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If you wanted to see how this once small industry enables reliable and safe travel to billions of people all across the world, this is the perfect place to do so. Descriptive statistics are the appropriate analyses when the goal of the research is to present the participants' responses to survey items in order to address the research questions. They are always the product of research. Suppose that we have a set of data with a total of n discrete points. , Carol Brady of The Brady Bunch was picked by 40% of survey respondents as the TV mom that would bake the best pie. 42 Incredibly Weird Facts You'll Want To Tell All Your Friends. Mother's Day Fun Facts. Because statistics can be used to mislead, an understanding of the topic can be helpful for more than just the mathematical skills that it imparts: knowledge of statistics theory can also be a strong defense against attempts by the unscrupulous to mislead others. How to calculate the certainty students fudge data to make chi-square "work right?". Lesson by Mark Liddell, animation. There are FOUR requirements/criteria that need to be met in order for a question to be classified as statistical: 1. If you're behind a web filter, please make sure that the domains *. If nine voters are selected at random, what is the probability that. Will never give your phone number or residence address to everyone those in whose personality looks distrustful to you personally. Facts About Media Violence and Effects on the American Family. Trick questions stimulate the brain and provide fun. What is another word for lexicon? Answer: Dictionary. It wasn't fun, though — other symptoms included crying, fainting, rashes, and pain. Practice and review questions follow each lesson. Color is a powerful tool. Weird and Wonderful Money Facts and Trivia Money Hungry From dirty money, to fake money, to fast money, to funny money - human beings are fascinated with the dollar bills, coins and cards that let us accumulate wealth and perhaps happiness. STA220 Final Exam 2001 has 30 questions from the 2001 final exam in an introductory Statistics service course at University of Toronto. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Quantitative Analysis. This texting statistics page has been one of our all-time best performing pages. Statistical Topics This topics list provides access to definitions, explanations, and examples for each of the major concepts covered in Statistics 101-103. 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In a country where so many desire status and wealth, petty annoyances can spark disproportionately violent behaviour. It involves asking questions about the world and finding answers to them in a scientific way. Everyday Mysteries. The investigation section of this site deals with the development and increasing sophistication of statistical investigations. Here are forklift accident statistics that we found around the web: There are roughly 85 forklift deaths and 34,900 serious injuries each year due to forklift accidents (data from 2007), according to the Occupational Safety and Health Administration (). Practice spotting the difference between statistical and non-statistical questions. Fact 10: The longest recorded volleyball game was in Kingston, North Carolina. Describes the importance of creating questions to guide research, provides insight on how to develop these questions, and includes many examples. Published and compiled by the Library & Information Statistics Unit, Loughborough University, Loughborough, Leicstershire. Show More Posts →. Statistics are a funny thing. • The sample consists of 50 customers from three local landscaping. The real job of an employee newsletter is to motivate workers to do positive things for customers and the community. However, growing evidence suggests that games can do more than keep us entertained, especially when it comes to learning in a classroom setting. COM Food Food Fun & News. Earth has only one satellite, the Moon. What Does Statistical Analysis Mean? What is the definition of statistical analysis? This is done in many ways such as: regression formulas, means, r-squared calculations, and ratio analysis. You will record your data and use graphs to represent your findings to the class. Friday the 13th Patterns Repeat in the Calendar. Lots of information to share? Making an infographic? 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If you select Gibnut from a menu, you’re about to dine on a rainforest rodent (Paca Cuniculus) which is a popular game meat in Belize also known as the Royal Rat. This is different from a question that anticipates a deterministic answer. Did you know 11% of people are left handed; Did you know August has the highest percentage of births; Did you know the average person falls asleep in 7 minutes; Did you know lemons contain more sugar than strawberries; Did you know 8% of people have an extra rib; Did you know 85% of plant life is found in the ocean; Did you know the Hawaiian alphabet has 13 letters. Catholic fun facts and statistics: Patron saint of single women, patron saint of cats, patron saint of dogs, patron saint against anger and catholic definitions. That is, set aside one to two hours at the same time each day for six days out of the week (Take the seventh day off) for studying statistics. 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Answer: Dom Perignon. However, growing evidence suggests that games can do more than keep us entertained, especially when it comes to learning in a classroom setting. To ask a question, go to a section to the right and select "Ask Free Tutors". There are no hypotheses in descriptive statistics. How Will I Show the Results? Go over each of the questions and think how you want the answers to go into your report: in a table, a bar graph, a pie chart, a dot plot, or just explained. Hundreds of silly and fun polls that you can take online. The population of children who play video games is growing, and has been boosted in large part by kids between the ages of 2 and 5. Survival Facts. Comprehending how to accumulate the best information will help you complete your assignment and have it in the best quality. Statistics question A manufacturer knows that their items have a normally distributed lifespan, with a mean of 13 years, and standard deviation of 1. I personally use this site. 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We everyday, normal people go about our business without ever really considering probability, despite the fact that if we did pay attention to these numbers, we'd have the closest thing we can get to a crystal ball. Note-taking is a way of remembering what you were taught or what you've read about. Who was the legendary Benedictine monk who invented champagne? Answer: Dom Perignon. 11 Interesting Facts – Architecture & Construction. Interesting Statistics - fun DRAFT. Pharmacist Fun Facts Coca Cola was originally a cough syrup made by a Pharmasist in the 1800s. The following is a look at some fun school facts that can make back to school more interesting: Fact one: People who work in schools do not get paid very well. Definition: Statistical analysis is the use of statistical data including varying variables, entities, and events to determine probabilistic or statistical relationships in quantitative manner. Questions :. Seeking truth, exposing fiction. A data set has no mode when all the numbers appear in the data with the same frequency. Bureau of Labor Statistics analysts then sort the information and group. Remember to leave plenty of space for open-ended questions. Regardless of how an answer can be derived, here is a selection of some of the most funny unanswerable questions that have been shared. See below:. This could make a fun starter/plenary activity … or even inspire students to create their own puzzle. If nine voters are selected at random, what is the probability that. The two-sample t-test is a hypothesis test for answering questions about the mean where the data are collected from two random samples of independent observations, each from an underlying normal distribution: The steps of conducting a two-sample t-test are quite similar to those of the one-sample test. SEO Statistics 2019. 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text:
# Find the value of $\sin25^° \sin35^° \sin85^°$

