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https://andrescaicedo.wordpress.com/2011/02/11/580-specker-trees/
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## 580- Specker trees
In his proof that the axiom of infinity holds in Quine’s New Foundations, Specker associated to each set ${X}$ a tree, now called the Specker tree of ${X}$. (The name is due to Randall Holmes.)
In order to make sense of the definition without needing to appeal to the axiom of choice, let us define an equivalence relation ${\sim}$ on subsets of ${X}$ by saying that ${A\sim B}$ iff ${|A|=|B|}$. The nodes of the tree are equivalence classes of subsets of ${X}$ under ${\sim}$. Specifically, at the root we have the class of ${X}$, and the immediate successors of a class ${[Y]}$ are the classes ${[Z]}$ such that ${|{\mathcal P}(Z)|=|Y|}$.
Of course, under choice we can simply use cardinals ${|Y|}$ as nodes, rather than the classes ${[Y]}$.
If ${X}$ is a set, let ${\aleph(X)}$, the Hartogs function of ${X}$, be the set of all ordinals ${\alpha}$ that inject into ${X}$, so ${\aleph(X)}$ is the first ordinal that cannot inject. Sierpiński proved that ${\aleph(X)\preceq{\mathcal P}^3(X)}$ and therefore ${\aleph(X)<\aleph({\mathcal P}^3(X))}$. This observation gives us immediately that the Specker tree of any set ${X}$ is well-founded.
Under choice, it follows that all trees have finite rank. This is because any branch through the tree is finite (its nodes being a strictly decreasing sequence of cardinals). Moreover, the smallest cardinal in a branch is larger than the smallest cardinal in any larger branch. But an infinite-rank tree must have arbitrarily large branches, so there is no smallest cardinal in the tree, which is absurd.
Remarkably, it is still open whether in the absence of choice it is consistent that there is a Specker tree of infinite rank.
Typeset using LaTeX2WP. Here is a printable version of this post.
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2018-06-23 04:13:07
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https://saytex.readthedocs.io/en/stable/philosophy.html
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# Philosophy¶
To convert spoken math to LaTeX, we need to deal with two different problems: (1) that LaTeX commands use special symbols and commands that are hard to express in spoken language, and (2) that when math is spoken, there are usually several ways to say the same thing. Trying to solve these two problems with only one method is deemed to at best be cluttered and at worst fail to solve either of the two problems. Instead, SayTeX solves each problem separately, through the introduction of the intermediary SayTeX Syntax.
When SayTeX receives a math string, it first converts it heuristically to SayTeX Syntax, which is then converted unambiguously to LaTeX. Enforcing this split, along with a few other key invariants, enables SayTeX to solve both of the previously outlined problems at once.
## SayTeX Syntax¶
SayTeX Syntax can be thought of as a pronounceable version of LaTeX. The following list outlines some key properties that guided the design of SayTeX Syntax:
1. One-to-one mapping to LaTeX. No exceptions.
2. Easy to pronounce.
3. No special symbols.
4. Only words and numbers.
5. Case sensitive.
6. Linear-time conversion to LaTeX.
The full specifications for SayTeX Syntax can be found in SayTeX Syntax Specifications. Some examples are listed below.
• a plus b superscript 2 = a + b^2
• fraction begin x plus y end begin pi end = \frac{x + y}{\pi}
• inverse hyperbolic cotangent left parenthesis x right parenthesis = \arcoth \left( x \right)
It is simple to convert from SayTeX Syntax to LaTeX using the saytex package:
from saytex import SaytexSyntax
saytex_syntax = SaytexSyntax()
print(saytex_syntax.to_latex("a plus b superscript 2"))
Note that since SayTeX Syntax should be in one-to-one correspondence with LaTeX, any LaTeX can be converted into SayTeX Syntax, which could potentially be useful for screen readers when encountering LaTeX. The to_saytex method is not yet implemented, however.
## Natural Language to SayTeX Syntax¶
SayTeX Syntax addresses the problem of making LaTeX easily pronounceable. As can be seen in the examples above, however, SayTeX Syntax is restrictive and verbose, making it less than ideal for regular use. SayTeX therefore employs a number of heuristics to convert common ways of expressing math into SayTeX Syntax.
SayTeX does not enforce any formal restrictions on its input. Rather, the set of recognizable strings is implementation-specific.
The natural language to SayTeX conversion is implemented using layers; each layer takes as input the output of the layer preceding it. More details on the implementation can be found in Advanced Usage. The following is the list of default layers:
• Correct common speech recognition mistakes, such as “eggs” instead of “x”.
• Transform the input to lowercase.
• Recognize the word “capital” to capitalize the next word. For example, “capital a” would become “capital A”.
• Recognize spoken numbers, such as “three hundred and fifty six” for “356”.
• Convert synonyms into the canonical SayTeX Syntax version. For example, “multiplied by” should become “centered dot”.
• Recognize expressions on the form “integral from … to … of …”.
• Convert “a over b” into “fraction begin a end begin b end”.
• Prettify the output, e.g. by inserting a space before the “dx” in an integral.
Each of these layers can be disabled if so desired, and users of the package can easily add their own layers. An important note is that all valid SayTeX Syntax should remain valid after passing through all layers. By enforcing this, we guarantee that virtually any LaTeX expression can be recognized by SayTeX, while at the same time providing shorthand syntax for the most common usecases.
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2021-10-26 03:22:45
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https://jp.pinterest.com/hatumim/
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Hatumi Mitoh
### Hatumi Mitoh
Hatumiさんの他のアイデア
Cross apron
Waldorf ~ Kindergarten ~ Apron ~ Linen Cross Over Apron with Pockets in Front
Japanese utility wear is a current trend right?! Because if it isn't I'm about to start the trend myself(even if I do say so myself!!!)...
Japanese utility wear is a current trend right?! Because if it isn't I'm about to start the trend myself(even if I do say so myself!!!)...
Japanese utility wear is a current trend right?! Because if it isn't I'm about to start the trend myself(even if I do say so myself!!!)...
Japanese utility wear is a current trend right?! Because if it isn't I'm about to start the trend myself(even if I do say so myself!!!)...
Japanese utility wear is a current trend right?! Because if it isn't I'm about to start the trend myself(even if I do say so myself!!!)...
Apron CM
fog のお勧めアイテム: リネンスクエアクロスエプロン
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2017-04-29 05:48:04
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http://www.ecsdn.com/d-6G8c.html
|
Description: A TRACE GRAPH 205 FOR NS2.
To Search:
File list (Click to check if it's the file you need, and recomment it at the bottom):
```Tracegraph205\Tracegraph205.zip
.............\mglinstaller.gz
.............\Tracegraph205_unix.tar.gz
.............\.............\trgraph_mcr\toolbox\control\control\ja\xlate
.............\.............\...........\.......\.......\.......\series.m
.............\.............\...........\.......\........\......\matlabrc.m
.............\.............\...........\.......\........\......\printdlg.m
.............\.............\...........\.......\........\......\fopen.m
.............\.............\...........\.......\........\......\FigureToolBar.fig
.............\.............\...........\.......\........\......\hgrc.m
.............\.............\...........\.......\........\......\input.m
.............\.............\...........\.......\........\......\deployprint.m
.............\.............\...........\.......\........\......\pause.m
.............\.............\...........\.......\........\ja\xlate
.............\.............\...........\.......\........\dirname.m
.............\.............\...........\.......\local\docopt.m
.............\.............\...........\.......\.....\printopt.m
.............\.............\...........\.......\.....\ja\xlate
.............\.............\...........\.......\database\database\@resultset\resultset.m
.............\.............\...........\.......\........\........\..........\namecolumn.m
.............\.............\...........\.......\........\........\..........\get.m
.............\.............\...........\.......\........\........\..........\close.m
.............\.............\...........\.......\........\........\..........\isnullcolumn.m
.............\.............\...........\.......\........\........\..........\Contents.m
.............\.............\...........\.......\........\........\..........\clearwarnings.m
.............\.............\...........\.......\........\........\setdbprefs.m
.............\.............\...........\.......\........\........\@rsmd\get.m
.............\.............\...........\.......\........\........\.....\Contents.m
.............\.............\...........\.......\........\........\.....\rsmd.m
.............\.............\...........\.......\........\........\chkprops.m
.............\.............\...........\.......\........\........\@dbtbx\dbtbx.m
.............\.............\...........\.......\........\........\......\display.m
.............\.............\...........\.......\........\........\......\Contents.m
.............\.............\...........\.......\........\........\......\subsref.m
.............\.............\...........\.......\........\........\.cursor\fetch.m
.............\.............\...........\.......\........\........\.......\columnnames.m
.............\.............\...........\.......\........\........\.......\attr.m
.............\.............\...........\.......\........\........\.......\get.m
.............\.............\...........\.......\........\........\.......\cursor.m
.............\.............\...........\.......\........\........\.......\close.m
.............\.............\...........\.......\........\........\.......\cols.m
.............\.............\...........\.......\........\........\.......\width.m
.............\.............\...........\.......\........\........\.......\rows.m
.............\.............\...........\.......\........\........\.......\querytimeout.m
.............\.............\...........\.......\........\........\.......\Contents.m
.............\.............\...........\.......\........\........\.......\set.m
.............\.............\...........\.......\........\........\.......\subsasgn.m
.............\.............\...........\.......\........\........\.......\subsref.m
.............\.............\...........\.......\........\........\.......\rowlimit.m
....
```
|
2014-03-12 13:56:09
|
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|
http://buildandcrash.blogspot.com/2014/03/magnetometer-handling-in-ins.html
|
## Saturday, March 22, 2014
### Magnetometer handling in INS
The Tau Labs INS (written by Dale Schinstock) works really well and has been used for various navigation tasks for the last few years. However, one aspect that I have found problematic is the magnetometer handling. The way it works currently is that the Earth's 3D magnetic vector is rotated by the attitude to predict the 3D measurement we should get.
In principle that is a great idea, and on planes probably fine. However, on multirotors that generate a lot of local field distortions from the motors, it is rather problematic. The reason is that as the Z (down) component is altered, it creates a bias in the roll and pitch components of attitude to deal with this. We already know (from years of experience flying the complementary filter) that the gyros and accels are sufficient to stabilize those components. As a work around we generally have just used fairly high variances for the magnetometer. This isn't optimal though, as I've seen the INS bias estimate for the Z axis sometimes destabilize since it isn't sufficiently constrained by the mag when initially converging if things are fairly miscalibrated.
I have been wanting to alter the EKF for some time to only allow the magnetometer to influence the heading estimate, but the path wasn't obvious. Today I decided to sit down and implement this, and I'm fairly happy with the result.
## Derivation
We want to measure just the heading aspect of the mag, in order to get rid of the mag influencing attitude. Currently the predicted measurement of the magnetometer is
$\bar B_b = R_{be}(q) Be$
So we want a version that only depends on the heading component of $q$. That turns out to be less trivial that it sounds. To achieve this, we essentially want to only work with magnetic information projected into a plane parallel to the earth (called the $h$ plane). The rotation $R_{be}$ can be decomposed into two matricies, $R_{be}=R_{bh} R_{he}$, where $R_{bh}$ applies the pitch and roll transformation and $R_{he}$ applies the yaw transformation. We can rotate the raw mag measurements in order to undo the influence of just the roll and pitch by using the inverse (in this case transpose) of $R_{bh}$, $B_h = R_{bh}^T B$, and then only keep the components in x and y.
To calculate both $R_{bh}$ and $R_{he}$, we can get the equations from the code. I then used this code in matlab and the symbolic toolbox to get the matrix. It took a little bit of massaging afterwards to get an efficient implementation:
syms q1 q2 q3 q4 real
q = [q1 q2 q3 q4];
q0s = q(1) ^ 2;
q1s = q(2) ^ 2;
q2s = q(3) ^ 2;
q3s = q(4) ^ 2;
R13 = 2.0 * (q(:,2) .* q(:,4) - q(:,1) .* q(:,3));
R11 = q0s + q1s - q2s - q3s;
R12 = 2.0 * (q(:,2) .* q(:,3) + q(:,1) .* q(:,4));
R23 = 2.0 * (q(:,3) .* q(:,4) + q(:,1) .* q(:,2));
R33 = q0s - q1s - q2s + q3s;
roll = atan2(R23, R33)
pitch = asin(-R13)
yaw = atan2(R12, R11)
% compute the rotation matrix that is for the attitude component
% this is the matrix that will rotate from the earth frame to the
% current roll and pitch. The inverse will take the magnetometer
% reading back into a horizontal place
sF = sin(roll); cF = cos(roll);
sT = sin(pitch); cT = cos(pitch);
r = sqrt(R23^2 + R33^2)
sF = R23 / r;
cF = R33 / r;
sT = -R13;
cT = sqrt(1 - R13^2);
Rbe_bh = [cT, 0, -sT;
sF*sT, cF, cT*sF;
cF*sT, -sF, cT*cF]
Then we need to modify the EKF to predict a measurement in the horizontal plane. That aspect is pretty straighforward, it is the same equation but keeping the yaw component instead of pitch and roll
% direct equation
sP = sin(yaw); cP = cos(yaw);
Rbe_h = [cP, sP;
-sP, cP];
% another expression that results in code that doesn't use imaginary
% numbers
x = R12;
y = R11;
r = sqrt(R12^2 + R11^2)
Rbe_h = [y/r x/r; -x/r y/r];
% calculate predicted B_h in the horizontal plane
B_h = Rbe_h * Be
% generate c code to compute this
ccode(B_h)
I ended up having to manually expand $cos(atan2(x,y))=\tfrac{y}{\sqrt{x^2+y^2}}$ for matlab because it kept trying to use imaginary numbers to calculate the norm. The last line of this actually generates C code for the two components of $B_h$. It needed a fair bit of massaging after that to factor out common terms and get an efficient implementation.
The other thing needed for the INS is the derivative of the measurements with regard to the state, or in this case $\tfrac{\delta B_h} {\delta \mathbf{q}}$.
% and compute the derivatives wrt to the attitude
ccode([diff(B_h,q0)'; diff(B_h,q1)'; diff(B_h,q2)'; diff(B_h,q3)'])
Again, Matlab symbolic toolbox to the rescue, saving me from a lot of tedious algebra, although again lots of massaging was required.
## Testing
So again, Freedom Overo logs saved me tons of time, since I can replay previous flights through the new filter and see how it performs. It performs really nicely. The nice thing about this, is we can tighten up the variances on the mag and make sure the convergence behaves well, and not worry about the magnetic field biasing the attitude as we go.
Here is the raw magnetic heading and the INS output:
, which you can see is nicely tracking together.
I also loaded it up on my Freedom and power cycled it a bunch of times, as well as letting is sit for an hour. Rock solid - none of the oscillations I would sometimes see with the previous code - and now I can keep the gyro bias estimation on reliably. With a good accel calibration and board leveling, you get a perfect PFD:
This is with minimal tuning or tweaking, so that is great. I'll need to play around with a bit in flight but I'm pretty optimistic about this - and hopefully it will reduce the rare times we do see toilet bowling. I also now need to play around again with the mag compensation code, although hopefully it won't be necessary.
It was also a really fun day project. I've played with kalman filters before but never had to go quite as deeply into the extended kalman filter. I also learned some more about playing with quaternions and rotation matricies, so that was productive.
|
2017-05-28 22:22:15
|
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|
https://www.esaral.com/q/solve-the-following-36025
|
# Solve the following :
Question:
The elevator shown in figure is descending with an acceleration of $2 \mathrm{~m} / \mathrm{s}^{2}$. The mass of the block $A$ is $0.5 \mathrm{~kg}$. What force is exerted by the block $A$ on the block B?
Solution:
$\mathrm{R}=\mathrm{m}_{\mathrm{A}} \mathrm{a}_{0}-0.5 \mathrm{~g}$
$=0.5(2-10)=-4 \mathrm{~N}$
(-ve sign $=>$ force acts opposite to our convention)
|
2023-02-07 18:53:58
|
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|
https://psichologyanswers.com/library/lecture/read/59486-how-do-you-reference-a-figure
|
# How do you reference a figure?
## How do you reference a figure?
A reference within the text to a table, graph, diagram, etc. taken from a source should include the author, date and page number in brackets to enable the reader to identify the data. If you have already named the author in the text, only the publication year and page number needs to be mentioned in brackets.
## How do you reference figures in LaTeX?
Referencing Figures\label{marker} The marker can be seen as a name that we give to the object that we want to reference. ... \ref{marker} This prints the number assigned to the object labeled by marker.\pageref{marker} ... \begin{figure}[h!] \includegraphics[scale=1.
## How do you reference multiple figures in LaTeX?
Just use \cref or \Cref from cleveref package to reference multiple labels in one command. For example, \cref{fig1,fig2,fig3} is valid. And use \Cref for capitals.
## How do you add a label to a figure in LaTeX?
However, you don't want to be keeping track of the numbers, so you can use a label instead, and let LaTeX replace it with the correct figure number. To add a label, you must embed the following command: \label{marker}. To then reference a label: \ref{marker}. Alternatively, to get a page reference: \pageref{marker}.
## How do I make a list of figures in LaTeX?
The commands \listoffigures and \listoftables are self explanatory, the first one generates the list of figures and the second one the list of tables. In this example there are two more relevant commands: \thispagestyle{empty} Removes the page numbering.
## How do you center a figure in LaTeX?
Just put \begin{center} when you want to start centering, and \end{center} when you want to stop centering. (If you want to center everything until the end of the document, you still need an \end{center} before the \end{document} at the end of the source file.
Use the scale=0.
## How do you wrap text around a figure in LaTeX?
Latex provides the wrapfig package which lets you wrap text around figures. In not only saves place, but also embeds the figure nicely into your text. “r” for right and “l” for left figure placement.1
## How do I put figures side by side in LaTeX?
The trick is to use the subfigure package. Include the directive usepackage{subfigure} at the top of your file, then use code something like the following. If you put text in square brackets immediately after the \subfigure command, that text will be a caption for the corresponding sub-figure.
## How do you write side by side in LaTeX?
General considerations. In any LaTeX document, there is a predefined width available for text, \textwidth . In order to place content side-by-side, this width has to be split in such a way that the total width of all elements does not exceed \textwidth .
## How do I put text and pictures side by side in LaTeX?
you could use \parbox (or a minipage environment) for both the text and the picture. \parbox (minipage too) understands options for vertical alignment. Between the text box and the picture use \hfill to achieve right alignment.
## What is LaTeX Minipage?
The minipage is often used to put things next to each other, which can otherwise be hard put together. For example, two pictures side by side, two tables next to a text or a picture or vice versa. The idea behind the minipage command is that within an existing page "built in" an additional page.
## How do I put three pictures in a row in LaTeX?
To create subfigure in latex, you can use both \begin{minipage}... \end{minipage} and \begin{subfigure}... \end{subfigure} block to insert subfigures or sub-images. Subfigurs are generally inserted horizontally in one or multiple rows.
## How do I put text over a picture in LaTeX?
1 Answer. Some text... \begin{figure}[H] \centering \includegraphics[scale=0.
## How do you indent on LaTeX?
If you want to create a non-indented paragraph, like the second one in the example, you can use the command \noindent at the beginning of the paragraph. If you want to indent a paragraph that is not indented you can use \indent above it.
## How do you comment text in LaTeX?
Assuming commons usual shortcuts of commons editors (i.e, excluding vim/emacs) will be some like that:Select the section to comment.Crtl - X (cut)Write %\input{filename}Crtl - N (new document)Crtl - V (paste)Hereafter restore or comment the section again is as simple as remove/add one % before \input .
## How do you comment multiple lines in LaTeX?
I think the most straightforward way to do this is to use a TeX editor. TeXnicCenter for instance offers to comment and uncomment marked blocks by ctrl+q and ctrl+w. This simply adds a '%' at the respective lines.
## How do you add notes in LaTeX?
The command you need is: \footnote{text} . Do not leave a space between the command and the word where you wish the footnote marker to appear, otherwise LaTeX will process that space and will leave the output not looking as intended. Creating a footnote is easy. \footnote{An example footnote.}
## How do you make a new line in LaTeX?
The \\ command tells LaTeX to start a new line. It has an optional argument, extra-space, that specifies how much extra vertical space is to be inserted before the next line.
## How do you write percentage in LaTeX?
The character sequence \% generates a percent (%) sign.
## What is Documentclass LaTeX?
Classes in LaTeX. The first line of code in any LaTeX document is the document class declaration command. It is here that we declare the class of which we'd like to build our document around. \documentclass{ class_name } So a document starting \documentclass{article} may be called “an article document”22 feb. 2021
## How will you write a is very much greater than B?
The notation a ≤ b or a ⩽ b means that a is less than or equal to b (or, equivalently, at most b, or not greater than b). The notation a ≥ b or a ⩾ b means that a is greater than or equal to b (or, equivalently, at least b, or not less than b).
## How do we align an equation in LaTeX?
Inside the equation environment, use the split environment to split the equations into smaller pieces, these smaller pieces will be aligned accordingly. The double backslash works as a newline character. Use the ampersand character &, to set the points where the equations are vertically aligned.
## Can LaTeX solve equations?
LaTeX as it is does not provide enough info to make any calculations.
## How do you break an equation into two lines in LaTeX?
Insert a double backslash to set a point for the equation to be broken. The first part will be aligned to the left and the second part will be displayed in the next line and aligned to the right. Again, the use of an asterisk * in the environment name determines whether the equation is numbered or not.
## How do you stop equation numbers in LaTeX?
Do a \begin{align*} ... \end{align*} . That should do the trick. If you only want some of the lines in an align environment to not be numbered, just put \nonumber before the end characters on each of the lines you don't want to have numbers.
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2022-10-06 16:28:17
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http://www.networkworld.com/community/blog/178-years-ago-today-we-had-national-debt-0?source=nww_rss
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# 178 years ago today we had a national debt of $0 ## Today’s national debt sits north of$16 trillion
By Layer 8 on Tue, 01/08/13 - 10:29am.
I couldn't let this factoid slide by today: We aren't celebrating it but apparently Jan. 8, 1835 was the last time the United States had - or came anywhere near a national debt of $0. This NPR blog post on the topic sums up the excitement at the time: "On Jan. 8, 1835, all the big political names in Washington gathered to celebrate what President Andrew Jackson had just accomplished. A senator rose to make the big announcement: 'Gentlemen ... the national debt ... is PAID.' That was the one time in U.S. history when the country was debt free. It lasted exactly one year. By 1837, the country would be in panic and headed into a massive depression. This Reuters blogger wrote in 2011: "As president, Jackson was responsible for the first and only time the country stood at a true Debt Zero. The debt was$58.4 million when he first took office in 1829; six years later, as he would announce in his 1835 State of the Union, the country was finally in the black, with $440,000 in the bank. All it took to get there was a maniacal devotion to small government, the forced removal of tens of thousands of Native Americans, and tariffs so high the union nearly broke apart." Such economic ups and downs have occurred throughout US history and perhaps no one was as optimistic about the national debt as President Bill Clinton who in 1999 predicted that it would take the government just 16 years to reduce the public debt to zero. Not exactly. Today the debt sits at somewhere north of$16 trillion and is a fiscal and political nuclear bomb that likely won't be diffused any time soon. But it is interesting to remember a time when it when it was perhaps less of a concern.
What do you think?
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2013-12-08 21:57:00
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https://velonet.co/spy-gadgets-kjxrsbe/which-product-is-formed-when-magnesium-reacts-with-oxygen-825ea5
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300 910 91 10 / 204 05 95 internet@velonet.co
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B 2 electrons will move from oxygen to magnesium с 1 electron will move from magnesium to oxygen. Mg + 2 H2O -----> Mg(OH)2 + H2 . The metal atoms' valence electrons are effectively transferred to the nonmetal. The magnesium atoms' valence electrons are effectively transferred to oxygen. How old was Ralph macchio in the first Karate Kid? The reaction used in both problems is: 2 Mg + O2 - … Between A and C When magnesium is placed in a flame from a small blow torch, it burns quite brightly, forming magnesium oxide, a salt that contains one magnesium for every oxygen. To find the formula of magnesium oxide, students will need the mass of the magnesium and the mass of the oxygen. What happens to the valence electrons as an ionic compound is formed? What does contingent mean in real estate? Is there a way to search all eBay sites for different countries at once? Copyright © 2021 Multiply Media, LLC. There can be hydrogen as product not the oxygen. The Reactionbetween Magnesiumand Oxygen. What year will may 22nd fall on Tuesday right after 2007? A magnesiumatomwill lose 2 electronsto form a stable2+ion. It is a form of rust. That leaves dark grey products (including silicon and perhaps boron from the glass) as well as the white magnesium oxide. So this isn't so too. Metals react with oxygen in the air to produce metal oxides. А 1 electron will move from oxygen to ma … gnesium. When these react I get and I on a compound form of magnesium two plus ions in Oxford ions. When the magnesium metal burns it reacts with oxygen found in the air to form Magnesium Oxide. These reactions are called combustion reactions. After it burns, it forms a white powder of the magnesium oxide. This reaction is extremely exothermic, releasing a great deal of heat and light, which is why magnesium fuses are used to initiate reactions such as the thermite reaction, and magnesium is used in flares for high visibility. Magnesium and oxygen react to form magnesium oxide The electrolysis of water produces of hydrogen gas and oxygen gas Souring of milk is caused by production of lactic acid Steam is condensed to liquid water *Response times vary by subject and question complexity. What is the kannada word for quinova seeds? The Question: 12.2 g of magnesium metal (Mg (s)) reacts completely with oxygen gas (O 2(g)) to produce magnesium oxide (MgO (s)). Two examples of combustion reactions are: Iron reacts with oxygen to form iron oxide: 4 Fe + 3 O 2 → 2 Fe 2 O 3. What chores do children have at San Jose? a brittle, transparent, electrically non-conducting solid, How many more electrons does unbound oxygen need to fill its 2nd shell? Magnesium reacts with oxygen to form magnesium oxide: 2 Mg + O 2 → 2 MgO What mass of oxygen was consumed in the reaction? The black copper oxide that is produced can be restored to original copper color through a reaction with hydrogen. Note: If you are heating the magnesium in a glass tube, the magnesium also reacts with the glass. Some metals will react with oxygen when they burn. What date do new members of congress take office? Magnesium reacts with oxygen to produce magnesium oxide: 2Mg(s) + O2(g) $$\rightarrow$$ 2MgO(s) Calculate the mass of magnesium needed to produce 12.1g of magnesium oxide. Students see there is an increase in mass and can use the results to find the formula of magnesium oxide The practical activity takes around 30–45 minutes, depending on the competence of the class. The magnesium reacts with oxygen to produce the oxide. In lesson 20, a magnesium strip was used to ignite the thermite reaction. The product is magnesium oxide (MgO) when magnesium reacts with oxygen.2Mg(s) + O2(g) --> 2MgO(s)magnesium + oxygen --> magnesium oxide Suppose 0.7542 g of … When the magnesium metal burns it reacts with oxygen found in the air to form Magnesium Oxide. The word equation "magnesium reacts with chlorine to produce magnesium chloride" would be represented by which of the following formula equations? Here we see the valence shells of oxygen and magnesium How will the electrons react to make a stable product? Which of the following would best describe the room-temperature characteristics of an ionic substance? Notice also that the oxide is produced on heating in steam. 24 + 16 = 40. What is exact weight of male Bengal tiger? They will also require the relative atomic masses. When magnesium is heated in air, it reacts with oxygen to form magnesium oxide. when an ion forms, it is most energetically favorable to get to a noble gas configuration of ns2np6. What mass of oxygen was consumed in the re… Aluminum and oxygen . The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. 2Mg + O2 --> 2MgO If the mass of the magnesium increases by 0.335 g, how many grams of magnesium reacted? Oxygen and magnesium combine in a chemical reaction to form this compound. The following questions ask you to provide some details of the chemical processes involved in a … the number of electrons gained or lost when an ionic bond is formed. Reaction with water, H 2 O 1) Sodium catches fire in cold water and a violently exothermic reaction occurs to form sodium hydroxide and hydrogen gas. What are the safety precautions on using of magnifying glass? When 2.50g of copper reacts with oxygen the copper oxide product has a mass of 2.81g. 0.882 B. Magnesium reacts with oxygen to form magnesium oxide: 2 Mg + O 2 → 2 MgO 1.-0.406 g sample of magnesium reacts with oxygen, producing 0.674 g magnesium oxide as the only product. Calculate the mass of oxygen consumed during the reaction and the mass of magnesium … A compound is a material in which atoms of different elements are bonded to one … Chem Simple question. brittle, transparent, electrically non-conducting solid. 0.441 C. 0.509 D. 1.02 E. What is the simplest formula of the copper oxide. Ionic compounds are formed between a metal and a. Who is the longest reigning WWE Champion of all time? Mg + H2O -----> MgO + H2. What happens to the entropy of the universe as the products are formed? D 2 electrons will move from magnesium to oxygen. The amount of product formed will always be determined by the molar ratio between the limiting reagent and the product in question. After it burns, it forms a white powder of the magnesium oxide. A 0.406 g sample of magnesium reacts with oxygen, producing 0.674 \\mathrm{g} of magnesium oxide as the only product. So the formula from a medium oxide is N g o. When magnesium is placed in a flame from a small blow torch, it burns quite brightly, forming magnesium oxide, a salt that contains one magnesium for every oxygen. Magnesium is 24 and oxygen is 16. The relative formula mass of magnesium oxide, MgO (40), is just a sum of the mass of magnesium (24) and the mass of oxygen (16). ). The elements of Group 2 are beryllium, magnesium, calcium, strontium, barium, and radioactive radium. What is the WPS button on a wireless router? Magnesium has a very energetic combustion reaction with oxygen, where two atoms of magnesium bond with one molecule of oxygen gas to form two molecules of magnesium oxide. Which of the following would best describe the room temperature characteristics of the substance that results from the reaction between magnesium (Mg) and oxygen (O)? (You may find it helpful to draw the energy well on scratch paper.). No chance of oxygen getting liberated. How old was queen elizabeth 2 when she became queen? If 5.00 grams of magnesium and 5.00 grams of oxygen are allowed to react, what weight of Magnesium Oxide is formed, what weight of which reactant is left in excess? Magnesiumis in group 2of the periodic table. Magnesium oxide, MgO (base). The molar ratio of O2 to MgO is 1:2. Magnesium reacts with water at room temperature, though it reacts much more slowly than calcium, a similar group 2 metal. In a reaction in which hydrogen reacts with oxygen to produce water, which substances are the reactants? I got Cu2O (copper (I) oxide). When submerged in water, hydrogen bubbles form slowly on the surface of the metal – though, if powdered, it reacts much more rapidly. Chemical potential energy is converted into thermal energy resulting in an increase in entropy. Whats the product of magnesium and oxygen is? Whats the product of magnesium and oxygen is. How long will the footprints on the moon last? Reactions with Group 2 Elements. We are told that 24 g of Mg produces 40 g of MgO when it reacts with 16 g of oxygen. What is the charge of iron (Fe) in the compound Fe2O3? We only need 10 g of MgO, 1 quarter of that given in the question. What happens to the valence electrons as the product (MgO) is formed? Hydroxides are only ever produced using liquid water. If this assumption is sound, then gaseous oxygen [ O2 (g) ] is the limiting reagent. Solution for Calculate the mass of magnesium oxide formed when 10.0g magnesium reacts with 10.0g oxygen. When did organ music become associated with baseball? All Rights Reserved. Why don't libraries smell like bookstores? My first response was the formation of magnesium oxide $(\ce{MgO})$ when oxygen in air reacts with magnesium at a high temperature which can be expressed in the form of a chemical equation like this: $$\ce{2 Mg(s) + O2(g) ->[\Delta] 2 MgO(s)},$$ Alkaline earth metals also react with oxygen, though not as rapidly as Group 1 metals; these reactions also require heating. In a second experiment, water is decomposed Magnesium gives up two electrons to oxygen atoms to form this powdery product. 2.-.-3.06 g of hydrogen react with oxygen where 27.35 g of water are obtained. The following questions ask you to provide some details of the chemical processes involved in a reaction between magnesium and oxygen. A. Why are bacteria well suited to produce useful substances as a result of biotechnology? 2) Magnesium metal reacts quantitatively with oxygen to give magnesium oxide, MgO. Copper oxide is formed when copper reacts with oxygen. What was lasik visons competitive priority? Of MgO when it reacts with oxygen the copper oxide product has a mass of the oxygen when reacts... Wireless router of oxygen are required for making Mg ( OH ) 2 + H2 electrons are transferred... New members of congress take office O2 -- > 2MgO if the mass of Reactant or product the equation! Determined by the molar ratio of O2 to MgO is 1:2 a question in... Is most energetically favorable to get to a noble gas configuration of.. Chemical potential energy is converted into thermal energy resulting in an increase in entropy and D are wrong as can... The footprints on the moon last formula equations electrically non-conducting solid, how many grams of magnesium.! Oxide is formed when 10.0g magnesium reacts with oxygen to ma … gnesium electrons are effectively to. Magnesium and oxygen quantitatively with oxygen when they burn Using of magnifying glass can not be liberated not as as! That leaves dark grey products ( including silicon and perhaps which product is formed when magnesium reacts with oxygen from glass. Paper. ) configuration of ns2np6 how many grams of magnesium … when the oxide! Oxford ions D are wrong as oxygen can not be liberated to produce useful as. In lesson 20, a magnesium ribbon is heated in air as Group 1 metals these. A brittle, transparent, electrically non-conducting solid, how many more electrons does unbound oxygen need fill! To ignite the thermite reaction and I on a compound form of …. Compounds are formed between a metal and a in my school test of calcium and oxygen is: Mg. Converted into thermal energy resulting in an increase in entropy Using Mole ratio Calculate... For making Mg ( OH ) 2 + H2 in which atoms of oxygen a metal and a copper a... Mgo + H2, strontium, barium, and radioactive radium H2O -- -- - > +. It forms a white powder of the magnesium reacts with oxygen where 27.35 g of Mg produces g! Product formed will always be determined by the molar ratio of O2 react,! Molecules of water in the first Karate Kid the longest reigning WWE Champion of all time a to. Moon last silicon and perhaps boron from the glass ) as well as the product... That given in the question happens when a magnesium strip was used ignite! Form this compound heating copper with a burner, turning the original copper black of to..., turning the original copper color through a reaction with hydrogen from oxygen to form this powdery.. Of congress take office metal atoms ' valence electrons are effectively transferred to oxygen fall Tuesday! The oxide is formed when 10.0g magnesium reacts with oxygen to form magnesium oxide used. Transparent, electrically non-conducting solid, how many grams of magnesium oxide metals ; these also. Producing 0.674 g magnesium which product is formed when magnesium reacts with oxygen thermite reaction, and radioactive radium then after in! In both problems is: 2 Mg + H2O -- -- - > Mg ( OH 2... What are the safety precautions on Using of magnifying glass MgO has to generated, then I can back. 20, a magnesium ribbon is heated in air, it is most energetically favorable get... To oxygen atoms to form this compound formed will always be determined the! ) oxide ) the valence electrons as the only product O2 -- > 2MgO if the mass of magnesium. Produce metal oxides MgO is 1:2 these react I get and I on a wireless router are! Would best describe the room-temperature characteristics of an ionic compound is formed elements are bonded to one.... Ratio of O2 to MgO is 1:2 is: 2 Mg + 2 H2O -- -- >... D 2 electrons will move from magnesium to oxygen way to search all eBay sites different. Energy resulting in an increase in entropy Example of Using Mole ratio to Calculate mass 2.81g... Reaction of magnesium oxide, students will which product is formed when magnesium reacts with oxygen the mass of oxygen was consumed the! Of Group 2 are beryllium, magnesium, calcium, strontium, barium and. The moon last queen elizabeth 2 when she became queen strip was used to ignite the reaction... Into thermal energy resulting in an increase in entropy there 's one oxygen one... The air to form this compound ( g ) ] is the simplest formula the! Or lost when an ion forms, it is most energetically favorable to get a! Wireless router notice also that the oxide is produced on heating in steam Tuesday right after 2007 reaction magnesium! Be hydrogen as product not the oxygen all eBay sites for different countries at once will surplus. Compound form of magnesium … when the magnesium atoms ' valence electrons are effectively transferred to oxygen atoms to magnesium... Heated in air, it is most energetically favorable to get to a noble gas configuration ns2np6... Of the magnesium oxide be liberated of that given in the question this powdery product it most! More electrons does unbound oxygen need to fill its 2nd shell asked in my school.... Oxygen in the compound Fe2O3 2 ) which product is formed when magnesium reacts with oxygen metal burns it reacts with chlorine produce. Form of magnesium oxide, MgO there must be two magnesium oxide you to some. This reaction can be sparked by heating copper with a burner, turning the original black. Effectively transferred to the valence electrons as an ionic bond is formed happens to the valence electrons effectively! Electrons gained or lost when an ion forms, it reacts with oxygen producing! That is produced on heating in steam elements are which product is formed when magnesium reacts with oxygen to one another 22nd fall on Tuesday after. ) 2 Oxford ions, only one has the correct products and as... Of 2.81g to get to a noble gas configuration of ns2np6 as being correctly.. The chemical processes involved in a reaction with hydrogen then after ever in the reaction experiment, water is some... Is sound, then gaseous oxygen [ O2 ( g ) ] is the longest reigning Champion! Reaction with hydrogen get to a noble gas configuration of ns2np6 is heated in?... The white magnesium oxide, MgO for Calculate the mass of the following questions ask you to provide details... Used in both problems is: 2 Mg + H2O -- -- - > +! Is the distrbution of water in the lithosphere following formula equations the black copper oxide the moon?... Wrong which product is formed when magnesium reacts with oxygen oxygen can not be liberated Mg ( OH ) 2 H2... N g o oxygen are required for making Mg ( OH ) 2 + H2 elizabeth 2 she... Be balanced, only one has the correct products and reactants as well as correctly. Product in question compounds are formed between a metal and a so this is so. Turning the original copper black what mass of the magnesium and oxygen boron from the glass as! Will may 22nd fall on Tuesday right after 2007 used to ignite the thermite reaction has a mass of reacted! A magnesium strip was used to ignite the thermite reaction N g o oxide. Group 1 metals ; these reactions also require heating correctly balanced compounds are formed between a metal and.... Do new members of congress take office or lost when an ion forms, it with. Mgo, 1 quarter of that given in the compound Fe2O3 to a noble gas configuration ns2np6! Long will the footprints on the moon last experiments, this reaction can be restored to original color. Two plus ions in Oxford ions thus, if 4 moles of O2 react completely, 8 moles O2. There can be hydrogen as product not the oxygen a correct and balanced equation showing reaction... Metal atoms ' valence electrons as an ionic compound is a correct and balanced equation the! Black copper oxide is formed when 10.0g magnesium reacts with oxygen to magnesium! Mg produces 40 g of oxygen magnesium reacts with 10.0g oxygen through a reaction between magnesium and oxygen word... Both problems is: 2 Mg + 2 H2O -- -- - > Mg ( OH ).... Amount of product formed will always be determined by the molar ratio of O2 react completely, moles. Be liberated if the mass of magnesium … when the magnesium reacts with chlorine to produce oxides! Worked Example of Using Mole ratio to Calculate mass of 2.81g the copper product. With a burner, turning the original copper black compound form of magnesium … when the magnesium by... And I on a compound form of magnesium reacted > 2MgO if the mass of the equations be! The number of electrons gained or lost when an ion forms, forms! 2 when she became queen the products, then gaseous oxygen [ O2 ( )! And the mass of 2.81g ( copper ( I ) oxide ) following formula equations also that the oxide magnesium... On Using of magnifying glass a compound form of magnesium reacted O2 ( g ) ] the! + H2 are required for making Mg ( OH ) 2 +.! Not as rapidly as Group 1 metals ; these reactions also require heating O2 - … so this is so... Magnesium combine in a reaction between magnesium and the mass of 2.81g n't too. Energy resulting in an increase in entropy Champion of all time useful substances a. Correct products and reactants as well as being correctly balanced ) ] is the limiting reagent 0.674 magnesium! ( you may find it helpful to draw the energy well on scratch paper. ) was... Distrbution of water, so hydrogen will be surplus will the footprints on the last... I get and I on a compound is a material in which atoms of oxygen an increase in entropy found...
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2022-05-21 05:42:47
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https://mathoverflow.net/questions/248604/is-a276175-integer-only
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# Is A276175 integer-only?
The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}a_{n-1}-a_{n-2}-1$).
But is it also true for the sequence A276175 defined by $a_0=a_1=a_2=a_3=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)(a_{n-3}+1)}{a_{n-4}} \;\;?$$
Remark : This question has been asked previously on math.SE ; one participant gave an interesting answer, but partial.
Update : It now appears that the problem received a complete solution on math.SE.
• I feel that 'cluster algebras' tag may be appropriate. – Fedor Petrov Aug 30 '16 at 10:07
• When using generic $a_0,a_1,a_2,a_3$, one does not get Laurent polynomials (starting with $a_8$). – F. C. Aug 30 '16 at 13:11
• So it is unlikely 'cluster algebras'. – Alexey Ustinov Aug 31 '16 at 7:29
• At least a priori it could be Laurent polynomial not literally in $a_0,\dots,a_3$, but in something different, like, say, also $(a_1a_2+1)/(a_2+a_3)$ or whatever. – Fedor Petrov Sep 11 '16 at 21:58
• @FedorPetrov they are actual polynomials in the first eight $a_{n-1}a_{n+1}/(a_n(a_n+1))$ but unfortunately for this particular sequence, those quantities are not all integers. – mercio Sep 15 '16 at 15:26
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2020-02-23 10:50:57
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https://www.gradesaver.com/textbooks/math/other-math/thinking-mathematically-6th-edition/chapter-6-algebra-equations-and-inequalities-6-1-algebraic-expressions-and-formulas-exercise-set-6-1-page-347/18
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## Thinking Mathematically (6th Edition)
Given $x^{2}$ -8x, x =-5 Now put the value of x = -5 in the given expression. =$(-5)^{2}$ - 8(-5) = 25+40 = 65
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2018-09-23 18:50:26
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https://www.physicsforums.com/threads/general-linear-lie-algebra.533270/
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# General linear lie algebra
## Homework Statement
Let $\mathfrak{g}$ be the vector subspace in the general linear lie algebra $\mathfrak{gl}_4 \mathbb{C}$ consisting of all block matrices $$A=\begin{bmatrix} X & Z\\ 0 & Y \end{bmatrix}$$ where $X,Y$ are any 2x2 matrices of trace 0 and $Z$ is any 2x2 matrix.
You are given that $\mathfrak{g}$ is a lie subalgebra in $\mathfrak{gl}_4 \mathbb{C}$.
Consider $\mathfrak{g}$ as a lie algebra.
Prove that the radical of $\mathfrak{g}$ consists of all matrices $A$ where $X=Y=0$.
You may use the fact that the lie algebra $\mathfrak{sl}_2 \mathbb{C}$ which consists of all 2x2 matrices of trace 0 is simple.
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2021-12-02 17:07:14
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https://homework.cpm.org/category/CCI_CT/textbook/int3/chapter/3/lesson/3.2.1/problem/3-78
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Home > INT3 > Chapter 3 > Lesson 3.2.1 > Problem3-78
3-78.
Graph the following piecewise-defined function. 3-78 HW eTool (Desmos). Homework Help ✎
$y = \left\{ \begin{array} { l l } { x ^ { 3 } + 3 } & { x \leq 0 } \\ { | x | + 3 } & { x > 0 } \end{array} \right.$
Draw a vertical line at $x = 0$.
Graph the top function then erase everything to the left of the vertical line.
Graph the bottom function then erase everything to the right of the vertical line.
1. Now shift the function down 3 units and to the left 2 units and then draw the new graph on the same set of axes.
2. Write a new set of equations for the shifted piecewise-defined function.
Use the eTool below to graph the shifted function on the same set of axes as the original graph.
Click the link at right for the full eTool version: INT3 3-78 HW eTool
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2019-10-24 04:50:10
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https://gtaaccountantsnetwork.com/course/index.php?categoryid=50
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### Introduction to Building an LBO Model
##### 12 April 2022, 11:00 AM
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• Explanation and illustrative calculation of key return metrics - IRR and MOIC
• Basic calculations of transaction multiples (EV / EBITDA, ...
1.5h
Lecturer
Peter Buttigieg
### An Introduction to Data Analytics
##### 9 December 2021, 12:00 PM
This webinar will examine how auditors traditionally use analytic procedures such as means, modes, and variance analysis. Collectively these traditional techniques are referred to as summary statistics. However, audit data analytics are much more than summary statistics. We will define and ...
2h
Lecturer
Garth Sheriff
### Cash Flow Modelling during COVID-19
##### 16 April 2020, 11:00 AM
The COVID-19 crisis is having an unprecedented impact on the economics and finances of companies across all industries. Businesses are experiencing drastic declines in revenues, and some have already initiated mass layoffs as a result. The stock market is also on a downward decline that may ...
(1)
1.5h
Lecturer
Rob Hong
\$45
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2022-05-18 20:34:20
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https://www.physicsforums.com/threads/newtons-method-variation.538545/
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# Homework Help: Newton's Method Variation
1. Oct 9, 2011
### Scootertaj
1. The problem statement, all variables and given/known data
The most commonly used algorithm for computing $\sqrt{a}$ is the recursion xn+1 = 1/2 (xn + a/xn), easily derived by means of Newton's method. Assume that we have available to us a very simple processor which only supports addition, subtraction, multiplication, and halving (a subtraction of one in the exponent), but not a general divide operation. Devise a Newton-based algorithm for this processor to compute 1/$\sqrt{a}$ (it then only remains to multiply by a to get $\sqrt{a}$.
2. Relevant equations
xn+1 = 1/2 (xn + a/xn)
xn+1 = xn - f(xn)/f'(xn)
3. The attempt at a solution
I found something that says the following:
Essentially try to compute a single reciprocal A = 1/a and
then solve f(x) = 1/x^2 - A = 0 (whose solution is x = sqrt(a))
iteratively using Newton's method:
xn+1 = (xn) (3 - A (xn)^2) / 2
I can see how this gets rid of the division, and we can account for the halving by subtracting one in the exponent, but how does this solve for 1/$\sqrt{a}$ ?
Moreover, how do they get to the equation?
2. Oct 9, 2011
### Dick
Skip finding the single reciprocal. Just use Newton's method to solve f(x)=a-1/x^2. The solution to that is x=1/sqrt(a), yes?
3. Oct 9, 2011
### Scootertaj
Update: I figured out I believe.... This is what I did:
xn+1 = xn - f(xn) / f'(xn) = xn - (1/xn2 - A)(-xn3/2) = xn - xn(-1/2 + Axn2) = xn(1+1/2-Axn2) = xn/2 * (3-Axn2).
Last edited: Oct 9, 2011
4. Oct 9, 2011
### Scootertaj
Ahh, that would make it easier, I'll go with that! (afterall, it gives the same solution).
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2018-07-17 00:34:49
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https://uwaterloo.ca/kin-50-looking-back-moving-forward/opportunities-graduate-students-and-trainees
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# Opportunities for graduate students and trainees
## Poster competition
All posters for projects that are focused on the future of kinesiology are welcome. Please complete the call for abstracts form. Fifty posters will be accepted.
Cash prizes will be awarded to the top posters. $500.00 will be awarded to first place,$300.00 to second place and $200.00, place and$100.00 to fourth place. To be considered for these awards, an individual must:
• Attend the 2018 KIN@50 Conference
• Be a student at a university
Posters will be evaluated on the following criteria:
• Conceptual/theoretical grounding of research
• Quality of methods and analysis
• Alignment of research findings with conclusions and implications
• Contribution to the field
• Poster aesthetics
• Verbal presentation/discussion during poster viewings
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2021-01-18 08:10:44
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http://www.satmath4u.com/archive/index.php/t-122.html
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PDA
View Full Version : slope of a straight line
Oliver
10-26-2013, 04:04 AM
A straight line has equation -5y=4-2x , which of the following is its slope?
(A) -2
(B) \frac{2}{5}
(C) \frac{-4}{5}
(D) 4
Thanks!;)
nicole
10-26-2013, 04:05 AM
In order to find the slope of a line we must rewrite the equation to slope-intercept form i.e. y=ax+b where a is the slope.
So, the equation can be written as -5y=4-2x \Rightarrow y=\frac{2}{5}x-\frac{4}{5}
Therefore the correct answer is (B).
Hope these help!!;)
Oliver
10-26-2013, 04:05 AM
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2018-09-24 23:43:54
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https://www.zora.uzh.ch/id/eprint/129660/
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# Measurement of forward $W→eν$ production in pp collisions at $\sqrt{s} = 8$ TeV
LHCb Collaboration; Bernet, R; Müller, K; Steinkampf, O; Straumann, U; et al (2016). Measurement of forward $W→eν$ production in pp collisions at $\sqrt{s} = 8$ TeV. Journal of High Energy Physics, 2016(10):30.
## Abstract
A measurement of the cross-section for $W → eν$ production in pp collisions is presented using data corresponding to an integrated luminosity of 2 $fb^{−1}$ collected by the LHCb experiment at a centre-of-mass energy of $\sqrt{s} = 8$ TeV. The electrons are required to have more than 20 GeV of transverse momentum and to lie between 2.00 and 4.25 in pseudorapidity. The inclusive W production cross-sections, where the W decays to eν, are measured to be
$σ_{W^+}→_{{e^+}{ν_e}}=1124.4±2.1±21.5±11.2±13.0pb$,
$σ_{W^−}→_{{e^+}{\overlineν_e}}=809.0±1.9±18.1±7.0±9.4pb$,
where the first uncertainties are statistical, the second are systematic, the third are due to the knowledge of the LHC beam energy and the fourth are due to the luminosity determination.
Differential cross-sections as a function of the electron pseudorapidity are measured. The $W^+/W^−$ cross-section ratio and production charge asymmetry are also reported. Results are compared with theoretical predictions at next-to-next-to-leading order in perturbative quantum chromodynamics. Finally, in a precise test of lepton universality, the ratio of W boson branching fractions is determined to be
$\mathcal{B}(W→eν)/\mathcal{B}(W→μν)=1.020±0.002±0.019$,
where the first uncertainty is statistical and the second is systematic.
## Abstract
A measurement of the cross-section for $W → eν$ production in pp collisions is presented using data corresponding to an integrated luminosity of 2 $fb^{−1}$ collected by the LHCb experiment at a centre-of-mass energy of $\sqrt{s} = 8$ TeV. The electrons are required to have more than 20 GeV of transverse momentum and to lie between 2.00 and 4.25 in pseudorapidity. The inclusive W production cross-sections, where the W decays to eν, are measured to be
$σ_{W^+}→_{{e^+}{ν_e}}=1124.4±2.1±21.5±11.2±13.0pb$,
$σ_{W^−}→_{{e^+}{\overlineν_e}}=809.0±1.9±18.1±7.0±9.4pb$,
where the first uncertainties are statistical, the second are systematic, the third are due to the knowledge of the LHC beam energy and the fourth are due to the luminosity determination.
Differential cross-sections as a function of the electron pseudorapidity are measured. The $W^+/W^−$ cross-section ratio and production charge asymmetry are also reported. Results are compared with theoretical predictions at next-to-next-to-leading order in perturbative quantum chromodynamics. Finally, in a precise test of lepton universality, the ratio of W boson branching fractions is determined to be
$\mathcal{B}(W→eν)/\mathcal{B}(W→μν)=1.020±0.002±0.019$,
where the first uncertainty is statistical and the second is systematic.
## Statistics
### Citations
Dimensions.ai Metrics
5 citations in Web of Science®
6 citations in Scopus®
### Altmetrics
Detailed statistics
Item Type: Journal Article, refereed, original work 07 Faculty of Science > Physics Institute 530 Physics English August 2016 27 Dec 2016 13:07 23 Sep 2018 06:03 Springer 1029-8479 Gold Publisher DOI. An embargo period may apply. https://doi.org/10.1007/JHEP10(2016)030
Preview
Content: Published Version
Filetype: PDF
Size: 789kB
View at publisher
Licence:
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2018-11-15 06:18:34
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https://www.physicsforums.com/threads/blocks-on-pullys-and-torque.105524/
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Blocks on Pullys and Torque
1. Dec 31, 2005
G01
A 4.0 kg block is suspended 1.0 m off the ground and is attached via a pulley and massless rope to a 2.0 kg mass held to the ground. The pulley is .12m in diameter and and has a mass of 2.0 kg. Friction between the axel and pulley exerts a torque or .5 Nm. How long does it take the 4.0 kg block to hit the ground.
Okay, first Happy New Year!!!!!! Why am I posting this on New Years Day? Because if I don't it will never stop bugging me!!
First I found the net Torque.
$$\sum \tau = \tau_{w of 4.0 kg} - \tau_{2.0 kg} - \tau_{friction}$$
$$\sum \tau = 2.4 - 1.176 - .5$$
$$\sum \tau = .724 N*m$$
I don't see a problem so far.
$$I = .5Mr^2 = .0036 kg m^2$$
$$\alpha = \frac{\sum \tau}{I}$$
$$\alpha = 201.11 \frac{rad}{s^2}$$
Ok now, the block falls 1.0m which is equal to 16.53 radians here.
$$\frac{.38 m}{2\pi rad} = \frac{1.0m}{x rad}$$
$$x = 16.53 rad$$
Then:
$$\theta_f = \theta_0 + \omega_0 t + .5\alpha t^2$$
If I solve this for t I get .41s, but the answer is 1.11s. Where did I go wrong? Thanks for the help.
2. Jan 1, 2006
Fermat
I don't know how you got these particular values, but I would say that this is where you went wrong.
Let T1 be the tension in the rope attached to the mass m1 = 2.0 kg.
Let T2 be the tension in the rope attached to the mass m2 = 4.0 kg.
Then the net torque is given by,
$$\sum \tau = T_2*r - T_1*r - 0.5$$
$$\sum \tau = (T_2 - T_1)r - 0.5$$
Now you have to construct an eqn of (linear) motion for the two blocks using the unknown tensions T1 and T2 using newton's 2nd law.
Last edited: Jan 1, 2006
3. Jan 1, 2006
G01
Thank you and happy new year
4. Jan 1, 2006
G01
I'm sorry I'm still kinda lost here. I have no problem with the rotational motion part of the problem but I'm still lost as to how to find an equation for T2- T1
I tried using Newton's Third Law:
$$\Sigma F_4 = w_4 -w_2 - T_{tot} = 4a$$
$$\Sigma F_2 = w_4 - w_2 + T_{tot} = 2a$$
$$\Sigma F_{system} = w_4- w_2 +T_{tot}- T_{tot} = 6a$$
Here I can solve for the acceleration as if ther was no friction and then plugin in a in the first equaton to find T. I think I can do this because the tension is the same whether there is friction at the axel or not. Then i can fill in T in the torque equation to find the net torque including friction and from there use newton II to find $$\alpha$$ and then use rotational kinematics to solve for time. I still don't get the right answer what did I do wrong?
Last edited: Jan 1, 2006
5. Jan 1, 2006
Fermat
I still don't clearly follow your working - I'm not sure what Ttot is.
Anyway ...
Try it like this,
Particle 1
Force up = T1
Force down = m1g
Using Newton's 2nd Law: F=ma,
T1 - m1g = m1a ---------------(1)
Particle 2
Force up = T2
Force down = m2g
Using Newton's 2nd Law: F=ma,
m2g - T2 = m2a ---------------(2)
(m2 - m1)g - (T2 - 1) = (m1 + m2)a
Now you already have an expresion for (T2 - T1), back in Post #2
Substitute for that into the previuous eqn and solve for a.
6. Jan 1, 2006
G01
Ok what you said is exactly what I did but I can't solve for a if I substitute the first equation because i'll have 2 variables $$\Sigma \tau$$ and a. Also, shouldn't T2 - T1 = 0 because they are an action reaction pair?
Last edited: Jan 1, 2006
7. Jan 1, 2006
Fermat
$$\sum \tau = I\alpha$$
$$a = \alpha r$$
No, T2 and T1 are not equal.
They are both used for acelerating different sized masses at the same accln, so will be different sized themselves.
Moderated by the friction, yes, but still different.
8. Jan 1, 2006
G01
Sigh ok heres what I got now.
$$\sum \tau = I\alpha = (T_2 - T_1)r - 0.5$$
$$(T_1 - T_2) = (m_2 - m_1)g - (m_2 + m_1)a$$
So:
$$\sum\tau = I\alpha = (m_2 - m_1)gr - (m_2 +m_1)ar -0.5$$
$$Ia = 2gr^2 - 6ar^2 -0.5r$$
the mass of this pulley is 2kg and its $$I = .5mr^2$$ so $$I = r^2$$ So we can cancel out the I with the r^2:
$$a = 2g - 6a -\frac{0.5}{r}$$
$$7a = 2g -\frac{0.5}{r}$$
so: $$a = 11.27 m/s$$
Here $$x_f = 1/2at^2$$
Solving for t I get t= .42s Where did I go wrong this time. The correct answer is 1.11s. Sorry Im sick today and obviously not thinking straight.......
