Risk and Trust and 2011 IEEE Third International Conference on Social Computing (2011), IEEE, pp. 724–727.
44. Kittur, A., Suh, B., Pendleton, B. A. & Chi, E. H. He says, she says: Conflict and coordination in Wikipedia. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (2007), pp. 453–462.
45. Borra, E. et al. Societal Controversies in Wikipedia Articles 193–196 (Association for Computing Machinery, 2015).
46. Brandes, U., et al. Network analysis of collaboration structure in Wikipedia. In WWW ’09 (New York, NY, USA, 2009), Association for Computing Machinery, pp. 731–740.
47. Garcia, D. & Rimé, B. Collective emotions and social resilience in the digital traces after a terrorist attack. Psychol. Sci. 30(4), 617–628 (2019).
48. Ciampaglia, G. L., & Taraborelli, D. MoodBar: Increasing new user retention in Wikipedia through lightweight socialization. In Proceedings of the 18th ACM Conference on Computer Supported Cooperative Work and Social Computing (2015), pp. 734–742.
49. Keegan, B. C., & Tan, C. A quantitative portrait of Wikipedia’s high-tempo collaborations during the 2020 coronavirus pandemic. arXiv:2006.08899 (arXiv preprint) (2020).
50. Hill, B. M. & Shaw, A. The Wikipedia gender gap revisited: Characterizing survey response bias with propensity score estimation. PLoS One 8(6), e65782 (2013).
51. Collier, B., & Bear, J. Conflict, criticism, or confidence: An empirical examination of the gender gap in Wikipedia contributions. In Proceedings of the ACM 2012 Conference on Computer Supported Cooperative Work (New York, NY, USA, 2012), CSCW ’12, Association for Computing Machinery, pp. 383–392.
52. Chang, S. et al. Mobility network models of COVID-19 explain inequities and inform reopening. Nature 2, 1–6 (2020).
53. Tsvetkova, M., García-Gavilanes, R., Floridi, L. & Yasseri, T. Even good bots fight: The case of Wikipedia. PLoS One 12(2), e0171774 (2017).
54. Wikimedia Foundation. Analytics datasets: Mediawiki history. https://dumps.wikimedia.org/other/mediawiki_history/readme.html. Accessed 13 Dec 2020 (2020).
55. Wikimedia Statistics. Active editors by country with 5 to 99 edits (2021).
56. Diego Sáez-Trumper. COVID-19 Wikipedia data. https://covid-data.wmflabs.org. Accessed 13 Dec 2020 (2020).
57. Vrandečić, D. & Krötzsch, M. Wikidata: A free collaborative knowledgebase. Commun. ACM 57(10), 78–85 (2014).
58. Wikimedia Foundation. REST API Documentation. https://wikimedia.org/api/rest_v1/. Accessed 1 Jan 2021 (2021).
59. Truong, C., Oudre, L. & Vayatis, N. Selective review of offline change point detection methods. Signal Process. 167, 107299 (2020).
60. Gruber, J. The incidence of mandated maternity benefits. Am. Econ. Rev. 2, 622–641 (1994).
61. Mian, A. & Sufi, A. House prices, home equity-based borrowing, and the US household leverage crisis. Am. Econ. Rev. 101(5), 2132–56 (2011).
62. Gelman, A. & Huang, Z. Estimating incumbency advantage and its variation, as an example of a before-after study. J. Am. Stat. Assoc. 103(482), 437–446 (2008).
63. Leys, C., Ley, C., Klein, O., Bernard, P. & Licata, L. Detecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median. J. Exp. Soc. Psychol. 49(4), 764–766 (2013).
64. Bilinski, A., & Hatfield, L. A. Nothing to see here? Non-inferiority approaches to parallel trends and other model assumptions (2020).
65. Olden, A. & Møen, J. The triple difference estimator. NHH Dept. of Business and Management Science Discussion Paper, 2020/1 (2020).
## Acknowledgements
Supported by TU Graz Open Access Publishing Fund. R.W.'s lab was partly funded by the Swiss National Science Foundation (grant 200021_185043), the European Union (TAILOR, grant 952215), Collaborative Research on Science and Society, and by generous gifts from Microsoft, Facebook, and Google.
## Author information
Authors
### Contributions
T.R. retrieved the dataset, processed the data, and performed the experiments. T.R. and M.H.R. wrote the code. T.R., M.H.R., and T.S. analyzed the data. T.R., M.H.R., T.S., F.L., M.S., R.W., and D.H. conceived and designed the experiments, developed the arguments, and wrote the paper.
### Corresponding author
Correspondence to Thorsten Ruprechter.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
### Publisher's note
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## Rights and permissions
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Ruprechter, T., Horta Ribeiro, M., Santos, T. et al. Volunteer contributions to Wikipedia increased during COVID-19 mobility restrictions. Sci Rep 11, 21505 (2021). https://doi.org/10.1038/s41598-021-00789-3
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• DOI: https://doi.org/10.1038/s41598-021-00789-3 | 2022-06-25 05:52:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "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": 0.5312350988388062, "perplexity": 4257.329388801002}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103034170.1/warc/CC-MAIN-20220625034751-20220625064751-00179.warc.gz"} |
https://www.cp2k.org/exercises:2014_ethz_mmm:t_melting | # Open SourceMolecular Dynamics
### Sidebar
#### For Developers
exercises:2014_ethz_mmm:t_melting
# Determination of the melting temperature of copper
In this exercise, we will use a slab geometry (without vacuum region, so without a surface) with full periodic boundary conditions to study the melting behavior of copper. As usual, connect to brutus and enter the following commands:
module load cp2k/trunk.2.5.13191
module load open_mpi/1.6.5 ! THIS IS NEEDED IF YOU WANT TO RUN AN MPI PARALLEL RUN --- NOT CONVENIENT IN THIS CASE
mkdir EX_5.1
cd EX_5.1
Copy into that directory the COMMENTED files that can be downloaded from the wiki: exercise_5.1.tar.gz
Now, run the first simulation, that should melt your system.
bsub cp2k.popt -i half.inp > half.out
It is a 3000 step molecular dynamics. During this time (about 20 minutes) you can complete the first assignments.
• A1) Take a look at the file 111.xyz with vmd. Visualize it on the screen, and try to reproduce the figure similar to the one on the last slide of the lectures of today. Include the pbc box, create a representation with vdw, periodic images, rotate the sample, etc. Produce a snapshot and include the file in your assignment.
• A2) Take a look at the half.inp file. How is the temperature controlled? Are all particles moving? Why? Which are the relevant sections for MD? Which kind of MD is it?
• A3) Plot the -growing- half*ener file with gnuplot. How is temperature changing? Is there a conserved quantity?
At the end of the first dynamics (hint: tail -f half*ener) , you can examine the half-pos-1.xyz file by performing z-profiles using the script doprof .
./doprof half-pos-1.xyz
The script calls the histogram script of last time, with a modification: a running window of configurations is averaged to produce a single frame. First, step 1-10, then step 10-20, and so on. At the end, the file movie.half-pos-1.xyz.gif, an animated gif is produced. If it works, you can run the command:
animate -loop 0 -delay 100 movie.half-pos-1.xyz.gif
or download the file to your local machine and open in your internet browser. It will run the animation.
A4) Describe the profile you have obtained. What do you see?
Now, starting from the restart of this simulation, we equilibrate the system in nve, and we move all particles:
bsub cp2k.popt -i 1400nve.inp > 1400nve.out
The resulting configuration (check) will be an equilibrated system (which profile?).
Now we have a file called 1400nve-1.restart
THIS WILL BE USED AS RESTART FILE FOR ALL SIMULATIONS! DO NOT DELETE IT!
## SIMULATIONS AT DIFFERENT TOTAL ENERGIES FOR DETERMINING THE MELTING TEMPERATURE
As explained in the class, we will run NPE (that is, constant energies but variable cell) simulations at energies which are above and below the supposed “melting energy” (energy corresponding to melting temperature).
THE TEMPERATURE WILL NOT BE CONTROLLED DURING THE RUN
For EACH temperature you have to:
• A5) Copy the files TEMPnpe.init.inp and TEMPnpe.inp into 1300npe.init.inp and 1300npe.inp (for T=1300) and then edit them in the appropriate points: PROJECT name, INITIAL temperature and RESTART filename.
• A6) Run the first simulation: bsub cp2k.popt -i 1300npe.init.inp > 1300npe.init.out ; This is a very short simulation to set the temperature using the old velocities. Why do you need it?
• A7) Run the second simulation: bsub cp2k.popt -i 1300npe.inp > 1300npe.out
• Observe the temperature and the z profile. Can you find the melting temperature? How do you choose temperatures?
Note that you can run several A5-A7 steps at the same time and in the same directory.
And finally…
A8) WHAT IS THE MELTING TEMPERATURE OF THIS POTENTIAL (APPROXIMATELY)?
exercises/2014_ethz_mmm/t_melting.txt · Last modified: 2014/10/15 13:29 by oschuett | 2019-06-20 23:32:12 | {"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": 0.6316922903060913, "perplexity": 4058.10720705618}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999291.1/warc/CC-MAIN-20190620230326-20190621012326-00479.warc.gz"} |
https://www.transtutors.com/questions/given-the-potential-field-v-100xz-x-2-4-v-in-free-space-a-find-d-at-the-surface-z-0--2007427.htm | # Given the potential field V = 100xz/(x 2 + 4) V in free space: (a) Find D at the surface z = 0.... 1 answer below »
Given the potential field V = 100xz/(x2 + 4) V in free space: (a) Find D at the surface z = 0. (b) Show that the z = 0 surface is an equipotential surface. (c) Assume that the z = 0 surface is a conductor and find the total charge on that portion of the conductor defined by 0
## 1 Approved Answer
Hoa
5 Ratings, (9 Votes)
a) We have
$$E=-\frac\partial V\partial xa_x-\frac\partial V\partial ya_y-\frac\partial V\partial za_z \\E=-100z\frac\partial\partial x(\fracxx^2+4)-0a_y-\frac10xx^2+4a_z$$
At z=0,
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Ask Now | 2021-04-20 04:38:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6156060695648193, "perplexity": 1970.003203024397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039375537.73/warc/CC-MAIN-20210420025739-20210420055739-00142.warc.gz"} |
http://math.stackexchange.com/questions/25867/the-number-of-young-tableaux-in-general | # the number of Young tableaux in general
From the wiki page Catalan number, we know the number of Young tableaux whose diagram is a 2-by-n rectangle given $2n$ distinct numbers is $C_n$. In general, given $m\times n$ distinct numbers, how many Young tableaux whose diagram is a $m\times n$ rectangle are there?
Also, what if these numbers can be repeated?
Many thanks.
-
For the answer to your main question, you need to use the hook-length formula.
OEIS A060854 gives the result $$(mn)! \prod_{i=0}^{n-1} \frac{i!}{(m+i)!} \textrm{ or equivalently } (mn)! \prod_{j=0}^{m-1} \frac{j!}{(n+j)!}$$ and some more information.
-
This is the number of standard Young tableaux. – Yuval Filmus Mar 9 '11 at 2:15
@Yuval: "Young tableaux" in the question really means "standard Young tableaux". To quote from the Wikipedia page linked to: "$C_n$ is the number of Young tableaux whose diagram is a 2-by-$n$ rectangle. In other words, it is the number ways the numbers $1, 2, \dots, 2n$ can be arranged in a 2-by-$n$ rectangle so that each row and each column is increasing. As such, the formula can be derived as a special case of the hook formula." – Hans Lundmark Mar 9 '11 at 7:27
...except that I've now edited the Wikipedia page so that it says standard tableaux, and has a link to the hook-length formula. – Hans Lundmark Mar 9 '11 at 7:38
It not quite clear what you mean by allowing repeated numbers, but what one usually considers in that case is so-called semi-standard Young tableaux, i.e., tableaux which are increasing (strict inequality) down each column, but only nondecreasing (equality allowed) along each row. The number of such arrangements on a given Young diagram, where the numbers $1,2,\dots,N$ are allowed, is counted as follows: define the "content" of box $(i,j)$ in the diagram to be $i-j$. Here's an illustration:
0 1 2 3 4
-1 0 1 2
-2 -1 0
-3 -2
-4 -3
-5
Hook lengths are defined as for the usual hook-length formula for counting standard Young tableaux:
10 8 5 3 1
8 6 3 1
6 4 1
4 2
3 1
1
To get the answer, take the product over all boxes of (($N$ plus the content of that box) divided by (the hook length for that box)).
(This is a special case of something called Stanley's hook-content formula.)
- | 2015-04-28 03:46:03 | {"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": 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": 0.88036048412323, "perplexity": 175.7152351976909}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246660628.16/warc/CC-MAIN-20150417045740-00221-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://datascience.stackexchange.com/questions/13323/minimum-number-of-trees-for-random-forest-classifier | Minimum number of trees for Random Forest classifier
I am searching for a theoretical or experimental estimation of the lower bound for the number of trees in a Random Forest classifier.
I usually test different combinations and select the one that (using cross-validation) provides the median best result.
However, I think that there may be a lower bound on the number of trees to use, given $m$ observations and $n$ features, to reduce the variance contribution to the error. Is there some test or reference I may check to correctly tune my classifier?
• I doubt it. But a suggestion: the area of machine learning that studies performance in machine learning is called empirical risk minimization. You can try to add those keywords to your searches. – Ricardo Cruz Aug 9 '16 at 10:10
This is not necessarily an answer to your question. Just general thoughts about cross-validating the number of decision trees within a random forest.
I see a lot of people in kaggle and stackexchange cross-validating the number of trees in a random forest. I have also asked a couple of colleagues and they tell me it is important to cross-validate them to avoid overfitting.
This never made sense to me. Since each decision tree is trained independently, adding more decision trees should just make your ensemble more and more robust.
(This is different from gradient boosting trees, which are a particular case of ada boosting, and therefore there is potential for overfitting since each decision tree is trained to weight residuals more heavily.)
I did a simple experiment:
from sklearn.datasets import load_digits
from sklearn.ensemble import RandomForestClassifier
from sklearn.grid_search import GridSearchCV
import numpy as np
import matplotlib.pyplot as plt
plt.ioff()
X = df['data']
y = df['target']
cv = GridSearchCV(
RandomForestClassifier(max_depth=4),
{'n_estimators': np.linspace(10, 1000, 20, dtype=int)},
'accuracy',
n_jobs=-1,
refit=False,
cv=50,
verbose=1)
cv.fit(X, y)
scores = np.asarray([s[1] for s in cv.grid_scores_])
trees = np.asarray([s[0]['n_estimators'] for s in cv.grid_scores_])
o = np.argsort(trees)
scores = scores[o]
trees = trees[o]
plt.clf()
plt.plot(trees, scores)
plt.xlabel('n_estimators')
plt.ylabel('accuracy')
plt.savefig('trees.png')
plt.show()
I am not saying your are committing this fallacy about thinking more trees can cause overfitting. You clearly are not since you have asked for a lower bound. This is just something that has been bugging me for awhile, and I think it is important to keep in mind.
(Addendum: Elements of Statistical Learning discusses this at page 596, and is in agreement on this with me. «It is certainly true that increasing B [B=number of trees] does not cause the random forest sequence to overfit». The author does make the observation that «this limit can overfit the data». In other words, since other hyperparameters may lead to overfitting, creating a robust model does not rescue you from overfit. You have to pay attention when cross-validating your other hyperparameters.)
To answer your question, adding decision trees will always be beneficial to your ensemble. It will always make it more and more robust. But, of course, it is dubious whether the marginal 0.00000001 reduction in variance is worth the computational time.
Your question therefore, as I understand, is whether you can somehow calculate or estimate the amount of decision trees to reduce the error variance to below a certain threshold.
I very much doubt it. We do not have clear answers for many broad questions in data mining, much less specific questions like that. As Leo Breiman (the author of random forests) wrote, there are two cultures in statistical modeling, and random forests is the type of model that he says has few assumptions, but is also very data-specific. That is why, he says, we cannot resort to hypothesis testing, we have to go with brute-force cross-validation.
• Mh, I thought that increasing the number of trees would lead to an increase of the bias component of the error (which is somehow counter-balanced by a decreasing variance of the model). – gc5 Aug 10 '16 at 11:06
• I always thought that adding trees could never hurt, but in The Elements of Statistical Learning page 596, they claim that too rich of a forest can introduce extra variability, so I think your answer is still correct, but with a tiny grain of salt :) – TBSRounder Aug 10 '16 at 11:43
• @TBSRounder Thanks for the link. The author is discussing too many things in such a short piece of text, so it's hard to distill him, especially confusing because he talks about tree depth and tree number at the same time. But, with regard to B (=#trees), he is saying that more trees makes the resulting model closer to the average tree, and, of course, the average tree may be biased, so you may end up with a biased model. I never claimed otherwise. I just said that B, by itself, does not increase bias. It just makes your model more robust (=reduces variance). – Ricardo Cruz Aug 10 '16 at 13:03
• Ah yes, It was a little confusing, thanks for the clarification. For some reason I thought the author was talking about B, not tree depth. – TBSRounder Aug 10 '16 at 13:12 | 2020-07-16 15:09:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.4828914403915405, "perplexity": 680.1908857669247}, "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-2020-29/segments/1593657169226.65/warc/CC-MAIN-20200716122414-20200716152414-00392.warc.gz"} |
https://math.stackexchange.com/questions/641490/stressing-out-because-im-not-sure-how-to-complete-the-square | # Stressing out because I'm not sure how to complete the square
Can someone please explain how to complete the square? Is there a specific formula you have to put the given equation in, or something? Every time I search how to complete the square each equation seems to employ a different method and it just confuses me. I'm not even understanding the concept at all. Can someone please help me? It feels like one of those things which I'll never be able to understand.
• ELI5? What does that mean? – Umberto Jan 17 '14 at 13:23
Suppose you have a quadratic expression $ax^2+bx+c$ with $a\ne 0$.
$ax^2+bx+c$
$=a(x^2+\frac{b}{a}x)+c$
$=a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})-\frac{b^2}{4a}+c$
$=a(x+\frac{b}{2a})^2-\frac{b^2}{4a}+c$
• Wait so we just add (b/2)^2 to both sides? – Michalo Jan 17 '14 at 11:06
• Here's how you can re-discover (in case you forget) that your add $\left(\frac{b}{2}\right)^2$ to both sides: You want the result to have the form $(x+R)^2$ for some number $R.$ Now look at the expanded form of $(x+R)^2$ (square first term, multiply the terms and double the result, square second term), which is $x^2 + 2Rx + R^2.$ Notice that the constant term, which is $R^2,$ is the square of half the coefficient of $x.$ That is, you can get $R^2$ from $2R$ (the coefficient of $x$) by taking half of $2R$ and then squaring the result. – Dave L. Renfro Jan 17 '14 at 14:55
A slightly different view, without dividing by $a$, but multiplying by $4a$, so the degree two term is easily a square and we also get the degree one term in good form to be twice a product: \begin{gather} ax^2+bx+c=0\\[2ex] 4a^2x^2+4abx+4ac=0\\[2ex] 4a^2x^2+4abx+b^2-b^2+4ac=0\\[2ex] (2ax+b)^2=b^2-4ac \end{gather}
• An $x^2$ missing in the second and third lines. – bubba Jan 17 '14 at 12:28
• @bubba Thanks, fixed. – egreg Jan 17 '14 at 13:20 | 2019-11-12 16:56:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8684800863265991, "perplexity": 250.47904466995737}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496665575.34/warc/CC-MAIN-20191112151954-20191112175954-00288.warc.gz"} |
https://plot.ly/scikit-learn/plot-polynomial-interpolation/ | Show Sidebar Hide Sidebar
# Polynomial Interpolation in Scikit-learn
This example demonstrates how to approximate a function with a polynomial of degree n_degree by using ridge regression. Concretely, from n_samples 1d points, it suffices to build the Vandermonde matrix, which is n_samples x n_degree+1 and has the following form:
[[1, x_1, x_1 2, x_1 3, ...],
[1, x_2, x_2 ** 2, x_2 ** 3, ...], ...]
Intuitively, this matrix can be interpreted as a matrix of pseudo features (the points raised to some power). The matrix is akin to (but different from) the matrix induced by a polynomial kernel. This example shows that you can do non-linear regression with a linear model, using a pipeline to add non-linear features. Kernel methods extend this idea and can induce very high (even infinite) dimensional feature spaces.
#### New to Plotly?¶
You can set up Plotly to work in online or offline mode, or in jupyter notebooks.
We also have a quick-reference cheatsheet (new!) to help you get started!
### Version¶
In [1]:
import sklearn
sklearn.__version__
Out[1]:
'0.18.1'
### Imports¶
This tutorial imports Ridge, PolynomialFeatures and make_pipeline.
In [2]:
import plotly.plotly as py
import plotly.graph_objs as go
import numpy as np
from sklearn.linear_model import Ridge
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline
### Calculations¶
In [3]:
def f(x):
""" function to approximate by polynomial interpolation"""
return x * np.sin(x)
# generate points used to plot
x_plot = np.linspace(0, 10, 100)
# generate points and keep a subset of them
x = np.linspace(0, 10, 100)
rng = np.random.RandomState(0)
rng.shuffle(x)
x = np.sort(x[:20])
y = f(x)
# create matrix versions of these arrays
X = x[:, np.newaxis]
X_plot = x_plot[:, np.newaxis]
colors = ['teal', 'yellowgreen', 'gold']
lw = 2
### Plot Results¶
In [4]:
data = []
p1 = go.Scatter(x=x_plot, y=f(x_plot),
mode='lines',
line=dict(color='cornflowerblue', width=lw),
name="ground truth")
p2 = go.Scatter(x=x, y=y,
mode='markers',
marker=dict(color='navy',
line=dict(color='black', width=1)),
name="training points")
data.append(p1)
data.append(p2)
for count, degree in enumerate([3, 4, 5]):
model = make_pipeline(PolynomialFeatures(degree), Ridge())
model.fit(X, y)
y_plot = model.predict(X_plot)
p3 = go.Scatter(x=x_plot, y=y_plot,
mode='lines',
line=dict(color=colors[count], width=lw),
name="degree %d" % degree)
data.append(p3)
layout = go.Layout(xaxis=dict(zeroline=False),
yaxis=dict(zeroline=False))
fig = go.Figure(data=data, layout=layout)
In [5]:
py.iplot(fig)
Out[5]:
Author:
Mathieu Blondel
Jake Vanderplas
BSD 3 clause | 2018-11-17 04:47:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49478474259376526, "perplexity": 11458.981358912355}, "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-2018-47/segments/1542039743282.69/warc/CC-MAIN-20181117040838-20181117062838-00398.warc.gz"} |
https://puzzling.stackexchange.com/questions/23958/what-is-a-beecumming-word/23965 | What is a Beecumming Word™?
This is in the spirit of the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles.
If a word conforms to a special rule, I call it a Beecumming Word™.
Use the following examples below to find the rule.
And, if you want to analyze, here is a CSV version:
Beecumming Words™,Not Beecumming Words™
CENTAUR,JACKALOPE
EDUCATING,INFORMING
GLACIER,ICEBERG
INTEGRALS,DERIVATIVES
MASTERING,PRACTICING
MIRACLES,WONDERS
MISCREANT,VAGABOND
SCREWING,HAMMERING
SECLUDED,HIDDEN
STRANGLED,CHOKED
TRIANGLE,HEXAGON
WETLANDS,SWAMP
WILDCAT,GRIZZLY
WRESTLING,FIGHTING
• That title seems dirty... – Deusovi Nov 13 '15 at 15:44
• @Deusovi It does look that way, yes, but it is a coincidence. The spelling is intentional and SFW. – Engineer Toast Nov 13 '15 at 15:46
• Does "beecumming" conform to the rule? Does "BEECUMMING"? – question_asker Nov 13 '15 at 15:51
• @question_asker Capitalization is not relevant. "BEECUMMING" does not conform to the rule. Neither does "question_asker" or "Engineer Toast" – Engineer Toast Nov 13 '15 at 16:30
• @question_asker It has nothing to do with jet fuel or steel beams so at least it won't be annoying in that way. – Engineer Toast Nov 16 '15 at 14:16
If you take the
Unique Letters from E. E. Cummings' full name (Edward Estlin Cummings)
then you get
acdegilmnrstuw
Which are the letters found in @Dr Xorile's answer as common letters! This means that the definition of a Beecumming Word™ is that it
is composed only of letters found in the name "Edward Estlin Cummings"
• oh my god, I thought about this, but I was so hung up on figuring out the rule that I had the exclusivity backwards. good job! – question_asker Nov 16 '15 at 20:41
I don't have an answer, but here's some basic analysis that may help (and may hinder):
Common letters: acdegilmnrstuw
No letters that appear in beecumming words only
Letters that appear in unbeecumming words only: bfhjkopvxyz
Letters that appear in neither group: q
I spent a while looking at the poetry of EE Cummings, but I was left none the wiser (at least for this puzzle - I'm sure the poetry was good for me in other ways)
• I think that u isn't as useful as it seems (sadly), in the question comments OP says "question_asker" is not becumming. – Raystafarian Nov 13 '15 at 18:24
• Good catch. I was kind of assuming that a subset of the unbeecumming words only was going to be the key – Dr Xorile Nov 13 '15 at 18:26
• I try to use every letter but I just couldn't find one I liked with a Q. The fact that it is left out is of no significance. – Engineer Toast Nov 13 '15 at 21:33
I think that a Beecumming word is
Used in the works of E.E. Cummings. That seems like what the OP is implying with the name Beecumming Word, but I haven't had any luck finding the list of words in several collections of his works. I'm not sure if I'm using incorrect or incomplete sources, but so far no word from either list has appeared in my searches. However, the implication seems so strong that I'm willing to hazard this as a guess.
• This is probably it, but you should add explanation - typically answers with no reasoning behind them are deleted. – Deusovi Nov 15 '15 at 20:39
• You got the clue in the title correct. However, you're still a little off-target. STARCRAFT is Beecumming™ but WARCRAFT is not. Dr Xorile has some useful analysis. – Engineer Toast Nov 16 '15 at 13:13 | 2020-01-25 05:54:13 | {"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": 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": 0.4993962347507477, "perplexity": 1794.674056393375}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251669967.70/warc/CC-MAIN-20200125041318-20200125070318-00017.warc.gz"} |
https://cstheory.stackexchange.com/questions/47768/mathitfo-times-seems-more-powerful-than-mathitdlogtime-uniform-m | # $\mathit{FO}[+,\times]$ seems more powerful than $\mathit{DLOGTIME}$-uniform $\mathit{AC}^0$?
I’ve been reading up on the connection between first order logic and small circuit complexity classes, and specifically Barrington, Immerman, and Straubing’s paper “On Uniformity Within $$\mathit{NC}^1$$”, where they prove $$\mathit{DLOGTIME}$$-uniform $$\mathit{AC}^0$$ = $$\mathit{FO}[+,\times]$$. However, it seems to me that $$\mathit{FO}[+,\times]$$ is much more powerful than $$\mathit{DLOGTIME}$$-uniform $$\mathit{AC}^0$$. For example, the following sentence:
$$\forall x \,(Qx \lor \exists a \exists b \,(a\times b = x \land a\ne1 \land b\ne1))$$
can be implemented as a circuit as an $$\mathrm{AND}$$ gate that is only connected to prime-numbered inputs (including input 1), and so defines the language $$\{1, 11, 110, 1100, 11001, 110010\}$$. However, $$\mathit{DLOGTIME}$$ only has time linear in the size of the gate indices to determine whether a connection exists between two gates. It seems to me that to recognize the connectivity language in linear time, one would have to be able to tell whether or not the input gate index was prime, and $$\mathrm{PRIMES}$$, although in $$P$$, is not computable in linear time. How can this language be $$\mathit{DLOGTIME}$$-uniform $$\mathit{AC}^0$$ then?
• Your sentence is equivalent to $\forall x\,Qx$, as you can always satisfy the disjunction by taking $a=1$ for the existential quantifier. Based on your description in words, you probably intended the sentence to read $$\forall x\,((\forall a\,\forall b\,(a\cdot b=x\to a=1\lor b=1))\to Qx),$$ or equivalently, $$\forall x\,(Qx\lor\exists a\,\exists b\,(a\cdot b=x\land a\ne1\land b\ne1)).$$ Oct 23, 2020 at 6:58
• Anyway, the root of your confusion is that you are prematurely eliminating connectives, making the circuit harder to describe. Just translate the quantifiers literally to propositional logic: you get an AND over all $x=0,\dots,n-1$ of binary ORs of $Q_x$ with ORs over all $a=0,\dots,n-1$ and $b=0,\dots,n-1$ of ANDs of constant $0/1$ circuits depending on if $ab=x$, $a\ne1$, or $b\ne1$. Thus, you don’t need a linear-time algorithm for primality, but only for the graph of multiplication. You still can’t do that, but you can fix it by adding more ANDs and ORs; the point is that ... Oct 23, 2020 at 7:17
• ... while multiplication (and primality) are presumably not computable in deterministic linear time, they are computable in the linear-time hierarchy. Oct 23, 2020 at 7:18
• Also, you are confusing the terms. You are not talking about $FO[+,\times]$-uniform anything, you are talking about just $FO[+,\times]$ itself. The statement is that the class $FO[+,\times]$ coincides with the DLOGTIME-uniform class $AC^0$. Oct 23, 2020 at 7:43 | 2023-03-21 13:27:26 | {"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": 15, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7167434096336365, "perplexity": 356.75218902162595}, "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/1679296943698.79/warc/CC-MAIN-20230321131205-20230321161205-00767.warc.gz"} |
http://mitloehner.com/lehre/rpa/RPA1RF.html | # Robotic Process Automation¶
Johann Mitloehner, 2022-10-13
## Definition¶
Robotic Process Automation (RPA) allows businesses to automate tasks that are typically carried out by employees.
• In contrast, RPA emulates human interaction with software systems, e.g. by using the web interface of an application instead of its API.
RPA is usually aimed primarily at repetitive and tedious tasks in order to
• free people for more intellectually satisfying work, or
• simply cut personel cost.
In order to mimick human interaction with applications RPA needs to
• work with the user interface provided by the system, such as
• enter data in web forms
• 'understand' the system responses, such as
• extract specific parts from the content of a web page
• assign meaning to those parts, such as success or failure of login
• perform actions corresponding to specific situations
• usually pre-defined in a programmatic way, but increasingly
• involving machine learning
• maybe interact with more than one system
• extract and process data from one system
• transfer data via a different interface to another system
#### Workflow Automation¶
This type of automation is usually understood to involve software that accesses the back-end of a system through its API (application programming interface). In contrast, RPA accesses the system via the front end, usually a GUI (graphical user interface), closely mimicking the human/computer interaction.
Since not all systems provide an API but need to provide a frontend for human users, there are situations where RPA is the only feasible approach to automation.
#### GUI Testing Tools¶
These are aimed at performing system testing with pre-defined test cases and expected results. The focus is usually not on interacting with multiple applications; however, sophisticated testing tools like the robot framework which will be used here allow for extracting data and processing it in more than one application.
#### Robot Software¶
This type of software is usually understood to control physical robots. RPA involves software robots i.e. programs that are not controlling a physical robot but perform actions by interacting with other software systems, not the physical world.
#### Artificial Intelligence¶
A confusing term that is hard to define, since even natural intelligence remains an ellusive concept. We will use the less misleading term machine learning instead to refer to a type of decision making that sometimes seems like AI but is really just another type of software.
#### Machine Learning¶
This usually refers to software that 'learns' from observation i.e. from data providing instances of situations and actions, e.g.
• credit card application data and corresponding decisions such as grant/reject
• measurements of petals and the corresponding species of Iris flowers (a classic dataset from 1936)
The system is usually understood to work on data from a large number of cases in an adaptive manner -- often, but not necessarily, working iteratively through the cases. It will (hopefully) adjust its behaviour towards optimal decisions, usually defined by minimising an error function.
Various approaches are used; connectionist models, especially deep learning neural nets, are currently very much in favour because of spectacular successes, particularly in image processing. However, there are a number of much simpler yet still useful approaches. Some of them will be discussed here, as it is feasible to apply them in RPA with limited resources in terms of time and coding skill.
#### Chatbots¶
RPA is usually aimed at back-office tasks while the automated servicing of customer request is increasingly offered using chatbot technology. Since chatbots automate front-office tasks they are often not seen as RPA; however, using the above definition chatbots qualify as RPA, since they are software robots that automate tasks otherwise carried out by employees.
### Benefits of RPA¶
There is always a lot of hype surrounding new concepts in management and technology. The following list of RPA benefits is somewhat conservative and also a little critical; just some food for thought.
• Cost savings. When robots do the work of people we can cut personel cost. Euphemistically this is described as freeing time for more creative work.
• Resilience. While the human workforce is limited there can always be more software robots, the only limit being the performance of the computing hardware. Therefore, if demand suddenly increases, the robot army is instantly ready.
• Accuracy. Once a process is defined any errors left are due to design, not the software robots. Humans make mistakes, particularly in tedious tasks; (software) robots do not. Physical robots are a different matter and can make disastrous mistakes.
• Compliance. An automated process can be defined so as to be fully compliant to some regulations at all times. Humans might make exceptions that can lead to trouble; robots make no exceptions, and no trouble.
• Productivity. When measured in terms of input/output relation robots are hard to beat, especially software robots who work 24/7 without any wear and tear. If needed they can be replicated at practically zero additional cost.
• Employee happiness. When freed from tedious tasks people can (in theory) do more creative and fulfilling things, and that may well make them happier (at least those who still have a job).
### RPA as Enterprise Software¶
This list is similar to the previous one but focuses on the development and deployment of software, which is often a huge and risky project. Fortunately, RPA is somewhat different from typical enterprise software projects:
• No Disruption. The RPA only uses the front-end of the system, so any problems caused must have been present already in day-to-day operations by human users (and have hopefully been wiped out). Deployment of RPA is less likely to cause disruption of service, whereas API-based process automation can access functions not available to users, or in a manner not possible when using the front-end GUI, and thereby causing unforeseen problems.
• Scalable. RPA software robots not only work faster than humans, they also run continously 24/7 all year long, thereby easily meeting with increased demand; and since they are just software programs we can have more than one instance running on one or more computers at the same time. The only limit is the performance of the computing hardware.
• Small Investment. Obviously this depends on the project. However, as we will see, at least some simple RPA projects can be cheap yet useful.
• Quick ROI. A simple RPA project can start generating return on investment relatively quickly since development and deployment tend to be less problematic compared to a similar process automation project based on API programming.
### Downsides of RPA¶
These are compared to traditional automation based on API programming i.e. using the back-end of the system. Most of these problems do not seem overwhelming or unsolvable, though.
• New Approach. Developers need to learn new methods, and in the case of the robot framework also a new language.
• Performance. Compared to API-based automation the RPA approach will tend to be slower, maybe even so much slower that it is not feasible for a particular project.
• Front-End Limitations. Remember that the front-end was not designed to be used by robots, but by humans. Problems may arise that were never faced before, because human users would know how to handle exceptional situations while robots just shamble on -- remember the paint robots in car factories that turn on each other
• Citizen Developers. Business units can now develop bots using simple end-user tools and without the need for support by an IT team, or any involvement (or even knowledge) of the IT department. This can be seen as a benefit or a nightmare, depending on where you stand.
### Examples¶
In the following we will look at examples of RPA in the following areas:
• Testing. While test suites can of course be run via the API (if one is available) there may be subtle differences to actual user interaction that are not easy to cover reliably. Using RPA and the GUI closely mimicks human interaction and (hopefully) bypasses those problems.
• Web Scraping. Since the Robot Framework uses XPath to access elements in HTML documents, and also allows for very simple integration of custom Python code in Robot test case files, it is easy to automatically extract content from web pages for further processing, also known as web scraping.
• Customer Service. Many customer requests fall into one of very few categories and are therefore prime candidates for automation. This is an area that applies concepts from chatbots and machine learning. We will look at a simple case study using robot testing for automation and open datasets for machine learning.
## Getting Started with the Robot Framework¶
The Robot Framework is available at robotframework.org. Its main purpose is automated testing; however, it is based on the Selenium library which can be used for general process automation in the interaction with web servers.
There are various types of testing frameworks; the Robot Framework uses keyword-driven testing: the idea is that the keywords
• describe the actions that need to be performed without too much detail
• are independent of the test framework being used
The approach somewhat resembles pseudo-code in algorithm design; it can be used for both manual and automated testing.
The following examples provide an introduction to the approach. The Robot Framework is written in Python, and we need to install some packages, and maybe Python itself as well.
### Installation¶
The installation can be tricky..