Trigo problem :

Find the value of $\sin25^° \sin35^° \sin85^°$.

My approach :

Using $2\sin A\sin B = \cos(A-B) -\cos(A+B)$

\begin{align} & \phantom{={}}[\cos10^{°} -\cos60^°] \sin85^° \\ & = \frac{1}{2}[2\cos10^°\sin85^° -2\cos60^° \sin85^°] \\ & = \frac{1}{2}[2\cos10^°\sin85^° -2 \frac{1}{2} \sin85^°] \\ & = \frac{1}{2}[2\cos10^°\sin85^° -\sin85^°] \\ & = \frac{1}{2} [\sin 95^° + \sin 75^° -\sin85^°] \end{align}

After this I am unable to solve further.... please guide... thanks.

Note that $\sin(95^\circ)=\sin(85^\circ)$.
We can get an explicit formula for $\sin(75^\circ)$ in various ways, probably most simply by writing it as $\sin(30^\circ+45^\circ)$ and using the Addition Law.
• oh! got it thanks.. I skipped that cancellation of sin $95^{\circ}$ and $sin85^{\circ}$ .. thanks once againthanks.. – sachin Sep 30 '13 at 3:41

format: plain


text:
# Limit point of a set

A point each neighbourhood of which contains at least one point of the given set different from it. The point and set considered are regarded as belonging to a topological space. A set containing all its limit points is called closed. The set of all limit points of a set $M$ is called the derived set, and is denoted by $M'$. If the topological space $X$ satisfies the first separation axiom (for any two points $x$ and $y$ in it there is a neighbourhood $U(x)$ of $x$ not containing $y$), then every neighbourhood of a limit point of a set $M\subset X$ contains infinitely many points of this set and the derived set $M'$ is closed. Every proximate point of a set $M$ is either a limit point or an isolated point of it.

#### References

 [1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian) [2] F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))


text:
# Suppress 'author' field when 'useauthor=false' in biblatex

I would like to prevent biblatex from printing by (name of author) in the bibliography in only those cases where I have put OPTIONS = "useauthor=false" in the entry in the .bib file. I assume I can put a relatively straightforward command in my .tex preamble that will take care of this. Below a silly example:

\documentclass{article}
\usepackage{csquotes}
\usepackage[
bibstyle = authoryear,
citestyle = authoryear-comp,
sorting = nyt,
language = american,
abbreviate = false,
backend = biber]{biblatex}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@BOOK{bible,
AUTHOR = "God",
EDITOR = "James, King",
TITLE = "The {H}oly {B}ible",
YEAR = "1611",
OPTIONS = "useauthor=false, useeditor=true"}
\end{filecontents}
\begin{document}
\noindent
Text \parencite{bible}.
\printbibliography
\end{document}

-

There are a bunch of ways you might do this; but a simple one (that doesn't involve fiddling with the bibliography drivers at all) is just to clear the author field if useauthor if false:

\AtEveryCitekey{\ifuseauthor{}{\clearname{author}}}
\AtEveryBibitem{\ifuseauthor{}{\clearname{author}}}
-

Redefine the byauthor bibmacro to do nothing.