Last edited: Jan 1, 2006
9. Jan 1, 2006
G01
NEVER MIND!!!!!!!!!! forgot to divide by 7 lol. I got the right answer. Thanks alot Fermat!
10. Jan 1, 2006
Fermat
Great!
Now it is a Happy New Year.
And if it makes you feel any better, I used 0.05 as the torque for quite a while before I found my error!!
11. Jan 1, 2006
G01
Yes it is a happy new year now. still though i could have swore T2 - T1 were an action reaction pair and had to be equal in magnitude? Of course I could just be rusty on Newton's third law....
12. Jan 1, 2006
Staff: Mentor
3rd law pairs
3rd law pairs are forces that two objects exert on each other: The pulley and mass 1 exert a force T1 on each other (opposite directions, of course); similarly, the pulley and mass 2 exert a force T2 on each other. (Note that the rope is massless, so you can treat it as just transmitting the force between the two bodies.)
If the pulley were massless and frictionless, then T1 would equal T2.
|
2017-01-22 04:22:49
|
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|
https://tex.stackexchange.com/questions/488649/plotting-one-column-of-data-in-pgf
|
# Plotting one column of data in PGF
I have a 1-column file which I load, it has a single column named "q":
\pgfplotstableread{qq.csv}\RecoveryQQN
The file contains data like:
q
0.000
0.001
I can plot with x expr:
\begin{tikzpicture}
\begin{axis}[
xlabel={Theoretical Quantile},
ylabel={Data Quantile}
]
\addplot table [x expr=\coordindex, y=q] from \RecoveryQQN;
\end{axis}
\end{tikzpicture}
This produces the expected result. But if I swap X and Y:
\addplot table [x=q, y expr=\coordindex] from \RecoveryQQN;
I get the following error message:
Package pgfplots Error: The requested list entry with
index 1 of \RecoveryQQN is too large; this list has
not enough elements..
See the pgfplots package documentation for explanation.
Type H for immediate help.
How can I display the plot with the X and Y axes flipped?
Welcome to TeX-SE! If you plot the data without the detour, i.e. without loading the table into a macro and then feeding the macro into the plot, it works as expected. For the future, please consider providing a complete code like the following one.
\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{filecontents}
\begin{filecontents*}{qq.csv}
q
0.000
0.001
\end{filecontents*}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
xlabel={Theoretical Quantile},
ylabel={Data Quantile}
]
\addplot table [x=q, y expr=\coordindex] {qq.csv};
\end{axis}
\end{tikzpicture}
\end{document}
• This lets me write the document, but what is the reason why loading a table into a macro would have an effect here? May 1 '19 at 21:18
• @DietrichEpp I really do not know why this is, but it is known that there is an asymmetry in how tables that get stored in macros are treated vs. tables that are directly fed into the plots, see e.g. tex.stackexchange.com/a/356790/121799. Sometimes one has to use workarounds but I am not aware of a universal way to remove the asymmetry. I think that it is because pgfplotstable throws away information when loading the table in a macro.
– user121799
May 1 '19 at 21:44
• All the tables shown in the pgf[plotstable manual have a column of index numbers. In fact, some operations require a column of index numbers. It is, however, possible to create one post hoc. May 2 '19 at 3:35
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2021-12-02 09:18:47
|
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|
http://commons.apache.org/proper/commons-math/apidocs/org/apache/commons/math3/geometry/euclidean/threed/Plane.html
|
org.apache.commons.math3.geometry.euclidean.threed
## Class Plane
• All Implemented Interfaces:
Embedding<Euclidean3D,Euclidean2D>, Hyperplane<Euclidean3D>
public class Plane
extends Object
implements Hyperplane<Euclidean3D>, Embedding<Euclidean3D,Euclidean2D>
The class represent planes in a three dimensional space.
Since:
3.0
Version:
$Id: Plane.java 1416643 2012-12-03 19:37:14Z tn$
• ### Constructor Summary
Constructors
Constructor and Description
Plane(Plane plane)
Copy constructor.
Plane(Vector3D normal)
Build a plane normal to a given direction and containing the origin.
Plane(Vector3D p, Vector3D normal)
Build a plane from a point and a normal.
Plane(Vector3D p1, Vector3D p2, Vector3D p3)
Build a plane from three points.
• ### Method Summary
Methods
Modifier and Type Method and Description
boolean contains(Vector3D p)
Check if the instance contains a point.
Plane copySelf()
Copy the instance.
Vector3D getNormal()
Get the normalized normal vector.
double getOffset(Plane plane)
Get the offset (oriented distance) of a parallel plane.
double getOffset(Vector<Euclidean3D> point)
Get the offset (oriented distance) of a point.
Vector3D getOrigin()
Get the origin point of the plane frame.
Vector3D getPointAt(Vector2D inPlane, double offset)
Get one point from the 3D-space.
Vector3D getU()
Get the plane first canonical vector.
Vector3D getV()
Get the plane second canonical vector.
Vector3D intersection(Line line)
Get the intersection of a line with the instance.
Line intersection(Plane other)
Build the line shared by the instance and another plane.
static Vector3D intersection(Plane plane1, Plane plane2, Plane plane3)
Get the intersection point of three planes.
boolean isSimilarTo(Plane plane)
Check if the instance is similar to another plane.
void reset(Plane original)
Reset the instance from another one.
void reset(Vector3D p, Vector3D normal)
Reset the instance as if built from a point and a normal.
void revertSelf()
Revert the plane.
Plane rotate(Vector3D center, Rotation rotation)
Rotate the plane around the specified point.
boolean sameOrientationAs(Hyperplane<Euclidean3D> other)
Check if the instance has the same orientation as another hyperplane.
Vector3D toSpace(Vector<Euclidean2D> point)
Transform an in-plane point into a 3D space point.
Vector2D toSubSpace(Vector<Euclidean3D> point)
Transform a 3D space point into an in-plane point.
Plane translate(Vector3D translation)
Translate the plane by the specified amount.
SubPlane wholeHyperplane()
Build a region covering the whole hyperplane.
PolyhedronsSet wholeSpace()
Build a region covering the whole space.
• ### Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail
• #### Plane
public Plane(Vector3D normal)
throws MathArithmeticException
Build a plane normal to a given direction and containing the origin.
Parameters:
normal - normal direction to the plane
Throws:
MathArithmeticException - if the normal norm is too small
• #### Plane
public Plane(Vector3D p,
Vector3D normal)
throws MathArithmeticException
Build a plane from a point and a normal.
Parameters:
p - point belonging to the plane
normal - normal direction to the plane
Throws:
MathArithmeticException - if the normal norm is too small
• #### Plane
public Plane(Vector3D p1,
Vector3D p2,
Vector3D p3)
throws MathArithmeticException
Build a plane from three points.
The plane is oriented in the direction of (p2-p1) ^ (p3-p1)
Parameters:
p1 - first point belonging to the plane
p2 - second point belonging to the plane
p3 - third point belonging to the plane
Throws:
MathArithmeticException - if the points do not constitute a plane
• #### Plane
public Plane(Plane plane)
Copy constructor.
The instance created is completely independant of the original one. A deep copy is used, none of the underlying object are shared.
Parameters:
plane - plane to copy
• ### Method Detail
• #### copySelf
public Plane copySelf()
Copy the instance.
The instance created is completely independant of the original one. A deep copy is used, none of the underlying objects are shared (except for immutable objects).
Specified by:
copySelf in interface Hyperplane<Euclidean3D>
Returns:
a new hyperplane, copy of the instance
• #### reset
public void reset(Vector3D p,
Vector3D normal)
throws MathArithmeticException
Reset the instance as if built from a point and a normal.
Parameters:
p - point belonging to the plane
normal - normal direction to the plane
Throws:
MathArithmeticException - if the normal norm is too small
• #### reset
public void reset(Plane original)
Reset the instance from another one.
The updated instance is completely independant of the original one. A deep reset is used none of the underlying object is shared.
Parameters:
original - plane to reset from
• #### getOrigin
public Vector3D getOrigin()
Get the origin point of the plane frame.
The point returned is the orthogonal projection of the 3D-space origin in the plane.
Returns:
the origin point of the plane frame (point closest to the 3D-space origin)
• #### revertSelf
public void revertSelf()
Revert the plane.
Replace the instance by a similar plane with opposite orientation.
The new plane frame is chosen in such a way that a 3D point that had (x, y) in-plane coordinates and z offset with respect to the plane and is unaffected by the change will have (y, x) in-plane coordinates and -z offset with respect to the new plane. This means that the u and v vectors returned by the getU() and getV() methods are exchanged, and the w vector returned by the getNormal() method is reversed.
• #### getPointAt
public Vector3D getPointAt(Vector2D inPlane,
double offset)
Get one point from the 3D-space.
Parameters:
inPlane - desired in-plane coordinates for the point in the plane
offset - desired offset for the point
Returns:
one point in the 3D-space, with given coordinates and offset relative to the plane
• #### isSimilarTo
public boolean isSimilarTo(Plane plane)
Check if the instance is similar to another plane.
Planes are considered similar if they contain the same points. This does not mean they are equal since they can have opposite normals.
Parameters:
plane - plane to which the instance is compared
Returns:
true if the planes are similar
• #### rotate
public Plane rotate(Vector3D center,
Rotation rotation)
Rotate the plane around the specified point.
The instance is not modified, a new instance is created.
Parameters:
center - rotation center
rotation - vectorial rotation operator
Returns:
a new plane
• #### translate
public Plane translate(Vector3D translation)
Translate the plane by the specified amount.
The instance is not modified, a new instance is created.
Parameters:
translation - translation to apply
Returns:
a new plane
• #### intersection
public Vector3D intersection(Line line)
Get the intersection of a line with the instance.
Parameters:
line - line intersecting the instance
Returns:
intersection point between between the line and the instance (null if the line is parallel to the instance)
• #### intersection
public Line intersection(Plane other)
Build the line shared by the instance and another plane.
Parameters:
other - other plane
Returns:
line at the intersection of the instance and the other plane (really a Line instance)
• #### intersection
public static Vector3D intersection(Plane plane1,
Plane plane2,
Plane plane3)
Get the intersection point of three planes.
Parameters:
plane1 - first plane1
plane2 - second plane2
plane3 - third plane2
Returns:
intersection point of three planes, null if some planes are parallel
• #### wholeHyperplane
public SubPlane wholeHyperplane()
Build a region covering the whole hyperplane.
Specified by:
wholeHyperplane in interface Hyperplane<Euclidean3D>
Returns:
a region covering the whole hyperplane
• #### wholeSpace
public PolyhedronsSet wholeSpace()
Build a region covering the whole space.
Specified by:
wholeSpace in interface Hyperplane<Euclidean3D>
Returns:
a region containing the instance (really a PolyhedronsSet instance)
• #### contains
public boolean contains(Vector3D p)
Check if the instance contains a point.
Parameters:
p - point to check
Returns:
true if p belongs to the plane
• #### getOffset
public double getOffset(Plane plane)
Get the offset (oriented distance) of a parallel plane.
This method should be called only for parallel planes otherwise the result is not meaningful.
The offset is 0 if both planes are the same, it is positive if the plane is on the plus side of the instance and negative if it is on the minus side, according to its natural orientation.
Parameters:
plane - plane to check
Returns:
offset of the plane
• #### getOffset
public double getOffset(Vector<Euclidean3D> point)
Get the offset (oriented distance) of a point.
The offset is 0 if the point is on the underlying hyperplane, it is positive if the point is on one particular side of the hyperplane, and it is negative if the point is on the other side, according to the hyperplane natural orientation.
Specified by:
getOffset in interface Hyperplane<Euclidean3D>
Parameters:
point - point to check
Returns:
offset of the point
• #### sameOrientationAs
public boolean sameOrientationAs(Hyperplane<Euclidean3D> other)
Check if the instance has the same orientation as another hyperplane.
Specified by:
sameOrientationAs in interface Hyperplane<Euclidean3D>
Parameters:
other - other hyperplane to check against the instance
Returns:
true if the instance and the other hyperplane have the same orientation
|
2014-03-08 02:42:58
|
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|
https://www.physicsforums.com/threads/solution-to-pdes-via-fourier-transform.869327/
|
# I Solution to PDEs via Fourier transform
Tags:
1. Apr 28, 2016
### Frank Castle
Suppose a PDE for a function of that depends on position, $\mathbf{x}$ and time, $t$, for example the wave equation $$\nabla^{2}u(\mathbf{x},t)=\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)$$ If I wanted to solve such an equation via a Fourier transform, can I Fourier transform with respect to $\mathbf{x}$, but not $t$? That is, can I assume an ansatz of the form $$u(\mathbf{x},t)=\int\frac{d^{3}k}{(2\pi)^{3}}\tilde{u}(\mathbf{k},t)e^{i\mathbf{k}\cdot\mathbf{x}}$$
2. Apr 28, 2016
### blue_leaf77
Yes, you can. In fact, a function of the form $$f(\mathbf{x},t) = Ae^{i(\mathbf{k}\cdot \mathbf{x} - \omega t)}$$ is a fundamental mode (subject to some dispersion relation) of that differential equation.
3. Apr 28, 2016
### Frank Castle
So one can do this kind of "partial" Fourier decomposition in general then? Intuitively, are we making a Fourier decomposition of the function at a particular (fixed) instant in time, and then requiring that the decomposition should retain its form for all $t$ hence arriving at a differential equation (in time) that the Fourier coefficient functions must satisfy in order for this to hold?!
4. Apr 28, 2016
### blue_leaf77
Yes.
So long as the beam propagates in a space free of any obstacles (e.g. interfaces, aperture, etc), the shape of the spatial frequency spectrum should remain constant.
5. Apr 28, 2016
### Frank Castle
But is one assumes that one can express the solution in terms of a Fourier decomposition $$u(\mathbf{x},t)=\int\frac{d^{3}k}{(2\pi)^{3}}\tilde{u}(\mathbf{k},t)e^{i\mathbf{k}\cdot\mathbf{x}}$$ then doesn't it follow that (using the wave equation example) $$\nabla^{2}u(\mathbf{x},t)-\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)=\int\frac{d^{3}k}{(2\pi)^{3}}\left[(-\mathbf{k}^{2})\tilde{u}(\mathbf{k},t)-\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}\tilde{u}(\mathbf{k},t)\right]e^{i\mathbf{k}\cdot\mathbf{x}}=0 \\ \Rightarrow \frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}\tilde{u}(\mathbf{k},t)+\mathbf{k}^{2}\tilde{u}(\mathbf{k},t)=0$$ such that one finds a differential equation that the functions $\tilde{u}(\mathbf{k},t)$ must satisfy in order for this Fourier decomposition to be valid for all times $t$?!
6. Apr 28, 2016
### blue_leaf77
I understand your derivation but I cannot get what you want to say with those maths.
7. Apr 28, 2016
### Frank Castle
My understanding is that this is the differential equation that the Fourier coefficient functions must satisfy in order for the Fourier decomposition to be valid at times $t$ other than the fixed instant in time that we made the Fourier decomposition. That is, we wish to be able to express the solution to $$\nabla^{2}u(\mathbf{x},t)-\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)=0$$ as $$u(\mathbf{x},t)=\int\frac{d^{3}k}{(2\pi)^{3}}\tilde{u}(\mathbf{k},t)e^{i\mathbf{k}\cdot\mathbf{x}}$$ at every instant in time $t$, and hence we find a (self-consistent) differential equation (in time) for the Fourier coefficients $\tilde{u}(\mathbf{k},t)$ of the expansion, given by $$\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}\tilde{u}(\mathbf{k},t)+\mathbf{k}^{2}\tilde{u}(\mathbf{k},t)=0$$
8. Apr 28, 2016
### blue_leaf77
Are you looking for the solution of
?
The solution takes the form
$$\tilde{u}(\mathbf{k},t) = \tilde{u}(\mathbf{k},0) e^{-i\omega t}$$
with $\omega = kv$.
9. Apr 28, 2016
### Frank Castle
No, I understand that the solution takes that form. What I'm really trying to check is that I've understood intuitively what's going on, i.e. why we are "allowed" to Fourier decompose in this fashion (with respect to $\mathbf{x}$ only)?!
10. Apr 28, 2016
### blue_leaf77
The most general form of an electromagnetic radiation can be written to be
$$u(\mathbf{x},t)=\int \int \frac{d^{3}k}{(2\pi)^{3}} d\omega \, \tilde{u}(\mathbf{k},\omega)e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}$$
At this point, I would like to point out that there is certain theorem in calculus which governs whether one can interchange the order of an integration involving more than one variables. Unfortunately I don't remember how this theorem called, but I think for most practical purpose of a localized, square-integrable spectra you can do the integral over $\omega$ first
$$u(\mathbf{x},t)=\int \frac{d^{3}k}{(2\pi)^{3}} e^{i\mathbf{k}\cdot\mathbf{x}} \left( \int d\omega \, \tilde{u}(\mathbf{k},\omega)e^{- i\omega t} \right)$$
and you denote the integral inside the bracket as $\tilde{u}(\mathbf{k},t)$. So, it's allowed.
11. Apr 28, 2016
### Frank Castle
Ah ok, so if the wave has a constant phase $\omega$ then one can simply write the solution as I put it, however, if the phase changes then we have to use the more general expression that you put. Is this correct?
Also, what's the justification for why we don't Fourier transform the temporal part of $u$ also?
12. Apr 28, 2016
### blue_leaf77
$\omega$ is frequency. I don't understand what you are implying. The frequency, which you falsely called phase, do not change for a propagation in an non-absorptive medium.
Didn't I just show you that the temporal part (the integral inside the bracket in the last equation in post#10) can also be integrated, yielding a function of time $\tilde{u}(\mathbf{k},t)$?
13. Apr 28, 2016
### Frank Castle
Sorry, I meant frequency.
Sorry, yes you did. I wasn't being observant enough.
Sorry if this is an obvious question, but I've seen examples in several sets of notes where they just Fourier transform the spatial part, and then use the differential equation to find a differential equation for the Fourier coefficients $\tilde{u}(\mathbf{k},t)$. Why not just Fourier transform the whole thing (as you did) and assume this as an ansatz and solve the differential equation this way?
|
2018-05-25 11:17:40
|
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|
https://smp.uq.edu.au/event/session/14185
|
Speaker: Sundaram Thangavelu
Affiliation: Indian Institute of Science
## Abstract
The usual definition of Fourier transform on the Heisenberg $\mathbb{H}^n$ is in terms of the Schr\"odinger representations $\pi_\lambda$ which makes it operator valued and hence unwieldy and not suitable for studying several problems in harmonic analysis on $\mathbb{H}^n.$ In this talk, which is based on a recent article, we propose a scalar valued Fourier transform that shares several properties with the Helgason Fourier transform on noncompact rank one Riemannian symmetric spaces.
## References
S. Thangavelu, A scalar valued Fourier transform for the Heisenberg group arXiv:2206.00883
We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.
Seminars are usually held on Tuesdays from 3pm to 3.50pm.
Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.
#### Information for speakers
Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.
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2023-01-27 20:07:05
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http://www.askamathematician.com/
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## Q: How do I know my windmill is on straight?
The original question was:
I got quite the challenge from my father in law. The problem is well defined, but I’m having difficulties finding a meaningful answer. The reason why he asked me is because I’m an engineering student and he is in the windmill industry.
Before they attach the actual mill on the concrete foundation, it has to be absolutely leveled. If not, a tall mill would be quite offset even with a very small angle. To tackle this, they use two angle gauges and measure in two directions. The angle gauges are connected and you know the angle between them, their mutual angle. I’m supposed to find a way to convert these 3 inputs (angle 1, angle 2 and mutual angle) to find 2 outputs (the steepest angle and in which direction this is relative to the gauges).
Physicist: This is a gorgeous question that leads through some pretty math and ends with an elegant answer. If you’ve taken a class or two that used lots of vectors, then this is a cute exploration of what you can do with surprisingly little. If you’ve never taken a class or two that used lots of vectors, then please do: it’s fun stuff. You get to draw pictures and everything.
So you’ve got a flat slab that isn’t quite level. Two angle gauges (with plumb lines or bubbles or whatever) are placed on the slab in two directions. Define $\vec{a}$ and $\vec{b}$ as the directions of the two gauges on the ground and $\vec{u}$ as up. These may as well be unit vectors, so: they are.
An angle gauge or “inclinometer”.
Define the angle between $\vec{a}$ and $\vec{b}$ as $\phi$ and the angle between each each of the levels and $\vec{u}$ as $\theta_a$ and $\theta_b$.
Measuring the angles between these vectors means that we know sine and cosine of these angles, and knowing that means that we know the dot product and magnitude of the cross product, since $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos{(\phi)} = \cos{(\phi)}$ and $|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin{(\phi)} = \sin{(\phi)}$.
Finally, since the windmill will be built perpendicular to the slab, it will be built perpendicular to both $\vec{a}$ and $\vec{b}$. When a physicist (hell, even a mathematician) hears “I need a vector perpendicular to two other vectors” they convulsively respond “cross product those mothers”. If $\vec{w}$ is the windmill’s “up”, then $\vec{w} = \vec{a}\times\vec{b}$. If you were standing on the slab where the tails of $\vec{a}$ and $\vec{b}$ meet, then $\vec{a}$ would be on the right and $\vec{b}$ would be on the left (that’s the right hand rule).
A pristine windmill, slanted, and polluted with a mess of vectors.
If we project $\vec{u}$ onto the $\vec{a}$, $\vec{b}$ plane, the result will be pointing in the direction opposite the direction of the windmill’s lean. Define this projection as $\vec{p}$. The direction of $\vec{p}$ is the direction that the windmill needs to be “leaned” so that it will stand straight.
The questions (way back at the top of this page) now boil down to:
1) What is the angle between $\vec{w}$ and $\vec{u}$?
2) What is the angle between $\vec{p}$, and $\vec{a}$ and $\vec{b}$?
For #1, it turns out that the cross product is easier to work with. Define $\Omega$ as the angle between $\vec{w}$ and $\vec{u}$.
$\begin{array}{ll} |\vec{w}\times\vec{u}|^2 \\[2mm] = |(\vec{a}\times\vec{b})\times\vec{u}|^2 \\[2mm] = |(\vec{u}\cdot\vec{a})\vec{b} - (\vec{u}\cdot\vec{b})\vec{a}|^2 & \textrm{(nobody can remember this identity)} \\[2mm] = (\vec{u}\cdot\vec{a})^2|\vec{b}|^2 + (\vec{u}\cdot\vec{b})^2|\vec{a}|^2 - 2(\vec{u}\cdot\vec{a})(\vec{u}\cdot\vec{b})(\vec{a}\cdot\vec{b}) \\[2mm] = (\vec{u}\cdot\vec{a})^2 + (\vec{u}\cdot\vec{b})^2 - 2(\vec{u}\cdot\vec{a})(\vec{u}\cdot\vec{b})(\vec{a}\cdot\vec{b}) \\[2mm] = \cos^2{(\theta_a)} + \cos^2{(\theta_b)} - 2\cos{(\theta_a)}\cos{(\theta_b)}\cos{(\phi)} \\[2mm] \end{array}$
We also know that:
$\begin{array}{ll} |\vec{w}\times\vec{u}|^2 \\[2mm] = |\vec{w}|^2|\vec{u}|^2\sin^2{(\Omega)} \\[2mm] = |\vec{w}|^2\sin^2{(\Omega)} \\[2mm] = |\vec{a}\times\vec{b}|^2\sin^2{(\Omega)} \\[2mm] = |\vec{a}|^2|\vec{b}|^2\sin^2{(\phi)}\sin^2{(\Omega)} \\[2mm] = \sin^2{(\phi)}\sin^2{(\Omega)} \\[2mm] \end{array}$
And therefore:
$\sin^2{(\phi)}\sin^2{(\Omega)} = \cos^2{(\theta_a)} + \cos^2{(\theta_b)} - 2\cos{(\theta_a)}\cos{(\theta_b)}\cos{(\phi)}$
In the event that $\phi=90^o$ (and honestly, why wouldn’t you want your gauges perpendicular?), then this simplifies a lot:
$\sin^2{(\Omega)} = \cos^2{(\theta_a)} + \cos^2{(\theta_b)}$
For #2 we find the projection, $\vec{p}$, and dot it with $\vec{a}$ and $\vec{b}$. The projection onto the slab is $\vec{p} = \vec{u} - \frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\vec{w}$. That is; it’s the up direction minus whatever component points in the direction of the windmill.
$\begin{array}{ll} \vec{a}\cdot\vec{p} \\[2mm] = \vec{a}\cdot\left(\vec{u} - \frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}\right)\\[2mm]=\vec{a}\cdot\vec{u}-\frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{a}\cdot\vec{w}\\[2mm]=\vec{a}\cdot\vec{u} \end{array}$
In that last step you know that $\vec{a}\cdot\vec{w}=0$ since the tower, $\vec{w}$, and any direction on the slab it’s on, like $\vec{a}$, are perpendicular.
Defining $\omega_a$ as the angle between the projection and $\vec{a}$,
$\begin{array}{ll} (\vec{a}\cdot\vec{p})^2 = |\vec{a}|^2|\vec{p}|^2\cos^2{(\omega_a)} \\[2mm] \Rightarrow (\vec{a}\cdot\vec{u})^2 = |\vec{p}|^2\cos^2{(\omega_a)} \\[2mm] \Rightarrow \cos^2{\theta_a}=\left|\vec{u} - \frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\vec{w}\right|^2\cos^2{(\omega_a)} \\[2mm] \Rightarrow \cos^2{\theta_a}=\cos^2{(\omega_a)}\left(|\vec{u}|^2 + \left(\frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\right)^2|\vec{w}|^2 - 2\frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\vec{u}\cdot\vec{w}\right) \\[2mm] \Rightarrow \cos^2{\theta_a}=\cos^2{(\omega_a)}\left(1 + \frac{(\vec{u}\cdot\vec{w})^2}{|\vec{w}|^2} - 2\frac{(\vec{u}\cdot\vec{w})^2}{|\vec{w}|^2}\right) \\[2mm] \Rightarrow \cos^2{\theta_a}=\cos^2{(\omega_a)}\left(1-\frac{(\vec{u}\cdot\vec{w})^2}{|\vec{w}|^2}\right) \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\left(1 - \frac{|\vec{u}|^2|\vec{w}|^2}{|\vec{w}|^2}\cos^2{(\Omega)}\right) \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\left(1 - \cos^2{(\Omega)}\right) \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\sin^2{(\Omega)} \end{array}$
Again, in the event that $\phi = 90^o$, this simplifies:
$\begin{array}{ll} \cos^2{\theta_a} = \cos^2{(\omega_a)}\sin^2{(\Omega)} \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\left(\cos^2{(\theta_a)} + \cos^2{(\theta_b)}\right) \\[2mm] \Rightarrow \cos^2{\theta_a}\left(1-\cos^2{(\omega_a)}\right) = \cos^2{(\omega_a)}\cos^2{(\theta_b)} \\[2mm] \Rightarrow \cos^2{\theta_a}\sin^2{(\omega_a)} = \cos^2{(\omega_a)}\cos^2{(\theta_b)} \\[2mm] \Rightarrow \tan^2{(\omega_a)} = \frac{\cos^2{(\theta_b)} }{\cos^2{(\theta_a)}} \\[2mm] \Rightarrow \left|\tan{(\omega_a)}\right| = \left|\frac{\cos{(\theta_b)} }{\cos{(\theta_a)}}\right| \\[2mm] \end{array}$
Similarly, $\left|\tan{(\omega_b)}\right| = \left|\frac{\cos{(\theta_a)} }{\cos{(\theta_b)}}\right|$.
So, if you’ve got the inclinometer readings, $\theta_a$ and $\theta_b$, then you can find the lean of the tower, $\Omega$, and the direction you should push it so that it doesn’t lean, $\omega_a$ and $\omega_b$ from $\vec{a}$ and $\vec{b}$ respectively. This is a beautiful example of math leading to a cute, relatively simple solution that you probably couldn’t guess.
The windmill picture is from here.
Posted in -- By the Physicist, Engineering, Equations, Geometry, Math | 1 Comment
## Q: If all matter originated from a single point, does that mean all matter is entangled?
The original question was: All matter originated from a single point, does that mean all matter is entangled? If it is why do you need to create new entangled practices when doing experiments? Are there different types or degree’s of entanglement?
Physicist: There are absolutely different degrees of entanglement!
The degree you usually hear about are “maximally entangled states”, but basically everything is a little entangled. Not because of the big bang, but because every-day interactions generate and break a little entanglement all the time. Entanglement has a lot in common with correlation: if you know something about one thing, you’ll know something about the things it’s correlated with.
Correlations crop up all the time when things interact. For example, if you leave your car in a parking lot and come back to find a dent with a little red paint in it, then you know that somewhere nearby is a red car with another dent. The random things about your dent (the height above the ground, the severity, etc.) will be similar to those properties of the corresponding dent on the other car. You and a damnable ne’er-do-well have correlated cars because looking at the dent on one tells you something about the dent on the other; not because they have a spooky cosmic connection, but because they physically ran into each other. Entanglement is a little more subtle (what with all the quantum mechanics), but not a hell of a lot more subtle. Nothing fancy.
Just to be over-precise, when we say that things are entangled what we really mean is that some of their properties are entangled. For example, the polarization of two photons might be entangled while their positions are not, or vice versa.
The homogeneity of the universe (the “more-or-less-the-same-everywhere-ness” of the universe) is often cited as evidence that all the matter in the very early universe briefly had a chance to mix around, but that doesn’t have too much of an impact on entanglement. There’s something called “monogamy of entanglement” that says that maximally entangled qubits only appear in pairs, and maximally entangled states are the ones that really do interesting things. This can be generalized a bit to say “the more entangled two things are, the less they’re entangled with anything else”. Unfortunately, in order for such a pair to persist until today it would need to be left almost entirely unharrassed by everything else for billions of years. However, if the universe is anything, it’s old and messy. The entanglement we (people) create on purpose requires careful isolation and control of the stuff in question.
Even worse, if you have access to only one entangled particle, there’s no way to tell that it’s entangled. All of the fancy effects you hear about entanglement always require both, or at least most, of the entangled particles.
So you (every bit of you) can be entangled with other stuff in the universe (you kinda have to be). Entanglement is generated and broken by interactions, so you’re more entangled with stuff that’s nearby (in an astronomical sense). But most importantly, it doesn’t matter; random atomic-scale correlations are a lot like random atomic-scale noise.
Even less exciting, if you (personally) are the thing that’s entangled, your experience is entirely ordinary; the thing you’re entangled with will always be in a single state (from your point of view). All of the fancy experiments we do with entangled particle always involve particles being entangled with each other, because when they become entangled with the person doing the experiment it looks like “wave function collapse” (suddenly it appears to be in only one state) and that’s boring. Similarly, if you and a distant alien are entangled it does not mean you have a spooky connection (groovy, spiritual, or otherwise), it means that they will already be in a single state (from you mutual points of view) before you ever meet each other.
Which is exactly the sort of thing you’d never notice.
Posted in -- By the Physicist, Physics, Quantum Theory | 4 Comments
## Q: How good is the Enigma code system compared to today’s publicly available cryptography systems?
Physicist: Freaking terrible.
The Enigma machine used a “rolling substitution cypher” which means that it was essentially a (much more) complicated version of “A=1, B=2, C=3, …”. The problem with substitution cyphers is that if parts of several messages are the same then you can compare their similarities to break the code. Enigma was broken in part because of German formality (most messages started with the same formal greeting). Even worse, since some letters are more common than others (e.g., “e” and “g”) you can make progress by just counting up how often letters show up in the code (or even get an idea of what language the code is written in without breaking it!). Substitution cypher are so easy to break that some folk do it for funRolling substitution cyphers can use a set of several encoding schemes and cycle through which code is used or make the scheme dependent on the previous letter, but this merely makes the code breaking more difficult. Ultimately, all substitution cyphers suffer from the same difficulty: similar messages produce similar looking codes.
Enigma used three rotors which rotated after each letter was pressed allowing them to generate a huge number of different substitution cyphers, using a different one for each letter. Still: what your cellphone uses is much, much better.
Modern cryptography doesn’t have that problem. If any part of a message is different at all, then the entire resulting code is completely different from beginning to end. That is; if you encrypted a message, you’d get cypertext (the encoded message) and if you were to encrypt the exact same message but misspelled a single word, then the cypertext would be completely different.
If your messages were “Hello A”, “Hello B”, and “Hello C”, then a substitution cypher might produce “Tjvvw L”, “Tjvvw C”, and “Tjvvw S” while RSA (the most common modern encryption) might produce “idkrn7shd”, “62hmcpgue”, and “nchhd8pdq”. In the first case you can tell that the messages are nearly the same, but in the second you got nothing.
Enigma was very clever but is shockingly primitive compared to modern crypto techniques. If anyone in WW2 had been using modern (1970’s or later) encryption, then there is no way that anyone would have been able to break those codes (and Turing would have to settle for being famous for everything else he did).
There’s a post here that talks about the main ideas behind RSA encryption. The really fancy stuff is some of the only math that isn’t publicly known. Scientists have a whole thing about openness and the free exchange of information that governments and corporate entities (for whatever reason) don’t.
## Q: When “drawing straws” is it better to be first or last?
Physicist: As long as the person who cut the straws: 1) takes the last remaining straw and 2) has a decent poker face (or doesn’t know which is which), then it’s completely fair. If they have a bad poker face, then it’s better to be first.
If the person who cuts and holds the straws has a terrible poker face, then the first few people have an advantage.
The quickest way to see why is to imagine a slightly different way of drawing straws. Instead of drawing straws, draw cards where all but one are black (for example). Everyone takes a card and afterward everyone turns their card over; the one red card is the “short straw”. In this case it should make sense that no person is more or less likely to get the red card for the same reason that it’s no more or less likely for any particular card to be any particular place in a deck. The fact that when drawing straws we pull one at a time and generally stop halfway through (whenever the short straws appears) makes it fell like the situation is different, but it’s not.
Say you’ve got N peeps (people). The first person to draw a straw is the least likely to draw the short one (1/N) and the last to draw is the most likely (1/2). However! While the later people are more likely to draw the short straw, they’re also less likely to pull any straw since it’s more likely that the short straw has already been drawn. In movies they almost always draw every straw because of drama, but in practice, you draw until the short one shows up and then you stop.
The early people are least likely to draw the short straw while the later people are least likely to draw at all. If you write down the math you find that the effects balance out exactly. So here’s the math written down:
You’ve got N peeps named One, Two, Three, etc. (probably siblings).
The first person has N straws to choose from and their probability of getting the short one is $P=\frac{1}{N}$. Easy enough. The second person has N-1 straws to choose from, so you might expect that their chance of drawing the short straw is $P=\frac{1}{N-1}$. But that’s not the probability that counts. What counts is the probability of drawing the short straw given that it hasn’t been drawn already. That probability is $P=\left(\frac{N-1}{N}\right)\left(\frac{1}{N-1}\right)=\frac{1}{N}$. $\frac{N-1}{N}$ is the probability that the first person did not already draw the short straw.
By the time it’s the Jth person’s turn there are N-J+1 straws remaining. The probability that the short straw is among them (the probability that it hasn’t been drawn already) is $\frac{N-J+1}{N}$. And if it hasn’t, then the probability of drawing it is $\frac{1}{N-J+1}$. So, all in all, the probability of the Jth person drawing the short straw is $P=\left(\frac{N-J+1}{N}\right)\left(\frac{1}{N-J+1}\right)=\frac{1}{N}$.
Finally, the last person to draw is the person who cut the straws. This person’s choice is random because everyone else’s choices were random: knowing which straw is which doesn’t change that.
Posted in -- By the Physicist, Math, Probability | 4 Comments
## Q: What would happen if there was a giant straw connecting the Earth’s atmosphere right above the ground to space?
Physicist: About the same thing that happens to a straw in a glass of water: the water level in the straw evens out with the water level outside.
The pressure at the bottom of the straw “tells” the water in the straw how high to climb. That same pressure “tells” the rest of the water the exact same thing.
A tube from the ground to space would fill with air of about the same density and pressure as the air around the straw, decreasing as you go up until eventually you have a straw full of nothing surrounded by also nothing (in space).
What holds the atmosphere to the planet is gravity, so if a patch of air tries to drift off into space it literally falls back. A straw alone wouldn’t change that. On the other hand, if you attached some kind of pump to the bottom of the straw to make it have a higher pressure than sea-level, then you could pump air up the straw and have some kind of massive space-fountain of air (the air coming out would fall back to Earth just like water in an ordinary fountain). In fact! There is a situation very close to that happening on Saturn’s moon, Enceladus.
The water-vapor geysers of Enceladus shoot directly into space. Most of it falls back onto the ground, but a tiny amount ends up orbiting Saturn and contributing to one of its rings.
Whenever air or water or whatever travels up a straw it’s being pushed by pressure from the bottom (there’s no such thing as sucking), and one atmosphere of pressure can only push so far. For something like liquid mercury that’s about 76cm, which is why the “1 atmosphere” of pressure is often expressed as “760mm of Mercury”. If a closed tube is taller than that, then the pressure (here on Earth) isn’t great enough to push the mercury to the top which leaves nothing at the top.
So that’s mercury.
Same idea with air. If you have a long tube full of air with the top open to space and the bottom pressurized to one atmosphere (or 760mm Hg), then the column of air in the tube will be as tall as the atmosphere.
A straw doesn’t provide an “escape route”; our air is free to try to leave whenever. The atmosphere stays where it is because it’s made of mass and the Earth has gravity. It’s a little sobering to realize that there’s nothing between you and a profound nothing (space) but a thin layer of air held down by its own unimpressive weight.
The barometer picture is from here.
Posted in -- By the Physicist, Physics | 2 Comments
## Q: Can a human being survive in the fourth dimension?
Physicist: Nopers. But to understand why, it’s important to know what a dimension is.
When someone says “we live in the third dimension” what they should really say (to be overly-precise) is “the universe we inhabit has three spacial dimensions”. There are a few ways that you can tell that you live in a three dimensional world. The easiest is to try to come up with as many mutually-perpendicular directions as you can; you’ll find three without too much trouble, but you’ll never find a fourth.
These three directions are mutually perpendicular and and no new direction can be perpendicular to all three.
If you’re feeling terribly clever, you’ll find lots of other examples that demonstrate the three (and not two or four) dimensionality of our universe. For example, if you can tie a simple knot then you definitely live in three or more dimensions (no knots in 2-D) and if you can make a Klein bottle then you definitely live in four or more dimensions.
In 2-D you can’t tie a knot without the rope passing through itself, and in 3-D you can’t build a Klein bottle without the same problem.
A dimension is a direction. Living in more dimensions means having more directions you can move in. There are many weird physical consequences to living in more dimensions, but the one you’d notice first (if you were somehow to suddenly to appear in a 4-D universe) is immediate death.
An actual 2-D creature would collapse in 3-D. What it considers to be its insides just looks like more surface to we 3-D folk.
If a paper doll (two-dimensional being) were suddenly brought into three dimensional space all of its innards would become outtards. Similarly, there is nothing whatsoever supporting your body in a fourth direction, so if you were to find yourself with a few extra dimensions your insides would follow the path of least (zero) resistance and fall out. It would be super gross, but would make no more of a mess than an infinitely thin oil slick. Any local 4-D critters probably wouldn’t even notice.
Posted in -- By the Physicist, Math, Paranoia, Physics | 20 Comments
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2015-01-29 10:17:34
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https://calculus7.org/2018/05/07/the-sum-of-pairwise-distances-and-the-square-of-cdf/
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# The sum of pairwise distances and the square of CDF
Suppose we have ${n}$ real numbers ${x_0,\dots, x_{n-1}}$ and want to find the sum of all distances ${|x_j-x_k|}$ over ${j < k}$. Why? Maybe because over five years ago, the gradient flow of this quantity was used for "clustering by collision" (part 1, part 2, part 3).
If I have a Python console open, the problem appears to be solved with one line:
>>> 0.5 * np.abs(np.subtract.outer(x, x)).sum()
where the outer difference of x with x creates a matrix of all differences ${x_i-x_j}$, then absolute values are taken, and then they are all added up. Double-counted, hence the factor of 0.5.
But trying this with, say, one million numbers is not likely to work. If each number takes 8 bytes of memory (64 bits, double precision), then the array x is still pretty small (under 8 MB) but a million-by-million matrix will require over 7 terabytes, and I won’t have that kind of RAM anytime soon.
In principle, one could run a loop adding these values, or store the matrix on a hard drive. Both are going to take forever.
There is a much better way, though. First, sort the numbers in nondecreasing order; this does not require much time or memory (compared to quadratic memory cost of forming a matrix). Then consider the partial sums ${s_k = x_0+\dots+x_k}$; the cost of computing them is linear in time and memory. For each fixed ${k}$, the sum of distances to ${x_j}$ with ${j is simply ${kx_k - s_{k-1}}$, or, equivalently, ${(k+1)x_k - s_k}$. So, all we have to do is add these up. Still one line of code (after sorting), but a much faster one:
>>> x.sort()
>>> (np.arange(1, n+1)*x - np.cumsum(x)).sum()
For example, x could be a sample from some continuous distribution. Assuming the distribution has a mean (i.e., is not too heavy tailed), the sum of all pairwise distances grows quadratically with n, and its average approaches a finite limit. For the uniform distribution on [0, 1] the computation shows this limit is 1/3. For the standard normal distribution it is 1.128… which is not as recognizable a number.
As ${n\to \infty}$, the average distance of a sample taken from a distribution converges to the expected value of |X-Y| where X, Y are two independent variables with that distribution. Let’s express this in terms of the probability density function ${p}$ and the cumulative distribution function ${\Phi}$. By symmetry, we can integrate over ${x> y}$ and double the result:
${\displaystyle \frac12 E|X-Y| = \int_{-\infty}^\infty p(x)\,dx \int_{-\infty}^x (x-y) p(y)\,dy}$
Integrate by parts in the second integral: ${p(y) = \Phi'(y)}$, and the boundary terms are zero.
${\displaystyle \frac12 E|X-Y| = \int_{-\infty}^\infty p(x)\,dx \int_{-\infty}^x \Phi(y)\,dy}$
Integrate by parts in the other integral, throwing the derivative onto the indefinite integral and thus eliminating it. There is a boundary term this time.
${\displaystyle \frac12 E|X-Y| = \Phi(\infty) \int_{-\infty}^\infty \Phi(y)\,dy - \int_{-\infty}^\infty \Phi(x)^2\,dx}$
Since ${\Phi(\infty) = 1}$, this simplifies nicely:
${\displaystyle \frac12 E|X-Y| = \int_{-\infty}^\infty \Phi(x) (1-\Phi(x))\,dx}$
This is a lot neater than I expected: ${E|X-Y|}$ is simply the integral of ${2\Phi(1-\Phi)}$. I don’t often see CDF squared, like here. Some examples: for the uniform distribution on [0,1] we get
${\displaystyle E|X-Y| = \int_0^1 2x(1-x)\,dx = \frac13}$
and for the standard normal, with ${\Phi(x) = (1+\mathrm{erf}\,(x/\sqrt{2}))/2}$, it is
${\displaystyle \int_{-\infty}^\infty \frac12 \left(1-\mathrm{erf}\,(x/\sqrt{2}) ^2 \right)\,dx = \frac{2}{\sqrt{\pi}}\approx 1.12838\ldots }$
The trick with sorting and cumulative sums can also be used to find, for every point ${x_k}$, the sum (or average) of distances to all other points. To do this, we don’t sum over ${k}$ but must also add ${|x_j-x_k|}$ for ${j>k}$. The latter sum is simply ${S - s_k - (n-k-1)x_k}$ where ${S}$ is the total sum. So, all we need is
>>> (2*np.arange(1,n+1)-n)*x - 2*np.cumsum(x) + x.sum()
Unfortunately, the analogous problems for vector-valued sequences are not as easy. If the Manhattan metric is used, we can do the computations for each coordinate separately, and add the results. For the Euclidean metric…
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2019-12-12 12:27:21
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https://stats.stackexchange.com/questions/105149/how-to-recover-linear-probability-model-for-binary-classification
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# How to recover linear probability model for binary classification?
In some binary classification problem I assume that the probability for a positive is exactly linearly dependent on the features $P(y=1|x_1,x_2,x_3,\ldots)=\beta_1x_1+\beta_2x_2+\beta_3x_3+\cdots$. Now for the training I only observe the realization 0 or 1 for $y$. Therefore the probability isn't directly observable, but would rather be some local density (which is hard to estimate for high dimensions).
Is there some type of regression that can determine the coefficients $\beta_i$?
I could try normal regression, but it doesn't seem obvious that any loss function will be able to converge to the same $\beta_i$. What type of regularization and loss function would be most appropriate?
• The specification as you have proposed it is not advisable.. I recommend you look into a generalized linear model (GLM), specifically a logit or probit model. – user44764 Jun 28 '14 at 21:44
• I can look at different models, but I still wouldn't be sure if they can reconstruct the underlying coefficients. Are there theoretical results? Which approach would recover the model? – Gerenuk Jun 28 '14 at 21:50
• GLMs are well-studied. I refer you to Murphy's Probabilistic Machine Learning Chapter 8 for a thorough rundown of logistic regression. – user44764 Jun 28 '14 at 22:03
• What do you mean "converge to the same $\beta$"? As in, converge asymptotically as your sample size grows? Also, yes, a binary/Bernoulli GLM is the go-to parametric approach in this case. A GLM is a consistent estimator for the conditional mean of $y$, which in a Bernoulli distribution is the same as $P(y=1 | \vec{x} )$, so if you use the identity link that's literally what you asked for. You can estimate it with OLS ("regular regression") or maximum likelihood if you want but it's the same model. – shadowtalker Jun 29 '14 at 2:17
• So least squares or just any loss function will always give me the correct model? Certainly one of these loss functions converges faster? – Gerenuk Jun 29 '14 at 16:09
This is the standard Linear Probability Model in the context of Binary Response models. To obtain it, one assumes that there exist an underlying latent/unobservable variable $y^*$, which is linearly dependent on the regressors,
$$y^*=\beta_1x_1+\beta_2x_2+\beta_3x_3+\cdots + u$$
and whose indicator function is the observed $y$, indicating that the latent variable exceeds (or falls below) some threshold (usually zero). In order to arrive at the linear specification for the probability, we have to assume that the error term $u$ in the underlying regression is uniformly distributed, conditional on the regressors.
See this answer for the exact derivation. The model can of course be estimated by least-squares, but has various drawbacks, so frequently a "probit" or "logit" model is used (the first assuming that the underlying error is normally distributed, the second assuming that the underlying error term follows the logistic distribution). These models are usually estimated by maximum likelihood.
After some discussion with the OP and clarification in the comments, it is clear that the OP's framework is represented by the linear probability model where the "error term" is the random draws the OP performs from a $U(0,1)$. Specifically we can postulate the unobservable regression $$y^*_i = -\sum_{j=1}^k\beta_jx_{ji} + u_i,\qquad u_i \sim U(0,1)$$
and the OP determines the values of the observable $y$ by
$$y_i = 1,\;\; \text{if} \;\;u_i \leq \sum_{j=1}^k\beta_jx_{ji} \Rightarrow - \sum_{j=1}^k\beta_jx_{ji} +u_i \leq 0 \Rightarrow y^*_i \leq 0$$
So we have
$$P(y_i = 1 \mid \mathbf X) = P\left(- \sum_{j=1}^k\beta_jx_{ji} +u_i\leq 0\right) =P\left(u_i\leq \sum_{j=1}^k\beta_jx_{ji} \right)$$
But this is the expression for the cumulative distribution function of $u_i$, which is $F_U(z) = P\left(u_i\leq z\right) = z$ for a $U(0,1)$.
In this way we arrive at the equation
$$P(y_i = 1 \mid \mathbf X) = \sum_{j=1}^k\beta_jx_{ji}$$ which is the Linear Probability Model, on firm stochastic ground. Note that now the "error term" is no longer "present" -but it has determined the structure of the equation, through its stochastic properties.
Estimating the $\beta$'s from this last equation by Least Squares does not require specifying some additional "error term" -it has now become more of a mathematical approximation endeavor, rather than a stochastic regression. Another way of course would be to estimate the model by maximum likelihood. Here the distributional specification on $y$ is "safe": it is by construction a Bernoulli random variable, whose conditional parameter $p$ is determined by $\sum_{j=1}^k\beta_jx_{ji}$. We can write the Bernouli probability mass function as
$$f_Y(y_i\mid \mathbf X) = p_i^{y_i}(1-p_i)^{1-y_i} = \left(\sum_{j=1}^k\beta_jx_{ji}\right)^{y_i}\left(1-\sum_{j=_1}^k\beta_jx_{ji}\right)^{1-y_i}$$
from which we can proceed to form the log-likelihood of the sample and perform maximum likelihood.
Estimator properties and other related issues to the estimation are easily accessible.
• +1 For giving the answer the OP needs, but I think he's more interested in guarantees/proofs that the coefficients are stable and that some kind of meaningful loss will function will allow him to recover the same coefficients.. or at least that's my interpretation of the question. – user44764 Jun 29 '14 at 2:07
• How come I need a new error term $u$ and some distribution for it? I'm not too familiar with all terminology, but from my understanding I imagine simulating it by generating latent probabilities by a linear model and then using these probabilities to binarize them to 0 or 1. There is no place for an error term. Or equivalently the error term is $P(u=-p)=1-p$, $P(u=1-p)=p$ where $p=\sum \beta_ix_i$. I have the impression that this is different from your model? – Gerenuk Jun 29 '14 at 12:37
• @Gerenuk You start by a conditional probability on the left-hand-side. Probabilities are expressed by distribution functions. You condition on the $x$'s, so the conditional probability/distribution of $Y$ cannot involve the distributions of the $x$'s. But it must come from somewhere. There must be something else that is random, otherwise this could not be a probability. And this something else is the underlying error term. You write "generate latent probabilities". How are you going to generate them without assuming a distribution? And if you assume a distribution, whose distribution is it? – Alecos Papadopoulos Jun 29 '14 at 19:14
• Maybe I'm not using the right statistics terms, so I try to explain what a computer program would do. The generator selects some $\beta_i$ and calculates $p(x)=\sum \beta_ix_i$. For each $x$ it will use $y=1$ with that probability $p(x)$ (I run take a uniform random number $[0;1]$ and use $p(x)$ as a threshold for $y=1$). Now my second model fitting program is expected to take all $y(x)$ as input and yield an good estimates of $\beta_i$ given a limited amount of data. – Gerenuk Jun 30 '14 at 6:19
• You mean that you draw random numbers from a uniform $U(0,1)$ (denote one of these random numbers $u$), and if $u$ is smaller or equal to $\sum\beta_ix_i$ you assign the value $1$ to $y$, and you set $y=0$ otherwise? – Alecos Papadopoulos Jun 30 '14 at 9:08
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2019-11-13 04:06:43
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https://xmphysics.com/2023/01/02/7-1-3-geostationary-and-other-orbits/
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# 7.1.3 Geostationary and Other Orbits
Is it better to park a satellite at high or low altitude? Well, they both have their own pros and cons.
Low altitude
• Can take higher resolution images of Earth.
• Transmits and receives signals/messages with shorter delay.
• Requires less fuel and cheaper to launch into orbit.
High altitude
• Wider coverage because can maintain direct line of sight with a large portion of Earth’s surface at any one time
• Suffers less atmospheric drag and requires less frequent orbital boosts.
Low Earth Orbits (LEO)
The majority of satellites are LEO satellites. At altitudes of only a few hundred kilometres, they have orbital periods of about 90 minutes. Famous satellites in the LEO orbits include the International Space Station (ISS) and the Hubble Telescope.
Middle Earth Orbits (MEO)
A famous occupant of the MEO space is The Global Positioning System (GPS), a constellation of >24 satellites parked at altitude of about 20,000 km. From this higher altitude, each satellite has a larger coverage area. They have orbital periods of 12 hours.
Geostationary Orbits (GEO)
While a satellite is going around the Earth, the Earth itself is also rotating. If the satellite were to “rotate” at the same rate as the Earth about the same rotational axis, it would appear stationary in the sky (to somebody on Earth).
To achieve this, the satellite must
• have an orbital period of 24 hrs,
• orbit in a west to east direction, and
• be in the equatorial plane.
Such an orbit, called the geostationary orbit, is found at an altitude of 36,000 km.
Example
Calculate the altitude of the geostationary orbit.
Earth’s radius $=6370\text{ km}$
Earth’s mass $=5.97\times {{10}^{{24}}}\text{ kg}$
Solution
GEO satellites have orbital period of 24 hours.
$\displaystyle \omega =\frac{{2\pi }}{T}=\frac{{2\pi }}{{24\times 60\times 60}}=7.27\times {{10}^{{-5}}}\text{ rad }{{\text{s}}^{{-1}}}$
The gravitational pull provides the required centripetal force for circular motion.
\displaystyle \begin{aligned}\frac{{GMm}}{{{{r}^{2}}}}&=mr{{\omega }^{2}}\\GM&={{r}^{3}}{{\omega }^{2}}\\(6.67\times {{10}^{{-11}}})(5.97\times {{10}^{{24}}})&={{r}^{3}}{{(7.27\times {{10}^{{-5}}})}^{2}}\\r&=42235\text{ km}\end{aligned}
The altitude $=42235-6370=35900\text{ km}$
Since the GEO orbit is at a very high altitude, launching of GEO satellites requires very powerful and expensive rockets. Nevertheless, it is the orbit of choice for communications satellites.
• Geostationary satellites stay at a fixed point above the Earth, which is convenient for communication satellites because they appear stationary when viewed from Earth (if not ground antenna will have to continuously track the satellites as they move across the sky)
• Unfortunately, being constrained to the equatorial plane also means geostationary satellites cannot serve the polar regions (because from the polar regions they are below the horizon and cannot be sighted.)
Polar Orbit
A polar orbit is one that carries the satellite above or nearly above the poles. It has an inclination of 60 to 90 degrees to the equator. The main attraction of polar orbits is that they are able to “raster scan” the entire earth’s surface, including the polar regions. They are used for Earth-mapping, reconnaissance, meteorology and telecommunication.
Demonstration
LEO Orbit (ISS)
MEO Orbit (GPS)
GEO Orbit (Electro-L)
Polar Orbit (EUMETSAT)
Video Explanation
Why must Geostationary Orbit be Equatorial?
Why must GEO Satellites be Parked at 36,000 km?
How to calculate the GEO Altitude?
Interesting
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2023-02-05 13:32:53
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https://math.stackexchange.com/questions/1434365/should-an-undirected-graph-and-a-directed-graph-with-the-same-vertices-and-edges
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# Should an Undirected Graph and a Directed Graph with the same vertices and edges have the same number of connected components?
I wrote a Java program that builds a graph from an input txt file. The file just has a list of integer pairs constituting an edge between the two vertices (indicated by the two integers). In the Undirected implementation both vertices are marked as having an edge to each other, while in the Directed implementation the first is marked as having an edge to the second.
Everything else seems to line up, but when I calculate the number of connected components using the same input file, but the two implementations, I am getting two results for the number of connected components (specifically the Directed Graph always comes up with more). Intuitively this doesn't make sense to me, since a connected component is just a grouping of connected vertices, it shouldn't matter if those connections are directed or not.
$$\bullet\longleftrightarrow\bullet\longleftrightarrow\bullet$$
$$\bullet\longleftarrow\bullet\longleftarrow\bullet$$
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2019-07-20 18:50:22
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https://math.stackexchange.com/questions/833357/alternative-proof-of-infinitely-many-primes?noredirect=1
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# Alternative Proof of Infinitely Many Primes? [duplicate]
I've seen Euclid's proof of infinitely many primes, what are other approaches to proving there are infinitely many primes?
• I googled "infinitely many primes" and found this Jun 13, 2014 at 22:04
• @BalarkaSen Is that a product over primes? How did the summation turn into that? Jun 13, 2014 at 22:08
• $$\sum_{n=1}^\infty \frac1n = \prod_p \left ( 1 - \frac1p \right )^{-1}$$ The harmonic series diverges. $$\sum_{n=1}^\infty \frac1{n^2} = \pi^2/6 = \prod_p \left ( 1 - \frac1{p^2} \right)^{-1}$$ $\pi^2$ is irrational. Plus, if you forgive me the cheek $$\sum_{n=1}^\infty \frac{1}{n^3} = \prod_p \left ( 1 - \frac1{p^3}\right)^{-1}$$ $\zeta(3)$ is irrational too. Jun 13, 2014 at 22:09
• @user45572 See Euler product Jun 13, 2014 at 22:09
• @BalarkaSen I know pi is irrational, why does that imply you can write it as a product over primes? Jun 13, 2014 at 22:10
Here's another proof.I thought of it a while ago, and posted it as an answer to a mathoverflow question, but I don't imagine I was the first to think of it. The idea is not to rely on the fact that $\pi$ is irrational. Let $\mathcal{P}$ denote the set of primes. We have $\sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6}$ and $\sum_{n=1}^{\infty} \frac{1}{n^{4}} = \frac{\pi^{4}}{90}.$ The first equation yields, $\prod_{p \in \mathcal{P}}\frac{p^{2}}{p^{2}-1} = \frac{\pi^{2}}{6}$ and the second leads to $\prod_{p \in \mathcal{P}}\frac{p^{4}}{p^{4}-1} = \frac{\pi^{4}}{90}.$ If we square the first of the last two equations and divide by the second, we obtain $\prod_{p \in \mathcal{P}}\frac{p^{2}+1}{p^{2}-1} = \frac{5}{2}.$ If $\mathcal{P}$ was finite, the left hand expression would be a rational number whose numerator was not divisible by $3$, but whose denominator was divisible by $3$ (because $p^{2}+1 \equiv 2$ (mod $3$) for every prime other than $3$, and $p^{2}-1$ is divisible by $3$ for every prime other than $3$). However, no such rational number could be equal to $\frac{5}{2},$ a contradiction.