The robotframework github site has detailled installation instructions for various operating systems.
When you install Python from the official source python.org make sure to check the little box "Add Python to the Path".
Adding python to the PATH means that you can start python on the command line. However, we also need the web driver scripts in the PATH. If you see something like
Driver copied to: C:\Users\ramen\bin\geckodriver.exe
WARNING: Path 'C:\Users\ramen\bin' is not in the PATH environment variable.
you need to add that users directory to the PATH. Exactly how this works depends on your operating system.
For related problems:
##### Python in Linux¶
It is recommended to use Python in one of the popular Linux distributions where it comes with the rest of the system. Many Linux desktop components rely on Python, so usually both Python 2 and Python 3 are part of the distribution.
Linux can easily be installed alongside an already existing operating system via dual boot i.e. you choose which system to use in this session when you start up your computer. The installation will take maybe half an hour or so, but it can save a lot more time and frustration later. All you need is a USB stick and at least about 20 GB of free space on your drive. Download the installation image, put it on your stick, and boot from that. Your favorite Linux distribution web site has all the details; this author prefers Mint, but there are many others, see distrowatch.
##### Python in Other Operating Systems¶
Python comes in a number of distributions for various needs and operating systems:
• The primary source is python.org. This is the standard and reference CPython implementation, and it works perfectly for our purposes; it uses the pip installer which you see in all the examples.
• Another option is the Anaconda distribution which uses its own installer conda instead of pip. This comes bundled with heaps of software, including flask (but not robotframework). Work with this if you have Anaconda already installed.
Once you have Python running you should be able to use pip (or conda) to install additional Python packages, and everything should work just fine. Fingers crossed, knock on wood.
☆ Depending on your distribution/setup, you may have to use the command py instead of python to run Python scripts.
##### Python 3 vs 2¶
Sadly the Python developers made a decision years ago to make the new Python 3 incompatible with the older version 2. The differences are few and small, but still more than enough to cause trouble. We will continue to suffer the consequences for many years to come.
We are using Python 3 here. Depending on your distribution and operating system this may be standard; however, make sure that when you enter on the command line
python
you actually get the Python 3 interpreter prompt, not the older Python 2. Leave the interactive interpreter by entering ctrl-d
You see python3 in all examples here since this gives us Python 3 on Linux. Otherwise, depending on your configuration, you might get Python 2.
Depending on your distribution and operating system you may not have a python3 command, so you have to use python instead.
☆ On Linux there are both versions available, since many system/desktop components depend on Python 2. Do not remove Python 2 from your Linux system.
#### Pip - Package Installer for Python¶
Open a terminal window and enter the following statements on the command line. This only needs to be done once for our setup.
First we make sure to have the current version of the package installer pip:
python3 -m pip install --upgrade pip
The pip module should be part of your Python distribution; otherwise you will get an error and you have to install pip: on Debian-based Linux systems enter
sudo apt install python3-pip
Now we can install Python packages:
python3 -m pip install selenium
python3 -m pip install --upgrade robotframework-seleniumlibrary
python3 -m pip install webdrivermanager
Linux: Note the message after the webdrivermanager install about .local/bin not being in our PATH environment variable! We will need to fix this.
Current pip versions should automatically switch to user install when root permission is missing. If you get errors about permissions then add --user at the end of all the install commands, such as
python3 -m pip install selenium --user
#### Fix the PATH¶
Depending on your operating system environment you need to put .local/bin on our PATH. The following refers to Linux.
1. On the command line use the editor by entering the following:
pico .bashrc
2. add another line at the end of the file (move the cursor to the end of the file, then press the Enter key for newline, and type or copy-paste the following):
export PATH=$PATH:$HOME/.local/bin
3. Save the file and exit by using the key strokes: ctrl-O, Enter, ctrl-X
4. Do not close this terminal window!
5. Open a new terminal and continue work there.
If you get weird error messages then you messed up the .bashrc file. Go back to the first terminal, start the pico editor again, and fix the problem.
Now we can configure Firefox as our web browser: On the command line enter
${BROWSER} Firefox *** Test Cases *** Valid Home Page - Keywords Open Browser To Home Page Page Should Contain Credit App [Teardown] Close Browser *** Keywords *** Open Browser To Home Page Open Browser${URL} ${BROWSER} PASS Valid Home Page - Keywords Things to note about the code above: • a keywords/argument format is used • default is two or more blanks for indent and separators • if blanks are inconvenient we can also use the pipe character | • start the line with | and then • use | with one space left and right for indent and separators • a trailing pipe at the end of the line is optional • in the Settings section we provide short documentation and request Selenium for web page processing • in the Variables section we define the URL and the browser • not strictly necessary, but good practise to do this here at the top, especially in longer files • the Test Cases section contains one or more cases • the first statement in that test case is defined in the Keywords section • the second statement contains the pre-defined keywords "Page Should Contain" • make sure to leave two or more blanks before the text to check for, otherwise • the Robot Framework will not recognise the text as a parameter and • try to treat it as another user-defined keyword phrase • The Teardown statement makes sure the browser windows is closed, instead of leaving it open • in the Keywords section we define what we mean by saying "Open Browser To Home Page" • we use the pre-defined phrase "Open Browser" • and supply it with the variables from above ### Expect to Fail¶ Let's try a test that we expect to fail. Use your text editor and put the following in a file t3.rob: In [21]: %%robot *** Settings *** Documentation Check for text in page, expect fail Library SeleniumLibrary *** Variables ***${LOGIN URL} http://localhost:8080
${BROWSER} Firefox *** Test Cases *** Valid Home Page - Expect Fail Open Browser To Home Page Page Should Contain The Credit App [Teardown] Close Browser *** Keywords *** Open Browser To Home Page Open Browser${LOGIN URL} ${BROWSER} FAIL Valid Home Page - Expect Fail Page should have contained text 'The Credit App' but did not. Note how we introduced a small change in the text: Our website says "Credit App", not "The Credit App" This test should fail. Our minimal app is performing its Hello function, but now we add some more features. We are slowly approaching a small application with database connection. Incremental development and testing will go side by side. Let's add a link to a list of clients. Note that at this point we do not yet need an actual procedure for listing anything, just the link to it. Use your text editor to change the content of the file credit.py to the following: from flask import Flask app = Flask(__name__) @app.route("/") def credit(): return """<h1>Credit App</h1> <p><a href=clients>Clients</a>""" Remember the option --reload which we added to the unicorn web server command. Take a look the terminal window that runs the gunicorn web server: you should see a line that looks like this: ... [INFO] Worker reloading: /home/.../credit.py modified The gunicorn web server has detected the change in the source file and restarted the web app. Our web app starts to get a little more elaborate. We now have a few distinct elements that we can check in our robot tests. We can use XPath expressions to find elements in the HTML and check their contents. Use your text editor to create a file t4.rob and put the following code into that file: In [22]: %%robot *** Settings *** Documentation Check shop page for header and links Library SeleniumLibrary *** Variables ***${LOGIN URL} http://localhost:8080
${BROWSER} Firefox *** Test Cases *** Valid Home Page - XPath Check Home Page [Teardown] Close Browser *** Keywords *** Check Home Page Open Browser${LOGIN URL} {BROWSER} Element Should Contain //h1 Credit App Element Should Contain //a Clients Element Should Contain //a[contains(text(), "Clients")] Clients PASS Valid Home Page - XPath Note that in the last line we need to supply the text twice although it is already obvious from the XPath expression. Run the robot again to see that the XPath expressions actually target the link as intended. robot t5.rob The text output should show PASS. The test works as intended. Now we can move on the actually providing procedures for the functions, such as the list of clients. For that purpose, we chose a somewhat more elaborate route by introducing a database connection to our little sample app. ### SQLite Database Connection for the Shop App¶ To make our approach extensible and reasonably realistic we add more functions to the sample app instead of just providing toy examples. Playing in the sandbox can only get us so far. Fortunately there is a free open-source database management system that we can easily use in our sample app: SQLite. This DBMS is widely used since it is so easy to install and apply. It should be noted that it is also lacking in several respects, such as in terms of implementing standard SQL numeric data types. We happily accept these deficiencies since they do not bother us here (much). See the documentation for more details about SQLite2 and its use in Python3: The new version of our shop app contains two new routes: • initialise the DB table for the clients and insert some data • list the clients Our application is now a handsome size, make sure you do not miss anything when copy and paste into the file credit.py: from flask import Flask, render_template, request import sqlite3 app = Flask(__name__) def getconn(): return sqlite3.connect("credit.db") @app.route("/") def credit(): return """<h1>Credit App</h1> <ul> <li><a href=clients>Clients</a></li> <li><a href=newclient>New Client</a></li> <li><a href=initdb>Init DB</a></li> </ul>""" @app.route('/initdb') def initdb(): conn = getconn() cur = conn.cursor() cur.execute("drop table if exists client") cur.execute("create table client " + "(id int primary key, lim int, sex int, edu int, mar int, age int)") conn.commit() conn.close() return "DB initialized." @app.route('/clients') def clients(): conn = getconn() cur = conn.cursor() rows = cur.execute("select id, lim from client") html = "<h3>Clients</h3><table>\n" for row in rows: html += "<tr><td align=right> %d <td align=right> %.2f\n" % row return html + "</table>\n" conn.close() @app.route('/newclient') def newclient(): return """<h3>New Client</h3> <form action=insertclient method=POST> <table> <tr><td>Client ID:<td><input type=text name=id> <tr><td>Credit Limit:<td><input type=text name=lim> <tr><td>Sex:<td><input type=text name=sex value=1> <tr><td>Education:<td><input type=text name=edu value=1> <tr><td>Marriage:<td><input type=text name=mar value=1> <tr><td>Age:<td><input type=text name=age value=30> </table><input type=submit value=OK></form>""" @app.route('/insertclient', methods=['POST']) def insertclient(): id = request.form['id'] lim = request.form['lim'] sex = request.form['sex'] edu = request.form['edu'] mar = request.form['mar'] age = request.form['age'] conn = getconn() cur = conn.cursor() cur.execute('insert into client (id, lim, sex, edu, mar, age) ' + ' values (?, ?, ?, ?, ?, ?)', (id, lim, sex, edu, mar, age)) conn.commit() conn.close() return "Client inserted." The SQLite3 connection module should be part of the Python3 distribution, we just need to import it. • we define a funtion getconn() to give a DB connection whenever we need it. SQlite stores the DB in a file in the current directory, as named in the connect() function. So, we should expect a file ending in ".db" after the initdb route is execute for the first time. • in the initdb route we use that getconn() function and get a cursor from the connection. With this cursor we can execute SQL statements. • we drop the table if it exists so we can run this code multiple times. Note that in this interface we do not end SQL statements with ";" • we create a simple table • we insert a few rows into our table. For the sake of simplicity we use integers for everything; SQLite does not worry about that at all, as we will see. • the commit() is necessary here since auto-commit is only default in interactive use. Without it all changes would be lost when the connection is closed: SQLite supports basic transaction logic. With the table initialised we can list its content: • again we get a cursor from the connection • for Select statements the execute() function returns the resulting rows • we use the printf() function to force two decimals in the currency values • without it we would see the actual float values as stored in the DB • we make a sporting attempt at presenting the content in a nice tabular layout • in the loop we go through the results and access the fields by index Now we can do quite a bit of testing! ### Including Python Code in Test Files¶ The following test will • insert a new client • list all clients and check for the newly inserted one A new client should have an ID and a credit limit; we can easily generate those in Python. We put our Python code into a file with the extension .py in the current directory. Let's call it mytools.py; create it with your text editor and put the following code into the file: import sqlite3 import random def get_new_client_id(): cur = sqlite3.connect("credit.db").cursor() row = cur.execute("select max(id)+1 from client").fetchone() return "%d" % row def get_new_client_limit(): return "%d" % (10000 * random.randint(1,5)) Do not try to run this file directly; it will not produce any useful results. It will work with the robot after the Library definition in the robot code file. The code above defines two functions for creating new values which are then available in our robot test files as user-defined phrases. The underscore character translates as blank in the robot code: get_new_client_id() becomes Get New Client Id get_new_client_limit() becomes Get New Client Limit Both functions return values which we can capture in the robot test file. ☆ Note that our simple solution is not thread-safe. SQLite DB files can be accessed be multiple processes; there is no locking for read access, but it employs database locking for write access. Two processes running at the same time will get the same value for max(id)+1 and therefore identical client records. The option AUTOINCREMENT set in the SQL table create statements should guarantee unique primary keys. Another clean solution here would be sequences. Sadly, SQLite does not support them. However, we could create a table in the DB init part: create table mycount(n int) insert into mycount values(0) And then use the following code via the Python API to get a new number: update mycount set n = n + 1 select n from mycount In theory, this solution should be thread-safe: • some process A starts the update • this should lock the DB • another process B wants to do the same update • since the DB is locked it has to wait • until the second statement in A has finished and the lock is released • new B can do its update and select A and B should always see different values in their select results. However, with respect to later development of the sample application we do not worry about this issue here, since we will bulk import external data. In the Settings section of the robot test file we use the keyword Library to include the code from our tools module. Now we can do a lot of testing! We need to call initdb first, otherwise everything else will fail, since the DB table would not yet exists. Let's put this into a file t6.rob: In [24]: %%robot *** Settings *** Documentation Check new client insert Library SeleniumLibrary Library mytools.py *** Variables ***{LOGIN URL} http://localhost:8080
${BROWSER} Firefox${ID} 1
*** Test Cases ***
Open Browser ${LOGIN URL}${BROWSER}
Valid Init DB
Go To ${LOGIN URL} Click Link //a[@href="initdb"] Page Should Contain DB initialized. Valid Insert${LIM}= Get New Client Limit
Go To ${LOGIN URL} Click Link //a[@href="newclient"] Page Should Contain New Client Input Text //input[@name="id"]${ID}
Input Text //input[@name="lim"] ${LIM} Click Element //input[@type="submit"] Page Should Contain Client inserted Set Global Variable${LIM}
Valid Listing
Go To ${LOGIN URL} Click Link //a[@href="clients"] Page Should Contain${ID}
Page Should Contain ${LIM} [Teardown] Close Browser PASS Valid Home Page PASS Valid Init DB PASS Valid Insert PASS Valid Listing In this test file we have introduced several test cases; divide and conquer. To access the generated client ID in more than one test we make it global. Sadly this does not work in the Variables section as one would expect. Instead, we do this in the first test case. The code above contains some other new features: • the user-defined phrase Get New Client Limit returns a value which we put into a variabel LIM • note the dollar sign and curly braces with variable names • We do not (yet) use our Get New Client Id • we just initialized the DB; the table is empty, and the max(id) expression would result in 'None' • instead we define the client ID in the Variables section • the XPath expression //a[@href="newclient"] finds the first a element with an attribute href equal to "newclient" • we follow this link by using the pre-defined phrase Click Link • the pre-defined phrase Input Text finds the form elements and enters the values • click on the submit button and check the response • now we could go straight to the client listing, but instead • we go back to the start page and then • follow the link to the client listing, much like a human user would • Now we check for the name of the new client in the listing This test will take a little longer; we will probably be able to see the new entry briefly showing in the form fields and the listing. Performance is not the strong point of this type of automated testing. However, it is still much faster than human testers. Run the robot: robot t6.rob and observe the results; you should see PASS for all tests. ### Tasks vs Tests¶ The Robot Framework can easily be used for robotic process automation. This can be made explicit by using the section header Tasks instead of Test Cases; everything else works in the same way as in tests. In order to facilitate our report processing later we will just continue to use "Test". We cannot use both tasks and tests in the same robot file. ### Arguments to User-Defined Keywords¶ When we create a task to initialise the DB and insert a few clients we do not want to go through all steps for inserting a new client again and again! DRY: Don't Repeat Yourself. Code duplication makes it harder to maintain code. It may well be the root of all evil (in software) We want to define a new user keywords with the required steps in one place and then use that code for repeated application, only supplying the necessary data in each call as arguments (parameters). Put the following into a file t7.rob: In [25]: %%robot *** Settings *** Documentation Init DB and insert some clients Library SeleniumLibrary Library mytools.py *** Variables ***${LOGIN URL} http://localhost:8080
${BROWSER} Firefox *** Test Cases *** Init DB Open Browser${LOGIN URL} ${BROWSER} Click Link //a[@href="initdb"] Page Should Contain DB initialized Insert Several Clients Insert Client id=1001 lim=4000 Insert Client id=1002 lim=8000 Insert Client id=2001 Insert Client id=2002 lim=6000 [Teardown] Close Browser *** Keywords *** Insert Client [Arguments]${id} ${lim}=5000 Go To${LOGIN URL}
Page Should Contain New Client
Input Text //input[@name="id"] ${id} Input Text //input[@name="lim"]${lim}
Click Element //input[@type="submit"]
Page Should Contain Client inserted
PASS Init DB
PASS Insert Several Clients
In the Keywords section we use named arguments with default values. This allows us to call the user-defined phrase Insert Client in the Test section. We can supply all arguments, some, or none.
Run the robot:
robot t7.rob
and observe the results; again, all tests should PASS.
### XML Reports¶
The Robot Framework provides tools for generating summaries from the XML reports for each test; however, they are somewhat cumbersome, and it is easier and more flexible to go through the reports using the XML package of plain Python.
After some study of the structure of the XML files we find that the last status element in each test contains the overall status of the test; we can get that element with the expression findall("status")[-1]
Put the following code into a file xmlrep.py (identical to optional section on Automating the RPA above):
import xml.etree.ElementTree as ET
import sys
def report(fn):
for t in ET.parse(fn).getroot().iter('test'):
s = t.findall("status")[-1]
res = s.get("status") + ' ' + t.get("name")
if s.get("status") == "FAIL": res = res + ' -- ' + s.text
print(res)
if __name__ == '__main__': report(sys.argv[1])
Here the last line is executed when the complete file is run as a script (instead of just importing the report function):
Run it on the command line to check the file output.xml in the current directory:
python3 xmlrep.py output.xml
The output should look like this:
PASS Init DB
PASS Insert Several Clients
### Running Several Test Case Files¶
The Robot Framework features the concept of a Test Suite for this purpose; however, with our XML reporting tools already available it is more convenient to run individual test case files and then process the reports.
To run several robot tests and direct the output to different XML files for later analysis we can add the XML output file to the robot call.
Use your text editor to create a script that runs several tests at once and then summarizes all the XML reports.
Put the following code into a file mytests.sh:
robot -o o4.xml t4.rob
robot -o o5.xml t5.rob
robot -o o6.xml t6.rob
for file in o4.xml o5.xml o6.xml
do
echo $file python3 xmlrep.py$file
done
Run this script by entering the following on the command line:
bash mytests.sh
### Python built-in web server¶
If for some reason you do not want to use Flask and run a very simple local web server that serves just HTML pages then you do not need to write any Python code: just enter the following on the command line:
python3 -m http.server 8080 --bind 127.0.0.1
This will start a very basic web server with the current directory as root for all HTML files. Note that the port must be free at this time; if your flask server still runs on 8080 then the above command will not work.
### Optional: Extracting Robot Code from Notebook cells¶
This notebook (the one whose HTML version you are reading now) is by default saved as a JSON file, which means that it is relatively easy to process the contents of Jupyter notebook cells with our own Python code.
Here is a little script to
• extract the cells containing robot test code from this notebook
• execute the tests with the --quiet option
• summarize the results from the XML files
import json
import sys
import os
f = open(sys.argv[1], 'r')
n = 1
cells = data['cells']
for x in cells:
src = x['source']
typ = x['cell_type']
if typ == 'markdown' and len(src) > 1:
if src[1].startswith('%%robot'):
fout = open('tmp.rob', 'w')
fout.write( '\n'.join([ s[:-1] for s in src[1:-1] ]) )
fout.close()
os.system('robot --quiet -o tmp' + str(n) + '.xml tmp.rob')
n = n + 1
for i in range(n):
os.system('python3 xmlrep.py tmp' + str(i) + '.xml')
## Extracting Data from Web Pages -- Web Scraping¶
The Robot Framework can also be used for web scraping i.e. extracting data from web pages using
• XPath locators
• test case syntax, and
• user-defined Python functions.
The Robot Framework provides functions for file access. However, it is much more convenient and versatile to define our own in plain Python.
Add the following code to the mytools.py library file:
def create_my_file(fn):
fp = open(fn, 'w')
fp.close()
def append_my_file(fn, txt):
fp = open(fn, 'a')
fp.write(txt + '\n')
fp.close()
Now we can use these two new keywords in the following robot test file.
As a real-world practical example let's assume that we are interested in the titles of all Bond movies ever made (although the exact definition of that list is somewhat fuzzy), and after a bit of search we find a web site that has a relatively simple structure:
https://www.pocket-lint.com/tv/news/148096-james-bond-007-best-movie-viewing-order-chronological-release
After exploring the page source (right mouse button, View Page Source in Firefox) we can create our robot file:
*** Settings ***
Library SeleniumLibrary
Library mytools.py
*** Variables ***
${url} https://www.pocket-lint.com/tv/news/148096-james-bond-007-best-movie-viewing-order-chronological-release *** Test Cases *** Start Movie List Open Browser${url} Firefox
Page Should Contain James Bond movies
Create My File list.txt
Iterate Through Movies
${elements}= Get WebElements //h3 FOR${element} IN @{elements}
Append My File list.txt \${element.text}
END
The robot code above
• uses the library mytools.py
• defines an URL with a conveniently simple HTML structure for our movie list: simply all level 3 headers
• in the first test case we create our output file
• in the second test case we
• find all h3 elements
• loop through the results and write each item to the file
## Exercises¶
• Further study the XPath syntax and examples, from sources such as
• Find external web pages with moderately complex structure
• Write more robot test files to check for elements and text content
• Specific challenges:
• relative paths
• position of elements
• wildcards
• Make sure not to call the robot too often; leave intervals of a minute or more between calls -- some websites apply automatic exclusion procedures when hit by too many requests from the same source.
In [ ]: | 2023-02-05 07:24:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.18720249831676483, "perplexity": 3699.7603172359795}, "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/1674764500250.51/warc/CC-MAIN-20230205063441-20230205093441-00544.warc.gz"} |
https://wiki2.org/en/127_(number) | To install click the Add extension button. That's it.
The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.
4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds
# 127 (number)
← 126 127 128 →
Cardinalone hundred twenty-seven
Ordinal127th
(one hundred twenty-seventh)
Factorizationprime
Prime31st
Divisors1, 127
Greek numeralΡΚΖ´
Roman numeralCXXVII
Binary11111112
Ternary112013
Quaternary13334
Quinary10025
Senary3316
Octal1778
DuodecimalA712
Vigesimal6720
Base 363J36
127 (one hundred [and] twenty-seven) is the natural number following 126 and preceding 128. It is also a prime number.
• 1/1
Views:
1 344
• ✪ Angel Number 127 - (MEANINGS & SYMBOLISM)
## In mathematics
• As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also an exponent for another Mersenne prime 2127 - 1, which was discovered by Édouard Lucas in 1876, and held the record for the largest known prime for 75 years - it is the largest prime ever discovered by hand calculations, as well as the largest known double Mersenne prime. Furthermore, 127 is equal to 27 - 1, and 7 is equal to 23 - 1, and 3 is the smallest Mersenne prime, this makes 7 the smallest double Mersenne prime and 127 the smallest triple Mersenne prime.
• 127 is also a cuban prime of the form ${\displaystyle p=(x^{3}-y^{3})/(x-y)}$, ${\displaystyle x=y+1}$.[1] The next prime is 131, with which it comprises a cousin prime. Because the next odd number, 129, is a semiprime, 127 is a Chen prime. 127 is greater than the arithmetic mean of its two neighboring primes, thus it is a strong prime.[2]
• 127 is a centered hexagonal number.[3]
• It is the 7th Motzkin number.[4]
• 127 is a palindromic prime in nonary and binary.
• It is the first nice Friedman number in base 10, since 127 = -1 + 27, as well as binary since 1111111 = (1 + 1)111 - 1 * 1.
• 127 is the sum of the sums of the divisors of the first 12 positive integers.[5]
• 127 is the smallest prime that can be written as the sum of the first two or more odd primes: 127 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29.[6]
• 127 is the smallest odd number that can't be written in the form p + 2x for p=prime number and x=integer, since 127 - 20 = 126, 127 - 21 = 125, 127 - 22 = 123, 127 - 23 = 119, 127 - 24 = 111, 127 - 25 = 95, and 127 - 26 = 63 are all composite numbers.[7]
• 127 is an isolated prime where both p-2 and p+2 isn't prime.
127 is also: | 2019-08-21 15:51:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "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": 0.5570284128189087, "perplexity": 742.4503343719593}, "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-2019-35/segments/1566027316075.15/warc/CC-MAIN-20190821152344-20190821174344-00272.warc.gz"} |
https://techutils.in/blog/2018/08/20/stackbounty-tikz-pgf-pgfplots-tikz-external-tabu-pgf-tikz-externalization-tabu-makefile-and-legend-to-name-problems/ | #StackBounty: #tikz-pgf #pgfplots #tikz-external #tabu pgf/tikz externalization + tabu: makefile and legend to name problems
Bounty: 50
I am having troubles with the combination of `tabu` and pgf/tikz exernalization.
Please consider the code snippet below which I try to compile using lualatex or pdflatex using texlive 2018.
``````documentclass{book}
usepackage{filecontents}
newcommand{legendname}{defaultlegendname}
begin{filecontents*}{test.tikz}
tikz{begin{axis}[
width=4cm,height=4cm,
]
table[row sep=crcr] {%
1 1\
2 4\
};
label{testAAA}
end{axis}
begin{axis}[
legend to name=legendname,
width=4cm,height=4cm,
at={(4.1cm,0cm)}
]
%
table[row sep=crcr] {%
1 2\
5 10\
};
end{axis}}
end{filecontents*}
usepackage{tikz,pgfplots,pgfplotstable,subcaption,tabu}
usetikzlibrary{external}
tikzexternalize[mode=list and make]
begin{document}
begin{figure}
begin{tabu} to linewidth {X}
renewcommand{legendname}{fig1-legend}ref{legendname}\
tikzsetnextfilename{fig1}subcaptionbox{}{input{test.tikz}}\
end{tabu}
end{figure}
end{document}
``````
My compilation procedure is as follows:
1) Lualatex which gives me the heads-up that there are figures to be updated.
The figfile and makefile are given as:
``````% cat test.figlist
fig1
% cat test.makefile
fig1.pdf:
lualatex -shell-escape -halt-on-error -interaction=batchmode -jobname "fig1" "deftikzexternalrealjob{test}input{test}"
fig1.pdf: fig1.md5
``````
2) `make -f test.makefile` which completes without errors
3) 2 times lualatex. Now there is no heads-up for figures to be updated (I would have expected so, because the legend is not yet externalized). Instead, I get the following updated makefile
``````% cat test.makefile
fig1.pdf:
lualatex -shell-escape -halt-on-error -interaction=batchmode -jobname "fig1" "deftikzexternalrealjob{test}input{test}"
fig1.pdf: fig1.md5
fig1.pdf: FORCEREMAKE
``````
4) I run the makefile again, to let latex know that there are references which are not yet up-to-date, but it gives me
``````% make -f test.makefile
make: *** No rule to make target 'FORCEREMAKE', needed by 'fig1.pdf'. Stop.
``````
5) Due to the bogus makefile, I run the command in the makefile manually. That works without problems.
6) I run lualatex again, to see whether he noticed that the legend still needs to be build. But still he doesn’t notices and, hence, also doesn’t update figlist or makefile
Consequently, I also see the actual figure with question marks for the legend.
Outside/without the `tabu` environment, it seems to work fine. But I have several multi-panel figures where `tabu` is very handy.
What am I doing wrong here?
Get this bounty!!!
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https://www.nakedcapitalism.com/2019/12/mariana-mazzucato-what-is-economic-value-and-who-creates-it.html | # Mariana Mazzucato: What Is Economic Value, and Who Creates It?
Yves here. We’re fans of Mariana Mazzucato’s work. Her book, The Entrepreneurial State, documented the critical role the US government has played in undertaking and sponsoring basic and other research that was critical to the development of new industries and products, and how, contrary to popular opinion, it supported projects in an adaptive and flexible way. Here, she kneecaps the idea of individuals as “value” or job creators by going though historical views on this question and describing how the rise of neoclassical economics radically changed conventional thinking.
Mazzucato is a skilled speaker so I think you will enjoy this TED talk. You’ll see the transcript provides time markers. You can also view this speech at TED.
Value creation. Wealth creation. These are really powerful words. Maybe you think of finance, you think of innovation, you think of creativity. But who are the value creators? If we use that word, we must be implying that some people aren’t creating value. Who are they? The couch potatoes? The value extractors? The value destroyers? To answer this question, we actually have to have a proper theory of value. And I’m here as an economist to break it to you that we’ve kind of lost our way on this question.
00:36
Now, don’t look so surprised. What I mean by that is, we’ve stopped contesting it. We’ve stopped actually asking really tough questions about what is the difference between value creation and value extraction, productive and unproductive activities.
00:50
Now, let me just give you some context here. 2009 was just about a year and a half after one of the biggest financial crises of our time, second only to the 1929 Great Depression, and the CEO of Goldman Sachs said Goldman Sachs workers are the most productive in the world. Productivity and productiveness, for an economist, actually has a lot to do with value. You’re producing stuff, you’re producing it dynamically and efficiently. You’re also producing things that the world needs, wants and buys. Now, how this could have been said just one year after the crisis, which actually had this bank as well as many other banks — I’m just kind of picking on Goldman Sachs here — at the center of the crisis, because they had actually produced some pretty problematic financial products mainly but not only related to mortgages, which saw many thousands of people actually lose their homes. In 2010, in just one month, September, 120,000 people lost their homes through the foreclosures of that crisis. Between 2007 and 2010, 8.8 million people lost their jobs. The bank also had to then be bailed out by the US taxpayer for the sum of 10 billion dollars. We didn’t hear the taxpayers bragging that they were value creators, but obviously, having bailed out one of the biggest value-creating productive companies, perhaps they should have.
02:11
What I want to do next is kind of ask ourselves how we lost our way, how it could be, actually, that a statement like that could almost go unnoticed, because it wasn’t an after-dinner joke; it was said very seriously.
02:25
So what I want to do is bring you back 300 years in economic thinking, when, actually, the term was contested. It doesn’t mean that they were right or wrong, but you couldn’t just call yourself a value creator, a wealth creator. There was a lot of debate within the economics profession. And what I want to argue is, we’ve kind of lost our way, and that has actually allowed this term, “wealth creation” and “value,” to become quite weak and lazy and also easily captured.
02:51
OK? So let’s start — I hate to break it to you — 300 years ago. Now, what was interesting 300 years ago is the society was still an agricultural type of society. So it’s not surprising that the economists of the time, who were called the Physiocrats, actually put the center of their attention to farm labor. When they said, “Where does value come from?” they looked at farming. And they produced what I think was probably the world’s first spreadsheet, called the “Tableau Economique,” and this was done by François Quesnay, one of the leaders of this movement. And it was very interesting, because they didn’t just say, “Farming is the source of value.”
03:27
They then really worried about what was happening to that value when it was produced. What the Tableau Economique does — and I’ve tried to make it a bit simpler here for you — is it broke down the classes in society into three. The farmers, creating value, were called the “productive class.” Then others who were just moving some of this value around but it was useful, it was necessary, these were the merchants; they were called the “proprietors.” And then there was another class that was simply charging the farmers a fee for an existing asset, the land, and they called them the “sterile class.” Now, this is a really heavy-hitting word if you think what it means: that if too much of the resources are going to the landlords, you’re actually putting the reproduction potential of the system at risk. And so all these little arrows there were their way of simulating — again, spreadsheets and simulators, these guys were really using big data — they were simulating what would actually happen under different scenarios if the wealth actually wasn’t reinvested back into production to make that land more productive and was actually being siphoned out in different ways, or even if the proprietors were getting too much.
04:31
And what later happened in the 1800s, and this was no longer the Agricultural Revolution but the Industrial Revolution, is that the classical economists, and these were Adam Smith, David Ricardo, Karl Marx, the revolutionary, also asked the question “What is value?” But it’s not surprising that because they were actually living through an industrial era with the rise of machines and factories, they said it was industrial labor. So they had a labor theory of value. But again, their focus was reproduction, this real worry of what was happening to the value that was created if it was getting siphoned out.
05:03
And in “The Wealth of Nations,” Adam Smith had this really great example of the pin factory where he said if you only have one person making every bit of the pin, at most you can make one pin a day. But if you actually invest in factory production and the division of labor, new thinking — today, we would use the word “organizational innovation” — then you could increase the productivity and the growth and the wealth of nations. So he showed that 10 specialized workers who had been invested in, in their human capital, could produce 4,800 pins a day, as opposed to just one by an unspecialized worker. And he and his fellow classical economists also broke down activities into productive and unproductive ones.
05:43
(Laughter)
05:44
And the unproductive ones weren’t — I think you’re laughing because most of you are on that list, aren’t you?
05:49
(Laughter)
05:51
Lawyers! I think he was right about the lawyers. Definitely not the professors, the letters of all kind people. So lawyers, professors, shopkeepers, musicians. He obviously hated the opera. He must have seen the worst performance of his life the night before writing this book. There’s at least three professions up there that have to do with the opera.
06:10
But this wasn’t an exercise of saying, “Don’t do these things.” It was just, “What’s going to happen if we actually end up allowing some parts of the economy to get too large without really thinking about how to increase the productivity of the source of the value that they thought was key, which was industrial labor.
06:26
And again, don’t ask yourself is this right or is this wrong, it was just very contested. By making these lists, it actually forced them also to ask interesting questions. And their focus, as the focus of the Physiocrats, was, in fact, on these objective conditions of production. They also looked, for example, at the class struggle. Their understanding of wages had to do with the objective, if you want, power relationships, the bargaining power of capital and labor. But again, factories, machines, division of labor, agricultural land and what was happening to it.
06:59
So the big revolution that then happened — and this, by the way, is not often taught in economics classes — the big revolution that happened with the current system of economic thinking that we have, which is called “neoclassical economics,” was that the logic completely changed. It changed in two ways. It changed from this focus on objective conditions to subjective ones.
07:21
Let me explain what I mean by that. Objective, in the way I just said. Subjective, in the sense that all the attention went to how individuals of different sorts make their decisions. OK, so workers are maximizing their choices of leisure versus work. Consumers are maximizing their so-called utility, which is a proxy for happiness, and firms are maximizing their profits. And the idea behind this was that then we can aggregate this up, and we see what that turns into, which are these nice, fancy supply-and-demand curves which produce a price, an equilibrium price. It’s an equilibrium price, because we also added to it a lot of Newtonian physics equations where centers of gravity are very much part of the organizing principle. But the second point here is that that equilibrium price, or prices, reveal value.
08:08
So the revolution here is a change from objective to subjective, but also the logic is no longer one of what is value, how is it being determined, what is the reproductive potential of the economy, which then leads to a theory of price but rather the reverse: a theory of price and exchange which reveals value.
08:26
Now, this is a huge change. And it’s not just an academic exercise, as fascinating as that might be. It affects how we measure growth. It affects how we steer economies to produce more of some activities, less of others, how we also remunerate some activities more than others. And it also just kind of makes you think, you know, are you happy to get out of bed if you’re a value creator or not, and how is the price system itself if you aren’t determining that?
08:52
I mentioned it affects how we think about output. If we only include, for example, in GDP, those activities that have prices, all sorts of really weird things happen. Feminist economists and environmental economists have actually written about this quite a bit. Let me give you some examples. If you marry your babysitter, GDP will go down, so do not do it. Do not be tempted to do this, OK? Because an activity that perhaps was before being paid for is still being done but is no longer paid.
09:21
(Laughter)
09:22
If you pollute, GDP goes up. Still don’t do it, but if you do it, you’ll help the economy. Why? Because we have to actually pay someone to clean it.
09:29
Now, what’s also really interesting is what happened to finance in the financial sector in GDP. This also, by the way, is something I’m always surprised that many economists don’t know. Up until 1970, most of the financial sector was not even included in GDP. It was kind of indirectly, perhaps not knowingly, still being seen through the eyes of the Physiocrats as just kind of moving stuff around, not actually producing anything new. So only those activities that had an explicit price were included. For example, if you went to get a mortgage, you were charged a fee. That went into GDP and the national income and product accounting. But, for example, net interest payments didn’t, the difference between what banks were earning in interest if they gave you a loan and what they were paying out for a deposit. That wasn’t being included.