\documentclass{article}
\usepackage{csquotes}
\usepackage[
bibstyle = authoryear,
citestyle = authoryear-comp,
sorting = nyt,
language = american,
abbreviate = false,
backend = biber]{biblatex}
\renewbibmacro*{byauthor}{}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@BOOK{bible,
AUTHOR = "God",
EDITOR = "James, King",
TITLE = "The {H}oly {B}ible",
YEAR = "1611",
OPTIONS = "useauthor=false, useeditor=true"}
\end{filecontents}
\begin{document}
\noindent
Text \parencite{bible}.
\printbibliography
\end{document}

Compare the original version of byauthor:

\newbibmacro*{byauthor}{%
\ifboolexpr{
test \ifuseauthor
or
test {\ifnameundef{author}}
}
{}
{\usebibmacro{bytypestrg}{author}{author}%


text:
# What are the software limitations in all possible subsets selection in regression?

If I have a dependent variable and $N$ predictor variables and wanted my stats software to examine all the possible models, there would be $2^N$ possible resulting equations.

I am curious to find out what the limitations are with regard to $N$ for major/popular statistic software since as $N$ gets large there is a combinatorial explosion.

I've poked around the various web pages for packages but not been able to find this information. I would suspect a value of 10 - 20 for $N$?

If anyone knows (and has links) I would be grateful for this information.

Aside from R, Minitab, I can think of these packages SAS, SPPS, Stata, Matlab, Excel(?), any other packages I should consider?

• @levon9 - I was able to generate all possible subsets using 50 variables in SAS. I do not believe there is any hard limitation other than memory and CPU speed. -Ralph Winters – Ralph Winters Mar 1 '11 at 14:56
• For what size of dataset? – M. Tibbits Mar 1 '11 at 16:17
• Thanks very useful information. Just curious, did this this take long? – Levon Mar 1 '11 at 21:34
• @levon9 This question generated a lot of sound answers and comments so that I've +1. But, please, forget about Excel for doing serious job in model selection... – chl Mar 1 '11 at 22:21
• @chl .. is that because Excel is slow, or just incapable (ie would give inaccurate results?). – Levon Mar 1 '11 at 23:56

I suspect 30--60 is about the best you'll get. The standard approach is the leaps-and-bounds algorithm which doesn't require fitting every possible model. In $R$, the leaps package is one implementation.

The documentation for the regsubsets function in the leaps package states that it will handle up to 50 variables without complaining. It can be "forced" to do more than 50 by setting the appropriate boolean flag.

You might do a bit better with some parallelization technique, but the number of total models you can consider will (almost undoubtedly) only scale linearly with the number of CPU cores available to you. So, if 50 variables is the upper limit for a single core, and you have 1000 cores at your disposal, you could bump that to about 60 variables.

• leaps is great, I like the plots from it, +1. In real applications some averaging techniques work faster (and better) than pretested estimators even found from all regression models. So I would suggest to go for Bayesian model averaging (BMA package) or the algorithm I like the most - Weighted average least squares (WALS[1]) developed by J.R.Magnus et al. The Matlab code is easily transformable to R code. The good thing for WALS is $N$ computational difficulty instead of $2^N$. [1]: tilburguniversity.edu/research/institutes-and-research-groups/… – Dmitrij Celov Mar 1 '11 at 11:14
• @Dmitrij, thanks for your comments. I've tried to remain fairly agnostic in my response regarding the utility of all-subsets regression. It seems to me that there is nearly always a better solution, but I felt that might seem like too trite of a response to the OP's question. – cardinal Mar 1 '11 at 12:34
• @Dmitrij, BMA over main-effects models would still have the same computational complexity as all-subsets regression. No? The main advantage in BMA seems to me to be in trying to figure out which covariates are likely to be influencing the response. BMA does this by essentially averaging the log likelihoods over $2^{n-1}$ submodels. – cardinal Mar 1 '11 at 12:38
• Thanks for the pointer to the leaps R package! I didn't know about it and it might come in handy in the future. If I could get some information on specific limitations for N for other popular packages it would be very helpful. – Levon Mar 1 '11 at 12:44
• @levon9, I doubt it will vary much at all by package. The algorithm that leaps uses has been state of the art for at least 20 years or so. Even if you found an implementation that was twice as fast, that would mean you got to increment the number of variables you can handle by one. For every doubling of the speed, you get one more variable. Hardware, not algorithmic, limitations are your bottleneck in this case. – cardinal Mar 1 '11 at 13:00

Just a caveat, but feature selection is a risky business, and the more features you have, the more degrees of freedom you have with which to optimise the feature selection criterion, and hence the greater the risk of over-fitting the feature selection criterion and in doing so obtain a model with poor generalisation ability. It is possible that with an efficient algorithm and careful coding you can perform all subsets selection with a large number of features, that doesn't mean that it is a good idea to do it, especially if you have relatively few observations. If you do use all subsets selection, it is vital to properly cross-validate the whole model fitting procedure (so that all-subset selection is performed independently in each fold of the cross-validation). In practice, ridge regression with no feature selection often out-performs linear regression with feature selection (that advice is given in Millar's monograph on feature selection).

