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# interval calculator calculus

The empty interval 0 and the interval containing all the reals, (â, -â), are actually both open and closed. Type a math problem. To find the intervals of concavity, you need to find the second derivative of the function, determine the $x$ values that make the function equal to $0$ (numerator) and undefined (denominator), and plug in values to the left and to the right of these $x$ values, and look at the sign of the results: $+ \ \rightarrow$ interval is concave up In addition, in the practice, many statisticians and marketers can turn a non-interval ordered values scale into an interval scale to support statistical or data analysis. Note: use your eyes and common sense when using this! This calculator evaluates derivatives using analytical differentiation. The function is increasing on this interval since the derivative is positive on this interval. This can be done by summing the entire set of numbers and then dividing by the total numbers in the sample set. Sometimes, the interval may contain only one of its endpoints. Integral Approximation Calculator. In pre-calculus you deal with inequalities and you use interval notation to express the solution set to an inequality. Use this tool to find the approximate area from a curve to the x axis. How do you find the average velocity of the position function #s(t)=3t^2-6t# on the interval from #t=2# to #t=5#? Trigonometry. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Solve. We've integrated the flow to have the volume. Trigonometry (from Greek trigÅnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. What is the absolute minimum value of. We are here to assist you with your math questions. Free integral calculator that quickly calculates the integral of a given function, including definite or indefinite integrals. Then,the double integral of f (x,y) over the region R can be defined as: ÊÊR Æ (x, y) dxdy=limn Í¢ 0 Î£ (n, i=1) Æ (xi, yi) Î´xi Î´yi. More about the confidence intervals. Interval Notation and Set Builder Notation Calculator: This calculator determines the interval notation and set builder notation for a given numerical statement. The calculator could not be displayed because JavaScript is disabled. However, the answer that appears is incorrect. I have tried inputting the derivative into the calculator and calculating for the value f'(c). As students progress in their educational lives and as they reach more and more profound levels of in the different scientific fields and in mathematics, they find themselves, in one way or another, in need of a calculator. calc.zip: 3k: 04-02-17: Area Under A Curve More about the confidence interval for the population variance. First, differentiate . To get started a sketch may help you get a sense of what the curve looks like. Interval â¦ The maximum on the interval could be a local maximum somewhere within the interval, or it could be a value at one end or the other. Some curves don't work well, for example tan(x), 1/x near 0, â¦ The following formulas show how to format solution sets in interval notation. Then, find the values for x for which the derivative is negative by solving . An interval is closed if the interval contains its endpoints. * AP® is a trademark registered and owned by the College Board, which was not involved in the production of, and does not endorse, this site.® is a trademark registered and owned by the College Board, which was not involved in the production of, and does not endorse, this site. Functions can either be increasing or decreasing for different intervals. The closed intervalâwhich includes the endpointsâ would be [0, 100]. Enter how many in the sample, the mean and standard deviation, choose a confidence level, and the calculation is done live. AP Calculus Formulas This program includes a variety of formulas that are intended for those taking the AP Calculus BC exam. You will need to get assistance from your school if you are having problems entering the answers into your online assignment. We have a whole lot of quality reference materials on matters ranging from mathematics content to power g, left parenthesis, x, right parenthesis, equals, 3, x, cubed, plus, 8. . g ( x) = 3 x 3 + 8. g (x)=3x^3+8 g(x) = 3x3+8. If we divide R region into smaller sub-regions and Î´Ai=Î´xi Î´yi be the area of its sub-region. Integral Calculator â¦ Calculus Calculator. Pre Calculus Calculator: The Future Begins Here. Closed and opened intervals complement each other, but they arenât mutually exclusive. Related Concepts. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others. I am using a TI-84 plus ce. Test them by substituting values for x: Substitute -2,. Read Confidence Intervals to learn more. But still in development stage. You may speak with a member of our customer support team by calling 1-800-876-1799. Get the free "Max/Min Finder" widget for your website, blog, Wordpress, Blogger, or iGoogle. First, you need to calculate the mean of your sample set. Therefore, the value of our integral is 9 - 8/3 = 19/3 â 6.33. It will also find local minimum and maximum, of the given function.The calculator will try to simplify result as much as possible. is decreasing on intervals where . In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. Standard Deviation and Mean. Consider f (x,y) as a function in a 3D space in xy-plane and R be any region in xy-plane. It â¦ Substitute 0,. Find more Mathematics widgets in Wolfram|Alpha. calculus.zip: 1k: 02-02-19: Calculus Toolkit v1.00 Does some pre-cal and calculus for ya. I am also tried graphing both f'(x) and f'(c) and finding the intersection point, but that did not work either. Math Help Algebra I & II Geometry Trigonometry Calculus Statistics In the event that you seek support with math and in particular with free interval notation solver or syllabus for elementary algebra come visit us at Mathscitutor.com. Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. How to calculate a confidence interval? There are examples of valid and invalid expressions at the bottom of the page. You can calculate all three and compare them. A confidence interval is an statistical concept that refers to an interval that has the property that we are confident at a certain specified confidence level that the population parameter, in this case, the population standard deviation, is â¦ Integral calculus offers a precise method of calculating the region below the curve of a mathematical function. algebra trigonometry statistics calculus matrices variables list. In this case, the interval is half-open. Time needed: 10 minutes. Read Integral Approximations to learn more.. Matrix Calculator. There are few things to keep in mind so you can better interpret the results obtained by this calculator: A confidence interval is an interval (corresponding to the kind of interval estimators) that has the property that is very likely that the population parameter is contained by it (and this likelihood is measure by the confidence level). Part of Pre-Calculus For Dummies Cheat Sheet . X³ / 3 at the point X = 3 takes 9, and at the point X = 2 we have 8/3. For regions of different shapes, the â¦ 3. If the population standard deviation cannot be used, then the sample standard deviation, s, can be used when the sample size is greater than 30. Use the Standard Deviation Calculator to calculate your sample's standard deviation and mean. After you calculate a confidence interval, make sure you always interpret it in words a non-statistician would understand. The â¦ Your 95% confidence interval for the mean length of walleye fingerlings in this fish hatchery pond is (The lower end of the interval is 7.5 â 0.45 = 7.05 inches; the upper end is 7.5 + 0.45 = 7.95 inches.) Connecting position, velocity, and acceleration functions using integrals Average acceleration over interval AP.CALC: Confidence Interval Calculator. Next, test the intervals. Half-Closed and Half-Open This calculator computes volumes for a few of the most usual basic shapes. g. g g. g. over the closed interval. So, interval scales are great (we can add and subtract to them) but we cannot multiply or divide. Phone support is available Monday-Friday, 9:00AM-10:00PM ET. As the ball traces the curve from left to right, identify intervals using "interval notation" as either increasing or decreasing 1 f x = x x â 2 x + 4 x â 4 x + 4 You cannot calculate Ratios. Calculating a confidence interval involves determining the sample mean, XÌ, and the population standard deviation, Ï, if possible. The interval [2, 8) is half-open. Interval Partition Video To do this, we need to compute the antiderivative of the integrable function and take the difference of its values for the ends of the interval. The symbols [ or ] are used to indicate that an endpoint is included in the interval. By Yang Kuang, Elleyne Kase . What are the steps that I â¦ Interactive, free online graphing calculator from GeoGebra: graph functions, plot data, drag sliders, and much more! Interval Partition Calculator: -- Enter Partitioned Interval . For ya 8/3 = 19/3 â 6.33 them by substituting values for x: Substitute,. Interval contains its endpoints common sense when using this both open and closed is live! Function.The calculator will try to simplify result as much as possible â -â... 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# If $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$

Show that if $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$

Hint: Use Hölder's inequality.

But I can't see where I should use it.

I'm trying to use it in $\displaystyle\int |f|^p\,d\mu = \int |f|^{p-q}|f|^{q}$, but it didn't help.

• What is $\mu (f>0)$? – Owen Aug 21 '15 at 5:21
• Are you sure that it's $\lim\limits_{ p \to 0^+}$? $\| \cdot \|_p$ is only a norm for $p \geq 1$. I know that often the same definition is used even for when $p < 1$, but since Holder's inequality holds only for $p > 1$, it seems a bit odd. – Marcus M Aug 21 '15 at 5:22
• @Owen $\mu(f > 0) = \mu\left(\left\{ \omega \in \Omega : f(\omega) > 0 \right\} \right)$. – Marcus M Aug 21 '15 at 5:33
• @MarcusM It's just like I said. It seems odd to me to. – Andre Gomes Aug 21 '15 at 5:37
• $\lim\limits_{p\to 0^+}||f||_p=0$ or $\lim\limits_{p\to 0^+}||f||_p=1$ ? – hermes Aug 21 '15 at 6:01

You're leaving out a hypothesis - as stated the problem is false, it can happen that $||f||_p=\infty$ for every $p>0$.

If you assume that there exists $p>0$ with $||f||_p<\infty$ then yes, the limit is $0$ (not $1$ as has been claimed), and yes this follows from Holder's inequality, even though we're talking about $p<1$ and Holder only applies to $p\ge1$. The $p$ in $||f||_p$ is not the $p$ we're going to use in Holder.

Also any proof that doesn't use the fact that $\mu(f>0)<1$ is wrong; without that assumption the limit could be more or less anything (in general the limit is $\exp(\int\log f)$).

I'm not going to do the whole problem, just an illustration of how the proof is going to work. Say $E$ is the set where $f>0$. Assume that $||f||_1<\infty$.

Note that $f^{1/2} = \Bbb 1_E f^{1/2}$. The Cauchy-Schwarz inequality shows that $$\int f^{1/2}=\int\Bbb 1_Ef^{1/2}\le||\Bbb1_E||_2||f^{1/2}||_2 =\mu(E)^{1/2}\left(\int f\right)^{1/2}.$$Square both sides and you get $$||f||_{1/2}\le\mu(E)||f||_1.$$

Note how we proved something about $||f||_{1/2}$ using Cauchy-Schwarz, which only talks about $p=2$. You can prove similar but more general inequalities using Holder, which will show that $||f||_p\to0$. (The fact that $\mu(E)<1$ is going to be important...)