10:15
And so the people doing the accounting started to look at some data, which started to show that the size of finance and these net interest payments were actually growing substantially. And they called this the “banking problem.” These were some people working inside, actually, the United Nations in a group called the Systems of National [Accounts], SNA. They called it the “banking problem,” like, “Oh my God, this thing is huge, and we’re not even including it.” So instead of stopping and actually making that Tableau Economique or asking some of these fundamental questions that also the classicals were asking about what is actually happening, the division of labor between different types of activities in the economy, they simply gave these net interest payments a name. So the commercial banks, they called this “financial intermediation.” That went into the NIPA accounts. And the investment banks were called the “risk-taking activities,” and that went in. In case I haven’t explained this properly, that red line is showing how much quicker financial intermediation as a whole was growing compared to the rest of the economy, the blue line, industry.
11:14
And so this was quite extraordinary, because what actually happened, and what we know today, and there’s different people writing about this, this data here is from the Bank of England, is that lots of what finance was actually doing from the 1970s and ’80s on was basically financing itself: finance financing finance. And what I mean by that is finance, insurance and real estate. In fact, in the UK, something like between 10 and 20 percent of finance finds its way into the real economy, into industry, say, into the energy sector, into pharmaceuticals, into the IT sector, but most of it goes back into that acronym, FIRE: finance, insurance and real estate. It’s very conveniently called FIRE.
11:55
Now, this is interesting because, in fact, it’s not to say that finance is good or bad, but the degree to which, by just having to give it a name, because it actually had an income that was being generated, as opposed to pausing and asking, “What is it actually doing?” — that was a missed opportunity.
12:13
Similarly, in the real economy, in industry itself, what was happening? And this real focus on prices and also share prices has created a huge problem of reinvestment, again, this real attention that both the Physiocrats and the classicals had to the degree to which the value that was being generated in the economy was in fact being reinvested back in. And so what we have today is an ultrafinancialized industrial sector where, increasingly, a share of the profits and the net income are not actually going back into production, into human capital training, into research and development but just being siphoned out in terms of buying back your own shares, which boosts stock options, which is, in fact, the way that many executives are getting paid. And, you know, some share buybacks is absolutely fine, but this system is completely out of whack. These numbers that I’m showing you here show that in the last 10 years, 466 of the S and P 500 companies have spent over four trillion on just buying back their shares. And what you see then if you aggregate this up at the macroeconomic level, so if we look at aggregate business investment, which is a percentage of GDP, you also see this falling level of business investment. And this is a problem.
13:27
This, by the way, is a huge problem for skills and job creation. You might have heard there’s lots of attention these days to, “Are the robots taking our jobs?” Well, mechanization has for centuries, actually, taken jobs, but as long as profits were being reinvested back into production, then it didn’t matter: new jobs appeared. But this lack of reinvestment is, in fact, very dangerous.
13:48
Similarly, in the pharmaceutical industry, for example, how prices are set, it’s quite interesting how it doesn’t look at these objective conditions of the collective way in which value is created in the economy. So in the sector where you have lots of different actors — public, private, of course, but also third-sector organizations — creating value, the way we actually measure value in this sector is through the price system itself. Prices reveal value. So when, recently, the price of an antibiotic went up by 400 percent overnight, and the CEO was asked, “How can you do this? People actually need that antibiotic. That’s unfair.” He said, “Well, we have a moral imperative to allow prices to go what the market will bear,” completely dismissing the fact that in the US, for example, the National Institutes of Health spent over 30 billion a year on the medical research that actually leads to these drugs. So, again, a lack of attention to those objective conditions and just allowing the price system itself to reveal the value.
14:46
Now, this is not just an academic exercise, as interesting as it may be. All this really matters [for] how we measure output, to how we steer the economy, to whether you feel that you’re productive, to which sectors we end up helping, supporting and also making people feel proud to be part of. In fact, going back to that quote, it’s not surprising that Blankfein could say that. He was right. In the way that we actually measure production, productivity and value in the economy, of course Goldman Sachs workers are the most productive. They are in fact earning the most. The price of their labor is revealing their value. But this becomes tautological, of course.
15:24
And so there’s a real need to rethink. We need to rethink how we’re measuring output, and in fact there’s some amazing experiments worldwide. In New Zealand, for example, they now have a gross national happiness indicator. In Bhutan, also, they’re thinking about happiness and well-being indicators. But the problem is that we can’t just be adding things in. We do have to pause, and I think this should be a moment for pause, given that we see so little has actually changed since the financial crisis, to make sure that we are not also confusing value extraction with value creation, so looking at what’s included, not just adding more, to make sure that we’re not, for example, confusing rents with profits. Rents for the classicals was about unearned income. Today, rents, when they’re talked about in economics, is just an imperfection towards a competitive price that could be competed away if you take away some asymmetries.
16:16
Second, we of course can steer activities into what the classicals called the “production boundary.” This should not be an us-versus-them, big, bad finance versus good, other sectors. We could reform finance. There was a real lost opportunity in some ways after the crisis. We could have had the financial transaction tax, which would have rewarded long-termism over short-termism, but we didn’t decide to do that globally. We can. We can change our minds. We can also set up new types of institutions. There’s different types of, for example, public financial institutions worldwide that are actually providing that patient, long-term, committed finance that helps small firms grow, that help infrastructure and innovation happen.
16:56
But this shouldn’t just be about output. This shouldn’t just be about the rate of output. We should also as a society pause and ask: What value are we even creating? And I just want to end with the fact that this week we are celebrating the 50th anniversary of the Moon landing. This required the public sector, the private sector, to invest and innovate in all sorts of ways, not just around aeronautics. It included investment in areas like nutrition and materials. There were lots of actual mistakes that were done along the way. In fact, what government did was it used its full power of procurement, for example, to fuel those bottom-up solutions, of which some failed. But are failures part of value creation? Or are they just mistakes? Or how do we actually also nurture the experimentation, the trial and error and error and error?
17:45
Bell Labs, which was the R and D laboratory of AT and T, actually came from an era where government was quite courageous. It actually asked AT and T that in order to maintain its monopoly status, it had to reinvest its profits back into the real economy, innovation and innovation beyond telecoms. That was the history, the early history of Bell Labs. So how we can get these new conditions around reinvestment to collectively invest in new types of value directed at some of the biggest challenges of our time, like climate change? This is a key question.
18:19
But we should also ask ourselves, had there been a net present value calculation or a cost-benefit analysis done about whether or not to even try to go to the Moon and back again in a generation, we probably wouldn’t have started. So thank God, because I’m an economist, and I can tell you, value is not just price.
18:41
Thank you.
18:42
(Applause)
1. Sound of the Suburbs
GDP tells us what real wealth creation is.
In the 1930s, they pondered over where all that wealth had gone to in 1929 and realised inflating asset prices doesn’t create real wealth, they came up with the GDP measure to track real wealth creation in the economy.
The transfer of existing assets, like stocks and real estate, doesn’t create real wealth and therefore does not add to GDP. The real wealth creation in the economy is measured by GDP.
Inflated asset prices aren’t real wealth, and this can disappear almost over-night, as it did in 1929 and 2008.
Real wealth creation involves real work, producing new goods and services in the economy.
Neoclassical economics predates the GDP measure and is all about the markets and capital accumulation.
They believed in the markets in the 1920s and after 1929 they had to reassess everything, which is when they invented the GDP measure.
The fictitious financial wealth in real estate keeps disappearing, but policymakers have been very slow to notice the evidence piling up around them.
1990s – UK, US (S&L), Canada (Toronto), Scandinavia, Japan
2000s – Iceland, Dubai, US (2008)
2010s – Ireland, Spain, Greece
Get ready to put Australia, Canada, Norway, Sweden and Hong Kong on the list.
Now policymakers know what GDP is, they can understand the problem.
1. Sound of the Suburbs
“Neoclassical economics predates the GDP measure and is all about the markets and capital accumulation”.
Zimbabwe has been so successful at capital accumulation they have got hyper-inflation.
That’s what happens with too much money.
You can just print money, the real wealth in the economy lies somewhere else.
Alan Greenspan tells Paul Ryan the Government can create all the money it wants and there is no need to save for pensions.
What matters is whether the goods and services are there for them to buy with that money. That’s where the real wealth in the economy lies.
Money has no intrinsic value; its value comes from what it can buy.
Zimbabwe has too much money in the economy relative to the goods and services available in that economy. You need wheelbarrows full of money to buy anything.
You can just print money, its real wealth creation that is the important thing and it’s measured by GDP.
2. Sound of the Suburbs
Messing about transferring financial assets around.
Watch this video of the S&L crisis, as the Americans learn the art of making money by just transferring financial assets around and inflating asset prices with bank credit.
Bank loans create money out of nothing.
https://www.bankofengland.co.uk/-/media/boe/files/quarterly-bulletin/2014/money-creation-in-the-modern-economy.pdf
Money and debt come into existence together and disappear together like matter and anti-matter.
The speculators pocket the money, and the debt builds up in the S&Ls until the ponzi scheme collapses.
The limited liability company is a key component.
The debt is owed by these limited liability companies to the S&Ls, so as asset prices collapse they go bust, and the S&L’s have the debt that won’t be repaid.
The speculators will have taken most of the money out of these companies as they go along.
Nothing appears to be wrong as the bubble inflates.
The S&Ls and limited liability companies have assets on their books covering their loans.
When the bubble bursts, asset prices collapse rapidly and those assets no longer cover those loans.
The limited liability companies go bust leaving the S&Ls to carry the can.
They S&L’s can’t repossess those assets and sell them to recoup the money they have lent out.
The S&L’s become insolvent.
US taxpayers then bail out the bust S&Ls.
They have had a long time to learn how this little game works by 2008.
“It’s nearly $14 trillion pyramid of super leveraged toxic assets was bult on the back of$1.4 trillion of US sub-prime loans, and dispersed throughout the world” All the Presidents Bankers, Nomi Prins.
When this little lot lost almost all its value overnight, the Western banking system became insolvent.
Wall Street can turn a normal asset price bubble into something that will take out the global economy using leverage.
2. cnchal
Too bad Mariana just didn’t come right out and tell it like it is, with regard to Goldman 666. It’s frauds were handsomely rewarded because it is the POLITICAL economy. Mr Market was prevented from doing his jawb of taking Goldman out behind the woodshed and shooting it dead to put it out of everone’s misery through political intervention.
Supposedly fraud is illegal, except for Wall Street where it is standard operating procedure. The price paid is high, the value is less than zero.
3. Samuel Conner
Don’t have time at the moment to read the whole thing, but I noticed the early mention of Marx’s labor theory of value. This is derided by present-day mainstream micro-founded economists (I don’t lump MM in that group) but it makes sense at an aggregate level. If people can’t purchase all the output, the unsold residue has very little value. In aggregate (modulo profit extraction, which admittedly is a bigger thing in recent decades), wages purchase output. That’s a labor theory of price.
4. Nels Nelson
This was an excellent presentation. Fortunately I was exposed to these issues some 40 years ago in my undergraduate and graduate environmental economic courses with Herman Daly. Since then whenever the opportunity would present itself I would interject some of what I had learned and Mariana discussed only to be treated to blank stares or get accused of being too philosophical.
Professor Daly explained to the class the shortcomings of GDP and discussed why it should be replaced with a measure he called GPI, Genuine Progress Indicator. Economic activity produces both goods and bads. Whereas now they are all added together, the bads must be subtracted from the goods. Daly promoted the concept of the “steady state economy’ given that infinite growth on a finite planet is unsustainable.
Capitalism and its process of accumulation is built on a foundation of first exclusion, privitizing public resources, then extraction and exploitation. Extraction is the process of pulling value out of a resource. For Marx, the last stage of the accumulation process was finacialization wherein finance functions through its own mode of extraction. During this mode financial instruments are tools of speculation and create merely ‘fictional’ values. Finance and debt relations are means to extract values that are produced socially, outside of finance capital’s direct management. Think about the term “data mining”. Traditional extractive operations have migrated to social domains. Accumulation by means of social-media platforms involve not only gathering and processing the data provided by users but creating algorithmic means to capitalize on the intelligence, knowledge and social relations they bring.
In my view, Mariana’s talk will fly over most heads but I will enthusiastically pass it along.
5. Ignacio
Though financial activities and their role in value creation vs rent extraction are often discussed here I miss other service whose role in the economy is dubious IMO: advertising. Advertising may help a company reaching consumers but by itself doesn’t create value except for what is called brand value. But is brand value a real one of nothing more than other way of rent extraction? According to the site statista global advertising accounted in 2018 for 560 billion dollars of which almost half was spent in US markets. Advertising spending as a % of GDP is not homogeneous around the world and in the US reaches levels well above any other market. Advertising must be more sophisticated and aggressive in the US including probably all kinds of phishing already invented. A lot of rent extraction and very small added value if any. So, there is not only over-financialization but also over-advertising and the more money spent on this, the less productivity IMO.
6. Susan the Other
So she’s Mazzucato. She’s really good. Thanks for this one. Pretty clear that it’s not capitalism if nobody can recapitalize their equipment or even make a profit if they do. She didn’t really talk about capitalism – just the creation of value. Value isn’t so hard to pin down. It isn’t the price of something because monopolies can extort the economy until it is bloodless. Or creating disasters because they are good for the economy. The FIRE sector has just been financing itself using their monopoly on money and soft targets. The great causes are few and far between. But they are the creators of value. Surviving the paleolithic was one. The Cold War and the Moon Shot and the race for technology was another. The reason being is that we wanted to survive – avoid another world war. But consequences, consequences. Now we have a devastated planet. Because we never did control externalizing and extracting. So that’s our next great cause. So we’ll mobilize to fix it and then we will learn to be good stewards of the planet (cease to externalize, profiteer and pollute) …. and then? Then we’ll really have to change finance altogether. Finally giving a definition to “value”.
7. lyman alpha blob
I see what she’s getting at and agree 100% with her conclusion that value is not just price. But I still have a hard time grokking this notion of value at all.
I’m about 1/3 of the way through Against the Gods: The Remarkable Story of Risk right now and the author discusses the how the theory of probability was developed by mathematicians, at first as a way to calculate odds for gaming. So far so good. Then he goes on to describe how the insurance industry developed by using statistics to determine life expectancy, etc. and how the idea of value and utility came into play. Now the author is not the most lucid writer I’ve ever run across, but it seems pretty clear that this idea of value is a largely subjective notion that can’t really be quantified, yet that hasn’t stopped economists from trying to do so for a couple centuries now, much to the detriment of 99% of us.
This talk explains what value isn’t or should not be, but I have to wonder why the concept is necessary at all. The example of the Goldman bankers’ “value” being tautological was a good one and it would seem to apply in all cases. The value of something is what it is because a person thinks it is, not because of any inherent objective qualities.
Or in the immortal words of Mr. Lebowski – “Yeah, well, you know, that’s just like your opinion, man.”
1. eg
I believe that part of Mazzucato’s project is to debunk the myth that “value” and “price” are the same thing — a myth propagated by the marginal revolutionists and their neoliberal fellow travellers. In order to do so, I think she is taking pains to redefine value in such a way as to expose orthodox economics for the sham that it is.
You can expect vicious resistance to her project from the usual suspects …
2. @pe
Entropy or Shannon information: S = \sum_i P_i log P_i
Probablity: P_i = exp(-E_i/kT) / \sum_j exp(-E_j / kT)
RIght? Probability, Energy and Entropy / Information are all linked together. No one would say that energy “is just subjective and unquantifiable” — it’s a real quantifiable thing. But the probability calculated here depends on the distinguishable states and thus the information depends on the history of the observer and it’s ability to make predictions about states.
Thus the linked equations are subjective but quantifiable.
I think everyone would agree that “value” has something to do with “energy” — and the concept of “socially useful labor” comes very close to pointing in the direction that you would measure value in the energy used, but measured in terms of relevant exploitable energy of the system after labor. We could also talk about it in terms of the change in entropy of the system (the world), as it can be measured by the computing machines inside of it (aka, humans).
So, in a fuzzy and imprecise bloggy way, I’m trying to say that the Lebowski quote is crap here — there is an objectively measurable kind of value that is relative to the subjective nature of the relevant computing machines. To the extent that these computing machines have a common informational history, there should even be fairly unique and globally agreeable measures about what is valuable. Not universally unique, but unique to all the machines that have shared information and capacity to interpret or decode a common environment.
3. Jason
The ephemeral nature of value was why the neoclassical economists of the late 1800s turned away from questions about objective value. Mazzucato laments this turn, but she also understands that there was a good epistemological reason for it. That’s why she’s not offering a solution for “what is value”, but is just imploring folks to think about the question.
8. inode_buddha
Value is created by actual productive work because a persons time is not free, nor are their living expenses. Anything else (the FIRE sector) is just rearranging the deck chairs on the Titanic.
9. Summer
RE: “Bell Labs, which was the R and D laboratory of AT and T, actually came from an era where government was quite courageous. It actually asked AT and T that in order to maintain its monopoly status, it had to reinvest its profits back into the real economy, innovation and innovation beyond telecom…”
Makes me think of financialization and Netflix. They received their financing and valuation by being conflated with “tech” companies and ATT is more tech than they’ve ever been and is getting more into streaming.
The over-financialization breeds hype.
10. JEHR
I know the speaker did not say it outright, but I think that all non-productive valuation in the financial sector should be omitted from GDP calculations. Make a little footnote on each financial model and every financial chart for the banks where they can laud it over each other and giggle over how many billionaires they have created today.
I would like to see my government start our banking system over again and have new banks created that are not given the opportunity to make money from gambling with deposited money. If the money is not loaned for productive purposes (r & d, labour training, infrastructure, etc), then it should never be included in GDP. Non-productive money can be put in the shadow banking system where it isn’t allowed to cross over into the productive sector. What if the stock market were put in the shadow part of our system too? Only farmers who need to hedge on their crops in order to stay in business should be able to buy insurance. We do not need any more AIGs insuring bad non-productive projects.
Where do we find bankers that are honest who want to work for the betterment of the human race? Are they gone forever?
1. inode_buddha
I was thinking about it earlier, wondering “Hey, what about all the money they’re supposed to be making off interest rates, transaction fees, and insurance premiums?”
I decided that I think the FIRE sector should be banned from issuing stock. If they can’t make it from the interest rate spreads, transaction fees, and insurance premiums, then maybe they shouldn’t be in business.
11. Anarcissie
I think value has to be examined very fundamentally, down to the level of biology, physics, sociology, and information theory (and whatever I’m missing). And certainly politics (by which I mean the theory and practice of determining whose will, in a community, will prevail, and how). There is no value without an (or actually, many an) evaluator and while evaluators differ something can be reasonably said about large sets of evaluators, for example, almost all of them think they need to eat, whereas only a subset think they need to possess gold or a good job.
Much of what I read slips past these questions into something the writer is comfortable with, for example, the Physiocrats were comfortable with farming, Smith with trading, and so forth, and thus economics, cast adrift, becomes this very labile slush of truth and propaganda we see today. I’m glad to see someone trying to get at the problem.
12. rosemerry
I have Mariana’s book from the Book Depository and it is called “The Value of Everything”, very clearly explaining why value is tagged with a moral judgment. This is so rare now but her work has been widely praised. The NC readers are those who would be most likely to applaud it. | 2020-01-18 12:05:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37476682662963867, "perplexity": 1728.8031556420212}, "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/1579250592565.2/warc/CC-MAIN-20200118110141-20200118134141-00518.warc.gz"} |
http://naniar.njtierney.com/reference/is_shade.html | This tells us if this column is a shade
is_shade(x)
any_shade(x)
## Arguments
x a vector you want to test if is a shade
## Value
logical - is this a shade?
## Examples
xs <- shade(c(NA, 1, 2, "3")) | 2018-10-18 06:45:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5979930758476257, "perplexity": 4649.490523707862}, "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-2018-43/segments/1539583511744.53/warc/CC-MAIN-20181018063902-20181018085402-00391.warc.gz"} |
https://dsp.stackexchange.com/questions/17725/mfccs-and-chord-recognition | # MFCCs and chord recognition
This is my first time working with Mel-frequency cepstral coefficients and I have two questions....
1. Are some of the 13 coefficients strictly related to some sound properties like frequencies, amplitude, pitch or something ? For example if I'm interested only in sound pitch, are there some coefficients that carry that information and I can exclude the rest of the coefficients that carry other information to reduce the size of the vector ? Or those coefficients means nothing individually and they must be used all together as a whole ?
2. I'm trying to learn Chord recognition methods. I was thinking for calculating MFCC from a short Wave file where chord is played. (It is just a chord played on piano, there is no ambient noise or other instruments). And after I get the 13 MFCCs, cluster them to a centroid which describes the appropriate chord. (24 centroids, 12 for minor chords, 12 for major). I'm just wondering, are the MFCCs good descriptor for the present pitches in the sound ? | 2020-08-12 21:54:22 | {"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": 0.803343653678894, "perplexity": 1201.483778676524}, "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/1596439738944.95/warc/CC-MAIN-20200812200445-20200812230445-00042.warc.gz"} |
https://shreevatsa.wordpress.com/2008/11/ | # The Lumber Room
"Consign them to dust and damp by way of preserving them"
## A “Möbius” inversion formula
[Not the Möbius inversion formula, but something similar.]
As usual, define the Möbius function μ on the natural numbers as
$\mu(n) = \begin{cases}0 \text{ if n is not square free,}\\ -1\text{ if n is squarefree with an odd number of prime factors,}\\ 1\text{ if n is squarefree with an even number of prime factors.}\\\end{cases}$
Let $f$ be any function defined on the natural numbers, and let $F$ be the function defined as $\displaystyle F(n) = \sum_{j=1}^{n}{f(\left\lfloor{n/j}\right\rfloor)}$.
Then it is true that $\displaystyle f(n) = \sum_{k=1}^{n}{\mu(k)F(\left\lfloor{n/k}\right\rfloor)}$.
Note that f need not be multiplicative; it can be any arbitrarily defined function. I have no idea why it is true. Help?
Written by S
Fri, 2008-11-28 at 18:55:44
Posted in Uncategorized
Tagged with , ,
## Asimov on ‘The Last Question’
I like tracking down quotes, but find it terribly hard to track down quotes by Asimov about his writing: there are so many anthologies, and so many comments he has made about a single story in different places. Here, for example, are two comments I remember having read about what is definitely one of his two most famous stories, The Last Question:
• The first was easier to track down; it’s on Wikipedia with a date. In The Best of Isaac Asimov, published in 1973, he says:
‘The Last Question’ is my personal favorite, the one story I made sure would not be omitted from this collection.
Why is it my favorite? For one thing I got the idea all at once and didn’t have to fiddle with it; and I wrote it in white-heat and scarcely had to change a word. This sort of thing endears any story to any writer.
Then, too, it has had the strangest effect on my readers. Frequently someone writes to ask me if I can write them the name of a story, which they think I may have written, and tell them where to find it. They don’t remember the title but when they describe the story it is invariably ‘The Last Question’. This has reached the point where I recently received a long-distance phone call from a desperate man who began, ‘Dr. Asimov, there’s a story I think you wrote, whose title I can’t remember–‘ at which point I interrupted to tell him it was ‘The Last Question’ and when I described the plot it proved to be indeed the story he was after. I left him convinced I could read minds at a distance of a thousand miles.
No other story I have written has anything like this effect on my readers—producing at once an unshakeable memory of the plot and an unshakeable forgettery of the title and even author. I think it may be that the story fills them so frighteningly full, that they can retain none of the side-issues.
Written by S
Fri, 2008-11-28 at 06:47:42
Posted in Uncategorized
## Fermat’s last theorem
Fermat’s last theorem has a long and exciting history. Which everyone knows, so I’ll not mention it here.1 What I suddenly find to be remarkable though, is the very first event. The fact that Fermat scribbled it in a margin of Diophantus’s Arithmetica. That Pierre de Fermat, in France in 1637, was reading an ancient book written by a Greek in the 3rd century. That he was reading it in such a manner that the book’s asking how to split a square into two squares should impel him to not only investigate the question of how to split a nth power into two nth powers, for all n, but to also do it until he believed he had a truly marvelous proof.
When was the last time you made margin notes in a book?
Off topic: The book only answers the question for 16(=4²). Wikipedia has pictures of the relevant page for a 1621 edition, and the 1670 edition that contains Fermat’s notes. (Fermat died in 1665.) I’m not sure I’ve deciphered the Latin correctly (the Greek is right out), but what it says is the following.
[BTW, in case you have been thinking so far and have the objection that 16 cannot be written as the sum of two squares, I should point that for Diophantus, “number” apparently meant “positive rational number”, there were no other kinds of numbers. Negative and irrational numbers were “useless”, “meaningless”, and “absurd”.]
Suppose one of the two squares that add up to 16 is Q=N². [“Q” because it is a square, “quadratum”.] The other square is 16-Q. If the other square is (2N-4)²=4Q+16-16N, [um, why should it be?] then we get 16-Q=4Q+16-16N so 5Q=16N, or N=16/5 and Q=N²=256/25 (which is misprinted as 256/52 in the 1670 edition), and the other square is 144/25, which add up to 400/25=16. So the (an) answer is that 16 = (16/5)² + (12/5)².
You might notice this is not really an answer; all that Diophantus has done is take 3²+4²=5² and multiplied it appropriately to make two “squares” add up to 16. We could do the same for any square, e.g. for 49=7², we could write (7×3)²+(7×4)²=(7×5)², then divide out by 5² to say (21/5)²+(28/5)²=49. For any x, we could take any a and b such that a²+b²=1 (e.g. 3/5 and 4/5) and write x²=(ax)²+(bx)².
1. I found today (2008-11-27) an anecdote. The setting is this: it was April 1994. Andrew Wiles had first announced his proof in June the previous year, and sent it off to a journal, but a hole had been found. It seemed at first it would take only a few hours, then weeks, to fix it, but months had dragged on without success. And on April 3 1994, Gian-Carlo Rota sent out an email announcing that Noam Elkies had found a counterexample to Fermat’s last theorem! So it seemed that the hole was unfixable after all. There was some disappointment all around before it was realised that the email was an April Fool’s joke, that had somehow got incorrectly dated :-) I found it on Wikibooks, but see Lance Fortnow’s blog post for the email.
Written by S
Fri, 2008-11-21 at 15:05:01
Posted in Uncategorized
Tagged with , ,
## Thank you for the music
with one comment
Just watched Mamma Mia! The Movie this week (twice!). It sucks, is completely ridiculous, Pierce Brosnan cannot sing to save his life, and there are far too many annoying characters, but because it’s ABBA, all is forgiven. Meryl Streep, perfect as always, appears to be having the time of her life, but maybe it’s just her acting. Some of it would have been better if they simply used “playback singing” the way only we in India seem used to, but it was mostly okay. Apparently there are even thoughts of a “sequel” (wouldn’t be the first time a movie with no plot has had a sequel, anyway), because “there are still plenty of ABBA songs left”.
Read the rest of this entry »
Written by S
Mon, 2008-11-10 at 07:01:58
Posted in Uncategorized
Tagged with ,
## Lattice points visible from the origin
[A test of LaTeX-to-Wordpress conversion. Bugs remain, point them out. Original PDF]
Here is a problem I love. It is simple to state, and it has a solution that is not trivial, but is easy to understand. The solution also goes through some beautiful parts, so I can promise it’s worth reading :-)
[The solution is not mine. Also, the question is just an excuse for the ideas in the solution. :P]
Question. Suppose you are standing on an infinite grid in the plane. You can see infinitely in all directions, but you cannot see through grid points: a point is hidden from view if some other grid point lies in your line of sight. What fraction of the grid points can you see?
Let us first imagine that we are standing at the origin, and that the grid is that of the lattice (integer) points.
The blue points are visible; the grey points are not
Read the rest of this entry »
Written by S
Fri, 2008-11-07 at 15:00:30
Posted in public
Tagged with , ,
## A timeline ends
with one comment
Sad news that was too easy to miss in the ongoing deluge: Michael Crichton passed away on Tuesday, at age 66. See my friend’s post.
Read the rest of this entry »
Written by S
Thu, 2008-11-06 at 07:08:52
Posted in Uncategorized
Tagged with , , | 2017-12-14 02:25:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 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": 0.5458324551582336, "perplexity": 1286.6693872841952}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948537139.36/warc/CC-MAIN-20171214020144-20171214040144-00301.warc.gz"} |
http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0963.58003 | Language: Search: Contact
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Zbl 0963.58003
Iwaniec, T.; Scott, C.; Stroffolini, B.
Nonlinear Hodge theory on manifolds with boundary.
(English)
[J] Ann. Mat. Pura Appl., IV. Ser. 177, 37-115 (1999). ISSN 0373-3114; ISSN 1618-1891/e
Authors' summary: The intent of this paper is first to provide a comprehensive and unifying development of Sobolev spaces of differential forms on Riemannian manifolds with boundary. Second, is the study of a particular class of nonlinear, first order, elliptic PDEs called Hodge systems. The Hodge systems are far reaching extensions of the Cauchy-Riemann system and solutions are referred to as Hodge conjugate fields. We formulate and solve the Dirichlet and Neumann boundary value problems for the Hodge systems and establish the ${\cal L}^p$-theory for such solutions. Among the many desirable properties of Hodge conjugate fields, we prove, in analogy with the case of holomorphic functions on the plane, the compactness principle and a strong theorem on the removability of singularities. Finally, some relevant examples and applications are indicated.
[A.P.Stone (Albuquerque)]
MSC 2000:
*58A14 Hodge theory (global analysis)
58J05 Elliptic equations on manifolds, general theory
Keywords: nonlinear Hodge theory; Dirichlet boundary value problems; Hodge systems; Neumann boundary value problems
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Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences | 2013-05-20 09:20:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.674524188041687, "perplexity": 2929.025844011291}, "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/1368698693943/warc/CC-MAIN-20130516100453-00025-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.experts-exchange.com/questions/26438859/Powershell-to-merge-Excel-columns.html | Solved
# Powershell to merge Excel columns
Posted on 2010-08-30
1,022 Views
I would like to use Powershell to create a username and email address from a csv file that contains 2 columns, first name and last name. The username is derived from the first initial of the first name plus last name.
0
Question by:acronie18
• 3
• 2
LVL 13
Accepted Solution
soostibi earned 500 total points
ID: 33560638
As a start look at this.
What kind of output would you like to have?
There might be problems, if you use such a character set, that include characters that are not accepted as username or email address. Also, your algorithm can result in multiple users with the same username and email address. If such occurs then tell me, I'll further improve the script.
Import-Csv c:\ee\names.txt |
Select-Object @{n="username"; e={$_.firstname[0]+$_.lastname}}, @{n="email";e={"$($_.firstname[0]+$_.lastname)@yourdomain.com"}} 0 Author Comment ID: 33561294 That works well. I need all to output all four columns. I added the below code and was able to get what I was looking for. Just so I can wrap my brain around it, are 'n' and 'e' just variables or do they mean something specific? Thanks for your help. @{n="firstname"; e={$_.firstname}}, @{n="lastname"; e={$_.lastname}} 0 LVL 13 Expert Comment ID: 33561825 If you need all four columns, you can simply use: Import-Csv c:\ee\names.txt | Select-Object firstname, lastname, @{n="username"; e={$_.firstname[0]+$_.lastname}}, @{n="email";e={"$($_.firstname[0]+$_.lastname)@yourdomain
So you do not have to use the complicated hashtable syntax.
0
LVL 13
Expert Comment
ID: 33561834
Some characters are missing:
Import-Csv c:\ee\names.txt |
Select-Object firstname, lastname, @{n="username"; e={$_.firstname[0]+$_.lastname}}, @{n="email";e={"$($_.firstname[0]+\$_.lastname)@yourdomain.com"}}
0
Author Comment
ID: 33562162
That makes much more sense. I was trying to do something similar, but I over complicated it. Thanks for your help.
0
## Featured Post
Utilizing an array to gracefully append to a list of EmailAddresses | 2016-12-08 06:04:57 | {"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": 0.17025743424892426, "perplexity": 8590.81297915692}, "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/1480698542414.34/warc/CC-MAIN-20161202170902-00399-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://www.j.sinap.ac.cn/nst/EN/10.1007/s41365-017-0261-4 | # Nuclear Science and Techniques
《核技术》(英文版) ISSN 1001-8042 CN 31-1559/TL 2019 Impact factor 1.556
Nuclear Science and Techniques ›› 2017, Vol. 28 ›› Issue (8): 109
• LOW ENERGY ACCELERATOR, RAY AND APPLICATIONS •
### Spin polarization and production rate studies of surface muons in a novel solenoid capture system based on CSNS
Ran Xiao1,2, Yan-Fen Liu1,2,3, Xiao-Jie Ni1,2, Zi-Wen Pan1,2, Bang-Jiao Ye1,2
1. 1 State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei 230026, China
2 Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
3 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
• Supported by:
Supported by the National Natural Science Foundation of China (No. 11527811).
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Ran Xiao, Yan-Fen Liu, Xiao-Jie Ni, Zi-Wen Pan, Bang-Jiao Ye. Spin polarization and production rate studies of surface muons in a novel solenoid capture system based on CSNS.Nuclear Science and Techniques, 2017, 28(8): 109
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Abstract:
A novel surface muon capture system with a large acceptance was proposed based on the China spallation neutron source (CSNS). This system was designed using a superconducting solenoid where a long graphite target was put inside it. Firstly, the spin polarization evolution was studied in a constant uniform magnetic field. As the magnetic field can interact with the spin of the surface muon, both the spin polarization and production rate of the surface muons collected by the new capture system were calculated by the G4beamline. Simulation results showed that the surface muons could still keep a high spin polarization (>90%) with different magnetic fields (0–10 T), and the larger magnetic field is, the more surface muons can be captured. Finally, the proton phase space, Courant–Snyder parameters, and intensities of surface muons of different beam fractions were given with magnetic fields of 0 and 5 T. The solenoid capture system can focus proton and surface muon beams and collect p_ and l_ particles. It can also provide an intense energetic positron source. | 2021-01-21 21:20:10 | {"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": 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": 0.24040648341178894, "perplexity": 2669.395335939775}, "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/1610703527850.55/warc/CC-MAIN-20210121194330-20210121224330-00081.warc.gz"} |
http://python-textbook.pythonhumanities.com/03_spacy/03_02_04_phrase_matcher.html | # 2.3. The PhraseMatcher#
## 2.3.1. Introduction#
Another rules-based component built into spaCy is the PhraseMatcher. Like the Matcher, the PhraseMatcher does not sit inside a spaCy pipeline. It does not, therefore, align with a spaCy extension, such as doc.ents, like the EntityRuler does. Instead, it is meant to run over a Doc container, just like the Matcher. Unlike the Matcher, however, the PhraseMatcher does not function a sequence of linguistic features at the token level, rather it is focused on matching at the phrase level.
In practice, you would use the Matcher when you need to rely on a sequence of linguistic features at the token level to extract data. This is powerful, but can sometimes be difficult to write robust patterns to match all instances of a the patterns you wish to match. The PhraseMatcher, on the other hand, should be used when you know relatively well how the data will appear in a text. It is easier to use the PhraseMatcher, but it is not as dynamic as the Matcher.
.## Basic Example
As with the Matcher, it is best to see the PhraseMatcher in action with a basic example. First, let’s import the PhraseMatcher class and load up the default small English pipeline.
import spacy
from spacy.matcher import PhraseMatcher
nlp = spacy.load("en_core_web_sm")
Now, let’s consider a basic example. Let’s consider the text below. Here, we wish to find and extract the instances where Harry Potter appears in the text. Harry appears in four different ways in the text: 1) Harry Potter, 2) Harry, 3) Potter, and 4) The Boy who Lived.
text = """
Harry Potter was the main character in the book.
Harry was a normal boy who discovered he was a wizard.
Ultimately, Potter goes to Hogwarts.
He is also known as the Boy who Lived.
The Boy who Lived has an enemy named Voldemorte who is known as He who Must not be Named.
"""
matcher = PhraseMatcher(nlp.vocab)
matcher.add("HARRY_POTTER", [nlp("Harry Potter"), nlp("Harry"), nlp("Potter"), nlp("the Boy who Lived")])
doc = nlp(text)
Again, we will create our matches.
matches = matcher(doc)
Let’s iterate over our matches.
for match in matches:
print(match)
(12243270181114079557, 1, 2)
(12243270181114079557, 1, 3)
(12243270181114079557, 2, 3)
(12243270181114079557, 12, 13)
(12243270181114079557, 27, 28)
(12243270181114079557, 38, 42)
And now let’s iterate over our matches and grab a bit more data, including the token spans and the sentence in which a match was found.
for match in matches:
lexeme, start, end = match
print(nlp.vocab[lexeme].text, doc[start:end])
print(f"Sentence: {doc[start].sent}")
HARRY_POTTER Harry
Sentence: Harry Potter was the main character in the book.