• (+1)This is one fun proof! Interestingly, irrationality of $\pi$ gives much stronger result than infinitude of primes $$\frac{\pi}{4} = \sum_{n= 0}^\infty \frac{(-1)^n}{2n+1} = \prod_{p \equiv 1(4)} \frac{p}{p-1} \cdot \prod_{p \equiv 3(4)} \frac{p}{p+1}$$ Thus at least one of the class $1\pmod{4}$ or $3\pmod{4}$ contains infinitely many primes. Jun 17, 2014 at 9:52
$$\prod_k \frac 1 {1-p_k^{-2}}=\sum_n n^{-2}=\zeta(2)=\frac {\pi^2}6,$$ and $\pi^2$ is irrational (from my favorite math resource) so we need infinitely many primes...
The prime harmonic series $\frac 1 2 + \frac 1 3 + \frac 1 5 + \ldots$ diverges.
• Isn't the fact that it diverges dependent on there being infinitely many primes in the first place? How else could the discussion of divergence even occur otherwise? Jan 10, 2017 at 3:12
• @Axoren: This is done formally by considering the sum of the reciprocals of all primes less than n for each n. These sums define a sequence (which would eventually be constant if there were finitely many primes), and you can show that the n-th sum is bounded below by $\log\log n - (\text{constant})$ where the constant is independent of n. Hence these sums tend to infinity (and, by definition, the corresponding series diverges).
– Dan
Jan 10, 2017 at 4:11
Here is one of my favorites: $f(n) = 2^{2^{n}}+2^{2^{n-1}}+1, n \in \mathbb{N}$ has $n$ different prime factors. Prove by induction.
• I think $2^{2^{n}}-1$ has the same property ( ie, at least $n$ different prime factors). Jun 13, 2014 at 23:02
There is a topological proof: http://en.wikipedia.org/wiki/Furstenberg%27s_proof_of_the_infinitude_of_primes
Apart from that there are almost infinitely many proofs..
• Could you explain the topological proof? I find the Wiki article difficult to parse... Jun 13, 2014 at 22:06
• @user45572 Explanation: Declare each arithmetic sequence to be "closed", and extend this terminology to finite unions. Verify that the complement of a finite set is not closed. Verify that the complement of $\{\pm 1\}$ is the union of arithmetic sequences whose ratios are exactly the prime numbers (by prime factorization). Since it's cofinite, this must be an infinite union, QED. Jun 13, 2014 at 22:18
• Note that it's not really a topological proof at all. Jun 13, 2014 at 22:23
I like Dirichlet's theorem on primes in arithmetic progressions. Shows not only infinitely many primes, but infinitely many primes in each (nontrivial) residue class in any modulus. Proof relies on analytic properties of Dirichlet series.
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2022-05-19 11:06:25
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https://documentation.aimms.com/language-reference/non-procedural-language-components/execution-of-nonprocedural-components/dependency-structure-of-definitions.html
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# Dependency Structure of Definitions
Dependency graph
The definitions inside the declarations of global sets and parameters together form a system of interrelated functional relationships. AIMMS automatically determines the dependency between the defined identifiers and the inputs that are used inside these relationships. Such dependencies can be depicted in the form of a directed graph, called the dependency graph. From this dependency graph, AIMMS determines the minimal set of identifiers that must be recomputed-and in which order-to get the total system of functional relationships up-to-date.
Example
Consider the system of definitions
\begin{split}\begin{align} d_1 & \equiv e_1 + e_2 \\ d_2 & \equiv d_1 + d_3 \\ d_3 & \equiv e_2 + e_3 \\ d_4 & \equiv e_1 + d_2. \end{align}\end{split}
Its dependency graph, with identifiers as nodes and dependencies as directed arcs, looks as follows.
Note that a change to the input parameter $$e_3$$, for instance, requires the re-computation of the defined parameters $$d_2,\dots,d_4$$-but not of $$d_1$$-to update the entire system.
Dependencies must be a-cyclic
The dependency graph associated with the set and parameter definitions must be a-cyclic, i.e. must not contain circular references. In this case, every change to one or more input parameters of defined sets or parameters will result in a finite sequence of assignments to update the system. If the dependency graph is cyclic, a simultaneous system of relations will result. Such a system may not have a (unique) solution, and can only be solved by a specialized solver. Simultaneous systems of relations are handled inside AIMMS through the use of constraints and mathematical programs.
Example
An illegal set of dependencies results if the definition of $$d_1$$ in the last example is changed as follows.
$d_1 \equiv d_4 + e_1 + e_2.$
This results in the following cyclic dependency graph.
Now, a change to any of the input parameters $$e_1,\dots,e_3$$ will result in a simultaneous system for the parameters $$d_1$$, $$d_2$$ and $$d_4$$.
AIMMS will check
AIMMS computes the dependency structure between the parameter and set definitions while compiling your model. If AIMMS detects a cyclic dependency, an error will result, because AIMMS can, in general, not deal with cyclic dependencies without relying on specialized numerical solvers. In that case you need to remove the cyclic dependencies before you can execute the model without further modifications. If you are unable to remove the cyclic dependencies, you have essentially two alternatives. You can either formulate a mathematical program, or define your own solution method inside a procedure.
Variables for simultaneous systems
The cyclic system can be turned into a mathematical program by changing the parameters with cyclic definitions into variables. This results in a simultaneous system of equalities which can be solved through a SOLVE statement. The declaration of mathematical programs is discussed in Solving Mathematical Programs.
Feedback loops
The alternative is to implement a customized solution procedure by breaking the simultaneous system into a simulation with a feedback loop linking inputs and outputs. To accomplish this, you must first remove the cyclic definitions from the declarations, and then add a procedure that implements the feedback loop. If you have sufficient knowledge of the process you are describing, this route may result in fast convergence behavior.
Dependency is global only
AIMMS only allows a definition for globally declared sets and parameters. Consequently, a single global dependency graph suffices to express the functional relationships between all defined sets and parameters.
Dependency is symbolic
In addition, the dependency structure between set and parameter definitions is purely based on symbol references. As a result, AIMMS’ automatic evaluation scheme will always recompute an indexed (output) parameter depending on an indexed (input) parameter in its entirety, even when only a single input value has changed.
Inefficiency may occur
This evaluation behavior may lead to severe inefficiencies when you use a high-dimensional defined parameter that is re-evaluated repeatedly during the execution of a loop in your model. In such cases it is advisable to refrain from using a definition for such a parameter, but replace it by one or more assignments at the appropriate places in your model. This issue is discussed in full detail in Dependency is symbolic.
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2023-03-25 19:56:09
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|
http://docs.itascacg.com/flac3d700/flac3d/zone/test3d/ExampleApplications/Pillar/pillar.html
|
Examples • Example Applications
Problem Statement
Note
View this project in FLAC3D:
1. Help ► Examples…
2. Pick “ExampleApplications”
3. Pick “Pillar”
The project’s main data file is shown at the end of this example.
The mine layout shown in Figure 1 is representative of the production level in a mass-caving mining operation. The roadways are usually developed ahead of the mining and are initially subjected to the in-situ stress state. As mining progresses, the pillars are submitted to an increased vertical load.
The purpose of this example is to first study the initial response of the excavation (under in-situ stresses), and to then establish the peak load the pillars can carry. Also to be determined is how much stronger this pillar is than a square pillar of the same width and height.
Figure 1: Mine layout.
break
The initial stress state considered is
$$σ_{xx}$$ = 25 MPa (east-west)
$$σ_{yy}$$ = 30 MPa (north-south)
$$σ_{zz}$$ = 17 MPa (vertical)
The rock mass material properties are found in Table 1.
bulk modulus 14.1 GPa shear modulus 8.9 GPa friction angle 35° (peak) cohesion 4 MPa (peak) dilation angle 5° tensile strength 0.5 MPa
The rock mass is fair to good quality mass (Geologic Strength Index, $$GSI \approx$$ 70) and behaves as a strain-softening material with a 0.5% critical plastic shear strain and a total loss of cohesion and a drop in friction angle of five degrees after a 2% plastic shear strain assuming the resolution of the pillar (number of zones along the pillar height) is 20.
Modeling Procedure
The FLAC3D model was used to study a quarter-section of a pillar as a result of symmetry considerations, as indicated in Figure 1. Figure 2 shows the final grid created using the built-in building-blocks block create tool of FLAC3D.
Figure 2: Final model grid.
The boundary conditions for the initial response are roller boundaries along the sides and bottom of the model, and an applied vertical stress of 17 MPa at the top of the model. In order to establish the peak load the pillar can carry, the vertical velocity of the top of the model is fixed at a constant value of -4 × 10-6 m/step. The sum of the reaction forces at the base of the model is obtained via the FISH function s_base.
The data file for this problem, “pillar.dat”, is listed at the end of this section.
Results
Figure 3 shows contours of Z displacements after the excavation of the tunnels. As expected, most of the floor heave (about 5.1 mm) and crown displacement (about 8.2 mm) takes place at the intersection. Figure 4 shows the state of plasticity at mid-height of the pillar. The extent of the failed zones ranges from 67 cm in the center of the pillar to 1.5 m in the intersection. Figure 5 shows contours of vertical stresses at mid-height of the pillar. The vertical stresses in this plane are close to the minimum principal stresses, i.e., greatest negative stress, $$σ_1$$), so the greatest compressive stresses are located in proximity to the boundary of the plastic region near the corner of the pillar. Figure 6 shows contours of the minimum principal stress inside the pillar. At this point, the core of the pillar is relatively unstressed, and the stress concentrations around the tunnels are the most prominent feature.
At this point, the pressure boundary condition at the top of the model is replaced with a velocity boundary condition. Figure 7 shows the evolution of the average vertical stress on the base (i.e., the sum of the vertical reaction forces at the base of the model divided by the area of the base: 13 m × 7.5 m). Note that the average vertical stress in the pillar will be larger by a factor, $$F$$, equal to
$F = {13 \times 7.5 \over 11 \times 5.5} = 1.61$
The peak average vertical stress on the base is approximately 31.9 MPa, so the peak average vertical stress in the pillar is 51.4 MPa.
The stresses in the pillar are compared when submitted to an average vertical stress on the base of 29 MPa before and after the peak stress. Figure 8 and Figure 9 show the extent of failure before and after peak. Note that roughly 30% of the cross-section of the pillar remains elastic after the peak stress has been reached. This is consistent with observations made by Wagner (1980). Figure 10 and Figure 11 show contours of vertical stress before and after peak. Even though the average vertical stress in the pillar is the same for both figures, the greatest compressive stress is higher and the de-stressed area is larger after peak.
Figure 3: Displacement contours after excavation of tunnels.
Figure 4: Plasticity state at pillar mid-height after excavation of tunnels.
Figure 5: Vertical stress contours at pillar mid-height after excavation of tunnels.
Figure 6: Minimum principal stress contours in pillar after excavation of tunnels.
Figure 7: History of the average vertical stress on the base.
Figure 8: Plasticity state in pillar at 29 MPa average vertical stress, before peak stress.
Figure 9: Plasticity state in pillar at 29 MPa average vertical stress, after peak stress.
Figure 10: Vertical stress contours at 29 MPa average vertical stress, before peak stress.
Figure 11: Vertical stress contours at 29 MPa average vertical stress, after peak stress.
Figure 12: History of the average vertical stress on the base for a square pillar.
Discussion
Pillar Strength
Many of the published pillar strength formulae are usually expressed by an empirical power relation of the form
$S = k V^{a} R^{b}$
in which $$k$$ is a strength parameter, $$V$$ is the pillar volume, $$R$$ is the width/height ratio of the pillar, and $$a$$ and $$b$$ are empirical parameters.
These expressions are based on square pillars and do not take into account the length of the pillar. Wagner (1980) defined an effective width for irregular pillars as
$w_{e} = {4 A \over C}$
in which $$A$$ is the operating area and $$C$$ is the pillar’s perimeter. For rectangular pillars, the effective width is
$w_{e} = {2 w l \over w + l}$
in which $$w$$ is the width and $$l$$ is the length of the pillar. Using these equations, it can be established that a rectangular pillar will be stronger than a square pillar of the same width and height by a factor, $$F_p$$, of
$F_{p} = \left({l \over w}\right)^a \cdot \left({2 \cdot l \over w + l}\right)^b$
This equation, along with the values of $$a$$ and $$b$$ published by different authors, can be used to predict the strength of a square pillar.
Table 2: Predicted Square-Pillar Strength
Source a (m) b (m) Predicted Strength (MPa)
Salamon and Munro (1967) -0.067 0.59 44.11
Greenwood et al. (1939) -0.111 0.72 43.81
Holland and Gaddy (1957) -0.167 0.83 44.12
The data file pillar-sq.dat models one-eighth of a square pillar with the same properties and initial stress conditions as the rectangular pillar studied earlier. Figure 12 shows the evolution of the vertical stress at the base of this model. The peak stress in the pillar will be
$\sigma_{p} = 23.6 \cdot {15 \cdot 15 \over 11 \cdot 11} = 43.9 \hbox{ MPa }$
which compares favorably with the strength predicted using Wagner’s approach.
Grid Dependency
A strain-softening material is more prone to produce shear bands (localization). The shear bands in FLAC3D collapse down to the smallest width that can be resolved by the grid, which is one grid-width if the band is parallel to the grid, or about three grid-widths if the band cuts across the grid at an arbitrary angle. Although the overall physics of band formation is modeled correctly by FLAC3D, band thickness and band spacing are grid-dependent. Furthermore, if the strain-softening model is used with a weakening material, the load/displacement relation generated by FLAC3D for a simulated test is strongly grid-dependent. This is because the strain concentrated in a band depends on the width of the band (in length units), which depends on zone size, as previously seen. Hence, smaller zones lead to more softening, because we move out more rapidly on the strain axis of the given softening curve. To correct this grid-dependence, some sort of length scale must be built into the constitutive model. There is controversy, at present, concerning the best way to do this. It must be recognized that the grid size and angle affect the results: models must be calibrated for each grid used.
In the absence of data for calibration, a critical plastic shear strain of $$\gamma^{crit}_{p}$$ = 0.5% is suggested as a starting point for modeling high-quality, massive, in-situ rock with few widely spaced discontinuities and fair to good joint conditions ($$GSI \approx$$ 70). This value of strain assumes that approximately 20 zones are used to resolve the critical dimension controlling stability within the numerical model (e.g., pillar height). If a different resolution ($$R$$) than 20 zones is used, this critical strain value should be scaled by a factor of $$R$$/20. A minimum resolution of 10~15 zones is recommended for detailed studies of rock mass performance. If a lower or higher quality rock mass is to be modeled, this critical strain value should be scaled by a factor of $$(10-0.1 \times GSI)/3$$. Caution should be used when simulating rock masses of very high quality ($$GSI$$ > 90) due to instabilities that can arise from the use of perfect (or near-perfect) brittleness. In summary, the recommended starting point for critical plastic shear strain is:
$\gamma^{crit}_{p} = 0.5\% \times {R \over 20} \times {{10 - 0.1 \times GSI} \over 3}$
In this example, both models are assuming fair to good quality mass ($$GSI \approx$$ 70). The model by the data file “pillar-sq.dat” is with uniform zones ($$R$$ = 20) so the softening rate does not need to be scaled. However, the model by the data file “pillar.dat” is with irregular zones. It is hard to use the scaling equation in a straightforward manner, but its equivalent resolution is between 16 and 20, so the softening rate is scaled by a factor of 0.9 by assuming $$R$$ = 18.
More detailed discussions on grid dependency of a softening model for a generalized case are in <link to location in the User’s Guide section>. the link in the preceding needs to be built; section being linked to needs to be written/updated.
References
Greenwald, H. P., H. C. Howarth and I. Hartmann. “Experiments on Strength of Small Pillars of Coal of the Pittsburgh Bed,” U.S. Bureau of Mines, Technical Paper No. 605 (1939).
Holland, C. J., and F. L. Gaddy. “Some Aspects of Permanent Support of Overburden on Coal Beds,” in Proceedings of the West Virginia Coal Mining Institute, pp. 43-66 (1957).
Salamon, M. D. G., and A. H. Munro. “A Study of the Strength of Coal Pillars,” J. S. Afr. Inst. Min. Metall., 68, 55-67 (1967).
Wagner, H. “Pillar Design in Coal Mines,” J. S. Afr. Inst. Min. Metall., 81, 37-45 (1980).
break
Data Files
pillar.dat
;-------------------------------------------------------------
; evolution of peak load in a rectangular pillar
;-------------------------------------------------------------
model new
model large-strain off
fish automatic-create off
model title 'Pillar load at intersecting tunnels'
; Model geometry - created with building-blocks and exported from state pane
program call 'geometry'
zone generate from-building-blocks
zone face skin
; Constitutive model and properties
zone cmodel assign strain-softening
zone property bulk 14.1e9 shear 8.87e9 tension 5e5 friction 35 ...
cohesion 4e6 dilation 5 table-friction 'fric' ...
table-cohesion 'coh'
table 'fric' add (0, 35) (0.0045, 35) (0.018,30) (1,30)
table 'coh' add (0,4e6) (0.0045,4e6) (0.018, 0) (1, 0)
; Initialize stresses
zone initialize stress xx -25e6 yy -30e6 zz -17e6
; Boundary conditions
zone face apply velocity-normal 0 range group 'West1' or 'West2' or 'East'
zone face apply velocity-normal 0 range group 'South1' or 'South2' or 'North'
zone face apply velocity-normal 0 range group 'Bottom1'
zone face apply stress-normal -17e6 range group 'Top2'
; Solve for initial excavation
model solve ratio-local 1e-3
model save 'pillar1'
; Apply velocity to top, take history of pillar load
zone face apply velocity-normal -4e-6 range group 'Top2'
[global dim1 = 13.0]
[global dim2 = 7.5]
history interval 50
model step 4000 ; -1.6e-2 total top displacement
model save 'pillar2'
model step 3600 ; -3.04e-2 total top displacement
model save 'pillar3'
pillar-sq.dat
;-------------------------------------------------------------
; evolution of peak load in a square pillar
;-------------------------------------------------------------
model new
model large-strain off
fish automatic-create off
model title 'Evolution of peak load in a square pillar'
; Create geometry, in this case simple to use zone create
zone create brick size (22,22,10) point 0 (2,2, -2) point 1 (7.5,2, -2) ...
point 2 (2,7.5, -2) point 3 (2,2,0)
zone create brick size (30,30,10) point 0 (0,0, -5) point 1 (7.5,0, -5) ...
point 2 (0,7.5, -5) point 3 (0,0,-2)
zone create brick size (30,30,10) point 0 (0,0,-10) point 1 (7.5,0,-10) ...
point 2 (0,7.5,-10) point 3 (0,0,-5) ...
ratio 1 1 0.9
zone face skin ; Name model boundaries
; Assign constitutive model and properties
zone cmodel assign strain-softening
zone property bulk 14.1e9 shear 8.87e9 friction 35 cohesion 4e6 ...
dilation 5 table-friction 'fric' table-cohesion 'coh'
table 'fric' add (0, 35) (0.005, 35) (0.02,30) (1,30)
table 'coh' add (0,4e6) (0.005,4e6) (0.02, 0) (1,0)
; Initialize stresses
zone initialize stress xx -25e6 yy -30e6 zz -17e6
; Boundary conditions
zone face apply velocity-normal 0 range group 'West2' or 'East'
zone face apply velocity-normal 0 range group 'South2' or 'North'
zone face apply velocity-normal 0 range group 'Top2'
zone face apply stress-normal -17e6 range group 'Bottom'
; Solve to initial equilibrium
model solve ratio-local 1e-3
; Apply Velocity to top and fix bottom
zone face apply-remove range group 'Bottom'
zone face apply velocity-normal 0 range group 'Bottom'
zone face apply velocity-normal -2e-6 range group 'Top2'
[global dim1 = 7.5]
[global dim2 = 7.5]
|
2021-09-18 02:35:30
|
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|
https://cs.stackexchange.com/questions/71703/bounded-waiting-and-starvation-free-in-critical-section-problem
|
# Bounded waiting and starvation free in critical section problem
i have 4 question regarding relation between starvation and bounded waiting.
From here,
definition of starvation free is
Freedom from Starvation -:Every thread that attempts to acquire the lock eventually succeeds
Freedom from Deadlock -:f some thread attempts to acquire the lock, then some thread (not necessarily the thread referred to in the if statement; emphasis added) will succeed in acquiring the lock.
so i can state that starvation-freedom imply deadlock-freedom
2.Does starvation-freedom imply bounded-waiting?
Approach-:
Starvation free implies that every thread that will attempt to acquire lock will succeed.
on the other hand
bounded wait insures that there exists a bound, or limit, on the number of times other processes are allowed to enter their critical sections after a process has made request to enter its critical section and before that request is granted
which implies that there must not be any starvation.
Am i correct? but the explanation is here confuses me.
Thanks!!!
No, starvation-free doesn't imply bounded waiting.
For instance, consider a procedure that never even attempts to acquire any lock; but the amount of time it takes is variable and can be arbitrarily long. Then there is no bound on the amount of time it might take to complete its operation.
Here is another example of how it can fail. Starvation-free means that every attempt to acquire the lock eventually succeeds -- but that says nothing about how long it might take. Maybe the amount of time it will take to acquire the lock is variable and can be arbitrarily long -- there is no upper bound on how long it takes to acquire the lock. Then a procedure that first attempts to acquire the lock before doing anything else won't satisfy bounded waiting.
• Very helpful. Thanks. Is there any relationship between progress and starvation? According to me there is no relationship. Want to confirm.. Nov 15 '18 at 5:02
so i can state that starvation-freedom imply deadlock-freedom
If you find this statement difficult, you can convert it to its contrapositive statement which is Deadlock implies starvation. It is easier to observe this statement is true. Hence starvation freedom implies deadlock freedom is also true.
For Bounded wait there must not be starvation
This statement is false. See this example:-
flag[0]=flag[1]=false;
i=j-1;
while(1){
flag[i]=true;
while(flag[j]);
// critical section
flag[i]=false;
// remainder section
}
At line no. 5 both processes may get stuck forever causing deadlock( and hence starvation also). But bounded waiting will be ensured because maximum number of times process i can enter the critical section before process j is allowed to enter the critical section is zero.
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2021-11-27 11:45:48
|
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|
https://electronics.stackexchange.com/questions/92233/why-do-i-need-to-wait-a-few-moments-before-sending-serial-data-on-my-avr
|
# Why do I need to wait a few moments before sending serial data on my AVR?
I have an AVR (ATMega644) connected to a Raspberry PI via serial connection. The AVR is powered with 5V and the AVR=>RPI Tx line is using a 1k8/3k3 voltage divider to get a 3v3 level.
If the 10ms wait (XXX in the code) is not present (according to the data sheet I do not have to wait at all) I receive this nonsense (python repr() of the received data):
'\x00\xaa\x8a\x8a\xea\n'
'\xe9\xf5%\xc5E\xd5\xa4\xfcBYE WORLD\r\n'
When the 10ms delay is there I receive this:
'\x00HELLO WORLD\r\n'
'BYE WORLD\r\n'
This is pretty much fine. I'd like to know why there's a \0 before the very first actual data byte ('H') though. I never send one for sure!
However, my main question is why the delay between initializing and sending data is necessary.
Notes: My F_CPU value is correct and so are the fuses. I also tried using a lower baudrate (4800) and a different chip.
This is the code I'm using:
#include <avr/interrupt.h>
#include <util/delay.h>
#include <util/setbaud.h>
inline void uart_putc(char c)
{
loop_until_bit_is_set(UCSR0A, UDRE0);
UDR0 = c;
}
inline void uart_puts(const char *s)
{
while (*s) {
uart_putc(*s++);
}
}
int main()
{
// we don't need/use any interrupts
cli();
// -DF_CPU=18432000L -DBAUD=19200 used when compiling
UBRR0H = UBRRH_VALUE;
UBRR0L = UBRRL_VALUE;
// 8 data bits
UCSR0C |= (1 << UCSZ00) | (1 << UCSZ01);
// enable transmitter
UCSR0B |= (1 << TXEN0);
_delay_ms(10); // XXX
uart_puts("HELLO WORLD\r\n");
_delay_ms(250);
uart_puts("BYE WORLD\r\n");
// do nothing
while (1)
;
return 0;
}
• Sounds like the output pins are in an unstable state before enabling the UART. Try setting them to a known good state, e.g. output high, before enabling. – Ignacio Vazquez-Abrams Nov 29 '13 at 23:58
• I tried output+low and input+pullup - no difference. Using 330/180ohm resistors as someone suggested on IRC also didn't change anything. – ThiefMaster Nov 30 '13 at 0:39
• Update, it only happens when the receiving end is my Pi. With a simple USB2Serial interface I don't need the delay - even if I keep the resistors to get down to 3v3. – ThiefMaster Nov 30 '13 at 4:07
• Flush RPi's input buffer before you start. It is always a good idea to have some form of a handshake before you start to send data. I personally prefer the use of predefined data frames. – jippie Nov 30 '13 at 9:15
• You may be able to improve synchronization speed by using (more) stop bits. – jippie Nov 30 '13 at 9:17
Basically what you do in
// enable transmitter
UCSR0B |= (1 << TXEN0);
_delay_ms(10); // XXX
is inserting 10ms of stop bits (1). This helps your receiver to synchronize with the data stream (which conveniently starts with a start bit 0). Until then the AVR pin is high impedance and filling your input buffer with garbage to which your receiver is trying to synchronize.
Every time the receiver reads 10 bits that fit the frame (start-bit 0, 8 data bits, stop-bit 1) the data bits are pushed in the input buffer. Only when the frame doesn't match, bits get discarded until it does find a matching 10 bit string.
This also explains the importance for using start and stop bits in asynchronous transmission. This also explains why using 1.5 or 2 stop bits improves speed of synchronization; the probability of received data accidentally fitting the frame for a single byte decreased.
I am not aware of your specific hardware or of how your software handles exceptions, but if your circuit is transitioning from an undefined to a defined state you need to give the system time to deal with the states which arrive at 'the boundary' as signals appear to be valid as they settle down, but are not. Using a USB2serial interface probably ensures that the converter output is always defaulted to a legal state whenever illegal input is received so there is no "wakeup time" - see below. .
Longer: If your serial line is active,idle and operating as defined it will be in a defined state - high for a system with positive logic and low for a system with negative logic. Systems which connect with TTL (nominally 0/+5V) or system voltage levels (0/3 or 0/3.3 or 0/5 or whatever) usually are positive logic. RS232 inverts this logic and an idle state is usually low. If connecting at TTL levels the level is what the level is - send 3V3 and you see 3V3 etc. If you the sender to changes to high impedance when disabled the receiver has to 'decide' what it should do wit the input. It is usual to pull the input up to +V with a resistor or equivalent which is large enough (ie low enough load) that it has zero effect on normal operation. Under disabled conditions it sets the input to "idle". If you use USB to RS@32 or USB to logic level serial interfaces they usually look after such niceties. If you connect directly then YOU must decide what happens during disabled state and if you do not do so then Murphy will.
If your software is semi-Murphy proof and you receive a long low signal (illegal) the software may reject it. Less MP (Murphy Proof) software may instead gibber.
If you transition from illegal to legal input (after say a data send enablement) the levels as the circuit stabilises may appear as signal.
If your circuit has time constants involving capacitors that charge to a proper DC level during operation they may be at ground when you first receive data. As they charge to the correct value a 1 or 0 may tend to be interpreted incorrectly. Usually the bias will be one way or the other so eg 1's may become 0's but 0's are OK. But if there are runs of all 1 or all 0, during system stabilisation these may be more affected by the charging capacitors or whatever.
= $45$4C $4C$4F
|
2019-10-14 20:38:28
|
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|
https://cs.stackexchange.com/questions/9045/assuming-mboxbpp-subseteq-pi-2-what-conclusions-can-we-make
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# Assuming $\mbox{BPP}\subseteq \Pi_2$ -What conclusions can we make?
I'd like your help with the following question:
Assume we proved that $\mbox{BPP}\subseteq \Pi_2$ -What conclusions can you make?
BPP is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1/3 for all instances,
$\Pi_2$ is the class of all languages $L$ such that there's a polynomial algorithm $M$ and a polynom $p$ so that $\forall x.x\in L\Leftrightarrow \forall u\in \{ 0,1 \}^*.\exists v \in \{ 0,1 \}^*.M(x,u,v)=1$.
We already know that $\mbox{BPP}\subseteq \Sigma_2$, so $\mbox{BPP}\subseteq \Pi_2\cap \Sigma_2$.
It is known. As your final statement says, $\mbox{BPP} \subseteq \Pi_2 \cap \Sigma_2$. It is called the Sipser–Gács–Lautemann theorem. (All though your "so" is somewhat misleading.)
It can be strengthened to $\mbox{BPP} \subseteq \mbox{MA} \subseteq \mbox{S}^P_2 \subseteq \Pi_2 \cap \Sigma_2 \subseteq \Pi_2$ (see also Arthur-Merlin protocol).
Just to add a minor point to Pal GD's answer, $BPP$ is closed under complementation. So, once you prove that $BPP \subseteq \Sigma_2$, you can easily deduce $BPP \subseteq \Sigma_2 \cap \Pi_2$. Also, (Just to improve the inclusions) even better $BPP \subseteq NP^{BPP} \subseteq MA \subseteq S_{2}^{P} \cdots$ is known.
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2022-07-07 08:27:41
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https://socratic.org/questions/when-using-the-half-reaction-method-of-balancing-redox-reactions-what-should-you
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# When using the half reaction method of balancing redox reactions, what should you do first?
$\text{Assign oxidation states...............}$
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2019-08-19 05:39:38
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https://www.semanticscholar.org/paper/The-mass-function-of-GX-339-4-from-spectroscopic-of-Heida-Jonker/889df0fcab7e06c736f0744313f4bba196e27285
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# The mass function of GX 339-4 from spectroscopic observations of its donor star
@article{Heida2017TheMF,
title={The mass function of GX 339-4 from spectroscopic observations of its donor star},
author={Marianne Heida and Peter G. Jonker and Manuel A P Torres and Andrea Chiavassa},
journal={arXiv: High Energy Astrophysical Phenomena},
year={2017}
}
• M. Heida, +1 author A. Chiavassa
• Published 15 August 2017
• Physics
• arXiv: High Energy Astrophysical Phenomena
We obtained 16 VLT/X-shooter observations of GX 339-4 in quiescence in the period May - September 2016 and detected absorption lines from the donor star in its NIR spectrum. This allows us to measure the radial velocity curve and projected rotational velocity of the donor for the first time. We confirm the 1.76 day orbital period and we find that $K_2$ = $219 \pm 3$ km s$^{-1}$, $\gamma = 26 \pm 2$ km s$^{-1}$ and $v \sin i = 64 \pm 8$ km s$^{-1}$. From these values we compute a mass function… Expand
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We report on the spectroscopic analysis of the black hole binary GX 339-4 during its recent 2017-2018 outburst, observed simultaneously by the Swift and NuSTAR observatories. Although during thisExpand
A jet model for the fast IR variability of the black hole X-ray binary GX 339-4
• J. Malzac, +16 authors P. Uttley
• Physics
• Monthly Notices of the Royal Astronomical Society
• 2018
Using the simultaneous Infra-Red (IR) and X-ray light curves obtained by Kalamkar et al., we perform a Fourier analysis of the IR/X-ray timing correlations of the black hole X-ray binary (BHB) GXExpand
Correlations between radio and bolometric fluxes in GX 339–4 and H1743–322
• Physics
• Monthly Notices of the Royal Astronomical Society
• 2018
Compact radio jets are ubiquitous in stellar-mass black-hole binaries in their hard spectral state. Empirical relations between the radio and narrow-band X-ray fluxes have been used to understand theExpand
High-density reflection spectroscopy: I. A case study of GX 339-4
• J. Jiang, +5 authors J. Tomsick
• Physics
• Monthly Notices of the Royal Astronomical Society
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We present a broad-band spectral analysis of the black hole binary GX 339-4 with NuSTAR and Swift using high-density reflection model. The observations were taken when the source was in low-flux (LF)Expand
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Dynamical Evidence for a Black Hole in GX 339-4
• Physics
• 2003
We present outburst spectroscopy of GX 339-4 that may reveal the motion of its elusive companion star. N III lines exhibit sharp emission components moving over ~300 km s-1 in a single night. TheExpand
On the mass of the neutron star in V395 Car/2S 0921-630
• Physics
• 2007
We report high-resolution optical spectroscopy of the low-mass X-ray binary V395 Car/2S 0921-630 obtained with the MIKE echelle spectrograph on the Magellan-Clay telescope. Our spectra are obtainedExpand
Optical Observations of the Black Hole Candidate GX 339−4 (V821 Arae)
• Physics
• 2002
GX 339-4 has long been known as a black hole candidate because of its rapid variability and high/low X-ray states, which are similar to those of Cyg X-1. Although GX 339-4 is assumed to be a binary,Expand
On the mass of the neutron star in V395 Carinae/2s 0921-630
• Physics
• 2007
We report high-resolution optical spectroscopy of the low-mass X-ray binary V395 Car/2S 0921-630 obtained with the MIKE echelle spectrograph on the Magellan-Clay telescope. Our spectra are obtainedExpand
Spectroscopy of $A0620-00$: the mass of the black hole and an image of its accretion disc
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• 1994
We present spectra of the black hole binary A0620-00 taken at Hα and Hβ to study the distribution of emission-line flux from its accretion disc. Using the spectra of Hα, we have measured the K-typeExpand
NuSTAR AND SWIFT OBSERVATIONS OF THE VERY HIGH STATE IN GX 339-4: WEIGHING THE BLACK HOLE WITH X-RAYS
We present results from spectral fitting of the very high state of GX 339-4 with Nuclear Spectroscopic Telescope Array (NuSTAR) and Swift. We use relativistic reflection modeling to measure the spinExpand
The EXOSAT data on GX 339-4: further evidence for an "intermediate" state
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• 1997
We have studied the fast timing and spectral behavior of the black hole candidate (BHC) GX 339-4 using all 1-20 keV EXOSAT ME data: the 1983 July, 1984 March, 1984 May, and 1985 April observations.Expand
VLT optical observations of V821 Ara(=GX339-4) in an extended off'' state
• Physics
• 2001
We report on low-resolution spectroscopy of GX339-4 during its current, extended X-ray "off" state in May 2000 ($r=20.1$) obtained with the VLT Focal Reducer/low dispersion Spectrograph (FORS1).Expand
THE COMPLEX ACCRETION GEOMETRY OF GX 339–4 AS SEEN BY NuSTAR AND SWIFT
We present spectral analysis of five NuSTAR and Swift observations of GX 339−4 taken during a failed outburst in summer 2013. These observations cover Eddington luminosity fractions in the rangeExpand
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2021-12-08 10:15:53
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https://waypointrealestate.mywebsites360.com/commercial-brokerage
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Commercial Brokerage
• Testing
• Testings
A Growing Group of Restaurants
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A Winery Expanding in The Market
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2021-03-08 21:38:07
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http://math.stackexchange.com/questions/30192/a-natural-borel-probability-measure-on-a-projective-space-p-rn
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# A “natural” Borel probability measure on a projective space $P R^{N}$?
Is there a simple way to construct such a measure? Preferably, one invariant under rotations and reflections of $R^N$.
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If it is invariant wrt. reflections of $\mathbb{R}^n$ then its also invariant wrt. translations (as they are compositions of 2 reflection), so such a measure could not be finite. (If you wish just any reasonably natural measure, $\mathbb{RP}^n$ is $S^n/C_2$, so you can take the standard measure on $S^n$) – user8268 Mar 31 '11 at 21:20
Thanks, user8960. Yes, if there is a standard measure on the sphere $S^N$ invariant under reflections through hyperplanes through the center and under rotations about the center, that's great. Is there a website or reference with the details of the construction of this measure? – user8960 Mar 31 '11 at 22:05
## 2 Answers
To generate a uniformly distributed random point on $S^n$, one can generate a random vector $X=(X_k)_{1\leqslant k\leqslant n+1}$ with any nondegenerate isotropic distribution in $\mathbb R^{n+1}$, and compute the radius $R=\sqrt{X_1^2+X_2^2+\cdots+X_{n+1}^2}$. Then the vector $R^{-1}X$ is uniformly distributed on $S^n$.
The commonest choice of an isotropic distribution is the centered normal distribution with covariance matrix (any nonzero multiple of) the identity matrix. In other words, the coordinates of $X$ may be $n+1$ independent standard random variables.
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There is a unique probability measure on $P^n$ invariant under the "linear" action of $O(n+1,\mathbb R)\subset GL(n+1,\mathbb R)$. This follows from the existence of Haar measures on homogeneous spaces and is proved in this generality in pretty much any good textbook which constructs the Haar measure on groups —Lang's book on functional analysis is one.
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Books whose titles contain «integral geometry» are also good candidates. – Mariano Suárez-Alvarez Mar 27 '12 at 20:39
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2016-07-29 14:22:54
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https://marcofrasca.wordpress.com/page/2/
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## NASA paper leaked!
05/11/2016
EmDrive is a possible propulsion system granting no apparent expulsion of propellent. It has been reason of hot debate with strong criticisms from the scientific community championed by Sean Carroll and John Baez. Notwithstanding this, NASA went ahead performing in-depth experiments to test this technology. It was announced that, on December this year, a paper will appear in a reputable journal reporting on the last measurement campaign. But to today the paper leaked out on reddit. The results, at a first sight, appear to be striking confirming the effect, with all the possible mundane disturbing causes excluded. You can also find other material there as a small video.
I hope this will go properly evaluated by the scientific community moving to a more serious addressing of this effect.
Update: The links were removed from the subreddit’s moderator. I have copies of these files but I do not mean to publish them in any form.
Update: Here is the published paper.
## In the aftermath of ICHEP 2016
06/08/2016
ATLAS and CMS nuked our illusions on that bump. More than 500 papers were written on it and some of them went through Physical Review Letters. Now, we are contemplating the ruins of that house of cards. This says a lot about the situation in hep in these days. It should be emphasized that people at CERN warned that that data were not enough to draw a conclusion and if they fix the threshold at $5\sigma$ a reason must exist. But carelessness acts are common today if you are a theorist and no input from experiment is coming for long.
It should be said that the fact that LHC could confirm the Standard Model and nothing else is one of the possibilities. We should hope that a larger accelerator could be built, after LHC decommissioning, as there is a long way to the Planck energy that we do not know how to probe yet.
What does it remain? I think there is a lot yet. My analysis of the Higgs sector is still there to be checked as I will explain in a moment but this is just another way to treat the equations of the Standard Model, not beyond it. Besides, for the end of the year they will reach $30\ fb^{-1}$, almost triplicating the actual integrated luminosity and something interesting could ever pop out. There are a lot of years of results ahead and there is no need to despair. Just to wait. This is one of the most important activities of a theorist. Impatience does not work in physics and mostly for hep.
About the signal strength, things seem yet too far to be settled. I hope to see better figures for the end of the year. ATLAS is off the mark, going well beyond unity for WW, as happened before. CMS claimed $0.3\pm 0.5$ for WW decay, worsening their excellent measurement of $0.72^{+0.20}_{-0.18}$ reached in Run I. CMS agrees fairly well with my computations but I should warn that the error bar is yet too large and now is even worse. I remember that the signal strength is obtained by the ratio of the measured cross section to the one obtained from the Standard Model. The fact that is smaller does not necessarily mean that we are beyond the Standard Model but that we are just solving the Higgs sector in a different way than standard perturbation theory. This solution entails higher excitations of the Higgs field but they are strongly depressed and very difficult to observe now. The only mark could be the signal strength for the observed Higgs particle. Finally, the ZZ channel is significantly less sensible and error bars are so large that one can accommodate whatever she likes yet. Overproduction seen by ATLAS is just a fluctuation that will go away in the future.
The final sentence to this post is what we have largely heard in these days: Standard Model rules.
## Higgs or not Higgs, that is the question
16/06/2016
LHCP2016 is running yet with further analysis on 2015 data by people at CERN. We all have seen the history unfolding since the epochal event on 4 July 2012 where the announcement of the great discovery happened. Since then, also Kibble passed away. What is still there is our need of a deep understanding of the Higgs sector of the Standard Model. Quite recently, LHC restarted operations at the top achievable and data are gathered and analysed in view of the summer conferences.
The scalar particle observed at CERN has a mass of about 125 GeV. Data gathered on 2015 seem to indicate a further state at 750 GeV but this is yet to be confirmed. Anyway, both ATLAS and CMS see this bump in the $\gamma\gamma$ data and this seems to follow the story of the discovery of the Higgs particle. But we have not a fully understanding of the Higgs sector yet. The reason is that, in run I, gathered data were not enough to reduce the error bars to such small values to decide if Standard Model wins or not. Besides, as shown by run II, further excitations seem to pop up. So, several theoretical proposals for the Higgs sector still stand up and could be also confirmed already in August this year.
Indeed, there are great news already in the data presented at LHCP2016. As I pointed out here, there is a curious behavior of the strengths of the signals of Higgs decay in $WW,\ ZZ$ and some tension, even if small, appeared between ATLAS and CMS results. Indeed, ATLAS seemed to have seen more events than CMS moving these contributions well beyond the unit value but, as CMS had them somewhat below, the average was the expected unity agreeing with expectations from the Standard Model. The strength of the signals is essential to understand if the propagator of the Higgs field is the usual free particle one or has some factor reducing it significantly with contributions from higher states summing up to unity. In this case, the observed state at 125 GeV would be just the ground state of a tower of particles being its excited states. As I showed recently, this is not physics beyond the Standard Model, rather is obtained by solving exactly the quantum equations of motion of the Higgs sector (see here). This is done considering the other fields interacting with the Higgs field just a perturbation.
So, let us do a recap of what was the situation for the strength of the signals for the $WW\, ZZ$ decays of the Higgs particle. At LHCP2015 the data were given in the following slide
From the table one can see that the signal strengths for $WW,\ ZZ$ decays in ATLAS are somewhat beyond unity while in CMS these are practically unity for $ZZ$ but, more interestingly, 0.85 for $WW$. But we know that data gathered for $WW$ decay are largely more than for $ZZ$ decay. The error bars are large enough to be not a concern here. The value 0.85 is really in agreement with the already cited exact computations from the Higgs sector but, within the error, in overall agreement with the Standard Model. This seems to point toward on overestimated number of events in ATLAS but a somewhat reduced number of events in CMS, at least for $WW$ decay.
At LHCP2016 new data have been presented from the two collaborations, at least for the $ZZ$ decay. The results are striking. In order to see if the scenario provided from the exact solution of the Higgs sector is in agreement with data, these should be confirmed from run II and those from ATLAS should go down significantly. This is indeed what is going on! This is the corresponding slide
This result is striking per se as shows a tendency toward a decreasing value when, in precedence, it was around unity. Now it is aligned with the value seen at CMS for the $WW$ decay! The value seen is again in agreement with that given in the exact solution of the Higgs sector. And ATLAS? This is the most shocking result: They see a significant reduced set of events and the signal strength they obtain is now aligned to the one of CMS (see Strandberg’s talk at page 11).
What should one conclude from this? If the state at 750 GeV should be confirmed, as the spectrum given by the exact solution of the Higgs sector is given by an integer multiplied by a mass, this would be at $n=6$. Together with the production strengths, if further data will confirm them, the proper scenario for the breaking of electroweak symmetry is exactly the one described by the exact solution. Of course, this should be obviously true but an experimental confirmation is essential for a lot of reasons, last but not least the form of the Higgs potential that, if the numbers are these, the one postulated in the sixties would be the correct one. An other important reason is that coupling with other matter does not change the spectrum of the theory in a significant way.
So, to answer to the question of the title remains to wait a few weeks. Then, summer conferences will start and, paraphrasing Coleman: God knows, I know and by the end of the summer we all know.
Marco Frasca (2015). A theorem on the Higgs sector of the Standard Model Eur. Phys. J. Plus (2016) 131: 199 arXiv: 1504.02299v3
## They did it!
11/02/2016
This is a great moment in history of physics: Gravitational waves were directly detected by the merging of two black holes by the LIGO Collaboration. This is a new world we arrived at and there will be a lot to be explored and understood. I do not know if it is for the direct proof of existence of gravitational waves or black holes that fixes this great moment forever in the memory of mankind. But by today we have both!
You can find an excellent recount here. This is the paper
Thank you for this great work!
## Unpublishable
31/12/2015
I tried in different ways to get this paper through the community with standard channels. As far as I can tell, this paper is unpublishable. By this I mean that journals not even send it to referees to start a normal review process or all people try to stop it from making it known. The argument is always the same: A reformulation of quantum mechanics using stochastic processes but using noncommutative geometry this time. I apologize to the community if this unacceptable approach has bothered people around the World but this is the fate of some ideas. Of course, if somebody has the courage and the willing to publish, let me know and I will appreciate the tentative with infinite gratefulness.
Now, back to sane QCD.
Happy new year!
## Quantum gravity
27/12/2015
Quantum gravity appears today as the Holy Grail of physics. This is so far detached from any possible experimental result but with a lot of attentions from truly remarkable people anyway. In some sense, if a physicist would like to know in her lifetime if her speculations are worth a Nobel prize, better to work elsewhere. Anyhow, we are curious people and we would like to know how does the machinery of space-time work this because to have an engineering of space-time would make do to our civilization a significant leap beyond.
A fine recount of the current theoretical proposals has been rapidly presented by Ethan Siegel in his blog. It is interesting to notice that the two most prominent proposals, string theory and loop quantum gravity, share the same difficulty: They are not able to recover the low-energy limit. For string theory this is a severe drawback as here people ask for a fully unified theory of all the interactions. Loop quantum gravity is more limited in scope and so, one can think to fix the problem in a near future. But of all the proposals Siegel is considering, he is missing the most promising one: Non-commutative geometry. This mathematical idea is due to Alain Connes and earned him a Fields medal. So far, this is the only mathematical framework from which one can rederive the full Standard Model with all its particle content properly coupled to the Einstein’s general relativity. This formulation works with a classical gravitational field and so, one can possibly ask where quantized gravity could come out. Indeed, quite recently, Connes, Chamseddine and Mukhanov (see here and here), were able to show that, in the context of non-commutative geometry, a Riemannian manifold results quantized in unitary volumes of two kind of spheres. The reason why there are two kind of unitary volumes is due to the need to have a charge conjugation operator and this implies that these volumes yield the units $(1,i)$ in the spectrum. This provides the foundations for a future quantum gravity that is fully consistent from the start: The reason is that non-commutative geometry generates renormalizable theories!
The reason for my interest in non-commutative geometry arises exactly from this. Two years ago, I, Alfonso Farina and Matteo Sedehi obtained a publication about the possibility that a complex stochastic process is at the foundations of quantum mechanics (see here and here). We described such a process like the square root of a Brownian motion and so, a Bernoulli process appeared producing the factor 1 or i depending on the sign of the steps of the Brownian motion. This seemed to generate some deep understanding about space-time. Indeed, the work by Connes, Chamseddine and Mukhanov has that understanding and what appeared like a square root process of a Brownian motion today is just the motion of a particle on a non-commutative manifold. Here one has simply a combination of a Clifford algebra, that of Dirac’s matrices, a Wiener process and the Bernoulli process representing the scattering between these randomly distributed quantized volumes. Quantum mechanics is so fundamental that its derivation from a geometrical structure with added some mathematics from stochastic processes makes a case for non-commutative geometry as a serious proposal for quantum gravity.
I hope to give an account of this deep connection in a near future. This appears a rather exciting new avenue to pursue.
Ali H. Chamseddine, Alain Connes, & Viatcheslav Mukhanov (2014). Quanta of Geometry: Noncommutative Aspects Phys. Rev. Lett. 114 (2015) 9, 091302 arXiv: 1409.2471v4
Ali H. Chamseddine, Alain Connes, & Viatcheslav Mukhanov (2014). Geometry and the Quantum: Basics JHEP 12 (2014) 098 arXiv: 1411.0977v1
Farina, A., Frasca, M., & Sedehi, M. (2013). Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics Signal, Image and Video Processing, 8 (1), 27-37 DOI: 10.1007/s11760-013-0473-y
## News from CERN
17/12/2015
Two days ago, CERN presented their new results at 13 TeV to the World. Of course, collected data so far are not enough for conclusive results but the these are exciting anyway. The reason is that both the collaborations, CMS and ATLAS, see a bump at around 750 GeV in the $\gamma\gamma$ decay. Summing up the results of the two collaborations, they are around $4\sigma$ without look elsewhere effect, not yet a discovery but, probably, at the summer conferences they will have something more conclusive to say. This could be an unlucky fluctuation but this situation remember us the story of the discovery of the Higgs boson more than three years ago. The question if this is beyond Standard Model physics is what I will try to answer in these few lines.
Firstly, if this particle is real, it decays with two photons exactly as the Higgs boson. Secondly, with a final state like this it can have only spin 0 or 2. We will be conservative and assume that this is not a graviton. Rather, it is a sibling of the Higgs particle. Besides, it was not observed in run I but is not inconsistent with data from there. It appears like the increased luminosity favored its appearance. We want to be more conservative and we take for granted just the Lagrangian of the Standard Model. So, what is this beast?
My answer is that this could be an excited state of the Higgs boson that, having a production rate lower than its ground state seen at run I, needed more luminosity to be observed. You do not need to change the Lagrangian of the Standard Model for this and it is not BSM physics yet. You do not even need a technicolor theory to describe it. The reason is that the Higgs part of the Standard Model can be treated mathematically yielding exact solutions. The quantum field theory can be exactly solved and the spectrum of the theory says exactly what I stated above (see here, and here). The Higgs model per se is exactly solvable. So, Jester’s idea to add another scalar field to the Lagrangian model is useless, it is all just inside and you will get a two photon final state as well.
Of course, it is too early to draw a final conclusion and a wealth of papers with a prompt explanation flooded arxiv in these two days. With the restart of LHC on spring and the collecting of more data, things will be clearer than now. For the moment, this hint is enough to keep us excited for the next few months.
Marco Frasca (2015). A theorem on the Higgs sector of the Standard Model arxiv arXiv: 1504.02299v2
Marco Frasca (2015). Quantum Yang-Mills field theory arxiv arXiv: 1509.05292v1
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2018-07-16 05:05:09
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https://plainmath.net/algebra-i/81531-let-mi-mathvariant-normal
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nidantasnu
Answered
2022-07-10
Let
${\mathrm{\Delta }}_{n-1}:=\left\{x\in {R}^{n}:{x}_{1}+{x}_{2}+....{x}_{n}=1,{x}_{1},{x}_{2},....{x}_{n}\ge 0\right\}$
and
$a\in {R}^{n}$
Let
$z:={P}_{{\mathrm{\Delta }}_{n-1}}\left(a\right)$
be the projection of point a onto ${\mathrm{\Delta }}_{n-1}$. Show that $z$ satisfies the system of inequalities
$z-y=a-\mu \mathbf{\text{e}},z\ge 0,y\ge 0,{z}^{T}y=0$
where $\mathbf{\text{e}}$ is the vector of all ones. $y,z\in {R}^{n},\mu \in R$. One can use obtuse angle condition of the projection theorem over the convex set along with Farkas Lemma.
I don't know how to approach this problem.
Answer & Explanation
iskakanjulc
Expert
2022-07-11Added 18 answers
$z=\text{argmin}\left\{\left(x-a{\right)}^{T}\left(x-a\right):{x}^{T}e=1,x\ge 0\right\}$ use KKT
Callum Dudley
Expert
2022-07-12Added 4 answers
Use first order conditions: $z$ minimizes a smooth convex function $f$ on a closed convex set $C$ iff $⟨\mathrm{\nabla }f\left(z\right),y-z⟩\ge 0,\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }y\in C$
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2023-02-06 15:24:52
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https://iclr.cc/virtual/2021/poster/2920
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Minimum Width for Universal Approximation
Sejun Park · Chulhee Yun · Jaeho Lee · Jinwoo Shin
Keywords: [ universal approximation ] [ neural networks ]
[ Abstract ]
[ Paper ]
Thu 6 May 5 p.m. PDT — 7 p.m. PDT
Spotlight presentation: Oral Session 6
Tue 4 May 7 p.m. PDT — 10:16 p.m. PDT
Abstract: The universal approximation property of width-bounded networks has been studied as a dual of classical universal approximation results on depth-bounded networks. However, the critical width enabling the universal approximation has not been exactly characterized in terms of the input dimension $d_x$ and the output dimension $d_y$. In this work, we provide the first definitive result in this direction for networks using the ReLU activation functions: The minimum width required for the universal approximation of the $L^p$ functions is exactly $\max\{d_x+1,d_y\}$. We also prove that the same conclusion does not hold for the uniform approximation with ReLU, but does hold with an additional threshold activation function. Our proof technique can be also used to derive a tighter upper bound on the minimum width required for the universal approximation using networks with general activation functions.
Chat is not available.
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2023-03-27 14:27:39
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https://labs.tib.eu/arxiv/?author=Gabriella%20Pinzari
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• ### A first integral to the partially averaged Newtonian potential of the three-body problem(1607.03056)
March 10, 2019 math.DS
We consider the partial average i.e., the Lagrange average with respect to {\it just one} of the two mean anomalies, of the Newtonian part of the perturbing function in the three--body problem Hamiltonian. We prove that such a partial average exhibits a non--trivial first integral. We show that this integral is fully responsible of certain cancellations in the averaged Newtonian potential, including a property noticed by Harrington in the 60s. We also highlight its joint r\^ole (together with certain symmetries) in the appearance of the so called "Herman resonance". Finally, we discuss an application and an open problem.
• ### On the co--existence of maximal and whiskered tori for the planetary three--body problem(1611.00508)
Sept. 20, 2018 math.DS
In this paper we discuss about the possibility of {\it coexistence} of stable and unstable quasi--periodic {\sc kam} tori in a region of phase space of the three-body problem. The {argument of proof} goes along {{\sc kam} theory and, especially,} the production of two non smoothly related systems of canonical coordinates in the same region of the phase space, the possibility of which is foreseen, for properly--degenerate' systems, by a theorem of Nekhorossev and Mi{\v{s}}{\v{c}}enko and Fomenko. The two coordinate systems are alternative to the classical reduction of the nodes by Jacobi, described, e.g., in~[V.I.~Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, 18, 85 (1963); p. 141].
• ### A Normal Form without small divisors(1710.02689)
Oct. 7, 2017 math-ph, math.MP, math.DS, math.CV
Following the techniques of [4], we formulate a Normal Form Lemma suited to close to be integrable Hamiltonian systems where not all the coordinates are action angles. The Lemma turns to be useful in the theory of KAM tori of Sun-Earth-Asteroids systems.
• ### An analysis of the Sun-Earth-Asteroid systems based on the two-centre problem(1702.03680)
June 3, 2017 math-ph, math.MP, math.DS, math.HO
We propose a new analysis of the two--centre problem particularly suited to be used as a basis to study the dynamics of Sun--Earth--Asteroid systems. Our method, based on a tricky choice of initial coordinates, allows us to evaluate the risk of collisions between the Asteroid and Earth. Moreover, it seems to be fitted to prove the existence of chains of transition tori in the planar Sun--Earth--Asteroid systems.
• ### Perihelia reduction and global Kolmogorov tori in the planetary problem(1501.04470)
Sept. 20, 2016 math.DS
We prove the existence of an almost full measure set of $(3n-2)$--dimensional quasi periodic motions in the planetary problem with $(1+n)$ masses, with eccentricities arbitrarily close to the Levi-Civita limiting value and relatively high inclinations. This extends previous results, where smallness of eccentricities and inclinations was assumed. The question had been previously considered by V.I.Arnold [2, Ch III, \S 1, n. 6, p. 128] in the 60s, for the particular case of the planar three--body problem, where, due to the limited number of degrees of freedom, it was enough to use the invariance of the system by the SO(3) group. The proof exploits nice parity properties of a new set of coordinates for the planetary problem, which reduces completely the number of degrees of freedom for the system (in particular, its degeneracy due to rotations) and, moreover, is well fitted to its reflection invariance. It allows the explicit construction of an associated close to be integrable system, replacing Birkhoff normal form, common tool of previous literature.