• @Dikran, (+1) good comments. I was trying to avoid going there since it didn't directly address the OP's question. But, I agree. All-subsets is rarely the way to go. And, if you do go that way, you need to understand all the implications. – cardinal Mar 1 '11 at 13:19
• @Dirkan thank you for the comments, I am a real stats newbie. I realize the danger of overfitting the model when too many variables are in play, so I am just looking at various automated ways (ie without much benefit of insight) such as the stepwise approach (which might get caught in a local maxima) and the exhaustive all subsets model - and the computational limits it faces (and the external limitations imposed by packages) – Levon Mar 1 '11 at 13:48
• @levon9, you can get over-fitting that is just as serious when you choose the features, so feature selection does not guard against over-fitting. Consider a logistic regression model used to predict the outcome of flipping a fair coin. The potential inputs are the outcome of flipping a large number of other fair coins. Some of these inputs will be positively correlated with the target, so the best all-subsets model will select inputs (even though they are useless) and you will get a model that appears to have skill, but in reality is no better than guessing. – Dikran Marsupial Mar 1 '11 at 14:39
• @Dikran (+1) the same as @cardinal, I first wrote a similar text, but then decided it is not what @levon9 asked, because he simply was curious about the complexity :) – Dmitrij Celov Mar 1 '11 at 16:06
• @Dikran +1 because I like such advice. – chl Mar 1 '11 at 22:22

I was able to generate all possible subsets using 50 variables in SAS. I do not believe there is any hard limitation other than memory and CPU speed.

### Edit

I generated the 2 best models for N=1 to 50 variables for 5000 observations.

@levon9 - No, this ran in under 10 seconds. I generated 50 random variables from (0,1)

-Ralph Winters

• For what size of dataset? – M. Tibbits Mar 1 '11 at 16:17
• Thanks very useful information. Just curious, did this this take long? – Levon Mar 1 '11 at 21:34
• I have undeleted this post (and merged another of your comments in an edit) because the OP found it useful and others might too. Thank you for your contribution; please keep it up! (If you really think it should be deleted, go ahead and do so; I won't contravene your wishes again.) – whuber Mar 2 '11 at 5:53
• It seems you are using two different unregistered accounts. I have merged them but you will still need to register. – chl Mar 2 '11 at 14:08

As $N$ gets big, your ability to use maths becomes absolutely crucial. "inefficient" mathematics will cost you at the PC. The upper limit depends on what equation you are solving. Avoiding matrix inverse or determinant calculations is a big advantage.

One way to help with increasing the limit is to use theorems for decomposing a large matrix inverse from into smaller matrix inverses. This can often means the difference between feasible and not feasible. But this involves some hard work, and often quite complicated mathematical manipulations! But it is usually worth the time. Do the maths or do the time!

Bayesian methods might be able to give an alternative way to get your result - might be quicker, which means your "upper limit" will increase (if only because it gives you two alternative ways of calculating the same answer - the smaller of two, will always be smaller than one of them!).

If you can calculate a regression coefficient without inverting a matrix, then you will probably save a lot of time. This may be particularly useful in the Bayesian case, because "inside" a normal marginalisation integral, the $X^{T}X$ matrix does not need to be inverted, you just calculate a sum of squares. Further, the determinant matrix will form part of the normalising constant. So "in theory" you could use sampling techniques to numerically evaluate the integral (even though it has an analytic expression) which will be eons faster than trying to evaluate the "combinatorical explosion" of matrix inverses and determinants. (it will still be a "combinatorical explosion" of numerical integrations, but this may be quicker).

This suggestion above is a bit of a "thought bubble" of mine. I want to actually test it out, see if it's any good. I think it would be (5,000 simulations + calculate exp(sum of squares) + calculate weighted average beta should be faster than matrix inversion for a big enough matrix.)

The cost is approximate rather than exact estimates. There is nothing to stop you from using the same set of pseudo random numbers to numerically evaluate the integral, which will again, save you a great deal of time.

There is also nothing stopping you from using a combination of either technique. Use exact when the matrices are small, use simulation when they are big. This is because in this part of the analysis. It is just different numerical techniques - just pick the technique which is quickest!