HARRY_POTTER Harry Potter
Sentence: Harry Potter was the main character in the book.
HARRY_POTTER Potter
Sentence: Harry Potter was the main character in the book.
HARRY_POTTER Harry
Sentence: Harry was a normal boy who discovered he was a wizard.
HARRY_POTTER Potter
Sentence: Ultimately, Potter goes to Hogwarts.
HARRY_POTTER the Boy who Lived
Sentence: He is also known as the Boy who Lived.
As we can tel, the results are good, but we are missing one case. The second example of boy who lived was not grabbed because The was not capitalized. We can account for this by changing the main attribute of the PhraseMatcher.
## 2.3.2. Setting a Custom Attribute#
Unlike the Matcher, the PhraseMatcher does not let us control how it reads each individual token in the pattern. The way the PhraseMatcher parses the phrase is as the sequence level. By default, the PhraseMatcher reads the entire pattern as ORTH, or raw text. In other words, it must be a precise match in order to be flagged and extracted. In some instances, however, it may be important for a pattern to be not just raw text, but also in all forms, both uppercase and lowercase. This is particularly important for phrases, like the boy who lived, where the word the may be capitalized if it is at the start of a sentence. In these instances, we can change the main way the PhraseMatcher works by using the attr argument. By using attr="LOWER", we can make our PhraseMatcher pattern case-agnostic.
matcher = PhraseMatcher(nlp.vocab, attr="LOWER")
matcher.add("HARRY_POTTER", [nlp("Harry Potter"), nlp("Harry"), nlp("Potter"), nlp("the Boy who Lived")])
doc = nlp(text)
matches = matcher(doc)
for match in matches:
lexeme, start, end = match
print(nlp.vocab[lexeme].text, doc[start:end])
print(f"Sentence: {doc[start].sent}")
HARRY_POTTER Harry
Sentence: Harry Potter was the main character in the book.
HARRY_POTTER Harry Potter
Sentence: Harry Potter was the main character in the book.
HARRY_POTTER Potter
Sentence: Harry Potter was the main character in the book.
HARRY_POTTER Harry
Sentence: Harry was a normal boy who discovered he was a wizard.
HARRY_POTTER Potter
Sentence: Ultimately, Potter goes to Hogwarts.
HARRY_POTTER the Boy who Lived
Sentence: He is also known as the Boy who Lived.
HARRY_POTTER The Boy who Lived
Sentence: The Boy who Lived has an enemy named Voldemorte who is known as He who Must not be Named.
Notice that now, we have grabbed all ways the phrase the boy who lived is expressed in our text.
## 2.3.3. Adding a Function with on_match#
In production, it can sometimes be difficult to deploy a spaCy-based solution that requires pasting a for loop each time you want to iterate over the results. Usually, you want to automate certain tasks so that when a match is found, some event occurs in your code. The PhraseMatcher allows you to pass an extra argument to your patterns: on_match. This keyword argument will take a function which will receive four arguments from the PhraseMatcher: matcher (the PhraseMatcher), doc (the doc container that the PhraseMatcher just passed over), id, and matches (the resulting matches from the PhraseMatcher).
Let’s create a basic function that will iterate over each match and print off the match, its label, and the sentence in which it was found.
def on_match(matcher, doc, id, matches):
for match in matches:
lexeme, start, end = match
print(nlp.vocab[lexeme].text, doc[start:end])
print(f"Sentence: {doc[start].sent}")
Just as before, we will create our PhraseMatcher.
matcher = PhraseMatcher(nlp.vocab, attr="LOWER")
This time, however, when we add our patterns to the PhraseMatcher, we will also add the keyword argument on_match that will point to the above function.
matcher.add("HARRY_POTTER", [nlp("Harry Potter")], on_match=on_match)
All that is left to do is then create the Doc container from the text and then run the PhraseMatcher over the Doc container.
doc = nlp(text)
matches = matcher(doc)
HARRY_POTTER Harry Potter
Sentence: Harry Potter was the main character in the book.
Just like the PhraseMatcher, the Matcher also can take the on_match keyword argument. | 2022-12-03 14:54:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4070528745651245, "perplexity": 3110.3182384323964}, "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/1669446710933.89/warc/CC-MAIN-20221203143925-20221203173925-00603.warc.gz"} |
http://mirror.math.ku.edu/tex-archive/help/Catalogue/entries/fihyph.html | Hyphenation patterns for Finnish language.
Two sets of hyphenation patterns are provided; fihyph.tex, and fi8hyph.tex (which is modified from fihyph.tex to make the Finnish accented letters to work with LaTeX2e, adding some \catcode, \uccode, and \lccode commands after the model used in the hyphenation files for the other European languages).
The package is superseded by the Finnish patterns available in the hyph-utf8 bundle.
The author is Timo Hellgren. | 2015-04-19 03:08:10 | {"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": 0.9370561242103577, "perplexity": 13567.147423767576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246637364.20/warc/CC-MAIN-20150417045717-00222-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.openfoam.com/documentation/guides/latest/api/pointMVCWeightI_8H_source.html | The open source CFD toolbox
pointMVCWeightI.H
Go to the documentation of this file.
1/*---------------------------------------------------------------------------*\
2 ========= |
3 \\ / F ield | OpenFOAM: The Open Source CFD Toolbox
4 \\ / O peration |
5 \\ / A nd | www.openfoam.com
6 \\/ M anipulation |
7-------------------------------------------------------------------------------
8 Copyright (C) 2011-2016 OpenFOAM Foundation
9-------------------------------------------------------------------------------
10License
11 This file is part of OpenFOAM.
12
13 OpenFOAM is free software: you can redistribute it and/or modify it
14 under the terms of the GNU General Public License as published by
15 the Free Software Foundation, either version 3 of the License, or
16 (at your option) any later version.
17
18 OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
19 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
20 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
21 for more details.
22
23 You should have received a copy of the GNU General Public License
24 along with OpenFOAM. If not, see <http://www.gnu.org/licenses/>.
25
26\*---------------------------------------------------------------------------*/
27
28#include "pointFields.H"
29
30// * * * * * * * * * * * * * * * Member Functions * * * * * * * * * * * * * //
31
32template<class Type>
34(
36) const
37{
38 const labelList& vertices = psip.mesh()().cellPoints()[cellIndex_];
39
40 Type t = Zero;
42 {
43 t += psip[vertices[i]]*weights_[i];
44 }
45
46 return t;
47}
48
49
50// ************************************************************************* //
const Mesh & mesh() const
Return mesh.
Generic GeometricField class.
const label cellIndex_
Cell index.
scalarField weights_
Weights applied to cell vertices.
bool interpolate() const noexcept
Same as isPointData()
pointField vertices(const blockVertexList &bvl)
static constexpr const zero Zero
Global zero (0)
Definition: zero.H:131
#define forAll(list, i)
Loop across all elements in list.
Definition: stdFoam.H:333 | 2022-11-28 11:51:39 | {"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": 0.9630091190338135, "perplexity": 6969.202715856442}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710503.24/warc/CC-MAIN-20221128102824-20221128132824-00263.warc.gz"} |
https://math.stackexchange.com/questions/1581513/existence-of-a-compactification-with-same-weight-for-a-tychonoff-space | # Existence of a compactification with same weight for a Tychonoff space
3.5.2. Engelking
Every Tychonoff space $X$ has a compactification $(Y,c)$, such that $w(X) = w(Y)$.
Any ideas on the proof of this?
I think I get it for the case of $m=w(X) \geq \mathbb{N}$ , as then I could embed $X$ in $I^m$ with some fuction $f$, and I would have $w(f(X)) = m$ (homeomorphic spaces have same weight), $w(I^m)=m$, so any closed subset of $I^m$ containing $f(X)$ will have weight $m$ - I think that should work. Does it seem right?
But I don't know how to show this for $m$ finite.
• The weight of a space $X$ is normally defined as $\omega+\min\{|\mathscr{B}|:\mathscr{B}\text{ is a base for }X\}$, so that $w(X)\ge\omega$ for all $X$. – Brian M. Scott Dec 19 '15 at 0:37
If $w(X)$ is finite its implies that $X$ is finite therefore it's obviously $X$ is compact.
When $X$ does not have a finite base and $w(X)=k$, note that if $A$ is a subspace of $B$ then $w(A)\leq w(B)$. So if $X\subset I^k$ we have $$k=w(X)\leq w(\bar X)\leq w(I^k)=k.$$ | 2019-11-14 09:58:18 | {"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": 0.9503072500228882, "perplexity": 112.47703756221118}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668334.27/warc/CC-MAIN-20191114081021-20191114105021-00323.warc.gz"} |
https://cs.stackexchange.com/tags/matrices/new | # Tag Info
You seem to be looking to enumerate (and/or randomly sample from) the set of isotropic two-state two-dimensional cellular automata on the Moore neighborhood. The general way to this is simply to find all the equivalence classes of the $2^{3\times3}$ possible neighborhoods, where two neighborhoods are considered equivalent if one can be transformed into the ...
I won't go into the details of your specific case but try to answer the general problem. In the unrestricted case there is a mapping from each of the $n$ (=512) input states to one of 2 output states and you want to restrict the function as follows: Create a partitioning of the initial $n$ input states into subsets. Your adapted function maps each of these ... | 2019-10-16 03:05:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5071492195129395, "perplexity": 249.41127763013867}, "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-2019-43/segments/1570986661296.12/warc/CC-MAIN-20191016014439-20191016041939-00050.warc.gz"} |
https://discourse.mc-stan.org/t/question-stan-modelspecification-for-hierarchical-linear-model-gelmans-radon-example/11986 | # Question: Stan Modelspecification for Hierarchical Linear Model (Gelman's Radon example)
Hello there.
Disclaimer: I am using the package rstan together with Stan.
I am currently working on the radon dataset of Gelman to build HLMs but I am new to Stan and am a bit confused about the syntax. I would be happy if somebody could help me out in my mistakes/if the intuition of my code is right.
The dataset contains household data of radon and an indication of where the measurement was taken (1 = floor, 0 = basement).
Furthermore, there are uranium measurements but only 1 per county (85 counties).
Let’s call the variables:
X = floor
U = uranium
N = number of observations (919)
J = number of counties (85)
I have three models:
1. Unpooled: (Varying Intercepts, Fixed Slopes)
J different models based only on the observations of each county.
Y_j = alpha_j + beta_j * X
I want to put a prior on mu_alpha, sigma_alpha, mu_beta, sigma_beta and sigma_y.
For this, I have the following Stan script:
data {
int<lower=1> N; // number of obs
int<lower=1> J; // Number of counties
vector[N] Y; // outcome data (log radon)
vector[N] X; // Inputa data (floor)
int<lower=1> county[N]; //county indices
}
parameters {
vector[J] alpha;
vector[J] beta;
real<lower=0> sigma_y;
}
model {
vector[N] mu_y;
for (i in 1:N)
mu_y[i] = beta[county[i]] * X[i] + alpha[county[i]];
// Priors
Y ~ normal(mu_y, sigma_y);
alpha ~ normal(0,10);
beta ~ normal(0,10);
sigma_y ~ normal(0,10);
}
But, I get the following error message:
Warning messages:
1: In readLines(file, warn = TRUE) :
incomplete final line found on 'C:\Users\..\unpool_1.stan'
Q1: Is this message neglectable? + Is this the right specification?
1. Hierarchical Linear Model with 1 Level (Varying Intercepts and Fixed Slopes):
J different models but now with information sharing, i.e. modelling the J different alphas as coming from the same distribution.
Y_j = alpha_j + beta * X
This is the Stan script I have for this model:
// HLM 1
data {
int<lower=1> N; // number of obs
int<lower=1> J; // Number of counties
vector[N] Y; // outcome data (log radon)
vector[N] X; // Inputa data (floor)
int<lower=1> county[N]; //county ID variable
}
parameters {
vector[J] alpha; // county j intercept
real beta;
real mu_alpha; // prior on alpha
real <lower=0, upper=100> sigma_alpha; // hyperparameter sigma of alpha
real<lower=0, upper=100> sigma_y;// prior sigma of y in county j
}
model { // Model specifications
vector[N] mu_y;
// Full Model
for(i in 1:N)
mu_y[i] = alpha[county[i]] + beta * X[i];
// Priors and Hyperparams
Y ~ normal(mu_y, sigma_y);
sigma_y ~ cauchy (0, 2.5);
alpha ~ normal(mu_alpha, sigma_alpha);
mu_alpha ~ normal(0, 10);
sigma_alpha ~ uniform(0, 10);
beta ~ normal(0, 10);
}
Here, I get this error message:
Warning messages:
1: There were 1 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded
2: Examine the pairs() plot to diagnose sampling problems
Q2: How can I fix this error + is the Stan specification right?
1. Hierarchical Linear Model with 2 Levels:
Well, this is actually the one which is bothering me the most.
Y_j = alpha_j + beta * X
alpha _j = gamma_0 + gamma_1 * U
I have no clue how to specificy the second layer of the hierarchical model.
Q3: How do I specify the model in Stan + is it better to have 85 obs for Uranium or is it better when each of the 919 observation gets its own uranium observation (however would be duplicates, since there is only 1 measurement per county).
Here’s the minimal code from R:
## Load Data
# County Categorical Variable
county_name <- as.vector(data$county) # create vector of county indications for each obs county_list <- unique(county_name) # create list of counties, drop duplicates J <- length(county_list) # number counties in Minessota county <- rep (NA, J) for (i in 1:J){ county[county_name==county_list[i]] <- i } # Seperate variables logradon <- data$log_radon
loguranium <- data$log_uranium loguranium <- loguranium[!duplicated(loguranium)] floorind <- data$floor
# Lists for STAN
X = floorind,
N = N)
X = floorind,
N = N,
J = J,
county = county)
X = floorind,
U = loguranium,
N = N,
J = J,
county = county)
Exclaimer:
I actually feel bad for posting such an extensive questionnary, since usually people don’t answer to that but I am really hopeless. I tried it now for 2 weeks and I don’t know whether my results are correct or not and how to specify the 2-level HLM.
I don’t know anybody how uses Stan and most tutorials in the internet use Python.
(There are R tutorials but with different packages, where they just use default priors which is not what I want to do. I want to properly specify the full model to also perform sensitivity analysis later on regarding the priors.)
I would really appreciate any kind of help. Thanks in advance.
Here is the original paper for further reference.
Shouldn’t this be “Varying intercepts, varying slopes”? Or maybe the “varying” terminology isn’t specific enough (since you can have fixed/fully-pooled, varying/partially-pooled, and varying/unpooled)?
I believe that message simply means that Stan wants the model file to have a final empty newline at the end. But I wasn’t able to replicate this message (Linux, rstan 2.19.2), so try adding an empty newline on your end and report back how it goes.
Your model estimates an intercept for the radon levels independently for each county as well as an effect of sampling location independently for each county. This is a completely unpooled model, so if that’s what you want, you specified it correctly.
Specification: this model estimates a single effect of measurement location (i.e. “fully pooled”; exact same effect magnitude in each county) and separate-but-not-independent (i.e. “partial-pooled”) county-level intercepts.
How to fix the error is pretty clear from the error message and the link it provides! You want to increase the value of the max_treedepth argument to be higher than the default (so, greater than 10). Note however that with hierarchical models this error often means you should switch to the non-centered parameterization.
You want to pass in as data a vector of the 83 observed uranium values as well as an index vector of which uranium value goes with which radon values. Then model uranium however you want and then use it as a predictor by indexing into the uranium vector.
2 Likes
Yes, at it’s not an error, but a warning. If you want it to go away just put an extra line in the end of the file.
People like @stijn would be best suited to tell you about model specification, but you can do two things to improve this: (i) reduce the scale of the Cauchy prior on sigma_y or change it to a normal or something with lighter tails and (ii) increase max_treedepth to 15 or something to cope with the difficult posterior geometry.
@martinmodrak, any strong feelings?
1 Like
Thanks for your reply! Knowing that the first 2 models are correct, really pushed my confidence.
Here’s (obviously wrong) minimal code for the 2nd level HLM, do you know what I have to adjust?
The part where it says 2nd layer is wrong but I don’t know if I also have to change the part of the 1st layer…
R Code:
## Load Data
# County Categorical Variable
county_name <- as.vector(data$county) # create vector of county indications for each obs county_list <- unique(county_name) # create list of counties, drop duplicates J <- length(county_list) # number counties in Minessota county <- rep (NA, J) for (i in 1:J){ county[county_name==county_list[i]] <- i } # Seperate variables logradon <- data$log_radon
loguranium <- data$log_uranium loguranium <- loguranium[!duplicated(loguranium)] floorind <- data$floor
county_i <- county
county_j <- 1:85
# Lists for STAN
X = floorind,
U = loguranium,
N = N,
J = J,
county_i = county_i,
county_j = county_j)
## HLM 2-Level - Varying Intercepts, Fixed Coefficients
hlm2 <- stan(file = 'C://Users//...//hlm_2level.stan', data = datalist,
iter = 5000, chains = 4)
print(hlm2)
Stan Code:
// HLM 2
data {
int<lower=1> N; // number of obs
int<lower=1> J; // Number of counties
vector[N] Y; // outcome data (log radon)
vector[N] X; // Inputa data Layer 1(room indication)
vector[J] U; // Input data Layer 2 (uranium)
int<lower=1> county_i[N]; //county ID variable for x
int<lower=1> county_j[J]; //county ID variable for u
}
parameters {
vector[J] alpha; // county j varying intercept
real beta; // fixed slope
real mu_alpha; // prior on alpha
real gamma_zero; // 2nd level fixed intercept
real gamma_one; // 2nd level fixed slope
real <lower=0> sigma_alpha; // hyperparameter sigma of alpha
real<lower=0> sigma_y;// prior sigma of y in county j
}
model { // Model specifications
vector[N] mu_y;
// 2nd Layer
for(j in 1:J)
alpha[county_i[i]] = gamma_zero + gamma_one * U[county_j];
// 1st Layer
for(i in 1:N)
mu_y[i] = alpha[county_i[i]] + beta * X[i];
// Priors and Hyperparams
Y ~ normal(mu_y, sigma_y);
sigma_y ~ cauchy (0, 10);
alpha ~ normal(mu_alpha, sigma_alpha);
mu_alpha ~ normal(0, 10);
sigma_alpha ~ cauchy(0, 10);
beta ~ normal(0, 10);
gamma_zero ~ normal(0, 10);
gamma_one ~ normal(0, 10);
}
Perfect, thanks. I will try it with smaller scale paramaters in the priors.
The 1st layer is correct. The 2nd layer should be
for(j in 1:J)
alpha[county_j[j]] = gamma_zero + gamma_one * U[county_j[j]];
(Although you have county_j[j] == j which allows additional simplification.)
alpha is defined, not sampled, so it cannot be declared in the parameters block. Move it to model block, right next to mu_y declaration. alpha also does not need a prior.
2 Likes
Thanks for your reply. It probably is close to the correct code, but there is still an error.
It says that alpha (left hand side) is real (scalar) but the right hand side is vector, so it’s not a “legal” statement. (I changed it according to your hint.)
Here’s what I have:
// HLM 2
data {
int<lower=1> N; // number of obs
int<lower=1> J; // Number of counties
vector[N] Y; // outcome data (log radon)
vector[N] X; // Inputa data Layer 1(room indication)
vector[J] U; // Input data Layer 2 (uranium)
int<lower=1> county_i[N]; //county ID variable for x
int<lower=1> county_j[J]; //county ID variable for u
}
parameters {
real beta; // fixed slope
real gamma_zero; // 2nd level fixed intercept
real gamma_one; // 2nd level fixed slope
real<lower=0> sigma_y;// prior sigma of y in county j
}
model { // Model specifications
vector[N] mu_y;
vector[J] alpha; // county j varying intercept
// 2nd Layer
for(j in 1:J)
alpha[county_j[J]] = gamma_zero + gamma_one * U[county_j];
// 1st Layer
for(i in 1:N)
mu_y[i] = alpha[county_i[i]] + beta * X[i];
// Priors and Hyperparams
Y ~ normal(mu_y, sigma_y);
sigma_y ~ cauchy (0, 10);
beta ~ normal(0, 10);
gamma_zero ~ normal(0, 10);
gamma_one ~ normal(0, 10);
}
That’s supposed to be U[county_j[j]] instead of U[county_j].
1 Like
I agree with @mike-lawrence, using the index variable is usually “nicer” (but hey, do what solves your problems).
Finally, I guess @razorlazor is doing this as an educational excersice to better understand Stan. Because if not, using rstanarm or brms would save you a lot of pain - they have formula syntax for hierarchical models, similar to the one used by lm :-)
Best of luck with your model!
2 Likes
Thanks it worked!
But the output shows the following:
print(hlm2)
Inference for Stan model: hlm_2level.
4 chains, each with iter=5000; warmup=2500; thin=1;
post-warmup draws per chain=2500, total post-warmup draws=10000.
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
beta -0.62 0.00 0.07 -0.75 -0.66 -0.62 -0.57 -0.49 8942 1
gamma_zero 1.47 0.00 0.03 1.41 1.45 1.47 1.49 1.52 8633 1
gamma_one 0.75 0.00 0.07 0.62 0.71 0.75 0.80 0.88 9623 1
sigma_y 0.74 0.00 0.02 0.71 0.73 0.74 0.75 0.78 9010 1
lp__ -185.23 0.02 1.44 -188.93 -185.91 -184.89 -184.18 -183.46 4355 1
Is there an easy method to compare the unpooled model, the HLM1 and the HLM2?
Not sure how to visualize the shrinkage effect…
You’re correct. I saw the easier versions but afaik it is terrible if you want to specify everything. Using the default priors it is really easy but if you want to go beyond that I think it’s equally cumbersome, so why not learn Stan right away especially if I can use it in future for other Bayesian projects.
1 Like
alpha and beta play the same role in pooled and unpooled model. You could compare pooled vs unpooled alpha for each county. These are the predicted mean radon levels in each county.
However, thinking more about this, the deterministic model of alpha as a linear function of uranium level only isn’t so reasonable. In the original paper you linked the hierarchical model in equation (1) is
\alpha_j \sim N\left(\gamma_0 + \gamma_1 u_j, \sigma_\alpha\right)
That means there’s additional noise and Stan code is
parameters {
...
real<lower=0> sigma_alpha;
vector[J] alpha;
}
model {
...
for(j in 1:J)
alpha[county_j[j]] ~ normal(gamma_zero + gamma_one * U[county_j[j]], sigma_alpha);
...
}
Btw, looking closer I noticed that you had upper case J in alpha[county_j[j]] but that would leave all alphas except the last one undefined.
1 Like
You are right! Thanks for the heads up.
Well, but now the model doesn’t converge…
// HLM 2
data {
int<lower=1> N; // number of obs
int<lower=1> J; // Number of counties
vector[N] Y; // outcome data (log radon)
vector[N] X; // Inputa data Layer 1(room indication)
vector[J] U; // Input data Layer 2 (uranium)
int<lower=1> county_i[N]; //county ID variable for x
int<lower=1> county_j[J]; //county ID variable for u
}
parameters {
real beta; // fixed slope
real gamma_zero; // 2nd level fixed intercept
real gamma_one; // 2nd level fixed slope
real<lower=0> sigma_alpha; // prior on noise of 2nd layer model
real<lower=0> sigma_y;// prior on noise of 1st layer model
}
model { // Model specifications
vector[N] mu_y;
vector[J] alpha; // county j varying intercept
// 2nd Layer
for(j in 1:J)
alpha[county_j[j]] ~ normal(gamma_zero + gamma_one * U[county_j[j]], sigma_alpha);
// 1st Layer
for(i in 1:N)
mu_y[i] = alpha[county_i[i]] + beta * X[i];
// Priors and Hyperparams
Y ~ normal(mu_y, sigma_y);
sigma_y ~ cauchy (0, 2.5);
sigma_alpha ~ cauchy(0,2.5);
beta ~ normal(0, 5);
gamma_zero ~ normal(0, 5);
gamma_one ~ normal(0, 5);
}
This is the error I get:
SAMPLING FOR MODEL 'hlm_2level' NOW (CHAIN 1).
Chain 1: Rejecting initial value:
Chain 1: Error evaluating the log probability at the initial value.
Chain 1: Exception: normal_lpdf: Random variable is nan, but must not be nan! (in 'model3ae07c904b8c_hlm_2level' at line 27)
Chain 1: Rejecting initial value:
Chain 1: Error evaluating the log probability at the initial value.
Chain 1: Exception: normal_lpdf: Random variable is nan, but must not be nan! (in 'model3ae07c904b8c_hlm_2level' at line 27)
..
edit:/ Do I Have to move the alpha back to parameters now again?
edit:/ After moving it back it works but I get a warning message. Not sure if it’s “important” or not…
Warning messages:
1: There were 4 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-low
2: Examine the pairs() plot to diagnose sampling problems
Also: How can I specify the Stan file so that I also calculate the log-likelihood of the models?
I need them to calculate the information criterion and train the model with LOO but it doesn’t work since I don’t provide them in the stanfit-objects. | 2020-01-28 16:26:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.6356969475746155, "perplexity": 8054.130474263191}, "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/1579251779833.86/warc/CC-MAIN-20200128153713-20200128183713-00544.warc.gz"} |
http://mathoverflow.net/revisions/77727/list | Here is one case: Suppose $A$ has a unique eigenvalue $\lambda$ of greatest absolute value that has algebraic multiplicity 1, with left and right eigenvectors $u^T$ and $v$ having all entries nonzero, normalized so $u^T v = 1$. Since $A$ is a real matrix, its complex eigenvalues come in complex-conjugate pairs, so $\lambda$ must be real. Then $A^q = \lambda^q v u^T + o(|\lambda|^q)$ as $q \to \infty$. If all entries of $u^T$ and $v$ have the same sign, then all entries of $A^q$ are positive for all sufficiently large $q$ (if $\lambda > 0$) or all sufficiently large even $q$ (if $\lambda < 0$). If some entries of $u^T$ or $v$ have different signs, there will be entries of $A^q$ with different signs for all sufficiently large $q$, and therefore for all positive integers $q$ (if the elements of $A^q$ all have the same sign, so do the elements of $A^{kq}$ for all positive integers $k$).
EDIT: Here's a partial converse. By the Perron-Frobenius theorem, if $A^q$ has all its entries strictly positive, then $A^q$ has a positive eigenvalue $\mu$ which is greater in absolute value than all other eigenvalues, and is simple, with left and right eigenvectors $u^T$ and $v$ having all entries strictly positive. Since the eigenvalues of $A^q$ are the $q$'th powers of eigenvalues of $A$, there must be one eigenvalue $\lambda$ of $A$ with $\lambda^q = \mu$, also having left and right eigenvectors $u^T$ and $v$. Since $A$ is real and $\mu$ is a simple eigenvalue, $\lambda$ must be real, and we are in the situation of the previous paragraph.
Matters can be somewhat more complicated if $A^q$ is nonnegative but never all strictly positive.
Here is one case: Suppose $A$ has a unique eigenvalue $\lambda$ of greatest absolute value that has algebraic multiplicity 1, with left and right eigenvectors $u^T$ and $v$ having all entries nonzero, normalized so $u^T v = 1$. Since $A$ is a real matrix, its complex eigenvalues come in complex-conjugate pairs, so $\lambda$ must be real. Then $A^q = \lambda^q v u^T + o(|\lambda|^q)$ as $q \to \infty$. If all entries of $u^T$ and $v$ have the same sign, then all entries of $A^q$ are positive for all sufficiently large $q$ (if $\lambda > 0$) or all sufficiently large even $q$ (if $\lambda < 0$). If some entries of $u^T$ or $v$ have different signs, there will be entries of $A^q$ with different signs for all sufficiently large $q$, and therefore for all positive integers $q$ (if the elements of $A^q$ all have the same sign, so do the elements of $A^{kq}$ for all positive integers $k$). | 2013-05-24 17:01:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9822852611541748, "perplexity": 51.63236920403031}, "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/1368704818711/warc/CC-MAIN-20130516114658-00081-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/537710/how-do-i-set-up-the-differential-equations-for-the-finite-length-iron-stick-heat | # How do I set up the differential equations for the finite length iron stick heat diffusion?
Consider an iron cylinder. The heat doesn't go through the cylinder surface, but at the circular cross-section in the endpoints. The temperature of the surroundings is $$15\ \mathrm{^\circ C}$$. The initial temperature of the iron stick is $$100\ \mathrm{^\circ C}$$. How to set up the differential equation? $$T(0,t)$$ and $$T(l,t)$$ doesn't fit anymore.
• Are you asking how the cylinder cools down, when its surface is insulated and heat is lost only through the cross sections at the ends of the cylinder? – Gert Mar 22 '20 at 15:48
• @Gert Yes, I cannot find an equation to describe the amount of heat into the cross-section. – yuanming luo Mar 22 '20 at 16:09
• OK, Will try and answer. – Gert Mar 22 '20 at 16:11
• The confusion is the endpoint of the iron cylinder loss heat and gets heat at the same point, I only know the rate of losing heat at a certain temperature but I cannot determine the rate of gaining heat. – yuanming luo Mar 22 '20 at 16:23
• Are you familiar with the transient heat conduction equation in 1D? – Chet Miller Mar 22 '20 at 17:40
So we have a cylinder of height $$H$$ and radius $$R$$:
Surrounding temperature is $$T_e=15\text{ C}$$ and initial, uniform temperature of the cylinder is $$T_0=100\text{ C}$$.
The cylinder is perfectly insulated, except for both circular ends, which lose heat through convection (only).
The go to equation for this kind of problem is Fourier's heat equation:
$$T_t=\alpha \nabla^2 T$$
Because of the geometry of the problem this equation begs for cylindrical coordinates:
$$\frac{\partial T}{\partial t}=\alpha \Big[\frac{1}{r}\frac{\partial}{\partial r}\Big(r\frac{\partial T}{\partial r}\Big)+\frac{1}{r^2}\frac{\partial^2 T}{\partial \phi^2}+\frac{\partial^2 T}{\partial x^2}\Big]$$
Due to symmetry considerations the $$\phi$$ term isn't needed and developing the partial derivatives, we get (using shorthand partials):
$$\frac{1}{\alpha}T_t=\frac{1}{r}T_r+T_{rr}+T_{xx}$$
$$T_r=T_{rr}=0$$
So:
$$\frac{1}{\alpha}T_t=T_{xx}$$
So we're looking for a function:
$$T(x,t)$$
which describes the spatial distribution of $$T$$, as well as its evolution in time $$t$$. Such a function would also allow to calculate the heat flux through the ends of the cylinder (as a function of time).
All real-world differential equations require boundary conditions and this case is no different.
Initial condition:
$$T(x,0)=100$$
Convection at ends:
$$T_t(r,+\frac{H}{2},t)=h(T-T_e)$$ and: $$T_t(r,-\frac{H}{2},t)=h(T-T_e)$$
where $$h$$ is the convection heat transfer coefficient.
It's advisable to make a small substitution:
$$u=T-T_e$$
So we have:
$$u(x,0)=85$$
$$u_t(+\frac{H}{2},t)=-hu$$
$$u_t(-\frac{H}{2},t)=-hu$$
Once $$u(x,t)$$ is found, we can convert it back to $$T(x,t)$$.
$$u=T-T_e$$
We assume (Ansatz):
$$u(x,t)=X(x)\Theta(t)$$ Inserting into the PDE we get:
$$X\Theta'=\alpha \Theta X''$$ Divide by $$u(x,t)=X(x)\Theta(t)$$:
$$\frac{\Theta''}{\alpha \Theta}=\frac{X''}{X}=-k^2$$
where $$k$$ is a Real number. So we have two ODEs:
$$\frac{\Theta''}{\alpha \Theta}=-k^2\tag{1}$$
$$\frac{X''}{X}=-k^2\tag{2}$$
$$(1)$$ solves to:
$$\Theta(t)=C \exp{(-k^2 \alpha t)}$$
And $$(2)$$ solves to:
$$X(x)=A \sin(kx)+B \cos(kx)\tag{3}$$
'All that is left to do' is determine the integration constants $$A$$, $$B$$ and $$C$$.
Unfortunately the BCs are of the Neumann type and thus non-homogeneous (non-zero). This generally creates an unsightly mess with no easy way from which to extricate $$A$$ and $$B$$.
Instead I'll try the much simpler case where the ends are kept at constant temperature $$u=0$$, so: $$u(-\frac{H}{2},t)=u(\frac{H}{2},t)=0$$ This also means: $$X(-\frac{H}{2})\Theta(t)=X(\frac{H}{2})\Theta(t)=0$$ Assume $$\Theta(t) \neq 0$$, thus: $$X(-\frac{H}{2})=X(\frac{H}{2})=0$$ Insert into $$(3)$$:
$$A \sin(k\frac{H}{2})+B \cos(k\frac{H}{2})\tag{4}=0$$
$$A \sin(-k\frac{H}{2})+B \cos(-k\frac{H}{2})=0$$ From the last equation: $$A \sin(k\frac{H}{2})-B \cos(k\frac{H}{2})=0\tag{5}$$ Now add $$(4)$$ to $$(5)$$:
$$2A\sin(k\frac{H}{2})=0$$
Assume $$A \neq 0$$, then:
$$\sin(k\frac{H}{2})=0$$
This happens for:
$$k\frac{H}{2}=n \pi$$ Or: $$k=\frac{2n\pi}{H}$$ For $$n=1,2,3,4,...$$
With $$\sin(k\frac{H}{2})=0$$, then $$B=0$$. So we get:
$$X_n(x)=A_n\sin\Big(\frac{2n\pi x}{H}\Big)$$
And:
$$u_n(x,t)=D_n \exp{\Big[-\Big(\frac{2n\pi}{H}\Big)^2 \alpha t\Big]}\sin\Big(\frac{2n\pi x}{H}\Big)$$
Applying the superposition principle:
$$u(x,t)=\sum_{n=1}^{+\infty}D_n \exp{\Big[-\Big(\frac{2n\pi}{H}\Big)^2 \alpha t\Big]}\sin\Big(\frac{2n\pi x}{H}\Big)$$
The coefficients $$D_n$$ are obtained with the initial condition and the Fourier series.
For $$t=0$$, $$u(x,0)=85$$ and:
$$85=\sum_{n=1}^{+\infty}D_n \sin\Big(\frac{2n\pi x}{H}\Big)$$
Thus:
$$D_n=\frac{2}{H}\int_{-H/2}^{+H/2}85 \sin\Big(\frac{2n\pi x}{H}\Big)\text{d}x$$
• Edit made: eliminated the confusing dual use o the symbol $T$. – Gert Mar 22 '20 at 18:12
• Now justify why the Ansatz yields a complete solution :p – user224659 Mar 22 '20 at 18:52
• I know that there is no easy justification for this in the general case.. Always left me kind of unsatisfied when first encountering this kind of problem in classical electrodynamics. – user224659 Mar 22 '20 at 18:55
• Trust me: the dreaded non-homogeneous BCs are a bigger problem! – Gert Mar 22 '20 at 19:48
• This is a 1D problem, and the temperature does not vary with r. It is only a function of x and t. – Chet Miller Mar 22 '20 at 21:45 | 2021-06-13 06:08:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 71, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8421282172203064, "perplexity": 560.7124243085094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487600396.21/warc/CC-MAIN-20210613041713-20210613071713-00083.warc.gz"} |
https://learn.careers360.com/ncert/question-find-the-product-into-brackets-4-p-power-2-plus-5p-plus-7-into-3p/ | Q
# Find the product : (4p^2 + 5p + 7) * 3p
Q: Find the product:
$(4p^2+5p+7)\times 3p$
$(4p^2+5p+7)\times 3p = 12p^3 + 15p^2 + 21p$ | 2020-02-20 19:18:16 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "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": 0.7665785551071167, "perplexity": 7504.397963675959}, "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/1581875145260.40/warc/CC-MAIN-20200220162309-20200220192309-00211.warc.gz"} |
https://www.nature.com/articles/s41598-020-60576-4?error=cookies_not_supported&code=92e277bc-6755-47af-acec-a1c6592e7eeb | # Temporal discounting correlates with directed exploration but not with random exploration
## Abstract
The explore-exploit dilemma describes the trade off that occurs any time we must choose between exploring unknown options and exploiting options we know well. Implicit in this trade off is how we value future rewards — exploiting is usually better in the short term, but in the longer term the benefits of exploration can be huge. Thus, in theory there should be a tight connection between how much people value future rewards, i.e. how much they discount future rewards relative to immediate rewards, and how likely they are to explore, with less ‘temporal discounting’ associated with more exploration. By measuring individual differences in temporal discounting and correlating them with explore-exploit behavior, we tested whether this theoretical prediction holds in practice. We used the 27-item Delay-Discounting Questionnaire to estimate temporal discounting and the Horizon Task to quantify two strategies of explore-exploit behavior: directed exploration, where information drives exploration by choice, and random exploration, where behavioral variability drives exploration by chance. We find a clear correlation between temporal discounting and directed exploration, with more temporal discounting leading to less directed exploration. Conversely, we find no relationship between temporal discounting and random exploration. Unexpectedly, we find that the relationship with directed exploration appears to be driven by a correlation between temporal discounting and uncertainty seeking at short time horizons, rather than information seeking at long horizons. Taken together our results suggest a nuanced relationship between temporal discounting and explore-exploit behavior that may be mediated by multiple factors.