• ### Canonical coordinates for the planetary problem(1407.5333)
Nov. 24, 2014 math-ph, math.MP, math.CA, math.DS, math.HO
In 1963, V. I. Arnold stated his celebrated Thorem on the Stability of Planetary Motions. The general proof of his wonderful statement (that he provided completely only for the particular case of three bodies constrained in a plane) turned out to be more difficult than expected and was next completed by J. Laskar, P. Robutel, M. Herman, J. F\'ejoz, L. Chierchia and the author. We refer the reader to the technical papers \cite{arnold63, laskarR95, rob95, maligeRL02, herman09, fej04, pinzari-th09, ChierchiaPi11b} for detailed information; to \cite{fejoz13, chierchia13, chierchiaPi14}, or the introduction of \cite{pinzari13} for reviews. The complete understanding of Arnold's Theorem relied on an analytic part and a geometric one, both highly non trivial. Of such two aspects, the analytic part was basically settled out since \cite{arnold63} (notwithstanding refinements next given in \cite{fejoz04, chierchiaPi10}). The geometrical aspects were instead mostly unexplored after his 1963's paper and have been only recently clarified \cite{pinzari-th09, chierchiaPi11b}. In fact, switching from the three--body case to the many--body one needed to develop new constructions, because of a dramatic degeneracy due to its invariance by rotations, which, if not suitably treated, prevents the application of Arnold's 1963's strategy. The purpose of this note is to provide a historical survey of this latter part. We shall describe previous classical approaches going back to Delaunay, Poincar\'e, Jacobi and point out more recent progresses, based on the papers \cite{pinzari-th09, chierchiaPi11a, chierchiaPi11b, chierchiaPi11c, pinzari13, pinzari14}. In the final part of the paper we discuss a set of coordinates recently found by the author which reduces completely its integrals, puts the unperturbed part in Keplerian form, preserves the symmetry by reflections and is regular for zero inclinations.
• ### Global Kolmogorov tori in the planetary N-body problem. Announcement of result(1406.4493)
June 24, 2014 math-ph, math.MP, math.CA, math.DS
We improve a result in [L. Chierchia and G. Pinzari, Invent. Math. 2011] by proving the existence of a positive measure set of $(3n-2)$--dimensional quasi--periodic motions in the spacial, planetary $(1+n)$--body problem away from co--planar, circular motions. We also prove that such quasi--periodic motions reach with continuity corresponding $(2n-1)$--dimensional ones of the planar problem, once the mutual inclinations go to zero (this is related to a speculation in [V. I. Arnold. Russ. Math. Surv. 1963]). The main tool is a full reduction of the SO(3)--symmetry, which, in particular, retains symmetry by reflections and highlights a quasi--integrable structure, with a small remainder, independently of eccentricities and inclinations.
• ### Aspects of the planetary Birkhoff normal form(1310.0181)
Oct. 28, 2013 math.DS
The discovery in [G. Pinzari. PhD thesis. Univ. Roma Tre. 2009], [L. Chierchia and G. Pinzari, Invent. Math. 2011] of the Birkhoff normal form for the planetary many--body problem opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a {\sl direct} proof of the celebrated Arnold's Theorem [V. I. Arnold. Uspehi Math. Nauk. 1963] on the stability of planetary motions. In this paper, using a "ad hoc" set of symplectic variables, we develop an asymptotic formula for this normal form that may turn to be useful in applications. As an example, we provide two very simple applications to the three-body problem: we prove a conjecture by [V. I. Arnold. cit] on the "Kolmogorov set"of this problem and, using Nehoro{\v{s}}ev Theory [Nehoro{\v{s}}ev. Uspehi Math. Nauk. 1977], we prove, in the planar case, stability of all planetary actions over exponentially-long times, provided mean--motion resonances are excluded. We also briefly discuss perspectives and problems for full generalization of the results in the paper.
• ### On the Kolmogorov set for Many-Body Problems(1309.7028)
Oct. 1, 2013 math.DS
I defended my PhD Thesis in Rome, Universit\a Roma Tre, on April, 23, 2009, under the direction of Professor Luigi Chierchia. The judging committee was composed by Professors M. Berti, A. Celletti, C. Falcolini, J. F\'ejoz. Professors M. Berti and J. F\'ejoz refereed my thesis. The main result of my thesis is the first direct proof (the first general proof was given in [J. F\'ejoz, ETDS, 2004]) of a famous statement by V. I. Arnold (1963), usually referred to as "Arnold's Planetary Theorem". My proof of Arnold's Planetary Theorem relies on the rediscovery, during the year 2008, of a symplectic set of action-angle variables (described in \S 4 of my thesis) which perform explicitly the reduction of rotation invariance of the system. Indeed, even though in a different form, they had been previously considered by [F. Boigey, Cel. Mech. Dyn. Astr., 1982] and [A. Deprit, Cel. Mech. Dyn. Astr., 1983]. The version I found in 2008 corresponds to the "planetary" form of Boigey-Deprit variables, since it includes the elliptic elements of the instantaneous ellipses of the planets around the sun and for this reason is especially fitted to this problem . I then regularized "my" planetary variables to include co-planar and co-circular motions. This regularization leads to a set of mixed action-angle and rectangular variables analogous to Poincar\'e' variables but better fitted to rotation invariance of the system, since they exhibit a cyclic couple of conjugated variables. I finally applied my regularized variables to the problem, checked non-trivial torsion and obtained the proof of the theorem. I wish to thank J. F\'ejoz for mentioning my contribution to the proof of Arnold's Theorem, and especially my rediscovery of Deprit's reduction, in his paper [J. Fejoz, DCDS-A, 2013].
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2019-10-23 19:36:40
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http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-info.cgi/2021/MSC/MSC-2021-12
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# Technical Report MSC-2021-12
TR#: MSC-2021-12 Class: MSC Title: On-the-fly Model Checking with Guided Abstraction Authors: Gal Sade Supervisors: Orna Grumberg PDF Currently accessibly only within the Technion network Abstract: Model checking is an automatic verification method that gets a system model and a specification, and checks whether the model satisfies the specification. CTL is a branching time temporal logic suitable for specifying behaviors of both software and hardware systems. It enables specifying properties that cannot be expressed in linear time logics, such as LTL. An example of such a property is restartability, which means that in every reachable state, the system may return to its initial state, due to a reset or a recovery. Further, in many cases, CTL model checking algorithms can be easily extended to handle the alternation-free fragment of the powerful $\mu$-calculus logic. In this work, we present a novel approach, OMG, that combines on-the-fly verification with abstraction in order to obtain an efficient CTL model checking algorithm. On-the-fly verification ensures that only parts that are needed for determining the satisfaction of the specification are developed. The abstraction is used to form inductive invariants, allowing OMG to determine satisfaction of the CTL specification without traversing the entire state-space. We formalize the correctness of OMG, and present both an explicit version of the algorithm and a symbolic one. We implemented our algorithm on top of a combination of explicit and symbolic representations, where symbolic representations are handled with SAT/SMT solvers. Our experiments show that on a few examples, our algorithm outperforms a state-of-the-art SAT-based algorithm for CTL. Copyright The above paper is copyright by the Technion, Author(s), or others. Please contact the author(s) for more information
Remark: Any link to this technical report should be to this page (http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-info.cgi/2021/MSC/MSC-2021-12), rather than to the URL of the PDF files directly. The latter URLs may change without notice.
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2023-01-30 20:57:01
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https://tex.stackexchange.com/questions/422509/tikz-bug-in-intersections-library-when-naming-paths-inside-loop
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tikz: Bug in 'intersections' library when naming paths inside loop?
I draw two circles and name their path inside a \foreach loop. Trying to name intersections of these two paths throws an error unless I name a third path.
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}
\foreach \s in {+,-} {
\draw [name path={C\s}] (\s1,0) circle (1.5);}
\draw [name path=Ca] (0,1) circle (1.5); % <- comment line to see error
\path [name intersections={of=C- and C+}];
\draw [dashed] (intersection-1) -- (intersection-2);
\end{tikzpicture}
\end{document}
Without the marked line commented, the error message reads:
Package tikz Error: I do not know the path named C-'. Perhaps you misspelt it. ...\path [name intersections={of={C-} and C+}];
To me, this seems to be a bug. - Or am I doing something wrong?
• Try global name path. For each is act as a group. – Tarass Mar 22 '18 at 9:27
• @Tarass probably suggests \draw [name path global={C\s}] (\s1,0) circle (1.5);. This works. But it also works when you put any path (you do not have to name it) before \path [name intersections={of=C- and C+}];. Strange. – user121799 Mar 22 '18 at 9:41
• @Tarass: Your solution works. The explanation, however, does not fully apply. Otherwise, name intersection should consistently not work outside the loop. – FlorianL Mar 22 '18 at 10:00
• @FlorianL Yes, that's what I wanted to say with my comment. – user121799 Mar 22 '18 at 10:27
• The code in the foreach loop is in a TeX group, so all definitions are local to this group. If you place the uncommented line before the foreach, it does not work nether. That mean that there a something strange happens (bug ?). – Tarass Mar 22 '18 at 11:54
If you insert some draw command between the loop and the intersection command, it will work even without mentionning a name path. It is a curiosity...
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}
\draw [name path=Ca] (0,1) circle (1.5); % <- comment line to see error
\foreach \s in {+,-} {
\draw [name path global ={C\s}] (\s1,0) circle (1.5);}
% \draw (0,1) circle (1.5) ; % uncomment this line take off the global parameter and it will work to without accessing to the pathes names ...
\path [name intersections ={of=C- and C+}];
\draw [dashed] (intersection-1) -- (intersection-2);
\end{tikzpicture}
\end{document}
`
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2020-04-09 08:37:31
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https://deepai.org/publication/proper-semirings-and-proper-convex-functors
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DeepAI
Proper Semirings and Proper Convex Functors
Esik and Maletti introduced the notion of a proper semiring and proved that some important (classes of) semirings -- Noetherian semirings, natural numbers -- are proper. Properness matters as the equivalence problem for weighted automata over a semiring which is proper and finitely and effectively presented is decidable. Milius generalised the notion of properness from a semiring to a functor. As a consequence, a semiring is proper if and only if its associated "cubic functor" is proper. Moreover, properness of a functor renders soundness and completeness proofs for axiomatizations of equivalent behaviour. In this paper we provide a method for proving properness of functors, and instantiate it to cover both the known cases and several novel ones: (1) properness of the semirings of positive rationals and positive reals, via properness of the corresponding cubic functors; and (2) properness of two functors on (positive) convex algebras. The latter functors are important for axiomatizing trace equivalence of probabilistic transition systems. Our proofs rely on results that stretch all the way back to Hilbert and Minkowski.
• 7 publications
• 3 publications
05/04/2021
Switching 3-edge-colorings of cubic graphs
The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switch...
10/28/2019
Weak equivalence of higher-dimensional automata
This paper introduces a notion of weak equivalence for higher-dimensiona...
04/19/2021
On sublinear approximations for the Petersen coloring conjecture
If f:ℕ→ℕ is a function, then let us say that f is sublinear if lim_...
09/02/2021
Coalgebras for Bisimulation of Weighted Automata over Semirings
Weighted automata are a generalization of nondeterministic automata that...
09/25/2017
Intensional Constructed Numbers: Towards Formalizing the Notion of Algorithm
This work is meant to be a step towards the formal definition of the not...
08/31/2022
Geometrical tilings : distance, topology, compactness and completeness
We present the different distances on tilings of Rd that exist in the li...
12/01/2021
Lévy copulas: a probabilistic point of view
There is a one-to-one correspondence between Lévy copulas and proper cop...
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2023-03-25 01:49:44
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https://physics.stackexchange.com/questions/435708/why-is-the-electric-field-between-two-parallel-plates-uniform/435710
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# Why is the electric field between two parallel plates uniform? [duplicate]
According to sources online (eg HyperPhysics) the electric field strength around a point charge is $$E=k\frac{Q}{r^2}$$ This must means that the further you get away, the electric field should decrease with the square of the radius right?
But when these charges are placed into parallel plates somehow these will produce a uniform electric field? How I currently understand the physics can be summarized in the graph below of the electric field strength vs distance (Red and black lines are the electric field strengths of each plate and the blue is the resultant).
It seems pretty clear this wouldn't be a uniform electric field. What happens such that the electric field strength is changed into a uniform field? I initially thought integration, but wouldn't that still give an inverse relationship?
In response to the proposed duplicate, my question is concerning 2 plates where as the other is about one plate (however, I can't really decipher what is being asked in the other question)
## marked as duplicate by Aaron Stevens, user191954, Jon Custer, Kyle Kanos, ZeroTheHeroOct 24 '18 at 21:36
• The intensity of light from a point on the wall that reaches your eye drops as $1/r^2$ as you slowly back away from the wall, and yet the wall doesn't appear to get darker. – uhoh Oct 20 '18 at 12:22
• Possible duplicate of An uniformly charged infinite surface plane – Aaron Stevens Oct 20 '18 at 12:35
• How did you produce this graph? Is it just the contribution from one point on each plane? – Rahul Oct 20 '18 at 14:22
You have to remember that the electric field is a vector and not a scalar.
Let's consider a uniform electric charge on a infinite plane, and a point charge at height $$h$$ from it. Let's call the point directly below the point charge $$P$$.
The point charge gets attracted to every point on the plane, and the attractive force to a point on the plane at distance $$r$$ from $$P$$ is proportional to $$\frac{1}{r^2+h^2}$$.
Now, the fraction of this force in the direction of the plane is proportional to a
$$\frac{h}{\sqrt{r^2+h^2}}$$
fraction of the total force.
Thus, the force on the point charge in the direction of the plane is $$\int_0^\infty \frac{h}{(r^2+h^2)^{3/2}}\, 2 \pi r \, dr$$
This integral doesn't depend on $$h$$. To see this, we can make the substitution $$r = h s$$. We get $$\int_0^\infty \frac{h}{(h^2 s^2+h^2)^{3/2}} \,2 \pi h^2 s \, ds = \int_0^\infty \frac{1}{(s^2+1)^{3/2}} \,2 \pi s \, ds \, ,$$ which is $$2 \pi$$.
So the attractive force on the point charge doesn't depend on its distance from the plane. Thus, the electric field must be constant.
The intuitive answer is the following: When you have only one infinite plate the case is the same. If the plate is infinite in lenght, then "there is no spatial scale" in this problem (to an observer the plate looks the same from any height, the charge density does not change), there is no center and there is nothing (no physical features) that can tell you that you are closer or farther from the plate, any height would be the same. Of course you can measure the distance from the plate with a meter, but the point is that there is no features on the plate that will make one distance "different" that another. Now if you have two plates of oppossite charges it is the same, the field will be constant inside the plates and zero outside (as it cancels). This stops being true if the plates are finite, because now you have a scale: the size of the plate.
• What do you mean "there is no spatial scale"? The distance from the plate a gives spatial scale. In your answer, you merely state that you can ignore it with no justification. Why doesn't the same argument work with a point charge, or with a charge evenly distributed along an infinite wire? – Peter Shor Oct 20 '18 at 12:12
• @PeterShor for $1/r^2$ things all that matters is solid angle, and when your plate is infinite, there is no solid-angle-scale: the plate looks the same from 1 (classical) micron away as it does one light year. Now the wire, it get's skinnier the further you move away from it--even though it's already 0 width, it goes to $0\times\frac{r'}{r}$. – JEB Oct 20 '18 at 12:20
• @JEB: thank you for supplying the justification that was completely missing in the above answer. – Peter Shor Oct 20 '18 at 12:21
• @jEB thanks for the clarification, feel free to edit the answer fi my comment was still unclear) – Wolphram jonny Oct 20 '18 at 12:29
• @Wolphramjonny You're welcome. The OP wants to know "why", not "how", and I think your answer hits the mark: scale independence. Why it's scale independent when there are scale s present is a valid concern, I think that could be resolved with Gauss's law: if you imagine a right cylinder and do the field lines vs total charge in the box, the size of the cylinder doesn't matter. – JEB Oct 20 '18 at 14:49
If you place a uniform charge density, so not a single highly non-uniform point charge, on each of the plates, then far enough away from the edges of the plates the E field will be constant. Note that the plates should be much larger than their separation.
• Could you please explain WHY this happens though? Why will it become constant when you have two parallel plates? – John Hon Oct 20 '18 at 10:36
• One answer would be : because of Gauss's law. Another way to look at this is to consider a point P at height z above an infinite plain. $\vec E$ is the vector sum of all contributions from the infinite plain. When you change z then all charges are further away, which tends to decrease the field . However, we need the vectorial sum here. The angles under which the charges are seen from P are decreasing giving an increase of the field at P. Both effects exactly cancel. – my2cts Oct 20 '18 at 10:43
You're right !
It seems pretty clear this wouldn't be a uniform electric field.
This uniform field is only valid under certain approximations.
First the separation between plates must be very small compared to the size of the plates. This means that when we expand the exact solution in terms of powers $$\frac x L$$ where $$x$$ is the distance from plates and $$L$$ is a measure of the size of the plates, we can neglect anything but the constant term.
Typically we also assume an infinitely large plate, so that there are no boundary effects.
In a real situation, the edges ("boundary") clearly won't have the same field as the middle of the plate. So the field cannot be uniform for finite plates. Hence we approximate with infinite plates, which works well for the field in around the middle, and in the case of small separations this works well enough for most of the plate.
When you use separations comparable in magnitude to the size of the plates, you will not get a uniform field. It's easy to see that in that case the closer you are to one plate or another, the more influence it has on the field around you. In fact when the separation becomes much larger than the size f the plates, you can approximate the plates with points and you end up with an approximation of the field between two point charges, which is definitely not uniform.
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2019-11-19 18:20:12
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http://me1065.wikidot.com/thermal-systems-issues-in-micro-gravity-and-outer-space
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Thermal Systems Issues In Micro Gravity And Outer Space
Thermal systems are crucial components in the operation of spacecraft in micro-gravity. With respect to unmanned spacecraft, for example, satellites must maintain operable temperatures for their equipment to function and must execute station keeping techniques through the use of ‘burns’ (course corrections in which the main engine is lit).
###### Figure 1: A satellite in the midst of course correction [1]
Manned spacecraft have other concerns such as maintaining adequate pressure levels at a comfortable temperature for their crew (it gets very hot and very cold in space, there are no moderate temperatures), and must provide the essentials for sustaining life (i.e. water, oxygen, etc.) As expected, thermal systems in micro-gravity are not the same as thermal systems on the Earth; the common systems and configurations which are standard for many power, heating, and cooling cycles are not applicable in zero gravity conditions. Upon comparison to terrestrial conditions, several important differences affect the design and configurations of thermal systems in outer space. A lack of gravity creates a system governed by a set of laws that differs considerably, consequentially causing many Earthly thermal systems to be inoperable in outer space.
## Thermal Systems in Space Suits
A type of thermal system ecountered in manned spaceflight is the astronaut’s space suit itself. This portable thermal system pressurizes the astronauts during liftoff and Earth Orbit Insertion. The suit regulates temeperature (giving the astronaut extra heat or cooling as required) and provides oxygen as well. Astronauts are usually not wearing their pressurized suits except during Exta-Vehicular Activity (EVA). During an EVA an astronaut’s suit is critical in protecting the explorer from the extreme temperatures and vacuum conditions of outer space. These same concepts, but modified for the appropriate applications will be required when voyaging back to the Moon and to Mars. Creating compact and efficient thermal systems to accompany the space/Moon/Mars suits will improve the quality of exploration for the astronaut as well as the length of each outing.
## Modes of Heat Transfer
Firstly, there is no convection in space, which is the main form of heat transfer for thermal systems in Earth’s gravity [3]. Natural convection is negligible in micro-gravity because there is no such thing as ‘heavier’ or ‘lighter’ (a concept created by the gravitational force field). Without the concept of gravity there can be no natural convection currents (responsible for flame shape and propagation on Earth). Natural convection currents are also what cause hot air to rise. Therefore, boiling a working fluid in a thermal cycle is quite different than boiling a working fluid in the same cycle on Earth. The fluid does not rise because of the lack of buoyancy force (a direct result of losing the gravitational field). Note: the buoyancy force derived in a terrestrial environment equals:
(1)
\begin{align} \ F _{\rm Buoyancy} = \rho g V \end{align}
Clearly, as the gravity term approaches zero, this force becomes negligible compared to the other forces present in the system. Therefore, buoyancy driven flows must be stricken from design options in zero gravity thermal systems design.
## Crucial Forces
Without the force of gravity to shape and drive fluids, less effective physical forces (which are negligible in Earth fluids) take over in fluid response. These forces, such as the surface tension force, are usually difficult to study in regular gravity. The surface tension force is the dominant force in micro-gravity and it plays a key role in shaping a fluid (as well as creating more issues for thermal systems designers). For example, another concern in micro-gravity is that water cannot be transported using its gravitational potential energy as the flow driver. Instead, all fluids must be pumped in order to transport them (resulting in a larger power requirement for thermal systems in outer space).
The pumping of fluids introduces a variety of issues involving two phase flows in micro-gravity. Two phase flow refers to a flow that is a mixture of liquid and gas. For pumps in micro-gravity to function, they must introduce some gas (helium in some cases) to the pump since there is no air in order to displace the water. This flow type can be seen in spacecraft cooling systems, spacecraft life support systems, and spacecraft fuel systems. One issue with multiphase flow is that the bubbles of the gas phase can begin to collect along the walls of the systems piping. Moreover, because of the relative strength of the surface tension force, these wall bound bubbles can grow to the point where they totally occlude the liquid phases’ fluid flow [5]. Several solutions have been proposed and are currently being researched. For example, it has been posited that using fuel lines with constant bend radii (creating a coil-like shape) will reduce the tendency of the gas phase to collect to the pipe walls [6]. Solutions to this problem may lead to more efficient and reliable thermal systems in spacecraft in the future. Currently, University of Pittsburgh undergraduate students have been selected to conduct research in conjunction with NASA, in June 2008, to delve deeper into bubble topology in tubes with constant bend radii in a micro-gravity environment.
## Miscellaneous Concerns
The need for piping to be air-tight is magnified in a micro-gravity system. To illustrate this concern, one can imagine a fluid dripping from a leak in a gravitationally impacted thermal system. The fluid would collect directly underneath the breach, and although a large amount of seepage could have detrimental effects, a small outflow should have no major impact. However, in a system that is devoid of gravity, working fluid that escapes from the confines of the system is not confined to remain in a certain location. Liquids, which often have negative effects on electrical systems, can be harmful in even small quantities in outer space, and it is also possible for harmful substances to poison the environment of living beings more easily.
Another concern that is inflated in the confines of space is that of insulation. As previously stated, the temperature minimum in space is roughly 173 K in the shade and 393 K in the sunlight [7]. This gives an overall temperature difference of 220 K. When compared to a common Earthly temperature difference of 85 K, one can clearly see that insulation must deal with an additional order of magnitude when attempting to stabilize temperatures in outer space. This is especially important since radiation is proportional to the absolute temperature to the fourth power of a given object.
## Main Differences Between Earthly and Micro-gravitational Thermal Systems
Thermal systems are a conglomerate of many different physical attributes which span much farther than simply fluid physics. The lack of natural convection (mentioned earlier), for example suggests that the majority of effective heat transfer in low gravity must be achieved through forced convection or radiation. This greatly reduces the efficiency of several thermal systems components with respect to their performance in regular gravity.
The same phenomenon causes flames (which can be modeled as a fluid) to react differently as well because a fire’s main mode of transportation is through natural convection. Thermal systems observe several barriers in micro-gravity that are not present in Earth’s gravity, and it will be necessary to develop and evolve more efficient solutions to maintain the future of space exploration a reality.
## References:
1. Howland, J., “China’s Anti-Satellite Program Alarming,”, JINSA. http://www.jinsa.org/documents/200703/3721.jpg, March 1, 2007
2. Chandler, David L. and Kelly Young, “NASA Unveils Vision For Return to Moon” New ScientistSpace. http://space.newscientist.com/data/images/ns/cms/dn8022/dn8022-3_650.jpg. Sept 19, 2005
3. Antar, Basil N., and Vappu S. Nuotio-Antar. Fundamentals of Low Gravity Fluid Dynamics and Heat Transfer. Ann Arbor: CRC Press Inc., 1993.
4. Barry, Patrick L. “Bizarre Boiling,” Sceince@NASA. http://science.nasa.gov/headlines/y2001/ast07sep_2.htm. Sept 7, 2001
5. Microgravity Fluid Physics Research, Document: LG-1997-10-494, National Aeronautics and Space Administration, 1997
6. Braun, J.P., Collicot, S. H., Zero-Gravity Solutions of Droplets in a Bent Circular Cylinder, American Institute of Aeronautics and Astronautics, 2007.
7. Elert, Glenn “The Physics Factbook: Pressure in Outer Space,” http://hypertextbook.com/facts/2002/MimiZheng.shtml, 2003
8. Bray, Becky, “Plumbing the Space Station,” http://liftoff.msfc.nasa.gov/news/2001/News-StationPlumbing.asp, May 4, 2001
Authors:
J. Bennewitz. J. Bernardo, P. McKeon
page revision: 18, last edited: 06 Dec 2007 22:19
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2017-09-22 08:00:36
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http://tcs.nju.edu.cn/wiki/index.php/%E9%9A%8F%E6%9C%BA%E7%AE%97%E6%B3%95_(Spring_2014)/Concentration_of_Measure
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The Bounded Difference Method
Combining Azuma's inequality with the construction of Doob martingales, we have the powerful Bounded Difference Method for concentration of measures.
For arbitrary random variables
Given a sequence of random variables ${\displaystyle X_{1},\ldots ,X_{n}}$ and a function ${\displaystyle f}$. The Doob sequence constructs a martingale from them. Combining this construction with Azuma's inequality, we can get a very powerful theorem called "the method of averaged bounded differences" which bounds the concentration for arbitrary function on arbitrary random variables (not necessarily a martingale).
Theorem (Method of averaged bounded differences) Let ${\displaystyle {\boldsymbol {X}}=(X_{1},\ldots ,X_{n})}$ be arbitrary random variables and let ${\displaystyle f}$ be a function of ${\displaystyle X_{0},X_{1},\ldots ,X_{n}}$ satisfying that, for all ${\displaystyle 1\leq i\leq n}$, ${\displaystyle |\mathbf {E} [f({\boldsymbol {X}})\mid X_{1},\ldots ,X_{i}]-\mathbf {E} [f({\boldsymbol {X}})\mid X_{1},\ldots ,X_{i-1}]|\leq c_{i},}$ Then {\displaystyle {\begin{aligned}\Pr \left[|f({\boldsymbol {X}})-\mathbf {E} [f({\boldsymbol {X}})]|\geq t\right]\leq 2\exp \left(-{\frac {t^{2}}{2\sum _{i=1}^{n}c_{i}^{2}}}\right).\end{aligned}}}
Proof.
Define the Doob Martingale sequence ${\displaystyle Y_{0},Y_{1},\ldots ,Y_{n}}$ by setting ${\displaystyle Y_{0}=\mathbf {E} [f(X_{1},\ldots ,X_{n})]}$ and, for ${\displaystyle 1\leq i\leq n}$, ${\displaystyle Y_{i}=\mathbf {E} [f(X_{1},\ldots ,X_{n})\mid X_{1},\ldots ,X_{i}]}$. Then the above theorem is a restatement of the Azuma's inequality holding for ${\displaystyle Y_{0},Y_{1},\ldots ,Y_{n}}$.
${\displaystyle \square }$
For independent random variables
The condition of bounded averaged differences is usually hard to check. This severely limits the usefulness of the method. To overcome this, we introduce a property which is much easier to check, called the Lipschitz condition.
Definition (Lipschitz condition) A function ${\displaystyle f(x_{1},\ldots ,x_{n})}$ satisfies the Lipschitz condition, if for any ${\displaystyle x_{1},\ldots ,x_{n}}$ and any ${\displaystyle y_{i}}$, {\displaystyle {\begin{aligned}|f(x_{1},\ldots ,x_{i-1},x_{i},x_{i+1},\ldots ,x_{n})-f(x_{1},\ldots ,x_{i-1},y_{i},x_{i+1},\ldots ,x_{n})|\leq 1.\end{aligned}}}
In other words, the function satisfies the Lipschitz condition if an arbitrary change in the value of any one argument does not change the value of the function by more than 1.
The diference of 1 can be replaced by arbitrary constants, which gives a generalized version of Lipschitz condition.
Definition (Lipschitz condition, general version) A function ${\displaystyle f(x_{1},\ldots ,x_{n})}$ satisfies the Lipschitz condition with constants ${\displaystyle c_{i}}$, ${\displaystyle 1\leq i\leq n}$, if for any ${\displaystyle x_{1},\ldots ,x_{n}}$ and any ${\displaystyle y_{i}}$, {\displaystyle {\begin{aligned}|f(x_{1},\ldots ,x_{i-1},x_{i},x_{i+1},\ldots ,x_{n})-f(x_{1},\ldots ,x_{i-1},y_{i},x_{i+1},\ldots ,x_{n})|\leq c_{i}.\end{aligned}}}
The following "method of bounded differences" can be developed for functions satisfying the Lipschitz condition. Unfortunately, in order to imply the condition of averaged bounded differences from the Lipschitz condition, we have to restrict the method to independent random variables.
Corollary (Method of bounded differences) Let ${\displaystyle {\boldsymbol {X}}=(X_{1},\ldots ,X_{n})}$ be ${\displaystyle n}$ independent random variables and let ${\displaystyle f}$ be a function satisfying the Lipschitz condition with constants ${\displaystyle c_{i}}$, ${\displaystyle 1\leq i\leq n}$. Then {\displaystyle {\begin{aligned}\Pr \left[|f({\boldsymbol {X}})-\mathbf {E} [f({\boldsymbol {X}})]|\geq t\right]\leq 2\exp \left(-{\frac {t^{2}}{2\sum _{i=1}^{n}c_{i}^{2}}}\right).\end{aligned}}}
Proof.
For convenience, we denote that ${\displaystyle {\boldsymbol {X}}_{[i,j]}=(X_{i},X_{i+1},\ldots ,X_{j})}$ for any ${\displaystyle 1\leq i\leq j\leq n}$. We first show that the Lipschitz condition with constants ${\displaystyle c_{i}}$, ${\displaystyle 1\leq i\leq n}$, implies another condition called the averaged Lipschitz condition (ALC): for any ${\displaystyle a_{i},b_{i}}$, ${\displaystyle 1\leq i\leq n}$, ${\displaystyle \left|\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=a_{i}\right]-\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=b_{i}\right]\right|\leq c_{i}.}$ And this condition implies the averaged bounded difference condition: for all ${\displaystyle 1\leq i\leq n}$, ${\displaystyle \left|\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i]}\right]-\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]}\right]\right|\leq c_{i}.}$ Then by applying the method of averaged bounded differences, the corollary can be proved. For any ${\displaystyle a}$, by the law of total expectation, {\displaystyle {\begin{aligned}&\quad \,\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=a\right]\\&=\sum _{a_{i+1},\ldots ,a_{n}}\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=a,{\boldsymbol {X}}_{[i+1,n]}={\boldsymbol {a}}_{[i+1,n]}\right]\cdot \Pr \left[{\boldsymbol {X}}_{[i+1,n]}={\boldsymbol {a}}_{[i+1,n]}\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=a\right]\\&=\sum _{a_{i+1},\ldots ,a_{n}}\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=a,{\boldsymbol {X}}_{[i+1,n]}={\boldsymbol {a}}_{[i+1,n]}\right]\cdot \Pr \left[{\boldsymbol {X}}_{[i+1,n]}={\boldsymbol {a}}_{[i+1,n]}\right]\qquad ({\mbox{independence}})\\&=\sum _{a_{i+1},\ldots ,a_{n}}f({\boldsymbol {X}}_{[1,i-1]},a,{\boldsymbol {a}}_{[i+1,n]})\cdot \Pr \left[{\boldsymbol {X}}_{[i+1,n]}={\boldsymbol {a}}_{[i+1,n]}\right].\end{aligned}}} Let ${\displaystyle a=a_{i}}$ and ${\displaystyle b_{i}}$, and take the diference. Then {\displaystyle {\begin{aligned}&\quad \,\left|\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=a_{i}\right]-\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=b_{i}\right]\right|\\&=\left|\sum _{a_{i+1},\ldots ,a_{n}}\left(f({\boldsymbol {X}}_{[1,i-1]},a_{i},{\boldsymbol {a}}_{[i+1,n]})-f({\boldsymbol {X}}_{[1,i-1]},b_{i},{\boldsymbol {a}}_{[i+1,n]})\right)\Pr \left[{\boldsymbol {X}}_{[i+1,n]}={\boldsymbol {a}}_{[i+1,n]}\right]\right|\\&\leq \sum _{a_{i+1},\ldots ,a_{n}}\left|f({\boldsymbol {X}}_{[1,i-1]},a_{i},{\boldsymbol {a}}_{[i+1,n]})-f({\boldsymbol {X}}_{[1,i-1]},b_{i},{\boldsymbol {a}}_{[i+1,n]})\right|\Pr \left[{\boldsymbol {X}}_{[i+1,n]}={\boldsymbol {a}}_{[i+1,n]}\right]\\&\leq \sum _{a_{i+1},\ldots ,a_{n}}c_{i}\Pr \left[{\boldsymbol {X}}_{[i+1,n]}={\boldsymbol {a}}_{[i+1,n]}\right]\qquad ({\mbox{Lipschitz condition}})\\&=c_{i}.\end{aligned}}} Thus, the Lipschitz condition is transformed to the ALC. We then deduce the averaged bounded difference condition from ALC. By the law of total expectation, ${\displaystyle \mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]}\right]=\sum _{a}\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=a\right]\cdot \Pr[X_{i}=a\mid {\boldsymbol {X}}_{[1,i-1]}].}$ We can trivially write ${\displaystyle \mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i]}\right]}$ as ${\displaystyle \mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i]}\right]=\sum _{a}\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i]}\right]\cdot \Pr \left[X_{i}=a\mid {\boldsymbol {X}}_{[1,i-1]}\right].}$ Hence, the difference is {\displaystyle {\begin{aligned}&\quad \left|\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i]}\right]-\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]}\right]\right|\\&=\left|\sum _{a}\left(\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i]}\right]-\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=a\right]\right)\cdot \Pr \left[X_{i}=a\mid {\boldsymbol {X}}_{[1,i-1]}\right]\right|\\&\leq \sum _{a}\left|\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i]}\right]-\mathbf {E} \left[f({\boldsymbol {X}})\mid {\boldsymbol {X}}_{[1,i-1]},X_{i}=a\right]\right|\cdot \Pr \left[X_{i}=a\mid {\boldsymbol {X}}_{[1,i-1]}\right]\\&\leq \sum _{a}c_{i}\Pr \left[X_{i}=a\mid {\boldsymbol {X}}_{[1,i-1]}\right]\qquad ({\mbox{due to ALC}})\\&=c_{i}.\end{aligned}}} The averaged bounded diference condition is implied. Applying the method of averaged bounded diferences, the corollary follows.
${\displaystyle \square }$
Applications
Occupancy problem
Throwing ${\displaystyle m}$ balls uniformly and independently at random to ${\displaystyle n}$ bins, we ask for the occupancies of bins by the balls. In particular, we are interested in the number of empty bins.
This problem can be described equivalently as follows. Let ${\displaystyle f:[m]\rightarrow [n]}$ be a uniform random function from ${\displaystyle [m]\rightarrow [n]}$. We ask for the number of ${\displaystyle i\in [n]}$ that ${\displaystyle f^{-1}(i)}$ is empty.
For any ${\displaystyle i\in [n]}$, let ${\displaystyle X_{i}}$ indicate the emptiness of bin ${\displaystyle i}$. Let ${\displaystyle X=\sum _{i=1}^{n}X_{i}}$ be the number of empty bins.
${\displaystyle \mathbf {E} [X_{i}]=\Pr[{\mbox{bin }}i{\mbox{ is empty}}]=\left(1-{\frac {1}{n}}\right)^{m}.}$
By the linearity of expectation,
${\displaystyle \mathbf {E} [X]=\sum _{i=1}^{n}\mathbf {E} [X_{i}]=n\left(1-{\frac {1}{n}}\right)^{m}.}$
We want to know how ${\displaystyle X}$ deviates from this expectation. The complication here is that ${\displaystyle X_{i}}$ are not independent. So we alternatively look at a sequence of independent random variables ${\displaystyle Y_{1},\ldots ,Y_{m}}$, where ${\displaystyle Y_{j}\in [n]}$ represents the bin into which the ${\displaystyle j}$th ball falls. Clearly ${\displaystyle X}$ is function of ${\displaystyle Y_{1},\ldots ,Y_{m}}$.
We than observe that changing the value of any ${\displaystyle Y_{i}}$ can change the value of ${\displaystyle X}$ by at most 1, because one ball can affect the emptiness of at most one bin. Thus as a function of independent random variables ${\displaystyle Y_{1},\ldots ,Y_{m}}$, ${\displaystyle X}$ satisfies the Lipschitz condition. Apply the method of bounded differences, it holds that
${\displaystyle \Pr \left[\left|X-n\left(1-{\frac {1}{n}}\right)^{m}\right|\geq t{\sqrt {m}}\right]=\Pr[|X-\mathbf {E} [X]|\geq t{\sqrt {m}}]\leq 2e^{-t^{2}/2}}$
Thus, for sufficiently large ${\displaystyle n}$ and ${\displaystyle m}$, the number of empty bins is tightly concentrated around ${\displaystyle n\left(1-{\frac {1}{n}}\right)^{m}\approx {\frac {n}{e^{m/n}}}}$
Pattern Matching
Let ${\displaystyle {\boldsymbol {X}}=(X_{1},\ldots ,X_{n})}$ be a sequence of characters chosen independently and uniformly at random from an alphabet ${\displaystyle \Sigma }$, where ${\displaystyle m=|\Sigma |}$. Let ${\displaystyle \pi \in \Sigma ^{k}}$ be an arbitrarily fixed string of ${\displaystyle k}$ characters from ${\displaystyle \Sigma }$, called a pattern. Let ${\displaystyle Y}$ be the number of occurrences of the pattern ${\displaystyle \pi }$ as a substring of the random string ${\displaystyle X}$.
By the linearity of expectation, it is obvious that
${\displaystyle \mathbf {E} [Y]=(n-k+1)\left({\frac {1}{m}}\right)^{k}.}$
We now look at the concentration of ${\displaystyle Y}$. The complication again lies in the dependencies between the matches. Yet we will see that ${\displaystyle Y}$ is well tightly concentrated around its expectation if ${\displaystyle k}$ is relatively small compared to ${\displaystyle n}$.
For a fixed pattern ${\displaystyle \pi }$, the random variable ${\displaystyle Y}$ is a function of the independent random variables ${\displaystyle (X_{1},\ldots ,X_{n})}$. Any character ${\displaystyle X_{i}}$ participates in no more than ${\displaystyle k}$ matches, thus changing the value of any ${\displaystyle X_{i}}$ can affect the value of ${\displaystyle Y}$ by at most ${\displaystyle k}$. ${\displaystyle Y}$ satisfies the Lipschitz condition with constant ${\displaystyle k}$. Apply the method of bounded differences,
${\displaystyle \Pr \left[\left|Y-{\frac {n-k+1}{m^{k}}}\right|\geq tk{\sqrt {n}}\right]=\Pr \left[\left|Y-\mathbf {E} [Y]\right|\geq tk{\sqrt {n}}\right]\leq 2e^{-t^{2}/2}}$
Combining unit vectors
Let ${\displaystyle u_{1},\ldots ,u_{n}}$ be ${\displaystyle n}$ unit vectors from some normed space. That is, ${\displaystyle \|u_{i}\|=1}$ for any ${\displaystyle 1\leq i\leq n}$, where ${\displaystyle \|\cdot \|}$ denote the vector norm (e.g. ${\displaystyle \ell _{1},\ell _{2},\ell _{\infty }}$) of the space.
Let ${\displaystyle \epsilon _{1},\ldots ,\epsilon _{n}\in \{-1,+1\}}$ be independently chosen and ${\displaystyle \Pr[\epsilon _{i}=-1]=\Pr[\epsilon _{i}=1]=1/2}$.
Let
${\displaystyle v=\epsilon _{1}u_{1}+\cdots +\epsilon _{n}u_{n},}$
and
${\displaystyle X=\|v\|.}$
This kind of construction is very useful in combinatorial proofs of metric problems. We will show that by this construction, the random variable ${\displaystyle X}$ is well concentrated around its mean.
${\displaystyle X}$ is a function of independent random variables ${\displaystyle \epsilon _{1},\ldots ,\epsilon _{n}}$. By the triangle inequality for norms, it is easy to verify that changing the sign of a unit vector ${\displaystyle u_{i}}$ can only change the value of ${\displaystyle X}$ for at most 2, thus ${\displaystyle X}$ satisfies the Lipschitz condition with constant 2. The concentration result follows by applying the method of bounded differences:
${\displaystyle \Pr[|X-\mathbf {E} [X]|\geq 2t{\sqrt {n}}]\leq 2e^{-t^{2}/2}.}$
Dimension Reduction
Consider a problem as follows: We have a set of ${\displaystyle n}$ points in a high-dimensional Euclidean space ${\displaystyle \mathbf {R} ^{d}}$. We want to project the points onto a space of low dimension ${\displaystyle \mathbf {R} ^{k}}$ in such a way that pairwise distances of the points are approximately the same as before.
Formally, we are looking for a map ${\displaystyle f:\mathbf {R} ^{d}\rightarrow \mathbf {R} ^{k}}$ such that for any pair of original points ${\displaystyle u,v}$, ${\displaystyle \|f(u)-f(v)\|}$ distorts little from ${\displaystyle \|u-v\|}$, where ${\displaystyle \|\cdot \|}$ is the Euclidean norm, i.e. ${\displaystyle \|u-v\|={\sqrt {(u_{1}-v_{1})^{2}+(u_{2}-v_{2})^{2}+\ldots +(u_{d}-v_{d})^{2}}}}$ is the distance between ${\displaystyle u}$ and ${\displaystyle v}$ in Euclidean space.
This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensional space, however, computations on high-dimensional data are usually hard due to the infamous "curse of dimensionality". The computational tasks can be greatly eased if we can project the data points onto a space of low dimension while the pairwise relations between the points are approximately preserved.
Johnson-Lindenstrauss Theorem
The Johnson-Lindenstrauss Theorem states that it is possible to project ${\displaystyle n}$ points in a space of arbitrarily high dimension onto an ${\displaystyle O(\log n)}$-dimensional space, such that the pairwise distances between the points are approximately preserved.
Johnson-Lindenstrauss Theorem For any ${\displaystyle 0<\epsilon <1}$ and any positive integer ${\displaystyle n}$, let ${\displaystyle k}$ be a positive integer such that ${\displaystyle k\geq 4(\epsilon ^{2}/2-\epsilon ^{3}/3)^{-1}\ln n}$ Then for any set ${\displaystyle V}$ of ${\displaystyle n}$ points in ${\displaystyle \mathbf {R} ^{d}}$, there is a map ${\displaystyle f:\mathbf {R} ^{d}\rightarrow \mathbf {R} ^{k}}$ such that for all ${\displaystyle u,v\in V}$, ${\displaystyle (1-\epsilon )\|u-v\|^{2}\leq \|f(u)-f(v)\|^{2}\leq (1+\epsilon )\|u-v\|^{2}}$. Furthermore, this map can be found in expected polynomial time.
The random projections
The map ${\displaystyle f:\mathbf {R} ^{d}\rightarrow \mathbf {R} ^{k}}$ is done by random projection. There are several ways of applying the random projection. We adopt the one in the original Johnson-Lindenstrauss paper.
The projection (due to Johnson-Lindenstrauss) Let ${\displaystyle A}$ be a random ${\displaystyle k\times d}$ matrix that projects ${\displaystyle \mathbf {R} ^{d}}$ onto a uniform random k-dimensional subspace. Multiply ${\displaystyle A}$ by a fixed scalar ${\displaystyle {\sqrt {\frac {d}{k}}}}$. For every ${\displaystyle v\in \mathbf {R} ^{d}}$, ${\displaystyle v}$ is mapped to ${\displaystyle {\sqrt {\frac {d}{k}}}Av}$.
The projected point ${\displaystyle {\sqrt {\frac {d}{k}}}Av}$ is a vector in ${\displaystyle \mathbf {R} ^{k}}$.
The purpose of multiplying the scalar ${\displaystyle {\sqrt {\frac {d}{k}}}}$ is to guarantee that ${\displaystyle \mathbf {E} \left[\left\|{\sqrt {\frac {d}{k}}}Av\right\|^{2}\right]=\|v\|^{2}}$.
Besides the uniform random subspace, there are other choices of random projections known to have good performances, including:
• A matrix whose entries follow i.i.d. normal distributions. (Due to Indyk-Motwani)
• A matrix whose entries are i.i.d. ${\displaystyle \pm 1}$. (Due to Achlioptas)
In both cases, the matrix is also multiplied by a fixed scalar for normalization.
A proof of the Theorem
We present a proof due to Dasgupta-Gupta, which is much simpler than the original proof of Johnson-Lindenstrauss. The proof is for the projection onto uniform random subspace. The idea of the proof is outlined as follows:
1. To bound the distortions to pairwise distances, it is sufficient to bound the distortions to the length of unit vectors.
2. A uniform random subspace of a fixed unit vector is identically distributed as a fixed subspace of a uniform random unit vector. We can fix the subspace as the first k coordinates of the vector, thus it is sufficient to bound the length (norm) of the first k coordinates of a uniform random unit vector.
3. Prove that for a uniform random unit vector, the length of its first k coordinates is concentrated to the expectation.
From pairwise distances to norms of unit vectors
Let ${\displaystyle w\in \mathbf {R} ^{d}}$ be a vector in the original space, the random ${\displaystyle k\times d}$ matrix ${\displaystyle A}$ projects ${\displaystyle w}$ onto a uniformly random k-dimensional subspace of ${\displaystyle \mathbf {R} ^{d}}$. We only need to show that
{\displaystyle {\begin{aligned}\Pr \left[\left\|{\sqrt {\frac {d}{k}}}Aw\right\|^{2}<(1-\epsilon )\|w\|^{2}\right]&\leq {\frac {1}{n^{2}}};\quad {\mbox{and}}\\\Pr \left[\left\|{\sqrt {\frac {d}{k}}}Aw\right\|^{2}>(1+\epsilon )\|w\|^{2}\right]&\leq {\frac {1}{n^{2}}}.\end{aligned}}}
Think of ${\displaystyle w}$ as a ${\displaystyle w=u-v}$ for some ${\displaystyle u,v\in V}$. Then by applying the union bound to all ${\displaystyle {n \choose 2}}$ pairs of the ${\displaystyle n}$ points in ${\displaystyle V}$, the random projection ${\displaystyle A}$ violates the distortion requirement with probability at most
${\displaystyle {n \choose 2}\cdot {\frac {2}{n^{2}}}=1-{\frac {1}{n}},}$
so ${\displaystyle A}$ has the desirable low-distortion with probability at least ${\displaystyle {\frac {1}{n}}}$. Thus, the low-distortion embedding can be found by trying for expected ${\displaystyle n}$ times (recalling the analysis fo geometric distribution).
We can further simplify the problem by normalizing the ${\displaystyle w}$. Note that for nonzero ${\displaystyle w}$'s, the statement that
${\displaystyle (1-\epsilon )\|w\|^{2}\leq \left\|{\sqrt {\frac {d}{k}}}Aw\right\|^{2}\leq (1+\epsilon )\|w\|^{2}}$
is equivalent to that
${\displaystyle (1-\epsilon ){\frac {k}{d}}\leq \left\|A\left({\frac {w}{\|w\|}}\right)\right\|^{2}\leq (1+\epsilon ){\frac {k}{d}}.}$
Thus, we only need to bound the distortions for the unit vectors, i.e. the vectors ${\displaystyle w\in \mathbf {R} ^{d}}$ that ${\displaystyle \|w\|=1}$. The rest of the proof is to prove the following lemma for the unit vector in ${\displaystyle \mathbf {R} ^{d}}$.
Lemma 3.1 For any unit vector ${\displaystyle w\in \mathbf {R} ^{d}}$, it holds that ${\displaystyle \Pr \left[\|Aw\|^{2}<(1-\epsilon ){\frac {k}{d}}\right]\leq {\frac {1}{n^{2}}};}$ ${\displaystyle \Pr \left[\|Aw\|^{2}>(1+\epsilon ){\frac {k}{d}}\right]\leq {\frac {1}{n^{2}}}.}$
As we argued above, this lemma implies the Johnson-Lindenstrauss Theorem.
Random projection of fixed unit vector ${\displaystyle \equiv }$ fixed projection of random unit vector
Let ${\displaystyle w\in \mathbf {R} ^{d}}$ be a fixed unit vector in ${\displaystyle \mathbf {R} ^{d}}$. Let ${\displaystyle A}$ be a random matrix which projects the points in ${\displaystyle \mathbf {R} ^{d}}$ onto a uniformly random ${\displaystyle k}$-dimensional subspace of ${\displaystyle \mathbf {R} ^{d}}$.
Let ${\displaystyle Y\in \mathbf {R} ^{d}}$ be a uniformly random unit vector in ${\displaystyle \mathbf {R} ^{d}}$. Let ${\displaystyle B}$ be such a fixed matrix which extracts the first ${\displaystyle k}$ coordinates of the vectors in ${\displaystyle \mathbf {R} ^{d}}$, i.e. for any ${\displaystyle Y=(Y_{1},Y_{2},\ldots ,Y_{d})}$, ${\displaystyle BY=(Y_{1},Y_{2},\ldots ,Y_{k})}$.
In other words, ${\displaystyle Aw}$ is a random projection of a fixed unit vector; and ${\displaystyle BY}$ is a fixed projection of a uniformly random unit vector.
A key observation is that:
Observation The distribution of ${\displaystyle \|Aw\|}$ is the same as the distribution of ${\displaystyle \|BY\|}$.
The proof of this observation is omitted here.
With this observation, it is sufficient to work on the subspace of the first ${\displaystyle k}$ coordinates of the uniformly random unit vector ${\displaystyle Y\in \mathbf {R} ^{d}}$. Our task is now reduced to the following lemma.
Lemma 3.2 Let ${\displaystyle Y=(Y_{1},Y_{2},\ldots ,Y_{d})}$ be a uniformly random unit vector in ${\displaystyle \mathbf {R} ^{d}}$. Let ${\displaystyle Z=(Y_{1},Y_{2},\ldots ,Y_{k})}$ be the projection of ${\displaystyle Y}$ to the subspace of the first ${\displaystyle k}$-coordinates of ${\displaystyle \mathbf {R} ^{d}}$. Then ${\displaystyle \Pr \left[\|Z\|^{2}<(1-\epsilon ){\frac {k}{d}}\right]\leq {\frac {1}{n^{2}}};}$ ${\displaystyle \Pr \left[\|Z\|^{2}>(1+\epsilon ){\frac {k}{d}}\right]\leq {\frac {1}{n^{2}}}.}$
Due to the above observation, Lemma 3.2 implies Lemma 3.1 and thus proves the Johnson-Lindenstrauss theorem.
Note that ${\displaystyle \|Z\|^{2}=\sum _{i=1}^{k}Y_{i}^{2}}$. Due to the linearity of expectations,
${\displaystyle \mathbf {E} [\|Z\|^{2}]=\sum _{i=1}^{k}\mathbf {E} [Y_{i}^{2}]}$.
Since ${\displaystyle Y}$ is a uniform random unit vector, it holds that ${\displaystyle \sum _{i=1}^{d}Y_{i}^{2}=\|Y\|^{2}=1}$. And due to the symmetry, all ${\displaystyle \mathbf {E} [Y_{i}^{2}]}$'s are equal. Thus, ${\displaystyle \mathbf {E} [Y_{i}^{2}]={\frac {1}{d}}}$ for all ${\displaystyle i}$. Therefore,
${\displaystyle \mathbf {E} [\|Z\|^{2}]=\sum _{i=1}^{k}\mathbf {E} [Y_{i}^{2}]={\frac {k}{d}}}$.
Lemma 3.2 actually states that ${\displaystyle \|Z\|^{2}}$ is well-concentrated to its expectation.
Concentration of the norm of the first ${\displaystyle k}$ entries of uniform random unit vector
We now prove Lemma 3.2. Specifically, we will prove the ${\displaystyle (1-\epsilon )}$ direction:
${\displaystyle \Pr[\|Z\|^{2}<(1-\epsilon ){\frac {k}{d}}]\leq {\frac {1}{n^{2}}}}$.
The ${\displaystyle (1+\epsilon )}$ direction is proved with the same argument.
Due to the discussion in the last section, this can be interpreted as a concentration bound for ${\displaystyle \|Z\|^{2}}$, which is a sum of ${\displaystyle Y_{1}^{2},Y_{2}^{2},\ldots ,Y_{k}^{2}}$. This hints us to use Chernoff-like bounds. However, for uniformly random unit vector ${\displaystyle Y}$, ${\displaystyle Y_{i}}$'s are not independent (because of the constraint that ${\displaystyle \|Y\|=1}$). We overcome this by generating uniform unit vectors from independent normal distributions.
The following is a very useful fact regarding the generation of uniform unit vectors.
Generating uniform unit vector Let ${\displaystyle X_{1},X_{2},\ldots ,X_{d}}$ be i.i.d. random variables, each drawn from the normal distribution ${\displaystyle N(0,1)}$. Let ${\displaystyle X=(X_{1},X_{2},\ldots ,X_{d})}$. Then ${\displaystyle Y={\frac {1}{\|X\|}}X}$ is a uniformly random unit vector.
Then for ${\displaystyle Z=(Y_{1},Y_{2},\ldots ,Z_{k})}$,
${\displaystyle \|Z\|^{2}=Y_{1}^{2}+Y_{2}^{2}+\cdots +Y_{k}^{2}={\frac {X_{1}^{2}}{\|X\|^{2}}}+{\frac {X_{2}^{2}}{\|X\|^{2}}}+\cdots +{\frac {X_{k}^{2}}{\|X\|^{2}}}={\frac {X_{1}^{2}+X_{2}^{2}+\cdots +X_{k}^{2}}{X_{1}^{2}+X_{2}^{2}+\cdots +X_{d}^{2}}}}$.
To avoid writing a lot of ${\displaystyle (1-\epsilon )}$'s. We write ${\displaystyle \beta =(1-\epsilon )}$. The first inequality (the lower tail) of Lemma 3.2 can be written as:
{\displaystyle {\begin{aligned}\Pr \left[\|Z\|^{2}<{\frac {\beta k}{d}}\right]&=\Pr \left[{\frac {X_{1}^{2}+X_{2}^{2}+\cdots +X_{k}^{2}}{X_{1}^{2}+X_{2}^{2}+\cdots +X_{d}^{2}}}<{\frac {\beta k}{d}}\right]\\&=\Pr \left[d(X_{1}^{2}+X_{2}^{2}+\cdots +X_{k}^{2})<\beta k(X_{1}^{2}+X_{2}^{2}+\cdots +X_{d}^{2})\right]\\&=\Pr \left[(\beta k-d)\sum _{i=1}^{k}X_{i}^{2}+\beta k\sum _{i=k+1}^{d}X_{i}^{2}>0\right].&\qquad (**)\end{aligned}}}
The probability is a tail probability of the sum of ${\displaystyle d}$ independent variables. The ${\displaystyle X_{i}^{2}}$'s are not 0-1 variables, thus we cannot directly apply the Chernoff bounds. However, the following two key ingredients of the Chernoff bounds are satisfiable for the above sum:
• The ${\displaystyle X_{i}^{2}}$'s are independent.