Of course this is all just a bit of "hand wavy" arguments, I don't exactly know the best software packages to use - and worse, trying to figure out which algorithms they actually use.

• @probabilityislogic, while your response is interesting, maybe it could be refocused to better address the OP's question. Also, no software for computing least-squares solutions performs matrix inversion, much less a determinant. Ever. Unless it's inverting a $1 \times 1$ matrix. – cardinal Mar 1 '11 at 12:28
• @probabilityislogic, handling the $2^n$ cases efficiently quickly far outstrips the $O(n^3)$ issues of an efficient least-squares solution. That's where the leaps-and-bounds algorithm comes in. – cardinal Mar 1 '11 at 12:30
• Thanks for the post. "Do the maths or do the time!" :-) .. I'm actually not even trying to figure out the underlying algorithms used by the packages (thought that is interesting to know), at this point really looking for specific information regarding the limitations of N by the major packages. – Levon Mar 1 '11 at 12:47
• @cardinal The updating and downdating algorithms also exist for various matrix decomposition procedures, I suspect that is what was meant by "matrix inverse" etc. – Dikran Marsupial Mar 1 '11 at 13:04
• @Dikran, several efficient and numerically stable approaches to least-squares exist, including methods for augmenting or reducing a design matrix by one column at a time. Sometimes it's good to understand what is happening under the surface, even if on most days you don't need to care. – cardinal Mar 1 '11 at 13:29


text:
Dataset Open Access

# Supplementary Data: Impact of vacuum stability, perturbativity and XENON1T on global fits of Z2 and Z3 scalar singlet dark matter (arXiv:1806.11281)

The GAMBIT Collaboration

### Citation Style Language JSON Export

{
"publisher": "Zenodo",
"DOI": "10.5281/zenodo.1298566",
"author": [
{
"family": "The GAMBIT Collaboration"
}
],
"issued": {
"date-parts": [
[
2018,
6,
29
]
]
},
"abstract": "

\n\n
Supplementary Data
\n\n

\n\n
Impact of vacuum stability, perturbativity and XENON1T on global fits of Z2 and Z3 scalar singlet dark matter arXiv:1806.11281
\n\n
The files in this record contain data for the scalar singlet dark matter models considered in the GAMBIT "Scalar singlet Mark II" paper.
\n\n
The files consist of
\n\n
\n\t
30 regular YAML files
\n\t
StandardModel_SLHA2_scan.yaml, a universal YAML fragment included from the other YAML files
\n\t
14 hdf5 files. 8 of these correspond to the complete set of combined samples for each fit. These 8 fits are generated from all binary permutations of three run properties: Z2 or Z3 model, with or without absolute vacuum stability demanded, and with constraints from the 2017 or 2018 XENON1T data. These 8 hdf5 files are used to generate the profile likelihood plots in the paper. The other 6 hdf5 files are the results of T-Walk runs, and are used to generate the posterior pdfs in the paper.
\n\t
Some example pip files for producing plots from the hdf5 files using pippi
\n\t
A tarball best_fits_yaml.tar.gz containing YAML files of the best-fit point in each of the 8 fits.
\n
\n\n
The files follow the naming scheme SingletDM_[model]_[slice]_[vacuum]_[xenon]_[prior]_[scanner].yaml.
\n\n
\n\t
model: Z2 or Z3
\n\t
slice: full, lowmass, neck or absent (for hdf5 files)
\n\t
vacuum: ms (metastable) or vs (absolute vacuum stability)
\n\t
prior: logmu3, flatmu3 or absent (for Z2 scans)
\n\t
scanner: TWalk or absent (implies Diver scans in the case of YAML files, and indicates merged samples potentially from both Diver and T-Walk in the case of hdf5 files)
\n
\n\n
A few caveats to keep in mind:
\n\n
\n\t
\n\t
The YAML files are designed to work with GAMBIT 1.2.0, commit e4d3f739, and the pip files are tested with pippi 2.1, commit c094b8c8. They may or may not work with later versions of either software (but you can of course always obtain the version that they do work with via the git history).
\n\t
\n\t
\n\t
The pip files are examples only. Users wishing to reproduce the more advanced plots in any of the GAMBIT papers should contact us for tips or scripts, or experiment for themselves. Many of these scripts are in multiple parts and require undocumented manual interventions and steps in order to implement various plot-specific customisations, so please don't expect the same level of polish as for files provided here or in the GAMBIT repo.
\n\t
\n
\n\n

",
"title": "Supplementary Data: Impact of vacuum stability, perturbativity and XENON1T on global fits of Z2 and Z3 scalar singlet dark matter (arXiv:1806.11281)",
"type": "dataset",
"id": "1298566"
}
198
569
views


text:
# Rewrite the following expressions without using radicals or negative exponents. Simplify when possible. root(5)(x^20y^-4)