## Introduction
The explore-exploit dilemma refers to a ubiquitous problem in reinforcement learning in which an agent has to decide between exploiting options it knows to be good and exploring options whose rewards are unknown1. For example, when ordering sushi at a favorite restaurant, should we exploit our usual favorite (the Rainbow Roll), which is guaranteed to be good, or explore the Burrito Roll, which could be delicious, disgusting or somewhere in between. As anyone who has agonized over a dining decision will know, making explore-exploit choices can be hard, and there is considerable interest in how these decisions are made by humans and other animals2.
Recently, a number of studies have shown that people make explore-exploit decisions using a mixture of two strategies: directed exploration and random exploration3,4,5,6,7,8. In directed exploration, choices are biased towards more informative options by an ‘information bonus,’ that increases the relative value of unknown options9. In random exploration, behavioral variability, perhaps driven by random noise processes in the brain, causes exploratory options to be chosen by chance1,10. Further work suggests these two types of exploration have different computational properties4, age dependence11, and may be controlled by different systems in the brain12,13,14,15.
Regardless of the type of exploration, the benefits of exploring over exploiting lie in the possibility of earning larger rewards in the future. For example, in our restaurant example, if the Rainbow Roll is an above-average item on the menu, then, in the short term, exploiting it will usually be best. In the longer term, however, if the Burrito Roll turns out to be sublime, then we could order this roll again and again for years to come. Thus, how much we care about future rewards, that is how we discount them relative to immediate rewards, should play a critical role in how we make our explore-exploit choice.
Optimal models of explore-exploit decision making formalize this relationship between temporal discounting and exploration, at least for directed exploration9. In these models, the explore-exploit choice is made by choosing the option that maximizes the expected discounted future reward. Because this maximizing behavior is deterministic (apart from rare cases in which options are tied), optimal models do not exhibit random exploration. Thus, while they predict a negative relationship between temporal discounting and directed exploration, they say nothing about the relationship with random exploration. Sub-optimal models of explore-exploit decision making do include random exploration, but most of them predict no relationship with temporal discounting1,10,16.
Thus, in theory, one might predict a negative relationship between temporal discounting and directed exploration, and no relationship between temporal discounting and random exploration. In practice, however, previous experimental work suggests a more nuanced picture because of how temporal discounting covaries with our attitudes toward risk. In particular, high temporal discounting is associated with greater impulsivity17, and higher impulsivity is associated with greater risk taking18. This suggests that more temporal discounting is associated with more risk seeking19,20 (However, by defining risk seeking in terms of probability discounting, some studies on the relationship between temporal and probability discounting have yielded ambiguous results on this suggestion21,22,23,24,25). In most explore-exploit paradigms, such increased risk taking would look a lot like increased directed exploration, because the more informative option is usually more uncertain, i.e. risky, too. Thus, while theory might predict a negative correlation between temporal discounting and directed exploration, this effect could be countered by a positive correlation between temporal discounting and risk taking.
In the current study, we investigated the correlation between temporal discounting and the two kinds of exploration using an individual differences approach. That is, we asked whether people with higher temporal discounting show less directed and/or random exploration. We used the 27-item Delay Discounting Questionnaire26 to measure temporal discounting. In this questionnaire, participants choose between between small but immediate amounts of money and a larger but delayed amounts of money (e.g. $11 now or$30 in two weeks). Based on participants’ pattern of choosing between immediate and delayed options, a parameter k27 is calculated for each participant which estimates their average discounting rate for delayed rewards.
We used the Horizon Task3 to measure directed and random exploration. In this task participants make a series of choices between two slot machines (one-armed bandits). When played, each machine pays out a reward from a Gaussian distribution. The average payout is different for each machine such that one option is always better on average. Thus, to maximize their rewards, participants need to exploit the option with the highest average payout, but can only find out which option is best by exploring both options first. By manipulating key parameters in this task (distribution of rewards, time horizon, and the amount of uncertainty for each bandit), the Horizon Task allows us to quantify directed and random exploration, and, crucially, to dissociate them from baseline risk seeking and behavioral variability.
Thus, by comparing individual differences in behavior on the Horizon Task with individual differences in temporal discounting, we aimed to quantify the relationship between the two types of exploration and temporal discounting.
## Methods
### Participants and sample size
We collected data from a total of 82 participants (ages 18–25, average = 19.10; Females = 47, Males = 35). To estimate the sample size, we chose the conventional level of significance at α = 0.05, and the typical power at P = 0.8. A priori power analysis provided by Cohen28 and implemented in G*Power 3 software29, estimated n = 82 as the appropriate sample size for a desired medium effect size of r = 0.3 at α = 0.05 and P = 0.8. We aimed to recruit around 100 participants but ended up with 82 which is sufficient for our desired level of significance and power. Participants were recruited through the Psychology subject pool at the University of Arizona and received course credit for their participation. All participants gave informed consent and the study was approved by the Institutional Review Board at the University of Arizona and all experiments were performed in accordance with relevant guidelines and regulations.
### Temporal discounting measure
To measure temporal discounting we used the Delay Discounting Questionnaire developed by30. In this instrument there are 27 questions asking participants’ preferences between two hypothetical monetary rewards: one of which pays immediately but is smaller, and the other pays more but is delayed. For example, one item asks: Do you prefer $11 today or$30 in 7 days? The amount of smaller-immediate reward (“today” option), larger-delayed reward (“later” option) and the delay (in terms of days) vary in those 27 questions (“today” reward between $11–$80; “later” reward between $25–$85; Delay between 7–186 days). The exact values are reported in30-Table 3.
One out of four participants were selected by chance (by drawing a card at the end of experiment) to receive the actual money according to their responses. If a participant drew a winning card (%25 chance), they then would proceed to draw a numbered chip from a bag (out of 27 chips numbered from 1 to 27 according to the number of items in the monetary choice questionnaire). The number on the chip corresponds to the number of the question we would look at for the actual pay-out. For example, if the winning participant picked the number 19 and they answered “later” on the question #19: “Do you prefer $33 now or$80 in 14 days?”, they need to come back to lab in 14 days and receive $80 in cash after signing a receipt form. To quantify temporal discounting we used a number of different measures. The simplest was just the number of today options chosen, with greater temporal discounting associated with larger number of “today” choices. More sophisticated measures of temporal discounting were obtained by fitting a hyperbolic discount factor to the data. In particular, we assume that future reward, A, arriving after a delay D, is discounted according to a hyperbolic discount factor31: $$V=A/(1+kD)$$ (1) where k is the subject-specific discount factor. Fitting k was done using the spreadsheet provided by32 based on the method described in27. In addition to computing an overall k using all 27 items, this approach also computes separate discount factors for small, medium and large reward items, based on the idea that delay discounting may be different for different range of rewards, and also the geometric mean of the small, medium and large ks. Based on the range of monetary values, the 27 choices are divided into three 9-item categories: small, medium and large ranges. Then, based on the hyperbolic discounting equation (Eq. 1), it finds a k value for each item as a point in which there is no difference between choosing “today” and “later” options for that item. Then for each participant based on his/her answers and the patterns of switches from “today” to “later” options and the reverse, it gives us a k-value for each 9-item category: Small k, Medium k, Large k. For example, in question 2 it asks: Would you prefer$55 today, or $75 in 61 days? The indifference point is when the$75 in 61 days worth as $55 today. We can calculate the k for the indifference point, in which the "today” and “later” choices look the same, by plugging V = 55, A = 75, D = 61 in Eq. (1): $$\begin{array}{lll}55 & = & 75/(1+61* k)\\ k & = & ((75/55)-1)/61\\ k & = & 0.00596125\end{array}$$ If a participant choose “today” for this question, they have a k > 0.00596125. Similarly, if the same participant answer “later” in question 7: Would you prefer$15 today, or \$35 in 13 days?, the indifference point would be k = ((35/15) − 1)/13 = 0.102564103 so our participant would have a k < 0.102564103. So for this participant given these two questions, we can estimate their k to be between 0.00596125 < k < 0.102564103. By adding more questions, we can obtain better estimates for k.
Thus we have six measures of temporal discounting for each subject: the fraction of “today” choices, overall k, small k, medium k, large k, and the geometric mean of small, medium and large ks.
The Horizon Task3 is a recently developed task that allows for the measurement of directed and random exploration. The key manipulation in the Horizon Task is the time horizon, the number of trials participants will make in the future. The idea being, that in a long time horizon, people should explore, while in a short time horizon, people should exploit. Thus the change in behavior between short and long horizons can be used to quantify directed and random exploration.
More specifically, in the Horizon Task participants choose between two one-armed bandits. When chosen, the bandits pay out rewards sampled from a Gaussian distribution whose standard deviation is always fixed at 8 points, but whose mean is different for each bandit and can change from game to game. Each game lasts for 5 or 10 trials and participants’ job is to make multiple choices between the two bandits to try to maximize their reward. Because they know nothing about the mean of each bandit at the start of each game, they can only find out which option is best by exploring.
To control the amount of information, the first four trials of each game are predetermined (Fig. 1B). Participants are instructed to pick either the left or right bandit during these four “forced trials”. By changing the number of forced choices for each bandit, we manipulate the amount of “uncertainty” or information participants have about the payoffs from each bandit. In the unequal uncertainty (or [1 3] condition) participants are forced to choose one option once and the other three times; whereas in the equal uncertainty (or [2 2] condition) participants play both options twice. After the forced-choice trials, the rest of trials are “free trials” in which participants make their own choice. The number of free trials varies between horizon conditions with 1 free choice in the horizon 1 condition and 6 free choices in the horizon 6 condition.
These two information conditions allow us to quantify directed and random exploration by looking at the first free choice in each game, immediately after the four forced choices (Fig. 1A). Because directed exploration involves information seeking, it can be quantified as the probability of choosing the more informative option in the [1 3] condition, p(high info). Conversely, because random exploration involves decision noise, it correlates with choosing the low mean option in the [2 2] condition, p(low mean). Computing these measures separately for each horizon condition allows us to quantify four key properties of explore-exploit behavior:
• uncertainty preference as p(high info) in horizon 1
• baseline behavioral variability as p(low mean) in horizon 1
• directed exploration as Δp(high info), the change in information seeking with horizon
• random exploration as Δp(low mean), the change in variability with horizon
In Supplementary Materials-4 you can find the onscreen instructions used to instruct participants at the beginning of the Horizon Task.
### Model-based analysis
In addition to the above-mentioned model-free parameters (p(high info) and p(low mean)) we also fit a logistic model that was previously shown to be adequate in capturing the basics of the Horizon Task3. With this model we estimate two main parameters: “information bonus” and “decision noise” which corresponds to the model-free measures of directed and random exploration, respectively. The description of the model is provided in the Supplementary Materials-1. The modeling will help us to disentangle directed and random exploration more clearly. However, since there was a high correlation between model-free and model-based parameters (Supplementary Materials-2 Fig. S1) and both the model-based and model-free parameters yielded the same relationships with the temporal discounting (Supplementary Materials-2 Fig. S2), and since the model-free approach requires less assumptions than the model-based approach, we chose to include the model-free analysis in the main article and move the modeling part to the Supplementary Materials.
### Statistical analysis
To evaluate the basic behavior on the Horizon Task, we used the paired (dependent) sample t-test. For directed exploration we looked to see whether there was a significant increase in the mean of p(high info) from horizon 1 to horizon 6 condition using the paired sample t-test. Similarly, for the random exploration we used paired sample t-test to see whether there was a significant increase in the p(low mean) between horizon 1 and horizon 6 condition.
To evaluate the relationship between measures of temporal discounting and the Horizon Task parameters, we simply calculated the Pearson correlation coefficients between the 6 measures of temporal discounting (the 5 k’s: overall k, small k, medium k, large k, geometric k and the total number of today items chosen) on one hand and the Horizon Task parameters (directed exploration, random exploration, p(high info) in horizon 1 and 6, p(low mean) in horizon 1 and 6, reaction time on the first free trial in horizon 1 and 6, and accuracy in horizon 1 and 6) on the other hand. Accuracy is defined as choosing the high mean option.
### Compliance with ethical standards
All procedures performed in experiments were in accordance with the ethical standards of the institutional research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.
### Informed consent
Informed consent was obtained from all individual adult participants included in the study.
## Results
### Behavior on the horizon task (Model-free)
Table 1 shows the range, mean and standard deviation (SD) for the basic task parameters in the sample. Figure 2 shows the distribution of basic task parameters in the sample (N = 82). Behavior on the Horizon Task was consistent with that previously reported in our studies3. Specifically we see a significant increase in p(low mean) with horizon (p(low mean)h1_average = 0.2883; p(low mean)h6_average = 0.3554; t(81) = 3.87; p < 0.001; and the effect size of d = 0.4033) (Fig. 3B) and we see a clear trend (but not significant) in p(high info) with horizon (p(high info)h1_average = 0.5146; p(high info)h6_average = 0.5486; t(81) = 1.75; p = 0.084; d = 0.24) (Fig. 3A), consistent with participants using both types of exploration in this paradigm. Figure 4 shows the scatter plots comparing p(high info) and p(low mean) for individual participants in horizon 1 and horizon 6 conditions. Out of 82 participants, 57 individuals showed random exploration (p(low mean) h6 > p(low mean) h1) and 47 individuals showed directed exploration (p(high info) h6 > p(high info) h1) on average.
### Behavior on the temporal discounting task
For the temporal discounting measure we obtained 5 different k values for each participant as a measure of how much they discount future reward. We also can simply estimate that measure just by counting the number of times participants chose the immediate versus delayed reward (Supplementary Materials-3). Table 2 shows the range, mean and standard deviations of temporal discounting indices (k’s and # today items) in 82 participants of our study which is similar to previous studies using the same measure26,30. Figure 5 shows the histogram of distribution of temporal discounting indices in the sample (N = 82).
In our research, it turned out that all of these indices are highly correlated with each other (Supplementary Materials-3) and all have very similar relationship with directed and random exploration. The more simple measure of # today items has a Pearson’s correlation coefficient between 0.89–1 with the more complicated k measures (Supplementary Materials-3 Fig. S3).
### Correlation between temporal discounting and explore-exploit behavior
Table 3 shows the correlations between a measures of temporal discounting (log k overall) and the horizon task parameters: directed and random exploration, p(high info) & p(low mean) at horizons 1 & 6, reaction times and accuracy (the percentage of times the “accurate” option (the higher mean option) was chosen for each horizon (1 & 6) conditions. We found a significant negative correlation between between temporal discounting and directed exploration, with more temporal discounting associated with less directed exploration. Closer inspection revealed that this negative correlation was driven by a positive correlation between temporal discounting and p(high info) at horizon 1 and a zero correlation between temporal discounting and p(high info) at horizon 6.
In contrast to directed exploration, temporal discounting did not correlate with random exploration. There was, however, a positive correlation between temporal discounting and overall behavioral variability, p(low mean) in both horizon conditions. This suggests that people with higher temporal discounting perform worse on the task overall.
Finally, to demonstrate that the significant correlations were not driven by outliers, we plot the correlations between measures of directed and random exploration and the number of today items chosen in Fig. 6.
### Model-based analysis
We also utilized a logistic model (further explained in the Supplementary Materials-1) to estimate two main parameters, “information bonus” and “decision noise”, which are assumed to correspond to p(high info) and p(low mean) in the model-free analysis, respectively. Figure S1 in the Supplementary Materials-2 shows that in fact there are high correlations between model-free and model-based parameters. Additionally, Fig. S2 shows that the correlations between temporal discounting and model-based parameters are similar to the correlations between temporal discounting and model-free parameters (Fig. 6).
## Discussion
In this study we investigated the correlation between temporal discounting measured by a monetary choice questionnaire30 and two types of exploration (directed and random) measured by the Horizon Task3. We found a negative correlation between temporal discounting and directed exploration that was driven by a positive correlation between temporal discounting and uncertainty seeking in horizon 1. Conversely, we found no correlation between temporal discounting and random exploration, although we did see a positive correlation between temporal discounting and overall behavioral variability.
While the negative correlation between temporal discounting and directed exploration (i.e. Δp(high info) is consistent with the theory, the correlation with p(high info) in each horizon condition is not. In particular, normative models predict a negative correlation between temporal discounting and p(high info) in horizon 6 and no correlation in horizon 1. Conversely, we found no correlation with horizon 6 behavior and a positive correlation with horizon 1 behavior.
One reason for this discrepancy could be the possible positive association between temporal discounting and risk taking19,20 (See21,22,23,24,25 for suggesting otherwise). In both horizon conditions in the Horizon Task, the more informative option is also the more uncertain, riskier option. Thus, by this account, people who discount more would show greater p(high info) in both horizon conditions, but this would be counteracted by a negative relationship between temporal discounting and directed exploration in horizon 6. That is, in horizon 1, directed exploration is not present, and so the positive association with temporal discounting is revealed. In horizon 6, directed exploration is present, and this negative relationship with temporal discounting counteracts the positive relationship with risk taking leaving no correlation overall. Testing this hypothesis requires a future study that includes appropriate measures of risk taking.
The fact that random exploration does not correlate with temporal discounting is also consistent with theories of random exploration1,10. Moreover, this apparent dissociation between directed and random exploration is consistent with other findings showing that directed and random exploration have different computational properties4, different age dependence11, and may rely on dissociable neural systems12,14,15. In this regard it is notable that directed exploration appears to rely on the same frontal systems thought to underlie temporal discounting5,12,14,34,35,36, while random exploration does not. Thus, an intriguing prediction is that the relationship between directed exploration and temporal discounting may be mediated by the integrity of frontal circuits, something that future neuroimaging studies could address.
There are several limitations in the current study. First, the chosen measures for both temporal discounting and exploratory behavior are very specific. This questions the generalizibility of our results. Although a strong correlation between different measures of temporal discounting has been demonstrated in several studies37,38, most of these measures are monetary which may have weak relationships with delay discounting in other domains39. Exploratory behavior also has been studied in different settings including foraging, repeated choice and sequential choice paradigms and it seems there is no shared factor underlying exploratory behavior in all of these tasks40. Replicating the current study using other measures of exploration and temporal discounting, will provide us with more evidence to better assess the generlizablilty of the current results.
Another important limitation of our study is recruiting university students as participants. Between all possible biases that such a selective sample may introduce in our study, age seems the most obvious one. It has been shown that temporal discounting41, exploratory behavior42 and risk-taking behavior43, all varies significantly through the lifespan. So it is unclear how the results of the current study would look like in different age groups. This would be an interesting topic for a future study.
Lastly, we hypothesised the mediating role of risk taking to explain the results while we haven’t included appropriate scales to measure it in the current study. A future study can shed more light on this hypothesis by adding measures of risk taking.
## Data availability
All the raw data and MATLAB codes for the analysis and plots are available at https://github.com/hashem20/temporal-discounting-explore-exploit.
## References
1. 1.
Sutton, R. S. and Barto, A. G. Reinforcement learning : an introduction (MIT press, 1998).
2. 2.
Daw, N. D., O’Doherty, J. P., Dayan, P., Seymour, B. & Dolan, R. J. Cortical substrates for exploratory decisions in humans. Nature 441, 876–879 (2006).
3. 3.
Wilson, R. C., Geana, A., White, J. M., Ludvig, E. A. & Cohen, J. D. Humans use directed and random exploration to solve the explore-exploit dilemma. Journal of experimental psychology. General 143, 2074–81 (2014).
4. 4.
Gershman, S. J. Deconstructing the human algorithms for exploration. Cognition 173, 34–42 (2018).
5. 5.
Frank, M. J., Doll, B. B., Oas-Terpstra, J. & Moreno, F. Prefrontal and striatal dopaminergic genes predict individual differences in exploration and exploitation. Nature Neuroscience 12, 1062–1068 (2009).
6. 6.
Schulz, E. & Gershman, S. J. The algorithmic architecture of exploration in the human brain. Current Opinion in Neurobiology 55, 7–14 (2019).
7. 7.
Wyart, V. & Koechlin, E. Choice variability and suboptimality in uncertain environments. Current Opinion in Behavioral Sciences 11, 109–115 (2016).
8. 8.
Wu, C. M., Schulz, E., Speekenbrink, M., Nelson, J. D. & Meder, B. Generalization guides human exploration in vast decision spaces. Nature Human Behaviour 2, 915–924 (2018).
9. 9.
Gittins, J. C. Bandit Processes and Dynamic Allocation Indices. Journal of the Royal Statistical Society. Series B (Methodological) 41, 148–177 (1979).
10. 10.
Watkins, C. Learning from delayed rewards. Ph.D. thesis, Cambridge University (1989).
11. 11.
Somerville, L. H. et al. Charting the expansion of strategic exploratory behavior during adolescence. Journal of experimental psychology. General 146, 155–164 (2017).
12. 12.
Zajkowski, W. K., Kossut, M. & Wilson, R. C. A causal role for right frontopolar cortex in directed, but not random, exploration. eLife 6 (2017).
13. 13.
Blanchard, T. C. & Gershman, S. J. Pure correlates of exploration and exploitation in the human brain. Cognitive, Affective and Behavioral Neuroscience 18, 117–126 (2018).
14. 14.
Gershman, S. J. & Tzovaras, B. G. Dopaminergic genes are associated with both directed and random exploration. Neuropsychologia 120, 97–104 (2018).
15. 15.
Warren, C. M. et al. The effect of atomoxetine on random and directed exploration in humans. PL0S One 12, e0176034 (2017).
16. 16.
Thompson, W. R. On the Likelihood that One Unknown Probability Exceeds Another in View of the Evidence of Two Samples. Biometrika 25, 285 (1933).
17. 17.
Wittmann, M. & Paulus, M. P. Decision making, impulsivity and time perception. Trends in Cognitive Sciences 12, 7–12 (2008).
18. 18.
Zuckerman, M. & Kuhlman, D. M. Personality and risk-taking: Common biosocial factors. Journal of Personality 68, 999–1029 (2000).
19. 19.
Madden, G. and Bickel, W. Impulsivity: The behavioral and neurological science of discounting. (2010).
20. 20.
Hill, E. M., Jenkins, J. & Farmer, L. Family unpredictability, future discounting, and risk taking. The Journal of Socio-Economics 37, 1381–1396 (2008).
21. 21.
Richards, J. B., Zhang, L., Mitchell, S. H. & de Wit, H. Delay or probability discounting in a model of impulsive behavior: effect of alcohol. Journal of the Experimental Analysis of Behavior 71, 121–143 (1999).
22. 22.
Green, L. & Myerson, J. A discounting framework for choice with delayed and probabilistic rewards. Psychological Bulletin 130, 769–792 (2004).
23. 23.
Reynolds, B., Richards, J. B., Horn, K. & Karraker, K. Delay discounting and probability discounting as related to cigarette smoking status in adults. Behavioural Processes 65, 35–42 (2004).
24. 24.
Myerson, J., Green, L., Scott Hanson, J., Holt, D. D. & Estle, S. J. Discounting delayed and probabilistic rewards: Processes and traits. Journal of Economic Psychology 24, 619–635 (2003).
25. 25.
Shead, N. W. & Hodgins, D. C. Probability discounting of gains and losses: Implications for risk attitudes and impulsivity. Journal of the Experimental Analysis of Behavior 92, 1–16 (2009).
26. 26.
Kirby, K. N. & Maraković, N. N. Delay-discounting probabilistic rewards: Rates decrease as amounts increase. Psychonomic Bulletin and Review 3, 100–104 (1996).
27. 27.
Kaplan, B. A. et al. Automating Scoring of Delay Discounting for the 21- and 27-Item Monetary Choice Questionnaires. Behavior Analyst 39, 293–304 (2016).
28. 28.
Cohen, J. Statistical Power Analysis for the Behavioral Sciences (Hillsdale, NJ: Erlbaum, 1988).
29. 29.
Faul, F., Erdfelder, E., Lang, A. G. & Buchner, A. G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods 39, 175–191 (2007).
30. 30.
Kirby, K. N., Petry, N. M. & Bickel, W. K. Heroin addicts have higher discount rates for delayed rewards than non-drug-using controls. Journal of Experimental Psychology: General 128, 78–87 (1999).
31. 31.
Mazur, J. E. An adjusting procedure for studying delayed reinforcement. In Commons, M. L., Mazur, J. E., Nevin, J. A. & Rachlin, H. (eds.) Quantitative analyses of behavior: vol. 5. The effect of delay and of intervening events on reinforcement value, 55–73 (Erlbaum, Hillsdale, New Jersey, USA, 1987).
32. 32.
Kaplan, B. A., Lemley, S. M., Reed, D. D. & Jarmolowicz, D. P. 21- and 27- Item Monetary Choice Questionnaire Automated Scorer. University of Kansas (2014).
33. 33.
Dunlap, W. P., Cortina, J. M., Vaslow, J. B. & Burke, M. J. Meta-analysis of experiments with matched groups or repeated measures designs. Psychological Methods 1, 170–177 (1996).
34. 34.
Doya, K. Metalearning and neuromodulation. Neural Networks 15, 495–506 (2002).
35. 35.
McClure, S. M., Laibson, D. I., Loewenstein, G. & Cohen, J. D. Separate Neural Systems Value Immediate and DelayedMonetary Rewards: EBSCOhost. Science 306, 503–507 (2004).
36. 36.
McClure, S. M., Ericson, K. M., Laibson, D. I., Loewenstein, G. & Cohen, J. D. Time Discounting for Primary Rewards. Journal of Neuroscience 27, 5796–5804 (2007).
37. 37.
Basile, A. G. & Toplak, M. E. Four converging measures of temporal discounting and their relationships with intelligence, executive functions, thinking dispositions, and behavioral outcomes. Frontiers in Psychology 6, 728 (2015).
38. 38.
Epstein, L. H. et al. Comparison between two measures of delay discounting in smokers. Experimental and Clinical Psychopharmacology 11, 131–138 (2003).
39. 39.
Weatherly, J. N., Terrell, H. K. & Derenne, A. Delay discounting of different commodities. Journal of General Psychology 137, 273–286 (2010).
40. 40.
von Helversen, B., Mata, R., Samanez-Larkin, G. R. & Wilke, A. Foraging, exploration, or search? On the (lack of) convergent validity between three behavioral paradigms. Evolutionary Behavioral Sciences 12, 152–162 (2018).
41. 41.
Green, L., Fry, A. F. & Myerson, J. Discounting of delayed rewards: A Life-Span Comparison. Psychological Science 5, 33–36 (1994).
42. 42.
Chin, J., Anderson, E., Chin, C. L. & Fu, W. T. Age differences in information search: An exploration-exploitation tradeoff model. In Proceedings of the Human Factors and Ergonomics Society 59th Annual Meeting, vol. 59, 85–89 (Sage CA: Los Angeles, 2015).
43. 43.
Rutledge, R. B. et al. Risk Taking for Potential Reward Decreases across the Lifespan. Current Biology 26, 1634–1639 (2016).
## Acknowledgements
The authors thank Shlishaa Savita and Kathryn Lui Kellohen for their help in collecting and organizing data.
## Author information
Authors
### Contributions
H.S., S.W., A.S. and R.C.W. designed the experiment. H.S., M.R.A. and H.M.K. ran the experiment. H.S. and S.W. analyzed the data with supervision from R.C.W. H.S. and R.C.W. wrote the manuscript with input from all other authors.
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Sadeghiyeh, H., Wang, S., Alberhasky, M.R. et al. Temporal discounting correlates with directed exploration but not with random exploration. Sci Rep 10, 4020 (2020). https://doi.org/10.1038/s41598-020-60576-4
• Accepted:
• Published: | 2020-08-03 18:38:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6446821093559265, "perplexity": 3848.5874968953144}, "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/1596439735823.29/warc/CC-MAIN-20200803170210-20200803200210-00464.warc.gz"} |
https://blogs.mathworks.com/pick/2014/01/03/fit-a-sphere/?s_tid=blogs_rc_3 | # Fit a Sphere!
Sean's pick this week is Sphere Fit by Alan Jennings.
### Fitting a Sphere to Data
I recently had some data and wanted to fit a sphere through it so that I could find the radius of this sphere. As I started writing out an objective function for one of the Optimization Toolbox optimizers (yes I was taking the way too big a hammer approach), a quick query on the File Exchange brought up sphereFit, a pleasant discovery.
Let's see it in action:
% Load mri of a human head
D = squeeze(S.D);
% Generate data points on the edge of the head
for ii = size(D,3):-1:1;
% Perimeter of convex hull of each slice
M(:,:,ii) = bwperim(bwconvhull(D(:,:,ii)>50));
end
% Find sub indices in our mask
idx = find(M); % find points in M
[rr,cc,pp] = ind2sub(size(M),idx); % sub indices
pp = pp.*floor(size(D,1)./size(D,3)); % rescale third dimension
% View data
figure;
scatter3(rr,cc,pp,10,pp);
daspect([1,1,1]);
view(-121,36);
axis tight;
% Fit the Sphere:
fprintf(1,'\nRadius of sphere is %3.1f\nIt is centered at [%3.1f %3.1f %3.1f]\n',radius,cent);
Radius of sphere is 54.3
It is centered at [72.9 63.4 51.7]
Use Alan's example code to show the sphere through the points
scatter3(cc,rr,pp,25,pp,'*'); %points
hold on; % hold
daspect([1,1,1]); % equal axis so a sphere looks like a sphere
[Base_rr,Base_cc,Base_pp] = sphere(20);
axis tight;
view(-121,36); | 2021-10-18 16:42:38 | {"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": 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": 0.45482027530670166, "perplexity": 5348.771506983069}, "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/1634323585204.68/warc/CC-MAIN-20211018155442-20211018185442-00463.warc.gz"} |
https://modelingwithdata.org/arch/00000135.htm | ### How to learn a new programming language
Part of a series of tips on POSIX and C. Start from the tip intro page, or get 21st Century C, the book based on this series.
This is a sequence of a dozen or so steps for getting to know a new programming language.
It came to mind a few weeks ago, during an informal job-type interview. There's always that point in the interview where the interviewer asks So what programming languages do you know? I know she was expecting something like Yeah, I took some SAS classes'; I told her I had reasonable facility and had done at least some nontrivial work in C, C++, Java, S-PLUS and R, Scheme, Matlab and Octave, Perl, Python, Ruby, FORTRAN 77, and I forget a few.
I find some people like my interviewer, who get by on what they learned in school; and others who find my list of a dozen languages to be average, who see that it's the same thing over and over but with the parens in different places. The intent of this paper is to point out the commonalities, and to help you make the jump to being a computing badass who sees across languages.
This paper is a checklist, a series of exercises, and a series of suggestions of where to look when you first find out that you're going to have to do work in some platform that you never thought about until somebody told you it's going to be your best friend for the duration of the next project. You'll need to work with the documentation and tutorials for your target language to answer all the questions I ask here--and you'll really have to delve. Introductory tutorials tend to be advertisements for the easiest features of a language, but you can't seriously work until you've gotten to know the ugly and the missing parts as well.
In writing this, I contend that this checklist applies to any mainstream programming language (thus excluding specialized languages like TEX, sed, gnuplot, or SAS). When you discover inevitable exceptions, that's great, because revealing those exceptions will reveal the character of the language and what makes it different from (and hopefully better than) all the others. So the questions are easy and lowest-common-demoninator, but once you find the right part of the manual that describes how to do the easy things, you're in the right place to see if your language does more.
OK, on to the checklist. I'll start with the basic platform, then move on to data types, functions, and larger blobs of code.
###### How do I print “Hello world” to the screen?
This was originally an exercise from Kernighan and Ritchie's 1978 C book, and the question it is really asking is the same in all languages: how do I set up the environment in which I have to work?
It is the question I can offer the least guidance on, because there are just so many ways in which these things happen: there are compilers, interpreters, integrated development environments (IDEs), a few languages that primarily run in your browser, and who knows what else.
Ex. Get Hello world to print on your screen.
###### How do I insert comments?
As essential as this is, some systems don't have an explicit comment mechanism, but just expect you to write a string or some other expression that evaluates to something innocuous.
Python. """This part prints Hello World:"""
###### Where's the documentation?
Your language may have a clever means of built-in documentation, which you will naturally want to get to know immediately. There may be manual pages, like the man perl set of pages or the POSIX-standard man pages for C libraries (try man 3 printf or man operator).
But the answer to this question also lies with your Web browser. The built-in and official documentation is probably a reference work, and learning from reference works is a bad idea: you also want didactic works that point out what's important in the reference, and you'll find those by checking in with your favorite search engine.
You will almost certainly be using external libraries or packages of some sort, so take some time to find the reference and/or tutorial documentation for those. They may or may not be related to the official documentation.
There's no explicit exercise for this one, because one of the goals of this tutorial is to get you to find and poke around the documentation, but if you want one, how about: Ex. Find the documentation for reading from/writing to a text file. The instructions may not be clear at this point, but you know where they are.
###### What are numbers like?
Your language has a data type that represents plain text (i.e. strings) and a type or two that represents numbers. We'll get to strings later, but there isn't much variation in number systems. The only serious point of variation is whether an operation on two integers always produces an integer--that is, does 8/3 turn out to be 2.66666 or just the truncated value of 2?
Ex. Check whether your language keeps a wall between integers and real numbers. What is 8/3? What about 8/3. or 8/(3+0.0)?
Python 2. 8/3 →2
Python 3. 8/3 →2.6666
###### What are the lists like?
I say list, but this could also be called an array or a vector. The big difference is that some of these types tend to have fixed length and some variable, but in all cases, they are an ordered collection of homogeneous items. Some languages don't even hold that requirement that all elements of the list/array/vector have the same type, but we'll stick to that for now.
Ex. Create a list of five numbers, 1, 2, 3, 4, 5 (don't bother with a for loop or such cleverness yet; just type it out). Print them to the screen.
Ex. Double all of the elements of your list; print.
Were you able to double the elements in place, or did you have to make a copy (perhaps a copy with the same name)? Did you have to double each element individually, or were you able to write something like 2*my_list?
###### How do I declare a new variable?
Your language may make some smug claims that it doesn't need you to declare variables, but you sometimes do need a means of indicating the type of an item. Here is a snippet of code for you to implement, which starts with an empty list, and then grows it by appending one item at a time:
my_list = [ ] #an empty list
for i=1 to 10:
my_list = [my_list, i]
That first line is a declaration, even if you don't want to call it that.
The problem here is that all languages have some set of rules to automtically convert one type to another. The more types you have, and the more do what I mean the language tries to be, the more conversion rules you need (and every language has at least a few). On the first step in the loop, when you append 1 to an empty list, you need to know if you can refer to a not-yet-existent list, whether that empty list will somehow get typecast away, and whether your one-element list [1] remains that way or gets cast into an integer.
Ex. Let a variable x be an arbitrary integer. Write code like the sample above to incrementally build the list, [1, 2, 3, ..., x].
Ex. Casting: assign the number ten to a text string ("10"), perhaps via a print-to-string function or a cast-to-string-type; convert the text string "10" to an integer variable.
C. char *str; asprintf(&str, "%s", 10); int n=atoi(str);
###### How are references handled?
Every operation that takes in a value and returns a value can either modify its input where it is, or make a copy of the input and mangle the copy to produce output. It is imperative that you know at every step of your program which is happening.
Some languages lack pointers/references/aliases, and so copy every single time. If your langauge does that, bear in mind that it will be slow for operations on large data sets, and when we get to function calls, remember that modifying the variable you passed in to a function is really just modifying a copy of the passed-in variable. You can sometimes use this to your advantage to write shorter functions that don't have to take care to prevent side-effects.
On the other end, some languages mostly lack copying, in the sense that they work with aliases/references/pointers by default. After you assign x = y, if you double x, then y will double along with it. But there is going to be some way to specify x = copy(y), which you should take note of for occasional cases when you'll need it.