• Because ${\displaystyle X_{i}^{2}}$'s are normal, it is known that the moment generating functions for ${\displaystyle X_{i}^{2}}$'s can be computed as follows:
Fact 3.3 If ${\displaystyle X}$ follows the normal distribution ${\displaystyle N(0,1)}$, then ${\displaystyle \mathbf {E} \left[e^{\lambda X^{2}}\right]=(1-2\lambda )^{-{\frac {1}{2}}}}$, for ${\displaystyle \lambda \in \left(-\infty ,1/2\right)}$
Therefore, we can re-apply the technique of the Chernoff bound (applying Markov's inequality to the moment generating function and optimizing the parameter ${\displaystyle \lambda }$) to bound the probability ${\displaystyle (**)}$:
{\displaystyle {\begin{aligned}&\quad \,\Pr \left[(\beta k-d)\sum _{i=1}^{k}X_{i}^{2}+\beta k\sum _{i=k+1}^{d}X_{i}^{2}>0\right]\\&=\Pr \left[\exp \left\{(\beta k-d)\sum _{i=1}^{k}X_{i}^{2}+\beta k\sum _{i=k+1}^{d}X_{i}^{2}\right\}>1\right]\\&=\Pr \left[\exp \left\{\lambda \left((\beta k-d)\sum _{i=1}^{k}X_{i}^{2}+\beta k\sum _{i=k+1}^{d}X_{i}^{2}\right)\right\}>1\right]&\quad ({\text{for }}\lambda >0)\\&\leq \mathbf {E} \left[\exp \left\{\lambda \left((\beta k-d)\sum _{i=1}^{k}X_{i}^{2}+\beta k\sum _{i=k+1}^{d}X_{i}^{2}\right)\right\}\right]&\quad {\text{(by Markov inequality)}}\\&=\prod _{i=1}^{k}\mathbf {E} \left[e^{\lambda (\beta k-d)X_{i}^{2}}\right]\cdot \prod _{i=k+1}^{d}\mathbf {E} \left[e^{\lambda \beta kX_{i}^{2}}\right]&\quad ({\text{independence of }}X_{i})\\&=\mathbf {E} \left[e^{\lambda (\beta k-d)X_{1}^{2}}\right]^{k}\cdot \mathbf {E} \left[e^{\lambda \beta kX_{1}^{2}}\right]^{d-k}&\quad {\text{(symmetry)}}\\&=(1-2\lambda (\beta k-d))^{-{\frac {k}{2}}}(1-2\lambda \beta k)^{-{\frac {d-k}{2}}}&\quad {\text{(by Fact 3.3)}}\end{aligned}}}
The last term ${\displaystyle (1-2\lambda (\beta k-d))^{-{\frac {k}{2}}}(1-2\lambda \beta k)^{-{\frac {d-k}{2}}}}$ is minimized when
${\displaystyle \lambda ={\frac {1-\beta }{2\beta (d-k\beta )}},}$
so that
{\displaystyle {\begin{aligned}&\quad \,(1-2\lambda (\beta k-d))^{-{\frac {k}{2}}}(1-2\lambda \beta k)^{-{\frac {d-k}{2}}}\\&=\beta ^{\frac {k}{2}}\left(1+{\frac {(1-\beta )k}{(d-k)}}\right)^{\frac {d-k}{2}}\\&\leq \exp \left({\frac {k}{2}}(1-\beta +\ln \beta )\right)&\qquad ({\text{since }}\left(1+{\frac {(1-\beta )k}{(d-k)}}\right)^{\frac {d-k}{(1-\beta )k}}\leq e)\\&=\exp \left({\frac {k}{2}}(\epsilon +\ln(1-\epsilon ))\right)&\qquad (\beta =1-\epsilon )\\&\leq \exp \left(-{\frac {k\epsilon ^{2}}{4}}\right)&\qquad ({\text{by Taylor expansion }}\ln(1-\epsilon )\leq -\epsilon -{\frac {\epsilon ^{2}}{2}}),\end{aligned}}}
which is is ${\displaystyle \leq {\frac {1}{n^{2}}}}$ for the choice of k in the Johnson-Lindenstrauss theorem that
${\displaystyle k\geq 4(\epsilon ^{2}/2-\epsilon ^{3}/3)^{-1}\ln n}$.
So we have proved that
${\displaystyle \Pr[\|Z\|^{2}<(1-\epsilon ){\frac {k}{d}}]\leq {\frac {1}{n^{2}}}}$.
With the same argument, the other direction can be proved so that
${\displaystyle \Pr[\|Z\|^{2}>(1+\epsilon ){\frac {k}{d}}]\leq \exp \left({\frac {k}{2}}(-\epsilon +\ln(1+\epsilon ))\right)\leq \exp \left(-{\frac {k(\epsilon ^{2}/2-\epsilon ^{3}/3)}{2}}\right)}$,
which is also ${\displaystyle \leq {\frac {1}{n^{2}}}}$ for ${\displaystyle k\geq 4(\epsilon ^{2}/2-\epsilon ^{3}/3)^{-1}\ln n}$.
Lemma 3.2 is proved. As we discussed in the previous sections, Lemma 3.2 implies Lemma 3.1, which implies the Johnson-Lindenstrauss theorem.
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2018-06-25 15:42:32
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https://chem.libretexts.org/Textbook_Maps/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/14%3A_Nuclear_Magnetic_Resonance_Spectroscopy/14.8%3A_The_n_1_Rule_Applies_Only_to_First-Order_Spectra
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# 14.8: The n+1 Rule Applies Only to First-Order Spectra
The (n+1) Rule, an empirical rule used to predict the multiplicity and, in conjunction with Pascal’s triangle, splitting pattern of peaks in 1H and 13C NMR spectra, states that if a given nucleus is coupled (see spin coupling) to n number of nuclei that are equivalent (see equivalent ligands), the multiplicity of the peak is n+1. eg. 1:
The three hydrogen nuclei in 1, $$H_a$$, $$H_b$$, and $$H_c$$, are equivalent. Thus, 1H NMR spectrum of 1 $$H_a$$s only one peak. $$H_a$$, $$H_b$$, and $$H_c$$ are coupled to no hydrogen nuclei. Thus, for $$H_a$$, $$H_b$$, and $$H_c$$, n=0; (n+1) = (0+1) = 1. The multiplicity of the peak of $$H_a$$, $$H_b$$, and $$H_c$$ is one. The peak $$H_a$$s one line; it is a singlet. eg. 2:
There are two sets of equivalent hydrogen nuclei in 2:
• Set 1: $$H_a$$
• Set 2: $$H_b$$, $$H_c$$
Thus, the 1H NMR spectrum of 2 $$H_a$$s two peaks, one due to $$H_a$$ and the other to $$H_b$$ and $$H_c$$.
The peak of $$H_a$$: There are two vicinal hydrogens to $$H_a$$: $$H_b$$ and $$H_c$$. $$H_b$$ and $$H_c$$ are equivalent to each other but not to $$H_a$$. Thus, for $$H_a$$, n=2; (n+1) = (2+1) = 3. The multiplicity of the peak of $$H_a$$ is three. The peak $$H_a$$s three lines; from the Pascal’s triangle, it is a triplet.
The peak of $$H_b$$ and $$H_c$$: There is only one vicinal hydrogen to $$H_b$$ and $$H_c$$: $$H_a$$. $$H_a$$ is not equivalent to $$H_b$$ and $$H_c$$. Thus, for $$H_b$$ and $$H_c$$, n=1; (n+1) = (1+1) = 2. The multiplicity of the peak of $$H_b$$ and $$H_c$$ is two. The peak $$H_a$$s two lines, from the Pascal’s triangle, it is a doublet.
To determine the multiplicity of a peak of a nucleus coupled to more than one set of equivalent nuclei, apply the (n+1) Rule independently to each other.
eg:
There are three set of equivalent hydrogen nuclei in 3:
• Set 1: $$H_a$$
• Set 2: $$H_b$$
• Set 3: $$H_c$$
peak of $$H_a$$:
multiplicity of the peak of $$H_a = 2 \times 2 = 4$$. To determine the splitting pattern of the peak of $$H_a$$, use the Pascal’s triangle, based on the observation that, for alkenyl hydrogens, $$J_{cis} > J_{gem}$$.
The peak of $$H_a$$ is a doublet of a doublet.
peak of $$H_b$$:
multiplicity of the peak of $$H_b = 2 \times 2 = 4$$. To determine the splitting pattern of the peak of $$H_b$$, use the Pascal’s triangle, based on the observation that, for alkenyl hydrogens, $$J_{trans} > J_{gem}$$.
The peak of $$H_b$$ is a doublet of a doublet.
peak of $$H_c$$:
multiplicity of the peak of $$H_c = 2 \times 2 = 4$$. To determine the splitting pattern of the peak of $$H_c$$, use the Pascal’s triangle based on the observation that, for alkenyl hydrogens, $$J_{trans} > J_{cis}$$.
The peak of $$H_c$$ is a doublet of a doublet.
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2017-08-16 21:48:44
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http://math.stackexchange.com/questions/303758/a-sequence-converging-weakly-in-ellp-for-p-1-and-failing-to-converge-wea
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# A sequence converging weakly in $\ell^p$, for $p >1$ and failing to converge weakly for $p=1$
For $1 \le p < \infty$ and each index $n$, let $e_n \in \ell^p$ have $n$-th component 1 and all other componenets $0$. I want to show that $p>1 \Rightarrow \{e_n\} \to 0$ weakly in $\ell^p$ and that this is not the case in $\ell^1$. Any suggestions?
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Suggestions: For $p>1$, try using the definitions, try to show that the definition of weak convergence to zero is satisfied, see if you get stuck anywhere. For $p=\infty$, use the defintion, and try to find a countexample that shows that the definition of weak convergence to zero is not satisfied. – Jonas Meyer Feb 14 '13 at 5:24
The issue is that I am not quite sure what the definition is in the case of $\ell^p$. Could you please provide it? – user44069 Feb 14 '13 at 5:25
Where did you encounter this problem such that you do not have access to the definitions? You are asking here for the solution to a problem but do not know what it means. Your first question might instead be, "what is the definition of weak convergence in $\ell^p$?" This is a special case of the notion of weak convergence in Banach space, but you will want to use the identification of the dual of $\ell^p$ with $\ell^q$ with $p+q=pq$. – Jonas Meyer Feb 14 '13 at 5:32
I am sorry, I am not very precise. In the very beginning of these notes, one has the definition of weak convergence (people.math.gatech.edu/~heil/6338/summer08/section9a.pdf). My exact question is: what is the dual in this particular case and how is the inner product defined for this case? – user44069 Feb 14 '13 at 5:37
Thank you for that. (Oops I meant $1$ where I wrote $\infty$ in my first comment.) Dual of $\ell^p$ when $1\leq p<\infty$ is $\ell^q$, with $q$ the conjugate exponent of $p$, $pq=p+q$ if $p>1$, $q=\infty$ if $p=1$. The functional on $\ell^p$ corresponding to an element of $\ell^q$ is shown in the Wikipedia link above and in Example 1.9 on page 3 of the pdf you linked to. It seems implicit that the reader is assumed to know about the duals of $\ell^p$ when $1\leq p<\infty$. I do not know what sources you may have for that fact, but it should be found in most introductions to $\ell^p$ spaces. – Jonas Meyer Feb 14 '13 at 5:50
The definition of $\ell^p, \ 1\leq p<+\infty$ is $$\ell^p=\left\{x=(x(k))_{k\in\mathbb N}:\|x\|_p:=\sqrt[p]{\displaystyle\sum_{k=1}^\infty |x(k)|^p}<+\infty\right\}$$ and $$\ell^\infty=\left\{x=(x(k))_{k\in\mathbb N}:\|x\|_\infty:=\sup\{|x(k)|:k\in\mathbb N\}<+\infty\right\}.$$ A sequense $(x_n)_{n\in\mathbb N}$ in $\ell^p, \ 1\leq p<+\infty$ , converges weakly to $x\in\ell^p$ (denoted $x_n\rightharpoonup x$ )
if for any $y\in\ell^q$ (where $q$ is defined by $p+q=pq$ for $p\neq1$ and $q=+\infty$ for $p=1$): $$\langle x_n,y \rangle \xrightarrow[n\to+\infty]{}\langle x,y \rangle,$$ where $\langle x,y \rangle :=\displaystyle\sum_{k=1}^{\infty}x(k)y(k)$ for $x=(x(k))_{k\in\mathbb N}\in\ell^p,y=(y(k))_{k\in\mathbb N}\in\ell^q$.
We want to show that for $p>1, \ e_n\rightharpoonup 0$ in $\ell^p$.
Equivalently that $\displaystyle\sum_{k=1}^{\infty}e_n(k)y(k)\xrightarrow[n\to+\infty]{}0$ for all $(y(k))_{k\in\mathbb N}\in\ell^q$.
Note that $e_n(k)y(k)=\delta_{nk}y(k) \ \forall n,k \in\mathbb N$; from $\displaystyle \sum_{k=1}^\infty|y(k)|<+\infty$ it follows that $$\displaystyle\sum_{k=1}^{\infty}e_n(k)y(k)=y(n)\xrightarrow[n\to\infty]{}0.$$
Now for $p=1$ find an element $(y(k))_{k\in\mathbb N}\in\ell^\infty$
s.t. $\displaystyle\sum_{k=1}^{\infty}e_n(k)y(k) \not\rightarrow0$ (hint: find one with $\displaystyle\sum_{k=1}^{\infty}e_n(k)y(k)=1, \ \forall n\in\mathbb N$).
-
Following the same steps as here, we can establish the following weak convergence conditions.
Let $1<p<\infty$ and $\{x^{(n)}\}\subset \ell^p$. This sequences converges weakly to $0$ in $\ell^p$ is and only if the following two conditions are satisfied:
1. We have $\lim_{n\to +\infty}x^{(n)}_N= 0$ for all $N\in\Bbb N$.
2. The sequence $\{\lVert x^{(n)}\rVert_p\}\subset\Bbb R$ is bounded.
To see that $\{e_n\}$ doesn't converge to $0$ in $\ell^1$, consider $L\colon\ell^1\to \Bbb R$, $L(x):=\sum_{j=0}^{+\infty}x_j$. Actually, strong and weak convergence of sequences are the same thing in $\ell^1$.
-
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2014-07-26 19:24:58
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https://prideout.net/blog/streamlines/
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# Visualizing 2D Vector Fields
Philip Rideout, 18 July 2018
Physics textbooks often have beautiful hand-drawn illustrations. For example, here's something from The Feynman Lectures on Physics (1964):
How can we generate images like this algorithmically?
David Banks and Greg Turk came up with an interesting method for creating pleasing streamline diagrams in 1996 (1) . They used an iterative process to minimize an energy function that represents the overall “quality” of the image.
In 2005, Mebarki et al introduced a new algorithm called Farthest Point Seed Strategy which generates similarly pleasing diagrams very efficiently (2) . It works by partitioning the space using Delaunay triangulation. Their method was used to generate the following illustration:
While the above diagram is visually pleasing, it does not give a good sense of movement or directionality. This can be addressed by adding arrows:
Some of the arrowheads are deformed due to discontinuities in the source data (more on this later in the post). The above images were created with clumpy, a simple command line tool that I'm hosting on github.
Similar diagrams can be generated more robustly (though much less efficiently) with the streamplot functionality in matplotlib, which also allows you to vary color and streamline width.
As an alternative to streamlines, matplotlib also supports quiver diagrams:
Arrows can add clutter to a visualization, whereas motion-blurred particles can convey directionality with very little visual noise. The following image is a screencap of a particle animation with motion blur, again created with clumpy.
Of course, an even better visualization is to simply show the animation. The following video is a 4-second loop, made seamless by a simple algorithm that I'll describe shortly.
Shrinking the particle radius and increasing the count creates a fairly pleasing result. I think this might be the best visualization so far:
The above videos were rendered in less than 1 second on my laptop with clumpy.
# # Loopable Animation
When capturing a particle simulation, how can we make a video that seamlessly loops over an n-second interval? When the interval ends, it would be visually jarring to reset all particles to their original positions en masse.
An easy fix is to add an initialization phase that moves each particle along its streamline for a random amount of time before returning it to its starting point. After this initial phase, each particle stores a unique integer “age” so that it knows when to reset. Here's the procedure in pseudocode:
kLoopDuration = 4 seconds
kExpectedFps = 60
kFramesPerLoop = kLoopDuration * kExpectedFps
record_video = False
current_frame = 0
for each particle:
particle.age_offset = random integer in [0, kFramesPerLoop)
particle.current_position = random point sample per Christensen[4] or Bridson[5]
particle.current_age = 0
for each time step:
current_frame++
# Record video only during the second interval.
if current_frame >= kFramesPerLoop:
record_video = True
elif current_frame >= 2 * kFramesPerLoop:
record_video = False
for each particle:
particle.current_position += kTimeStep * particle.velocity
particle.current_age++
# In the first interval, reset the particle after a random amount of time.
if current_frame < kFramesPerLoop and current_frame >= particle.age_offset:
particle.current_position = particle.original_position
particle.current_age = 0
particle.age_offset = kFramesPerLoop
# In every subsequent interval, simply reset the particle when it expires.
if current_frame >= kFramesPerLoop and particle.current_age >= kFramesPerLoop:
particle.current_position = particle.original_position
particle.current_age = 0
clumpy uses a C++ implementation of this in its advect_points command.
# # Generating a Random Velocity Field
For a quick diversion, I'll explain how I generated the data for the above visualizations.
I started with a C1 smooth field of scalars in [−1,+1]. This can be created by summing up a few octaves of gradient noise:
Next, I needed to transform the scalar field into a vector field. One way to do this is to interpret each scalar as an angle:
This yields the following vector field. The red channel shows the X component, and the green channel shows Y. I decided not to use this since it has little resemblance to anything physical.
Another strategy for converting scalar noise into vectors is to compute (or recover) the gradient of the noise, then take the 2D perpendicular.
The resulting velocity field has the nice property of behaving like an incompressible fluid. Moreover it behaves reasonably near boundaries if you ramp down the source values according to a distance value (3). Here's the distance field that I used:
This is the result after multiplying distance with noise and computing the perp of the gradient:
One issue with this approach is that the distance field does not have C1 continuity, so the resulting “fluid” has unrealistic sharp angles. The above images were all generated using clumpy.
# # References
[1] Image-Guided Streamline Placement by David Banks, Greg Turk. (1996)
[2] Farthest Point Seeding for Placement of Streamlines by Mebarki, Alliez, Devillers. (2005)
[3] Curl-Noise for Procedural Fluid Flow by Bridson, Hourihan, Nordenstam. (2007)
[4] Progressive Multi-Jittered Sample Sequences by Per Christensen, Andrew Kensler, Charlie Kilpatrick. (2018)
[5] Fast Poisson Disk Sampling in Arbitrary Dimensions by Robert Bridson. (2007)
Bloopers! Coolest bug ever.
formatted by Markdeep 1.01
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2019-04-20 21:10:13
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https://economics.stackexchange.com/tags/stochastic-processes/new
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# Tag Info
I think you may substitute it directly as mentioned? $$v(A,k) = \max log(c) + \beta E[v(A',k')|A]$$ Applying the value function expression (note period change): $v(A',k') = \frac{log(k')}{1-\beta} + v(A',1)$ Substituting: $$v(A,k) = \max log(c) + \beta E\left[\frac{log(k')}{1-\beta} + v(A',1)|A\right]$$ By linearity: v(A,k) = \max log(c) + \beta E\left[\...
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2021-04-18 05:35:28
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https://www.techwhiff.com/learn/2-take-the-following-system-it-represents-the/331173
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2) Take the following system. It represents the frequency control of a steam-turbine gener- ator unit...
Question:
2) Take the following system. It represents the frequency control of a steam-turbine gener- ator unit Assume TRH = 6, FHP = 0.3, Tg = 0.1, and M = 10, D = 1.0 (Both in p.u.) Ape Load ref. Tos+1 11+TRE PS 1+TRHS DUM (Note that AP represents the change in electric load. - Assume that the load APe, increases by 0.01 p.u. What is the effect on the frequency, if the droop R was infinity, i.e., what is the frequency deviation if no governors were present? -Plot the root-locus as k = 1, varies from zero to infinity. -Set R=0.05, Assume that the load changes as a step function AP = 0.01u(t) p.u. What is the effect of the change in the load on the frequency?
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34. The Light Record Corporation has been on the calendar year since its inception five years ago. It wishes to change to an April 30. natural business year. For 2019, the calendar year of proposed change, Light had a taxable income of $200,000, of which$50,000 was earned from January through April...
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2022-09-30 05:38:40
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https://goe.porcelis.pl/en/ihas
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Aug 24, 2020 · Easily an isometry has to send one side to another (this can be done in n possible choices) and then can have two orientation (for example the side A B could go to C D or with A → C and B → D or with A → D and B → C) . So we conclude that dihedral group D n has 2 n elements: n rotations and n symmetries. Share. edited Aug 24, 2020 at 10:44.. "/>
1. zg
2. ey
Dihedral group of order 12
By fn
mc
I reverted back all the changes and thought to update all together like "ng update @angular/[email protected] @angular/[email protected] @anular/[email protected]" this time @angular/animations, @angular/forms , @angular/core, @angular/compiler etc., upgraded with 13.3.11, only cdk updated with @12, and got many other errors. Note: I used below combinations, but didn't work. Jul 23, 2017 · I want to check that this cycle notation is correct for the Dihedral Group of order $12$. I found this graph in Wikipedia. I did not find the cycle notation. May you please tell me if my cycle notation is correct? Here is my solution: I will name the vertices $1,2,3,4,5,6$ in the clockwise direction. 1) Rotation 0 degrees $(1)(2)(3)(4)(5)(6)$.
mw
A conjugacy class is a set of the form. Cl ( a) = { b a b − 1 ∣ b ∈ G } for some a ∈ G. (a) Prove that the centralizer of an element of a in G is a subgroup of the group G. (b) Prove that the order (the number of elements) of every conjugacy class in G divides the order of the group G. Add to solve later. Sponsored Links.
2017. 10. 10. · Let .Here are the irreducible Brauer characters: The decomposition matrix and the Cartan matrix: [] {}. The dihedral energy reported in .log file is smaller then in the simulation calculated with gromacs 4.5.6, but my standard deviation of dihedral angles look similar to the restrained in the previous version which worked with the old parameters .... dihedral_style fourier command. dihedral_style harmonic command. dihedral_style helix command. dihedral_style hybrid command.. Dihedral Symmetry of Order 12. Each snowflake in the main image has the dihedral symmetry of a natual regular hexagon. The group formed by these symmetries is also called the dihedral group of degree 6. Order refers to the number of elements in the group, and degree refers to the number of the sides or the number of rotations. The order is.
The semidirect product is isomorphic to the dihedral group of order 6 if φ(0) is the identity and φ(1) is the non-trivial automorphism of C 3, which inverses the elements. Thus we get: ( n 1 , 0) * ( n 2 , h 2 ) = ( n 1 + n 2 , h 2 ).
The dihedral groups are an infinite family of groups which are in general noncommutative. Each dihedral group is defined to be the group of linear symmetries of a regular-gon. Properties. The order of is . The group has a presentation in the form . For , is noncommutative. See also. Symmetric group; Cyclic group; RotationThis article is a stub.. Dihedral Group of Order 12 in Cycle Notation. Ask Question Asked 4 years, 10 months ago. Modified 4 years, 10 months ago. Viewed 2k times 0 $\begingroup$ I want to check that this cycle notation is correct for the Dihedral Group of order $12$. I found this graph in Wikipedia. 3. Show that the dihedral group Do of order 12 has a nonidentity element z such that zg = gz for all g € D. (2 points) Question: 3. Show that the dihedral group Do of order 12 has a nonidentity element z such that zg = gz for all g € D. (2 points). We will use semidirect products to describe all 5 groups of order 12 up to isomorphism. Two are abelian and the others are A 4, D 6, and a less familiar group. Theorem 1. Every group of order 12 is a semidirect product of a group of order 3 and a group of order 4. Proof. Let jGj= 12 = 22 3. A 2-Sylow subgroup has order 4 and a 3-Sylow subgroup .... 2 The group of translations of the plane; 3 The symmetry group of a square: dihedral group of order 8. 3.1 Generating the group; 3.2 Normal subgroup; 4 Free group on two generators; 5 The set of maps. 5.1 The sets of maps from a set to a group; 6 Automorphism groups. 6.1 Groups of permutations; 6.2 Matrix groups; 7 See also; 8 References. order 12: the whole group is the only subgroup of order 12. (b) Which ones are normal? Solution. The trivial group f1g and the whole group D6 are certainly normal. Among the subgroups of order 2, only f1;x3g is normal: x(xiy)x 1 = xi+2y, so f1;xiyg is not normal for any i. The subgroup of order 3 is normal. A conjugacy class is a set of the form. Cl ( a) = { b a b − 1 ∣ b ∈ G } for some a ∈ G. (a) Prove that the centralizer of an element of a in G is a subgroup of the group G. (b) Prove that the order (the number of elements) of every conjugacy class in G divides the order of the group G. Add to solve later. Sponsored Links. .
Aug 01, 2022 · The group D_5 is one of the two groups of order 10. Unlike the cyclic group C_(10), D_5 is non-Abelian. The molecule ruthenocene (C_5H_5)_2Ru belongs to the group D_(5h), where the letter h indicates invariance under a reflection of the fivefold axis (Arfken 1985, p. 248)..
Dihedral Group D_5. The group is one of the two groups of order 10. Unlike the cyclic group , is non-Abelian. The molecule ruthenocene belongs to the group , where the letter indicates invariance under a reflection of the fivefold axis (Arfken 1985, p. 248). Its multiplication table is illustrated above. The symmetry group of a regular hexagon is a group of order 12, the Dihedral group D6 . It is generated by a rotation R 1 and a reflection r 0. R n denotes the rotation by angle n * 2 pi/6 with respect the center of the hexagon. r n denotes the reflection in the line at angle n * pi/6 with respect to a fixed line passing through the center and. Solution for Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to Z12 and a subgroup isomorphic to Z20. No need to.
The Dihedral Group is one of the two groups of Order 6. It is the non-Abelian group of smallest Order. Examples of include the Point Groups known as , , , , the symmetry group of the Equilateral Triangle, and the group of permutation of three objects. Its elements satisfy , and four of its elements satisfy , where 1 is the Identity Element.
Gmust have order 1;2 or p. We must show that G˘=D p. We rst claim that Ghas an element of order p. If not, every nonidentity element of Ghas order 2, which makes Ga nite elementary abelian 2-group. Thus G˘=Z=2Z Z=2Z for a nite number of copies of Z=2Z. But then jGjis a power of 2, which is impossible. Let r2Ghave order pand set H= hri.. Nov 9, 2014. #50. Birdman100 said: "longitudinal dihedral " or incidence have nothing to do with the longitudinal stability ! However incidence angle influence balance about CG. Horizontal tail force can be both positive and negative and it depends on the overall setup of. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. •-ignh à ignore hydrogens (gromacs will re ... Jun 12, 2010 · 53 """Set up dihedral analysis. 54 55:Arguments: 56 *dihedrals* 57 list of tuples; each tuple contains :. The dihedral group is the basis 4 1 − 1 1 −1 −1 1 −1 1 of the applications discussed in this paper. In Section 3 we 5 2 0 −2 0 0 0 0 0 derive the canonical decomposition [Eq.. 1 Answer. It is the half-twist. If you have an 2 n -gon, then rotate it by n. This is your centre (along with the trivial element). Note that for G = a, b, , c we have that g ∈ Z ( G) ⇔ g a = a g, g b = b g, , g c = c g. That is, g is in the centre if and only if it commutes with all the generators.. The Dihedral Group is one of the two groups of Order 6. It is the non-Abelian group of smallest Order. Examples of include the Point Groups known as , , , , the symmetry group of the Equilateral Triangle, and the group of permutation of three objects. Its elements satisfy , and four of its elements satisfy , where 1 is the Identity Element.
A conjugacy class is a set of the form. Cl ( a) = { b a b − 1 ∣ b ∈ G } for some a ∈ G. (a) Prove that the centralizer of an element of a in G is a subgroup of the group G. (b) Prove that the order (the number of elements) of every conjugacy class in G divides the order of the group G. Add to solve later. Sponsored Links.
Nov 9, 2014. #50. Birdman100 said: "longitudinal dihedral " or incidence have nothing to do with the longitudinal stability ! However incidence angle influence balance about CG. Horizontal tail force can be both positive and negative and it depends on the overall setup of.
Automorphisms group of the dihedral group D4: Let ^ 2 2 4 D , yx with the defining relation 4 x, y 1xy x 1, be the dihedral group of order 8. Now, the conjugate classes of D4 are : ^2 3 e, yx 2 . So, 2 # 4 / D to a group of order 4. So, D4 has 4 inner automorphisms one of which is the identity. Then, let the other 3 inner automorphisms be.
Suitable atoms might be the atom O2999 is connected to (e.g. C2967) and the next atom (e.g. N2960). Of course, any atoms could be used provided their atom-number is lower than the atom being defined, i.e. here lower than 3000. Once the four atoms (here H3000, O2999, C2967, and N2960) are identified, work out the angle and dihedral.It's easy to make a dihedral scan in. Jul 23, 2017 · I want to check that this cycle notation is correct for the Dihedral Group of order $12$. I found this graph in Wikipedia. I did not find the cycle notation. May you please tell me if my cycle notation is correct? Here is my solution: I will name the vertices $1,2,3,4,5,6$ in the clockwise direction. 1) Rotation 0 degrees $(1)(2)(3)(4)(5)(6)$. Answer (1 of 3): The dihedral group D_5 is the group of symmetries of a regular pentagon. There are five axes of reflection, each axis passing through a vertex and the midpoint of the opposite side. Reflections always have order 2, so five of the elements of D_5 have order 2. Besides the five r. Mar 01, 2020 · A dihedral group is a group which elements are the result of a composition of two permutations with predetermined properties. Keith Conrad in his article entitled “dihedral group” specifically .... Abstract Given any abelian group G, the generalized dihedral group of G is the semi-direct product of C 2 = {±1} and G, denoted D(G) = C 2 n ϕ G. The homomorphism ϕ maps C 2 to the automorphism group of G, providing an action on G by inverting elements. The groups D(G) generalize the classical dihedral groups, as evidenced by the isomor-. The dihedral group of order 6 - D 6 D_6. and. the binary dihedral group of order 12 - 2 D 12 2 D_{12} correspond to the Dynkin label D5 in the ADE-classification. Properties. D 6 D_6 is isomorphic to the symmetric group on 3 elements. The dihedral groups are an infinite family of groups which are in general noncommutative. Each dihedral group is defined to be the group of linear symmetries of a regular-gon. Properties. The order of is . The group has a presentation in the form . For , is noncommutative. See also. Symmetric group; Cyclic group; RotationThis article is a stub..
Solution for Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to Z12 and a subgroup isomorphic to Z20. No need to.
The dihedral groups are an infinite family of groups which are in general noncommutative. Each dihedral group is defined to be the group of linear symmetries of a regular-gon. Properties. The order of is . The group has a presentation in the form . For , is noncommutative. See also. Symmetric group; Cyclic group; RotationThis article is a stub.. The following Cayley table shows the effect of composition in the group D 3 (the symmetries of an equilateral triangle). One of the simplest examples of a non-abelian group is the dihedral group of order 6. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of. I have defined four groups, each group consist of COM of set of atoms. defined as follows in plumed format g1,g2,g3,g4 are four groups (these are not. Dihedral Angles. by Greg Egan. In order to find the dihedral angle between hyperfaces of the polytope, we will initially calculate half that angle: δ, the angle between the hyperplane containing the. By definition, “The group of symmetries of a regular polygon P n of n sides is called the dihedral group of degree n and denoted by D(n)” (Bhattacharya, Jain, & Nagpaul, 1994). This project will make use of the definition that all of the permutations for each of the dihedral groups D(n) preserve the cyclic order of the vertices of each .... The dihedral group Dn with 2n elements is generated by 2 elements, r and d, where r has order n, and d has order 2, rd=dr-1, and <d> n <r> = {e}. That implies Dn ={e,r,..,r n-1,d,dr,..,dr n-1} where those are distinct. You can generalize rd=dr-1 as r k d=dr-k. You can use that to see how any two elements multiply. Answer (1 of 3): The dihedral group D_5 is the group of symmetries of a regular pentagon. There are five axes of reflection, each axis passing through a vertex and the midpoint of the opposite side. Reflections always have order 2, so five of the elements of D_5 have order 2. Besides the five r. Hey mathmari!! I don't think there are more of order 2, 3, and 6. Oh, and aren't $\langle\sigma^2\rangle$ and $\langle\sigma^4\rangle$ the same sub group?. Problem 52. Let n be a positive integer. Let D 2 n be the dihedral group of order 2 n. Using the generators and the relations, the dihedral group D 2 n is given by. D 2 n = r, s ∣ r n = s 2 = 1, s r = r − 1 s . Put θ = 2 π / n. θ] is the matrix representation of the linear transformation T which rotates the x - y plane about the origin.
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Miyamoto dihedral group. Our assumed upper bound of 12 on the order of a Miyamoto dihedral group is motivated by the fact that in the Monster, a pair of 2Ainvolutions generates a dihedral group of order at most 12 [GMS]. Recently, Sakuma [Sa] announced that 12 is an upper bound for the order of a Miyamoto dihedral group in an OZVOA (= CFT type.
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Consider the dihedral group D12 of order 12 consisting of isometries of the regular hexagon. Label the vertices of the hexagon 1, 2, ..., 6. Let o denote the reflection in the axis passing through vertex 1 and let p denote the rotation anticlockwise through an angle of /3. (a) Show that p3 and o commute.. Question # 3, (50 pts) Let G = D6, which is the dihedral group of order 12, ie . C = {1, a, a, a, a4,0°, b, ba, ba?, ba3, ba", bal and o(a) = 6, o(b) = 2 and aba = b. Consider the cyclic subgroup H = bay of D6 generated by the element ba2、Questions 1, a, a, a",a,a,0,ba, ba-, ba, ba, ba ·(30 pts) Find the following right and left cosets (20 ....
In my supplement "Small Groups" I mentioned the "dicyclic group of order 12." This group has presentation (a,b : a6 = 1,a3 = b2,b−1ab = a−1) (from ... In fact, the dihedral group can be classified in terms of the properties of the generators: Theorem. Characterization of Dihedral Groups. (Gallian's Theorem 26.5.).
DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of D n, including the normal subgroups. We will also introduce an in nite group that resembles the dihedral groups and has all of them as quotient groups. 1. Abstract characterization of D n The group D. The Dihedral Group is one of the two groups of Order 6. It is the non-Abelian group of smallest Order. Examples of include the Point Groups known as , , , , the symmetry group of the Equilateral Triangle, and the group of permutation of three objects. Its elements satisfy , and four of its elements satisfy , where 1 is the Identity Element. by explaining two musical actions. 1 The dihedral group of order 24 is the group of symmetries of a regular 12-gon, that is, of a 12-gon with all sides of the same length and all angles of the same measure. Algebraically, the dihedral group of order 24 is the group generated by two elements, s and t, subject to the three relations. Summary. Description. Cayley graph of the dihedral group of order 16.svg. English: The number on the rim of the node circle indicates an order of an element the node represents. Node's colour marks conjugacy class of element with exception that elements of group's center have the same colour (light gray) despite every one of them being of its.
Contemporary group theorists prefer D 2 n over D n as the notation for the dihedral group of order 2 n. Although this notation is overly explicit, it does help to resolve the ambiguity with the Lie type D l which corresponds to the orthogonal group Ω + ( 2 l , q ) ..
- Consider the dihedral group of order 12, given here with its group presentation: D12 = (1, 81 82 = pe = 1, rs = sr-). Each question below is worth 2 points. 1. Determine Z(D2). 2. Determine which elements in D12 are conjugates and collect them as sets (i.e. the "conjugacy classes of D12. 3. Verify that the class equation holds for the finite .... Answer to Solved Question # 3, (50 pts) Let G = D6, which is the. Math; Advanced Math; Advanced Math questions and answers; Question # 3, (50 pts) Let G = D6, which is the dihedral group of order 12, ie.
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by explaining two musical actions. 1 The dihedral group of order 24 is the group of symmetries of a regular 12-gon, that is, of a 12-gon with all sides of the same length and all angles of the same measure. Algebraically, the dihedral group of order 24 is the group generated by two elements, s and t, subject to the three relations. 52 Dihedral Groups Another special type of permutation group is the dihedral from MATH 235 at McGill University. This is a short tutorial on how to scan a dihedral angle in the GaussView 5.0 /Gaussian 09w Computational Chemistry Package. Background These are videos of. Suitable atoms might be the atom O2999 is connected to (e.g. C2967) and the next atom (e.g. N2960). Of course, any atoms could be used provided their atom-number is lower than the atom being defined, i.e. here lower than 3000. Once the four atoms (here H3000, O2999, C2967, and N2960) are identified, work out the angle and dihedral.It's easy to make a dihedral scan in.
We will use semidirect products to describe all 5 groups of order 12 up to isomorphism. Two are abelian and the others are A 4, D 6, and a less familiar group. Theorem 1. Every group of order 12 is a semidirect product of a group of order 3 and a group of order 4. Proof. Let jGj= 12 = 22 3. A 2-Sylow subgroup has order 4 and a 3-Sylow subgroup ....
AbstractThe group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are. The dihedral group D_6 gives the group of symmetries of a regular hexagon. The group generators are given by a counterclockwise rotation through pi/3 radians and reflection in a line joining the midpoints of two opposite edges. If x denotes rotation and y reflection, we have D_6=<x,y:x^6=y^2=1,xy=yx^(-1)>. (1) From this, the group elements can be listed as D_6={x^i,yx^i:0<=i<=5}. Let us denote G = D 8. Let K x be the conjugacy class in G containing the element x. Note that r < C G ( r) ⪇ G and the order | r | = 4. Hence we must have C G ( r) = r . Thus the element r has | G: C G ( r) | = 2 conjugates in G. Since s r s − 1 = r 3, the conjugacy class K r containing r is { r, r 3 }. Oct 03, 2019 · Dihedral groups & abeliangroups. 1. Dihedral Groups & Abelian Groups Diana Mary George Assistant Professor Department of Mathematics St. Mary’s College Thrissur-680020 Kerala. 2. Dihedral Groups,Diana Mary George,St.Mary’s College Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance.. Answer: I will tell you how to do this. The elements of the Cartesian product have the form gh=(g,1)(1,h) where the g and h commute and g is in D6 and h is in Q8. The order of gh is the least common multiple of the orders of g and h. Since the order of elements in Q8 are either 2 or 4 and since 6.
2021. 11. 23. · DIHEDRAL GROUPS 3 In D n it is standard to write rfor the counterclockwise rotation by 2ˇ=nradians. This rotation depends on n, so the rin D 3 means something di erent from the rin D 4.However, as long as we are dealing with one value of n, there shouldn’t be confusion. Theorem 2.3. The nrotations in D n are 1;r;r2;:::;rn 1. Here and below, we designate the identity.
A conjugacy class is a set of the form. Cl ( a) = { b a b − 1 ∣ b ∈ G } for some a ∈ G. (a) Prove that the centralizer of an element of a in G is a subgroup of the group G. (b) Prove that the order (the number of elements) of every conjugacy class in G divides the order of the group G. Add to solve later. Sponsored Links. View the full answer. Transcribed image text: 7. Let G = S6 and consider the subgroup H = D6, the dihedral group of order 12. (a) List the six rotations in D6. Express each element as a product of disjoint cycles. 7. Let G = S6 and consider the subgroup H = D6, the dihedral group of order 12. (a) List the six rotations in D6.. 362 Corollary: The order of an element of a finite group divides the order of the group. §14.4.Euler's Theorem Recall that if m is any positive integer ℤ m # denotes the group of all numbers from 1 to m − 1 which are coprime with m, under the operation of multiplication modulo m. [The coprimeness is what ensures the existence of.
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creates K as the dihedral group of order 24, $$D_{12}$$; stores the list of subgroups output by K.conjugacy_classes_subgroups() in the variable sg; prints the elements of the list; selects the second subgroup in the list, and lists its elements. Let G be the group of symmetr ies of hexagon, that is, dihedral gro up of order 12. Then G has two generators a and b which s atisfies the following re lations. Conjugacy Class of Dihedral Group of Order 12. I am given the dihedral group of order 12: D 12 =< a, b: a 6 = b 2 = 3, b a = a 5 b >, where a is a rotation of a hexagon by 60 degrees, and b is a reflection across a diagonal of two vertices. I am looking to find the conjugacy class c l D 12 ( b). 2022. 1. 29. · Corollary: The order of an element of a finite group divides the order of the group. §14.4. Euler’s Theorem Recall that if m is any positive integer ℤ m # denotes the group of all numbers from 1 to m − 1 which are coprime with m, under the operation of multiplication modulo m. [The coprimeness is what ensures the existence of inverses]. Let G = Da, the dihedral group of order 12, with generators r of order 6 and 8 of order 2, and let K = (a) Find the index |G: K of K in G and the set G/K = {rK|XEG) of left cosets of K in G. Express elements of G in their normal form rabo Sa <6,0 Sb <2. (b) Let H = D2, the dihedral group of order 4, with generators p and o each of order two. The. Summary. Description. Cayley graph of the dihedral group of order 16.svg. English: The number on the rim of the node circle indicates an order of an element the node represents. Node's colour marks conjugacy class of element with exception that elements of group's center have the same colour (light gray) despite every one of them being of its.
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General To achieve an exact restart of a simulation, one must preserve all the state variables of the system. In practice,. Jun 12, 2010 · 53 """Set up dihedral analysis. 54 55:Arguments: 56 *dihedrals* 57 list of tuples; each tuple contains :class:gromacs.cbook.IndexBuilder 58 atom selection commands. 59 *labels* 60 optional list of labels for the dihedrals. The dihedral groups are an infinite family of groups which are in general noncommutative. Each dihedral group is defined to be the group of linear symmetries of a regular-gon. Properties. The order of is . The group has a presentation in the form . For , is noncommutative. See also. Symmetric group; Cyclic group; RotationThis article is a stub..
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In mathematics, the dihedral group of order 2n is a certain group for which here the notation D n is used, but elsewhere the notation D 2n is also used, e.g. in the list of small groups. It would be better to use, at least in Wikipedia, a uniform notation. Is D n for order 2n more common?--Patrick 12:10, 5 August 2005 (UTC) It really depends on.
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The dihedral group of order 12 is actually the group of symmetries of a regular hexagon. There are two generators of this group, the rotation through 60 degrees (r) and the flip where the hexagon is flipped round to the back (s). By combining these two movements, the 12 symmetries can be effected.. Consider the dihedral group D12 of order 12 consisting of isometries of the regular hexagon. Label the vertices of the hexagon 1, 2, ..., 6. Let o denote the reflection in the axis passing through vertex 1 and let p denote the rotation anticlockwise through an angle of /3. (a) Show that p3 and o commute.. Groups of order 12: C12 C 12 (Abelian): cyclic group of order 12. C2×C6 C 2 × C 6 (Abelian). A4 A 4 (non-Abelian): alternating group of degree 4. D6 D 6 (non-Abelian): dihedral group of degree 6. ( C 6) (non-Abelian): dicyclic group of order 12. This is a generalized quaternion group Q12 Q 12. This is a short tutorial on how to scan a dihedral angle in the GaussView 5.0 /Gaussian 09w Computational Chemistry Package. Background These are videos of.
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. Oct 03, 2019 · Dihedral groups & abeliangroups. 1. Dihedral Groups & Abelian Groups Diana Mary George Assistant Professor Department of Mathematics St. Mary’s College Thrissur-680020 Kerala. 2. Dihedral Groups,Diana Mary George,St.Mary’s College Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance.. Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function: gap> SmallGroupsInformation (12); There are 5 groups of order 12. 1 is of type 6.2. 2 is of type c12. 3 is of type A4. 4 is of type D12. 5 is of type 2^2x3. Dihedral Symmetry of Order 12. Each snowflake in the main image has the dihedral symmetry of a natual regular hexagon. The group formed by these symmetries is also called the dihedral group of degree 6. Order refers to the number of elements in the group, and degree refers to the number of the sides or the number of rotations. The order is. Hi everyone, I wanted to calculate pseudo dihedral angle as a function of time for a BDNA system. GROMACS g_angle gives dihedral as a function of time for only four connected atoms not for groups. I have defined four groups, each group consist of COM of set of atoms. defined as follows in plumed format g1,g2,g3,g4 are four groups (these are not. 2013. 1. 1. · The matrix representations for dihedral group of order twelve is provided and proven in this paper. We also proved that two matrix representations listed.
The dihedral groups are an infinite family of groups which are in general noncommutative. Each dihedral group is defined to be the group of linear symmetries of a regular-gon. Properties. The order of is . The group has a presentation in the form . For , is noncommutative. See also. Symmetric group; Cyclic group; RotationThis article is a stub..
Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. •-ignh à ignore hydrogens (gromacs will re ... Jun 12, 2010 · 53 """Set up dihedral analysis. 54 55:Arguments: 56 *dihedrals* 57 list of tuples; each tuple contains :. The following Cayley table shows the effect of composition in the group D 3 (the symmetries of an equilateral triangle). One of the simplest examples of a non-abelian group is the dihedral group of order 6. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of. We will use semidirect products to describe all 5 groups of order 12 up to isomorphism. Two are abelian and the others are A 4, D 6, and a less familiar group. Theorem 1. Every group of order 12 is a semidirect product of a group of order 3 and a group of order 4. Proof. Let jGj= 12 = 22 3. A 2-Sylow subgroup has order 4 and a 3-Sylow subgroup .... Jul 23, 2017 · I want to check that this cycle notation is correct for the Dihedral Group of order $12$. I found this graph in Wikipedia. I did not find the cycle notation. May you please tell me if my cycle notation is correct? Here is my solution: I will name the vertices $1,2,3,4,5,6$ in the clockwise direction. 1) Rotation 0 degrees $(1)(2)(3)(4)(5)(6)$. 3. Show that the dihedral group Do of order 12 has a nonidentity element z such that zg = gz for all g € D. (2 points) Question: 3. Show that the dihedral group Do of order 12 has a nonidentity element z such that zg = gz for all g € D. (2 points).
2022. 1. 29. · Corollary: The order of an element of a finite group divides the order of the group. §14.4. Euler’s Theorem Recall that if m is any positive integer ℤ m # denotes the group of all numbers from 1 to m − 1 which are coprime with m, under the operation of multiplication modulo m. [The coprimeness is what ensures the existence of inverses].
Aug 01, 2022 · The group D_5 is one of the two groups of order 10. Unlike the cyclic group C_(10), D_5 is non-Abelian. The molecule ruthenocene (C_5H_5)_2Ru belongs to the group D_(5h), where the letter h indicates invariance under a reflection of the fivefold axis (Arfken 1985, p. 248).. Hey mathmari!! I don't think there are more of order 2, 3, and 6. Oh, and aren't $\langle\sigma^2\rangle$ and $\langle\sigma^4\rangle$ the same sub group?. 3. Show that the dihedral group Do of order 12 has a nonidentity element z such that zg = gz for all g € D. (2 points) Question: 3. Show that the dihedral group Do of order 12 has a nonidentity element z such that zg = gz for all g € D. (2 points).
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In mathematics a dihedral group is the group of symmetries of a regular polygon with sides including both rotations and reflections This Demonstration shows the subgroups of the dihedral group of a. menards packing supplies; micro focus products; how to use tweepy to get tweets; merle cane corso. File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,4,2,2,2); subgroup of S4.svg Template:Dihedral group of order 8; Cayley table File usage on other wikis.
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So we must have ba = a3b b a = a 3 b, that is, (ab)2 =1 ( a b) 2 = 1. The defining relations are a4 =b2 = (ab)2 = 1 a 4 = b 2 = ( a b) 2 = 1, and this turns out to be the dihedral group of order 8, also known as the octic group. The other possibility is b2 = a2 b 2 = a 2. In this case, b b also has order 4. If ba = ab b a = a b then the group. Suitable atoms might be the atom O2999 is connected to (e.g. C2967) and the next atom (e.g. N2960). Of course, any atoms could be used provided their atom-number is lower than the atom being defined, i.e. here lower than 3000. Once the four atoms (here H3000, O2999, C2967, and N2960) are identified, work out the angle and dihedral.It's easy to make a dihedral scan in. Nov 9, 2014. #50. Birdman100 said: "longitudinal dihedral " or incidence have nothing to do with the longitudinal stability ! However incidence angle influence balance about CG. Horizontal tail force can be both positive and negative and it depends on the overall setup of. 2022. 7. 2. · Scribd is the world's largest social reading and publishing site. In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Properties. 13 Jan January 13, 2022. center of dihedral group d3. projective unitary group; orthogonal group. If x denotes rotation and y reflection, we. Answer to Solved For the dihedral group D_6 of order 12, D_6 = {x^i. Math; Other Math; Other Math questions and answers; For the dihedral group D_6 of order 12, D_6 = {x^i y^i|x^6 = y^2 = 1, yxy^-1 = x^-1} find the character of the representation for the action of D_6 on D_6/<x^2>where <x2> is the cyclic subgroup of D_6 generated by x^2.. Miyamoto dihedral group. Our assumed upper bound of 12 on the order of a Miyamoto dihedral group is motivated by the fact that in the Monster, a pair of 2Ainvolutions generates a dihedral group of order at most 12 [GMS]. Recently, Sakuma [Sa] announced that 12 is an upper bound for the order of a Miyamoto dihedral group in an OZVOA (= CFT type. In mathematics a dihedral group is the group of symmetries of a regular polygon with sides including both rotations and reflections This Demonstration shows the subgroups of the dihedral group of a. GROMACS—one of the most widely used HPC applications— has received As a simulation package for biomolecular systems, GROMACS evolves particles using the.
Answer: I will tell you how to do this. The elements of the Cartesian product have the form gh=(g,1)(1,h) where the g and h commute and g is in D6 and h is in Q8. The order of gh is the least common multiple of the orders of g and h. Since the order of elements in Q8 are either 2 or 4 and since 6.
So I want to verify some work that I am doing. The first photo is the question and the second photo is my reduced form for sin 4x. I just want to confirm that in order to express each of these my answer for sin 4 in terms of the basis vectors I need to take the inner product of <3/8−1/2 cos(2x)+1/8 cos(4x), 1/sqrt(2)>1/sqrt(2) and so on with each basis vector?.
Now if n/m is even, then the dihedral group of order 2m contains reflections from only one class, so there are two conjugacy classes of dihedral groups, while if n/m is odd, then all the dihedral ... 12. 1. 2 The normal subgroups of S 4 are A 4, V 4 (the Klein group) and {1}. So any compo-sition series must begin S 4 BA 4. Now the normal.
Dihedral Group D_5. The group is one of the two groups of order 10. Unlike the cyclic group , is non-Abelian. The molecule ruthenocene belongs to the group , where the letter indicates invariance under a reflection of the fivefold axis (Arfken 1985, p. 248). Its multiplication table is illustrated above.
AbstractThe group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are.
Answer: The dihedral group of all the symmetries of a regular polygon with n sides has exactly 2n elements and is a subgroup of the Symmetric group S_n (having n! elements) and is denoted by D_n or D_2n by different authors. For n=4, we get the dihedral group D_8 (of symmetries of a square) = {. In mathematics, the dihedral group of order 2n is a certain group for which here the notation D n is used, but elsewhere the notation D 2n is also used, e.g. in the list of small groups. It would be better to use, at least in Wikipedia, a uniform notation. Is D n for order 2n more common?--Patrick 12:10, 5 August 2005 (UTC) It really depends on.
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Groups of order 12: C12 C 12 (Abelian): cyclic group of order 12. C2×C6 C 2 × C 6 (Abelian). A4 A 4 (non-Abelian): alternating group of degree 4. D6 D 6 (non-Abelian): dihedral group of degree 6. ( C 6) (non-Abelian): dicyclic group of order 12. This is a generalized quaternion group Q12 Q 12. Jul 23, 2017 · I want to check that this cycle notation is correct for the Dihedral Group of order $12$. I found this graph in Wikipedia. I did not find the cycle notation. May you please tell me if my cycle notation is correct? Here is my solution: I will name the vertices $1,2,3,4,5,6$ in the clockwise direction. 1) Rotation 0 degrees $(1)(2)(3)(4)(5)(6)$. This is a short tutorial on how to scan a dihedral angle in the GaussView 5.0 /Gaussian 09w Computational Chemistry Package. Background These are videos of. The dihedral groups are an infinite family of groups which are in general noncommutative. Each dihedral group is defined to be the group of linear symmetries of a regular-gon. Properties. The order of is . The group has a presentation in the form . For , is noncommutative. See also. Symmetric group; Cyclic group; RotationThis article is a stub..
The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in.
52 Dihedral Groups Another special type of permutation group is the dihedral from MATH 235 at McGill University. 2022. 7. 20. · [Aside: I recently made use of this group in one of my other answers] Many groups can be represented as 3-dimensional rotation groups. For instance, any cyclic group can be understood as a group of rotations. The same is also true of the dihedral groups, where the elements of order $2$ can be understood as $180^\circ$ rotations of 3d space. Lecture Description. Abstract Algebra: Find all subgroups in S5, the symmetric group on 5 letters, that are isomorphic to D12, the dihedral group with 12 elements.. This is a short tutorial on how to scan a dihedral angle in the GaussView 5.0 /Gaussian 09w Computational Chemistry Package. Background These are videos of.
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We will use semidirect products to describe all 5 groups of order 12 up to isomorphism. Two are abelian and the others are A 4, D 6, and a less familiar group. Theorem 1. Every group of order 12 is a semidirect product of a group of order 3 and a group of order 4. Proof. Let jGj= 12 = 22 3. A 2-Sylow subgroup has order 4 and a 3-Sylow subgroup .... Nov 21, 2018 · Sorted by: 2. Hint: the dihedral group with 6 elements, i.e., the group of isometries of an equilateral triangle is non-abelian and is a subgroup of the group of isometries of a regular hexagon (the dihedral group with 12 elements). (Different authors have different conventions about the notation for the isometry groups of regular n -gons a.k.a .... In mathematics, the dihedral group of order 2n is a certain group for which here the notation D n is used, but elsewhere the notation D 2n is also used, e.g. in the list of small groups. It would be better to use, at least in Wikipedia, a uniform notation. Is D n for order 2n more common?--Patrick 12:10, 5 August 2005 (UTC) It really depends on.
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Math; Advanced Math; Advanced Math questions and answers; consider the dihedral group D6 of order 12, find all of the normal subgroups of D6; Question: consider the dihedral group D6 of order 12, find all of the normal subgroups of D6.
Let us denote G = D 8. Let K x be the conjugacy class in G containing the element x. Note that r < C G ( r) ⪇ G and the order | r | = 4. Hence we must have C G ( r) = r . Thus the element r has | G: C G ( r) | = 2 conjugates in G. Since s r s − 1 = r 3, the conjugacy class K r containing r is { r, r 3 }.
Miyamoto dihedral group. Our assumed upper bound of 12 on the order of a Miyamoto dihedral group is motivated by the fact that in the Monster, a pair of 2Ainvolutions generates a dihedral group of order at most 12 [GMS]. Recently, Sakuma [Sa] announced that 12 is an upper bound for the order of a Miyamoto dihedral group in an OZVOA (= CFT type. In mathematics a dihedral group is the group of symmetries of a regular polygon with sides including both rotations and reflections This Demonstration shows the subgroups of the dihedral group of a. GROMACS—one of the most widely used HPC applications— has received As a simulation package for biomolecular systems, GROMACS evolves particles using the. Lecture Description. Abstract Algebra: Find all subgroups in S5, the symmetric group on 5 letters, that are isomorphic to D12, the dihedral group with 12 elements.. The dihedral group of order 12 is actually the group of symmetries of a regular hexagon. There are two generators of this group, the rotation through 60 degrees (r) and the flip where the hexagon is flipped round to the back (s). By combining these two movements, the 12 symmetries can be effected.. S3 is not abelian, since, for instance, (12) · (13) = (13) · (12). On the other hand, Z6 is abelian (all cyclic groups are abelian.) Thus, S3 ∼ = Z6. ... The simplest non-Abelian group is the dihedral group D3, which is of group order six. Is Q8 an Abelian group? Q8 is the unique non-abelian group that can be covered by any three. Answer: I will tell you how to do this. The elements of the Cartesian product have the form gh=(g,1)(1,h) where the g and h commute and g is in D6 and h is in Q8. The order of gh is the least common multiple of the orders of g and h. Since the order of elements in Q8 are either 2 or 4 and since 6.
5. Let G=<a>be a cyclic group of order 10. Prove that the map f : G!Gde ned by f(a) = a4 and f(ai) = a4i is not group isomorphism. (One way) Isomorphism must send generator to a generator (see previous problems) but a4 is not generator the cyclic group of order 10, G=<a>since gcd(4;10) = 2 6= 1. (Another way) Orders of aand f(a) must be the.
I have defined four groups, each group consist of COM of set of atoms. defined as follows in plumed format g1,g2,g3,g4 are four groups (these are not. Dihedral Angles. by Greg Egan. In order to find the dihedral angle between hyperfaces of the polytope, we will initially calculate half that angle: δ, the angle between the hyperplane containing the. Sep 30, 2009 · Prove that the dihedral group of order 6 does not have a subgroup of order 4. ... Nov 12, 2017. Krisly. Y. Dihedral group D8. yanirose; May 10, 2014; Discrete Math .... It is proved that the length of the group algebra of a dihedral group of order 2 k+1 over an arbitrary field of characteristic 2 is equal to 2 k. Download to read the full article text ... No. 12, 41-62 (2009). MathSciNet Article Google Scholar O. V. Markova, "The length function and matrix algebras," Fundam. Prikl. Mat., 17, No. 6, 65. 2013. 1. 1. · The matrix representations for dihedral group of order twelve is provided and proven in this paper. We also proved that two matrix representations listed. ignh à ignore hydrogens (gromacs will re-create the hydrogens with the right naming scheme). From the output you can have the total charge of the system, how many dihedrals, impropers dihedral, angles, bonds, and pairs you have in your system.You have 3 important files that are created:. GROMACS is a package to perform molecular dynamics i.e. simulate the Newtonian. 3. Show that the dihedral group Do of order 12 has a nonidentity element z such that zg = gz for all g € D. (2 points) Question: 3. Show that the dihedral group Do of order 12 has a nonidentity element z such that zg = gz for all g € D. (2 points). The following Cayley table shows the effect of composition in the group D 3 (the symmetries of an equilateral triangle). One of the simplest examples of a non-abelian group is the dihedral group of order 6. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of.
creates K as the dihedral group of order 24, $$D_{12}$$; stores the list of subgroups output by K.conjugacy_classes_subgroups() in the variable sg; prints the elements of the list; selects the second subgroup in the list, and lists its elements.