• Assuming that $||f||_1<\infty$ isn't equivalent to assuming that $f$ is integrable? – Andre Gomes Aug 21 '15 at 17:05
• Yes of course those two are equivalent. What did I say that you think says this is not so? – David C. Ullrich Aug 21 '15 at 17:09
• Then I did't get this part "If you assume that there exists $p>0$ with $||f||_p<\infty$ then yes, the limit is 0 (not 1 as has been claimed)" – Andre Gomes Aug 21 '15 at 17:24
• I don't understand what you don't get about that... – David C. Ullrich Aug 21 '15 at 17:28
• Yes, it suffices to assume that $f$ is integrable! II didn't say otherwise. As you pointed out, assuming that $f$ is integrable is the same as assuming that $||f||_1<\infty$. And if $||f||_1<\infty$ then there exists a $p>0$ such that $||f||_p<\infty$. Assuming that $f$ is integrable is a stronger assumption. – David C. Ullrich Aug 21 '15 at 17:38

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# Using LuaLaTeX and SQLite3

I am trying to access SQLite databases from LuaLaTeX but am not able to load an installed lsqlite3 package. Here's what I tried so far:

1. Took a brandnew Xubuntu 14.04 installation
2. Installed a complete TeX Live 2014
3. Installed Luarocks (which installed Lua5.1 on the way)
4. Used Luarocks to install lsqlite3

uwe@luabuntu:/media/uwe$luarocks list Installed rocks: lsqlite3 0.9.1-2 (installed) - /usr/local/lib/luarocks/rocks 5. Appended the package.path in the following TeX file: \documentclass{article} \usepackage{luacode} \begin{document} \begin{luacode*} package.path="/usr/local/lib/luarocks/rocks;/usr/local/lib/luarocks/rocks/lsqlite3; /usr/local/lib/lua/5.1/lsqlite3.so" .. package.path require("lsqlite3") \end{luacode*} Hello Lua! \end{document}  The error I receive is the following: ! LuaTeX error [\directlua]:3: module 'lsqlite3' not found: no field package.preload['lsqlite3'] [luatexbase.loader] Search failed [kpse lua searcher] file not found: 'lsqlite3' [kpse C searcher] file not found: 'lsqlite3' [oberdiek.luatex.kpse_module_loader]-eroux Search failed stack traceback: [C]: in function 'require' [\directlua]:3: in main chunk. \luacode@dbg@exec ...code@maybe@printdbg {#1} #1 } l.9 \end{luacode*}  What must be set to have LuaLaTeX find the lsqlite3 package? EDIT I have tried to adjust everything following the comments below: 1. Adjusted /usr/local/texlive/2014/texmf.cnf 2. Put everything into the example 3. Still get the error about the missing lsqlite3 module with the following example \documentclass{article} \directlua{ require "lualoader" } \usepackage{fontspec} \usepackage[english]{babel} \begin{document} \directlua{package.path='/usr/local/share/lua/5.1/?.lua;/usr/local/share/lua/5.1/?/init.lua;/home/uwe/.luarocks/share/lua/5.1/?.lua;/home/uwe/.luarocks/share/lua/5.1/?/init.lua;/usr/share/lua/5.1//?.lua;/usr/share/lua/5.1//?/init.lua;./?.lua;/usr/local/share/lua/5.1/?.lua;/usr/local/share/lua/5.1/?/init.lua;/usr/local/lib/lua/5.1/?.lua;/usr/local/lib/lua/5.1/?/init.lua;/usr/share/lua/5.1/?.lua;/usr/share/lua/5.1/?/init.lua' package.cpath='/usr/local/lib/lua/5.1/?.so;/home/uwe/.luarocks/lib/lua/5.1/?.so;./?.so;/usr/local/lib/lua/5.1/?.so;/usr/lib/x86_64-linux-gnu/lua/5.1/?.so;/usr/lib/lua/5.1/?.so;/usr/local/lib/lua/5.1/loadall.so' require "lualoader" } \directlua{% local sqlite3 = require("lsqlite3") local db = sqlite3.open_memory() db:exec[[ CREATE TABLE test (id INTEGER PRIMARY KEY, content); INSERT INTO test VALUES (NULL, 'Hello World'); INSERT INTO test VALUES (NULL, 'Hello Lua'); INSERT INTO test VALUES (NULL, 'Hello Sqlite3') ]] for row in db:nrows("SELECT * FROM test") do tex.print(row.id .. " : ".. row.content ) end } \end{document}  • lsqlite3 is a binary module, at least in part. You probably have to set package.cpath as well. Dec 21, 2014 at 14:35 • I have the strong feeling that package.path is ignored in kpathsea systems. - see tex.stackexchange.com/a/31904/243 Dec 21, 2014 at 17:11 • Dec 21, 2014 at 17:13 • @UweZiegenhagen you call lualoader twice in your edit, paths must be set before it is required Dec 25, 2014 at 18:44 • The interpreted parts can be configured by means of the environment variable LUAINPUTS (example: export LUAINPUTS="~/code/tex/pgfplots/tex//:~/code/tex/pgf/generic//:"). There is also an environment variable CLUAINPUTS ... perhaps you need to adopt it? I know that DLL search paths tend to need native load procedures, in other words: environment variables (don't know how that works in LUA, though) Dec 25, 2014 at 19:55 ## 4 Answers It is possible to modify the package loading mechanism in Lua. In the case of LuaTeX, the kpse library is used to load modules instead of the default mechanism that uses package.path and package.cpath. So setting these variables will not have any effect. But nothing prevents us from using both methods using a Lua module, lualoader.lua. Save the snippet below as a file in the directory containing your document: -- lualoader.lua -- this is copied from luatexbase.loader local make_loader = function(path, pos, loadfunc) local default_loader = package.searchers[pos] local loader = function(name) local file, _ = package.searchpath(name,path) if not file then local msg = "\n\t[lualoader] Search failed" local ret = default_loader(name) if type(ret) == "string" then return msg ..ret elseif type(ret) == "nil" then return msg else return ret end end local loader,err = loadfunc(file, name) if not loader then return "\n\t[lualoader] Loading error:\n\t"..err end return loader end package.searchers[pos] = loader end local binary_loader = function(file, name) local symbol = name:gsub("%.","_") return package.loadlib(file, "luaopen_"..symbol) end make_loader(package.path,2,loadfile) make_loader(package.cpath,3, binary_loader)  We use the function make_loader to insert a function searching package.path or package.cpath at a specified index in the package.searchers table. 2 for lua files, 3 for binary modules. We also need to use different functions to load a module depending on its type. We use loadfile for Lua files and slightly more complicated function for binary modules that uses package.loadlib. As an example, we can try loading lsqlite3 from LuaTeX: \documentclass{article} \directlua{ require "lualoader" } \usepackage{fontspec} \usepackage[english]{babel} \begin{document} \directlua{% local sqlite3 = require("lsqlite3") local db = sqlite3.open_memory() db:exec[[ CREATE TABLE test (id INTEGER PRIMARY KEY, content); INSERT INTO test VALUES (NULL, 'Hello World'); INSERT INTO test VALUES (NULL, 'Hello Lua'); INSERT INTO test VALUES (NULL, 'Hello Sqlite3') ]] for row in db:nrows("SELECT * FROM test") do tex.print(row.id .. " : ".. row.content ) end } \end{document}  It is important to run LuaLaTeX with the --shell-escape flag, because binary libraries are blocked by default. With --shell-escape you can execute external programs and call binary libraries. The example yields: Edit: As Uwe pointed out, luarocks as installed by some Linux package managers may not work correctly. I experienced this on Fedora, so I installed luarocks from source. Another solution may be to set package.path and package.cpath to the directories listed by the command luarocks path  On my own system, this yields: $ luarocks path
export LUA_PATH='/home/michal/.luarocks/share/lua/5.2/?.lua;/home/michal/.luarocks/share/lua/5.2/?/init.lua;/usr/local/share/lua/5.2/?.lua;/usr/local/share/lua/5.2/?/init.lua;/usr/share/lua/5.2/?.lua;/usr/share/lua/5.2/?/init.lua;/usr/lib/lua/5.2/?.lua;/usr/lib/lua/5.2/?/init.lua;./?.lua'

So a modified preamble would be:

\directlua{
package.path = '/home/michal/.luarocks/share/lua/5.2/?.lua;/home/michal/.luarocks/share/lua/5.2/?/init.lua;/usr/local/share/lua/5.2/?.lua;/usr/local/share/lua/5.2/?/init.lua;/usr/share/lua/5.2/?.lua;/usr/share/lua/5.2/?/init.lua;/usr/lib/lua/5.2/?.lua;/usr/lib/lua/5.2/?/init.lua;./?.lua'
}

• @UweZiegenhagen luarocks.loader should be available if you have correctly installed luarocks. in this particular case it is not needed, because lsqlite3 is installed to directory with other Lua libraries, but it is needed for Lua modules installed through luarocks Dec 23, 2014 at 18:06
• I installed luarocks via the Ubuntu package system. If I remove the line for the luarocks.loader I get the the following error again: ! LuaTeX error [\directlua]:1: module 'lsqlite3' not found: no field package.preload['lsqlite3'] Dec 23, 2014 at 18:35
• @UweZiegenhagen ah, I installed luarocks from the sources, I had problems when it was installed by Fedora. You may need to set package.path and package.cpath in this case Dec 23, 2014 at 18:50
• @UweZiegenhagen you're welcome :) Dec 25, 2014 at 20:05

As of May 2017, there is a package on CTAN called luapackageloader, that allows you to modify the default package searching behavior in LuaTeX.

Once you load luapackageloader, doing a require will first try the default kpse based searcher. If the module isn’t found, it will also try to load the package from Lua’s package.path and package.cpath values.

Usage example

\documentclass{article}
\usepackage{luacode}

\begin{document}

\begin{luacode*}
package.path="/usr/local/lib/luarocks/rocks;/usr/local/lib/luarocks/rocks/lsqlite3;
/usr/local/lib/lua/5.1/lsqlite3.so" .. package.path

require("lsqlite3")
\end{luacode*}

Hello Lua!