Ex. [Impossible in some languages] You have a list [1 2 3 ...9 10]; give the list an alias as per the x = y example above. Verify that the alias works by changing the second element in the aliased list to 100, then print the original list.
Ex. Copy your list to a new variable with a new name. Change the second element in your new variable to 100; verify that the original list didn't change.
###### How do I handle strings of text?
If you stop to think about how they are handled in memory, you quicky realize that strings are hard. A number has a fixed memory footprint: you don't need twice as much memory to write 20 or 4 as you need to write 2. But Hello needs five slots in memory while Hi only needs two. If you wrote HHello, then fixing your typo means moving every item in the string over by one in the little array that is the sequence of letters.
C is famously terrible about hiding these details from you, to the point that you might want to check GLib for its smarter string library. Many languages (C family included) treat them like arrays, so you can do array-like operations. Some languages make heavy use of regular expressions for string manipulation, and if you're already good with regexes, then great--you can reduce your problem count with them.
Ex. Fix the typo: put HHello in a string and replace it with the string with the extra H lopped off. Were you able to do it in place, or did you have to copy to a new variable?
Perl. $x = "Hhello";$x = s/Hh/h/;
###### What are the structs like?
Arrays are for homogeneous items; structures, dictionaries, or hashes can be used for heterogeneous collections, where each item is a named element of the whole.
If your preferred language uses formally declared structures, you may one day find yourself in a language that uses a dictionary or hash--an array with names instead of numeric indices--to serve as your structure. Conversely, if you're used to dictionaries or hashes, bear in mind that some languages require small collections of heterogeneous items be declared in structures. For a long list of homogeneous items that happen to have the same type, you may have to just use a numeric index, generate a hash-type device using an array of key/value structs, or find a key/value system in the libraries.1
Ex. Write a structure, which we'll call the rational structure, with three parts representing a fraction: an integer numerator, an integer denominator, and a text name (like "5/6"). For now, just declare the type if necessary, fill an instance of such a structure for 5/6, and print the elements.
###### How do I write and call a function? Are function arguments copied in or pointed to?
The form of a function call doesn't change much from language to language. You define the language in one place, and call it with a form like, new_fn(x) (or (for LISP-inspired languages) a form like (new_fn x)).
Ex. Reusing the code you wrote above, write a function that takes in an integer x, and returns a list [1, 2, ..., x].
Some languages allow you to write inline functions--nameless little routines for throwaway transformations of a list [a, b, c] into [f (a), f (b), f (c)]. This may be in the index under list comprehension, lambda functions, or anonymous functions. Those languages that allow you to do this kind of thing tend to rely heavily on the facility, so check that it's possible, and if it is, redo the above example about doubling your list using that feature.
Scheme. (define L (list 1 2 3 4 5))
(map (lambda (x) (* 2 x)) L)
###### How do I debug a function?
Here are the sort of things I mean by debugging:
• pausing your program at a certain point,
• getting the current value of any variables that exist in that function;
• jumping to a parent function and checking variable values there;
• stepping past the point where you paused, one line of code at a time.
There's diversity here: for C, you might use the GNU debugger or your IDE might have a built-in hook, some interpreted languages have the facility built in to the interpreter, some have a debugging library that you import like any other library, javascript has some browser plugins, bash has a verbose mode which doesn't do all of the above tasks but is at least a start.
Ex. Write a program/script that calls your function to produce a list 100 elements long. Set a breakpoint that stops when your list is 34 items long and print the list as it looks at that moment.
The worst case is inserting print statements where you need to know a variable's value. It takes a few seconds to set up each print statement, and is annoying when you run and then find out that you needed one more variable's value or the same variable's value three lines down. If the debugging facilities I enumerated above really are not available for for your language of choice, bear in mind that large projects may be a pain relative to how they'd work in other languages.
###### What are the scoping rules?
At the least, you need to know how to declare variables that are global to the entire program, and variables that are local to a function. Some languages go crazy from there; I count four different scoping systems that you'd have to bear in mind for C++ (file, curly brace delimited, object, namespace).
Ex. Rewrite your function to have an explicit iterator (i.e., use a for i=1 to N sort of loop). After calling the function, try to print the value of your iterator (i), and verify that it is not defined outside the function's context.
Ex. Make the iterator global, so that its value is N after the function is called. [Then revert to the prior version, because having an iterator as a global variable is absurdly bad form.]
In lexical scoping, variables that are not explicitly defined in a function, and so would normally be looked up in the global environment, are looked up in the environment as it looked when the function was first called. If you're in a lexically-scoped language, you can do some tricks to generate on-the-fly specialized functions, but well before you get creative with those methods you'll need to make sure you don't get confused and presume that you are looking at a global variable as it is now when you are actually looking at the version that was bound to the function on first call.
###### How do I maintain continuity across function calls?
Ex. Write a function so that on the first call, count() returns one, on the next call, count() returns two, and so on to infinity.
The cheap way of making this work is to use a global variable. C goes one step further with the static keyword. Some have an explicit syntax for continuations.
C. int count(){static int i=0; return i++;}
Ex. Write a function that takes in a list or NULL/nil/0/whatever. If it gets a list, it returns the first item in the list and stores the list internally; if it gets a blank marker, then it returns the next item in the stored list; use the mod (or %) operator to cycle back to the beginning of the list if you hit the end.
Ex. If your language has lexical scoping, write a function that takes in a list and returns a function. The returned function is as above: on each call, it will return the next item in the list, cycling back to the beginning as needed. Here's a sample use of the function you're going to write:
next_prime = generate_list_step_function([1 2 3 5 7 11 13])
non_prime = generate_list_step_function([4 6 8 9 10 12 14])
next_prime()
next_prime()
next_prime()
non_prime()
`
which will print 1 2 3 4. In some languages without lexical scoping, implementing this requires some creativity; in some it is impossible (in which case you'll probably wind up sending in the list every time).
###### Can I do text substitutions (i.e., macros)?
Your typical language mostly focuses on functions that generate their own space in which to work (as per the scope section); the text substitution simply replaces a blob of text with another blob of text.
There are some languages that have no macro processing abilities, some that have a preprocessor that takes the text that is your program file and convert pieces of text into other pieces of text, some that are sufficiently self-aware to convert a text string to operational code in real time via an eval-type function, and some that use lazy evaluation to leave your text as text until you want it to be evaluated.
Ex. Write a macro you'd call like this: call_function(your_function, 10) or like this: call_function("your_function", 10). The macro would then print Calling the your_function function to the screen and then execute your_function(10). Use this to call the list-generating function you wrote above.
OK, to this point, everything has been about the details of the language: how do I deal with types? Can I do clever tricks with scope and persistent variables? The rest of this is going to be about design: how can I introduce new nouns, and the verbs that those nouns can do? How do I package them so I can easily use them for the next project, and how do I use already packaged elements for today's project?
###### How do I load libraries/packages so I don't have to reinvent wheels?
The most basic sort of library inclusion is to simply have a mechanism for including a text file with some code verbatim at the head of the text file you're working with now.
Let me give you a few examples, because the point of the package concept is that not everybody has similar needs.
Ex. A: Make a hundred draws from a standard Normal distribution.
Ex. B: Load an XML document into memory, and print all of one type of element.
Ex. C: Open an empty SQLite database and create a table.
###### How do I set up functions that act on a specific structure?
So very much of the code in the world involves a single specialized data structure, and a set of functions that manipulate that struture. This, of course, is the basis of object-oriented coding, wherein those functions to manipulate a structure are a part of the structure itself. But before that happened, there were simple naming conventions, which worked just fine but weren't as attractive. Some languages have no discernable naming convention, in which case there's a null answer to this question: just put the functions that act on a structure wherever.
Ex. Write a rational_set function that takes in two integers and outputs a rational structure, a get_value function that returns the value of the fraction, a get_name function that returns the fraction as a text string, and an add or + function that adds two rationals. If appropriate, also define a free or destroy function. Add those functions to the object itself if appropriate.
###### How do I get an auxiliary structure that builds upon a base structure?
OK, so you wrote a structure above that has a fixed list of elements, but would now like to extend the structure in some way. As above, some systems will require that you declare the structure beforehand, so you will need to extend either via a structure-extending mechanism (new struct is a version of old struct), or a new structure that consists of some elements and an embedded old structure (new struct has a version of old struct). The wrapping-around approach works everywhere; the object-inheritance method of directly extending a structure is pretty common, and when it exists people will expect you to make some use of it.2 Even in languages where extending a structure just means tacking another element onto the list, there are reasons for an object-extension grammar.
Ex. Use accepted custom to create a new structure, a signed rational, where all of the elements are positive, but there is another element sign, which is 1 if the fraction is positive, -1 if it is negative. Write new set/get/add functions that makes use of the new structure. If you are comfortable with irrational numbers, make this exercise more interesting by extending your rationals to complex rationals.
###### How do I package my own stuff?
Since you already loaded a package/library/file above, you have some idea of how packages work in your new language. There may or may not be extra steps to packaging your own.
Ex. Package the rational structures' declarations (if any) and the get/set/add functions in a separate file (or files if you need a header or manifest or what-have-you). To make all this work, you wrote tests that made sure that the functions worked as planned; make sure those tests still work when you import/include the structure and functions as a package or library.
#### Footnotes
... libraries.1
Dictionary, hash, or struct--they're all about as good, but there are some older languages that have absolutely no way to bundle a set of heterogeneous variables. They are obsolete, and should be used only when the situation really gives you no other choice.
... it.2
This is a claim about custom across communities, and therefore can only be roughly true. Some languages have a bolted-on object model that never received wide acceptance--some even have two bolted-on object models. | 2019-10-23 18:03:39 | {"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": 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": 0.3464524447917938, "perplexity": 919.6094651220242}, "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-2019-43/segments/1570987835748.66/warc/CC-MAIN-20191023173708-20191023201208-00095.warc.gz"} |
https://physics.stackexchange.com/questions/567201/a-single-spatial-mode-whose-associated-harmonic-oscillator-is-in-its-nth-excit | # A single spatial mode whose associated harmonic oscillator is in its $n$th excited state unambiguously contains $n$ photons?
I am reading the introduction to the textbook The Quantum Theory of Light, third edition, by Louden. When discussing the photon, the author says the following:
The idea of the photon is most easily expressed for an electromagnetic field confined inside a closed optical resonator, or perfectly-reflecting cavity. The field excitations are then limited to an infinite discrete set of spatial modes determined by the boundary conditions at the cavity walls. The allowed standing-wave spatial variations of the electromagnetic field in the cavity are identical in the classical and quantum theories but the time dependences of each mode are governed by classical and quantum harmonic-oscillator equations, respectively. Unlike its classical counterpart, a quantum harmonic oscillator of angular frequency $$\omega$$ can only be excited by integer multiples of $$\hbar \omega$$, the integers $$n$$ being eigenvalues of the oscillator number operator. A single spatial mode whose associated harmonic oscillator is in its $$n$$th excited state unambiguously contains $$n$$ photons.
This part isn't clear to me:
A single spatial mode whose associated harmonic oscillator is in its $$n$$th excited state unambiguously contains $$n$$ photons.
Why does the single spatial mode unambiguously contain $$n$$ photons? This is my first exposure to quantum optics, so I would greatly appreciate it if people would please take the time to explain this.
• A bit hazy to me, but it looks like the author's saying "look at what happens when you add the waveforms of n photons of energy $\hbar\omega$ in phase..." and also have exactly one spatial mode. Jul 21, 2020 at 12:56
The idea of a photons dates back to 1905, when Einstein discussed the photo-electric effect. According to Einstein a photon is a quantised light package, and therefore an elementary excitation of the electromagnetic field. However, it is not immediately clear how we could formulate this conceptional idea mathematically.
The mathematical description of a photon becomes much simple if we consider an optical cavity (two mirror). The mirrors impose boundary conditions similar to the ones given by an oscillating string fixed at both ends: The allowed energies becomes quantised. The different oscillation "types" are called spatial modes:
The important point here is that each of these spatial modes can be thought of being occupied by "excitations of the electromagnetic field", and that these excitations are well-defined mathematical quantities. E.g. let's consider only the fourth mode. We can think of it to be ...
• not excited at all, $$|0\rangle_4$$
• excited once, $$|1\rangle_4$$,
• excited twice, $$|2\rangle_4$$,
• ...
According to Einstein the electromagnetic excitations are called photons.
• Oh, wow. So photons are considered to be the excitations themselves? That's really interesting. Thanks for the illuminating answer. Jul 25, 2020 at 15:50
Actually, this is the rigorous definition of a photon in quantum field theory: the excitation of a mode (or, in the langauge of the cited book, the excitation of a harmonic oscillator associated with the mode).
The problem with this definition is that it is at odds with more intuitive notions of a photon as a particle. These are not without their merits: e.g., one may claim that, due to the energy-time uncertainty relation $$\Delta E \Delta t \geq \frac{\hbar}{2}$$ one can never have a mode with a well defined energy/number of excitations, and thus any electromagnetic excitation is a wave packet.
• So I should just accept it as a definition and move on? Jul 23, 2020 at 13:59
• This is what I would do - I also was struck by how this is manifestly "not a particle" when I heard about it for the first time. But be prepared that in less mathematical context people may use word photon in rather vague sense. Jul 23, 2020 at 14:04
• It also closely relates to the second quantization formalism, which is in fact first quantization for EM field - see my answer here: physics.stackexchange.com/a/542483/247642 Jul 23, 2020 at 14:05
The Hamiltonian of the quantum harmonic oscillator is given by $$H = (N+\frac{1}{2}) \hbar \omega$$ N corresponds to the number operator and its eigen values corresponds to number of photons in the cavity. The author has ignored the zero point energy which is quite common thing to do. Hence the Hamiltonian becomes $$H = N \hbar \omega$$ Now consider an eigenstate of $$N$$ having n photons with frequency $$\omega$$ denoted by $$|n \rangle$$ $$H|n \rangle = n\hbar \omega | n \rangle$$
which gives the energy $$n \hbar \omega$$ hence it has $$n$$ photons.
• Can you please explain which part is the zero-point energy? Jul 23, 2020 at 14:23
• @ThePointer Zero point energy is the energy of the harmonic oscillator when there are no photons which is $\frac{1}{2} \hbar \omega$. Jul 23, 2020 at 14:26
• How did you get $H \mid n \rangle = n\hbar \omega$? $n$ is a vector, and $H = N \hbar \omega$, so the how does the inner product equal $n \hbar \omega$? Jul 23, 2020 at 23:21
• @ThePointer my bad its not an inner product but an eigen Value equation $| n \rangle$ is an eigen vector of $N$ with eigen value $n$ Jul 23, 2020 at 23:54
• @ThePointer Ok I will derive Jul 24, 2020 at 0:17 | 2022-05-27 00:14:57 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7678173780441284, "perplexity": 315.3115999268211}, "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/1652662627464.60/warc/CC-MAIN-20220526224902-20220527014902-00543.warc.gz"} |
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# Each side of a square substends an angle of 60^@ at the top of a tower h metres h metres high standing in the centre of the square . If a is the length of each side of the square , then
Khareedo DN Pro and dekho sari videos bina kisi ad ki rukaavat ke!
2 a^2 = h^2 2h^2 = a^23a^2 =2h^22h^2 =3a^2 | 2023-01-31 07:18:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.35195204615592957, "perplexity": 5518.81986559685}, "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/1674764499845.10/warc/CC-MAIN-20230131055533-20230131085533-00725.warc.gz"} |
https://mathsgee.com/tag/divisible | # Recent questions tagged divisible
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Prove by induction that for every natural number $n$, $n^{3}+2 n$ is divisible by 3
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$a(x)=2 x^{4}+x^{3}-x^{2}+5$ and $b(x)=x^{2}+4 x-3 .$ Find $a(x) \div b(x)$
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$a(x)=18$ and $b(x)=7 .$ Find $a(x) \div b(x)$
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Divide $a(x)$ by $b(x)$ and find the quotient $q(x)$ and remainder $r(x)$. Check your answer by showing that $q(x) b(x)+r(x)=a(x)$.
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Simplify $\dfrac{x}{x^{2}-5 x+6}-\dfrac{6}{x^{2}+2 x-8}$
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Simplify $\left(1+\frac{p+q}{p-q}\right) \div\left(1+\frac{p-q}{p+q}\right)$
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Simplify $\dfrac{7}{x^{2}-5 x+6}-\dfrac{6}{x^{2}+2 x-8}$
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Use the Remainder Theorem to find linear factors for the polynomials below. Once you are down to a quadratic, you can use the quadratic formula instead.
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How many numbers between 7500 and 7800 are divisible by 5 but not divisible by 7 using venn diagram?
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How many numbers less than 400 but greater than 10 are divisible by 7?
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How many positive numbers less than 1000 are divisible by 6 but not by 5?
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How many positive numbers less than 300 are divisible by 7 but not 5?
Find the largest integer that divides all terms of the sequence ${a_n}$, where $a_n = n^5 −n, n ≥ 1$.
Find the largest integer that divides all terms of the sequence ${a_n}$, where $a_n = n^5 −n, n ≥ 1$.Find the largest integer that divides all terms of the sequence ${a_n}$, where $a_n = n^5 −n, n ≥ 1$. ... | 2022-01-20 08:39:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5381825566291809, "perplexity": 561.651996149686}, "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/1642320301730.31/warc/CC-MAIN-20220120065949-20220120095949-00456.warc.gz"} |
https://stacks.math.columbia.edu/tag/07A4 | Lemma 21.18.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism
$Lf^*( \mathcal{F}^\bullet \otimes _{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet ) = Lf^*\mathcal{F}^\bullet \otimes _{\mathcal{O}}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet$
for $\mathcal{F}^\bullet , \mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O}'))$.
Proof. By our construction of derived pullback in Lemma 21.18.2. and the existence of resolutions in Lemma 21.17.11 we may replace $\mathcal{F}^\bullet$ and $\mathcal{G}^\bullet$ by complexes of $\mathcal{O}'$-modules which are K-flat and have flat terms. In this case $\mathcal{F}^\bullet \otimes _{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet$ is just the total complex associated to the double complex $\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet$. The complex $\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet )$ is K-flat with flat terms by Lemma 21.17.5 and Modules on Sites, Lemma 18.28.12. Hence the isomorphism of the lemma comes from the isomorphism
$\text{Tot}(f^*\mathcal{F}^\bullet \otimes _{\mathcal{O}} f^*\mathcal{G}^\bullet ) \longrightarrow f^*\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet )$
whose constituents are the isomorphisms $f^*\mathcal{F}^ p \otimes _{\mathcal{O}} f^*\mathcal{G}^ q \to f^*(\mathcal{F}^ p \otimes _{\mathcal{O}'} \mathcal{G}^ q)$ of Modules on Sites, Lemma 18.26.2. $\square$
There are also:
• 2 comment(s) on Section 21.18: Derived pullback
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | 2022-07-03 05:26:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9758462309837341, "perplexity": 570.9898890970803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104215790.65/warc/CC-MAIN-20220703043548-20220703073548-00545.warc.gz"} |
http://mathoverflow.net/questions/127622/tame-morphism-from-a-curve-to-mathbbp1 | # Tame morphism from a curve to $\mathbb{P}^1$
Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $C$ be a smooth projective curve over $k$. Is it possible to find a map $C \to \mathbb{P}^1$ that is tamely ramified at every point of $C$, i.e. such that the ramification index at every point of $C$ is prime to $p$?
A result of Fulton says that, if $k$ is (algebraically closed) of characteristic $p\ne 2$, then it is possible to find a morphism $C \to \mathbb{P}^1$ that is a simple cover: only double points may appear and at most one in every fiber. (This is theorem 8.1 in "Hurwitz schemes and the irreducibility of moduli of algebraic curves", Ann. of Math. 90, 1969. He says it is classical and dates back to Severi.)
Fulton's result gives a positive answer for fields of characteristic $p\ne 2$. But what about characteristic 2? Does the result still hold? I would already be interested in answers in particular cases (elliptic curves for instance).
EDIT: I added the hypothesis that the field is algebraically closed in order to focus on what I am really interested in. Still, I would also appreciate comments on how relevant this hypothesis is.
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Consider a degree $4$ cover of $\mathbb P^1$ that is ramified over $4$ points, of degree $3$ at each. It is easy to see via the group theory of $S^4$ that such exists. Then by Riemann-Hurwitz, this is an elliptic curve. The moduli space of such covers is one-dimensional, and one just has to check that the map to the moduli space of elliptic curves is nonconstant, which I think is true but haven't checked. Similar explicit constructions might work for other genuses. – Will Sawin Apr 15 '13 at 16:30
Thanks Will. There are a few things I don't understand. What is the link with $S^4$? How do you know the dimension of the moduli space of covers? (Since $M_{0,4}$ has dimension 1, I would have thought it would be bigger, but this will not cause any trouble anyway.) I don't have time to check the details right now but will come back to it next week. – Jérôme Poineau Apr 16 '13 at 9:17
@Will: $S^4$ is meant to be the symmetric group on $4$ letters? Anyway, in this characteristic $2$ context without Riemann's existence theorem, it isn't clear to me that this cover exists. I would be more convinced if one would write it down explicitly for an elliptic curve in Weierstrass form. – Peter Mueller Apr 16 '13 at 9:29
@Peter: Thanks for your comment. This $S^4$ makes more sense but I don't know how to make the argument work in characteristic 2 either. On the other hand, I may be able to do it using patching techniques from inverse Galois theory (maybe if $k$ is algebraically closed but I don't mind too much). – Jérôme Poineau Apr 16 '13 at 9:40
1) The condition that $k$ be algebraically closed is certainly not necessary in the result from Fulton's paper, it suffices to assume that $k$ be infinite. The result probably holds over finite fields as well, but would require some more arguments. 2) An example of an elliptic curve in characteristic $2$ where there is a tamely ramified map is the elliptic curve with an automorphism of order $3$: the quotient of the curve modulo the group generated by this automorphism is $\mathbb{P}^1$. – ulrich Apr 25 '13 at 10:51
show 1 more comment
this is silly but if you do not assume the ground field is algebraically closed, then the answer is no. Namely, suppose that C is the generic curve of genus g in char 2 where g is large. Then every divisor class on C is a multiple of K_C. (This is a highly nontrivial theorem.) Now if we had a tame morphism C ---> P^1 then the ramification indices would all be even, hence the ramification divisor would be 2E for some effective divisor. Then K_C = -2H + 2E where H is the pullback of the ample divisor from P^1. Contradiction.
Edit: Actually, now I just got a little bit worried about the difficult thing referenced above, as I don't know a reference and it is possible that one can take the square root of the canonical divisor on a general curve in characteristic 2. Namely, there is that weird thing where d(x^2 + x^3) = x^2dx in characteristic 2. So if you take a general Lefschetz pencil on that curve then it seems that the ramification has degree 2 everywhere and the x^3 term is also present at every point and then you'd actually get a square root of K_C. So I am sorry but my argument is faulty! Please take away the upvotes for this answer! Thanks!
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It seems that your answer is correct. The argument is given in [Stefan Schroer, The strong Franchetta conjecture in arbitrary characteristics, Theorem 6.1]. If I understand well, the reason why the argument in your edit is not a problem is that the ramification divisor might be geometrically divisible by 2 but not divisible by 2 if its support contains a point whose residue field is inseparable over the base field. – Olivier Benoist Mar 28 at 10:06
Thank you very much for this clarification and the reference. – answer_bot Mar 29 at 0:09 | 2014-04-17 07:00:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9227071404457092, "perplexity": 146.20066844230786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00536-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/4279-trig-substitution-print.html | # Trig substitution
• Jul 23rd 2006, 10:00 AM
c_323_h
Trig substitution
I've tried to substitute x in but seem to get the integration wrong.
1. $\int \frac{x^3}{\sqrt{x^2+9}} dx$ let $x=3tan\theta$
2. $\int \frac{1}{x^2\sqrt{25-x^2}}dx$ let $x=5sin\theta$
• Jul 23rd 2006, 11:36 AM
galactus
Your choice of substitution is good. Don't forget dx. That can be overlooked
For instance, #2. $x=5sin{\theta}\;\ dx=5cos{\theta}d{\theta}$
$\int{(25sin^{2}{\theta})\sqrt{25(1-sin^{2}{\theta})}5cos{\theta}d{\theta}}$
$625\int{sin^{2}{\theta}cos^{2}{\theta}d{\theta}$
• Jul 23rd 2006, 01:10 PM
Soroban
Hello, c_323_h!
Don't be discouraged . . . the first one takes forever
. . and there are a dozen places to make errors (I've made them all).
Quote:
$1)\;\;\int \frac{x^3}{\sqrt{x^2+9}}\,dx\qquad\text{Let }x = 3\tan\theta$
Let: $x = 3\tan\theta\quad\Rightarrow\quad dx = 3\sec^2\!\theta\,d\theta$
. . and $\sqrt{x^2 + 9}\:=\:\sqrt{9\tan^2\!\theta + 9} \:=\:\sqrt{9(\tan^2\!\theta + 1)} \:=$ $\sqrt{9\sec^2\!\theta} \:= \:3\sec\theta$
Substitute: . $\int\frac{(3\tan\theta)^3}{3\sec\theta}(3\sec^2\! \theta\,d\theta) \;= \;27\!\int\tan^3\!\theta\sec\theta\,d\theta$
We have: . $27\!\int\tan^2\!\theta(\sec\theta\tan\theta\,d \theta) \;= \;27\!\int(\sec\theta - 1)(\sec\theta\tan\theta\,d\theta)$
Let: $u = \sec\theta\quad\Rightarrow\quad du = \sec\theta\tan\theta\,d\theta$
Substitute: . $27\!\int(u^2 - 1)\,du \;= \;27\left(\frac{u^3}{3} - u\right) + C \;= \;9u(u^2 - 3) + C$
Back-substitute
. . Since $u = \sec\theta$, we have: . $9\sec\theta(\sec^2\!\theta - 3) + C$
Back-substitute: Since $\tan\theta = \frac{x}{3}$, then $\sec\theta = \frac{\sqrt{x^2+9}}{3}$
We have: . $9\cdot\frac{\sqrt{x^2+9}}{3}\left[\left(\frac{\sqrt{x^2+9}}{3}\right)^2 - 3\right] + C \;=$ $3\sqrt{x^2 + 9}\left(\frac{x^2 + 9}{9} - 3\right) + C$
. . $= \;3\sqrt{x^2+9}\left(\frac{x^2+9-27}{9}\right) + C \;= \;\frac{1}{3}\sqrt{x^2+9}\left(x^2 - 18\right) + C$
I need a nap . . .
• Jul 23rd 2006, 03:42 PM
ThePerfectHacker
Quote:
Originally Posted by c_323_h
$
\int \frac{1}{x^2\sqrt{25-x^2}}dx
$
I do these problems in a formal way.
Ignore this, if you do not understand I do not want to confuse you.
------
Let,
$u=\sin^{-1}(5x)$
Then,
$\frac{du}{dx}=\frac{1}{\sqrt{25-x^2}}$--->Denominator.
Express integrand as,
$\int \frac{1}{x^2}\cdot \frac{1}{\sqrt{25-x^2}}dx$
Also, we have,
$\sin u=5x$
Thus,
$\csc u=\frac{1}{5x}$
Thus,
$\csc^2 u=\frac{1}{25x^2}$
Thus,
$\frac{1}{x^2}=25\csc^2 u$
Thus, you finally have,
$\int 25\csc^2u \frac{du}{dx}dx=25\int \csc^2udu=-25\cot u+C$
Substitute back,
$-25\cot (\sin^{-1}(5x))+C$
Simplify the inverse trigonometric with trigonometric, to get,
$-25\cdot \frac{\sqrt{1-25x^2}}{5x}+C$
Thus,
$-\frac{5\sqrt{1-25x^2}}{x}+C$ | 2017-12-12 10:29:49 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 34, "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": 0.995510995388031, "perplexity": 6868.3781997145825}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948515313.13/warc/CC-MAIN-20171212095356-20171212115356-00744.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-8-polynomials-and-factoring-cumulative-test-prep-gridded-response-page-530/25 | ## Algebra 1
The tree was 4 feet tall when planted and grew 2 feet in 4 years. Growing 2 feet in 4 years is the same as growing six inches in a year (half a foot). If the tree was 5 feet tall when planted, and the growth rate didn't change over 6 years, then the tree would be 8 feet tall. $5+.5*6 = 5+3 = 8$ | 2019-07-19 01:52:47 | {"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": 0.36065852642059326, "perplexity": 1097.996544783904}, "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/1563195525973.56/warc/CC-MAIN-20190719012046-20190719034046-00102.warc.gz"} |
https://brilliant.org/problems/how-can-you-cut-a-square-like-this/ | # How To Cut This Square?
Geometry Level 2
A square is cut into 37 squares, of which 36 have area $$1\text{ cm}^{ 2 }$$. What is the side length of the original square?
× | 2017-11-22 10:54:11 | {"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": 0.5884018540382385, "perplexity": 756.0977104178631}, "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/1510934806569.66/warc/CC-MAIN-20171122103526-20171122123526-00195.warc.gz"} |
https://www.islamicclimatedeclaration.org/lup4eznk/bebc26-how-many-square-feet-in-a-quarter-acre | Measurements in various units of area measurement equalling 1 acre. For area conversion from square meters to other area units, multiply the square meter value by the conversion factor. There are 43560 square feet in an acre so one quarter of an acre is square feet. How many square feet are in a 12×16 room? Based upon the International yard and pound agreement of 1959, an acre may be declared as exactly 4,046.8564224 … There are exactly 640 acres in a square mile. One important thing he does with the data is show that the “quarter acre lot” (1,000 sq m) of suburban legend is in reality a very minor player in today’s Melbourne. a acre is 43,560 square feet.... a quarter acre is 10890 square feet. Traditionally, Australians and New Zealanders aspired to own a 3- or 4-bedroom house or bungalow on a section of around a quarter of an acre (about 1,000 square metres), also known locally as the Australian Dream or the New Zealand dream. 0.75 acre equals 3.267×10 4 square feet because 0.75 times 4.356×10 4 (the conversion factor) = 3.267×10 4 All In One Unit Converter Please, choose a physical quantity, two units, then type a value in any of the boxes above. This site is associated with unit conversion metrics that meet the daily needs of the public. The other imperial area units are perch (1 rod * 1 rod), rood (1 rod * 1 furlong), square inch, square mile and square yard. Acres and square footage are terms that are applied to the area of a piece of land. The global symbol used to denote an acre is ac, and the international acre is the commonly used measurement. 1 Acre = 43,560 Square Feet (exact result) Display result as. 1 acres to sq ft conversion. um quarto acre equals 1.089×10 4 square feet because 0.25 times 4.356×10 4 (the conversion factor) = 1.089×10 4 All In One Unit Converter Please, choose a physical quantity, two units, then type a value in any of the boxes above. We know how to calculate square footage which is length times width. Type in your own numbers in the form to convert the units! Anonymous. Here is the area conversion factors list from acres to other common area units: 1 Acre = 0.404686 hectare; 160 perches; 4 rood; 43560 square feet; 6272640 square inches; … A quarter acre is one-fourth ( 1/4 = 0.25 ) of an acre. So, if the 1/4 acre was a square plot, it would be about 104.36 feet square, so perimeter would be 417.42 feet. The ideal, well-rounded, quarter acre farm should have fruits, vegetables and livestock that are easy to … That is a square approximately 104.36 feet on a side. Definition: An acre (symbol: ac) is a unit of land area used in the imperial and US customary systems. How much should it cost to fence 1/2 acre??? An Acre is approximately equal to 1/2.5 of a Hectare. In this case, 0.5 acres is multiplied by 43,560 square feet. To convert acres into square feet, the number of acres is multiplied by the conversion factor. Use the calculator above to convert between acres and square feet. You can view more details on each measurement unit: acre or square feet The SI derived unit for area is the square meter. In mathematical expression, 0.25 Acre (ac) = 10890 Square Feet (sq ft). An acre is measured … But your land could be shaped like a square, or more like a rectangle. Thus an acre is 10 square chains, 160 square rods, 43,560 square feet, 4840 square yards, 1.184 arpent or 0.4047 hectares. About 43,560 square feet make one acre. (remember we only produce 10% of our fruit right now) 200 square feet for 50 meat chickens. That means 5,280 x 5,280 is 27,878,400 and that means that that all sections are 27,878,400 square feet. A quarter of an acre is slightly larger than a house lot. One Acre consists of 43,560 square feet A quarter of an acre would thus consist of 10890 square feet Acres. The precise meaning of this depends on the exact definition adopted for a foot: the international acre is 4 046.856 422 4 m² (for the UK, see). You also may have to measure acres or feet if you are planning a landscaping or building project. Since there are 4,840 square yards in an acre, a quarter-acre is also defined as 1,210 square yards of land. It is defined as the area of 1 chain (66 feet) by 1 furlong (660 feet), which is exactly equal to 1/640 of a square mile, 43,560 square feet, or exactly 4046.8564224 square meters. Unit of Measure Amount Equaling 1 Acre; Hectares.4047 hectares: Meters: 4047 m²: Miles: 0.0015625 miles² (640 acres = 1 mile²) Yards: 4,840 yds²: Square Feet: 43,560 ft²: Square Inches: 6,272,640 in²: Perches: 160 perches (1 perch = 1 rod) Rods: 160 rod² (1 rod = 16 ½ feet) Thus F^2 = 10890 and F = \sqrt {10890} = 104.35 feet. Your email address will not be published. A quarter acre is 10890 square feet. Date Time: 2020-12-18 16:04:52, My IP: 71.178.255.4. This measurement is roughly equivalent to 40 per cent of a hectare. It is defined as the area of 1 chain (66 feet) by 1 furlong (660 feet), which is exactly equal to 1/640 of a square mile, 43,560 square feet, or exactly 4046.8564224 square meters. Type in unit You can do the reverse unit conversion from For the best answers, search on this site https://shorturl.im/avdLh. Don't forget to talk with your builder about how much space you want for a backyard, front yard, garage and fencing when setting the property lines before the building starts and … Well, you'll have to start with square footage to get an idea. Traditionally, Australians and New Zealanders aspired to own a 3- or 4-bedroom house or bungalow on a section of around a quarter of an acre (about 1,000 square metres), also known locally as the Australian Dream or the New Zealand dream. An entire football field measured comes to about 350 feet by 160 feet, which equals 57,600 square feet. 1 Acre = 43560 Square Feet This means that one acre is made up of 43560 square feet. conversion calculator for all types of measurement units. How much is 0.29 acre in square feet? One Acre is equal to 43560 square feet, 4840 square yards, or 4046.8564 square meters. If you’re wondering how big an acre is in length and width, it’s not quite that simple. It is defined as the area of 1 chain (66 feet) by 1 furlong (660 feet), which is exactly equal to 1/640 of a square mile, 43,560 square feet, or exactly 4046.8564224 square meters. How many square feet in an acre of land. A half acre equals to 21,780 square feet. Note that rounding errors may occur, so always check the results. Physics Chemistry Recipes ⇌ Please, choose a physical quantity, two units, then type a value in … A quarter acre = 10,890 square feet. In word, decimal point two five Acre (ac) is equal to ten thousands eight hundred and ninety Square Feet (sq ft) in Survey System. Use this page to learn how to convert between square feet and quarter acre. Quarter acre is 10,890. 1 acre. 2 1. We multiply the value in acres by the conversion factor to find out how many square feet are in an acre. Type in unit How many square foot in 1 quarter acre? Jul 6, 2018 - A quarter acre is one-fourth ( 1/4 = 0.25 ) of an acre. Did I translate "How Many Square Feet In An Acre" correctly in your language? How many acres in 29000 square feet? Acres can be abbreviated as ac; for example, 1 acre can be written as 1 ac. Acre. Acre is an This helps convert viewer from one unit to other unit. A quarter acre is 10890 square feet. Anonymous. Did you know that there are 1,742,400 square feet in a standard 40 acre parcel. One acre is 43,560 feet. An acre is a measure of surface area. ›› It answers questions like: * 100 square feet to acres * how many square meters in an acre Go back to Conversion category Suggested. Yahoo fait partie de Verizon Media. T That's equal to 43,560 square feet for those unfamiliar with those units of measure. The land was frequently put to use with vegetable gardens, fruit trees, or … Both square feet and cents are commonly used units of measurement of land. Tomatoes 600 sq ft; Potatoes 400 sq ft; Carrots 400 sq ft; … It is a part 1/640 of a Square Mile, 4840 Square Yard. 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Translate how many square feet are 43,650 square feet are in roll... Own numbers in the United States, divisions of land commonly employed within North America and the.... Many Decimal in Bihar, Gunta ( guntha ) to square feet, the of. Thing, including an two gates large enough for vehicles to drive through… customary and imperial unit of.. Ip: 71.178.255.4 a section '' for the goats =1340 square feet the! Quantities is a job unto itself as land is approximately equal to 0.00024710538146717 acre 10,300 square feet exact!, i.e both units used to denote an acre is in length and width 1/2 acre??. 10890 square feet in an acre is 10,890 one foot on a side, quarter-acre. With sq km, it 's a quadrilateral, i.e you that if any wrong information is found let... English, a quarter acre is equal to 1/640 th of a square mile, 4840 square yards of is! Know how to convert between acres and square feet in a 12×16 room search! 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Quick results 2.29568411386595 is equal to 43,560 square feet to 10.76391041671 square feet the! 16:04:52, My IP: 71.178.255.4 give you 8 such plots with access. Our fruit right now ) 200 square feet, multiply the acre square! Acre value by the conversion factor ac ; for example, 1 =! Math Central is supported by the conversion factor 2.29568411386595 'll have to measure area Chart Near 0.14 acre,. Have to start with square footage which is the answer helps convert viewer from one to. That historically has been used to measure area rectangular shaped 1/2 acre sized lot like this livestock. 'Ll have to start with square footage are terms that are easy grow... Conversion Chart Near 0.14 acre tomatoes 600 sq ft ) definition: an acre is 10890 square feet now! Tracts of land information is found please let me know and that means that one acre 10,890... Square rods divide ; 4 = 10,890 square feet is an irrational number or square feet me to a. 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Calculations or integration techniques about 350 feet by 160 feet, which equals square... | 2021-09-18 00:41:34 | {"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": 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": 0.6170848608016968, "perplexity": 3206.5737816781398}, "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/1631780056120.36/warc/CC-MAIN-20210918002951-20210918032951-00434.warc.gz"} |
http://physics.tutorcircle.com/thermodynamics/third-law-of-thermodynamics.html | Sales Toll Free No: 1-855-666-7446
# Third Law of Thermodynamics
Top
Sub Topics There are three laws of thermodynamics. The first law of thermodynamics tells about the work and energy. The second law gives the concept of entropy. The third law is in fact more interesting that tells all about the concept of absolute zero. Lets study more about it.