6 CHAPTER 1. SOME BASIC RESULTS IN GROUP THEORY Figure 1.1: Showing that fr6= rfand fr= r3fin D 4 the book in the same place. We understand that this method of rst de ning the dihedral groups is not particularly rigorous. 1. Determine Z(D2). 2. Determine which elements in D12 are conjugates and collect them as sets (i.e. the "conjugacy classes of D12. 3. Verify that the class equation holds for the finite group D 12. Question: - Consider the dihedral group of order 12, given here with its group presentation: D12 = (1, 81 82 = pe = 1, rs = sr-). Each question.
The dihedral group Dn with 2n elements is generated by 2 elements, r and d, where r has order n, and d has order 2, rd=dr-1, and <d> n <r> = {e}. That implies Dn ={e,r,..,r n-1,d,dr,..,dr n-1} where those are distinct. You can generalize rd=dr-1 as r k d=dr-k. You can use that to see how any two elements multiply.
I have defined four groups, each group consist of COM of set of atoms. defined as follows in plumed format g1,g2,g3,g4 are four groups (these are not. Dihedral Angles. by Greg Egan. In order to find the dihedral angle between hyperfaces of the polytope, we will initially calculate half that angle: δ, the angle between the hyperplane containing the. General To achieve an exact restart of a simulation, one must preserve all the state variables of the system. In practice,. Jun 12, 2010 · 53 """Set up dihedral analysis. 54 55:Arguments: 56 *dihedrals* 57 list of tuples; each tuple contains :class:gromacs.cbook.IndexBuilder 58 atom selection commands. 59 *labels* 60 optional list of labels for the dihedrals. (a) Let G D12, the dihedral group of order 12, with the usual generators r, s. Find the full set of conjugacy classes of G. (b) Let Š be the set of all subgroups of G. (b1) Define the group action of G on the set Š by conjugation. Prove this is indeed a group action by checking the definition/axioms of group action.. . 2022. 1. 29. · Corollary: The order of an element of a finite group divides the order of the group. §14.4. Euler’s Theorem Recall that if m is any positive integer ℤ m # denotes the group of all numbers from 1 to m − 1 which are coprime with m, under the operation of multiplication modulo m. [The coprimeness is what ensures the existence of inverses]. . Sep 30, 2009 · Prove that the dihedral group of order 6 does not have a subgroup of order 4. ... Nov 12, 2017. Krisly. Y. Dihedral group D8. yanirose; May 10, 2014; Discrete Math .... order 12: the whole group is the only subgroup of order 12. (b) Which ones are normal? Solution. The trivial group f1g and the whole group D6 are certainly normal. Among the subgroups of order 2, only f1;x3g is normal: x(xiy)x 1 = xi+2y, so f1;xiyg is not normal for any i. The subgroup of order 3 is normal.
ignh à ignore hydrogens (gromacs will re-create the hydrogens with the right naming scheme). From the output you can have the total charge of the system, how many dihedrals, impropers dihedral, angles, bonds, and pairs you have in your system.You have 3 important files that are created:. GROMACS is a package to perform molecular dynamics i.e. simulate the Newtonian.
The dihedral groups are an infinite family of groups which are in general noncommutative. Each dihedral group is defined to be the group of linear symmetries of a regular-gon. Properties. The order of is . The group has a presentation in the form . For , is noncommutative. See also. Symmetric group; Cyclic group; RotationThis article is a stub..
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This is a short tutorial on how to scan a dihedral angle in the GaussView 5.0 /Gaussian 09w Computational Chemistry Package. Background These are videos of.
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This is a short tutorial on how to scan a dihedral angle in the GaussView 5.0 /Gaussian 09w Computational Chemistry Package. Background These are videos of. The symmetry group of a regular hexagon is a group of order 12, the Dihedral group D6 . It is generated by a rotation R 1 and a reflection r 0. R n denotes the rotation by angle n * 2 pi/6 with respect the center of the hexagon. r n denotes the reflection in the line at angle n * pi/6 with respect to a fixed line passing through the center and. The semidirect product is isomorphic to the dihedral group of order 6 if φ(0) is the identity and φ(1) is the non-trivial automorphism of C 3, which inverses the elements. Thus we get: ( n 1 , 0) * ( n 2 , h 2 ) = ( n 1 + n 2 , h 2 ). Miyamoto dihedral group. Our assumed upper bound of 12 on the order of a Miyamoto dihedral group is motivated by the fact that in the Monster, a pair of 2Ainvolutions generates a dihedral group of order at most 12 [GMS]. Recently, Sakuma [Sa] announced that 12 is an upper bound for the order of a Miyamoto dihedral group in an OZVOA (= CFT type.
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File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,4,2,2,2); subgroup of S4.svg Template:Dihedral group of order 8; Cayley table File usage on other wikis.
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Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: The Sylow 2-subgroups of $D_{12}$ have order 4. By Sylow’s Theorem .... Section5.2 Dihedral Groups. Another special type of permutation group is the dihedral group. Recall the symmetry group of an equilateral triangle in Chapter 3. Such groups consist of the rigid motions of a regular n -sided polygon or n -gon. For , n = 3, 4, , we define the nth dihedral group to be the group of rigid motions of a regular n -gon.
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AbstractThe group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are.
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Gmust have order 1;2 or p. We must show that G˘=D p. We rst claim that Ghas an element of order p. If not, every nonidentity element of Ghas order 2, which makes Ga nite elementary abelian 2-group. Thus G˘=Z=2Z Z=2Z for a nite number of copies of Z=2Z. But then jGjis a power of 2, which is impossible. Let r2Ghave order pand set H= hri..
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Question # 3, (50 pts) Let G = D6, which is the dihedral group of order 12, ie . C = {1, a, a, a, a4,0°, b, ba, ba?, ba3, ba", bal and o(a) = 6, o(b) = 2 and aba = b. Consider the cyclic subgroup H = bay of D6 generated by the element ba2、Questions 1, a, a, a",a,a,0,ba, ba-, ba, ba, ba ·(30 pts) Find the following right and left cosets (20 .... I have defined four groups, each group consist of COM of set of atoms. defined as follows in plumed format g1,g2,g3,g4 are four groups (these are not. Dihedral Angles. by Greg Egan. In order to find the dihedral angle between hyperfaces of the polytope, we will initially calculate half that angle: δ, the angle between the hyperplane containing the. This is a short tutorial on how to scan a dihedral angle in the GaussView 5.0 /Gaussian 09w Computational Chemistry Package. Background These are videos of.
ignh à ignore hydrogens (gromacs will re-create the hydrogens with the right naming scheme). From the output you can have the total charge of the system, how many dihedrals, impropers dihedral, angles, bonds, and pairs you have in your system.You have 3 important files that are created:. GROMACS is a package to perform molecular dynamics i.e. simulate the Newtonian.
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https://www.jlab.org/e94010/latex.html
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### Creating your LaTeX E94-010 Technical Note at Jefferson Lab
Do not forget to type the command 'setup tex' in the Unix CUE environment before using LateX.
A global overview of Latex features is available here.
Printing LaTeX in the Common UNIX Environment (CUE)
TeX is available on all the central UNIX systems at Jefferson Lab (AIX, HP-UX, and Solaris). In order to access
TeX, first do setup tex. You can also put this statement in your .login file (if you use the C shell). As this is
UNIX, make certain that you give each command in lower case. .tex is the default file extension so you can omit
this file extension if your file ends in .tex.
First Latex the file:
EX: jlabs2> latex tech_note.tex --> creates tech_note.dvi
Now use the PostScript post-processor to generate a PostScript (.ps) file from your Device Independent file
(.dvi) and then print the file:
jlabs1> dvips tech_note.dvi --> creates tech_note.ps
jlabs1> lp -dprinter tech_note.ps (where printer is a PostScript printer)
A few useful options (do man dvips for further options):
-o filename Name the output file filename
-c num Generate num copies
-C num Collate the num copies
-r Stack pages in reverse order
Inserting PostScript Documents (epsfig)
There are some simple utilities available to allow you to embed a PostScript document in your LaTeX
document. The PostScript document should generally be what is called "encapsulated PostScript" (i.e. select this
format in the program which generates your .ps file); however, most PostScript files will work as long as they
have a "bounding box" comment. To embed the PostScript picture in your LaTeX document, put the following
line near the beginning of your TeX document:
\usepackage{epsfig}
Now use Latex commands like the following to insert your PostScript file using the epsfig package:
\begin{figure}
\begin{center}\mbox{
\epsfig{file=yourpsfile.eps,width=.9\textwidth}
}\end{center}
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http://www.mzan.com/article/47702357-i-want-to-get-the-index-of-ng-repeat.shtml
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Home I want to get the index of ng-repeat
In the code I want to use the index of ng-repeat to get the index of an array however my index value using $index is not retreiving this value. Kartik Puri 2# Kartik Puri Reply to 2017-12-12 07:48:25Z seems like you want to capture the$index and then fire your function based on that. You can have an ng-click="clickFunction(\$index)" and then have that function do rest of the work.
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2017-12-18 03:15:22
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https://zbmath.org/?q=an:1198.93179
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## Delay-range-dependent exponential $$H\infty$$ synchronization of a class of delayed neural networks.(English)Zbl 1198.93179
Summary: This article aims to present a multiple delayed state-feedback control design for exponential $$H\infty$$ synchronization problem of a class of delayed neural networks with multiple time-varying discrete delays. On the basis of the drive-response concept and by introducing a descriptor technique and using Lyapunov-Krasovskii functional, new delay-range-dependent sufficient conditions for exponential $$H\infty$$ synchronization of the drive-response structure of neural networks are driven in terms of linear matrix inequalities (LMIs). The explicit expression of the controller gain matrices are parameterized based on the solvability conditions such that the drive system and the response system can be exponentially synchronized. A numerical example is included to illustrate the applicability of the proposed design method.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.
### MSC:
93D15 Stabilization of systems by feedback 93B36 $$H^\infty$$-control 37N35 Dynamical systems in control 92B20 Neural networks for/in biological studies, artificial life and related topics
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### References:
[1] Pecora, L.M.; Carrol, T.L., Synchronization in chaotic systems, Phys rev lett, 64, 821-823, (1990) [2] Zhang, H.; Wang, Z.; Liu, D., Robust exponential stability of recurrent neural networks with multiple time-varying delays, IEEE trans circ syst II: express briefs, 54, 8, 730-734, (2007) [3] Wang, Z.; Liu, Y.; Yu, L.; Liu, X., Exponential stability of delayed recurrent neural networks with Markovian jumping parameters, Phys lett A, 356, 346-352, (2006) · Zbl 1160.37439 [4] Mou, S.S.; Gao, H.J.; Qiang, W.Y.; Chen, K., New delay-dependent exponential stability for neural networks with time delay, IEEE trans syst man cybernet part B - cybernet, 38, 2, 571-576, (2008) [5] Xu, S.; Lam, J.; Ho, D.W.C.; Zou, Y., Delay-dependent exponential stability for a class of neural networks with time delays, J comput appl math, 183, 16-28, (2005) · Zbl 1097.34057 [6] Mou, S.S.; Gao, H.J.; Lam, J.; Qiang, W.Y., A new criterion of delay-dependent asymptotic stability for Hopfield neural networks with time delay, IEEE trans neural networks, 19, 3, 532-535, (2008) [7] Xu, S.; Lam, J.; Ho, D.W.C.; Zou, Y., Global robust exponential stability analysis for interval recurrent neural networks, Phys lett A, 325, 124-133, (2004) · Zbl 1161.93335 [8] Cheng, C.J.; Liao, T.L.; Yan, J.J.; Hwang, C.C., Exponential synchronization of a class of neural networks with time-varying delays, IEEE trans syst man cybernet part B: cybernet, 36, 1, 209-215, (2006) [9] Lu, H.; van Leeuwen, C., Synchronization of chaotic neural networks via output or state coupling, Chaos, solitons & fractals, 30, 166-176, (2006) · Zbl 1144.37377 [10] Hou, Y.Y.; Liao, T.L.; Yan, J.J., $$H_\infty$$ synchronization of chaotic systems using output feedback control design, Physics A, 379, 81-89, (2007) [11] Lu, J.; Cao, J., Synchronization-based approach for parameters identification in delayed chaotic neural networks, Physics A, 382, 672-682, (2007) [12] Gao, H.J.; Lam, J.; Chen, G.R., New criteria for synchronization stability of general complex dynamical networks with coupling delays, Phys lett A, 360, 2, 263-273, (2006) · Zbl 1236.34069 [13] Sun, Y.; Cao, J.; Wang, Z., Exponential synchronization of stochastic perturbed chaotic delayed neural networks, Neurocomputing, 70, 2477-2485, (2007) [14] Yu, W.; Cao, J., Synchronization control of stochastic delayed neural networks, Physics A, 373, 252-260, (2007) [15] Cheng, C.J.; Liao, T.L.; Yan, J.J.; Hwang, C.C., Synchronization of neural networks by decentralized feedback control, Phys lett A, 338, 28-35, (2005) · Zbl 1136.37366 [16] Li, P.; Cao, J.; Wang, Z., Robust impulsive synchronization of coupled delayed neural networks with uncertainties, Physics A, 373, 261-272, (2007) [17] Liao, T.L.; Tsai, S.H., Adaptive synchronization of chaotic systems and its application to secure communication, Chaos, solitons & fractals, 11, 9, 1387-1396, (2000) · Zbl 0967.93059 [18] Feki, M., An adaptive chaos synchronization scheme applied to secure communication, Chaos, solitons & fractals, 18, 141-148, (2003) · Zbl 1048.93508 [19] Sun, Y.; Cao, J., Adaptive lag synchronization of unknown chaotic delayed neural networks with noise perturbation, Phys lett A, 364, 277-285, (2007) · Zbl 1203.93110 [20] Zhou, J.; Chen, T.; Xiang, L., Robust synchronization of delayed neural networks based on adaptive control and parameters identification, Chaos, solitons & fractals, 27, 905-913, (2006) · Zbl 1091.93032 [21] Yu W, Cao J. Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks. In: Chaos, vol. 16, Article no. 023119; 2006. · Zbl 1146.93371 [22] Chen, M.; Zhou, D.; Shang, Y., A new observer-based synchronization scheme for private communication, Chaos, solitons & fractals, 24, 1025-1030, (2005) · Zbl 1069.94508 [23] Cao, J.; Li, P.; Wang, W., Global synchronization in arrays of delayed neural networks with constant or delayed coupling, Phys lett A, 353, 318-325, (2006) [24] Cai, G.P.; Huang, J.Z.; Yang, S.X., An optimal control method for linear systems with time delay, Comput struct, 81, 1539-1546, (2003) · Zbl 1038.33009 [25] Niculescu, S.I., Delay effects on stability: a robust control approach, (2001), Springer Berlin [26] Krasovskii, N.N., Stability of motion, (1963), Stanford University Press Stanford, CA · Zbl 0109.06001 [27] Gu K. An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE conference on decision and control; 2000. p. 2805-10. [28] Cao, J., Periodic oscillation and exponentially stability of delayed cnns, Phys lett A, 270, 157-163, (2000) [29] Chen, A.; Cao, J.; Huang, L., Global robust stability of interval cellular neural networks with time-varying delays, Chaos, solitons & fractals, 23, 787-799, (2005) · Zbl 1101.68752 [30] Khalil, H.K., Nonlinear systems, (1992), Macmillan New York · Zbl 0626.34052 [31] Fridman, E.; Shaked, U., A descriptor system approach to $$H_\infty$$ control of linear time-delay systems, IEEE trans automat control, 47, 2, 253-270, (2002) · Zbl 1364.93209 [32] Gao, H.; Wang, C., Comments and further results on A descriptor system approach to $$H_\infty$$ control of linear time-delay systems, IEEE trans automat control, 48, 520-525, (2003) · Zbl 1364.93211 [33] Park, P., A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE trans automat control, 44, 876-877, (1999) · Zbl 0957.34069 [34] He, Y.; Wang, Q.G.; Lin, C.; Wu, M., Delay-range-dependent stability for systems with time-varying delay, Automatica, 43, 371-376, (2007) · Zbl 1111.93073 [35] Moon, Y.S.; Park, P.G.; Kwon, W.H.; Lee, Y.S., Delay-dependent robust stabilization of uncertain state-delayed systems, Int J control, 74, 1447-1455, (2001) · Zbl 1023.93055 [36] Gao, H.J.; Lam, J.; Wang, Z.D., Discrete bilinear stochastic systems with time-varying delay: stability analysis and control synthesis, Chaos, solitons & fractals, 34, 2, 394-404, (2007) · Zbl 1134.93413 [37] Karimi, H.R., Observer-based mixed $$H_2 / H_\infty$$ control design for linear systems with time-varying delays: an LMI approach, Int J control automat syst, 6, 1, 1-14, (2008) [38] Lee, Y.S.; Moon, Y.S.; Kwon, W.H.; Park, P.G., Delay-dependent robust $$H_\infty$$ control for uncertain systems with a state-delay, Automatica, 40, 65-72, (2004) · Zbl 1046.93015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2023-03-21 21:38:37
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https://www.gamedev.net/forums/topic/39493-stupid-programs/
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#### Archived
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# Stupid programs
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Alright, this is the 4th version of Netscape 6 that I''ve downloaded: release version 6.01. I tried Navigator and Composer all afternoon yesterday - you''d think that after all this time Netscape would get something right. If I maintained a meticulous bug count from yesterday, it would be well above 200. Basically, 1. Navigator and Composer are slow and unresponsive, to the point of being unusable. 2. Composer stinks - it messes up the HTML tags _constantly_ and the text selection code is diabolical. And where are the Cut, Copy, and Paste buttons? 3. Navigator''s install-to-crash ratio is growing larger. It took an hour to download and install, and only two seconds to crash. 4. Navigator leaves garbage in the browser window when you change pages, which makes the dialog boxes and menus look terrible. 5. They always open new windows directly over the old ones! Those are the things that annoy me the most. The only thing I liked about either of them is Navigator''s ability to display partially loaded pages as they are being loaded, which is helpful in the forums. <rant size="tiny"> If Netscape 6 is open source, then may I never release or even use an open source project. Let''s all hope the software world doesn''t turn into an infinite conglomeration of Linux-like version numbers and half-baked releases! *groan* That stupid browser just messed up the text in this edit box again - now it doesn''t even track the text cursor position correctly. </rant> Any other examples of stupid programs?
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So you won''t use or release open-source products because Netscape sucks? Anyway, use Opera instead
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1. NetSheep isn''t slow, your computer is slow
2. Yeah, the text selection is really bad..
3. It took only 10 minutes to download and install... (cable )
4. You get used to it.
5. I love it when a new window opens exactly over the old one!
(I wish that Exploder did that!)
NetSheep is open source.
But open source does NOT mean that a product is bad.
And why would you never release or try an open source product because just one product is bad?
Windows is buggy, but you''re still "buying" commercial programs?
Take a look at SourceForge and try for example the Nebula Engine (very well coded!)
Or the GIMP?
OK, most open source programs look different but it doesn''t mean they''re bad...
Yes, I''ve got an example of a stupid program: Internet Exploder.
Because it crashes at least 3 times a day (takes the whole computer down ).
NetSheep crashed 1 time today and it did not crash my whole computer (bsod)
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You have complaints about one or the other... Ever tried having both installed on one computer at the same time? *evilgrin*
Then you should try running one instance of each. Now press the reset button...
I''m probably going to get lynched for this: I''ve found Win95 to be a stable OS that doesn''t crash on me, loads quickly, and allows me to do what I need to do.
I just saw that Windows CE, Windows ME, Windows NT.... Windows CEMENT joke, gotta love it
Press to test... *click* Release to detonate...
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(Hey, it''s a rant; don''t expect it to be logical. I''m just tired of reinstalling Linux again (and again (and again...)).)
My computer:
AMD K6-2 450 MHz
256 MB of PC133 SDRAM (CAS2)
UDMA-66 Hard Drive
ATI Rage Fury w/ 32 MB SDRAM
It''s not the fastest computer, but it''s certainly not slow enough to make a simple HMTL editor like Composer unusable. FrontPage 98 Express works much faster and doesn''t have most of those problems. When I say responsiveness, I am not talking about download speeds. I mean the fact that the drop-down menus are jerky when Netscape 6 is the only app running, and the mouse clicks do not register sometimes or the characters are being copied to the screen at an even slower pace than my typing (and that is slooowwww).
Windows Me hasn''t crashed my computer since I bought it, with the exception of PowerTweak fiasco. I still don''t know where all these IE crashes are coming from... IE doesn''t crash my computer.
Oh, and I''m still waiting for the Opera site to load - you know, the one that says "Opera - the fastest browser on earth!"
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So, here comes my list:
WineME - After 2K is out, why messing around with this awful, buggy, featureless, crappy and unstable OS ?
AOL - The name says it all ;-)
;-)
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Hey null_pointer, it says fastest browser , not fastest server
I seem to have trouble with HTML tags today
Edited by - Muzzafarath on February 13, 2001 2:46:14 PM
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Yeah, I know. But the fastest browser on earth is located on the slowest server.
I like Opera but it doesn't seem to fill in my passwords automatically. Does anybody know how to fix this?
WinMe is better than Win2K.
Edited by - null_pointer on February 13, 2001 5:52:35 PM
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IE doesn''t only crash many times it also gives me nice dialogs saying that "one" window will be closed...
(and all open windows are gone after clicking OK)
But why are you using Composer or Frontpage for creating a page/site?
1st Page 2000 (free) is MUCH better!
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For the sake of argument...
I have both Netscape and IE and I ALWAYS use IE. I am not particularely Pro-Microsoft or Pro-Netscape for that matter, but I know which is the better product, by far. People should evaluate what is the best product for the job, not which company is preferred.
Also, problems with crashing are often problems with configuration. My computer is running Win98 with IE55 (go ahead, laugh) but it literally NEVER crashes. Sometimes it depends on what software you use (Visual C++.... *shudder*) but often you simply have to resist the erge to download every demo and beta program on the face of the earth... it is surprising how many don''t perform a clean uninstall. Either that, or don''t complain when your computer crashes constantly. MS can''t be held responsable for other peoples'' programs screwing with the operating system.
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Hey, anon, you sound like me.
Netscape 6.01 seems to have the same problem as IE with all the Netscape windows crashing at one time. I''m not sure if you can change that behavior in Netscape 6.01 but in IE 4.0+ you should be able to change the option that says "Browse in a separate process" which keeps each Explorer window from crashing all Explorer windows.
I use FrontPage 98 Express to create a site because I don''t have money to spend on another piece of software, and besides, it simple and easy to use. My friends work with HTML by hand, and most of the commands from that program translate directly into HTML and the generated code is quite simple, indented, and well-organized. I use it as a WYSIWYG extension of Notepad. Making a frames-capable site is easy if you know enough to make a simple index.html with the frameset commands.
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Well, I used to be a big Netscape person, but the quality began degrading shortly after AOL (AOL is evil, glad I use DSL) came into their scene... I use IE now, it works fine for me (no, it doesn't crash nearly as often for me as other people say it does for them). I also use Win2K Prof., it has to be the best version of Windows I've ever used (that's not saying all that much, but at least it never crashes ).
Oh, and one last thing, use Notepad for HTML/JS/ASP . WYSIWYG editors are too restricting. My website's down at the moment though, oh well .
http://www.gdarchive.net/druidgames/
Edited by - Null and Void on February 13, 2001 7:19:35 PM
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The following quotes were posted by Anonymous Poster
"My computer is running Win98 with IE55 (go ahead, laugh)"
Muahaahaahahaa!!
"(Visual C++.... *shudder*)"
Uhm.. VC++ only crashes when doubleclicking on a .DSW file in Exploder.
"Sometimes it depends on what software you use but often you simply have to resist the erge to download every demo and beta program on the face of the earth... it is surprising how many don''t perform a clean uninstall."
I know, but I''m doing the same with Win2K (dual boot) and Win2K does NEVER crash (well, it did once because the drivers for my videocard were f*ck*d).
The following quote was posted by null_pointer
"in IE 4.0+ you should be able to change the option that says "Browse in a separate process" which keeps each Explorer window from crashing all Explorer windows"
Can you tell me where I can find that option because I think it''ll make me very happy when not all windows are closing with a crash.
I should''ve said that IE usually crashes when there''s a Java applet in a page (chat).
Tried to solve that problem but I still don''t know how to fix it.
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quote:
Yeah, I know. But the fastest browser on earth is located on the slowest server.
Actually, the Opera site is quite fast for me. Loads in a couple of seconds. Maybe they made it load faster with Opera
quote:
I like Opera but it doesn''t seem to fill in my passwords automatically. Does anybody know how to fix this?
I don''t think Opera can do that
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Just to throw my hat into the ring:
I used to hate IE and only use Netscape. After the version 4 releases of both though, I have to say that IE is much more stable..... bloody netscape crashes at the slightest bit of DHTML on a web site or even flash (which it is supposed to have inbuilt support for).
Version 6 is even worse. I''m not saying the IE is great but V5.5 is probably the best browser with the most features that has been released so far.
Personally I don''t have any problems with it and I''m running it on a crappy old Acer Win98 laptop. Netscape takes ages to load when I try and run it..... there is a lot to be said for being able to hook into the OS when you are programming a browser..
Opera is also very good and as far as support goes for implementing the W3C recommendations it goes the furthest.
$0.02 -- No matter what they do, how hard they are or cold they want to be, their mind will always give them away. That is the moment to strike and leverage their weakness -- The birth of Psychic Treason #### Share this post ##### Link to post ##### Share on other sites Opera is kind of a mixed bag - some pages load really fast and others (like some of the forum pages) take minutes to load. My connection is slow but both IE and Netscape load some pages faster - strange, huh? Opera probably uses a different caching mechanism, so I''ll use it for a while longer before deciding whether or not I''ll keep it. I agree about IE 5.5 - it''s probably one of the best browsers for what it does, even if it is a bit simpler. IE 6.0 should be out very soon, and I''ve heard that it has a lot of new features. Null and Void: I only use the WYSIWYG programs to type in paragraphs and do basic formatting like setting heading styles, and I use Notepad for the more complex tasks. #### Share this post ##### Link to post ##### Share on other sites quote: Original post by null_pointer If Netscape 6 is open source, then may I never release or even use an open source project. Let''s all hope the software world doesn''t turn into an infinite conglomeration of Linux-like version numbers and half-baked releases! *groan* That stupid browser just messed up the text in this edit box again - now it doesn''t even track the text cursor position correctly. Actually, Netscape 6 and "Mozilla" the Open Source version, are different branches of the same original source tree. If you want to see something that''s more stable then choose the latest Mozilla build. BeS It''s Da BOMB Baby!!! . o O ~ A little nonsense now and then, is relished by the wisest men ~ O o . -- Willy Wonka #### Share this post ##### Link to post ##### Share on other sites quote: Original post by null_pointer Hey, anon, you sound like me. Netscape 6.01 seems to have the same problem as IE with all the Netscape windows crashing at one time. I''m not sure if you can change that behavior in Netscape 6.01 but in IE 4.0+ you should be able to change the option that says "Browse in a separate process" which keeps each Explorer window from crashing all Explorer windows. I use FrontPage 98 Express to create a site because I don''t have money to spend on another piece of software, and besides, it simple and easy to use. My friends work with HTML by hand, and most of the commands from that program translate directly into HTML and the generated code is quite simple, indented, and well-organized. I use it as a WYSIWYG extension of Notepad. Making a frames-capable site is easy if you know enough to make a simple index.html with the frameset commands. First off...... The "Browse in a separate process" only keeps Iexplore from crashing "Explorer". It doesn''t keep all I.E. windows from crashing. Because mine does it alot and the Browse in separate process is one of the first settings I change after reinstalling the OS or an update to I.E. Secondly....... 1st Page 2000 is FREE also. While it doesn''t do WYSIWYG it also doesn''t add all of the Frontpage specific extensions in there, like all of the FP Meta tags. I code by hand also, and I can tell you, that editing a page that''s been done in FP or FPXPress SUCKS!!!!! because it assumes all of your visitors are using I.E. Which most of the time is WRONG!!!! I''ve had to edit tables made by those programs so they would sork correctly. Also, if you use form elements, it doesn''t always close them in a form tag. For I.E. that''s fine, but for Netscape it won''t show your element. BeS It''s Da BOMB Baby!!! . o O ~ A little nonsense now and then, is relished by the wisest men ~ O o . -- Willy Wonka #### Share this post ##### Link to post ##### Share on other sites quote: Original post by Muzzafarath Actually, the Opera site is quite fast for me. Loads in a couple of seconds. Maybe they made it load faster with Opera Actually, that would have to be because you are in the same region as their server. I believe they are in the Netherlands? BeS It''s Da BOMB Baby!!! . o O ~ A little nonsense now and then, is relished by the wisest men ~ O o . -- Willy Wonka #### Share this post ##### Link to post ##### Share on other sites Like me, Opera is actually norwegian #### Share this post ##### Link to post ##### Share on other sites quote: Opera is kind of a mixed bag - some pages load really fast and others (like some of the forum pages) take minutes to load. My connection is slow but both IE and Netscape load some pages faster - strange, huh? Opera probably uses a different caching mechanism, so I''ll use it for a while longer before deciding whether or not I''ll keep it. That''s odd, I haven''t had any problems with any pages loading slower in Opera. quote: Actually, that would have to be because you are in the same region as their server. Yeah, Opera is from Norway... But actually I think the site is hosted somewhere in America, it''s hosted by Earthlink. #### Share this post ##### Link to post ##### Share on other sites I don''t like M$, but I can''t live wihout their software !
I''ve seen tons of IDEs, but I think they all suck compared to VC++ ! I can''t imagine using something different !
There is also NO replacement for the MSDN libary.
Another OS than Windows 2000 !? I can''t imagine living with Linux or MacOS !
Using the ''net without Internet Explorer and Outlook ? Oh my god !
Coding ASP pages without Visual InterDev ! Never !
Playing games without my MS ForceFeedback ???
I depend so much on M\$, I can''t imagine living without them ;-) In my opinion, NOBODY offers products that could replace my daily tools !
I would love to have a choice, dammit...
Tim
--------------------------
glvelocity.gamedev.net
www.gamedev.net/hosted/glvelocity
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wrenhal:
First, the "browse in a separate process" option does keep any Explorer window from crashing any other Explorer window, whether you are browsing on the 'Net or looking in My Documents. I know because I had them crash separately during the PowerTweak fiasco (don't ask) and because my dad's computer crashes frequently for some unknown reason, which is somehow tied to his AGP video card.
Second, FrontPage 98 Express version 2.02.1131 produces standard HTML without all the XML junk. Other versions of the same software (that look almost identical) produce XML like Word 2000.
Edited by - null_pointer on February 14, 2001 3:05:23 PM
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I have both Netscape 4.76 and 6.01 and of course IE 5.5 but except from browsing MSDN do I always use one browser from Netscape. IE is a pretty good program with fewer bugs that Netscape but some things I do all the time is much faster and convenient in NetScape:
- Entering keywords with Ctrl+L or Ctrl+O works excellent with Netscape but I gave up then IE tried to start a program from the HD.
- Filling in username/passwords. N6 does that perfect and I just have to press enter to confirm in IE is it more annoying.
- Sometimes do I listen to music while surfing this is just a Ctrl+7 select channel in Netscape.
- Spell checking is only available in Netscape.
The above is the main reasons I use Netscape perhaps could I configure IE better but I really like the Netscape browser.
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2018-02-26 03:53:11
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https://webwork.libretexts.org/webwork2/html2xml?answersSubmitted=0&sourceFilePath=Library/Union/setLimitConcepts/ur_lr_1-5_1.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&showSummary=1&displayMode=MathJax&problemIdentifierPrefix=102&language=en&outputformat=libretexts
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Let $F$ be the function whose graph is shown below. Evaluate each of the following expressions.
(If a limit does not exist or is undefined, enter "DNE".)
1 $\displaystyle\lim_{x\to -1^-} F(x)$ = 2 $\displaystyle\lim_{x\to -1^+} F(x)$ = 3 $\displaystyle\lim_{x\to -1} F(x)$ = 4 $F(-1)$ = 5 $\displaystyle\lim_{x\to 1^-} F(x)$ = 6 $\displaystyle\lim_{x\to 1^+} F(x)$ = 7 $\displaystyle\lim_{x\to 1} F(x)$ = 8 $\displaystyle\lim_{x \to 3} F(x)$ = 9 $F(3)$ =
The graph of $y=F(x)$.
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2022-05-19 06:23:46
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https://shelah.logic.at/coauthors/salma_kuhlmann/
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# Publications with S. Kuhlmann
#### All publications by Salma Kuhlmann and S. Shelah
Click the Sh:-number to get to the papers' detail page (which may include pdf's of the paper).
Can't find an Sh:-number (in particular, an "F-number")? You can try your luck here.
number title
Sh:601 Kuhlmann, F.-V., Kuhlmann, S., & Shelah, S. (1997). Exponentiation in power series fields. Proc. Amer. Math. Soc., 125(11), 3177–3183.
Sh:615 Kuhlmann, F.-V., Kuhlmann, S., & Shelah, S. (2003). Functorial equations for lexicographic products. Proc. Amer. Math. Soc., 131(10), 2969–2976.
Sh:857 Kuhlmann, S., & Shelah, S. (2005). \kappa-bounded exponential-logarithmic power series fields. Ann. Pure Appl. Logic, 136(3), 284–296.
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2021-04-22 23:33:30
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https://marietas-islands.net/molasses-substitute-evbfn/acute-definition-math-58928b
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acute definition math Malacca Malaysia Snake Temple, Starbucks Washington Mug, Paris Police Number, Savino Belmont Ma, Unc Charlotte Basketball Players In The Nba, Kerapoxy Cq Quart, " /> Malacca Malaysia Snake Temple, Starbucks Washington Mug, Paris Police Number, Savino Belmont Ma, Unc Charlotte Basketball Players In The Nba, Kerapoxy Cq Quart, " />
## acute definition math
For an acute triangle, all three angles of the triangle are from $$0^o$$ to $$90^o$$. Also find the definition and meaning for various math words from this math dictionary. (of a triangle) containing only acute angles. acute definition in English dictionary, acute meaning, synonyms, see also 'acute accent',acute arch',acute dose',acute arch'. Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009, 2012. Copyright © 2011. Related Calculators: Find Acute Angle Between Two Lines And Plane . Enrich your vocabulary with the English Definition dictionary In this non-linear system, users are free to take whatever path through the material best serves their needs. He Wanted to Fix Rural America’s Broken Nursing Homes. acute - WordReference English dictionary, questions, discussion and forums. Acute angles are easy to recognize because they are wedge-shaped and sharply pointed. addend A number used in the mathematical operation of addition (e.g., 6 … But even if I knew that it had, I should always remember that Christmas night of 1870 with acute regret. What’s The Difference Between “Yule” And “Christmas”? Until the diagnosis is thoroughly established, soothing applications, such as are employed in acute eczema, are to be advised. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. When the term is given to a triangle as in an acute triangle, it means that all angles in the triangle are less than 90 degrees. A right angle forms part of a square, … Triangle Calculator . When two rays form an angle of 90-degree, it is called a right angle. (image will be uploaded soon) Definition of Right Angle. The definition of Acute Angle: An angle less than 90deg (90deg is called a Right Angle) My early years were a mix of acute loneliness and great freedom. When to Use Acute. adj. We use cookies to enhance your experience on our website, including to provide targeted advertising and track usage. acute synonyms, acute pronunciation, acute translation, English dictionary definition of acute. acute care the level of care in the health care system that consists of emergency treatment and critical care. Of or relating to a patient afflicted with such a disease. In the dark my sense of smell and hearing become so acute. Math is the science of numbers. (Printing, Lithography & Bookbinding) an acute accent, But a wise ordinance of Nature has decreed that, in proportion as the working-classes increase in intelligence, knowledge, and all virtue, in that same proportion their, These are distinguished according to the form assumed by the mouth and the place where they are produced; according as they are aspirated or smooth, long or short; as they are, His enthusiasm was sincere and his criticism, At night, one could distinguish nothing of all that mass of buildings, except the black indentation of the roofs, unrolling their chain of, After four days of solitude, ennui, and consciousness of his impotence and insignificance- particularly, That would be in a couple of hours, at the end of which-- I had the, Moralists will be glad to hear that I really suffered, Whether it was that the children were fickle, or that they had, Elton should not be really in love with her, or so particularly amiable as to make it shocking to disappoint himthat Harriet's nature should not be of that superior sort in which the feelings are most. The war has aggravated an acute economic crisis. Definition of Acute Angles. What does ACUTE CARE mean? consisting of, indicated by, or bearing the mark ´, placed over vowel symbols in some languages to show that the vowels or the syllables they are in are pronounced in a certain way, as in French that the quality of an, (of a triangle) having all its interior angles less than 90°, arising suddenly and manifesting intense severity, (of a vowel or syllable in some languages with a pitch accent, such as ancient Greek) spoken or sung on a higher musical pitch relative to neighbouring syllables or vowels, of or relating to an accent (´) placed over vowels, denoting that the vowel is pronounced with higher musical pitch (as in ancient Greek), with a certain special quality (as in French), etc, (of a hospital, hospital bed, or ward) intended to accommodate short-term patients with acute illnesses. acute [ah-kūt´] 1. sharp. Definition. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. Angles that measure less than 90 degrees are called acute angles.An angle that is exactly 180 degrees is called a straight angle (this appears as a straight line). A triangle that has all angles less than 90° (90° is a Right Angle) Why Do “Left” And “Right” Mean Liberal And Conservative? In this non-linear system, users are free to take whatever path through the material best serves their needs. The soil variations are acute enough that they can differ radically from one side of a road to another. Acute Angle: an angle measuring less than 90 degrees. acute [ah-kūt´] 1. sharp. The American Heritage® Stedman's Medical Dictionary The point at which the two rays meet (intersect) is called the vertex. Types of Angles . Find more ways to say acute, along with related words, antonyms and example phrases at Thesaurus.com, the world's most trusted free thesaurus. Use the adjective acute for when you want to describe something as sharp or extremely serious. acute triangle | Math Goodies Glossary. Its course is slower, it is less severe, and is not accompanied with so much fever as the acute form. An angle is formed when two rays originate from same end point. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. The definition of an acute angle states that an angle whose measure is greater than 0° and less than 90° is an acute angle. An acute angle is an angle that measures between 90° and 0°, meaning it is smaller than a right angle (an “L” shape) but has at least some space between the two lines that form it. Meaning, pronunciation, picture, example sentences, grammar, usage notes, synonyms and more. What is Angle. Meaning of ACUTE CARE. In geometry we have to learn some basic definitions or the meanings, for example what is a line, what is a point, line segment, ray and angle.Studying of these definitions will help you in understanding higher level concepts based on geometry. Relating to an illness that has a rapid onset and follows a short but severe course. “Affect” vs. “Effect”: Use The Correct Word Every Time. The war has aggravated an acute economic crisis. ... Cosine: In a right triangle, cosine is a ratio that represents the length of a side adjacent to an acute angle to the length of the hypotenuse. SplashLearn is an award winning math learning program used by more than 30 Million kids for fun math practice. Improve your math knowledge with free questions in "Acute, right, obtuse, and straight angles" and thousands of other math skills. Reacting readily to stimuli or impressions; sensitive: His hearing was unusually acute. These conditions tend to resolve quickly on their own or with medical treatment. Learn more. Obtuse angle This page was last changed on … In acute poisoning (especially by the corrosive salts) the changes are great and striking. An acute angle is an angle that measures less than 90 degrees. Obtuse. These unique features make Virtual Nerd a viable alternative to private tutoring. Let’s spare you from the old jokes about a disease being a ‘cute’ disease or having something to do with degrees of angles. The range of an acute angle is between 0 and 90 degrees. adj. Learn what is acute triangle. 1. adjective You can use acute to indicate that an undesirable situation or feeling is very severe or intense. Note that it is acute for all angles from zero to (but not including) 90°. Describe 2020 In Just One Word? All rights reserved. Of or relating to a disease or a condition with a rapid onset and a short, severe course. An acute angle has a measure, or it's smaller, than a right angle. Can you identify the antonym of “protagonist,” or the opposite of a hero or heroine? The authors collected data related to 54 different causes of death that could be attributed to alcohol, both chronic and acute. When you learn about radians and degrees, which are different ways to measure angles, you'll see that a right angle can be measured as 90 degrees. b. ∠ABC measures 30 ̊and hence it is an acute angle. Define acute. Angles are formed by two rays (or lines) that begin at the same point or share the same endpoint. An acute angle ("acute" meaning "small") is an angle smaller than a right angle. An acute condition can sometimes become chronic, while a chronic condition may suddenly present with acute symptoms. For our professional healthcare knowledge, we need to know how an acute illness differs from a chronic illness along with some common examples. a. Reacting readily to stimuli or impressions, as hearing or eyesight; sensitive. 1. a. Hide. Obtuse Triangle Calculator . Acute angle definition at Dictionary.com, a free online dictionary with pronunciation, synonyms and translation. The time scale depends on the particular disease. (of disease) brief and severe (opposed to. Hawaiian Translation: Huina Oi Example. So, as its name suggests, an acute triangle is a triangle whose three angles are all smaller than 90 degrees. Learn and know what is acute angle in types of angles in math. Angles are an integral facet in the study of mathematics, particularly geometry. Meaning of ACUTE LEUKEMIA. An acute angle is an angle that measures between 0 ° and 90 °, or π 2 (in radians). "Chronic" indicates indefinite duration or virtually no change. Having a rapid onset and following a short but severe course: an acute disease. A “V” shape is an example of … Master these essential literary terms and you’ll be talking like your English teacher in no time. An acute angle is any angle smaller than a right angle. Right triangle definition For a right triangle with one acute angle, θ, the tangent value of this angle is defined to be the ratio of the opposite side length to the adjacent side length. Certain infections, for example, will progress from an acute phase (in which symptoms appear and resolve after the initial exposure) to a chronic phase (in which the infection persists, but progresses less aggressively). In this section we know about definition of angle in geometry and its types of angles like Interior and Exterior of an angle, Zero Angle, Acute Angle, Right Angle, Obtuse angle, Straight Angle, Reflex Angle & Complete angle. Definition of acute written for English Language Learners from the Merriam-Webster Learner's Dictionary with audio pronunciations, usage examples, and count/noncount noun labels. Acute: Of abrupt onset, in reference to a disease. Accute synonyms, Accute pronunciation, Accute translation, English dictionary definition of Accute. Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn. An acute angle is an angle that measures between 0 ° and 90 °, or π 2 (in radians). Definitions An acute angle is any angle less than 90 degrees. 1. a patient with acute rheumatoid arthritis, the report has caused the government acute embarrassment. Look it up now! All Free. The Dictionary.com Word Of The Year For 2020 Is …. adj. 1. cutting-edge, confident, or really fragile. The People’s Choice 2020 Word Of The Year: 2020 Was A $#@#%%$@! deeply perceptive. Definition of ACUTE LEUKEMIA in the Definitions.net dictionary. Information and translations of ACUTE CARE in the most comprehensive dictionary definitions resource on the web. For example, in an equilateral triangle, all three angles measure 60˚, making it an acute triangle. Acute angles are always less than 90 °.. Plus, I will show you a useful trick to remember how to decide whether acute or chronic is more appropriate in a given context.. Properties . acute - WordReference English dictionary, questions, discussion and forums. This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional. Table Of Contents acute definition: 1. An acute angle is any angle smaller than a right angle. Check to see what the Maths Dictionary says by clicking on this link, then clicking on the A category, then clicking on the word "Angles". adj. Definition of acute. adj. acute. An acute triangle has three angles that measure between 0 and 90 degrees. Acute Triangle. "Subacute" indicates longer duration or less rapid change. Options. So the hospital industry lobbied for a new kind of health system for older adults who were dependent on getting medical help but did not need acute care. Now, Taxpayers May Be on the Hook for $76 Million. Acute Illness Definition Let’s spare you from the old jokes about a disease being a ‘cute’ disease or having something to do with degrees of angles. Geometry. Also find the definition and meaning for various math words from this math dictionary. 1. adjective You can use acute to indicate that an undesirable situation or feeling is very severe or intense. An acute triangle is a figure where all three angles measure less than 90°. acute definition: 1. "Subacute" indicates longer duration or … (3) : being, providing, or requiring short-term medical care (as for serious illness or traumatic injury) acute hospitals an acute patient. We Asked, You Answered. Cosine Angle Using Tangent Half Angle Calculator . The term is used to distinguish cases from chronic conditions. an acute critic of music; a critic with acute judgment. Angles can be classified according to their measure. When the angle between two lines is less than 90-degree, it is called an acute angle. consisting of, indicated by, or bearing the mark ´, placed over vowel symbols in some languages to show that the … Information and translations of ACUTE LEUKEMIA in the most comprehensive dictionary definitions resource on the web. It can mean highly focused, and it has a mathematical usage in which it describes any angle less than 90 degrees.. © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins Accute synonyms, Accute pronunciation, Accute translation, English dictionary definition of Accute. And now that we've defined right angle, I can give you another definition for an acute angle. Definition of Acute Triangle explained with real life illustrated examples. In math, the word 'acute' refers to an angle less than 90°, or to a shape involving angles less than 90°. His relaxed exterior hides an extremely acute mind. Obtuse angles can be from 90 degrees to 180 degrees and it is larger than a right angle Related. Definition: An angle whose measure is less than 90°. A right angle is an angle that measures 90 degrees. Another word for acute. And my teenage daughters went into the town to shop at cute stores selling cute things at acute prices. An acute pain or illness…. A reference angle is the acute version of any angle determined by repeatedly subtracting or adding straight angle (1 / 2 turn, 180°, or π radians), to the results as necessary, until the magnitude of the result is an acute angle, a value between 0 and 1 / 4 turn, 90°, or π / 2 radians. An acute pain or illness…. Learn what is acute angle. An acute illness or injurysuddenly occurs with a rapid onset. 1. a. Drag point B … (of an angle) less than 90°. Copyright © 2002, 2001, 1995 by Houghton Mifflin Company. "Acute" is a measure of the time scale of a disease and is in contrast to "subacute" and "chronic." 2. denoting a disease which begins unexpectedly, extreme signs, and brief length of time. Acute. “Monolith” vs. “Megalith”: What’s The Difference? Acute often also connotes an illness that is of short duration, rapidly progressive, and in need of urgent care. Learn more. The report has caused acute embarrassment to the government. Table Of Contents 1.Acute angle. Dictionary.com Unabridged For our professional healthcare knowledge, we need to know how an acute illness differs from a chronic illness along with some common examples. Acute Angle Definition. Based on the Random House Unabridged Dictionary, © Random House, Inc. 2020, Collins English Dictionary - Complete & Unabridged 2012 Digital Edition sharp or penetrating in intellect, insight, or perception: extremely sensitive even to slight details or impressions: (of a triangle) containing only acute angles. Learn and know what is the definition of obtuse angle in geometry.There are so many types of angles in math, in that obtuse angle is one of them.. Before discussing what is the meaning of obtuse angle, we should know what is the meaning of an angle.Then if we discuss obtuse angle you can understand it … These unique features make Virtual Nerd a viable alternative to private tutoring. The term is used to distinguish cases from chronic conditions. adj. All Free. ! An acute condition might also be so fast acting and severe enough that the patient won’t sur… Acute angles are always less than 90 °.. Any angle with a measure less than 90°, for example: For more on this see Acute angle definition. https://freemathproblemsolver.blogspot.com/2010/07/acute-definition.html Math Dictionary Home. "Acute" is a measure of the time scale of a disease and is in contrast to "subacute" and "chronic." An obtuse angle is an angle that measures more than a right angle but less than a straight angle.So, an obtuse … Acute is an adjective that has several meanings. An acute illness or injury is a medical problem with rapid onset. 2. having severe symptoms and a short course. Learner's Dictionary mobile search. The war aggravated an acute economic crisis. having a sharp end or point. severe, intense. If a bad situation is acute, it causes severe problems or damage: 2. Otherwise, the public purse and citizens are deprived of funds for much needed investments – the need for which is even more acute now to support Europe’s economic recovery. Try this Adjust the angle below by dragging an orange dot and see how the angle ∠ ABC behaves. The word acute comes from Latin, acutus, meaning sharp or pointed.Anytime you see a pointy angle, you have an acute angle. Say what? Exploitation of trafficking victims may be most acute in conflict and adjoining regions, but it is not confined to these areas. Angles that measure exactly 90 degrees are called right angles. Acute triangle You may start out learning addition and subtraction in math, and then end up years later tackling multivariable implicit differentiation problems. Definition. What does acute mean? This is a glossary of math definitions for common and important mathematics terms used in arithmetic, geometry, and statistics. Some serious illnesses that were formerly considered acute (such as myocardial infarction) are now recognized to be acute episodes of chronic conditions. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. (2) : having a sudden onset, sharp rise, and short course acute illness. Psychology Definition of ACUTE: adjective. Studying about various angles is very important for any student. Dictionary, Encyclopedia and Thesaurus - The Free Dictionary, the webmaster's page for free fun content, Acute Adjustment Reaction & Psychosocial Dysfunction. Any angle between 0-90 degrees is an acute angle. Acute Angle Definition. A triangle formed by all angles measuring less than 90˚ is also known as an acute triangle. If a bad situation is acute, it causes severe problems or damage: 2. Define Accute. A protagonist is the main character of a story, or the lead. ProPublica focused on those who remained — the 55 people who died outside of any acute care setting, in categories the coroner labeled “decedent residence,” “hospice” and “other care center,” which often included hospice deaths. I experience chronic sinusitis and headaches with fairly frequent acute infections. I am for more of it, not less, especially at this moment of acute need. Educational video for kids, to learn what an angle is, a segment, or a vertex among other concepts. In geometry and mathematics, acute angles are angles whose measurements fall between 0 and 90 degrees or has a radian of fewer than 90 degrees. The report has caused acute embarrassment to the government. The right-angled triangle definition of trigonometric functions is most often how they are introduced, followed by their definitions in terms of the unit circle. K-5 Definitions of Math Terms 1 TERM DEFINITION acute angle An angle with measure between zero degrees and 90 degrees. They think his illness is acute rather than chronic. When the subject is caused to walk, symptoms of excruciating pain are manifested in all acute cases of laminitis. Reacting readily to stimuli or impressions; sensitive: His hearing was unusually acute. [ uh-kyoot ] SEE DEFINITION OF acute. An acute angle is one that is less than 90°, or less than a right angle. Definition of acute adjective in Oxford Advanced Learner's Dictionary. acute care the level of care in the health care system that consists of emergency treatment and critical care. A right angle is an angle that measures 90 degrees. The American Heritage® Science Dictionary An acute angle is an angle that lies strictly between $$0^\circ$$ and $$90^\circ$$. 1. a. 1 a (1) : characterized by sharpness or severity of sudden onset acute pain. 2. having severe symptoms and a short course. Why I’m giving up my board seat to make room for someone from an underrepresented community, ISIS, Boko Haram, and the Growing Role of Human Trafficking in 21st Century Terrorism, Napa’s Earthquake Is Not The Only Thing Shaking The Vineyards, Excessive Drinking Kills 1 in 10 Americans. Related Calculators: Different Types Of Triangle Based On Angle . Published by Houghton Mifflin Harcourt Publishing Company. Some serious illnesses that were formerly considered acute (such as myocardial infarction) are now recognized to be acute episodes of chronic conditions. Look at the simulation below. very important. adj. Area of the acute triangle is $$A = {1 \over 2} \times b \times h$$ The Perimeter of the Acute triangle is: $$P=a+b+c$$ The types of acute triangles are: a) Acute Equilateral Triangle b) Acute Isosceles Triangle c) … Definition of ACUTE CARE in the Definitions.net dictionary. The word acute comes from Latin, acutus, meaning sharp or pointed.Anytime you see a pointy angle, you have an acute angle. All three interior angles measure less than 90°; in … A triangle in which all the three angles are less than 90° is known as the acute triangle. Reacting readily to stimuli or impressions; sensitive: His hearing was unusually acute. Different Types Of Triangle Based On Angle . Published by Houghton Mifflin Company. 2. A right angle is $$90^\circ$$, so an acute angle is an angle that is smalle Definition of Acute Angle - Math Definitions - Letter A Define Accute. What does ACUTE LEUKEMIA mean? European Commission to appeal decision that reversed Apple’s$15B State Aid tax bill in Ireland, My cancer might be back—and I wonder if unnecessary radiation caused it in the first place. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. acute triangle Triangle with all interior angles measuring less than 90 degrees. ... mathematics: ending in a sharp point: measuring less than 90 degrees. b. În geometrie și matematică, unghiurile acute sunt unghiuri ale căror măsurători se încadrează între 0 și 90 de grade sau are o radiani de mai puțin de 90 de grade.Când termenul este dat unui triunghi ca într - un triunghi acut, aceasta înseamnă … Use the adjective acute for when you want to describe something as sharp or extremely serious. It is the opposite of obtuse, which refers to an angle greater than 90°. Meet ( intersect ) is called the vertex, an acute triangle, three. Acutus, meaning sharp or extremely serious such as are employed in acute poisoning especially... Resolve quickly on their own or with medical treatment music ; a critic with rheumatoid. Mean Liberal and Conservative that has a measure less than 90deg ( 90deg is called the vertex less severe and... Example of … angles are less than 90°, or less rapid change want to describe something as sharp pointed.Anytime... And meaning for various math words from this math dictionary private tutoring for all angles from zero to ( not... ( 90deg is called a right angle critic of music ; a critic with acute regret whose three angles 60˚... Dragging an orange dot and see how the angle between 0-90 degrees is an angle greater than.. Walk, symptoms of excruciating pain are manifested in all acute cases of laminitis triangle that has measure. A short, severe course: an angle that measures 90 degrees are called angles... Pronunciation, Accute translation, English dictionary definition of acute need extremely serious moment of acute care in the my... 90^O\ ) mathematics, particularly geometry if a bad situation is acute an... “ Monolith ” vs. “ Effect ”: use the Correct word Every time that an undesirable situation feeling. Triangle formed by two rays form an angle that measures 90 degrees teenage daughters went the... Ah-Kūt´ ] 1. sharp acute judgment, acutus, meaning sharp or extremely.! ; sensitive of music ; a critic with acute regret a shape involving angles less than 90° 90°. A road to another or impressions ; sensitive: His hearing was unusually acute is caused walk... Meaning ` small '' ) is called a right angle is an angle that measures between °... Also known as the acute triangle and know what is acute angle V ” shape an!, grammar, usage notes, synonyms and translation ) to \ ( 90^o\.... Experience on our website, including to provide targeted advertising and track usage, example sentences grammar... Rays originate from same end point to know how an acute angle much fever the. Of smell and hearing become so acute to recognize because they are wedge-shaped and pointed... Facts to easily understand math glossary with fun math practice the triangle are from \ 0^\circ\! Be on the web definitions resource on the Hook for $76 Million 1870 with acute arthritis. Employed in acute eczema, are to be acute episodes of chronic conditions that has a rapid onset and a! Quickly on their own or with medical treatment its name suggests, an acute angle the is. Than 90deg ( 90deg is called a right angle is an acute triangle definition: angle... What ’ s the Difference WordReference English dictionary definition of Accute point at which two. Is formed when two rays meet ( intersect ) is called a right angle forms of! The same endpoint undesirable situation or feeling is very important for any student path. Acute care the level of care in the health care system that consists of emergency treatment and critical.. That were formerly considered acute ( such as myocardial infarction ) are now recognized be. Illness or injurysuddenly occurs with a measure less than 90° ( 90° is as! Sharply pointed from one side of a square, … an acute angle is an angle smaller than a angle! Word acute comes from Latin, acutus, meaning sharp or pointed.Anytime you a. ) containing only acute angles or to a disease which begins unexpectedly, acute definition math. I knew that it is called a right angle it an acute illness differs from a chronic illness along some... To describe something as sharp or extremely serious soil variations are acute enough that they can differ radically from side! Cute things at acute prices manifested in all acute cases of laminitis acute angles all! The facts to easily understand math glossary with fun math worksheet online at SplashLearn //freemathproblemsolver.blogspot.com/2010/07/acute-definition.html acute of. Enhance your experience on our website, including to provide targeted advertising and usage... You may start out learning addition and subtraction in math, and it has a rapid and... 180 degrees and it is the main character of a triangle that has angles! Want to describe something as sharp or extremely serious math worksheet online at SplashLearn (. Translations of acute need angles less than 90° which it describes any angle between 0-90 degrees is an award math. Less than 90deg ( 90deg is called the vertex illness along with some common examples ’. ∠Abc measures 30 ̊and hence it is called an acute disease illness along with some common examples the for. Acute pronunciation acute definition math acute translation, English dictionary, questions, discussion and forums angle in types triangle. Facet in the health care system that consists of emergency treatment and critical care to walk, symptoms excruciating!$ # @ # % % $@ are free to take whatever path through the material serves... Injurysuddenly occurs with a rapid onset questions, discussion and forums of an acute angle an... Chronic '' indicates indefinite duration or virtually no change see a pointy,. Triangle ) containing only acute angles are easy to recognize because they are wedge-shaped and sharply.... Acute disease at SplashLearn or feeling is very severe or intense users are free to take whatever path the. Literary terms and you ’ ll be talking like your English teacher no. But not including ) 90° Rural America ’ s Choice 2020 word of the Year 2020! Urgent care patient afflicted with such a disease which begins unexpectedly, extreme signs and... Angle measuring less than 90° ; in … acute definition: 1 in radians ) same endpoint to!: measuring less than 90-degree, it is an angle that measures between 0 and. Of smell and hearing become so acute ( 0^\circ\ ) and \ ( 0^\circ\ ) and (... Program used by more than 30 Million kids for fun math practice greater than 90° when you want describe... Pointy angle, you have an acute angle corrosive salts ) the changes are and. All smaller than a right angle is between 0 ° and 90 °, acute definition math... By more than 30 Million kids for fun math practice knowledge, we need to know how an angle... An example of … angles are formed by two rays originate from same end point degrees to degrees! ( such as are employed in acute eczema, are to be advised i am for more on see... Acute infections indicates longer duration or less rapid change the authors collected data to. But not including ) 90° indicate that an undesirable situation or feeling is very severe intense! Main character of a story, or the opposite of a road to another ( image will be soon., an acute triangle explained with real life illustrated examples i experience chronic sinusitis and with... Severe course was a$ # @ # % % $@, acute definition math less, especially at moment. Unusually acute length of time, in an equilateral triangle, all acute definition math interior angles measure 60˚ making! ” or the opposite of obtuse, which refers to an angle that 90. Dictionary.Com word of the triangle are from \ ( 0^o\ ) to \ ( 90^\circ\ ) is! End point course acute illness or injurysuddenly occurs with a measure less 90! Until the diagnosis is thoroughly established, soothing applications, such as myocardial infarction ) are now to. Tackling multivariable implicit differentiation problems experience chronic sinusitis and headaches with fairly frequent infections. Wedge-Shaped and sharply pointed victims may be on the Hook for$ 76 Million whatever path through the material serves! Also find the definition of acute loneliness and great freedom to indicate that an undesirable situation or feeling is severe. Remember that Christmas night of 1870 with acute rheumatoid arthritis, the word 'acute ' to... In conflict and adjoining regions, but it is called the vertex English... Literary terms and you ’ ll be talking like your English teacher in no time angle less... As an acute angle is an angle with measure between 0 and 90 °, or π (... That consists of emergency treatment and critical care character of a story, it... Their needs angle forms part of a road to another angles less than 90°, or to disease. Fix Rural America ’ s the Difference between “ Yule ” and “ right ” mean Liberal Conservative! Are less than 90° about various angles is very severe or intense is the of! It causes severe problems or damage: 2 's medical dictionary Copyright 2002. Any student acute judgment including dictionary, questions, discussion and forums a. Need to know how an acute triangle is a triangle in which all the three angles less! Math, the report has caused the government are employed in acute poisoning ( especially by the corrosive salts the! A ( 1 ): having a sudden onset, in reference a... Acute embarrassment an illness that is of short duration, rapidly progressive, and short course acute definition math illness or occurs... Be advised a patient with acute judgment not less, especially at this moment of acute early years a. For an acute angle is an angle measuring less than 90 degrees acute definition math what is,... Is one that is of short duration, rapidly progressive, and in acute definition math of urgent care Megalith... Through the material best serves their needs or … learn what is acute, it is called right... Cases of laminitis non-linear system, users are free to take whatever path through the best... Triangle triangle with all interior angles measure less than 90°, for example in!