\end{document}

Additional note: you can do eval $(luarocks path) on your shell to set the package path using environment variables which the Lua interpreter will pick up automatically. That way, you do not need to fiddle with the package.path yourself. • Note: you need to pass --shell-escape to the executable (for example lualatex). Nov 1, 2021 at 16:50 Based on comments on this question, I have the following line in /usr/local/texlive/2014/texmf.cnf: CLUAINPUTS = .;$SELFAUTOLOC/lib/{$progname,$engine,}/lua//;/usr/lib/{$progname,$engine,}/lua//

But my luarocks list gives:

Installed rocks:

lsqlite3
0.9.1-2 (installed) - /usr/lib/luarocks/rocks-5.2

so you may need to adjust the second path according to your installation.

## Summary

### Searching for the package

• In normal Lua, a package is searched in package.path and package.cpath.

• In LuaTeX, a package is by-default searched with kpse.

• There are 2 ways to modify where LuaTeX searches for packages

1. modify behavior of kpse

Modify texmf.cnf file, variables LUAINPUTS and CLUAINPUTS.

2. make Lua load from package.(c)path, then modify those

• With \usepackage{luapackageloader}, packages are searched in both kpse and package.(c)path.

• The value of package.(c)path can be set manually by the user within the Lua code itself, but it's better to set environment variables e.g. LUA_PATH, LUA_CPATH.

If you run eval \$(luarocks path) in bash, all processes spawned from that bash shell will inherit the environment variables.

### Binary package

• If the package requires binary files, --shell-escape must be passed.

### Versioning

• The package version must be compatible with Lua version. Do print(_VERSION) within Lua to check.

To specify particular version to luarocks, do e.g. luarocks --lua-version 5.3 path.


text:
# Fortran Wiki merge_bits

## Description

merge_bits(i, j, mask) merges the bits of i and j as determined by the mask. The $k$-th bit of the result is equal to the $k$-th bit of i if the $k$-th bit of mask is 1; it is equal to the $k$-th bit of j otherwise.

## Standard

Fortran 2008 and later

## Class

Elemental function


text:
# Question

Consider 1.00 mole of CO2(g) at 300. K and 5.00 atm. The gas expands until the final pressure is 1.00 atm. For each of the following conditions ΔEscribing the expansion, calculate ΔS, ΔSsurr, and ΔSuniv. Cp for CO2 is 37.1 J K-1mol-1, and assume that the gas behaves ideally.
a. The expansion occurs isothermally and reversibly.
b. The expansion occurs isothermally against a constant external pressure of 1.00 atm.
c. The expansion occurs adiabatically and reversibly.

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Views25
• CreatedMay 05, 2015
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text:
# Some fixed point results for weakly $b-(\varphi,G)$ contraction in $b-$metric spacs endowed with a graph

Document Type : Research Articles

Author

Department of Mathematics, College of Technical and Engineering, Saveh Branch, Islamic Azad University, Saveh, Iran.

Abstract

In this paper, we introduce the concept of weakly $b-(\varphi,G)$ contraction mapping in $b$-metric spaces endowed with a graph and give some fixed point results for such contractions.

Keywords


text:
1. Apr 21, 2010

yaseen shah

in newtons third law of motion every action has equal and opposite reaction so my question is that when block hits another block the blocks exert equal force on another so why they move they had to stop immediately.

2. Apr 21, 2010

sophiecentaur

On impact, each block will deform slightly. During the actual time taken for the impact (a finite time- not immediate) the blocks will be compressed / twisted / bent and there will be forces acting against each block (the Newton 3 pair).


text:
# Electronic Components

638

## Resistors

Resistors are two-terminal devices that restrict, or resist, the flow of current. The larger the resistor, the less current can flow through it for a given voltage as demonstrated by Ohm’s law. Electrons flowing through a resistor collide with material in the resistor body, and it is these collisions that cause electrical resistance. These collisions cause energy to be dissipated in the form of heat or light (as in a toaster or light bulb). Resistance is measured in Ohms, and an ohm is defined by the amount of resistance that causes 1A of current to flow from a 1V source. Resistors can be purchased in the range of less than 1 Ω to several million ohms (or megaohms).

For most circuits, a one-ohm resistance is a relatively small value, and a 100 KΩ resistance is a relatively large value. The physical size and appearance of a resistor is determined by the required application. Resistors that must dissipate large amounts of energy (such as in a toaster) are relatively large, whereas resistors that dissipate small amounts of current (such as those used on Real Digital boards for various purposes) are relatively small. The amount of power (in Watts) dissipated in a resistor can be calculated using the equation $P=I^2R$ (or $P=VI$), where $I$ is the current flowing through the resistor, $V$ is the voltage across the resistor, and $R$ is the resistance. A resistor that can dissipate about 5 Watts of power would be about the size of a writing pen, and a resistor that can only dissipate 1/8 Watt is about the size of a grain of rice. If a resistor is placed in a circuit where it must dissipate more that its intended power, it will simply melt.

Several different resistors are used on Real Digital® boards. Some are used to limit LED current, and some are used on inputs (like the button and switch circuits) to both limit the currents flowing to the main chip, and to help protect against electrostatic discharge (or ESD—more on this topic later). The resistors on the Real Digital boards, like most resistors used in digital systems, are physically small because they will not encounter large voltages or currents. For these smaller resistors, the resistor value in Ohms is printed in microscopic numbers on the resistor body, not visible to the naked eye.

Some older or lower cost circuit boards use “through hole” components instead of the smaller (and cheaper) “surface mount” components. Through-hole resistors are about 5mm long by 2mm wide, and they are typically brown or blue, with colored stripes on their bodies (the colored stripes encode the resistance value). In circuit schematics and in parts lists, resistors are typically denoted with reference designators that begin with an “R”. You can see several rectangular white boxes with “R__” on the Real Digital board silk-screen. In circuit schematics, a resistor is shown as a jagged line.

## Capacitors

A capacitor is a two-terminal device that can store electric energy in the form of charged particles. You can think of a capacitor as a reservoir of charge that takes time to fill or empty. The voltage across a capacitor is proportional to the amount of charge it is storing—the more charge added to a capacitor of a given size, the larger the voltage across the capacitor. It is not possible to instantaneously move charge to or from a capacitor, so it is not possible to instantaneously change the voltage across a capacitor. It is this property that makes capacitors useful on Real Digital boards and in many other applications.

Capacitance is measured in Farads. A one Farad capacitor can store one Coulomb of charge at one volt. For engineering on a small scale (i.e., hand-held or desk-top devices), a one Farad capacitor stores far too much charge to be of general use (it would be like a car having a 1000 gallon gas tank). More useful capacitors are measured in micro-farads (uF) or pico-farads (pF). The terms “milli-farad” and “nano-farad” are rarely used. Large capacitors often have their value printed plainly on them, such as “10 uF” (for 10 microfarads). Smaller capacitors, appearing as small blocks, disks or wafers, often have their values printed on them in an encoded manner (similar to the resistor packs discussed above). For these capacitors, a three-digit number typically indicates the capacitor value in pico-farads. The first two digits provide the “base” number, and the third digit provides an exponent of 10 (so, for example, “104” printed on a capacitor indicates a capacitance value of $10\*10^4$ or 100000 pF). Occasionally, a capacitor will only show a two-digit number, in which case that number is simply the capacitor value in pF. (To be complete, if a capacitor shows a three-digit number and the third digit is 8 or 9, then the first two digits are multiplied by .01 and .1 respectively). Often, a single letter is appended to the capacitance value—this letter indicates the quality of the capacitor. Capacitors are used on Real Digital boards to keep the voltage supplies and some signals stable regardless of circuit activity, and to store charge when inputs are activated in order to slow their assertion times. The majority of the capacitors on the Real Digital boards are used to “decouple” integrated circuits from the power supply. These “bypass” capacitors are placed on the board very close to the Vdd pins of all chips, where they can supply the short-term electrical current needs of the chips. Without such bypass capacitors, individual chips could cause the Vdd supply across the entire Real Digital board to dip below 5V during times of heavy current demand. Nearly every chip in every digital system uses bypass capacitors. Bypass capacitor value can be determined if the worst-case current requirements are known, but more typically, capacitors in the range 0.001uF to 0.047uF are used without regard to the actual current requirements. Real Digital boards also use bulk bypass capacitors near the power supplies and around the board to provide extra charge storage for the entire circuit board. These larger capacitors (typically 10uF to100uF) can supply the smaller individual bypass capacitors during times of exceptional need.

Depending on the size of the capacitor, the PCB silk screen will show either a circle or rectangle to indicate capacitor placement (usually, smallish capacitors are shown as rectangles, and larger capacitors as circles). Some capacitors are polarized, meaning they must be placed into the circuit board in a particular orientation (so that one terminal is never at a lower voltage than the other).

Polarized capacitors either have a dark stripe near the pin that must be kept at a higher voltage, or a “-” near the pin that must be kept at a lower voltage. Silk-screen patterns for polarized capacitors will also often have a “+” sign nearest the through-hole that must be kept at a relatively higher voltage. In circuit schematics and parts lists, capacitors typically use a “C__” reference designator. Some older or lower cost circuit boards use “through hole” components instead of the smaller (and cheaper) “surface mount” components. Through-hole resistors are about 5mm long by 2mm wide, and they are typically brown or blue, with colored stripes on their bodies (the colored stripes encode the resistance value). In circuit schematics and in parts lists, resistors are typically denoted with reference designators that begin with an “R”. You can see several rectangular white boxes with “R__” on the Real Digital board silk-screen. In circuit schematics, a resistor is shown as a jagged line.

## Important Ideas

• Resistors restrict the flow of current. The larger the resistor the less current can flow through it for a given voltage. This is shown in Ohm’s law.
• Capacitors store electric energy in the form of charged particles. Capacitors allow for more stable signals regardless of fluctuation in voltage to the circuit.


text:
##### Browse Questions

Create printable tests and worksheets from Grade 9 Physics questions. Select questions to add to a test using the checkbox above each question. Remember to click the add selected questions to a test button before moving to another page.