Question
Rewrite the following expressions without using radicals or negative exponents. Simplify when possible. $$\displaystyle{\sqrt[{{5}}]{{{x}^{{20}}{y}^{{-{{4}}}}}}}$$

2021-01-18
The given radical can be rewritten as,
$$\displaystyle{\sqrt[{{5}}]{{{x}^{{20}}{y}^{{-{{4}}}}}}}={\left({x}^{{20}}{y}^{{-{{4}}}}\right)}^{{\frac{{1}}{{5}}}}{\left\langle.{\sqrt[{{n}}]{{{a}}}}={\left({a}\right)}^{{\frac{{1}}{{n}}}}\right.}$$
The radical can be further simplified as,
$$\displaystyle{\sqrt[{{5}}]{{{x}^{{20}}{y}^{{-{{4}}}}}}}={\left({x}^{{20}}{y}^{{-{{4}}}}\right)}^{{\frac{{1}}{{5}}}}={\left({x}^{{20}}\right)}^{{\frac{{1}}{{5}}}}\cdot{\left({y}^{{-{{4}}}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{4}}{y}^{{-\frac{{4}}{{5}}}}={\left({x}{y}^{{-\frac{{1}}{{5}}}}\right)}^{{4}}$$

### Relevant Questions

Simplify each of the following expressions. Be sure that your answer has no negative or fractional exponents. $$a*(1/81)^(-1/4)b*x^(-2)y^(-4)c*(2x)^(-2)(16x^2y)^(1/2)$$
Simplify the following. Write answers without negative exponnents. $$\displaystyle{\left({4}{x}^{{2}}{y}^{{-{{3}}}}\right)}^{{-{{2}}}}$$
Write each radical expression using exponents and each exponential expression using radicals. Radical expression Exponential expression $$\sqrt[3]{y^{4}}$$
Rewrite the following without the absolute value symbols.
|x-11|<6
x<5
x>17 or x<5
x<17 or x> 5
x>17
x>17 and x<5
Radical and Exponents Simplify the expression $$\frac{8r^(1/2)s^-3}{2r^-2 x^4}$$
Simplify: $$\displaystyle{\left({7}^{{5}}\right)}{\left({4}^{{5}}\right)}$$. Write your answer using an exponent.
Explain in words how to simplify: $$\displaystyle{\left({153}^{{2}}\right)}^{{7}}.$$
Is the statement $$\displaystyle{\left({10}^{{5}}\right)}{\left({4}^{{5}}\right)}={14}^{{5}}$$ true?
Radical and Exponents Simplify the expression $$\frac{(ab^2 c^-3}{2a^3 b^-4)}^{-2}$$
$$\displaystyle{\frac{{\sqrt{{{4}}}{\left\lbrace{x}^{{{7}}}\right\rbrace}}}{{\sqrt{{{4}}}{\left\lbrace{x}^{{{3}}}\right\rbrace}}}}$$
$$\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}$$
Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. NKS $$\displaystyle\sqrt{{{x}^{{{5}}}}}{x}{5}$$


text:
# Assignment 5 - Cloth Simulation

Due Wednesday, 11/4 Monday, 11/9 at 23:59:59. You must work individually.

## Goals

In this assignment, we’re going to simulate cloth with linearly implicit Euler, penalty collision forces, and sparse matrices.

## Starting Point

Your main task is to complete the function Cloth::step(). I have written a lot of code for you that deal with setting up the scene and rendering the cloth. Study the code carefully to see what is happening.

• main()

• The simulation thread is separate from the rendering thread. The thread is defined and started on line 282, right before the main thread enters the render loop. The space bar turns on/off the simulation, and the h key runs one simulation step.
• Scene

• This class is responsible for holding the things in the scene. Currently it holds a pointer to a cloth, and an empty vector of pointers to spheres.

• To change the number of rows and columns of the cloth, modify Scene::load(). You should start with a 2 by 2 cloth.

• Particle

• The particle class is used to represent both cloth nodes and spheres.

• It has position x and velocity v.

• The radius r is used for collisions. A cloth particle has a small radius that is noticeably larger than the thickness of the cloth.

• Cloth

• Cloth::Cloth()

• The particles and the springs are created. The particles are created from a grid with corners x00, x01, x10, and x11. There are ‘x’ springs, ‘y’ springs, two sets of ‘shear’ (or diagonal) springs, and two sets of ‘bending’ springs.

• Each particle knows its starting index into the global vectors and matrices (lines 64 and 67). For example, if the starting index of a particle is 3, then the particle’s indices are {3,4,5}. If a particle is fixed to the world, it gets the index of -1.