## Third law of thermodynamics definition
Rudolf Clausius developed the concept of entropy. It is nothing but the dispersal of energy. The greater the dispersal or spreading the larger is the entropy.
The Third Law of Thermodynamics states that,
The entropy of a system approaches a constant value as temperature approaches zero. It states that absolute zero, the lowest possible temperature can never be reached. In short the entropy of a system is zero at absolute zero.
## Third law of thermodynamics equation
The Third Law of Thermodynamics is also called Nernst law given as
$lim_{T\to 0}$ S = 0
If the entropy is zero at temperature T = 0 then the absolute entropy Sab of a substance at temperature Sab of a substance at temperature T and pressure P is given by expression
$S_{ab}$ = $\int_{0}^{T}$ $\frac{(\Delta Q)_rev}{T}$
where,
$\Delta$ Qrev is the amount of heat supplied to the system,
T is the temperature | 2017-03-30 02:42:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.780228853225708, "perplexity": 488.6866550233448}, "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/1490218191984.96/warc/CC-MAIN-20170322212951-00276-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://pi3d.github.io/html/FAQ.html | ## Graphics Memory¶
### assert s >= 0¶
When starting any demo I get an AssertionError in DisplayOpenGL assert s >= 0
This is generally caused by the graphics memory allocation on the Raspberry Pi being too low (less than 32)
### EGL_NO_SURFACE¶
When starting any demo I get an AssertionError in DisplayOpenGL assert self.surface != EGL_NO_SURFACE
This is generally caused by the graphics memory allocation on the Raspberry Pi being too low (less than 64) It has also been caused by extra shared libraries such as /usr/lib/arm-linux-gnueabihf/libEGL.so.1 and symbolic links being added by other programs see five questions below…
### Start then shut down¶
When running ConferenceHall (or other program with a large number of textures) it appears to start ok then shuts down (byebye 3 message) just before the main display loop should appear.
This is generally caused by the graphics memory allocation on the Raspberry Pi being too low (less than 128)
Is there a way of using memory more efficiently when using large images?
There are two features added to the Texture class from v2.9 The first allows ‘normal’ CPU memory to be released after the GPU texture sampler has been created: use the argument pi3d.Texture(free_after_load=True)
The second feature allows more highly compressed texture formats to be used for the GPU storage. The default is GL_RGB or GL_RGBA for jpg or png style textures respectively which take three or four bytes. pi3d.GL_RGBA4, pi3d.GL_RGB5_A1, pi3d.GL_RGB565 are alternatives that use only two bytes and give quite acceptable colours for most things. Use for instance pi3d.Texture(i_format=pi3d.GL_RGB565)
## _tkinter.TclError¶
When starting TigerTank, ConferenceHall or MarsStation demos I get an error _tkinter.TclError: couldn't connect to display ":0"
tkinter is trying to use a display provided by the X-server. Run startx from the command line.
## PiStore install¶
### taking too long¶
I am trying to install using PiStore but it’s been running for hours with no sign of completing.
The PiStore install adds pip and Pillow which take quite a bit of the cpu resources. It could be that you have set the graphics memory share to the higher level needed to run pi3d. In the long run it will be quicker to abort the installation, remove the half installed pi3d, use raspi-config to set the graphics memory low, re-install pi3d, then set the graphics memory back up again!
### Doing more with PiStore installation¶
I have installed using PiStore and run the demos from the Menu. Now I would like to play around for myself.
There is a PiStore button next to launch that lets you see the source code. It should open a file browser window to /usr/local/indiecity/InstalledApps/skillmanmedia/Full/pi3d_demos which is a very obscure location and also protected against modification. To play around with the code you should either copy the whole of this directory to your home directory (i.e. so you have /home/pi/pi3d_demos/) or clone or download the demos from http://github.com/pi3d/pi3d_demos (which will include a couple of other demos excluded from PiStore because they use very large resource files.)
Before going too far it would be a good idea to ReadMe
## GLX DRI2 not supported or failed to authenticate¶
When I try and run the demos I just get a load of error messages such as libEGL warning: GLX/DRI2 is not supported/failed to authenticate etc
The chances are this is because ‘something’ (such as gedit) has installed mesa which added its own versions of libEGL and libGLESv2. If you run:
$sudo find / -name libEGL*$ sudo find / -name libGLESv2*
on the Raspberry Pi you should just get /opt/vc/lib/libEGL.so and /opt/vc/lib/libGLESv2.so if other ones turn up i.e. /usr/lib/arm-linux-gnueabihf/libEGL.so.1 you could try creating symbolic links for them all like this:
$sudo ln -fs /opt/vc/lib/libEGL.so /usr/lib/arm-linux-gnueabihf/libEGL.so$ sudo ln -fs /opt/vc/lib/libEGL.so /usr/lib/arm-linux-gnueabihf/libEGL.so.1
$sudo ln -fs /opt/vc/lib/libGLESv2.so /usr/lib/arm-linux-gnueabihf/libGLESv2.so$ sudo ln -fs /opt/vc/lib/libGLESv2.so /usr/lib/arm-linux-gnueabihf/libGLESv2.so.2
NB on other systems such as Debian, ubuntu etc the “good” files will not be in this location, they will be somewhere like /usr/lib/x86_64-linux-gnu/mesa-egl/ so you have to alter the commands accordingly Use the actual paths as listed by find. After creating new symbolic links you will need to run:
$sudo ldconfig This issue is being looked into by the maintainers of Raspbian so, hopefully, it will be fixed in later releases. ## Runtime errors¶ ### Crashing¶ My RPi Crashes or reboots when I try and run a demo. Any program using a loadable texture, which includes nearly all the demos, requires the Python Imaging module (PIL). Additionally, some demos require tk. See the ReadMe for information on how to install these packages. ### white screen¶ I see nothing but a white screen. Possibly something has gone wrong with an opengles function, sometimes there might be an error message in the terminal such as failed to create display. This could be caused by running out of gpu memory (see ReadMe for how to set up memory split). Occasionally multi-threaded applications can cause this problem if an opengles function call is made not from the main thread. If you encounter this, please contact the pi3d team so we can protect against this in future. ### black screen¶ I see nothing but a black screen. This can be caused by running out of GPU memory, especially if it’s set to 64 in which case some things will work fine then stop after the addition of just one seemingly insignifican component. From v2.20 there will be a Logger message to this effect. See below for questions using pi3d.Log Possibly something has gone wrong in a shader, such as using a shader requiring texture coords (i.e. mat_relfect) on a Model exported with no uv mapping. There may be a line number reference output by the shader compiler in the terminal window. It is great fun experimenting with shaders but they are ?£#% taciturn beasts to debug! The problem could be caused by sending some bad render setting to a shader. ### only background¶ I see nothing but the background. You will need to set background to non-transparent and a color not equal to black or white to determine if this is happening. Either the shape is behind the camera, too far away, is outside the field of view, is too small, too large or the polygons are facing away from the camera. Often this is because you are actually inside the object. Try using the Camera.point_at([x,y,z]) function (see pi3d_demos/ClashWalk.py for use) or move and rotate the object and camera. Sprite and ImageSprite shapes are one sided so cannot be seen from behind, try using a Plane instead Prior to v2.9 of pi3d there was a bug that stopped shapes with very large numbers of vertices rendering at all (c. 25000). Upgrade to a version post 2.9 to see if this fixes the problem. ### unlit silhouettes¶ I see only black silhouettes against the background. You may be trying to use a shader that requires light but there isn’t any, or it’s turned down too low. Try switching to a ‘flat’ shader to check. Alternatively, if it’s a shape you have generated such as a Lathe or a Model, the normal vectors might be pointing in the wrong direction. Try re-generating the shape, the path you use for the Lathe needs to start at the top of the object and there are functions in most 3D modeling applications to recalculate normals, or force them to point outwards. ### Mouse not working¶ The demo loads but the mouse doesn’t move the camera as it’s supposed to. If this only happens on demos using the event library (such as pi3d_demos/Silo.py) then it could be the hardware configuration is pretending to be something it isn’t. It’s not uncommon for keyboards to say they are mice or joysticks. If you have a mouse combined with a keyboard (to save on USB slots) then you might need to use get_mouse_movements(1). If you have problems with a device or inputs using the event system it’s a good idea to run python FindDevices.py from pi3d/event/ - this will give you lots of additional information. There is also an application pi3d_demos/TestEvents.py that you can run to find what information is being returned by your input devices. In some circumstances you might need to modify the values returned by the pi3d/event/Event.py InputEvents methods. TODO at the moment this involves hacking the file but it will use a lookup table. When running on my laptop (lenovo T420, ubuntu 13.10), occasionally, the mouse doesn’t work with the event input, but starts to do after running pi3d_demos/TestEvents.py and changing the number in get_mouse_movements() a few times. It’s not clear what causes this but it might be when the USB mouse is plugged in after the computer has been booted up. See also in the section Permission denied below ### cbreak error¶ When I try to run a demo I get an error ending curses.cbreak() error: cbreak() returned ERR The chances are that you are trying to run python directly from the IDLE editor. Although IDLE works fine for most things, it seems to have issues with the curses module that is used for keyboard input and is incorporated in pi3d. Try running the demo from the command line. Geany seems to run ok on the Raspberry pi and has python highlighting and context suggestions. ### scrambled terminal after Ctrl-C or crash¶ With some of the demos the terminal seems to stop working after a break out of the program. Sometimes typing doesn’t appear on the screen, sometimes carriage return doesn’t work. This is due to curses being used for the keyboard input system. It needs to do some tidying up by calling pi3d.Keyboard.close(). You can return the terminal to a normal state with$ stty sane or with $reset. A more complete solution that allows programs to be stopped with Ctrl+c and the required closing down code still to run would be a variation of: keys = pi3d.Keyboard() ... try: while DISPLAY.loop_running(): ... if keys.read() == 27: break finally: #can also except KeyboardInterrup: for ctrl c specific things keys.close() ... alternatively from release v2.15 onwards there is a KeyboardContext that can be used: with KeyboardContext() as keys: while DISPLAY.loop_running(): sprite.draw() if keys.read() == 27: break ### No video output running remotely¶ I am using the Raspberry Pi through a remote terminal with ssh or putty but when I run pi3d applications there isn’t any video output. Well, when I look at the screen connected to the Raspberry Pi I can see that the video output is appearing there. How can I see it on a remote terminal? To pipe the images from the local display to a remote terminal requires slightly specialist software and takes quite a bit of processing time. So you won’t be able to see programs run full speed but you can do development and testing. An easy way is to install realvnc on the Raspberry Pi (it’s already there in the latest raspbian but needs ‘activating’ https://www.realvnc.com/docs/raspberry-pi.html#raspberry-pi-setup) and on the machine you want to connect from. There are explicit instructions on the realvnc website. To get things working on my machine I had to reduce the screen resolution quite a bit and enable the experimental direct capture mode. The animation speed is significantly faster with the xserver desktop not running and executing the pi3d programs from the command line. To do this you will have to change the default behavior using raspi-config. ## Optional arguments¶ It appears from the demos that there are some arguments that are optional. For example, can a Shape be drawn without specifying a shader and a texture? There are (almost too) many ways to set Shapes up to draw. The draw method needs to have a Shader, a Light and a Camera specified but if you neglect to create a Light and Camera when you first draw a Shape it will generate ‘default instances’ which most of the time are just what you want. (These default instances can be accessed to change settings such as color or direction for a Light or field of view for a Camera by using the syntax: Camera.instance(). However the default instance of Shader is mat_light which uses the ‘self color’ of the Shape (defaulting a neutral (0.5, 0.5, 0.5)) as it would be messy to try to figure out if or what Textures to use. Generally you choose the Shader to do the kind of rendering you want, but you can feed that in by various means, many of which also cater for specifying the Texture(s) to use at the same time: Set them directly in the Buffer array - the other methods are really just wrappers for this i.e.: myshape.buf[0].shader = myshader myshape.buf[0].textures = [mytex, normtex, refltex] Include them at draw time: myshape.draw(myshader, [mytex, normtex, refltex], 1.0, 0.1) Set them beforehand (probably the most usual way): myshape.set_draw_details(myshader, [mytex, normtex, refltex], 1.0, 0.1) For Model objects the ambient texture or material shade will normally be defined in the 3D object file (egg or obj/mtl) In these cases you could use: myshape.set_shader(myshader) ... myshape.set_normal_shine(normtex, ntiles..) # leaves the first texture if there ... myshape.set_material(mtrl) ## Blending¶ How can I blend objects, why do objects vanish when they go behind a transparent object and other questions to do with transparency (or apha property) Transparency of Shapes can be altered by 1. the set_alpha() method 2. the alpha value of pixels in a png type image file 3. alpha value of the fog. The blending of the pixels with alpha less than 1.0 is controlled by setting Texture.blend to True or False. The way that transparency is handled is quite hard to understand. Here is some good information http://www.opengl.org/wiki/Transparency_Sorting The graphics processor has a global setting to enable blending that is switched on or off as each Shape is drawn, allowing or preventing the pixels to be blended with whatever’s behind them. In pi3d this can be controlled by setting the blend=True argument when the Texture is created or at a later point by mytexture.blend = True In addition to this setting there is a check in the draw() method so that blend is enabled when alpha is set to less than 1.0. When the gpu is rendering an object there is a depth buffer that holds information on how far from the camera each pixel has been drawn. Because of this it is normally optimal to draw foreground objects first as there is then less of the background to fill in. If the background was drawn first then the same pixel might have to be redrawn several times as the gpu found something else nearer to the view point. However the gpu doesn’t take into account the transparency of the pixel when it’s deciding if something is nearer or further away, so for blending you have to draw things on top of other things… Which sounds obvious but to give an example; if a slideshow tries to blend between two images, one drawn in front of the other: If you first draw the canvasFront (z=0.1) with alpha=0.1 then draw the canvasBack (z=0.2) with alpha=0.9 the result will be a very faint image on canvasFront and nothing on canvasBack. Wrong! i.e. canvasBack always has to be drawn first and if the application is purely fading from one image to another it can leave canvasBack at apha=1.0 (i.e. default value) and just increase then decrease the alpha of canvasFront In addition to blending, when the Shader is rendering an object it discards some pixels without drawing anything at all. The decision is based on the alpha value of the pixel as read from the Texture. If blend is True then pixels with alpha < 0.05 are discarded if blend is False then pixels with alpha < 0.6 are discarded. This allows objects to be drawn after nearer objects but still be seen through ‘holes’ in the image. i.e. the trees in ForestWalk I want to use pi3d on the Raspberry Pi at the same time as other applications that use the dispmanx display surfaces (omxplayer, wayland, kivy etc) how can I set the layer to be in front or behind. There is an argument to Display.create(...layer=0) that you can alter to change the order of layers. To draw behind the X11 desktop on the Raspberry Pi Raspbian setup you need to set the layer to -128 see pi3d_demos/PyQtCube.py ## Materials¶ All the demos use images to create the surface patterns for shapes. Is it possible to define a material color. The method myshape.set_material((0.9, 0.4, 0.0)) can be used (the default is (0.5, 0.5, 0.5)) but to render using this you need to use an appropriate mat_shader: myshape.set_draw_details(shader, []) # shader = Shader('mat_flat') uses no lighting myshape.set_draw_details(shader, []) # mat_light uses a light myshape.set_draw_details(shader, [bumptex], 4.0) # mat_bump uses light and normal map myshape.set_draw_details(shader, [bumptex, shinetex], 4.0, 0.2) # mat_shine uses light, normal map, reflection texture and one demo does use material color: Shapes.py look at the code for the wine glass. Also, there is now a default instance for Shader so if you try to draw a Shape without specifying a Shader it will load and use mat_light which gives 3D shading but requires no Textures. ## joysticks etc¶ How do I use a joystick, gamepad, xbox controller etc with a pi3d application? Often these will just work with the event module when plugged into the USB, sometimes you may need to use a different InputEvents method, for instance with an xbox 360 you get the left joystick from get_joystickB3d() Also you would need to install the driver and start it running first: sudo apt-get install xboxdrv sudo xboxdrv -s -i 0 See also in the section Permission denied below ## 3D models¶ ### Making 3D models¶ How do I make my own 3D model to load into pi3d? You will need to ‘make’ one on a bigger computer using 3D software such as blender. This falls outside the scope of this FAQ but your best option is to export the model as an obj file. In Bl2.6 options I specify: Apply Modifiers (default) Include Edges (default) Include Normals (tick this) <<<<<<<<<<<<<<<<<<<<< * Include UVs (default but see below) Write Materials (default) Object as OBJ Objects (default) Forward -Z Forward (default) Up Y Up (default) these last two will mean that.. Blender.x=>pi3d.x, Blender.y=>pi3d.z, Blender.z=>pi3d.y with no reflection of whatever you design * If you export without getting blender to Include Normals then pi3d will have to generate them when the model is loaded. This is not a good idea for several reasons: It will be slower to do on the pi then on a ‘big’ computer, it will have to be done every time the model is loaded rather than just once, it will not give the fine control available in blender to define the sharpness of edges. NB You will need to define uv mapping even if you define a material color and don’t intend to use a texture but might want to use a normal mapping shader. To do this in blender you need to tab to edit mode, select all vertices (a), unwrap (u, Unwrap). If the model has multiple objects you will need to do this for each one. After you export you may need to edit the mtl file so the relative path to the image is correct for their locations on the pi. In programs such as blender it is also possible to use a more detailed (high polygon) model to create a ‘normal map’ image that can be used to give surface detail to the model in pi3d. Quite technical but lots of instructional videos on youtube! ### Imported model doesn’t appear¶ I have a 3D object in obj format which I tried to import into a scene, but I do not see it! I used the file ‘LoadModelObj.py’ which I modified. For the model, I just took a logo I created a 3D extrusion in Photoshop and I exported the layer in OBJ format [but seel also any other method of creating 3D obj files]. The fact of not seeing it must be related to the size of my model and the parameter of depth z in ‘Pi3D.model’. But I do not know how to get the correct settings apart from trying random values. The first issue is with the x,y,z values of the vertices. You can check these by opening the obj file in a text editor. In this case the y and z values are very large: around -2100 and 1840. You can edit them to bring the model nearer the origin by globally replacing using regular expressions such as Search For "( -21)(..)" Replace with " \2". Or write a simple python program to go through all the lines of the obj file, parse out the v lines and re-write them with altered values. Once the x,y,z values are reasonable you can also import the model and adjust its position manually on blender and re-export it. If needed you can also alter the scale of the model. If the model is very large you will need to move it further from the camera to see it all. The second issue is that the uv mapping of textures in obj files is flipped compared with the default pi3d sense so if you want to apply your own texture you need to set the flip=True argument to Texture(): tex = pi3d.Texture('Logo_IUT.png', flip=True) bumptex = pi3d.Texture("textures/floor_nm.jpg") shinetex = pi3d.Texture("textures/stars.jpg") mymodel = pi3d.Model(file_string='Logo_IUT_3D_mod1.obj', name='teapot', z=40.0) mymodel.set_draw_details(shader, [tex, bumptex, shinetex], 16.0, 0.5) Alternatively in blender open the UV editor and specify the correct image file to use as a Texture. When you export this it will create an associated mtl file which will be picked up when you import the model in pi3d. ## 2D images¶ ### How to use 2D images¶ Can I use pi3d for 2D images? There are various ways of doing this. The easiest way is to use the image to texture a simple rectangle. The simplest shape to do this is the Sprite which is also utilised by the ImageSprite shape to allow the texture to be specified as it is created. The Plane object is similar but is two sided. The advantage and disadvantage of this method is that images will be different when viewed from different locations. If you specify an orthogrphic camera (set the argument is_3d=False) then there will be no perspective (the image will not get smaller as it moves away from the camera) and each unit of the dimensions of the object will be one pixel on the screen. With both these methods the shape can be rotated, moved and scaled in all dimensions. You can also use the shader 2d_flat which takes pixels from an image and maps them to the screen, see below. The advantage of this method is that it can use the even simpler Canvas object and it always stays in the same place relative to the camera so you only need one camera, which can be the default one that you don’t have to bother creating. See below. From v1.14 the Buffer.re_init() method (see below under Is it possible to change the shape...) allows vertices to be moved around quite easily. These vertices can be rendered as points and the Shader can be made to draw an image or part of an image at each point. This technique allows much larger numbers of sprites to be drawn per frame, especially if the fast array processing power of numpy is used as well. See pi3d_demos/SpriteBalls.py and pi3d_demos/SpriteMulti.py ### nearness of 2D relative to 3D¶ How do I display 2D images in front of a 3D scene? (or behind, for that matter) create two cameras one 3D and one 2D and assign the relevant camera to the types of objects you want to be drawn by each method. You can move the 3D camera around the scene but leave the 2D one stationary, that way you won’t have to keep moving and rotating the 2D objects to keep them in front of the camera. Orthographic (2D) cameras will render objects with a z value that is severely non linear and does not relate in a simple way to the z values for the perspective camera. Generally 2D objects will be in front of objects rendered by perspective (3D) cameras unless you assign z values in the thousands. Too large a z value (> 10000), though, and they will disappear beyond the ‘far plane’. If z_o is the z value of a Shape viewed with an orthographic camera and z_p is the z value of a Shape viewed with a perspective camera then their relative distances during rendering by the Shader (i.e. which obscures the other) follows: z_p = 10000 / (10000 - z_o) # so z_o of 9000 gives z_p of 10 z_o = 10000 * (1 - 1/z_p) # so z_p of 500 gives z_o of 9980 If you create a camera it will become the default instance so if you need more than one you need to explicitly create them, and it’s a good idea to assign the one you want to each object as an argument while the object is being created. Before the development of the orthographic Camera matrix system there was a system of drawing onto a Canvas object using the 2d_flat shader. This method is really deprecated though ### Nearness of PointText and TextBlock relative to other 2D objects¶ Even setting the z value of my TextBlock to a smaller value than an ImageSprite z value it still gets drawn behind! The PointText/TextBlock system developed by Matt Coleman is very fast and flexible but also quite complicated, especially hard to follow as the demo tries to showcase all the features! One subtlety that isn’t obvious is the fact that ‘extra’ info is being shoehorned into the array passed to the shader, critically the z value of each vertex (i.e. each character) contains both the depth and the size. Coupled with the use of medium precision floats this means that the behaviour of the depth and size values of TextBlock seem very strange. Depth as specified in TextBlock doesn’t match up with the normal depth of 2D objects such as ImageSprite (this is different from the 2D v 3D depth difference mentioned above) Keep the TextBlock z values small and the other objects large (i.e. 0.5 may be in front of normal 2D z of 6.0 but behind 5.9 so play safe with a value of 50.0) Size varies from 0.05 to 0.99 (subject to float precision) however if you use a size of more than 1.0 it will effect the spacing! Try using 0.99, 1.99, 2.99 etc. ### PointText stops working if too many TextBlocks added¶ The text seemed to be drawing OK but when I added an extra TextBlock or increased the amount of characters in a TextBlock it all stopped working. When you create a PointText object you have to specify the max_chars argument and it will go horribly wrong if you exceed this as the speed of this text display method relies on updating an existing buffer. There is probably not too much of a penalty in making it a bit bigger than you expect to use. ### Default fog distance¶ I’ve moved my yellow plane behind other objects by setting z=9900 and viewing it with an orthographic camera. But it has become grey and slightly transparent! The default Fog distance was set up before the orthographic camera had been implemented. It is mid grey and ramps up to full strength at z=5000. From v1.12 This will be increased but in the mean time you can: myshape.set_fog((0.5, 0.5, 0.5, 1.0), 30000) I want to change the distance at which the fog effect starts. It seems to default to one third of the distance (in the above answer fog would start at 10000 and be full by 30000). From pi3d v.2.22 it is possible to define the fraction by adding it to the fog distance. i.e. 30000.99 would use 0.99 rather than one third, it would start at 29700.98 and be full by 30000.99 ### Pixel perfect¶ How do I display an image exactly without anti-aliasing or smoothing i.e. pixel perfect? This can be done by using a Camera with argument is_3d=False and specifying, when the Texture is loaded, that mipmap=False. Because this is a global setting it will be overwritten by whichever Texture is the last to be loaded. Use the uv_flat shader and the Sprite or ImageSprite Shape. It is also important that the image is one of the standard widths used by the GPU (see the FAQ section on Texture blurring below) also; that the dimensions of the image are the same as the sprite. ### anti-aliasing¶ Where I have one shape in front of another with contrasting colors can the diagonal line be anti-aliased to prevent ‘steppyness’? This can be done when the Display is created by setting the samples argument to 4: DISPLAY = pi3d.Display.create(x=150, y=150, samples=4) Generally the edges don’t look too bad, there is a small processing cost associated with this sampling and there is a recorded instance of the sampling causing an error when running pi3d on vmware on a mac or when running on Windows. ### Texture blurring¶ Some of my Textures look a bit blurred or pixely. Early GPUs had to have image sizes of powers of 2 pixels. i.e. 2,4,8..1024,2048 because of the algorithm used for texture sampling, but modern ones can manage with any dimensions. With the raspberry pi we have found that some widths can cause rows of pixels to be offset unless they fall on certain sizes (below). If the image width is a value not in this list then it will be rescaled with a resulting loss of clarity Allowed widths 4, 8, 16, 32, 48, 64, 72, 96, 128, 144, 192, 256, 288, 384, 512, 576, 640, 720, 768, 800, 960, 1024, 1080, 1920 ### Display size greater than 2048¶ Is there any way to use monitors with higher resolution than the 2048 limit of the OpenGL ES implementation on the Raspberry Pi? This is not straightforward but it is possible to use the python multiprocessing module to run two instances of pi3d in different “slices” of the screen. Because the processes are running in parallel there is a problem with synchronisation, however for slow operations such as a slideshow this can be reduced to an ammount that is indescernable. The code would be something like: import pi3d import numpy as np import ctypes from multiprocessing import Process from multiprocessing.sharedctypes import RawArray, RawValue from PIL import Image W, H = 2160, 1440 L_X, L_Y, L_W, L_H = 0, 0, 985, 1440 R_X, R_Y, R_W, R_H = 985, 0, W - 985, 1440 img = ['temp10.png', 'temp11.png'] #images 2160x1440 img_i = 0 img_n = 2 def leftScreen(arr, flag): L_display = pi3d.Display.create(x=L_X, y=L_Y, w=L_W, h=L_H) L_camera = pi3d.Camera(is_3d=False) L_shader = pi3d.Shader('uv_flat') L_sprite = pi3d.Sprite(w=L_W, h=L_H, camera=L_camera) L_tex_arr = np.frombuffer(arr, dtype=np.uint8) L_tex_arr.shape = (H, W, 4) while L_display.loop_running(): while flag.value == 0: pass if flag.value == 3: break L_tex = pi3d.Texture(L_tex_arr[:,:L_W,:].copy()) L_sprite.set_draw_details(L_shader, [L_tex]) flag.value = 0 L_sprite.draw() L_display.destroy() flag = RawValue(ctypes.c_int, 0) shared_arr = RawArray(ctypes.c_uint8, H * W * 4) # alternatively, Array automatically does locking in which case tex_arr = np.frombuffer(shared_arr, dtype=np.uint8) # you need to call Array.get_obj() method here tex_arr.shape = (H, W, 4) p = Process(target=leftScreen, args=(shared_arr, flag)) p.start() display = pi3d.Display.create(x=R_X, y=R_Y, w=R_W, h=R_H) camera = pi3d.Camera(is_3d=False) shader = pi3d.Shader('uv_flat') sprite = pi3d.Sprite(w=R_W, h=R_H, camera=camera) tm = 0 while display.loop_running() and tm < 800: if (tm % 100) == 0: tex_arr[:] = np.array(Image.open(img[img_i])) flag.value = 1 img_i = (img_i + 1) % img_n tex = pi3d.Texture(tex_arr[:,R_X:,:].copy()) sprite.set_draw_details(shader, [tex]) while flag.value == 1: pass sprite.draw() tm += 1 flag.value = 3 shared_arr = None display.destroy() ## Log messages¶ When the demos start there is sometimes a message in the terminal looking like: 2013-08-19 15:36:46,232 INFO: __main__: Starting CollisionBalls Where does that come from and what does it mean? The Log module used to be started by several of the basic classes (Buffer, EventStream, Display, Loadable, Mouse, parse_mtl, Shader, Screenshot) However there were issues with this adding event handlers into the python logger hierarchy. In version v2.17 the logging within the pi3d module classes was switched to use the normal python logging. The example logger message above is shown because the CollisionBalls demo makes use of the pi3d.Log class. ### How to use logging¶ How do I use pi3d.Log to gather or display useful information in my application? pi3d.Log is a wrapper for python logging. You use it by making an instance of this class in your application then calling the methods debug() info() warning() etc. Whether the message is logged depends on the level set and whether it appears on screen or is sent to file and the formatting or additional info can also be controlled. See the documentation here. ### Printed messages not visible on Raspberry Pi¶ Sometimes I can’t get printed messages to show in the terminal when running on the Raspberry Pi and it makes it hard to debug the program. Sometimes they show but the carriage returns just do line feeds so the text works its way across the terminal window. This is a side effect of using ncurses for the pi3d.Keyboard which makes it fairly universal where the program is exited by watching for the ESC key. A solution is to use the pi3d.Log class and send the output to a file. See the Blur.py demo and the pi3d.Log module documentation (link above) ## Moving shapes together¶ How do I keep two components (Shapes) ‘joined together’ as they pitch, roll and rotate (yaw), like the TigerTank does with its body, turret and gun? This can be done automatically by adding Shapes to other Shapes.children lists which can be done using the Shape.add_child() method. All transformation applied to a Shape will then be relative to its parent and will be inherited its children. See pi3d_demos/TigerTank.py. One problem with an arrangement of parent -> child -> grandchild -> greatgrandchild etc. (as generally used by game-engines) is that the resultant orientations are not trivial to determine. However the draw() method does calculate a transformation matrix for each Shape that is available. There are two convenience methods transform_direction() and rotate_to_direction() that are used in pi3d_demos/TigerTank.py to find the direction that the gun is pointing and to rotate a missile in line with this. ## Angle of bank¶ I want to give my shape an angle of bank (z-axis rotation) which it maintains as it turns (y-axis rotation) - like an aeroplane. However the z-rotation is always relative to the absolute frame of reference so the shape pitches backwards and forwards as it turns. How do I make the frame of reference rotate with the shape? This is because of the order of the transformations done prior to redrawing the scene (z, then x, then y). You have to work out what the pitch and roll would have to be prior to rotating them about their own y axis! To see what I mean watch the behaviour of the tanks in pi3d_demos//TigerTank.py You have to figure out the ‘slope of the ground’ so that when your aeroplane (or boat) is rotated it ends up with the correct pitch and roll. For a shape with zero pitch you can use something like: absheel = degrees(asin(sin(radians(heel)) * cos(radians(heading)))) abspitch = degrees(asin(-sin(radians(heel)) * sin(radians(heading)))) hull.position(xm, ym, zm) hull.rotateToX(abspitch) hull.rotateToY(-heading) hull.rotateToZ(absheel) And see the pi3d_demos/DogFight.py version which has an extra degree of freedom. Generally problems like this can be done most easily by using the parent child structure as described above in Moving shapes together ## Moving vertices of existing Shape¶ Is it possible to change the shape of an object once it’s been made? The most efficient way is to use the scale(sx, sy, sz) method. However, this obviously limits the shape changing that can take place. If the shape needs to be changed more than this then it can be remade as a new instance to replace the old one. (At one stage it was necessary to clear the previous opengles buffers using the unload_opengl() method before destroying the old shape to stop a graphics memory leak. This issue seems to be fixed but if you run into memory problems it might be worth trying this. Plus, obviously, report it to us!) The alternative (faster, better) way of doing it is to use the Buffer.re_init() method which takes arguments to set new values for pts (i.e. vertices), texcoords and normals. These are passed as lists of xyz or uv lists or tuples or better, two dimentional numpy arrays. An offset argument can also be passed to allow only a section of vertices (normals or texcoords) to be modified. re_init() can’t change the number of vertices, just move them around. See the demos pi3d_demos/IceGrow.py and pi3d_demos/ProceduralTerrain.py. ## Slow animation¶ Sometime, when I move the mouse or the program is loading a file from disk, everything slows down or freezes. The Display has a frames_per_second argument and if you set this lower than the flat out rate it will give the processor some ‘slack’ to accomplish other jobs. To do things like file loading in the background (for instance, preloading an image or Shape so that it can instantly appear later) you need to use Python’s threading - pi3d_demos/Slideshow_2d.py is an example. ### Slow on non-raspberry pi machine¶ I am running pi3d on a non-raspberry pi Linux machine but it’s running at a very slow frame rate. Probably the GPU can’t run the OpenGL2+ code that mesa interprets from the pi3d OpenGLES2 commands. Check the specification for the graphics card. lspci -v and feedback.wildfiregames.com/report/opengl/ ### unresponsive mouse movement¶ Using python3 and the InputEvents mouse input (Silo and DogFight demos) I get very ragged and unresponsive camera movment. This should be fixed as of v1.5, try upgrading to the latest version of pi3d ## Permission denied¶ Some of the demos on a non-raspberry pi Linux machine work fine but other don’t run and give an error: IOError: [Errno 13] Permission denied: u'/dev/input/mice' what is the cause of this The default Mouse gets its info from the operating system file described in the error message. This requires it to be run from root, you can do this by sudo python ForestWalk.py. Alternatively, from v2.7, there is an argument to Display.create() use_pygame=True which will use mouse and keyboard input from a pygame display - the system that is used on Windows. See also below… NB A better fix for the access to /dev/input/ on laptops etc it to add your user to the input group. On this ubuntu 14.04 computer I did:$ getent group # to see if there was an existing group 'input' which there wasn't
$sudo groupadd -f input$ sudo gpasswd -a USERNAMEHERE input