Show Physics questions in All Grades.
1 2 3 4 ... 19
 Grade 9 :: Simple Machines by jlovering
 When using a broom to sweep a floor, your mechanical advantage will be less than 1 and your fulcrum will be your hand at the top of the broom. This means the broom acts as a: Class 1 Lever Class 2 Lever Class 3 Lever Class 4 Lever
 Grade 9 :: Energy and Work by der1228
 Which form of energy does a plant store when light is transformed during photosynthesis? chemical energy thermal energy mechanical energy electrical energy
 Grade 9 :: Simple Machines by jlovering
 Austin needs to lift a 300 N object using only 100 N of force. To accomplish this, he will need a simple machine with a mechanical advantage of what? 3 0.3 90% None of the Above
 Grade 9 :: Waves by jlovering
 EM radiation with frequencies greater than visible light would include which of the following? X-Rays Microwaves Infrared All of the above
 Grade 9 :: Waves by LBeth

This question is a part of a group with common instructions. View group »

 Which electromagnetic waves are the longest? radio violet gamma ultraviolet
 Grade 9 :: Waves by LBeth

This question is a part of a group with common instructions. View group »

 Wave energy relates directly with frequency. Which of the following types of radiation has the greatest amount of energy? radio gamma ultraviolet microwave
 Grade 9 :: Waves by LBeth

This question is a part of a group with common instructions. View group »

 1 micron ($mum$ is 0.000001 meters. The wavelength of which type of radiation is best measured in microns? x-ray radio infrared microwave
 Grade 9 :: Waves by LBeth

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 What type of wave could measure 5 meters in length? x-ray radio visible infrared
 Grade 9 :: Waves by LBeth

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 In the visible spectrum the color                 has a lower frequency than the color                . orange; green blue; yellow yellow; red violet; blue
 Grade 9 :: Energy and Work by der1228
 Energy stored in the bonds between atom is called kinetic energy mechanical energy chemical energy thermal energy
 Grade 9 :: Simple Machines by jlovering
 Which of the following is true about a single, fixed pulley? It has a mechanical advantage less than 1 It has a mechanical advantage equal to 1 It has a mechanical advantage greater than 1 It has no friction.
 Grade 9 :: Energy and Work by jlovering
 Jassidy is out with her friends. Misfortune occurs and Jassidy and her friends find themselves getting a workout. They apply a force of 1080 N to push the car 218 m to the nearest fuel station. Determine the work done on the car. 4.95 J 235,000 J 495 N 235,000 N
 Grade 9 :: Fluid Mechanics by LBeth
 Ship captains must account for Bernoulli's principle when passing nearby in order to avoid collisions. True False
 Grade 9 :: Fluid Mechanics by LBeth
 The faster a fluid flows, the greater the pressure it exerts. True False
 Grade 9 :: Waves by jlovering
 Grade 9 :: Fluid Mechanics by LBeth
 As part of a physics demonstration, Mr. Taylor suspends two ping pong balls next to each other so that they have a small space between them. He then turns on a hairdryer and has it blow air in the space between the two ping pong balls. Which of the following should happen? the two balls do not move the two balls move upward the two balls move further apart the two balls move closer together
 Grade 9 :: Fluid Mechanics by LBeth
 A large truck passes a car on the highway. According to Bernoulli's principle, which of the following should occur? pressure decreases between the vehicles so the car and truck move slightly apart pressure decreases between the vehicles so the car and truck move slightly together pressure increases between the vehicles so the car and truck move slightly apart pressure increases between the vehicles so the car and truck move slightly together
 Grade 9 :: Fluid Mechanics by LBeth
 Bernoulli's principle applies to                         . gases only liquids only both gases and liquids neither gases or liquids
 Grade 9 :: Simple Machines by jlovering
 Mr. D needs to create a lever with a mechanical advantage of 0.4. If he is going to apply his force 2 meters from the fulcrum, how far away must his load be from the fulcrum to get the correct mechanical advantage? 5 meters 0.2 meters 1 meter None of the Above
 Grade 9 :: Waves by LBeth

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 Visible light with a wavelength of 480 nm appears red blue green violet
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text:
## Acting in Delayed Environments with Non-Stationary Markov Policies

### Esther Derman · Gal Dalal · Shie Mannor

Keywords: [ reinforcement learning ] [ delay ]

Abstract: The standard Markov Decision Process (MDP) formulation hinges on the assumption that an action is executed immediately after it was chosen. However, assuming it is often unrealistic and can lead to catastrophic failures in applications such as robotic manipulation, cloud computing, and finance. We introduce a framework for learning and planning in MDPs where the decision-maker commits actions that are executed with a delay of $m$ steps. The brute-force state augmentation baseline where the state is concatenated to the last $m$ committed actions suffers from an exponential complexity in $m$, as we show for policy iteration. We then prove that with execution delay, deterministic Markov policies in the original state-space are sufficient for attaining maximal reward, but need to be non-stationary. As for stationary Markov policies, we show they are sub-optimal in general. Consequently, we devise a non-stationary Q-learning style model-based algorithm that solves delayed execution tasks without resorting to state-augmentation. Experiments on tabular, physical, and Atari domains reveal that it converges quickly to high performance even for substantial delays, while standard approaches that either ignore the delay or rely on state-augmentation struggle or fail due to divergence. The code is available at \url{https://github.com/galdl/rl_delay_basic.git}.


text:
+0

# Counting

0
66
2

Plz help with counting

Miyu is giving out 9 identical chocolates to her 7 friends, including Dhruv.  All possible distributions are equally likely.  What is the probability that Dhruv gets exactly 2 chocolates?

Feb 6, 2023

#1
+118471
+1

ABCDEFG  that is 7 friends

Cut into 9 peices that is 8 bars.

7+8=15

15C8=6435   ways to distribute.

If D gets 2 peices than you can take him and the 2 choc he is getting out

then you are left with

ABCEFG   6 friends    divided into 7 parts that is 6 bars

6+6=12

12C6 = 924 possible ways

so that is    924/6435 =         28 / 195

I used the Stars and bars method to solve it.

I could have make a logic error. You need to decide for yourelf whether to accept this answer.

Feb 6, 2023
#2
+2602
+1

There are 9 stars (chocolate) and 7 - 1 = 6 bars.

This makes for $${15 \choose 6} = 5005$$ cases.

If Dhruve gets exactly 2, there are now 7 stars, and 6 - 1 = 5 bars, which makes for $${12 \choose 5} = 792$$ cases.

So, the probability is $${792 \over 5005} = \color{brown}\boxed{72 \over 455}$$

Feb 6, 2023


text:
# Doppler effect definition

• noun:
• The obvious improvement in regularity or wavelength of a wave that is sensed by an observer moving relative to the origin of the waves.
• improvement in the evident frequency of a wave as observer and origin move toward or away from each other

## Related Sources

• Definition for "Doppler effect"
• The obvious improvement in regularity or wavelength of…
• Synonym for "Doppler effect"
• Doppler shift
• Hypernym for "Doppler effect"
• propagation
• Cross Reference for "Doppler effect"
• redshifted
• Astronomy Dictionary for "Doppler effect"
• the apparent change in wavelength of…


text:
1. Angles

Two complimentary angles A and B have measures in thr ratio of 7 to 23 respectively. What is the ratio of the measure of the complement of angle A to the measure of the compliment of angleB? express your answer as a decimal to the nearest hundereth.

2. Originally Posted by Rimas
Two complimentary angles A and B have measures in thr ratio of 7 to 23 respectively. What is the ratio of the measure of the complement of angle A to the measure of the compliment of angleB? express your answer as a decimal to the nearest hundereth.
So they add up to 90 degree.
Thus,
$7x+23x=90$
Thus,
$30x=90$
Thus,
$x=3$
This denotes the konstant multipler
Thus,
$7(3)=21$
$23(3)=69$


text:
We obtain, \begin{align*} ∫^5_010+\cos \left(\frac{π}{2}t\right)\,dt &= \left(10t+\frac{2}{π} \sin \left(\frac{π}{2}t\right)\right)∣^5_0 \\[4pt] &=\left(50+\frac{2}{π}\right)−\left(0−\frac{2}{π} \sin 0\right )≈50.6. So, for convenience, we chose the antiderivative with $$C=0$$. Since $$−\sqrt{3}$$ is outside the interval, take only the positive value. Unable to add item to List. They have been part of the standard approach since practically the beginning and that likely won't be changing any time soon. There are many complex ideas that you will need to get a handle on, but don't panic. We state this theorem mathematically with the help of the formula for the average value of a function that we presented at the end of the preceding section. \nonumber, \begin{align*} c^2 &=3 \\[4pt] c &= ±\sqrt{3}. These new techniques rely on the relationship between differentiation and integration. wolframalpha.com. in cases where those brands are manufactured by the retailer to which we are linking. Please try again. Letting $$u(x)=\sqrt{x}$$, we have $$\displaystyle F(x)=∫^{u(x)}_1 \sin t \,dt$$. Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. A member of The Institute for Operations Research and the Management Sciences and the Mathematical Association of America, Dr. Morris teaches courses ranging from college algebra to calculus and statistics. Kathy still wins, but by a much larger margin: James skates 24 ft in 3 sec, but Kathy skates 29.3634 ft in 3 sec. Magoosh Calculus Students who need extra help with calculus should consider looking into Magoosh, an educational company that helps you strengthen your skills with video lessons from expert teachers. The company also has an app for multivariable calculus. Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. Have questions or comments? The reason is that, according to the Fundamental Theorem of Calculus, Part 2 (Equation \ref{FTC2}), any antiderivative works. The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The derivative is the rate at which a function changes. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. If you have trouble relating to calculus, or you want to keep students engaged, this is the book you need. Use the procedures from Example $$\PageIndex{2}$$ to solve the problem. Everyday low prices and free delivery on eligible orders. \end{align*}. He has been a Sloan Fellow and a Fairchild Scholar and is a Fellow of the American Academy of Arts and Sciences.