• The global matrices and vectors are allocated (line 121).

• The vertex buffers for rendering the cloth are allocated (line 127).

• Cloth::updatePosNor()

• Whenever the simulation step changes the particle positions, we have to resend the position and normal data to the GPU. This function fills the position buffer from the particles and recomputes the normals using the cross product.
• Cloth::draw()

• Sends the updated position and normal buffers to the GPU and then draws the cloth mesh using triangle strips.

At each time step, you must fill in $$M, K, v$$, and $$f$$ with appropriate values, and then solve the linear system

$A x = b,$

where

\begin{aligned} A &= M - h^2 K\\ b &= M v + h f. \end{aligned}

The solution, $$x$$, of the linear system contains the new velocities of the particles.

## Setting up Eigen

For solving a linear system, we are using a C++ library called Eigen. You’ll have to install this in the same way you installed glm in lab 0. You’ll need to create a new environment variable called EIGEN3_INCLUDE_DIR that points to the Eigen directory. Once you have it installed, you may want to look at this quick tutorial.

## Explicit Integration

We’ll start with symplectic Euler, which boils down to the same thing as above but with $$A=M$$, which means you don’t need to form $$K$$ yet.

1. In Cloth::Step(), loop through the particles to fill in $$M$$ and $$v$$, and fill in $$f$$ with gravity. (Note the const Vector3d &grav argument of Cloth::step().) Use the block() and segment() functions in Eigen. Remember to not fill in these values for fixed particles. (I.e., only add these values for free particles.) The indices of the particles are stored inside the particle class Particle::i. This is the starting index of the particle into $$M,$$ $$v,$$ and $$f$$. For example, if the index of a particle is 6, then you should store its velocity into the $$v$$ vector at indices 6, 7, and 8.

2. In the same function, loop through the springs to add spring forces to $$f$$. Remember to add, not set. Each spring object contains pointers to the two particles attached to the spring. The formula for the spring force is

\begin{aligned} & f_s = E (l-L) \frac{\Delta x}{l},\\ & f = \begin{pmatrix} f_s \\ -f_s \end{pmatrix}, \end{aligned}

where $$E$$ is the stiffness constant, $$\Delta x = x_1 − x_0$$ is the vector between the two particle positions, $$l = \| \Delta x \|$$ is the current spring length, and $$L$$ is the spring’s rest length. (Each spring is between particle 0 and particle 1.) This force, $$f_s$$, is the force for particle p0 of the spring. The force for the other particle p1 is the negation, $$-f_s$$. Again, remember to add this force only to free particles.

1. Solve the linear system $$A x = b$$. There are many linear solvers provided by Eigen. The LDLT solver is a good choice for now: A.ldlt().solve(b). The resulting solution $$x$$ contains the new velocities of the particles.

2. Loop through the particles to extract the velocities from the solution vector.

3. Loop through the particles one last time to integrate the positions using the new velocities.

With symplectic Euler, you should be able to simulate a $$2 \times 2$$ system with h=5e-3 and stiffness=1e1. You should see a cloth that looks like a flat sheet that does not bend. Try changing these values - you’ll see that the simulation quickly becomes unstable. One of the best ways to make it stable is to implement implicit integration, which is our next task.

## Implicit Integration

Now implement implicit Euler by filling in and using the stiffness matrix, $$K$$. The formula for the $$2 \times 2$$ block stiffness matrix for a spring is

\begin{aligned} & K_s = \frac{E}{l^2} \left( \left(1 - \frac{l-L}{l} \right) \left( \Delta x \Delta x^T \right) + \frac{l-L}{l} \left( \Delta x^T \Delta x \right) I \right),\\ & K = \begin{pmatrix} -K_s & K_s \\ K_s & -K_s \end{pmatrix}. \end{aligned}

Note that $$\Delta x^T \Delta x$$ is a scalar, and $$\Delta x \Delta x^T$$ is a $$3 \times 3$$ matrix. $$I$$ is the $$3 \times 3$$ identity matrix.

This $$2 \times 2$$ block matrix, $$K$$, should be added to the global stiffness matrix into the appropriate rows/cols using the indices of the two particles. Again, remember to not add these values for fixed particles. The negated, off-diagonal values should be added only if both particles of the spring are free.