$sudo nano /etc/udev/rules.d/pi3d.rules # new file to which just had this line SUBSYSTEM=="input", MODE="666" # restart computer This should also get the input events system working as used in Silo and allow joysticks and xbox controllers to be used. Thanks to Piotr Bednarski for sorting this out. ## Full Screen¶ I would like to have a fullscreen frameless/borderless window for pi3d when running under x/mesa. It should looks just like it does for the RPi. If the use_pygame=True argument is used for Display.create() and no w, h, x, y values are given then the pygame supplied drawing surface will be full screen without borders. ## Post processing¶ How do I do post-rendering processing on a scene, such as blurring, edge detection or fancier effects such as oil painting. There is a class PostProcess that can be used to render a scene to a texture. The Post.py demo shows a simple 3x3 convolution matrix shader and there are a host of post process filter shaders that are in the pi3d_demos/shaders directory. These wll be loaded in turn by pi3d_demos/FilterDemo.py but the pi will run out of graphics memory if you leave the full list in. For more complicated effects it’s over to you! ### PostProcess class¶ OK the example for post processing (pi3d_demos/Post.py) is quite hard to follow how exactly does the PostProcess class work. PostProcess inherits from Texture (via OffScreenTexture) so you can use an instance of it anywhere you would use a texture, i.e. you could uv map it onto any other shape or use it as a bump or reflection map. Or use it with your own shader to do something I haven’t thought of. PostProcess.sprite is a Sprite shape that can be used just as any other Shape in your program, you could rotate it or change its alpha value or z location to draw it in front of other objects. There is also a 2D camera created in PostProcess which is used to draw the sprite at full screen using the saved texture and the shader you supply in the constructor or post_base if you don’t supply one. PostProcess.draw({48:1.1414, 49:2013, 50:0.0}) will set the unif array in PostProcess.sprite as unif[48] = 1.1414 unif[49] = 2013 unif[50] = 0.0 you can then access these values as uniform variables in your shader as vec3 unif[16][0] unfi[16][1] unfi[16][2]. If the array indices are contiguous you could do the same thing using PostProcess.sprite.set_custom_data(48, [1.1414, 2013, 0.0]) or even PostProcess.sprite.unif[48] = 1.1414 etc I see no reason why you shouldn’t do something like: render the scene to a texture once a second draw it off-screen using a shader to extract edges as dayglo on white, blur them to a second texture, draw this onto a foreground sprite fading from alpha 0 to 1 back to 0 over 1s cycle. Use a different shader to draw the original texture onto a spherical surface that gradually changes shape in the background. etc etc. ### Is it possible to access the PostProcess image as a numpy array¶ In order to get the pixels ‘out of’ the GPU memory into CPU space, the only way I have found is to do something like: import numpy as np import ctypes ... ntex = np.zeros((post.iy, post.ix, 4), dtype=np.uint8) # make an empty array of the correct size ... # inside the drawing loop. If you are offscreen rendering then you need to do this # BEFORE you switch back to the normal view with end_capture pi3d.opengles.glReadPixels(0, 0, post.ix, post.iy, pi3d.GL_RGBA, pi3d.GL_UNSIGNED_BYTE, ntex.ctypes.data_as(ctypes.POINTER(ctypes.c_short))) But glReadPixels is relatively slow compared with rendering to and then drawing with a renderbuffer object so don’t expect a fantastic framerate. ### python v. shader unif arrays¶ And why does python set Shape.unif[48] but the shader use vec3 unif[16][0]. On the shader side it’s really efficient to define variables as vec3, vec4, mat4 etc. and at one stage I tried doing a lot of the matrix manipulation in the vertex shader. There were pros and cons but in the end I found that using python’s numpy library was the best bet. But in the mean time I had started storing much of the shape information in a form that allowed it to be accessible by the shader i.e. location x,y,z was vec3 unif[0] in the shader, rotation was vec3 unif[1], scale unif[2], origin offset unif[3] etc. Although I no longer needed these for normal rendering I thought that they may come in useful for someone at some stage so I just left them. I only needed to pass one array pionter so there was no cost to having 60 floats available! Meanwhile back in the python description of the Shape I had to make the unif array a ctypes.c_float array and that seemed to have to be one-dimensional. So after a long story unif[16][0] in the shader is (same name but different) unif[16*3 + 0] in python ## Texture UV mapping¶ The individual vertex UV coordinates are used to map Texture sampler calls from image (or numpy array) locations varying from 0,0 in one corner to 1,1 in the opposite. However the Buffer instance holds uniform variables that can be added or multiply these values. Buffer.unib[6:8] holds (umult, vmult) and Buffer.unib[9:10] holds (u_off, v_off). The default values are (1.0, 1.0) and (0.0, 0.0) respectively. ### Move Textures¶ Is it possible to ‘slide’ a Texture over the surface of a Shape to give the impression of movement, say? This can be done using Shape.set_offset((u_off, v_off)) which is a wrapper for Buffer.set_offset((u_off, v_off)) This is used in pi3d_demos/Water.py ### Scale Textures¶ How do I adjust the scale of a texture and the number of repeats across the width and height of a Shape? umult and vmult are arguments to Shape.set_draw_details and can be used to increase the number of repeats of a texture. These are used ‘behind the scenes’ in the Building class in the draw_details argument, see the Silo demo. Can I use part of a Texture to render onto a Shape? More specifically, is it possible to use MergeShape to combine several objects into one for efficient drawing but draw different parts with different textures? It’s possible to use part of a texture by combining offset and mult, for instance to use one quarter of a texture you could set umult, vmult to (0.5, 0.5) and u_off, v_off to (0.0, 0.0), (0.5, 0.0), (0.0, 0.5) or (0.5, 0.5) To draw different parts of one Buffer with sampling from different parts of a Texture sampler (as above) then the UV texture coordinates have to be modified as shown in the listing posted in this forum discussion ForumMergeShapes (commented out section under #1.Textures) However the method of adding child objects as in the Molecule1 demo is more straightforward and, from pi3d v.2.22, it is possible to construct a MergeShape with multiple Buffers each with its own material, texture, shader etc. as in the Molecule2 demo. ## Blend shaders¶ How do the blend shaders work as used in the PictureFrame demo These shaders are based on the 2d_flat shader (as mentioned above) that uses the screen coordinates of each pixel, rather than the interpolated coordinates of 3D polygon uv values, to look up the color values. The main differences from 2d_flat are 1. There are two textures passed to the shader 2. There are two sets of x, y, w, h and screen height values passed to the shaders (one for each texture) 3. There is a time value passed to the shader varying from 0.0 to 1.0 to control the proportion of blending 4. There is a blending function! If you look at the code for blend_bump.vs (and the other blend vertex shaders) you will see that it calculates two vec2 varying values that are passed to the fragment shader. The sole reason for doing this is relative expense of dividing by a variable compared with multiplication in the fragment shader. The values are used to scale the pixel locations to texture lookup locations. All of the fragment shaders then operate in a fairly similar way: pick up the fading factor (tm = unif[14][2]), define coord as the pixel location on the screen, for the foreground and background textures check if the pixel falls outside the texture, if it doesn’t then look up the RGBA value from the texture. Having got the foreground and background pixel values there is then a process of combining them which generally involves calculating a factor dependent on some or all of a) pixel values b) x,y location on the screen c) tm. Using the factor in a mix() function. bump: generates a factor as if the background texture was a normal map to modify the foreground as it blends from one to the other burn: compares the brightness of the background pixel with a sliding threshold to determine how much to mix the foreground and background false: creates a false middle color using factors acting on the foreground and background RGB values and blends to and from the mid color holes: uses the distance from a grid of points to determine the proportion of mixing star: calculates the pixel position in polar coordinates (angle and radius) then does some trig to determine the blend proportion ## Points¶ How can I render points like a star field or sparks from an explosion. If you use the method set_point_size() on a Shape to a value other than 0.0 then the vertices of the Shape will be rendered as points. The size will actually vary with distance but will be the size you specified at 1 unit of distance from the camera. pi3d.Points can be used to render points using the mat_flat shader or special shaders as used in the demo pi3d_demos/SpriteMulti.py ## Lines¶ How can I render lines such as graphs or axes or the ‘wireframe’ version of a Shape. If you use Shape.set_line_width() then the the Buffer objects in the Shape will have their draw_method set to GL_LINE_STRIP which will join all the vertices as point on a line. There is an optional argument closed that defaults to False which can be used to join the last vertex back to the first (by setting draw_method to GL_LINE_LOOP). To create your own lines you would need to make a list of (x,y,z) vertices and an element array to join them together and pass them to the Buffer constructor. The pi3d.Lines class does this for you. It is possible to create multiple Buffers within a Shape and set some as faces (draw_method set to GL_TRIANGLES) and some as points or lines. You can set different Shaders for different Buffers from v2.6 ## Using numpy arrays for Textures¶ I would like to do fast array processing with numpy and use the results directly as Textures in pi3d. How do I do this? You can pass either a PIL Image or a numpy array directly to the Texture class as it creates an instance. The numpy array will need to have shape (H, W, N) where N is 4, 3 or 1 and the array must be dtype=np.uint8 If you attempt to use a slice of another array the resultant texture will be scrambled unless you make an explicit copy i.e.: ar = np.array([[[2 * i, 2 * j - i, i + j] for i in range(128)] for j in range(128)], dtype=np.uint8) tex = pi3d.Texture(ar[:50,:100].copy()) # try without copy() To refresh the Texture with a constantly changing numpy array use the Texture.update_ndarray() method. This can be passed the new array as an argument, alternatively and faster, the Texture.image ndarray can be modified in situ and update_ndarray() called without any arguments. ## Assigning different Textures to vertex groups¶ I want an ElevationMap with more detailed textures mapped to different areas. i.e. rock, grass, heather, swamp. From v2.19 there are ready made shaders included to do this as well as an argument to specify how up to four different diffuse textures along with four different normal maps can be allocated on a vertex by vertex basis - see pi3d_demos/TigerTank.py. ## Without PIL (Pillow)¶ Can I use pi3d without installing PIL - for instance trying to run on a different platform or to have an ultra small SD image? Yes as of v2.15 you can do this but some of the classes will not work i.e. FixedString, Font, Building The Pngfont will still work for rendering text. HOWEVER normal image files cannot be imported (jpg, png, gif) instead you have to convert these into compressed saved numpy files (npz) including any png files used as fonts. This can be done using a simple script on a machine that does have PIL installed: from PIL import Image import numpy as np import os dirctry = 'textures' for f in os.listdir(dirctry): f = f.split('.') if f[-1].lower() in ['jpg','png','gif']: print(f) im = np.array(Image.open('{}/{}.{}'.format(dirctry, f[0], f[1]))) np.savez_compressed('{}/{}'.format(dirctry, f[0]), im) ## Minimal SD card¶ How can I set up an SD card without all of Raspbian’s clutter that will boot quickly and allow me to run a dedicated pi3d application. This is what I did to get a version of the PictureFrame demo running on a Raspberry Pi that I wanted to just do this job without the need for Wolfram, Scratch or even the X11 desktop system. 1. The Raspberry Pi Foundation hosts a ‘stripped down’ version of Raspbian Jessie LITE avaliable at https://www.raspberrypi.org/downloads/raspbian/ which can be downloaded and burned to SD following the instructions there. 2. Start the Raspberry Pi logging in as user pi, password raspberry then:$ sudo raspi-config
- boot options: console login as pi automatically
- internationalisation options: select relevant city
- expand file system
- advanced: increase graphics memory to 128
3. Install just the software to run the application:
$sudo apt-get update$ sudo apt-get upgrade
$sudo apt-get install python3$ sudo apt-get install python3-numpy
$sudo apt-get install python3-pillow$ sudo apt-get install python3-pip
$pip install pi3d 4. Download the modified project from github:$ wget https://github.com/paddywwoof/pi3d_pictureframe/archive/master.zip
$unzip master.zip$ rm master.zip
$mv pi3d_pictureframe-master pi3d_pictureframe$ cd pi3d_pictureframe
5. Checkout the excellent documentation by Tathros on which I have based these instructions. The main differences are getting it to work in console mode, without the need for a desktop environment:
### screen application needed to show output run from crontab ######
$sudo apt-get install screen ### turn off screensaver ###########################################$ sudo nano /etc/kbd/config
... BLANK_TIME=0
...
... POWERDOWN_TIME=0
CtrlX,Y,Rtn
### set up wifi (you will need dongle if RPi < 3) ##################
$sudo nano /etc/wpa_supplicant/wpa_supplicant.conf ... ctrl_interface=DIR=/var/run/wpa_supplicant GROUP=netdev ... update_config=1 ... ... network={ ... ssid="YOUR_SSID" ... psk="YOUR_PASSWORD" ... } CtrlX,Y,Rtn ### auto start and stop - only do after checking everything works ##$ crontab -e
... # turn off screen at 21:00
... 00 21 * * * touch /home/pi/pi3d_pictureframe/stop; /opt/vc/bin/tvservice -o
... # turn on screen 07:00
... 00 07 * * * /opt/vc/bin/tvservice -p; /bin/chvt 2; /bin/chvt 1; screen -dmS PICFRAME /usr/bin/python3 /home/pi/pi3d_pictureframe/PictureFrame.py
... # kill any extra python processes that might have crept in
... 00 04 * * * killall python3
... # switch on at start up
... @reboot screen -dmS PICFRAME /usr/bin/python3 /home/pi/pi3d_pictureframe/PictureFrame.py
CtrlX,Y,Rtn
In these instructions the “…” at the start of lines represents the fact this is text inside a file and shouldn’t be actually typed in! Also you need to change the WiFi credentials to match your router, NB the SSID and PASSWORD need to be inside quotes. In the picture_getter.py script you will need to put in the email server, user and password for picking up images.
## pypy¶
Does pi3d work with pypy
pi3d relies on some of the functionality and speed of numpy and this only really became usable as of pypy-2.2 and the recent versions of pypy-numpy need more recent versions of pypy (>4.0.1 or 4.4 see below and bitbucket.org/pypy/numpy)
As at 2017-04-24 I have installed and tested pypy on a Raspberry Pi 3 and on my ThinkPad i5-2410Mx4core laptop running ubuntu. It pretty much all works though there are a couple of features of numpy that still don’t. Fixes to Font and FixedString will be in pi3d from v.20 (in the develop branch from now). NB it’s generally a good idea to do all this inside virtualenv if you are thinking of using normal python at the same time
On the Laptop thefollowing steps seemed to get pypy working:
1. in a terminal install pypy from ubuntu:
sudo apt-get install pypy
sudo apt-get install pypy-dev
sudo apt-get install pypy-pkg-resources
sudo apt-get install pypy-setuptools # needed for Pillow install from source
2. download and install pypy-numpy I did this then changed to that directory:
git clone https://bitbucket.org/pypy/numpy.git --depth 5
cd numpy
sudo pypy setup.py install
# you can then delete this directory
3. download Pillow source from https://pypi.python.org/pypi/Pillow and extract it, cd to that directory and install as above:
cd Pillow-4.x.x
sudo pypy setup.py install
# you can then delete this directory
On Raspberry Pi raspbian stretch (Aug 2017) pypy v5.6.0 is already installed however, for some reason, pypy pip has not been, which limits use to core or pure python modules. Also I found that neither the latest version of numpy or pillow worked:
1. Get hold of pip for pypy:
cd ~
sudo curl -O https://bootstrap.pypa.io/get-pip.py
sudo pypy get-pip.py
pypy -mpip install -U wheel
2. Pip install numpy NB using pypy -mpip ..:
pypy -mpip install cython
sudo apt-get install pypy-dev
sudo pypy -mpip install numpy==1.12.1
3. Pip install PIL agian pypy -mpip ..:
sudo apt-get install libjpeg-dev #zlib1g-dev libpng12-dev libfreetype6-dev # seemed to cause errors
sudo pypy -mpip install Pillow==4.0.0
Prior to stretch (i.e. jessie) I found that the pypy installed is v4.0 and pypy-numpy demands > v4.4 so this is what I did (NB I noticed subsequently that there is an option to install numpypy for pypy4.0.1 - see the bitbucket/pypy/numpy page if you want to try this first):
2. to save confusion I overwrote the symlink to the previously installed pypy-4.0. In a terminal:
sudo ln -s -f /home/pi/pypy2-v5.7.1-linux-armhf-raspbian/pypy /usr/bin/pypy
3. I tried the three apt-get install instructions as per ubuntu but only pypy-dev seemed to work. It wasn’t clear how to use pip with pypy but in the end I did the following in a terminal:
cd /home/pi/pypy2-v5.7.1-linux-armhf-raspbian/
pypy -m ensurepip
bin/pip install -U pip wheel
bin/pip install pkg_resources
I think there was an error but import pkg_resources seemed to work after this.
4. At one stage I installed all of the following, many may not be needed but some might. You could try without first I suppose, but it didn’t take too long.:
sudo apt-get install gcc make libffi-dev pkg-config libz-dev libbz2-dev \
libsqlite3-dev libncurses-dev libexpat1-dev libssl-dev libgdbm-dev tk-dev \
libgc-dev python-cffi liblzma-dev
5. For Pillow to work you need these (that’s a ‘one’ in zlib1g-dev by the way):
sudo apt-get install libjpeg-dev zlib1g-dev libpng12-dev libfreetype6-dev
6. Download, extract and install pypy-numpy. I used git, as with ubuntu above.:
git clone https://bitbucket.org/pypy/numpy.git --depth 5
cd numpy
sudo pypy setup.py install
# you can then delete this directory
7. Pip install Pillow:
cd /home/pi/pypy2-v5.7.1-linux-armhf-raspbian/
bin/pip install Pillow
For both these installations you will need to pip install pi3d (using the pypy pip) or download (or git clone) pi3d and use sys.path.insert(1, '/home/pi/pi3d') before import pi3d. The pi3d_demo files do this automatically.
At the moment the PexParticles and Pong demos fail mainly due to non-implemented features of numpy (einsum, interp, remainder and outer). The speed benefit of running pypy won’t improve the GPU functionality or the numpy calculations but in some areas where there is a lot of looping in the python code there could be gains (pi3d_demos/CollisionBalls.py for instance).
## DIY environments¶
### Cube¶
How can I make my own EnvironmentCube images using pictures of my garden or school playground?
Option 1. Using an EnvironmentCube (as the question says) but see below for using a Sphere, which is probably easier.
There are lots of ways of doing this and different software as well as special cameras. However this is the method I have followed using freely available software: gimp and blender (running on a ‘normal’ computer rather than the pi at this stage).
The first half of the job is to get a set of images into a ‘seamless’ band. Obviously you need to have taken a set of pictures that overlap 25% to 50%. In gimp make a new image that is higher and wider than you will need to paste all the images side by side. You will need to have the same image repeated at the left end and the right end.
Open each image in gimp then copy it, go to the new ‘wide strip’ image and paste as new layer. Use the four headed arrow to position each layer so it ‘joins up’. When you put the duplicate left most image at the right end you need to make sure that it is at exactly the same vertical position as it is on the left.
Working down from the top layer add layer masks (default white, full opacity) then using gradient fill tool make the mask fade from transparent to opaque across the overlapping portion. You might need to slightly rotate some images to make them join up nicely from one side to the other.
When it looks perfect (!) merge the layers down then crop the image so there are no gaps at the top and bottom and so the left and right edges join seamlessly. You will probably have to zoom to maximum and choose an easily identifiable pixel. The rectangular selection tool in gimp allows the edges to be dragged to fine tune it. Export the image to jpg or png possibly after reducing to a reasonable size. Have some suitable sky only image to patch into the top of the sphere you will create in blender…
I used blender 2.69, it’s not a trivial application if you’ve not used it before and it might take a bit of effort to figure out what I’m referring to [tab] means tab key, otherwise it’s probably a menu item or an icon in the right hand. Lots of youtube videos to look at. In blender:
1. [del] delete the startup cube
3. [s] to scale up to about 10x
4. [tab] to edit mode [a] to deselect all vertices. R-click on top vertex the Ctrl-numpad+ to select vertices down to about 45 degrees north (or use [b] and box select) [del] delete vertices. You should now have a sphere with the top cut off
5. [tab] back to object mode then create another sphere at the same location but scale it up very slightly bigger and chop off the bottom but so they overlap just a little.
6. [tab] back to object mode then Add Empty Cube at the same location (NB if you accidentally left click on the view window you will move the starting point marker where new things appear). You should be able to zoom in with the mouse wheel and see this cube inside the spheres.
7. still in object mode right click to select the bottom (inner and larger) sphere. The edge should go yellow to indicate it’s been selected.
8. on the right properties window click the Materials icon (CofG circle 4th from right), then + new.
9. then click the Textures icon (red/white check 3rd from right), then + new, Type Image or movie, Image New browse to the wide horizon image you made, Mapping Projection Tube
10. still in object mode right click on the top sphere, add material and texture exactly as for the bottom sphere but select the patch of sky image mentioned above and choose Mapping Projection Flat
11. in object mode right click on the Empty Cube and add a new Texture (you should see a reduced list of options so it’s 2nd from right in the list)
12. select under Type Environment Map then under Environment Map Static, Mapping Cube and Viewpoint Object Empty
13. in the properties icons select render (camera left most) then under Render press the render button. This should flash up a series of six smaller images then go black!
14. re-select the Texture icon (all of these steps should have the Empty Cube as the selected object) and the little down arrow under Environment Map should produce a drop-down menu with an option to save the image.
The texture can then be used in pi3d with EnvironmentMap type BLENDER. However there will be a sharp line where the edge of the bottom sphere fell. You can smooth this out using clone, repair, blur and blend tools in gimp; be careful not to blur the boundaries between the six images.
### Sphere¶
How do I make an Environment Sphere (such as can use the Photo Sphere images created by later versions of Android)
First you need an image very much like the one outlined in the previous question. If you have the software on your phone or tablet to do a Photo Sphere that’s going to be a lot easier but you can do something similar with a series of panoramas as modern cameras can make. The image needs to be twice as wide as it is high using a standard cylindrical projection http://en.wikipedia.org/wiki/Equirectangular_projection
This image is used for a Texture uv mapped to a standard pi3d.Sphere but the Texture needs to have the argument flip=True and the Sphere needs the argument invert=True
If the same image is used as the reflection with uv_reflect or mat_reflect shaders then the correct part of the scenery will be rendered - i.e. behind the camera and transposed left-right, see demo pi3d_demos/EnvironmentSphere.py.
## pickling¶
How can I speed up loading Models. Even quite low polygon counts seem to take ages on the Raspberry Pi
Thanks to Avishay https://github.com/avishorp it is possible to use the python pickle functionality to serialise pi3d Shapes including Model.
There is an example on github.com/pi3d/pi3d_demos pi3d_demos/LoadModelPickle.py which shows the process but basically:
load the models once normally, create a file (has to be binary for python3) to write to, then pickle.dump(mymodel, f)
subsequently open the file to read from and mymodel = pickle.read(f) the loaded file will have any required Textures included automatically including bump and reflection maps. However the shader will still need to be set with set_shader()
Loading from a pickle file is significantly faster than parsing a wavefront obj file but (because of the less efficient image compression) the disk space used will be much higher.
## Strings¶
### quick change¶
How to have lots of rapidly changing text on the screen (such as location game-status readouts etc) whithout having to create new String objects all the time (with associated processor load)
This can be done using the String.quick_change() method.
When you first create the string you need to make it big enough to fit in any additional characters you may send to quick_change() subsequently. At the moment it doesn’t cope with multi-line Strings.
There is an example in pi3d_demos/ForestQuickNumbers.py
Why do I get an error when I try call the quick_change() method in my program.
If you get AttributeError: 'Buffer' object has no attribute 'vbuf' then this could be because you are calling quick_change() before the first draw() of the String object. Unfortunately you can’t do this (as at pi3d v2.10) and you will have to alter your code to ensure the draw happens before the change.
### FixedString¶
How to have a large amount of text without creating hundreds of extra polygons for the gpu to render?
The String object has a little rectangle for each letter, each of which needs four vertices and two triangles. If the text does not need to be changed then it is better to use the FixedString class. The object inherits from Texture with the provided text drawn onto it. It also creates a simple sprite with four vertices and two triangles that can be used to draw the texture. There are filters that can produce effects such as blurring, outlining and normal map generation.
### PointText¶
I want lots of text changing in real-time, scrolling, rotating, fading or changing colour.
For complicated things like this then the PointText class should be used (with TextBlock components) see pi3d_demo/StringMulti.py
How can I make my text show up against different coloured backgrounds (such as when displaying the score as 2D text)
There are two ways to do this in pi3d. The first is to define a background_color in the Font or FixedString with a low alpha value (say 20 out of 255). This will produce partially transparrent background for your letters which will be a rectangle if you are using a FixedString but will be uneven edged if you are using a normal String with Font. If you set the alpha to zero you will end up with a subtle line around the outside of each letter, which may be enough.
To get a larger outline around letters you can specify a shadow_radius and a shadow RGBA value (default black with no transparrency). This produces a gaussian blurred version ‘underneath’ each letter. See the Shapes demo for how this might look.
## Normal Map generation¶
I have a low polygon model that I want to appear more detailed. I know I can do that if I have a normal map to supply to the .._bump or .._reflect shaders. Is there a way to generate these automatically.
Yes. If you simply use the same image file for the texture and the normal map then this will alter the lighting of the surface. However this will often give rather strange effects due to the interpretation of surface normals from RGB values. To get a better result you can create a version that uses the lightness of the image as a height map, for instance in pi3d_demos/Water.py:
shapeimg = pi3d.Texture("textures/straw1.jpg")
shapebump = pi3d.Texture("textures/straw1.jpg", normal_map=-6.0)
The size and sign of the normal_map argument can be used to compensate for the contrast and positive/negative nature of the image.
Alternatively you can make a copy of the colour texture image and edit it with GIMP (or similar) to make a greyscale version that matches the surface geometry you want.
## Texture animation¶
Is it possible to change a texture every frame at a reasonable frame rate? i.e. for displaying a video, a feed from a web cam or an image manipulation program such as OpenCV, Scipy or numpy?
This became much more feasible after v1.4 and more so after v2.1 The Texture class now accepts a numpy array (size (H,W,N) where N is 3 for RGB or 4 for RGBA), remember C arrays are row,col,pixel)
There is also a method Texture.update_ndarray(new_array) that can efficiently switch the image to the new array. See the demo pi3d_demos/VideoWalk.py which maps a movie onto a shape using ffmpeg.
A much faster system can be used on the Raspberry Pi that utilises the C code used in the demos in /opt/vc/src/hello_pi. This is tricky programming but a working version has been created by @swhner https://github.com/swehner/foos/blob/opengl_replay/foos/ui/opengl_replay.py and https://github.com/swehner/foos/blob/opengl_replay/egl_replay/video_helper.c
## Profiling¶
How to profile code to find where the bottlenecks are? For example to find if it’s worth doing something complicated with numpy or ‘blitting’ small areas of the screen as in the NumpyBalls demo?
The python profiler cProfile is very easy to use but I have found it struggles to find directories from the code and gives quirky information unless I do something like:
$cd ~/pi3d_demos$ python -m cProfile ~/pi3d_demos/NumpyBalls.py > result.txt
## Desktop or laptop¶
Is it possible to use pi3d on my laptop or desktop computer ideally running windows?
If your computer has a suitable graphics card then you should be able to do this.
Raspbian Pixel is probably the easiest to set up from scratch and most closely matches behaviour on the Raspberry Pi, if you are happy booting onto a USB stick see ReadMe Linux
windows requires a version of pi3d v2.0 or later see ReadMe Windows
linux is more similar to the Raspberry Pi, also see ReadMe Linux
mac ought to be possible following a very similar procedure to linux but I havn’t tried (let me know if you do!)
On windows or mac it is also possible to use something like VMWare:
Setup:
VMWare Client
3d Accel.. activated!
LinuxMint Installation (Ubundu-based and Debian base version works)
very important
mesa-utils-extra
python-numpy
and the rest as described in the Pi3D documentation
Important: pi3d scripts must be started with sudo
e.g. sudo python ./Pi3D2.py
In the VM it does not run very smooth, but it works without errors.
After testing this setup I've installed the setup to a partition... runs like a charm :-)
## Android¶
Is is possible to run pi3d on my Android phone or tablet which uses a chip and operating system very similar to the Raspberry Pi?
Is is possible; but you have to compile an apk package using python-for-android from a linux machine. There are instructions here http://pi3d.github.io/html/AndroidUse.html
## Targetting¶
### Identify what’s in front of the Camera¶
How to find the name of the object in the camera’s target (pi3D 3d world) example:https://www.youtube.com/watch?v=0u91pcNXtcI&feature=autoshare sun, world, astroid123?
One approach would be to look at the size of the angle between the direction to each possible target and the direction the camera is pointing. You would then choose objects where that angle was smaller than a threshold then choose the nearest if more than one. Say in pi3d_demos.ForestWalk.py you had kept a dict of different objects as obj_dict, you could add a print() when you press the ‘h’ key:
obj_dict = {'monument':monument, 'trees1':mytrees1, 'trees2':mytrees2, 'trees3':mytrees3}
THRESHOLD = 0.95
...
elif k == ord('h'):
nearest = None
target_name = None
for o_name in obj_dict:
o = obj_dict[o_name]
v1 = o.unif[0:3] - CAMERA.eye # numpy vector from camera to object
dist = (v1 ** 2).sum() ** 0.5
v1 /= dist # divide by length
v2 = CAMERA.get_direction() # already length = 1
cos_angle = v1.dot(v2) # 1 straigt at it, 0 at right angles, -1 opposite direction
if cos_angle >= THRESHOLD:
if nearest is None or dist < nearest:
nearest = dist
target_name = o_name
print('nothing' if target_name is None else target_name)
However this only works for smallish objects. If you want to see if the cross-hairs are hitting the Earth on the meteorize game then you would either have to include a size factor into the obj_dict above instead of using an overall threshold i.e.:
obj_dict = {'monument':[monument, 4],
'trees1':[mytrees1,15],
'trees2':[mytrees2,20],
'trees3':[mytrees3, 20]}
o = obj_dict[o_name][0]
sz = obj_dict[o_name][1]
if cos_angle > (1 - (sz / dist) ** 2) ** 0.5
Though this method would have issues with large objects such as the merged ‘tree’ objects in ForestWalk, where it is possible to be ‘inside’ the object.
Or look at each face of the object and see if the vector intercepts each triangle - but this would be very slow. See the Pong demo and the ElevationMap.clashTest() method.
Finally a fairly efficient but technical method would be a variation of the Clashtest system using glScissor to draw a single pixel of each object to an offscreen texture with different RGB values for each then checking what if any had been drawn.
### Bullets¶
How to create bullets and shoot targets? (like counter strike)
The bullets in Counter Strike just look like streaks of light with some smoke so the best way is probably something like the bullets in DogFight. This uses two Plane objects merged at 90 degrees to each other with a series of images that can be mapped onto them in sequence (there are two sets of guns on the Aeroplane which is why there are four Planes).
In the pi3d_demos/TigerTank.py (from v2.23) there are missiles (purple beer bottles) and you can aim (mouse rotates, u, j raise lower gun) and fire (f)
How you animate the target being shot depends on the effect you want. In the meteorize game I use a part transparent texture and scale the meteors up so they look to be bursting soap bubbles.
### Laser sight¶
How to create a laser point for gun?
This is the kind of thing you can do. For example to modify pi3d_demos/ForestWalk.py to include a laser dot:
1.Copy the shaders you want to get lasers on to the pi3d_demos/shaders directory and rename i.e. uv_bump_laserdot.vs, uv_bump_laserdot.fs
#include std_bump.inc
float radial = distance(vec2(gl_FragCoord), unif[15].xy); // 1 --- distance from centre of screen
texc.rgb += vec3(5.0, -2.0, -2.0) / (dist * radial); // 2 --- increase R, decrease GB reduce size in distance
gl_FragColor = (1.0 - ffact) * texc + ffact * vec4(unif[4], unif[5][1]); // ------ combine using factors
2.change ForestWalk.py to use the new shaders: | 2018-10-18 20:41:11 | {"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": 0.18495143949985504, "perplexity": 3602.6465654929593}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583512014.61/warc/CC-MAIN-20181018194005-20181018215505-00022.warc.gz"} |
https://ilyasergey.net/YSC2229/week-01-intro.html | # 1.1. Introduction¶
Data represents information and computations represent data processing, i.e., obtaining new information from what we already know about the world.
An algorithm is any well-defined computational procedure that takes some data value, or a set of values, as input and produces some value, or set of values, as output, always terminating with a result. An algorithm is thus a sequence of computational steps that transform the input data into the output data.
In this course, we will take a look at some of the problems that can be solved by algorithms, learn how to approach challenges that require algorithmic solution, and, most important, will learn how to reason about crucial properties of algorithms: correctness, termination, and complexity.
## 1.1.2. What problems are solved by algorithms?¶
Algorithms describe a problem in a way that it could be implemented as a computer program and solved automatically. You can think of an algorithm as of a description of a procedure that achieves that.
For example, we might need to sort a sequence of numbers in a non-decreasing order. This problem arises frequently in practice and provides fertile ground for introducing many standard design techniques and analysis tools. Here is how we formally define the sorting problem:
• Input: a sequence of numbers a1, a2, …, an.
• Output: a permutation (or, reordering) of the initial sequence b1, b2, …, bn, such that b1 <= b2 <= ... <= bn.
The description above presents a specification of a problem, but does not describe how to come from an input to an output. Indeed, there might be many ways to approach the same specification, and, as a characteristic example, there are multiple algorithms for solving the sorting problem, some of which we will see in this course.
Obviously, sorting sequences is not the only class of problems that can be solved by means of an algorithm. Some other problems include:
• Searching – quickly finding an element satisfying certain requirements in a large collection of other elements. This is a problem frequently appearing in the context of e-commerce (think about searching for a book to buy on Amazon).
• Data compression/decompression – representing an information so it would take the least memory, making it easier to store and transmit, while retaining the ability to restore it without any losses. As an example, think of video and audio encoders and decoders, whose goal is to minimise the size of a file to be streamed, while keeping its high quality.
• Path finding – navigating from a point A to a point B, in a quickest way, given a fixed map.
• Optimisations in manufacturing and other commercial enterprises. For instance, you can think of an algorithm that computes an arrangement of cables allowing to connect a number of call centers in the least expensive way.
• Various geometric problems: for instance, locating the closest facility to one’s position on a map, or installing a set of security cameras in an art gallery in the most efficient way.
Some real-world problems might require multiple algorithms to solve. As instance of such a problem, during the lectures we discussed a room furnishing challenge, in which one need to cover the maximal room surface with furniture. We will discuss more problems of that kind and will learn how to approach them in a class.
## 1.1.3. Data structures¶
Obviously, some of the tasks outlined above can benefit from a data arranged in a more convenient way. For example, if one is interested in finding the largest element in a collection, it would help to have this collection first sorted. As another example, if one wants to represent a map of roads of different length, which is suitable for calculating the shortest paths, it is wise to invest some time into thinking on how to arrange this information to facilitate retrieving the relevant topographical properties.
Data structures are means to organise information in ways that make it conceptually simpler and faster to retrieve the bits we might need in the future. Some of the data structures you have already seen in the past are lists, trees, and arrays. However, more intricate problems will require more sophisticated data structures.
Therefore, one of the goals of this course is to show how to pick a right data structure for a corresponding problem.
## 1.1.4. What is analysis of algorithms?¶
When we solve a problem algorithmically, we need the solution (i.e., an algorithm providing an answer) to satisfy the following hard criteria:
• Correctness: Does the algorithm really do what it’s supposed to do?
• Termination: Does the algorithm terminate for any given input?
• Complexity: Why the algorithm consumes so much resource (time, memory) and how can we improve it?
An algorithm that is simply not correct is not worth much (although the notion of correctness is, as we will see, in the eye of the beholder). An algorithm that does not terminate, but only on very special cases, might be actually useful. Finally, the analysis of computational complexity tells us how slow will be an algorithm in certain cases or how much memory will it consume.
In terms of time consumption, problems that can be solved algorithmically can be themselves partitioned to several classes:
• tractable problems - admit solutions that run in “reasonable” time: sorting, searching, compression/decompression.
• possibly intractable — probably don’t have reasonable-time algorithmic solutions: SAT, graph isomorphism
• practically intractable — definitely don’t have such solutions: the Towers of Hanoi
• non-computable — can’t be solved algorithmically at all: the halting problem
A good programmer should be able to guess correctly whether the problem she is trying to solve algorithmically belongs to one of those classes.
There are two ways to analyse algorithms for correctness, termination and complexity:
• Empirical – repeatedly run algorithm with different inputs to get some idea of behaviour on different sizes of input. This approach is very practical (and we are going to rely on it a lot), but has certain shortcomings:
• Was our selection of inputs representative?
• This process consumes the very resource (time) we are trying to conserve!
• Theoretical – a mathematical analysis of a “paper” version of the algorithm:
• Can deal with all cases (even impractically large input instances);
• Machine-independent;
• Sometimes requires non-trivial mathematical reasoning. | 2021-09-18 17:47:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5124620795249939, "perplexity": 558.9384713669456}, "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-2021-39/segments/1631780056548.77/warc/CC-MAIN-20210918154248-20210918184248-00025.warc.gz"} |