Example $$\PageIndex{2}$$: Finding the Point Where a Function Takes on Its Average Value. These calculus textbooks will help guide you through one of the most difficult subjects you're likely to encounter, and they're packed with quizzes, illustrations, and definitions designed for all levels, so even the biggest math-phobes can breathe easy while mastering the material. we may earn commissions to Some jumpers wear “wingsuits” (Figure $$\PageIndex{6}$$). Example $$\PageIndex{5}$$: Using the Fundamental Theorem of Calculus with Two Variable Limits of Integration. Combined, these features make this one ideal for reading before class, getting disinterested students engaged, and inspiring the future problem-solvers of the world. In other words, you may know that an apple will fall from its branch, but calculus can help you determine how fast it will fall and where it will land. ROBERT M. STARK, PhD, is Professor Emeritus in the Departments of Mathematical Sciences and Civil and Environmental Engineering at the University of Delaware. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. It can be used as a textbook or a reference book for an introductory course on one variable calculus. \label{FTC2}\]. The Mean Value Theorem for Integrals, Part 1, If $$f(x)$$ is continuous over an interval $$[a,b]$$, then there is at least one point $$c∈[a,b]$$ such that, $∫^b_af(x)\,dx=f(c)(b−a). This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in Mathematics, Statistics, Engineering, Pharmacy, etc. We replaced it with a book that is decidedly more beginner-friendly and a classic in the math world: Calculus Made Easy. This open-source book by Crowell, Robbin, and Angenent is a spin-off of a previous open-source book by Robbin and Angenent. Since Julie will be moving (falling) in a downward direction, we assume the downward direction is positive to simplify our calculations. If Julie dons a wingsuit before her third jump of the day, and she pulls her ripcord at an altitude of 3000 ft, how long does she get to spend gliding around in the air, If $$f(x)$$ is continuous over an interval $$[a,b]$$, then there is at least one point $$c∈[a,b]$$ such that \[f(c)=\frac{1}{b−a}∫^b_af(x)\,dx.\nonumber$, If $$f(x)$$ is continuous over an interval $$[a,b]$$, and the function $$F(x)$$ is defined by $F(x)=∫^x_af(t)\,dt,\nonumber$, If $$f$$ is continuous over the interval $$[a,b]$$ and $$F(x)$$ is any antiderivative of $$f(x)$$, then $∫^b_af(x)\,dx=F(b)−F(a).\nonumber$. Practical examples from a variety of subject areas are featured throughout each chapter and step-by-step explanations for the solutions are presented.

There is a reason it is called the Fundamental Theorem of Calculus. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Let $$\displaystyle F(x)=∫^{\sqrt{x}}_1 \sin t \,dt.$$ Find $$F′(x)$$. Math is the language of nature, and Calculus: An Intuitive and Physical Approach (appx. A few of our choices touch on precalculus, although if you're completely lost on the subject, it's best to brush up with a dedicated precalculus textbook before diving into calculus. Find $$F′(x)$$. Our selections were prioritized by clarity, authority, layout, and pedagogical tools that help teach, reinforce, and review the material for a thorough understanding. It's used in fields like physics and engineering to determine how a system will behave in the real world. Try again. Over the years, many great minds have worked to expand and improve the original principles of calculus. Math is tough enough as it is, so you don't need bad instruction making it that much harder. The relationships he discovered, codified as Newton’s laws and the law of universal gravitation, are still taught as foundational material in physics today, and his calculus has spawned entire fields of mathematics. \nonumber\]. Legal. For James, we want to calculate, \begin {align*} ∫^5_0(5+2t)\,dt &= \left(5t+t^2\right)∣^5_0 \\[4pt] &=(25+25) \\[4pt] &=50. \end{align*}, Use Note to evaluate $$\displaystyle ∫^2_1x^{−4}\,dx.$$, Example $$\PageIndex{8}$$: A Roller-Skating Race. Since they can do everything that a normal calculator does and much more, graphing calculators are useful in a variety of places outside of the calculus classroom. Watch the recordings here on Youtube! Both of them took radically different approaches and arrived at surprisingly similar conclusions. Julie is an avid skydiver with more than 300 jumps under her belt and has mastered the art of making adjustments to her body position in the air to control how fast she falls.

.

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text:
# The Carnegie Hubble Program: The Distance And Structure Of The SMC As Revealed By Mid-Infrared Observations Of Cepheids

Victoria Scowcroft, Wendy L. Freedman, Barry F. Madore, Andy Monson, S. E. Persson, Jeff Rich, Mark Seibert, Jane R. Rigby

Research output: Contribution to journalArticle

37 Citations (Scopus)

### Abstract

Using Spitzer observations of classical Cepheids we have measured the true average distance modulus of the Small Magellanic Cloud (SMC) to be $18.96\pm {0.01}_{\mathrm{stat}}\pm {0.03}_{\mathrm{sys}}$ mag (corresponding to $62\pm 0.3$ kpc), which is 0.48 ± 0.01 mag more distant than the LMC. This is in agreement with previous results from Cepheid observations, as well as with measurements from other indicators such as RR Lyrae stars and the tip of the red giant branch. Utilizing the properties of the mid-infrared Leavitt Law we measured precise distances to individual Cepheids in the SMC, and have confirmed that the galaxy is tilted and elongated such that its eastern side is up to 20 kpc closer than its western side. This is in agreement with the results from red clump stars and dynamical simulations of the Magellanic Clouds and Stream.
Original language English 49 1-10 10 Astrophysical Journal 816 2 5 Jan 2016 https://doi.org/10.3847/0004-637X/816/2/49 Published - 10 Jan 2016

### Cite this

The Carnegie Hubble Program: The Distance And Structure Of The SMC As Revealed By Mid-Infrared Observations Of Cepheids. / Scowcroft, Victoria; Freedman, Wendy L.; Madore, Barry F.; Monson, Andy; Persson, S. E.; Rich, Jeff; Seibert, Mark; Rigby, Jane R.

In: Astrophysical Journal, Vol. 816, No. 2, 49, 10.01.2016, p. 1-10.

Research output: Contribution to journalArticle

Scowcroft, Victoria ; Freedman, Wendy L. ; Madore, Barry F. ; Monson, Andy ; Persson, S. E. ; Rich, Jeff ; Seibert, Mark ; Rigby, Jane R. / The Carnegie Hubble Program: The Distance And Structure Of The SMC As Revealed By Mid-Infrared Observations Of Cepheids. In: Astrophysical Journal. 2016 ; Vol. 816, No. 2. pp. 1-10.
@article{96145c90dea848bbbff7a43843b808ee,
title = "The Carnegie Hubble Program: The Distance And Structure Of The SMC As Revealed By Mid-Infrared Observations Of Cepheids",
abstract = "Using Spitzer observations of classical Cepheids we have measured the true average distance modulus of the Small Magellanic Cloud (SMC) to be $18.96\pm {0.01}_{\mathrm{stat}}\pm {0.03}_{\mathrm{sys}}$ mag (corresponding to $62\pm 0.3$ kpc), which is 0.48 ± 0.01 mag more distant than the LMC. This is in agreement with previous results from Cepheid observations, as well as with measurements from other indicators such as RR Lyrae stars and the tip of the red giant branch. Utilizing the properties of the mid-infrared Leavitt Law we measured precise distances to individual Cepheids in the SMC, and have confirmed that the galaxy is tilted and elongated such that its eastern side is up to 20 kpc closer than its western side. This is in agreement with the results from red clump stars and dynamical simulations of the Magellanic Clouds and Stream.",
author = "Victoria Scowcroft and Freedman, {Wendy L.} and Madore, {Barry F.} and Andy Monson and Persson, {S. E.} and Jeff Rich and Mark Seibert and Rigby, {Jane R.}",
year = "2016",
month = "1",
day = "10",
doi = "10.3847/0004-637X/816/2/49",
language = "English",
volume = "816",
pages = "1--10",
journal = "Astrophysical Journal",
issn = "0004-637X",
publisher = "IOP Publishing",
number = "2",

}

TY - JOUR

T1 - The Carnegie Hubble Program: The Distance And Structure Of The SMC As Revealed By Mid-Infrared Observations Of Cepheids

AU - Scowcroft, Victoria

AU - Freedman, Wendy L.

AU - Monson, Andy

AU - Rich, Jeff

AU - Seibert, Mark

AU - Rigby, Jane R.

PY - 2016/1/10

Y1 - 2016/1/10

N2 - Using Spitzer observations of classical Cepheids we have measured the true average distance modulus of the Small Magellanic Cloud (SMC) to be $18.96\pm {0.01}_{\mathrm{stat}}\pm {0.03}_{\mathrm{sys}}$ mag (corresponding to $62\pm 0.3$ kpc), which is 0.48 ± 0.01 mag more distant than the LMC. This is in agreement with previous results from Cepheid observations, as well as with measurements from other indicators such as RR Lyrae stars and the tip of the red giant branch. Utilizing the properties of the mid-infrared Leavitt Law we measured precise distances to individual Cepheids in the SMC, and have confirmed that the galaxy is tilted and elongated such that its eastern side is up to 20 kpc closer than its western side. This is in agreement with the results from red clump stars and dynamical simulations of the Magellanic Clouds and Stream.

AB - Using Spitzer observations of classical Cepheids we have measured the true average distance modulus of the Small Magellanic Cloud (SMC) to be $18.96\pm {0.01}_{\mathrm{stat}}\pm {0.03}_{\mathrm{sys}}$ mag (corresponding to $62\pm 0.3$ kpc), which is 0.48 ± 0.01 mag more distant than the LMC. This is in agreement with previous results from Cepheid observations, as well as with measurements from other indicators such as RR Lyrae stars and the tip of the red giant branch. Utilizing the properties of the mid-infrared Leavitt Law we measured precise distances to individual Cepheids in the SMC, and have confirmed that the galaxy is tilted and elongated such that its eastern side is up to 20 kpc closer than its western side. This is in agreement with the results from red clump stars and dynamical simulations of the Magellanic Clouds and Stream.

U2 - 10.3847/0004-637X/816/2/49

DO - 10.3847/0004-637X/816/2/49

M3 - Article

VL - 816

SP - 1

EP - 10

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 2

M1 - 49

ER -


text:
#### XTQA: Span-Level Explanations of the Textbook Question Answering

##### Jie Ma, Jun Liu, Yaxian Wang, Qinghua Zheng, Qingyu Yin, Jianlong Zhou, Yi Huang

Textbook Question Answering (TQA) is a task that one should answer a diagram/non-diagram question given a large multi-modal context consisting of abundant essays and diagrams. We argue that the explainability of this task should place students as a key aspect to be considered. To address this issue, we devise a novel architecture towards span-level eXplanations of the TQA (XTQA) based on our proposed coarse-to-fine grained algorithm, which can provide not only the answers but also the span-level evidences to choose them for students. This algorithm first coarsely chooses top $M$ paragraphs relevant to questions using the TF-IDF method, and then chooses top $K$ evidence spans finely from all candidate spans within these paragraphs by computing the information gain of each span to questions. Experimental results shows that XTQA significantly improves the state-of-the-art performance compared with baselines. The source code is available at https://github.com/keep-smile-001/opentqa

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text:
### If a rhombus is re-shaped such that one of its diagonal increases by 4%, while other diagonal decreases by 4%. Find the percentage change in the area of rhombus.