With the following settings in Scene::Scene():

h = 5e-3;
grav << 0.0, -9.8, 0.0;
int rows = 2;
int cols = 2;
double mass = 0.1;          // total mass of the cloth
double stiffness = 1e1;     // E

the $$A$$ matrix and the $$b$$ vector should be the following for the very first step.

cout << A << endl;
0.0254         0    0.0001   -0.0003         0         0
0    0.0250         0         0         0         0
0.0001         0    0.0254         0         0         0
-0.0003         0         0    0.0254         0   -0.0001
0         0         0         0    0.0250         0
0         0         0   -0.0001         0    0.0254

cout << b << endl;
0
-0.0012
0
0
-0.0012
0

A common bug is that the sign of the stiffness matrix $$K_s$$ is flipped. Try negating it!

Try increasing the cloth resolution to $$10 \times 10$$ or higher. You may have to play around with the time step and stiffness values to make the cloth stable.

You should try running your code in release mode (See lab 0) so that you can simulate more particles.

## Sphere Collision

If you uncomment line 91 in Scene::draw(), you’ll see a collision sphere moving, but it won’t be affecting the cloth yet. We need to write some collision code first.

Implement collisions in Cloth::step(), right before the linear solve. You must check all the cloth particles against all the collision spheres (which are also particles) in the scene, which in this case is just a single sphere. For each particle, if there is a collision, add a collision force to the RHS and the corresponding implicit term to the LHS. Unlike cloth spring forces, these collision forces act on a single particle.

Let $$x$$ be the position of the particle, and $$x_s$$ be the position of the sphere. Then the vector from the sphere to the particle is $$\Delta x = x - x_s.$$ Let $$l = \| \Delta x \|$$ be the length of $$\Delta x.$$ If we subtract $$l$$ from the sum of the radii of the sphere and the particle, we obtain the penetration depth: $$d = r + r_s - l.$$ If this distance, $$d,$$ is positive, then we have detected a collision. The collision force should be computed as:

$f_c = c d \frac{\Delta x}{l},$

where $$c$$ is the collision stiffness constant. The stiffness matrix entry can be approximated simply as:

$K_c = c d I,$

where $$I$$ is the $$3 \times 3$$ identity matrix.

## Sparse Matrices (bonus)

If you’re planning to use cloth simulation for your project, I highly recommend converting your code to use sparse matrices instead of dense matrices. On my computer, for a $$10 \times 10$$ cloth, sparse is twice as fast as dense. For a bigger system, the difference will be even more significant. See http://eigen.tuxfamily.org/dox-devel/group__TutorialSparse.html.

You have to modify your matrix filling code and matrix solving code. To fill a sparse matrix, you must first create an array of triplets of the form row, col, val. As an example, the code to create a sparse $$2 \times 2$$ identity matrix is

typedef Eigen::Triplet<double> T;
std::vector<T> A_;
A_.push_back(T(0, 0, 1));
A_.push_back(T(1, 1, 1));
Eigen::SparseMatrix<double> A(2,2);
A.setFromTriplets(A_.begin(), A_.end());

To solve using a sparse matrix, we must use a specialized sparse solver. There are many options, but for animation purposes, running Conjugate Gradients for several iterations is often sufficient.

ConjugateGradient< SparseMatrix<double> > cg;
cg.setMaxIterations(25);
cg.setTolerance(1e-6);
cg.compute(A);
VectorXd x = cg.solveWithGuess(b, x0);

For Conjugate Gradients, it is important to give the initial guess, x0. In our case, x represents the new particle velocities, so you should use the previous particle velocities as the initial guess.

Constructing a sparse matrix is expensive. Therefore, it is not efficient to create the mass and stiffness matrices separately and add them to form the final matrix $$A$$. Instead, it is better to create $$A = M - h^2 K$$ and fill it as you loop through the particles and the springs. You should also create the right-hand-side vector $$b = M v + h f$$ as you loop through the particles and springs.

With sparse matrices implemented, try increasing the resolution to $$30 \times 30$$.

## Point breakdown

• [30] Explicit integration
• [40] Implicit integration
• [20] Sphere collision
• [10] Coding style and general execution
• [+15] Sparse matrices

Total: 100 points + 15 bonus points

## What to hand in

If you’re using Mac/Linux, make sure that your code compiles and runs by typing:

> mkdir build
> cd build
> cmake ..
> make
> ./A5 <SHADER DIR>

If you’re using Windows, make sure your code builds using the steps described in Lab 0.

• Put your name on the program window.
• Include an ASCII README that includes
• Do not hand in the executable, old save files (*.~), or object files (*.o). You should hand in the minimum set of files you need to compile plus the README file.
• Create a single zip file of all the required files. The filename of this zip file should be USERNAME.zip (e.g., sueda.zip). The zip file should extract everything into a folder named USERNAME/ (e.g. sueda/).
• When you unzip this file, it should extract src/, resources/, CMakeLists.txt, and your README file to the USERNAME/ directory.
• Use the standard .zip format (not .gz, .7z, .rar, etc.).