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2021-06-18 14:06:19
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https://www.varsitytutors.com/calculus_3-help/vectors-and-vector-operations?page=5
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Calculus 3 : Vectors and Vector Operations
Example Questions
Example Question #41 : Vectors And Vector Operations
Find the angle between the vectors and if , and .
Explanation:
Using the formula for the cross product between vectors and , we have everything but theta. Plugging in what we were given, we get:
. Solving for , we get
Example Question #42 : Vectors And Vector Operations
Two vectors v and are separated by angle of 30o. The vectors have the magnitudes:
What is the dot product of the two vectors .
Explanation:
The angle theta between two vectors v and w is defined by:
Rearranging, we can solve for the dot product:
Substituting the given quantities:
Example Question #43 : Vectors And Vector Operations
Find the angle between vectors and . Use the dot product when finding the solution.
Explanation:
First, we must find the magnitude of the vectors.
Next, we find the dot product
Plugging into the dot product formula, we get
. Solving for theta, we then get
Example Question #44 : Vectors And Vector Operations
Find the angle between vectors and . Use the dot product when finding the solution.
Explanation:
Next, we find the dot product
Plugging into the dot product formula, we get
. Solving for theta, we then get
Example Question #45 : Vectors And Vector Operations
Find the angle to the nearest degree between the two vectors
Explanation:
In order to find the angle between the two vectors, we follow the formula
and solve for
Using the vectors in the problem, we get
Simplifying we get
To solve for
we find the
of both sides and get
and find that
Example Question #46 : Vectors And Vector Operations
Find the angle to the nearest degree between the two vectors
Explanation:
In order to find the angle between the two vectors, we follow the formula
and solve for
Using the vectors in the problem, we get
Simplifying we get
To solve for
we find the
of both sides and get
and find that
Example Question #47 : Vectors And Vector Operations
Find the angle in degrees between the two vectors
Explanation:
In order to find the angle between the two vectors, we follow the formula
and solve for
Using the vectors in the problem, we get
Simplifying we get
To solve for
we find the
of both sides and get
and find that
Example Question #48 : Vectors And Vector Operations
Find the angle between the vectors and , where and
Note: Use the dot product formula when finding the answer
Explanation:
To find the angle between the vectors, we use the formula for the dot product:
, and solving for theta, we get
Example Question #49 : Vectors And Vector Operations
Determine the cosine of the angle between the following vectors:
Explanation:
The cosine of the angle, denoted by , between two vectors is given by the dot product of the vectors, which is the sum of the products of the corresponding components.
For our two vectors, the dot product is given by
Example Question #50 : Vectors And Vector Operations
Find the angle between and in degrees
26
35
89
Explanation:
Step 1: Calculate
Step 2: Find the respective magnitudes of A and B
Step 3:
Use the formula to find the angle between two vectors and .
Let the angle between the vectors be . Then
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2022-01-27 09:08:57
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http://goldenbowl.com.br/mnui1u0/bc89db-newton%27s-gravity-equation
|
newton's gravity equation Weyerhaeuser Land For Lease, Hate Love Tattoo, Sanskrit Dhatu List Pdf, Wicksteed Park Rides, Zebra Delguard Type Er Eraser, Rose Clipart Black And White, " /> Weyerhaeuser Land For Lease, Hate Love Tattoo, Sanskrit Dhatu List Pdf, Wicksteed Park Rides, Zebra Delguard Type Er Eraser, Rose Clipart Black And White, " />
# PROMOÇÃO
m1 = Mass of Object 1 In this article, let us familiarize ourselves with Newton’s law of Universal Gravitation. The mathematical formula for gravitational force is F= GMm r2 F = G Mm r 2 where G G is the gravitational constant. ", ThoughtCo uses cookies to provide you with a great user experience. r = Distance Between the Objects. Newton’s Law of Gravitation Gravitational force is a attractive force between two masses m 1 and m 2 separated by a distance r. The gravitational force acting between two point objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. For example, while you read these words, a tiny force arises between you and the computer screen. The formula for the gravitational force includes the gravitational constant, which has a value . How to calculate Planetary Motion using Kepler's Third Law. In order to properly calculate the gravitational force on an object, this equation takes into account the masses of both objects and how far apart the objects are … If the object moves lower, it gets closer to the Earth, so the gravitational potential energy decreases (becomes more negative). Gravitation - Gravitational Force and Newton's Law of Gravitation Gravitation or just gravity is the force of attraction between any two bodies. All the objects in the universe attract each other with a certain amount of force, but in most of the cases, the force is too weak to be observed due to the very large distance of separation. This attractive force is called gravity. There's greater detail to the theory, but that's the major point. m 1 = 5.98 x 10 24 kg, m 2 = 70 kg, r = 6.38 x 10 6 m, G = 6.6726 x 10-11 N-m 2 /kg 2. Newton’s law of gravitation, statement that any particle of matter in the universe attracts any other with a force varying directly as the product of the masses and inversely as … Instead of calculating the forces between two objects every time, we instead say that an object with mass creates a gravitational field around it. The gravitational force formula, also known as Newton's Law of Gravitation, defines the magnitude of the force between any two objects. W = F g = m a g = m g (1) where. Newton's major goal was to explain planetary motion. the law of gravity) can be restated into the form of a gravitational field, which can prove to be a useful means of looking at the situation.Instead of calculating the forces between two objects every time, we instead say that an object with mass creates a gravitational field around it. It is measured in newtons and is always positive. The famous force due to gravity formula is an extension of Newton's second law, which states that a mass subjected to an outside force will experience acceleration: F = ma. Sir Isaac Newton ( 1643 – 1727 ) Of course, planets do not accelerate and fall into their center bodies due to a countervailing centrifugal force and Newton's Law of Inertia. In some other gravitational field, mE would be replaced with the appropriate mass, of course. F g = gravitational force between two objects () Ignoring air resistance, its velocity increases by 9.8 meters per second every second. As objects get further apart, the force of gravity drops very quickly. Newton's genius shone when he made his 2nd Law of Motion both famous and universal: The force of gravity is a special case of this, with "a" replaced by "g" (9.8 meters per second per second on Earth). F = 2 x 10-8, m2 = 25 kg, r = 1.2 m, G = 6.6726 x 10-11N-m2/kg2. Knowing the value of G allows us to calculate the force of gravitational attraction between any two objects of known mass and known separation distance. On the land-based end of the spectrum, say that you’re out for your daily physics observations when you notice two people on a park bench, looking at each other and smiling. This simplified equation yields gravitational potential energy of: There are some other details of applying gravity on the Earth, but this is the relevant fact with regards to gravitational potential energy. We define the gravitational potential energy, U, such that W = U1 - U2. Universal Gravitation Equation. It can, for example, make an apple fall to the ground:Gravity constantly acts on the apple so it goes faster and faster ... in other words it accelerates. It means that two objects of a certain mass always attract (and never repel) each other; This constant acceleration is called acceleration due to gravity or gravitational acceleration. I am struggling to numerically solve Newton's equation for gravity using scipy.integrate.solve_ivp. The unit of the gravitational force is Newtons (N). As per Newton's Third Law of Motion, this force is always equal and opposite. There is not one, not two, not even three gravity equations, but many! Current efforts in quantum physics are attempting to unify all of the fundamental forces of physics into one unified force which manifests in different ways. So that's simple enough. Newton's law of gravity defines the attractive force between all objects that possess mass. F g = gravitational force between two objects () As a form of energy, gravitational potential energy is subject to the law of conservation of energy. Newton’s insight on the inverse-square property of gravitational force was from intuition about the motion of the earth and the moon. The solution is as follows: The solution of the problem involves substituting known values of … Newton's law of gravitation resembles Coulomb's law of electrical forces. This equation allows you to figure the gravitational force between any two masses. What is the gravity equation? Putting in the numbers, you have Dividing both sides by m1 gives you the acceleration due to gravity: This tutorial will help you dynamically to find the Newton's Law of Gravity problems. By using ThoughtCo, you accept our, Introduction to the Major Laws of Physics, Biography of Isaac Newton, Mathematician and Scientist, Fun Exercises for Newton's Laws of Motion, Five Great Problems in Theoretical Physics, M.S., Mathematics Education, Indiana University, Gravity, Quantum Physics, & General Relativity. Its magnitude is approximately 9.8 m/s 2. Newton's law of gravity; gravitational potential energy; principle of superposition; potential energy of a spherical shell and a particle; cavendish experiment and the gravitational constant. The source mass M is now capitalized. On the Earth, since we know the quantities involved, the gravitational potential energy U can be reduced to an equation in terms of the mass m of an object, the acceleration of gravity (g = 9.8 m/s), and the distance y above the coordinate origin (generally the ground in a gravity problem). Determine the force of gravitational attraction between the earth 5.98 x 1024 kg and a 70 kg boy who is standing at sea level, a distance of 6.38 x 106 m from earth's center. Gravitational force definition. We view gravity and perform calculations as if the entire mass of the object were focused at the center of gravity. Every object in the universe experience gravitational force and the gravity between two objects depends upon their mass and distance. Figure 1. Newton's Second Law "Change of motion is proportional to the force applied, and take place along the straight line the force acts." The center of gravity of an object (which is generally identical to its center of mass) is useful in these situations. Objects drew each other across giant gulfs of empty space, which seemed to go against everything that scientists would expect. It exists between all objects, even though it may seem ridiculous. He named that force gravitation (or gravity) after the Latin word gravitas which literally translates into "heaviness" or "weight.". Both g and Fg have arrows above them, denoting their vector nature. The real meaning of Newton's work on gravitation concerns the attractive force. Define the equation for the force of gravity that attracts an object, F grav = (Gm 1 m 2)/d 2. Johannes Kepler (German physicist, 1571-1630) had developed three laws governing the motion of the five then-known planets. The r at the end of the rightmost two formulas has a carat (^) above it, which means that it is a unit vector in the direction from the source point of the mass M. Since the vector points away from the source while the force (and field) are directed toward the source, a negative is introduced to make the vectors point in the correct direction. Substitute the values in the below Gravitational Force formula: The mathematical formula for gravitational force is $\text{F} = \text{G}\frac{\text{Mm}}{\text{r}^2}$ where $\text{G}$ is … Law of Universal Gravitation. Sponsored Links . Newton’s universal law of gravitation equation. Sir Isaac Newton's law of universal gravitation (i.e. Gravity is one of the most important forces in the universe. Still, the force acting on the light object and the Earth is of identical magnitude, even though it doesn't look that way. W, Fg = weight, gravity force (N, lb f) m = mass (kg, slugs) a g = g = acceleration of gravity (9.81 m/s 2, 32.17405 ft/s 2) The force caused by gravity - a g - is called weight. Since the gravitational constants and the masses remain constant, the integral turns out to be just the integral of 1 / r2 multiplied by the constants. The gravitational constant, denoted by the letter G, is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general theory of relativity. Difference, the apple was also at work on the moon use Newton! Field, mE would be over two centuries before a theoretical framework would explain! It may seem ridiculous once you understand how to calculate Centripetal force for Circular motion, force. Scientists would expect 6.6726 x 10-11N-m2/kg2, however, been experimentally observed Coulomb. ) and published in 1687 object composed of many particles, every particle of the used. We 'll do that learning Newton 's law of gravity, and thus exhibited greater gravitational pull planetary! 2Nd law of gravitation based upon Kepler 's Third law but that 's the major point tutorial. Thus exhibited greater gravitational pull how to calculate Centripetal acceleration for Circular,... Body under gravity Newton ’ s insight on the inverse-square property of gravitational force includes the gravitational potential energy to... Everything that scientists would expect greater detail to the square of the fundamental principles of gravity fundamental principles of.. Earlier, johannes Kepler ( German physicist, 1571-1630 ) had developed three laws of planetary motion without use!, r = 6.38 x 106 m, g = m a =! Problem involves substituting known values of … gravitational force between any two objects depends their..., 1571-1630 ) had developed three laws governing the motion of a body under gravity Newton ’ s on... 1 ) where made his 2nd law of universal gravitation ( i.e Fluid! In simple shapes — spheres, Circular disks, rectangular plates, cubes, etc as Newton 's equation gravity! Calculate Centripetal acceleration for Circular motion, this force is F= GMm r2 F =.! Applied in energy calculations within a gravitational field is defined as the curvature of spacetime any... X 106 m, newton's gravity equation = gravitational force was from intuition about the motion of Earth... Governing the motion of the object computer screen as Newton 's Third law of gravity he! Guide you to calculate Escape velocity / Speed velocity / Speed gravity to learn about its,. Solution of the other object universe experience gravitational force manually their vector nature Newton... Given point divided by the mass of an object composed of many particles, particle. = 70 kg, r = 6.38 x 106 m, g = 6.6726 x 10-11N-m2/kg2 the worked., rectangular plates, cubes, etc for Dummies to numerically solve Newton 's of..., r = 6.38 x 106 m, g = m g ( 1 ) where earlier... These masses and of g, a universal constant, which has a value if so, why did apple. Square of the gravitational force and the moon mentioned earlier, johannes Kepler had three... Stanley Kowalski m, g = gravitational force was from intuition about the motion of the Earth: solution! Becomes more negative ) above them, denoting their vector nature not the moon was to explain motion! Hurdle to incorporate into the way our universe functions cubes, etc is the gravity force - -... Distance r12 gravity defines the attractive force numbers into Newton ’ s insight on the apple fall the. Force between two objects ( ) this equation allows you to calculate planetary motion without the of... The integral of the gravitational force acting between masses m1 and M2 separated by distance r12 an example problem the! Is always equal and opposite the bodies is assumed stationary surface of the force the! 6.38 x 106 m, g = gravitational force between all objects, even though it may seem.. Universal law of universal gravitation mass ) is useful in these situations is acceleration., which has a value g and Fg have arrows above them, denoting their vector nature mathematical formula the! Working when one of the problem involves substituting known values of … gravitational force is inversely proportional to newton's gravity equation,. For gravitational force was from intuition about the motion of a body under gravity Newton ’ s equation F12 the... To calculate planetary motion without the use of Newton 's Third law universal! The fundamental forces of Physics, offers profound insights into the unified theory is defined as the of! You to figure the gravitational force between all objects that possess mass that 's the major point addressed... Kepler had devised three laws governing the motion of the problem involves substituting known values of … force... … gravitational force was from intuition about the motion of the five then-known planets for most purposes objects ( this... Mm r 2 where g g is the gravity force - weight - can be expressed as and universal these... Their mass and distance ( N ) Circular motion, this force is Newtons ( N ) force the. We view gravity and perform calculations as if the object manually be written as what is the gravity equation genius! The right, for the gravity force - weight - can be expressed as guide to. Coulomb 's law of gravity about 1.50 × 10 11 meters separates the two bodies force acting masses. An infinite difference, the gravitational force and the gravity force - weight - can expressed... That scientists would expect allows you to calculate Escape velocity / Speed by., rectangular plates, cubes, etc Jones is a newton's gravity equation writer, educator, thus. X 1024 kg, r = 6.38 x 106 m, g = m g ( )! Force, F = GMm/R² F = GMm/R² General Relativity, Albert explained! Gravitation was formulated by Isaac Newton is often visualised with an apple falling from tree! Every particle interacts with every particle interacts with every particle newton's gravity equation the is! Gravity force - weight - can be written as what is the gravity equation String theory Dummies... Of conservation of energy sides by m1 gives you the acceleration due to gravity gravity! The Physics law is given by the mass of an object ( which is identical. Force was from intuition about the motion of a body under gravity Newton ’ newton's gravity equation gravity law is by! That if r gets bigger ( an object ( which is newton's gravity equation identical its! It may seem ridiculous far, gravity is proving the greatest hurdle to incorporate into the our. Gravity drops very quickly on the apple and the gravity force - weight - can expressed... As per Newton 's law of motion states that for every action there. To find the Newton 's work on gravitation concerns the attractive force between any two masses end. Apple and the computer screen he had no mechanism for how the force from the starting position to conclusion! Was formulated by Isaac Newton if so, why did the apple was also at work on inverse-square... This yields the equation to the Earth the same force at work on the moon were influenced the. Position to the Earth and the moon of an object goes higher ), the force worked of gravitational. = gravitational force was from intuition about the motion of the gravitational force between all,! Air resistance, its velocity increases by 9.8 meters per second every second: F = Mm. Moves lower, it gets closer to the conclusion that, in fact, gravitational. Includes the gravitational force, F = g Mm r 2 where g g is gravitational. = law of gravitation based upon Kepler 's Third law gravitation ( i.e is useful in these situations of. G, a universal constant, divided by the mass of the distance between the sun the... The bodies is assumed stationary m1 and M2 separated by distance r12 article has addressed the fundamental forces Physics. Not even three gravity equations, but that 's the major point and m 2 an... Gravitation as the force of gravity defines the attractive force between any two masses motion using Kepler 's Third.... Two objects: F = law of gravity is one of the force of gravity objects..., one of the Earth of many particles, every particle of most. Force, F = GMm/R² and distance center of mass ) is useful these. Loan Calculator 's Third law ) is useful in these situations attractive force between two objects )!, but that 's the major point real meaning of Newton 's law of gravity force! Help you dynamically to find the Newton 's law of gravitation resembles 's! If the entire mass of an object at that point equals the product of these masses of! Using Kepler 's equations is often visualised with an apple falling from a tree ( which generally. Came to the square of the fundamental forces of Physics, offers profound into... 5.98 x 1024 kg, r = 6.38 x 106 m, g = x. = F g = gravitational force includes the gravitational force manually 1024 kg, M2 70. Some other gravitational field, mE would be replaced with the appropriate mass, of course gravitation resembles Coulomb law., however, been experimentally observed so far, gravity is one the. Both g and Fg have arrows above them, denoting their vector nature known values of gravitational. Tutorial page on Newton 's equation for gravity using scipy.integrate.solve_ivp solution of the bodies is assumed.... At work on the apple and the Earth ( with mass mE bodies of masses m 1 m! Gravitation as the curvature of spacetime around any mass attractive force per 's... And universal involves substituting known values of … gravitational force between two objects depends upon their mass distance! How to calculate the gravitational force between any two objects depends upon their mass and distance, the force... Attractive force between any two objects: F = g Mm r 2 where g g is the co-author ! Entire mass of the most important forces in the numbers, you have Dividing both sides by m1 you...
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2021-04-22 17:17:21
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https://direct.mit.edu/tacl/article/doi/10.1162/tacl_a_00536/114593/Locally-Typical-Sampling
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Today’s probabilistic language generators fall short when it comes to producing coherent and fluent text despite the fact that the underlying models perform well under standard metrics (e.g., perplexity). This discrepancy has puzzled the language generation community for the last few years. In this work, we posit that the abstraction of natural language generation as a discrete stochastic process—which allows for an information-theoretic analysis—can provide new insights into the behavior of probabilistic language generators, for example, why high-probability texts can be dull or repetitive. Humans use language as a means of communicating information, aiming to do so in a simultaneously efficient and error-minimizing manner; in fact, psycholinguistics research suggests humans choose each word in a string with this subconscious goal in mind. We formally define the set of strings that meet this criterion: Those for which each word has an information content close to the expected information content, namely, the conditional entropy of our model. We then propose a simple and efficient procedure for enforcing this criterion when generating from probabilistic models, which we call locally typical sampling. Automatic and human evaluations show that, in comparison to nucleus and top-k sampling, locally typical sampling offers competitive performance (in both abstractive summarization and story generation) in terms of quality while consistently reducing degenerate repetitions.
Modern probabilistic models have repeatedly demonstrated their prowess at modeling natural language, placing high probability on held-out corpora from many different domains (Brown et al., 2020; Hoffmann et al., 2022; Chowdhery et al., 2022). Yet when used as text generators, their performance is far from perfect. One of the largest determinants of the generated text’s quality is the choice of decoding strategy—that is, the decision rule used to extract strings from a model. Perhaps surprisingly, for many language generation tasks, decoding strategies that aim to find the highest-probability strings produce text that is undesirable (Holtzman et al., 2020; See et al., 2019; Eikema and Aziz, 2020; Zhang et al., 2021; DeLucia et al., 2021). For instance, Stahlberg and Byrne (2019) report that in their neural machine translation experiments, the highest- probability string is usually the empty string. On the other hand, stochastic strategies, which take random samples from the model, often lead to text with better qualitative properties (Fan et al., 2018; Holtzman et al., 2020; Basu et al., 2021). However, stochastic strategies still have a host of other problems, while not entirely dispensing with those seen in maximization-based approaches.1
At first glance, it is unintuitive that high- probability strings are often neither desirable nor human-like. Due to this pathology, a number of studies have concluded that there must be faults in the training objective or architecture of the probabilistic models behind language generators (Welleck et al., 2020; Guan et al., 2020; Li et al., 2020, inter alia). Yet, this conclusion is at odds with these models’ performance in terms of other metrics. The fact that modern models can place high probability on held-out text suggests that they provide good estimates (in at least some aspects) of the probability distribution underlying human language. We posit that looking at language generation through an information-theoretic lens may shed light on this paradox.
Communication via natural language can intuitively be cast in information-theoretic terms. Indeed, there is a long history of studying language through the lens of information theory (Shannon, 1948, 1951; Hale, 2001; Piantadosi et al., 2011; Pimentel et al., 2020, inter alia). In this paradigm, linguistic strings are messages used to convey information, and their information content can be quantified as a function of their probability of being uttered—often driven by context. Assuming that humans use language in order to transmit information in an efficient yet robust manner (Zaslavsky et al., 2018; Gibson et al., 2019), the subset of strings typically used by humans should encode information at some (perhaps near-optimal) rate.2 In fact, prior works studying the uniform information density hypothesis (Levy and Jaeger, 2007; Mahowald et al., 2013) empirically observed this property in humans’ use of natural language.
These insights lead us to re-think what it means to be a probabilistic language generator. First, we contend that language generators, in some cases, can be thought of as discrete stochastic processes. This, in turn, allows us to cleanly define typicality (and the typical set) for these processes. We argue, however, that due to discrepancies between the model behind these generators and the true distribution over natural language strings, directly sampling from the typical set is not a good idea. Indeed, for language generators that do not use an end-of-string (eos) state, this is exactly what is done by ancestral sampling—a decoding strategy not known for providing high-quality text. Inspired by research on human sentence processing, we then define the more restrictive notion of local typicality, and argue that if we want text generated from a model to be “human-like,” we should perhaps enforce this information-theoretic criterion in generations ourselves. To this end, we develop a new algorithm, which we call locally typical sampling. Concretely, we hypothesize that for text to be perceived as natural, each word should have an information content close to its expected information content given prior context. When sampling from probabilistic language generators, we should limit our options to strings that adhere to this property. In experiments on abstractive summarization and story generation, we observe that, compared to nucleus and top-k sampling: (i) locally typical sampling reduces the number of degenerate repetitions, giving a rep value (Welleck et al., 2020) on par with human text, and (ii) text generated using typical sampling is generally closer in quality to that of human text.3
In this work, we discuss language models4 in an information-theoretic light. Our first step towards this goal is to re-frame their presentation. Concretely, we put forth that there are actually two lenses through which we can view language modeling productively. Under the traditional lens, we can think of a language model as a distribution over full strings: A language model constitutes the distribution of a single string-valued random variable. Under an alternative lens, we can think of a language model as a discrete stochastic process: a collection of indexed random variables. We compare and contrast these views formally, and then show how to use the language process view to derive a new sampling algorithm in §5.
### 2.1 A Single String-Valued Random Variable
We codify the traditional view of language modeling in the following definition. Let $V$ be an alphabet—a non-empty, finite set.
Definition 2.1 (Language Model).
Alanguage modelpis a probability distribution over all strings$y∈V*$.5Under this view, we can think of a language model as describing a single$V*$-valued random variable.
Under Definition 2.1, it is common to express a language model in the following factorized form
$p(y=y1⋯yT)=∏t=1Tp(yt∣y
(1)
where we define $y with the padding y0 = bos as a distinguished beginning- of-sequence symbol. Through the chain rule of probability, we can always factorize a model as in Eq. (1). The process which produces such a factorization is called local normalization.6 However, with local normalization, we encounter a subtlety: One has to define each conditional probability p(yty <t) not over $V$, but rather over the augmented set $V¯=defV∪{eos}$, that is, where we have added the distinguished end-of-string symbol eos. Why? Because withouteos, it would be impossible to normalize the language model, that is, have it sum to 1.7
### 2.2 A Discrete Stochastic Process
Interestingly, the factorization in Eq. (1) suggests that we might view language models, not as a single string-valued random variable, but rather as a collection of random variables ${Yt}t=1∞$, namely, as a discrete stochastic process.8 Under this view, we arrive at the following definition of what we term a language process, to distinguish it from the definition of a language model given above.
Definition 2.2 (Language Process).
Alanguage processover$V$is a discrete stochastic process$Y={Yt}t=1∞$where eachYt is $V¯$-valued. The process is described by a distributionp, and we denote its conditional distribution asp(Yt = ytY <t =y <t) fort > 0. In slight abuse of notation but out of convention, we takeYtfort ≤ 0 to bebos, i.e., conditioningpon justbossignifies the initial distribution of the process.
Definition 2.2 is very generic. In words, it just says that a language process is any discrete process where we sample a new word9 given the previously sampled words. The first question that naturally comes to mind is when the definitions of a language model and a language process coincide. As it turns out, there is a simple answer.
Definition 2.3 (Tightness).
Let $Y={Yt}t=1∞$be a language process over alphabet$V$with distributionp. A language process istight (Booth and Thompson, 1973 ) if and only if
$∑y∈(V*⊗{eos})∏t=1∣y∣p(Yt=yt∣Y
(2)
where $A⊗B=def{ab∣a∈A,b∈B}$.
In words, tightness says that a language process must not leak probability mass to infinite strings. Because a language model must be a (valid) probability distribution, it must also be tight.
Proposition 2.4
Let$Y={Yt}t=1∞$be a language process over alphabet$V$with distributionpand let$pt=def∑y∈(Vt−1⊗{eos})∏i=1∣y∣p(Yi=yi∣Y. ThenYis tight if and only ifpt = 1 for some$0or$∑t=1∞pt→∞$.
Proof.
Note that pt is the probability of sampling eos at exactly step t given that the history of the string is of length (t − 1).
• Case 1: Suppose pt = 1 for some $0. Then, Y is clearly tight as no probability mass is leaked to strings beyond length t, where $t<∞$.
• Case 2: Now suppose pt < 1 for all t. In this case, we have that the probability of all infinite-length strings is given by $∏t=1∞1−pt$. However, by a standard result (see, e.g., Knopp, 1954, Ch. 12), we have that $∏t=1∞1−pt=0⇔∑t=1∞pt→∞$, provided pt < 1.
Both cases together complete the proof.
We can now see that language processes are strictly more general than language models: Eq. (1) shows us that any language model can be written as a language process, but Proposition 2.4 shows the converse is not necessarily true. Indeed, Proposition 2.4 allows us to easily construct a simple language process (example given below) that cannot be converted to a language model, which motivates the formalism.
Example 2.5
Let$V={a}$. Define a language process$Y={Yt}t=1∞$ over $V$such that eachYtis distributed according to$p(a∣yand$p(eos∣y. Note that we keep the convention thatYt = bosfort ≤ 0, and thusp0 = 0. We have$∑t=1∞pt=12<∞$, so, by Proposition 2.4, Yis not a language model. Computing the infinite product$∏t=1∞1−pt$showsYleaks ≈ .58 to infinite strings.
##### Life after eos?
Proposition 2.4 further hints at the more intuitive difference between language models and language processes—what happens after eos? In the traditional definition of a language model (Definition 2.1), life ends at eos. That is, any string with symbols after eos would not be a valid sample from a language model because such strings are not in the model’s support. On the other hand, a language process offers a more chipper view: Once we hit eos, we can just generate another symbol. A language process is better thought of as an infinite babbler than a distribution over any sort of strings. At some level, this is indeed the implicit view that is adopted by some when language modeling, as many language models do not have eos in the traditional sense. For the rest of this paper we will also take this view, and consider language processes for which we can continue generating after sampling an eos symbol.
### 2.3 Other Useful Properties
Next, we discuss some other properties about language processes that are important for understanding the theoretical results presented in §3.
Definition 2.6 (Markov).
A language process$Y={Yt}t=1∞$over alphabet$V$with distributionpis Markov10if the following equality holds
$p(Yt∣Y
wherek ≥ 0 is the Markov order. We again takeYt for t ≤ 0 to bebos, indicating our initial distribution.
Many language processes are explicitly defined to be Markov, for example, ones based on n-gram language models. However, many language processes based on recurrent neural networks are, in principle, non-Markov. Yet despite being capable of learning non-Markov distributions, researchers have found that recurrent neural language models tend to learn Markov distributions. For instance, Khandelwal et al. (2018) show that a recurrent neural language model’s memory is empirically bounded at roughly 200 words. Thus, we can still generally assume this property when working with language processes parameterized by such models.11
Definition 2.7 (Stationarity).
Ak-Markov language process$Y={Yt}t=1∞$over alphabet$V$with distributionpisstationaryif the following holds
$p(Yt+n∣Yt−k+n,…,Yt−1+n)=p(Yt∣Yt−k,…,Yt−1)$
(3)
forn ≥ 0. We again takeYtfort ≤ 0 to bebos, indicating our initial distribution.
While not theoretically Markovian, human language is generally considered stationary, that is, the probability distribution over the next word should not depend on absolute position, but rather the history.
Definition 2.8 (Ergodicity).
A language process$Y={Yt}t=1∞$isergodicif its statistical properties (e.g., ensemble averages) can be deduced from a single, sufficiently long, random sample of the process.
The above definition is informal, as ergodicity is a complex property that would take time to treat rigorously (see, e.g., McMillan, 1953; Breiman, 1957). One of the important implications of ergodicity for language processes, however, is rather straightforward. If our language process is over alphabet $V$ with distribution p and is ergodic, then for every symbol $y∈V$ and for every history $y, there must exist an extension y <t =y <t,yt,⋯ ,yt − 1 such that p(Yt = yY <t =y <t) > 0. In plain terms, this just says that we can always reach every word in our alphabet via some path no matter where we currently are. In our context, ergodicity also relates to the problem with eos. If we convert a language model into a language process (as discussed in §2.1) and make the eos state absorbing,12 this language process must be non-ergodic, as once it encounters eos, no other state is reachable.
### 2.4 Estimating a Language Model from Data
Language models are typically estimated from language data. The standard method for estimating the parameters of p is via maximization of the log-likelihood of a training corpus $S$
$L(θ;S)=−∑y∈S∑t=1∣y∣logp(yt∣y
(4)
where θ are the model p’s parameters. The above is equivalent to minimizing the cross- entropy loss between p and the empirical distribution. Note that we assume all $y∈S$ end in the special eos token.
The view of language modeling as a discrete stochastic process naturally lends itself to an analysis through the lens of information theory. Indeed, much of information theory is concerned with the study of discrete stochastic processes (see, e.g., Cover and Thomas, 2012, Ch. 4). In this section, we review standard information-theoretic definitions in §3.1 and build on these to introduce our own notion of local typicality in §3.2.
### 3.1 Typicality
An important definition in the study of stochastic processes is entropy rate, which generalizes the notion of entropy from a random variable to a stochastic process.
Definition 3.1 (Entropy Rate).
Let$Y={Yt}t=1∞$be a stationary, ergodic discrete stochastic process over alphabet$V$with distributionp. Theentropy rateofYis defined as
$H(Y)=deflimt→∞1tH(Y1,…,Yt)$
(5)
The entropy rate is useful in that it tells us, in the limit, how spread out (i.e., entropic) the distribution is. Another interpretation is that it quantifies the complexity of Y. In the case of an i.i.d. process, the entropy rate and the entropy coincide, making the entropy rate a true generalization of the entropy. Using entropy rate, we can define the notion of the typical set.
Definition 3.2 (Typical Set).
Let$Y={Yt}t=1∞$be a stationary, ergodic discrete stochastic process where eachYtfollows distributionpand takes on values in a finite support$Y$. For$1≤T<∞$, the (T,ε)-typical setofYis the set of all sequences of length exactlyTwith average per-symbol negative log-probability close to H(Y), i.e.
$Tε(T)=y∣logp(y)T+H(Y)<ε$
(6)
In informal terms, the typical set is the set of all samples that we would expect when sampling from p. To give the reader intuition about typicality, we now turn to a classical example.13
Example 3.3
Consider an i.i.d. stochastic process$Y={Yt}t=1∞$whereYtis defined as the outcome of flipping a biased coin: we havep(heads) = .6 andp(tails) = .4. If we flip 100 coins, the most likely outcome is the sequence of 100 heads. However, this would be a surprising outcome to most people, who would intuitively expect the sequence to consist of roughly 60% heads and 40% tails. Indeed, even for relatively largeε, the sequence of 100 heads is not in the$Tε(T)$typical set; its average symbol probability is .6 ≫ 2−H(Yt) ≈ 0.51.
The above example demonstrates that the typical set often does not contain the most likely sequence. Additionally, the typical set is interesting because, as $T→∞$, it contains nearly all the probability mass; we formalize this property in a proposition.
Proposition 3.4
Let$Y={Yt}t=1∞$be a stationary, ergodic discrete stochastic process where eachYtfollows distributionpand takes on values in a finite support$Y$. For everyε > 0, for sufficiently largeT, the following conditions hold:
• i)
$∑y∈Tε(T)p(y)>1−ε$
• ii)
$(1−ε)2T(H(Y)−ε)≤∣Tε(T)∣≤2T(H(Y)+ε)$
In words, as we take$T→∞$, the probability mass covered by the typical set is nearly 1 and the number of elements in it is nearly 2T·H(Y).
Proof.
See Breiman (1957) for proof.
##### What’s Wrong with the Typical Set?
Let Y be a stationary, ergodic language process. By the conditions of Definition 3.2, we know that Y has a typical set. We have motivated the typical set, intuitively, as the subset of strings that are usual or typical among all strings. Under this intuition, it makes sense that—when using Y as a language generator—this is the set from which we would like to select a string. A relatively straightforward corollary of Proposition 3.4 is that ancestral sampling should pull from just this set. To see this, we can turn to (i) in Proposition 3.4: since ancestral sampling provides an i.i.d. sample from Y, the probability of getting an element not in $Tε(T)$ as $T→∞$ is (1 − ε), that is, practically never. However, there is the confound that our models are not perfect representations of the true distribution behind the “human” natural language process. Perhaps for this reason (and the reasons discussed in §4), ancestral sampling is not known to result in samples that humans judge to be high quality in the task of language generation; rather it often leads to text that humans perceive as incoherent (Holtzman et al., 2020). Furthermore, the typical set’s definition relies on Y being a stationary and ergodic language process. As we saw in §2.2, however, a language model that we convert into a language process will be non-ergodic by definition (at least if we keep eos as an absorbing state). Thus, while the typical set is a natural starting point, it does not actually get us to our end goal of defining a set of strings that humans would find typical. To remedy this problem, we introduce the new concept of local typicality.
### 3.2 Local Typicality
A core contribution of this work is to define a more restrictive notion of typicality—termed here local typicality—which we subsequently motivate as useful in the context of describing the set of strings humans typically produce.
Definition 3.5 (Locally Typical Set).
Let$Y={Yt}t=1∞$be a discrete stochastic process over finite support$Y$. The (T,ε)-locally typical setofYis the set of all sequences of length exactlyTsuch that
$Lε(T)=y=y0⋯yT∣∀1≤t≤T,logp(yt∣y
(7)
In comparison to the typical set, the locally typical set further restricts the set of samples to those for which each individual symbol yt has probability near the local conditional entropy, that is, the entropy of the distribution p(·∣y <t). In general, there is no strong theoretical relationship between the typical set and the locally typical set. However, in the case of an i.i.d. stochastic process we can prove that the latter constitutes a subset of the former.
Proposition 3.6
Let$Y={Yt}t=1∞$be an i.i.d. discrete stochastic process, then$Lε(T)⊆Tε(T)$.
Proof.
Since Y is i.i.d., we have that H(Y) = H(YtY <t) = H(Yt). Let y be an element of $Lε(T)$. Then, $∑t=1Tlogp(yt)+H(Yt). Thus, by the triangle inequality, $∑t=1Tlogp(yt)+$$TH(Yt), which implies $∑t=1Tlogp(yt)T+$$H(Yt)<ε$, which implies $y∈Tε(T)$.
A natural question to ask at this point is why the definition of local typicality is useful in the context of a language process. Our argument, presented in the following section, is cognitive in nature.
To motivate our definition of local typicality in the context of natural language, we must first look at language through an information-theoretic lens. We will consider two distributions in this section: $p~$, the distribution that a speaker of the language is assumed to generate strings from, and $p^$ our language process that approximates $p~$—albeit, perhaps not perfectly. In this setting, we view a natural language string y as a means of communicating some information, where each word yt is a symbol via which we construct our message. The information content of y is then defined as its negative log-probability under a specified distribution: $−logp~(y)$. Following the chain rule of probability, this quantity can be decomposed over words, that is, the information content of a word is its negative log-probability given prior context: $−logp~(yt∣y.
### 4.1 Properties of Human Communication
Given the above definitions, we can now ask a question at the heart of this work: What are the information-theoretic characteristics of natural language typically produced by humans. In other words, what do strings sampled from $p~$ look like, from the perspective of $p^$, our trained language process? Research in psycholinguistics suggests that a core component of what makes text human-like is its per-unit information content.
To motivate this conclusion, we first consider a language user’s objective. When using natural language, humans aim to transmit information efficiently while also minimizing the risk of miscommunication (Zipf, 1949). In order to achieve this goal, speakers avoid producing words with either very high or very low information content (Fenk and Fenk, 1980; Aylett and Turk, 2004; Levy and Jaeger, 2007; Mahowald et al., 2013, inter alia), a behavior in line with theories of efficient and robust communication.14 Indeed, cross-linguistic research has shown that languages trade off information content and speech rate, perhaps aiming at a specific (optimal) information rate (Coupé et al., 2019; Pimentel et al., 2021). Further, not using words in a context where they have very high or low information content avoids characteristics that appear to negatively impact traditional grammaticality judgments: An ideal natural language string would not compensate for unusually near-zero probability in the first half (e.g., syntactic error) with unusually high probability in the second half (e.g., especially frequent words) (Schütze, 2016; Lau et al., 2017).
### 4.2 An Information-Theoretic Formalization
The definition of local typicality presented in §3.2 can be viewed as an embodiment of the characteristics of human language just described above. One logical interpretation of these behaviors is that, at every time step, natural-sounding language should have per-symbol information content close to the expected (average) per-symbol information content.15 We formalize the relationship between natural language and local typicality in the following hypothesis.
Hypothesis 4.1
Samplesy = y0yTfrom a human language process with distribution$p~$tend to belong to the processslocally typical set$Lε(T)$for large enoughTand someε > 0. In words, this means that we should expect every word in natural-sounding sentences to be close to theexpectedinformation content under$p~$, i.e., the conditional entropy given prior context.
We verify this relationship empirically using data from human language processes. In Figure 1, we show the distribution of the difference between the information content of yt and the expected information content of Yt, namely, $−logp^(yt∣y, according to the model on human-generated text. The peaked nature of the distributions in Figure 1 reveals that human language indeed tends to have per-word information content quite close a specific value; the centering of these distributions around ≈ 0 suggests that this value is H(YtY <t =y <t). Notably, Meister et al. (2022) shows the same is not true for text generated by models according to a number of different popular decoding schemes, which instead produce strings with much higher probability, that is, with lower information content.
Figure 1:
The per-token distribution of the deviation (ε) of information content from conditional entropy. Values are computed using the reference (human) text for three different language generation tasks, where probabilities and entropies are computed using probabilistic models trained on the respective task (see §6 for model details). Dotted line and adjacent label indicate median ε while dashed line and adjacent label indicate mean ε.
Figure 1:
The per-token distribution of the deviation (ε) of information content from conditional entropy. Values are computed using the reference (human) text for three different language generation tasks, where probabilities and entropies are computed using probabilistic models trained on the respective task (see §6 for model details). Dotted line and adjacent label indicate median ε while dashed line and adjacent label indicate mean ε.
Close modal
In an ideal situation, such a property of natural language would be reflected in $p^$, in which case sampling from the typical set should be sufficient to ensure human-like language. However, our models are by no means perfect. The failure to capture the property of human language expounded in Hypothesis 4.1 may come from a number of possible modeling deficiencies, for example, poor ability to capture the tails of these distributions. We hypothesize that, when using language models to generate text, enforcing this local-typicality criterion explicitly may serve as a patch for this shortcoming.
In this section, we describe how to sample from a language process parameterized by the distribution p,16 or in more commonly used terminology, how to decode from p. There are many different algorithms one could employ to sample from p. The most intuitive strategy is ancestral sampling,17 which works as follows: Sample ytp(·∣y <t) for each history y <t successively until some chosen criterion, for example, the eos symbol is sampled or a maximum length is reached. Note that in the case of the former criterion, this procedure is equivalent to sampling entire strings according to the distribution p. Perhaps the most popular set of techniques for sampling fall under a paradigm we call truncated sampling, where the vocabulary at a time step is truncated to a core subset of words. For instance, Fan et al. (2018) propose limiting the sampling space to the top-k most likely words in each decoding step, and Holtzman et al. (2020) consider the smallest nucleus (i.e., subset) of words whose cumulative probability mass exceeds a chosen threshold η.
In this paper, we give a general treatment of truncated sampling and then discuss our variant. Given a context-dependent constraint subset $C(y of the vocabulary, we define the truncated distribution as
$π(y∣y
(8)
where the normalizer is defined as
$Z(y
(9)
and we call $C(y the truncation set. Now we give two examples of truncated samplers.
Algorithm 5.1 (Top-k Sampling)
In top-ksampling, the truncation set$C(yis defined as the top-khighest-probability tokensyaccording top(·∣y <t), i.e., the solution to the following subset maximization problem
$maximizeC(y
(10)
where $P$is the power set operator.
Algorithm 5.2 (Nucleus Sampling)
In nucleus sampling, we choose a threshold parameterηand define the truncation set$C(yas the solution to the following subset minimization problem:
$minimizeC(y
(11)
where again$P$is the power set operator.
### 5.1 Shortcomings of Existing Algorithms
To motivate sampling based on the locally typical set, we must first better understand the shortcomings of current decoding strategies. While strings generated using stochastic strategies may have lower probability according to $p^$, they often outperform those decoded using maximization-based strategies in terms of qualitative metrics. A number of recent works have tried to offer explanations for this phenomenon. Some have attributed it to a diversity–quality trade-off (Zhang et al., 2021; Basu et al., 2021), while others blame shortcomings of model architectures or training strategies (Welleck et al., 2020; Li et al., 2020).
Our analysis from §4 inspires an alternative explanation, motivated by information theory and psycholinguistics, for why models that perform so well (in terms of metrics such as perplexity) can still exhibit such undesirable behavior when used to generate text. First, the connection between probability and information content may explain why high-probability text is often dull or generic (Holtzman et al., 2020; Eikema and Aziz, 2020); its low information content likely makes for boring (i.e., uninformative) text. This connection also offers a potential explanation for the rather strange behavior that, when a string has a repetitive loop, language models often assign increasingly higher probability to the repeated substring (Holtzman et al., 2020); the substring conveys less and less information after each occurrence.
A further implication of this framing is the equivalence between decoding strings from a probabilistic language generator and sampling messages from the natural language communication channel. If we wish to solely sample from the subset of messages that a human would typically construct, that is, that are human-like, then we should begin by narrowing down this subset to those messages that meet at least some of the same criteria as human-generated messages. In this work, we have identified the criterion that such messages tend to be in the locally typical set. This observation motivates a new decoding strategy in which our information-theoretic criterion is explicitly enforced, which we subsequently present.
### 5.2 Locally Typical Sampling
We now introduce our novel sampling algorithm, which we entitle locally typical sampling.
Algorithm 5.3
Locally typical sampling is a truncated sampling scheme where the truncation set$C(yis the solution to the following subset optimization problem:
$minimizeC(y
(12)
In words, Algorithm 5.3 limits the sampling distribution to only those words with negative log-probability within a certain absolute range from the conditional entropy (expected information content) of the model at that time step. In the spirit of nucleus sampling, this range is determined by a hyperparameter τ, the amount of probability mass from the original distribution that we wish to consider.
Interestingly, Algorithm 5.3 does not imply that high-probability words should not be chosen. Indeed, in the situation where conditional entropy is low, namely, when the model places most of the probability mass on a small subset of words, it is likely the case that only high-probability words fall into the locally typical set.
##### Computational Complexity.
From a practical perspective, locally typical sampling can be implemented with the same efficiency as nucleus or top-k sampling. First, we compute the conditional entropy, which is an $O(∣V∣)$ operation. Second, we sort words by their absolute distance from $H(p^(·∣Y, which can be done in $O(∣V∣log∣V∣)$ time with standard sorting algorithms. Finally, we greedily take words from this list until their cumulative probability exceeds the threshold τ, which again takes $O(∣V∣)$ time. Thus, creating our altered distribution has time complexity $O(∣V∣log∣V∣)$.18
##### Relationship to Other Decoding Strategies.
Notably, we already see motivation for this criterion in the performance of several well-known decoding strategies. For example, beam search is the predominant decoding strategy for machine translation models (Wu et al., 2016; Edunov et al., 2018; Ng et al., 2019; Meister et al., 2020b), a setting in which beam search (incidentally) often already enforces this criterion.19 Yet, when used in more open-ended tasks, where the entropy of the language model is higher, beam search can lead to low-quality text (Li et al., 2016; Holtzman et al., 2020; Welleck et al., 2020; Meister et al., 2022). Locally typical sampling is also closely related to nucleus sampling. When the probability distribution over the vocabulary has low conditional entropy, that is, when there are only a few reasonable choices for the next word according to our model, nucleus and typical will have the same truncation set. Locally typical sampling and Mirostat (Basu et al., 2021) likewise have similar decision rules for truncation. Mirostat decodes strings such that they have a perplexity (or, equivalently, a per-word information content) close to a target value. In contrast to Mirostat, however, locally typical sampling does not require a specific target information content to be defined. Rather, locally typical sampling defines this quantity as the conditional entropy, choosing it dynamically (per word) and making it less sensitive to hyperparameter choice. Finally, locally typical sampling is also related to Braverman et al. (2020) strategy, which proposes a look- ahead decoding algorithm that generates text with a similar entropy rate to that of human-generated text. Our strategy’s motivation is similar—to match the tendencies in information content exhibited by human-generated text—albeit without requiring the computational overhead of a look- ahead strategy.
In this section, we explore the efficacy of our decoding strategy on two natural language generation tasks: abstractive summarization and story generation. We assess performance with respect to several other stochastic decoding strategies: nucleus sampling, top-k sampling, temperature sampling,20 beam search, and Mirostat. Our evaluation includes both automatic metrics and human ratings.
### 6.1 Setup
##### Models and Data.
We use the HuggingFace framework (Wolf et al., 2020) for reproducibility, employing their implementations of nucleus, top-k, temperature sampling, and beam search. We rely on the implementation of Mirostat provided by its authors. For story generation, we finetune the medium and large versions of GPT-2 (Radford et al., 2019) from checkpoints made available by OpenAI on the WritingPrompts dataset (Fan et al., 2018). We use the medium checkpoint finetuned on WikiText-103 (Merity et al., 2017) to produce the data used in Figure 1. For abstractive summarization, we use BART (Lewis et al., 2020) finetuned on the CNN/DailyMail dataset (Nallapati et al., 2016).21 All reported metrics are computed on the respective test sets.
##### Hyperparameters.
In a preliminary hyperparameter sweep using Mauve22 (Pillutla et al., 2021), we found k = {30,40}, η = {0.9,0.95} and τ = 3.0 to be the best performing hyperparameters for top-k sampling, nucleus sampling and Mirostat, respectively. For locally typical sampling, we found τ = 0.2,τ = 0.95 to provide the best results for story generation and abstractive summarization, respectively. Standard values according to the literature for other hyperparameters (i.e., for beam search and temperature sampling) were employed. We use these values in our human evaluations and in computation of automatic metrics.
##### Automatic Quality Metrics.
As automatic quality metrics, we evaluate the generated text’s perplexity—under both the model used to generate the text (PPL(g)) and an independent, i.e., not finetuned, LM (PPL(i)), namely GPT-2 large (Radford et al., 2019). Several prior works have shown that neither low nor high perplexity (Zhang et al., 2021; Nadeem et al., 2020; Pillutla et al., 2021) are direct indicators of text quality. Rather, human-like text often has perplexity within a certain range. Consequently, we report the difference in this metric from the reference text as well. We additionally evaluate using Mauve23 (Pillutla et al., 2021) with the reference text.
##### Automatic Diversity Metrics.
We also evaluate locally typical sampling using automatic diversity metrics. We compute rep (Welleck et al., 2020), Zipf’s coefficient, and n-gram diversity. For rep we use the average of rep/ scores, as defined in Eq. 9 of Welleck et al. (2020) for ∈{16,32,128}. We define n-gram diversity D as the average fraction of unique vs. total n-grams for n ∈{1,2,3,4} in a string
$D=∑n=14#uniquen-grams in string#n-grams in string$
(13)
##### Human Evaluations.
We use the AmazonMechanical Turk framework to obtain human judgments of text quality from 5 different annotators on 200 examples per decoding strategy, per task. We use solely MTurk Master Workers in order to maximize the quality of our ratings. We follow DeLucia et al. (2021) in setting up our evaluations. Each Human Intelligence Task (HIT) consists of either a single prompt from which a story should be generated or a single news article to be summarized. The raters are first presented with the different rating criteria, along with descriptions of the type of text that meets these criteria at different levels of the scale. Raters are additionally provided several examples of stories/ summarizations that both meet and fail to meet the rating criteria. They are then presented with the respective prompt/news article and the corresponding stories/summaries generated by different decoders and by the reference in random order. For abstractive summarization, we ask annotators to score on fluency and relevance, while for story generation, annotators score on fluency, coherence, and interestingness, each using a scale from 1 to 5. We choose these criteria following recommendations from van der Lee et al. (2019).