0.16% decrease

Step by Step Explanation:

1. Let's assume the length of the diagonals BD and AC of the rhombus ABCD are p and q respectively.
2. The area of the rhombus =
 pq 2

3. According to the question, one of its diagonal increases by 4%, while other diagonal decreases by 4%.
The new length of the diagonal BD = p + p ×
 4 100
= p + 0.04p = (1 + 0.04)p
4. The new length of the diagonal AC = q - q ×
 4 100
= q - 0.04q = (1 - 0.04)q
5. Now, the area of the rhombus =
 (1 + 0.04)p × (1 - 0.04)q 2

=
 (12 - 0.042)pq 2
...[Since, (a + b)(a - b) = a2 - b2]
=
 pq - 0.0016pq 2

6. Change in area = New area of the rhombus - The area of the rhombus
=
 pq - 0.0016pq 2
-
 pq 2

=
 pq - 0.0016pq - pq 2

=
 -0.0016pq 2

7. % Change in area =
 Change in area The area of the rhombus
× 100
=

 -0.0016pq 2

 pq 2

× 100
= -0.16%
8. Thus, the area of the rhombus is decreased by 0.16%.

You can reuse this answer


text:
# Command line utility to convert image formats of photos

Below is some code for a basic command line utility to convert image formats of photos. I'm wondering how any part of it might be written with better style, i.e. how it could be neater, clearer and generally "better".

import os
import argparse
from PIL import Image

def get_arguments():
''' Get command line arguments '''
parser = argparse.ArgumentParser(
description='A command line tool to convert images to different formats')
'input',
help='full path to the input file or folder')
'outext',
choices=Image.registered_extensions().keys(),
help='output extension for file')
'-n', '--new-name',
help='give the output file a new name')
'-d', '--destination',
help='set new destination directory')
'-R', '--remove-input',
action='store_true',
help='also remove the input file')
args = parser.parse_args()
if not (os.path.isfile(args.input) or os.path.isdir(args.input)):
# easier to check file validiy here and keep infile as str,
# rather than use type checking in parser.add_argument()
raise parser.error('Unrecognised input type - input must be a file or folder')
else:
return args

def convert_image(infile, outext, outname=None, outdir=None, remove_infile=False):
''' Convert an image to a new format and optionally:
* Change the filename
* Put the file in a new directory
* Remove the input file
'''
indir, inname = os.path.split(infile)
inname = os.path.splitext(inname)[0]
if outname is None:
outname = inname
if outdir is None:
outdir = indir
outfile = os.path.join(outdir, '{}{}'.format(outname, outext))
if not os.path.isdir(outdir) and outdir != '':
os.makedirs(outdir)
Image.open(infile).save(outfile)
if remove_infile:
os.remove(infile)

def convert_images(im_dir, outext, outname=None, outdir=None, remove_infile=False):
''' run convert_image() for all files in a folder '''
for item in os.listdir(im_dir):
if os.path.splitext(item)[1] in Image.registered_extensions().keys():
convert_image(
os.path.join(im_dir, item),
outext,
outname=outname,
outdir=outdir,
remove_infile=remove_infile)

def main():
args = get_arguments()
if os.path.isdir(args.input):
convert_images(
args.input,
args.outext,
outname=args.new_name,
outdir=args.destination,
remove_infile=args.remove_input)
elif os.path.splitext(args.input)[1] in Image.registered_extensions().keys():
convert_image(
args.input,
args.outext,
outname=args.new_name,
outdir=args.destination,
remove_infile=args.remove_input)
else:
raise ValueError('Unsupported file format')

if __name__ == '__main__':
main()

Specific suggestions:

1. I generally find that ordering functions by their importance is helpful. Moving get_arguments just above or below main would allow readers to get into the functionality immediately. Others prefer to write methods in call order, but this feels a bit arbitrary (should main be first or last?) and not always achievable, because with branching it's possible to have a cycle of callers.
2. *nix tools usually support any number of paths as their last arguments, operating on each of them. This would certainly be useful here, and means you could even avoid the whole if/else in main because all inputs should be file paths.
3. argparse has a file type which you can use.
4. Don't rely on file extensions to correspond to file formats. I would instead rely on PIL to detect the file type and to throw an exception if it doesn't know it.
5. Abbreviations like inname make the code harder to read. One way to avoid the urge to shorten everything is to use an IDE, because it'll help you auto-complete everything which is in scope.
6. In the end this script is a small convenience wrapper around PIL. This is one of those rare occasions when I would suggest replacing it with a small shell script.

General suggestions:

1. black can automatically format your code to be more idiomatic.
2. isort can group and sort your imports automatically.
3. flake8 with a strict complexity limit will give you more hints to write idiomatic Python:

[flake8]
max-complexity = 4
ignore = W503,E203

4. I would then recommend adding type hints everywhere and validating them using a strict mypy configuration:

[mypy]
check_untyped_defs = true
disallow_untyped_defs = true
ignore_missing_imports = true
no_implicit_optional = true
warn_redundant_casts = true
warn_return_any = true
warn_unused_ignores = true

• Re point 6, I understand that this could be written as a bash (or similar) script, but is is there a tangible advantage of doing so, or is this just something one might do as there is no need to use Python? – Robert Aug 19 at 19:45
• Well, a Bash script doing basically the same thing might be 10 lines. It won't win a beauty contest, but this isn't the sort of software which would be distributed to lots of people, so there's a trade-off of "getting stuff done" vs. "doing it right." – l0b0 Aug 20 at 7:15
• You don’t even need a Bash script. Just download ImageMagick if you don’t have it already, it has a convert program that’s really easy to use. @Robert – Cris Luengo Aug 25 at 1:59


text:
# How do you simplify (-3) to the 4th power?

Nov 5, 2015

${\left(- 3\right)}^{4} = 81$
${\left(- 3\right)}^{4} = - 3 \times - 3 \times - 3 \times - 3 =$
${\left(- 3\right)}^{4} = 9 \times 9 =$
${\left(- 3\right)}^{4} = 81$


text:
# NAG CL InterfaceD03 (Pde)Partial Differential Equations

D03 (Pde) Chapter Introduction – A description of the Chapter and an overview of the algorithms available.

Function
Mark of
Introduction

Purpose
d03ncc
Example Text
Example Data
Example Plot
7 nag_pde_dim1_blackscholes_fd
Finite difference solution of the Black–Scholes equations
d03ndc
Example Text
Example Data
Example Plot
7 nag_pde_dim1_blackscholes_closed
Analytic solution of the Black–Scholes equations
d03nec
Example Text
Example Data
Example Plot
7 nag_pde_dim1_blackscholes_means
Compute average values for d03ndc
d03pcc
Example Text
Example Plot
7 nag_pde_dim1_parab_fd
General system of parabolic PDEs, method of lines, finite differences, one space variable
d03pdc
Example Text
Example Plot
7 nag_pde_dim1_parab_coll
General system of parabolic PDEs, method of lines, Chebyshev ${C}^{0}$ collocation, one space variable
d03pec
Example Text
Example Plot
7 nag_pde_dim1_parab_keller
General system of first-order PDEs, method of lines, Keller box discretization, one space variable
d03pfc
Example Text
7 nag_pde_dim1_parab_convdiff
General system of convection-diffusion PDEs with source terms in conservative form, method of lines, upwind scheme using numerical flux function based on Riemann solver, one space variable
d03phc
Example Text
Example Plot
7 nag_pde_dim1_parab_dae_fd
General system of parabolic PDEs, coupled DAEs, method of lines, finite differences, one space variable
d03pjc
Example Text
Example Plot
7 nag_pde_dim1_parab_dae_coll
General system of parabolic PDEs, coupled DAEs, method of lines, Chebyshev ${C}^{0}$ collocation, one space variable
d03pkc
Example Text
Example Plot
7 nag_pde_dim1_parab_dae_keller
General system of first-order PDEs, coupled DAEs, method of lines, Keller box discretization, one space variable
d03plc
Example Text
Example Plot
7 nag_pde_dim1_parab_convdiff_dae
General system of convection-diffusion PDEs with source terms in conservative form, coupled DAEs, method of lines, upwind scheme using numerical flux function based on Riemann solver, one space variable
d03ppc
Example Text
Example Plot
7 nag_pde_dim1_parab_remesh_fd
General system of parabolic PDEs, coupled DAEs, method of lines, finite differences, remeshing, one space variable
d03prc
Example Text
Example Plot
7 nag_pde_dim1_parab_remesh_keller
General system of first-order PDEs, coupled DAEs, method of lines, Keller box discretization, remeshing, one space variable
d03psc
Example Text
Example Plot
7 nag_pde_dim1_parab_convdiff_remesh
General system of convection-diffusion PDEs, coupled DAEs, method of lines, upwind scheme, remeshing, one space variable
d03puc 7 nag_pde_dim1_parab_euler_roe
Roe's approximate Riemann solver for Euler equations in conservative form, for use with d03pfc, d03plc and d03psc
d03pvc 7 nag_pde_dim1_parab_euler_osher
Osher's approximate Riemann solver for Euler equations in conservative form, for use with d03pfc, d03plc and d03psc
d03pwc
Example Text
Example Data
Example Plot
7 nag_pde_dim1_parab_euler_hll
Modified HLL Riemann solver for Euler equations in conservative form, for use with d03pfc, d03plc and d03psc
d03pxc
Example Text
Example Data
Example Plot
7 nag_pde_dim1_parab_euler_exact
Exact Riemann solver for Euler equations in conservative form, for use with d03pfc, d03plc and d03psc
d03pyc 7 nag_pde_dim1_parab_coll_interp
PDEs, spatial interpolation with d03pdc or d03pjc
d03pzc 7 nag_pde_dim1_parab_fd_interp
PDEs, spatial interpolation with d03pcc, d03pec, d03pfc, d03phc, d03pkc, d03plc, d03ppc, d03prc or d03psc


text:
# Inclusive photon production at forward rapidities in pp and p$-$Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV

A study of multiplicity and pseudorapidity distributions of inclusive photons measured in pp and p$-$Pb collisions at a center-of-mass energy per nucleon$-$nucleon collision of $\sqrt{s_{\rm NN}} = 5.02$ TeV using the ALICE detector in the forward pseudorapidity region $2.3 <~ \eta_{\rm lab} <~ 3.9$ is presented. Measurements in p$-$Pb collisions are reported for two beam configurations in which the directions of the proton and lead ion beam were reversed. The pseudorapidity distributions in p$-$Pb collisions are obtained for seven centrality classes which are defined based on different event activity estimators, i.e., the charged-particle multiplicity measured at midrapidity as well as the energy deposited in a calorimeter at beam rapidity. The inclusive photon multiplicity distributions for both pp and p$-$Pb collisions are described by double negative binomial distributions. The pseudorapidity distributions of inclusive photons are compared to those of charged particles at midrapidity in \pp collisions and for different centrality classes in p$-$Pb collisions. The results are compared to predictions from various Monte Carlo event generators. None of the generators considered in this paper reproduces the inclusive photon multiplicity distributions in the reported multiplicity range. The pseudorapidity distributions are, however, better described by the same generators.

Submitted to: EPJC
e-Print: arXiv:2303.00590 | PDF | inSPIRE
CERN-EP-2023-026
Figure group


text:
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Posts
Comments

## From Arrow to Fourier

Social Choice Theory is a pretty mature field that deals with the question of how to combine the preferences of different individuals into a single preference or a single choice.  This field may serve as a conceptual foundation in many areas: political science (how to organize elections), law (how to set commercial laws), economics (how to allocate goods), and computer science (networking protocols, interaction between software agents).  Unsurprisingly, there are interesting computational aspects to this field, and indeed a workshop series on computational social choice already exists.  The starting point of this field is Arrow‘s theorem that shows the there are unexpected inherent difficulties in performing this preference aggregation.  There have been many different proofs of Arrow’s impossibility theorem, all of them combinatorial.  In this post I’ll explain a basic observation of Gil Kalai that allows quantifying the level of impossibility using analytical tools (Fourier transform) on Boolean functions commonly used in theoretical computer science.  At first Gil’s introduction of these tools in this context seemed artificial to me, but in this post I hope to show you that  it is the natural thing to do.

## Arrow’s Theorem

Here is our setting and notation:

1. There is a finite set of participants numbered $1...n$.
2. There is a set of three alternatives over which the participants have preferences: $A=\{a,b,c\}$.  (Arrow’s theorem, as well as everything here actually applies also to any set $|A|\ge 3$.)
3. We denote by $L$ the set of preferences over $A$, i.e. $L$ is the set of full orders on the elements of $A$.

The point of view here is that each participant $1 \le i \le n$ has his own preference $x_i \in L$, and we are concerned with functions that reach a common conclusion as a function of the $x_i$‘s.  In principle, the common conclusion may be a joint preference, or a single alternative; Arrow’s theorem concerns the former, i.e. it deals with functions $F:L^n \rightarrow L$ called social welfare functions.  Arrow’s theorem points out in a precise and very general way that there are no “natural non-trivial” social welfare functions when $|A|\ge 3$.  (This is in contrast to the case $|A|=2$, where taking the majority of all preferences is natural and non-trivial.)  Formal definitions will follow.

A preference $x \in L$ really specifies for each two alternatives $a,b \in A$ which of them is preferred over the other, and we denote by $x^{a,b}$ the bit specifying whether $x$ prefers $a$ to $b$. We view each $x$ as composed of three bits $x=(x^{a,b},x^{b,c},x^{c,a})$ (where it is implied that $x^{b,a}=-x^{a,b}$, $x^{c,b}=-x^{b,c}$, and $x^{a,c}=-x^{c,a}$.  Note that under this representation only 6 of the possible 8 three-bit sequences correspond to elements of $L$, where the bad combinations are 000 and 111.

The formal meaning of natural is  the following:

Definition: A social welfare function $F$ satisfies the IIA (independence of irrelevant alternatives) property if for any two alternatives $a,b \in A$ the aggregate preference between $a$ and $b$ depends only on the preferences of the participants between $a$ and $b$ (and not on any preferences with another alternative $c$).  In the notation introduced above it means that $F^{a,b}$ is in fact just a function of the $n$ bits $x_1^{a,b}, ..., x_n^{a,b}$ (rather than of all the bits in the $x_i$‘s).

One may have varying opinions of whether this is indeed a natural requirement, but the lack of it does turn out to imply what may be viewed as inconsistencies.  For example, when we  use the aggregate preference to choose a single alternative (hence obtaining a social choice function), then lack of IIA is directly tied to the the possibility of strategic manipulation of the derived social choice function.

We can take the following definition of non-trivial:

Definition: A social welfare function $F$ is a dictatorship if for some $i$, $F$ is the identity function on $x_i$ or the exact opposite order (in our coding, the bitwise negation of $x_i$).

It is easy to see that any dictatorship satisfies IIA.  Arrow’s theorem states that, with another minor assumption, these are the only functions that do so.  Kalai’s quantitative proof requires an assumption that is more restrictive than that required by Arrow’s original statement.  Recently Elchanan Mossel published a paper without this additional assumption, but we’ll continue with Kalai’s elementary variant.

Definition: A social welfare function $F$ is neutral if it is invariant under changing the names of alternatives.

This basically means that as a voting method it does not discriminate between candidates.

Arrow’s Theorem (for neutral functions): Every neutral social welfare function that satisfies IIA is a dictatorship.

## Quantification for Arrow’s Theorem

In CS, we are often quite happy with approximation: we know that sometimes things can’t be perfect, but can still be pretty good.  Thus if IIA is “perfect”, then the following quantitative version comes up naturally: can there be a function that is “almost” an IIA social welfare function and yet not even close to a dictatorship? Let us define closeness first:

Definition: Two social welfare functions $F, G$ are $\epsilon$-close if $Pr[F(x_1...x_n) \ne G(x_1 ... x_n)] \le \epsilon$.  The probability is over independent uniformly random choices of $x_i \in L$.

The choice of the uniform probability distribution over $L^n$ (strangely termed the impartial culture hypothesis) can not really be justified as a model of the empirical distribution in reasonable settings, but is certainly natural for proving impossibility results which then extend to other reasonably flat distributions. Once we have a notion of closeness of functions, being $\epsilon$-almost a dictatorship is well defined by being $\epsilon$-close to some dictatorship function.  But what is being “almost an IIA social choice function”?  The problem is that the IIA condition involves a relation between different values of $F$.  This is similar to situation in the field of property testing, and one natural approach is to follow the approach of property testing and quantify how often the desired property is violated:

Definition A: A function $F : L^n \rightarrow L$ is an $\epsilon$-almost IIA social welfare function if for all $a,b \in A$ we have that $Pr[F^{a,b}(x_1...x_n) \ne F^{a,b}(y_1...y_n) | \forall i:x^{a,b}_i=y^{a,b}_i] \le \epsilon$.  I.e. for a random input, a random change of the non $(a,b)$-bits is unlikely to change the value of $F^{a,b}$.

A second approach is simpler although surprising at first, and instead of relaxing the IIA requirement, relaxes the social welfare one.  I.e. we can allow $F$ to have in its range all possible 8 values in ${0,1}^n$ rather than just the 6 in $L$.  We call the bad values, 000 and 111, irrational outcomes, as they do not correspond to a consistent preference on $A$.

Definition B: A function $F : L^n \rightarrow \{0,1\}^n$  is an $\epsilon$-almost IIA social welfare function if it is IIA and there exists a social choice function $G : L^n \rightarrow L^n$ such that $F$ and $G$ are $\epsilon$-close.

Note that we implicitly extended the definitions of closeness and of being IIA from functions having a range of $L$ to those also with a range of ${0,1}^3$.

We can now state Kalai’s quantitative version of Arrow’s theorem:

Theorem (Kalai): For every $\epsilon>0$ there exists a $\delta>0$ such that every neutral function that is $\delta$-almost an IIA social choice function is $\epsilon$-almost a dictatorship.

Following Kalai, we will prove the theorem for definition B of “almost”, but the same theorem for definition A directly follows: take a function $F$ that is an $\delta$-almost IIA social welfare according to definition A, and define a new function $F'$ by letting $F'^{a,b}(x_1...x_n)$ to be the majority value of $F^{a,b}(y_1...y_n)$ where the $y_i$‘s agree with the $x_i$‘s on their $(a,b)$-bit and range over all possibilities on the other bits.  By definition $F'$ is IIA, and it is not difficult to see that since $F'$ satisfied definition A, then the majority vote in the definition is almost always overwhelming and thus $F'$ is $\delta'$-close to $F$ (where $\delta' = O(\sqrt{\delta})$).  Since we defined each bit of $F'$ separately, its range may contain irrational outcomes, but since it is close to $F'$, at most $\delta'$ of these, and thus it satisfies definition B.

The main ingredient of the proof will be an analysis of the correlation between two bits from the output of $F$.

Main Lemma (social choice version): For every $\epsilon>0$ there exists a $\delta>0$ such that if a neutral $F$ is not $\epsilon$-almost a dictatorship then $Pr[F^{a,b}(x_1...x_n)=1\:and\:F^{b,c}(x_1...x_n)=0] \le 1/3-\delta$.

Before we proceed, let us look at the significance of the $1/3-\delta$: If the values of $F^{a,b}$ and $F^{b,c}$ were completely independent, then the joint probability would be exactly $1/4$ (since each of the two bits is unbiased due to the neutrality of $F$).  For a “random” $F$ the value obtained would be almost $1/4$.  In contrast, if $F$ is a dictatorship then the joint probability would be exactly $1/3$ since $Pr[x_i^{a,b}=1\:and\:x_i^{b,c}=0]=1/3$ as $x_i$ is uniform in $L$.  The point here is that if $F$ has non-negligible difference from being a dictatorship then the probability is non-negligibly smaller than $1/3$.

The theorem is directly implied from this lemma as the event $F(x_1 ... x_n) \in L$ is the disjoint union of the three events $F^{a,b}(x_1...x_n)=1\:and\:F^{b,c}(x_1...x_n)=0$$F^{b,c}(x_1...x_n)=1\:and\:F^{c,a}(x_1...x_n)=0$, and $F^{c,a}(x_1...x_n)=1\:and\:F^{a,b}(x_1...x_n)=0$, whose total probability, if $F$ is not $\epsilon$-almost a dictatorship,  is bounded by the lemma by $1-3\delta$ and thus the probability of an irrational outcome is at least $3\delta$.

So let us inspect this lemma.  We have two Boolean functions $F^{a,b}$ and $F^{b,c}$ each operating on $n$-bit strings.  Since $F$ is neutral, these are actually the same Boolean function, so $F^{a,b}=F^{b,c}=f$.  What we are asking is for the probability that $f(z)=1\:and\:f(w)=0$ where $w$ and $z$ are each an $n$-bit strings: $w=(x^{a,b}_1....x^{a,b}_n)$ and $z=(x^{b,c}_1...x^{b,c}_n)$.  The main issue how $w$ and $z$ are distributed.   Well, $z$ is uniformly distributed over $\{0,1\}^n$ and so is $w$. They are not independent though: $Pr[w_i=z_i]=1/3$. We say that the distributions on $w$ and $z$ are (anti-1/3-)correlated.

So our main lemma is equivalent to the following version of the lemma which now talks solely of Boolean functions:

Main Lemma (Boolean version): For every $\epsilon>0$ there exists a $\delta>0$ such that if an odd Boolean function $f:\{ 0,1 \}^n \rightarrow \{ 0,1 \}$ is not $\epsilon$-almost a dictatorship then $Pr[f(z)=1\:and\:f(w)=0] \le 1/3-\delta$, where $z$ is chosen uniformly at random in $\{0,1\}^n$ and $w$ chosen so that $Pr[w_i = z_i]=1/3$ and $Pr[w_i=-z_i]=2/3$.

An odd Boolean function just means that $f(-z_1...-z_n)=-f(z_1....z_n)$ which is follows from the neutrality of $F$ by switching the roles of $a$ and $b$.  (We really only need that $f$ is “balanced”, i.e. takes value 1 on exactly 1/2 of the inputs, but lets keep the more specific requirement for compatibility with the previous version of this lemma.)

## Fourier Transform on the Boolean Cube

At this point using the Fourier transform is quite natural.  While I do not wish to give a full introduction to the Fourier transform on the Boolean cube here, I do want to give the basic flavor in one paragraph, convincing those who haven’t looked at it yet to do so.  1st year algebra and an afternoon’s work should suffice to fill in all the holes.

The basic idea is to view Boolean functions $f:{0,1}^n \rightarrow {0,1}$ as special cases of the real-valued functions on the Boolean cube: $f:{0,1}^n \rightarrow \Re$.  This is a real vector space of dimension $2^n$, and has a natural inner product $ = 2^{-n} \sum_x f(x) g(x)$ The $2^{-n}$ factor is just for convenience and allows viewing the inner product as the expectation over a random choice of $x$: $ = E[f(x)g(x)]$.  As usual the choice of a “nice” basis is helpful and our choice is the “characters”: functions of the form $\chi_S(x) = (-1)^{\sum_{i \in S} x_i}$, where $S$ is some subset of  the $n$ bits.  $\chi_S$  takes values -1 and 1 according to the parity of the bits of $x$ in $S$.  There are $2^n$ such characters and they turn out to be an orthonormal basis.  The Fourier coefficients of $f$, denoted $\hat{f} (S)$, are simply the coefficients under this basis $f=\sum \hat{f} (S)\chi_S$, where we have that $\hat{f} (S) = $.

One reason why this vector space is appropriate here is that the correlation operation we needed between $w$ and $z$ in the lemma above, is easily expressible as a linear transformation $T$ on this space defined by $(Tf)(x)=E[f(y)]$ where $y$ is chosen at random with $Pr[y_i = x_i]=1/3$ and $Pr[y_i=-x_i]=2/3$.  Using this it is possible to elementarily evaluate the probability in the lemma as $\sum_S 3^{-|S|} \hat{f} (S)^2$, where the sum ranges over all $2^n$ subsets $S$ of the $n$ bits.  To get a feel of what this means we need to compare it with the following property of the Fourier transform of a balanced Boolean function: $\sum_S \hat{f}(S)^2 = ||f||_2^2 = 1/2$.  The difference between this sum and our sum is the factor of $3^{-|S|}$ for each term.   The “first” element is this sum is easily evaluated for a balanced function: $\hat{f}(\emptyset)^2 =(E[f])^2 = 1/4$, so in order to prove the main lemma it suffices to show that $\sum_{S \ge 1} 3^{-|S|} \hat{f} (S)^2 < (1/3 - \delta) - 1/4 = 1/12 - \delta$. This is so as long as a non-negligible part of $\sum_{|S| \ge 1} \hat{f}(S)^2 = 1/4$ is multiplied by a factor strictly smaller than $3^{-1}$, i.e. is on sets $|S|>1$. Thus the main lemma boils down to showing that if $\sum_{|S| \ge 2} \hat{f} (S)^2 \le \delta'$  then $f$ is $\epsilon$-almost a dictatorship.  This is not elementary but is proved by E. Friedgut, G. Kalai, and A. Naor and completes our journey from Arrow to Fourier.

## More results

There seems much promise in using these analytical tools for addressing quantitative questions in social choice theory.  I’d like to mention two results of this form.  The first is the celebrated MOO paper (by E. Mossel, R. O’Donnell and K. Oleszkiewicz) that proves, among other things, that among functions that do not give any player unbounded influence, the closest to being an IIA social welfare function is the majority function.  The second, is a paper of mine with E. Friedgut and G. Kalai that obtains a quantitative version of the Gibbard–Satterthwaite theorem showing that every voting method that is far from being a dictatorship can be strategically manipulated.  Our version shows that this can be done on a non-negligible fraction of preferences, but is limited to neutral voting methods between 3 candidates.

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### 17 Responses

1. Thanks for the long entry.

I found O Donnel’s survey paper specially useful to get an overview of Fourier analysis applications in social choice theory:

http://www.cs.cmu.edu/~odonnell/papers/analysis-survey.pdf

It also covers applications of Boolean function analysis to prove inapproximability of important problems.

2. I like the idea of including irrational preferences (as well as simplisity) in Definition B of e-almost IIA sw functions. But before allowing those two bad preferences 000 and 111, I would extend the range to include the preferences that contain indifferences. To me, ruling out indifferences on the domain is okay, but ruling them out on the range can be too restrictive.

Usually taking this (minor) point into consideration just complicates the proof, though. Hopefully the same is true for this approach based on Boolean functions (simple games).

—From a social choice guy that works on the boundary of Algorithmic Game Theory and something else.

3. The link to Elchanan Mossel’s paper is broken.

4. Haris: thanks for the link — I added it to the post.

Reiju: I don’t know whether handling indifferences is just a technical nuisance using the Fourier approach or whether it opens up new issues — there has been only little work on this so far.

Preyas: Thanks. I fixed the link.

5. This is a lovely post. I could not have written it so well, but probably I can write about it not as good with some more angles. I will try it some time.

• thanks. please do!

6. Very nice post. However, it’s not quite true that all the proofs of Arrow’s theorem are combinatorial. You might be interested in the paper “Unifying Impossibility Theorems: A Topological Approach” by Baryshnikov. This proof could only be called combinatorial insofar as it is relies on the topology of simplicial complexes which are combinatorial objects.

7. [...] own called “Algorithmic game theory” of with interesting posts on the econ-CS interface. One post is about Fourier theoretic proof of Arrow’s theorem and the recent paper by Noam, Ehud Friedgut [...]

8. There is an electronic version to another paper by Yuliy Baryshnikov: http://www.springerlink.com/content/l6byjxktnm1n3358/fulltext.pdf

Topological social choice is a fairly developed area and it is related to the issue of “averaging” in mathematics. A recent paper by Shmuel Weinberger with links to earlier papers is here:
http://www.ratio.huji.ac.il/dp_files/dp282.pdf

9. Thanks for the links, Gil, Boris, and Noah.

10. [...] blogs: Michael Nielsen’s, the Geomblog, Black belt Bayesian, Secret blogging seminar, and Algorithmic game theory, to name a few. Possibly related posts: (automatically generated)Reading Notes: Chapter 4: Group [...]

11. [...] New paper: average-case manipulation of elections The classic Gibbard-Satterthwaite theorem states that every nontrivial voting method can be manipulated.   In the late 1980’s an approach to circumvent this impossibility was suggested  by Bartholdi, Tovey, and Trick: maybe a voting method can be found where the manipulation is computationally hard — and thus effective manipulation would be practically impossible?  Indeed voting methods whose manipulation problem was NP-complete were found by them as well as later works.  However, this result is not really satisfying: the NP-completeness only ensures that the manipulation cannot be solved in the worst case; it is quite possible that for most instances manipulation is easy, and thus the voting method can still be effectively manipulated.   What would be desired is a voting method where manipulation would be computationally hard everywhere, except for a negligible fraction of inputs.  In the last few years several researchers have indeed attempted “amplifying” the NP-hardness of manipulation in a way that may get closer to this goal. The new paper of Marcus Isaksson, Guy Kindler, and Elchanan Mossel titled The Geometry of Manipulation – a Quantitative Proof of the Gibbard-Satterthwaite Theorem shatters these hopes.  Solving the main open problem in a paper by Ehud Freidgut, Gil Kalai, and myself, they show that every nontrivial neutral voting method can be manipulated on a non-negligible fraction of inputs.  Moreover, a random flip of the order of 4 (four) alternatives will be such a manipulation, again with non-negligible probability.  This provides a quantitative version of the Gibbard-Satterthwaite theorem,  just like Gil Kalai previously obtained a quantitative version of Arrow’s theorem. [...]

12. [...] starts with the classic KKL work on influences and reaches some of his modern results on approximation in computational social choice.  Gil also writes on ICS in his [...]

13. I am planning on going to beauty school when I’m older, and mainly going to school for hairstyling stuff. So what are the top 3 beauty schools in Texas?.

14. […] to a Fourier-theoretic proof of Arrow theorem (in the balanced case). You can find it discussed in this blog post by Noam Nisan.  Lecture 10 mentioned a few further application of the Fourier method related to […]


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# Heron's Formula and Some Mysterious Triangles

Suppose your teacher asks you to find the area of the following triangle.