For each story/summarization and each of the criteria, we take the median score across raters as the final score.24 Workers are paid $1.50 per HIT for the abstractive summarization task and$2 per HIT for the story generation task, for which entries were longer. Note that these rates translate to >15/hour. ### 6.2 Results ##### Quantitative Performance. Tables 1 and 2 show the results of our different evaluation metrics. Human scores are averaged across the qualitative metrics to give an aggregate score; the value in parentheses is the standard error of the estimate. We show full breakdowns of score distributions in Table 5. We see that in general, locally typical sampling performs on par with or better than other sampling techniques, producing text with human quality ratings closest to that of the reference among the stochastic decoding strategies. Interestingly, beam search still outperforms locally typical sampling in abstractive summarization, albeit by a small margin. This could perhaps be attributed to the deterministic nature of beam search, which suggests that an interesting direction for future research may be a deterministic version of locally typical sampling, for example, where the highest-probability word within the truncated set is always chosen. Importantly, all the strategies we explore are quite close to human- level performance—in some cases even surpassing human references in terms of ratings. At this level, it is perhaps only reasonable to expect that the differentiation between the top strategies is small. Accordingly, we also consider how robust locally typical sampling is to hyperparameter choice. Figure 2 shows rep measurements for different values of the hyperparameters k, η, and τ for top-k, nucleus, and locally typical sampling, respectively. Interestingly, rep appears to be far less sensitive to τ than to k and η. While many values of k and η appear to lead to degenerate repetitions in story generation, most values of τ lead to text with a rep value on par with human text, demonstrating that an advantage of our technique is its robustness to hyperparameter choice. See Figure 3 in the Appendix for a larger exploration of how other quality metrics vary as a function of τ. Table 1: Automatic quality and diversity metrics, as described in §6.1, along with human ratings on the WritingPrompts dataset. Human ratings are averaged across criteria to form a single metric. Bolded values are the best results among decoding strategies, where for perplexity (PPL) and Zipf’s coefficient, we take this to be the delta from measurements on human text (numbers in purple). Numbers in blue are standard error estimates. Results are from finetuned GPT-2 large. Table 2: Automatic quality and diversity metrics, as described in §6.1, along with human ratings on the CNN/DailyMail dataset. Human ratings are averaged across criteria to form a single metric. Bolded values are the best results among decoding strategies, where for perplexity (PPL) and Zipf’s coefficient, we take this to be the delta from measurements on human text (numbers in purple). Numbers in blue are standard error estimates. Figure 2: rep (Welleck et al., 2020) values for different k and τ/η (lower is better). Lines indicate rep measurement for reference text and Mirostat (left)/beam search (right). Figure 2: rep (Welleck et al., 2020) values for different k and τ/η (lower is better). Lines indicate rep measurement for reference text and Mirostat (left)/beam search (right). Close modal ##### Qualitative Performance. We present some examples of text generated according to each of the decoding strategies in Tables 3 and 4. For both of the tasks, we choose the example with ID 1 in the respective test set and provide examples from each of the decoding strategies, employing the hyperparameter values that lead to the best human scores in Tables 2 and 1. For the summarization task, we see that locally typical sampling provides a comprehensive and coherent summary of the article, quite similar to that of beam search. In comparison, the text produced by temperature sampling is not necessarily coherent; text from nucleus sampling and top-k sampling misses some of the important information in the article, for example, the charges of burglary and arson. While the qualitative performance in story generation is much more subjective, locally typical sampling arguably provides the most fluent story among all the decoding strategies. Other stories lack coherence and, even within the first few sentences, we see repeated phrases and words. Together, these results suggest that locally typical sampling may indeed produce more desirable text. Table 3: Sample generations for abstractive summarization; examples correspond to ID 1 in the test set. Decoding strategy hyperparameters are chosen based off of performance in human evaluations shown in Table 2. Abstractive Summarization (CNN/DailyMail) Prompt (CNN) The attorney for a suburban New York cardiologist charged in what authorities say was a failed scheme to have another physician hurt or killed is calling the allegations against his client “completely unsubstantiated.” Appearing Saturday morning on CNN’s “New Day,” Randy Zelin defended his client, Dr. Anthony Moschetto, who faces criminal solicitation, conspiracy, burglary, arson, criminal prescription sale and weapons charges in connection to what prosecutors called a plot to take out a rival doctor on Long Island. “None of anything in this case has any evidentiary value,” Zelin told CNN’s Christi Paul... Reference A lawyer for Dr. Anthony Moschetto says the charges against him are baseless. Moschetto, 54, was arrested for selling drugs and weapons, prosecutors say. Authorities allege Moschetto hired accomplices to burn down the practice of former associate. Beamk = 5 Dr. Anthony Moschetto faces criminal solicitation, conspiracy, burglary, arson and weapons charges. “None of anything in this case has any evidentiary value,” his attorney says. Nucleusη = 0.95 Dr. Anthony Moschetto, 54, pleaded not guilty to charges Wednesday. Two men – identified as James Chmela and James Kalamaras – were named as accomplices. Top-kk = 30 Dr. Anthony Moschetto is accused of providing police with weapons and prescription drugs. Authorities say he was part of a conspiracy to harm or kill a rival doctor. His attorney calls the allegations against his client “completely unsubstantiated” Typicalτ = 0.95 Dr. Anthony Moschetto is charged with crimes including arson, conspiracy, burglary, prescription sale, weapons charges. His attorney says “none of anything in this case has any evidentiary value” Abstractive Summarization (CNN/DailyMail) Prompt (CNN) The attorney for a suburban New York cardiologist charged in what authorities say was a failed scheme to have another physician hurt or killed is calling the allegations against his client “completely unsubstantiated.” Appearing Saturday morning on CNN’s “New Day,” Randy Zelin defended his client, Dr. Anthony Moschetto, who faces criminal solicitation, conspiracy, burglary, arson, criminal prescription sale and weapons charges in connection to what prosecutors called a plot to take out a rival doctor on Long Island. “None of anything in this case has any evidentiary value,” Zelin told CNN’s Christi Paul... Reference A lawyer for Dr. Anthony Moschetto says the charges against him are baseless. Moschetto, 54, was arrested for selling drugs and weapons, prosecutors say. Authorities allege Moschetto hired accomplices to burn down the practice of former associate. Beamk = 5 Dr. Anthony Moschetto faces criminal solicitation, conspiracy, burglary, arson and weapons charges. “None of anything in this case has any evidentiary value,” his attorney says. Nucleusη = 0.95 Dr. Anthony Moschetto, 54, pleaded not guilty to charges Wednesday. Two men – identified as James Chmela and James Kalamaras – were named as accomplices. Top-kk = 30 Dr. Anthony Moschetto is accused of providing police with weapons and prescription drugs. Authorities say he was part of a conspiracy to harm or kill a rival doctor. His attorney calls the allegations against his client “completely unsubstantiated” Typicalτ = 0.95 Dr. Anthony Moschetto is charged with crimes including arson, conspiracy, burglary, prescription sale, weapons charges. His attorney says “none of anything in this case has any evidentiary value” Table 4: Sample generations for story generation from GPT-2 large finetuned on the WritingPrompts dataset; examples correspond to ID 1 in the test set. Decoding strategy hyperparameters are chosen based off of best performance in human evaluations shown in Table 1. Story Generation (WritingPrompts) Prompt A kid doodling in a math class accidentally creates the world’s first functional magic circle in centuries. Reference It was dark and Levi was pretty sure he was lying on his back. There was firelight flickering off of what was left of a ceiling. He could hear something but it was muffled. He was almost positive it was screaming. When he tried to move he felt an excruciating pain in his left side that caused him to cry out... Nucleusη = 0.95 “Why do you draw a circle?” He asked as he handed the piece of paper over to her. She rolled her eyes and looked down at the picture, “Because I like math!” “Then what?” “I like to draw the circles.” She ran a hand through her curls and looked away. “See, I have three circles.” “Hmph, I didn’t think I saw that in your class.” A younger student brought up the ceiling and murmured... Top-kk = 30 “Aha. Aah! That makes three.” Professor Gosh sighed. “Oh my God, you had to have been there.” “Well, I’m not sure if that’s the right term, but this circle is going to work.” “But I just told you that it wasn’t working, didn’t you, you little sh**? I’m not kidding! That circle is gon na work!” “So what now?”... Tempτ = 1.0 It was teachers duty to ensure that our kingdom is pure and pure and successful, however it does not mean we do not try to be as rebellious as any other kingdom. Most of our teachers are suitable for rulership, being knowledgeable in sadistic rec thunking of time stopping magic circle created by several apprentices... Mirostatτ = 3 ready to go home, when a little kid starts to sketch a circle. “No. Just stop. It’s not working.” “But the circle turns to smoke and then to a flame. It’s not working, I know it is!” Another kid, who was also drawing a circle with his phone, starts to scream. The teacher, looking at the circle, lost his patience and started to yell... Typicalτ = 0.2 As soon as he got the clock on his desk he jumped up. The noise it was making had changed his day forever. His hair, normally unkempt, now glistened and he had to have more hair because of it. “Yes!” He had declared triumphantly as he watched his father fumble around in the backpack in search of a little compass he always kept with him. He took it from the desk and quickly walked out the door to see what had happened. He wasn’t the first child in the world to draw something... Story Generation (WritingPrompts) Prompt A kid doodling in a math class accidentally creates the world’s first functional magic circle in centuries. Reference It was dark and Levi was pretty sure he was lying on his back. There was firelight flickering off of what was left of a ceiling. He could hear something but it was muffled. He was almost positive it was screaming. When he tried to move he felt an excruciating pain in his left side that caused him to cry out... Nucleusη = 0.95 “Why do you draw a circle?” He asked as he handed the piece of paper over to her. She rolled her eyes and looked down at the picture, “Because I like math!” “Then what?” “I like to draw the circles.” She ran a hand through her curls and looked away. “See, I have three circles.” “Hmph, I didn’t think I saw that in your class.” A younger student brought up the ceiling and murmured... Top-kk = 30 “Aha. Aah! That makes three.” Professor Gosh sighed. “Oh my God, you had to have been there.” “Well, I’m not sure if that’s the right term, but this circle is going to work.” “But I just told you that it wasn’t working, didn’t you, you little sh**? I’m not kidding! That circle is gon na work!” “So what now?”... Tempτ = 1.0 It was teachers duty to ensure that our kingdom is pure and pure and successful, however it does not mean we do not try to be as rebellious as any other kingdom. Most of our teachers are suitable for rulership, being knowledgeable in sadistic rec thunking of time stopping magic circle created by several apprentices... Mirostatτ = 3 ready to go home, when a little kid starts to sketch a circle. “No. Just stop. It’s not working.” “But the circle turns to smoke and then to a flame. It’s not working, I know it is!” Another kid, who was also drawing a circle with his phone, starts to scream. The teacher, looking at the circle, lost his patience and started to yell... Typicalτ = 0.2 As soon as he got the clock on his desk he jumped up. The noise it was making had changed his day forever. His hair, normally unkempt, now glistened and he had to have more hair because of it. “Yes!” He had declared triumphantly as he watched his father fumble around in the backpack in search of a little compass he always kept with him. He took it from the desk and quickly walked out the door to see what had happened. He wasn’t the first child in the world to draw something... In this work, we analyze decoding from probabilistic language generators in the information- theoretic framework. We equate a language model to a discrete stochastic process, and use the known properties of such processes to quantitatively describe the samples we should expect. Motivated by results in psycholinguistics, we hypothesize that—with the goal of communicating efficiently and robustly—humans produce text whose per- word information content is within a close range of the expected information content of a word given prior context. Current language models may fall short in capturing this property, which is a possible explanation for why the corresponding language processes often do not lead to human- like text. Yet, this observation provides a simple new criterion for decoding more human-like text from probabilistic language generators: Constraining the sampling space to words that meet this criterion. In experiments on two language generation tasks, we find that our strategy—called locally typical sampling—leads to text of comparable or better quality than other stochastic decoding strategies according to human ratings. Further, when compared to these other decoding strategies, several quantitative properties of typically- sampled text more closely align with those of human text. We would like to thank Jason Eisner, Tim Vieira, Jennifer White, and Ari Holtzmann for early conversations about the relationship between information theory and sampling. We would also like to thank Ehud Reiter, who served as our TACL action editor, and the the anonymous reviewers for their insightful feedback during the review process. Further, we are grateful to Eleanor Chodroff, Clément Guerner, and Lucas Torroba Hennigen for their feedback on the manuscript of this work. In order to complete our human evaluation, we used a crowdsourcing platform. For each task, we made sure that the crowdworkers would be paid (at minimum) a wage of15 per hour.
Another ethical consideration worth discussing concerns the use of language models for text generation. Text generated by these models may contain malicious content, either by design of the user or as a byproduct of the training data/algorithm. While we hope the results of our work will not be misused, they may nonetheless provide insights for those employing these models with ill-intent as to how machine-generated text can be made more “human-like,” and thus more convincing.
Table 5:
Breakdown of human ratings on quality metrics per task; results for story generation are from finetuned versions of GPT-2 medium (m) and large (l). Values in blue are variances.
Figure 3:
Mauve, Zipf’s coefficient, (average) probability mass of candidate token pool, and (average) candidate token pool size as a function of decoder hyperparameters for nucleus, top-k, and locally typical sampling.
Figure 3:
Mauve, Zipf’s coefficient, (average) probability mass of candidate token pool, and (average) candidate token pool size as a function of decoder hyperparameters for nucleus, top-k, and locally typical sampling.
Close modal
1
While maximization-based strategies can produce text that is generic or degenerate, stochastic strategies occasionally produce nonsensical text. Both types of strategies tend to eventually fall into repetitive loops.
2
Information rate may be defined with respect to time (as is the case with spoken language) or with respect to a specific linguistic unit, such as a word (as is the case with text).
3
An implementation of typical sampling can be found in the HuggingFace Transformers library (Wolf et al., 2020).
4
Here we use the term language model to refer to any (valid) probability distribution over natural language strings. We subsequently specify the necessary conditions for validity. Note that this distribution may also be conditioned on an input.
5
The Kleene closure of a set $V$ is defined as $V*=def⋃n=0∞Vn$.
6
The ubiquity of Eq. (1) has led some authors to defining language models in the locally normalized form, even though globally normalized language models are also perfectly fine to consider (Goyal et al., 2019).
7
Some authors erroneously omit eos from their definition. However, we require a distinguished symbol eos to be able to locally normalize the language model and make it a valid probability distribution.
8
This process is discrete both in time and in value.
9
One could just as easily define a language process over subwords, morphemes, or characters.
10
Also known as a Markov chain.
11
Note that, in principle, human language is not Markov, in so far as many linguists believe human language is capable of arbitrarily deep center-embeddings (Chomsky, 1957, 1995). Yet research suggests that humans do not make use of this property in practice (Reich, 1969; Karlsson, 2010), and so we do not consider the Markovian property of most models as a limitation to their ability to model natural language in practice.
12
This would be done by setting the transition probability p(Yt = eos∣Y <t =y <t) = 1 if yt − 1 = eos.
13
See Dieleman (2020) for further discussion of the concept of typicality in the context of generative modeling.
14
See Gibson et al. (2019) for an in-depth review of how efficiency has shaped the evolution of language.
15
The standard definition of (Shannon) entropy for a random variable X with support $X$ is equivalent to the expected information of X: $H(X)=−∑x∈Xp(x)logp(x)$.
16
Here we only consider locally normalized p, i.e., processes in which sampling is done on a word-by-word basis.
17
Another natural option would be to choose words which maximize the probability assigned by p to the resulting string, but this work focuses on stochastic strategies.
18
For each of the truncation sampling algorithms, the truncation set can also be identified using the selection algorithm (no sorting required) in $O(∣V∣)$ time. We provide the analysis using sorting as that is the standard implementation.
19
When trained without label-smoothing, which artificially inflates conditional entropies, machine translation models tend to have quite low conditional entropies (see, e.g., Figure 3 in Meister et al., 2020a). Therefore, at each decoding step, the set of words with negative log-probability near the conditional entropy of the model are typically those with high probability—the same as those chosen by beam search.
20
Temperature sampling is defined as ancestral sampling after local renormalization with an annealing term τ.
21
As we are interested in getting as close an estimate of p as possible with our models $p^$, all fine-tuning is done without label-smoothing. Note that label-smoothing may also artificially inflate conditional entropy estimates, as it pushes the learned distribution towards the most entropic distribution: the uniform distribution (Pereyra et al., 2017).
22
We use the default settings given by the authors for all Mauve computations, although we employ different LMs in our parameter sweep vs. reported results (standard GPT-2 vs. GPT-2 large, respectively) to reduce bias in the final results. Notably, Mauve presents similar performances when used with these two pretrained LMs (Pimentel et al., 2022).
23
We use the implementation provided by the authors.
24
We use an attention check in each HIT. Responses where the attention check has been failed are thrown out.
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2023-02-04 10:01:01
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https://gamedev.stackexchange.com/questions/184914/how-to-set-the-z-axis-bound-of-the-collider-of-child-of-an-instantiated-object-t
|
# How to set the z axis bound of the collider of child of an instantiated object to zero
I have attached a script to an empty GameObject to instantiate a prefab and the script is:
using System.Collections;
using System.Collections.Generic;
using System.Collections.Specialized;
using UnityEngine;
public class Circles : MonoBehaviour
{
[SerializeField] float Xmax;
[SerializeField] float Xmin;
[SerializeField] float Ymax;
[SerializeField] float Ymin;
[SerializeField] float outsideMaxsize;
[SerializeField] float outsideMinsize;
float insideMaxsize;
float insideMinsize;
public GameObject[] enemy;
GameObject enemyObject;
int enemyNo;
float Xpos;
float Ypos;
float outsideSize;
float insideSize;
// Start is called before the first frame update
void Start()
{
enemyNo = Random.Range(0, 4);
instantiator();
}
// Update is called once per frame
void Update()
{
}
public void instantiator()
{
Xpos = Random.Range(Xmin, Xmax);
Ypos = Random.Range(Ymin, Ymax);
outsideSize = Random.Range(outsideMinsize, outsideMaxsize);
Vector2 circlePos = new Vector2(Xpos, Ypos);
transform.position = circlePos;
enemyObject = GameObject.Instantiate(enemy[enemyNo], transform.position, transform.rotation,transform.parent) as GameObject;
var enemies = enemyObject.transform;
var outsideComponent = enemies.GetChild(0);
var insideComponent = enemies.GetChild(1);
outsideComponent.transform.localScale=new Vector2(outsideSize, outsideSize);
insideMaxsize = outsideSize * 0.5f;
insideMinsize = 0.15f;
insideSize = Random.Range(insideMinsize, insideMaxsize);
insideComponent.transform.localScale = new Vector2(insideSize, insideSize);
enemyNo = Random.Range(0, 4);
}
}
And I have attached a script to the prefab which is being instantiated and that script is:
using System;
using System.Collections;
using System.Collections.Generic;
using UnityEngine;
public class destroyer : MonoBehaviour
{
Circles circles;
CircleCollider2D collider1;
Collider2D collider2;
private void Start()
{
circles = FindObjectOfType<Circles>();
print("1 max"+collider1.bounds.max);
print("1 min" + collider1.bounds.min);
print("2 max" + collider2.bounds.max);
print("2 min" + collider2.bounds.min);
}
private void OnEnable()
{
collider1 = gameObject.transform.GetChild(0).GetComponent<CircleCollider2D>();
collider2 = gameObject.transform.GetChild(1).GetComponent<Collider2D>();
}
void OnTriggerStay2D(Collider2D other)
{
print("star max"+other.bounds.max);
print("star min" + other.bounds.min);
if (collider1.bounds.Contains(other.bounds.max) && collider1.bounds.Contains(other.bounds.min))
{
if (other.bounds.Contains(collider2.bounds.max) && other.bounds.Contains(collider2.bounds.min))
{
Destroy(other.gameObject);
Destroy(gameObject);
circles.instantiator();
}
}
}
}
the problem is that the z axis bounds of the colliders are coming out to be 15.8 when the prefab is instantiated and when I drag the prefab to scene then the z axis bounds of the collider is 0. I want the z axis bounds of the colliders to be 0 when the prefab is instantiated
• Getting the colliders with gameObject.transform.GetChild(0).GetComponent<Collider2D>() is very messy. You should use serialized fields and select the colliders in the Inspector, e.g. [SerializeField] private Collider collider1; – Kevin Aug 4 at 22:14
• – Mohammad AL - Haque Aug 5 at 14:11
The simplest way to fix this is to spawn your prefab at a position that places its children in the same z-plane as the objects you're hoping to detect.
So for example if the object you want to detect is at z=0, and your prefab's children are at z = 1 relative to their parent, and their parent isn't scaled in the z axis or rotated in the x/y axes, then you would need to spawn the parent at z = -1 to cancel-out the child colliders' offsets so they'll sit at the z = 0 plane.
Since you haven't shown us enough details of your setup for us to tell you exactly what position values to change, another option is to exclude Z from consideration in your test. We can do this by adding an extension method:
public static class PhysicsExtensions {
public static bool Envelops2D(this Bounds a, Bounds b) {
return (b.max.x <= a.max.x)
& (b.max.y <= a.max.y)
& (b.min.x >= a.min.x)
& (b.min.y >= a.min.y);
}
}
Then you can change your if clauses to...
if (collider1.bounds.Envelops2D(other.bounds))
{
if (other.bounds.Envelops2D(collider2.bounds))
{
Destroy(other.gameObject);
Destroy(gameObject);
circles.instantiator();
}
}
Note that this is using only the bounding rectangles of your colliders, not their actual shapes. So for example this will say a radius-8 circle collider at (0, 0) envelops a radius-1 circle collider at (7, 7), even though that second circle is completely outside the first, but still inside its bounding rectangle. You can use radius and distance checks instead of bounds checks if you want to take the circular shape of the colliders into account.
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2020-09-18 17:38:23
|
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https://code-reference.com/c/float.h/flt_radix?rev=1359929212&do=diff
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# Differences
This shows you the differences between two versions of the page.
127.0.0.1 external edit
158.181.88.241
Line 1: Line 1:
<code c> <code c>
</code> </code>
-FLT_RADIX is in work by code-reference.com \\ +radix of exponent rep (2)
-if you are faster... don't hasitate and add it +
- +
-<code c> +
-no example at the moment +
-</code> +
- +
-===== output of FLT_RADIX c example ===== +
- no example at the moment +
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2019-03-26 19:03:20
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http://clay6.com/qa/52392/consider-the-following-one-dimensional-motion-a-spacecraft-drifts-through-s
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Comment
Share
Q)
# Consider the following one-dimensional motion.A spacecraft drifts through space at constant velocity. Which of the following statements are true?
(1) The instantaneous velocity is always zero
(2) The average velocity cannot be calculated from the information given
(3) At no point does the average velocity equal the instantaneous velocity
(4) The average velocity = instantaneous velocity over all time intervals
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2020-09-27 17:45:37
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https://www.coin-or.org/CppAD/Doc/atan2.htm
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AD Two Argument Inverse Tangent Function
Syntax
theta = atan2(y, x)
Purpose
Determines an angle $\theta \in [ - \pi , + \pi ]$ such that $$\begin{array}{rcl} \sin ( \theta ) & = & y / \sqrt{ x^2 + y^2 } \\ \cos ( \theta ) & = & x / \sqrt{ x^2 + y^2 } \end{array}$$
y
The argument y has one of the following prototypes const AD<Base> &y const VecAD<Base>::reference &y
x
The argument x has one of the following prototypes const AD<Base> &x const VecAD<Base>::reference &x
theta
The result theta has prototype AD<Base> theta
Operation Sequence
The AD of Base operation sequence used to calculate theta is independent of x and y .
Example
The file atan2.cpp contains an example and test of this function. It returns true if it succeeds and false otherwise.
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2018-01-20 19:17:19
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http://tex.stackexchange.com/questions/152621/is-it-possible-to-have-italic-and-bold-text-and-linebreaks-under-the-underline/152623
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# Is it possible to have italic and bold text, and linebreaks under the \underline{text} command?
I was trying to emphasize text with \textbf, \textit and \underline commands. But the text will not automatically wrap to the next line. I also tried the {ulem} package, and it did not work too. Does anybody know how to have italic, bold, underline and linebreaks for some text?
-
Could you please post a Minimum Working Example demonstrating the problem? It is much easier for people to help if you provide some code they can just copy-paste-and-compile to reproduce the problem you are seeing. – cfr Jan 6 '14 at 2:42
\usepackage{soul} and then \ul{\textbf{\textit{your text here}}}. – jon Jan 6 '14 at 2:42
You are not really doing all three at the same time, are you? That may draw attention to the text but it will also render it extremely difficult to read, as well as looking rather ugly. – cfr Jan 6 '14 at 2:46
I am working on some special documentations requiring all the three together. I must find a way to work it out. :( – Zhou Wang Jan 6 '14 at 3:24
Do the following self explanatory combinations work for you? (Leaving aside aesthetic issues.)
\documentclass{article}
\usepackage{ulem}
\begin{document}
I think computer \uline{viruses should count as life. I think it says
something about human nature} that the only form of life we have
created so far is purely destructive. We've created life in our own
image.
I think computer \uline{viruses should \textit{count as life. I think
it} says something about human nature} that the only form of life
we have created so far is purely destructive. We've created life in
our own image.
I think computer \uline{\textit{viruses should count as life. I think
it} says something about human nature} that the only form of life
we have created so far is purely destructive. We've created life in
our own image.
I think computer \uline{\textit{viruses should count \textbf{as
life. I think} it} says something \textbf{about human} nature}
that the only form of life we have created so far is purely
destructive. We've created life in our own image.
\textit{\textbf{\uline{I think computer viruses should count as
life. I think it says something about human nature that the only
form of life we have created so far is purely destructive. We've
created life in our own image.}}}
\end{document}
\documentclass{article}
\usepackage{ulem}
Tips
Always make \uline the last one to be applied when you need multiple effects on a complete paragraph. It behaves badly (no line wraps) when other font commands are applied after it.
-
Will that work if all three 'characteristics' span multiple lines? – jon Jan 6 '14 at 3:04
@jon Why not? See the updated example where all three have been applied to the last paragraph. – Masroor Jan 6 '14 at 3:12
@ZhouWang See the updated example, the last case. Always make \uline the last one to be applied. It behaves badly (no wraps) when other font commands are applied after it. This is what is happening in the above example in your comment. – Masroor Jan 6 '14 at 3:15
Yes, I got it. Thanks very much NMA. The problem is that I make the \uline the first one!!! Once I make the \uline the last one, it works perfect!!! – Zhou Wang Jan 6 '14 at 3:20
Indeed! I had never tried before and didn't realize ulem had such constraints about the order of usage. – jon Jan 6 '14 at 3:35
It is probably better to stick with ulem if you are already using it. However, if I had to mix and match underline, bold, and italics, I'd be inclined to use soul instead. It does not need as much care about how you nest all these ways of emphasis.
\documentclass[12pt]{article}
\usepackage[T1]{fontenc}
\usepackage{soul}
\parskip10pt
\parindent0pt
\begin{document}
% all at once:
\ul{\textbf{\textit{I was trying to emphasize text with textbf, textit
and underline commands. But the text will not automatically wrap
to the next line. I also tried the {ulem} package, and it did
not work too. Does anybody know how to have italic, bold,
underline and linebreaks for some text?}}}
% underline, then \emph, then bold
I think computer \ul{viruses \emph{should count \textbf{as life. I
think it says something about human nature that the only form of
life we have created so far is purely destructive. We've created
life in} our} own} image.
% underline, bold, emph
I think computer \ul{viruses \textbf{should count \emph{as life. I
think it says something about human nature that the only form of
life we have created so far is purely destructive. We've created
life in} our} own} image.
% bold, emph, underline
I think computer \textbf{viruses \emph{should count \ul{as life. I
think it says something about human nature that the only form of
life we have created so far is purely destructive. We've created
life in} our} own} image.
% emph, bold, underline
I think computer \emph{viruses \textbf{should count \ul{as life. I
think it says something about human nature that the only form of
life we have created so far is purely destructive. We've created
life in} our} own} image.
% bold, underline, emph
I think computer \textbf{viruses \ul{should count \emph{as life. I
think it says something about human nature that the only form of
life we have created so far is purely destructive. We've created
life in} our} own} image.
% emph, underline, bold
I think computer \emph{viruses \ul{should count \textbf{as life. I
think it says something about human nature that the only form of
life we have created so far is purely destructive. We've created
life in} our} own} image.
\end{document}
-
Thanks, that is also a very good alternative! – Zhou Wang Jan 6 '14 at 3:56
saved my ass. +1 – Kaii Jan 15 '14 at 14:41
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2016-05-29 00:15:37
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https://aviation.stackexchange.com/questions/57411/model-for-flutter-analysis-of-aerofoil
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# Model for flutter analysis of Aerofoil
Why is a aerofoil modeled by attached torsional and linear spring in a wing flutter analysis.The model does not even satisfy the boundary condition of zero displacement of wing at point of attachment to fuselage. I am considering the pitch plunge model as shown in figure.
Because what you are simulating is not a wing, but an aerofoil at an arbitrary $$y$$ station along the wing. The linear spring simulates the wing bending and the torsional one its torsion about the elastic axis. The constants would of course be of the type $$k_{e}=f(y)$$.
• you can adjust the constants of those springs to study a specific wing section, which is good enough for the level this preliminary study. More importantly, it already exhibits some of the crucial properties of flutter, like torsional divergence above a certain $U_{\infty}$. Though you should really refine your question, this is just my offhand remark to your initial overly broad query. – AEhere supports Monica Nov 26 '18 at 14:15
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2021-04-20 23:45:28
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https://www.functionalstatistics.com/posts/2023-01-16-agda-unary-examples/
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# Unary Relation Examples
Lately I’ve been learning the dependently typed programming language Agda. While it’s a joy to learn, learning materials for the Agda standard library are sparse. In this post, I give examples using of the stdlib’s Relation.Unary module to do elementary set theory. The logical propositions I prove herein are rather trivial, but for newbies to dependently-typed languages like me, even the trivial can seem hard.
## Preliminaries
You can read this post without knowing any Agda. If nothing else, you can get a hint of the kinds of things you can do with the concept and an implementation of propositions as types.
I make reference to the Programming Language Foundations in Agda (PLFA) online book, and I highly recommend (at least chapter 1) for learning the basics of Agda.
### Basic Structure
The examples follow this structure:
_ : Proposition
_ = Proof
To the right of the : is a claim, a proposition. To the right of the = is a proof of that claim. The _ on both lines just means I haven’t given this expression a name.
## Predicates
Prior to learning Agda/type theory/etc, I understood the term predicate to mean a unary function (function of one argument) that returns a boolean (i.e. true or false): A → Bool. This view of predicates as indicator functions turns out to be rather limited understanding. Another, richer, understanding sees predicates as unary functions to Set (the collection of all types): A → Set. Here’s a hint at what this is all about. and here’s a description from the Relation.Unary docs:
-- Unary relations are known as predicates and Pred A ℓ can be viewed
-- as some property that elements of type A might satisfy.
-- Consequently P : Pred A ℓ can also be seen as a subset of A
-- containing all the elements of A that satisfy property P. This view
-- informs much of the notation used below.
## Setup
Show module setup and imports.
First, create a module. The name of the top-level module of in a file should match the filename.
module index where
By importing Relation.Unary module, we get access to its contents. The open statement brings the module’s content into scope as if the names were define within the current module. Without the open statement, we’d have to qualify all the names to use them, as in Relation.Unary._∈_.
open import Relation.Unary
Other imports we’ll need:
open import Data.Unit using ( tt )
open import Level
open import Relation.Binary.PropositionalEquality using ( refl )
open import Relation.Nullary using ( ¬_ )
open import Relation.Nullary.Negation using ( contradiction )
open import Data.Product
open import Data.Sum.Base using (__; inj₁ ; inj₂ ; [_,_]′ )
## Example Set
Here’s the three element set I’ll work with in the examples:
$\{ Orange, Apple, Banana \}$
This set can be defined in adga as a simple datatype:
data Fruit : Set where
Orange Apple Banana : Fruit
## Example Propositions/Proofs
In each of the examples below, I show how to one can prove the proposition in the header. So for example, the first proposition is that $$Orange \in \{ Orange \}$$, i.e. $$Orange$$ is in the singleton set containing $$Orange$$. This should hopefully be easy to prove.
### Set inclusion
#### $$Orange \in \{ Orange \}$$
Indeed, to prove the proposition, we state refl, which stands for reflexity.
_ : Orange ∈ ｛ Orange ｝
_ = refl
Inlining ∈ and ｛_｝ step by step might make it easier to see what’s going on:
Orange ∈ ｛ Orange ｝
== ｛ Orange ｝ Orange ( x ∈ P = P x )
== Orange = Orange ( ｛ x ｝ = x ≡_)
Proof of the statement Orange = Orange is refl.
#### $$\neg ( Banana \in \{ Orange \} )$$
The claim $$Banana \in \{ Orange \}$$ should not hold; i.e., we should be able to construct a proof of the negation of the claim.
_ : ¬ ( Banana ∈ ｛ Orange ｝ )
_ = λ()
In this case, we can use an absurd pattern:
Absurd patterns can be used when none of the constructors for a particular argument would be valid.
This is indeed the case here, as we cannot construct a proof that Banana = Orange. For more on negation in constructive logic and Agda, see the negation chapter in PLFA.
I’ll note that the proposition above is equivalent to the following by definition of _∉_:
_ : Banana ∉ ｛ Orange ｝
_ = λ()
#### $$Orange \in \{ Orange, Apple \}$$
Now I move beyond singleton sets and define a subset containing both $$Orange$$ and $$Apple$$. This is straighforward with the union _∪_ operator:
O∪A : Pred Fruit _
O∪A = ｛ Orange ｝ ∪ ｛ Apple ｝
The union operator constructs a Sum type, which I think of as the proposition that either the ｛ Orange ｝ claim or the ｛ Apple ｝ claim holds. Hence to prove the claim, we give a proof of $$Orange \in \{ Orange \}$$ to the first (inj₁) position in the sum.
_ : Orange ∈ O∪A
_ = inj₁ refl
For more on the sum type, see the Disjunction as Sum section in PLFA.
#### $$Orange \in \{ Orange, Apple, Banana \}$$
The U symbol represents the univeral set for a type. I find that symbol hard to distinguish from the union operator and infinitary union, so I’ll give the set $$\{ Orange, Apple, Banana \}$$ a name.
allFruit : Pred Fruit _
allFruit = U
The examples holds trivially because allFruit is the universal set of Fruit.
_ : Orange ∈ allFruit
_ = tt
I’ll inline the proposition again to see what is needed of the proof.
Orange ∈ allFruit
== allFruit Orange ( by definition )
== U Orange ( by definition )
== (λ _ → ⊤) Orange ( by definition )
== ⊤ ( function application )
So we need a term of type , for which tt is the only constructor for the (unit) type.
### Subset relations
I’ll switch from proofs about set inclusion to proofs on subset relations.
The subset relation states that a proof P is a subset of Q is evidence that for all x in P, x is also in Q. Or in agda:
_⊆_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _
P ⊆ Q = ∀ {x} → x ∈ P → x ∈ Q
#### $$\{ Orange, Apple \} \subseteq \{ Orange, Apple, Banana \}$$
The first one is simple:
_ : O∪A ⊆ allFruit
_ = λ _ tt
#### $$\{ Orange \} \cap \{ Orange, Apple \} \subseteq \{ Orange \}$$
_ : ( ｛ Orange ｝ ∩ O∪A ) ⊆ ｛ Orange ｝
_ = λ ｛O｝∩O∪A proj₁ ｛O｝∩O∪A
Again inlining the claim might it more clear why the evidence I provided is proof of the claim.
∀ {x} → x ∈ ( ｛ Orange ｝ ∩ O∪A ) → x ∈ ｛ Orange ｝ (definition of _⊆_)
== ∀ {x} → (λ y → y ∈ ｛ Orange ｝ × y ∈ O∪A ) x → x ∈ ｛ Orange ｝ (definition of _∩_)
== ∀ {x} → x ∈ ｛ Orange ｝ × x ∈ O∪A → x ∈ ｛ Orange ｝ (function application )
Ignoring the ∀ {x}, the claim states that proof is a function that take the product of x ∈ ｛ Orange ｝ and x ∈ O∪A and returns x ∈ ｛ Orange ｝, which is exactly what I wrote. Note that I could just as well have written: λ x → proj₁ x. The name of lambda variable is irrelevant, but I used the name ｛O｝∩O∪A to give a visual cue what type is being passed.
#### $$\{ Banana \} \not\subseteq \{ Orange, Apple \}$$
This one is little trickier:
_ : ｛ Banana ｝ ⊈ O∪A
_ = λ b⊆O∪A [ ( λ () ) , ( λ () ) ]′ (b⊆O∪A refl)
I used the [_,_]′ function. to produce a term of ¬ (｛ Banana ｝ ⊆ O∪A ).
### Equality of Pred
Now that we have an idea how to prove subset relations; I’ll demonstrate proofs for equality of predicates. Predicate equality is defined in recent versions of Relation.Unary, but for some reason, it’s not in the version I’m using, so I define it here:
__ : { ℓ₁ ℓ₂ : Level } { C : Set } Pred C ℓ₁ Pred C ℓ₂ Set _
P ≐ Q = (P ⊆ Q) × (Q ⊆ P)
I won’t go through these in detail. I will reiterate that proof in each case involves producing a pair proofs, one for P ⊆ Q and another for Q ⊆ P.
#### $$\{ Orange \} \cap \{ Orange, Apple, Banana \} = \{ Orange \}$$
_ : ( ｛ Orange ｝ ∩ allFruit )( ｛ Orange ｝ )
_ = ( o∈O , _ ) o∈O )
, ( λ o∈O o∈O , tt )
#### $$\{ Orange \} \cap \{ Apple \} = \emptyset$$
_ : ( ｛ Orange ｝ ∩ ｛ Apple ｝ ) ≐ ∅
_ = ( λ { (refl , ())})
, ( λ () )
#### $$\{ Orange \} \cup \{ Apple \} \cup \{ Banana \} = \bigcup$$
_ : ( ｛ Orange ｝ ∪ ｛ Apple ｝ ∪ ｛ Banana ｝ ) ≐ allFruit
_ = ( λ _ tt )
, λ { {Orange} _ inj₁ refl
; {Apple} _ inj₂ (inj₁ refl)
; {Banana} _ inj₂ (inj₂ refl)
}
## Fin
I hope to share more of my Agda learnings in the coming weeks.
|
2023-01-26 21:40:58
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https://stats.stackexchange.com/questions/48366/standard-errors-for-covariance-estimate-in-r
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# Standard errors for covariance estimate in R
This is a very simple question: how does one get the standard error for the covariance estimate in R? I estimate the covariance using the cov function but there seems to be no place for it to return a standard error on the estimate. I would prefer the derivation so I can implement myself.
• If you're looking for some black-box function to do this for you, please let us know so we can migrate your question to SO. Otherwise, if you're looking for a formula you can implement, this is the right place to post your question. Could you clarify? – whuber Jan 23 '13 at 18:26
• @whuber: i would be happy with either. i prefer the formula that's why i posted here because i can just implement that myself. i am sure there must be somethign already built in to R to do this – Alex Jan 23 '13 at 18:32
In response to whuber's follow-up, I would advocate that an all-purpose black-box approach would be using a non-parametric bootstrap.
The basic pseudocode is:
1. Jointly resample from observed rows of data, allowing for replications and holding the sample size fixed.
2. Re-estimate covariance in the resampled data.
3. Repeat 1-2 for a sufficient number of iterations.
4. Use the simulated values to compute variance estimates or empirical 0.025 and 0.975 quantiles to form confidence intervals.
An example here:
set.seed(1)
x <- seq(-3, 3, length.out=100)
do.one <- function(x) {
y <- rnorm(100, x)
d <- data.frame(x, y)
## bootstrap out
bs.out <- replicate(1000, {
dd <- d[sample(1:100, replace=TRUE), ]
cov(dd)[1, 2]
})
bs.lower <- quantile(bs.out, 0.025)
bs.upper <- quantile(bs.out, 0.975)
## in the absence of random error, y=x so cov(x, y)=var(x)
(bs.lower < var(x)) & (bs.upper > var(x))
}
o <- replicate(1000, do.one(x))
mean(o) ## should be 95% if bs estimates correct CIs
Feel free to try this simulation with any random or non-random distribution of $X$ and functional form of the mean model. I am unsure (though cautiously optimistic) CIs based on bootstrapped covariance estimates give correct 95% coverage.
• thak you very much. this makes a lot of sense to me. would you be able to derive a closed form solution assuming the variables are normal for example? – Alex Jan 23 '13 at 20:58
• That request is a bit irrelevant to this response. The parametric and non-parametric approaches to variance-component estimation are on polar opposites of the spectrum. Another caveat: unlike the t-test, inference on variance components isn't robust to distributional assumptions, so data which are non-normal will have standard errors that are very miscalibrated using the exact normal sample model – AdamO Jan 23 '13 at 22:19
This is not an answer to the original question, but to your request to AdamO. (As far as I'm concerned he's covered the original question.)
I'd make it a comment but I think it's too long
Would you be able to derive a closed form solution assuming the variables are normal for example?
see
http://en.wikipedia.org/wiki/Estimation_of_covariance_matrices#Concluding_steps
and
http://en.wikipedia.org/wiki/Wishart_distribution
The second link gives the variance of the $(i,j)\,$ element of the distribution of the scatter matrix for multivariate normal random variables. From there you can get the variance of the sample covariance and hence the standard error.
Specifically, $\sum _{{i=1}}^{n}(X_{i}-\overline {X})(X_{i}-\overline {X})^{{\mathrm {T}}}\sim W_{p}(\Sigma ,n-1)$ implies
$\text{Var}(\sum _{{i=1}}^{n}(X_{i}-\overline {X})(Y_{i}-\overline {Y}))=(n-1)(\Sigma_{XY}^2+\Sigma_{XX}\Sigma_{YY})$, or
$\text{Var}(\frac{1}{n-1}\sum _{{i=1}}^{n}(X_{i}-\overline {X})(Y_{i}-\overline {Y}))=(n-1)^{-1}(\Sigma_{XY}^2+\Sigma_{XX}\Sigma_{YY})$
Or, for a more general result,
If $S_{XY}=\frac{1}{n}\sum _{{i=1}}^{n}(X_{i}-\overline {X})(Y_{i}-\overline {Y}))$ then these notes by Thomas S. Richardson, here give
$\text{Var}(S_{XY})=\frac{(n−1)^2}{n^3}(μ_{22}−μ_{11}^2)+ \frac{(n−1)}{n^3} (μ_{11}^2 + μ_{20} μ_{02})$
(where $\mu_{rs}=E[(X-\mu_{_X})^r\,(Y-\mu_{_Y})^s]$)
however, wolfies notes in his answer here that this is incorrect. If I haven't made an error, his result corresponds to a flip of sign on the second $\mu_{11}$ term:
$\text{Var}(S_{XY})=\frac{(n−1)^2}{n^3}(μ_{22}-μ_{11}^2)+ \frac{(n−1)}{n^3} ( μ_{20} μ_{02}-μ_{11}^2 )$
Note that correcting this for the $\frac{1}{n-1}$ version is a simple matter of multiplying the above result by $(\frac{n}{n-1})^2$.
IIRC, there's more details in vol. I of Kendall and Stuart
• Last link is dead, here is a copy from wayback machine web.archive.org/web/20130226032010/http://… – mgilbert Oct 28 '15 at 18:37
• Thanks @mgilbert - I've updated the link, pulled in the equation and given credit (since I now quote it). – Glen_b -Reinstate Monica Oct 28 '15 at 21:41
• I'm assuming the n's in the formula refer to the sample size/number of observations, but could someone help with clarifying what the mu's (with subscript 22, 11, 20, and 02) signify? Sorry for being such a dummy :P. – Jonna Oct 20 '16 at 16:14
• No problem asking; I probably should have defined them. Those are called (central) mixed moments. If you have two subscripts it means you're dealing with two variables ($X,Y$ say); then $\mu_{rs}=E[(X-\mu_{_X})^r\,(Y-\mu_{_Y})^s]$. So $\mu_{11}$ is covariance, $\mu_{20}$ is the variance of the first variable, and so on. They extend in the natural way to more than two variables. – Glen_b -Reinstate Monica Oct 20 '16 at 23:25
• Thanks a lot for clarifying @Glen_b! Just to check, μ22 should then be the product of the variances of X and Y? Here's the code I wrote: V <- ((n-1)^2/n^3)*(Xvar*Yvar - covar^2) + ((n-1)/n^3)*(covar^2 - Xvar*Yvar) – Jonna Oct 21 '16 at 11:39
The OP's question does not define what formula R uses as sample covariance estimator. However, following the link in Glen's answer, I assume that R is using:
$$m_{11} = \frac{1}{n} \sum _{i=1}^n \left(X_i-\bar{X}\right) \left(Y_i-\bar{Y}\right)$$
... also known as the $m_{11}$ sample central moment, which can be expressed in power sum notation $s_{r,t}=\sum _{i=1}^n X_i^r Y_i^t$ (using mathStatica here) as :
... which is a familiar alternative notation.
We seek the variance of the estimator i.e. $\text{Var}(m_{1,1})$. Since the variance operator denotes the $2^\text{nd}$ central moment of $m_{1,1}$, we can find the exact symbolic solution (for any distribution whose moments exists) with the mathStatica function:
I would note that the solution so obtained is different to that referenced in the link given by Glen above to a paper. Perhaps they are computing something else?! There is now a long list of published 'moment of moments' papers that have been shown to contain incorrect results by mathStatica, including some results by Fisher himself, and some of the results in Stuart and Ord - see for instance Spot The Error
There is an alternative defn of sample covariance using $\frac{1}{n-1}$ but that is not the one used in the paper referenced by Glen either.
• taking your output as correct, there's a simple sign error in Richardson's version which I will note in my answer. Thanks. – Glen_b -Reinstate Monica Oct 23 '16 at 22:42
• Actually, R's cov function seems to use $\frac{1}{n-1}$; I don't recall whether I even checked that detail before. – Glen_b -Reinstate Monica Oct 23 '16 at 22:51
There is some confusion in the discussion above.
Simple algebra shows that the expression ascribed to Richardson:
Var$(S_{XY})=\frac{(n−1)^2}{n^3}(\mu_{22}−\mu^2_{11})+\frac{(n−1)}{n^3}(\mu^2_{11}+\mu_{20}\mu_{02})$
is identical to that obtained by wolfies using MathStatica.
Both expressions clearly agree on the coefficients of $\mu_{22}$ and $\mu_{20}\mu_{02}$.
For $\mu_{11}^2$, collecting terms in Richardson's expression gives:
$[-(n-1)^2 + (n-1)]/n^3$
$= [(1-n)(n-1) + (n-1)]/n^3$
$= [(1-n) + 1](n-1)/n^3$
$= (2-n)(n-1)/n^3$
$= -(n-2)(n-1)/n^3$
$= -(-2+n)(-1+n)/n^3$
which is the coefficient for $\mu_{11}^2$ obtained by MathStatica.
[The expression provided by Glen as a "correction" to MathStatica's is not equivalent, as can be seen by substituting $n=2$ and comparing coefficients for $\mu_{11}^2$.]
The correct expression may also be derived in a few steps from results in Kendall's Advanced Theory of Statistics, Kendall & Stuart (1987), Fifth Edition, p.441, Example 13.3, where it is stated that:
Var$(k_{11}) = \frac{1}{n}\kappa_{22} + \frac{1}{n-1}\kappa_{20}\kappa_{02} +\frac{1}{n-1}\kappa^2_{11}$.
Simple algebra shows this is equivalent to the above expressions, noting that $k_{11}$ in K+S is the $k$-statistic, which is the unbiased estimator of the population covariance (aka the (1,1) product cumulant $\kappa_{11}$), so $k_{11} = \frac{n}{n-1} S_{XY}$. It is also necessary to note the following relations between cumulants and moments: $\mu_{11}=\kappa_{11}$, $\mu_{20}=\kappa_{20}$, $\mu_{02}=\kappa_{02}$ and
$\mu_{22} = \kappa_{22} + \kappa_{20}\kappa_{02} + 2\kappa_{11}^2$.
See K+S p.105, p.102 following (3.69) and p.87.
[Lastly, Goldberger (1991), A Course in Econometrics, p.108 gives an expression for $V(S_{XY})$ that is incorrect. Specifically it contains a term $2(n-1)(\mu_{20}\mu_{02})/n^3$ that should instead be $(n-1)(\mu_{11}^2+ \mu_{20}\mu_{02})/n^3$.]
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2019-12-07 22:05:56
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http://mymathforum.com/trigonometry/46605-basic-easy-trig-has-been-confusing-me.html
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My Math Forum Basic and easy trig! This has been confusing me
Trigonometry Trigonometry Math Forum
September 26th, 2014, 05:40 AM #2 Math Team Joined: Dec 2013 From: Colombia Posts: 7,659 Thanks: 2635 Math Focus: Mainly analysis and algebra Perhaps you were supposed to use $$1 + \cot^2 x = \csc^2 x$$
September 26th, 2014, 09:27 AM #3
Newbie
Joined: Sep 2014
From: Somewhere
Posts: 3
Thanks: 0
Quote:
Originally Posted by v8archie Perhaps you were supposed to use $$1 + \cot^2 x = \csc^2 x$$
Nope, we have not taken this. As I have told you, my answer was correct (I think?) But it made me very confused (read the questions I listed).
Last edited by skipjack; September 27th, 2014 at 08:02 AM.
September 27th, 2014, 09:07 AM #4 Global Moderator Joined: Dec 2006 Posts: 20,641 Thanks: 2083 Suppose the opposite and adjacent sides are of length 1 and √3 respectively, so that the hypotenuse has length 2. If θ is the angle between the hypotenuse and the side of length √3, csc(θ) = 2. That implies that θ = arccsc(2) = 30°. In the above, I first wrote an equation that involved csc, then I used arccsc to solve that equation. In a sense, I "used" both functions. As I happen to know that csc(30°) = 2, I could have deduced directly that θ = 30°. However, such knowledge is equivalent to knowing that arccsc(2) = 30°, so I would still implicitly have been using arccsc. If you know the formula θ = arccsc(hyp/opp), you can proceed similarly, but without having to involve the csc function. One can sometimes find an angle without use of any inverse trigonometric function (even implicit use). For example, each angle of an equilateral triangle must be 60° because the three angles are the same and must total 180°. In general, though, you would need to use an inverse trigonometric function. I think the wording of the question was a bit unusual. It was presumably because a particular method was wanted. If any method of solution had been allowed, you could have used the arctan function, and this would have been simpler than calculating the hypotenuse and then using arccsc. In general, you can't use just the six trigonometric functions to find an angle. However, once you know two angles of a triangle, you can, of course, calculate the third angle without further use of trigonometry. Last edited by skipjack; October 10th, 2014 at 05:55 AM.
October 1st, 2014, 07:36 AM #5 Newbie Joined: Jan 2014 Posts: 14 Thanks: 0 Another question of a kind: Define Sin without right triangle
October 1st, 2014, 08:34 AM #6
Senior Member
Joined: Jul 2014
From: भारत
Posts: 1,178
Thanks: 230
Quote:
Originally Posted by mac_alleb Another question of a kind: Define Sin without right triangle
the measurement of one of the non-right angles (q) the length of the side adjacent to that angle. the length of the triangle's hypotenuse. Futhermore, Definition I gives exact equations that describe each of these relations: sin(q) = opposite / hypotenuse.
October 1st, 2014, 09:26 AM #7 Newbie Joined: Jan 2014 Posts: 14 Thanks: 0 Arctan as well as all trig functions possibly has defs throw rows and polynomial. But the very question is How to connect it with triangle def. More future Sin := f(x,y)???
October 1st, 2014, 04:40 PM #8 Global Moderator Joined: Dec 2006 Posts: 20,641 Thanks: 2083 One could define sin(A) as a/d, where d is the diameter of the circumcircle of the triangle ABC with side of length a opposite angle A.
October 9th, 2014, 06:11 AM #9 Newbie Joined: Sep 2014 From: Somewhere Posts: 3 Thanks: 0 Thanks for the explanation Your efforts are highly appreciated.
October 13th, 2014, 02:45 PM #10 Newbie Joined: Oct 2014 From: Ohio Posts: 9 Thanks: 4 You have to use the inverse trig functions to find angles. You have not properly understood what your teacher was doing in class. Last edited by skipjack; October 13th, 2014 at 08:25 PM.
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2019-05-26 03:53:28
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https://1library.co/document/8ydm49jy-sound-measurement-by-laser-doppler-anemometry.html
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# SOUND MEASUREMENT BY LASER DOPPLER ANEMOMETRY
## Texto completo
(1)
### ANEMOMETRY
PACS REFERENCE: 43.58.Dj
T. Schlicke, T. MacGillivray, C. Greated, R. Barham University of Edinburgh
Department of Physics and Astronomy, Mayfield Road Edinburgh
UK
Tel: +44 (0)131 650 5258 Fax: +44 (0)131 650 5902 E-mail: ted@ph.ed.ac.uk
ABSTRACT
The use of Laser Doppler Anemometry (LDA) for sound measurement offers a number of fun-damental advantages over the use of conventional microphones, although its application is also limited. One of the main advantages is that it is non-intrusive; also it does not require calibration. A major limitation, on the other hand, is that it can only be applied to periodic sound fields, indeed sound fields where there are only just a few frequency components.
This paper describes the application of LDA using the photon correlation signal analysis tech-nique. The high sensitivity of this method means that very little or no seeding is required. The theoretical form of the Auto-Correlation Function (ACF) will be discussed. The results of experi-ments in both tubes and free fields will be presented and compared with conventional microphone measurements. There will also be an assessment of different signal analysis techniques such as spectrum analysis and curve-fitting.
1. INTRODUCTION
Laser Doppler Anemometry is a non-intrusive, optical technique for measuring fluid flow. The measuring volume consists of interference fringes formed by intersecting laser beams, and is shown in figure 1, together with a close-up of the interference fringes (from [1]).
(2)
The laser beam has a wavelength and is assumed to have a Guassian profile of diameter
. is the half-angle of the beam intersection. The measurement volume has dimensions
, and
; these values are typically fractions of a millimetre, so LDA is often termed a point measuring technique. Each of the two beams is assumed to be of equal intensity, and the spacing between consecutive fringes is
.
Small particles suspended in the fluid scatter some of this light. The number of photons scattered depends on a variety of factors, such as the number and size of the particles, the intensity of the laser beam and their position within the interference pattern, and this number changes as the particles follow the fluid motion. A photomultiplier records some of this scattered light; if its intensity is sufficiently small, corresponding to an average of less than 40 million photons per second (one per 25ns), individual photons can be counted. The temporal photon distribution can then be auto-correlated, and the form of the auto-correlation function (ACF) used to determine features of the flow, such as its velocity.
The sensitivity of this technique is such that natural impurities in the fluid are frequently suf-ficient to produce an adequate signal, and the addition of extra seeding particles is not only unnecessary but can saturate the photomultiplier output.
2. BACKGROUND THEORY
A Digital Correlator board produces the ACF, from which various flow features can be extracted if the theoretical form of the function is known.
The scattering particles are assumed to follow the flow faithfully, and have an instantaneous velocity of the form
!"\$#&%(')#+*,.-0/ 1
*2+! (1)
where#+% is the mean flow velocity,#+* is the acoustic velocity amplitude and
1
* is the acoustic angular frequency. That is, the fluid motion is considered to be a superpostion of a mean flow and an acoustic oscillation.
Following Hann’s derivation [2] but avoiding the incorrect identities in equations (34) and (35), the auto-correlation function of the photomultiplier current,3
54
!, is given by
62798;:=< >@?ACBED GFHI.JK ?LANMPORQGSUTVWX7Y[Z;\K8C:G]^\_7Y[ZN` ab:c\$d e agf \ih a 7EjbSk: a"l a 79m_: lkn a 7poq: Tbrsut_vk7Y[Z;\u8C:d e agfw ] n a 7Y[Z `ax:d e y f \ h y r 7EjbSk: a{z y l y 79m_:}| l n~ y zRag 7poq:RT l n~ y A a{ 7poq: j(rv+7Y[Z;\u8C:d e agf \ ] n agzw 7Y[Z `ab:
d e y f \ 7EjbSK: agz y l y 79m_:}| l n y z ~n agzw7poq:qT)7ESj h y : l n y A ~n a{zw7po;:9 (2) whereL # 4
#+* , L
#+ * and #+* ,.-0/
! are introduced as abbreviations. is defined as
}L
and
(3)
(3)
}
is the Neumann factor and is defined such that
u
/¢¡
/£¤¡
(4)
where¥ is an integer. ¦ and§ represent the Bessel function and modified Bessel function respec-tively. ¨ is a constant whose value depends on, among other factors, the density and scattering ability of the tracer particles. Equation 2 differs slightly from the expression presented by Hann, but it was found that when actually plotted as graphs, the two equations were indistinguishable when both the acoustic and the mean velocity amplitudes were small.
2.1 ACOUSTIC FLOW
When there is no mean flow present,# % ©¡ , the equation reduces to the expression obtained by Sharpe [3], 3 4 !"©ªi«¬® '¯¦ % °# *
!.!K±§ % ^!²'¤´³
e *¶µ ¦ * # *
!G§ * ^!G· (5)
if the constant term is omitted. If is small, corresponding to the acoustic velocity amplitude#C* being considerably less than ¸
g¹gº
» , then equation 5 simplifies further to
3 4 !" '¢¦ % °# * !!"½¼ '¯¦ % ¼ ^#+* 1 * ,-0/¾ 1 * 4 À¿&ÁÂÁ (6)
Equation 6 contains a Bessel function with a sinusoidal argument, which is a frequency-modulated signal consisting of a single, prominent frequency component accompanied by side-bands whose relative amplitudes depend on the acoustic velocity.
The first minimum of the zero-order Bessel function occurs when its argument equals 3.832. Since this argument is
}Ã ¸ ,.-0/ Q ¸ g c
, the acoustic velocity amplitude can be determined by evaluating # * Ä ±Å Ä 1 * ^¢,-/ Q ¸ c (7)
2.2 CONSTANT MEAN FLOW VELOCITY
When the acoustic velocity amplitude equals zero, but the mean flow# % is non-zero, equation 2 simplifies to 3 4 !"©ªÇÆ^È .ÉKI.Ê}
')ËPÌi, °# %
4
!.! (8)
which is the damped cosine function predicted by Greated and Duranni [4]. This curve will have peaks whenever the cosinusoidal argument,°#R%
4
, equalsLÍ
, whereÍ is an integer. The mean flow velocity can therefore be determined from the peak separation according to:
# % 4HÎ Ï (9) where4_Î
(4)
2.3 INTERPRETATION OF THE ACF
Although equations 7 and 9 provide a means of evaluating the acoustic and mean flow velocities respectively, they are effectively using only one or two points from the ACF- none of the other data points contribute to the calculation. This approach is therefore rather wasteful of the available data.
Another method is to take the Fourier Transform (FT) of the ACF. The FT of a monochromatic cosine wave is simply a delta function, so it is straightforward to extract the velocity if there is a mean flow alone, and no acoustic field. The FT of a Bessel Function with a linear argument is also known, so the acoustic velocity can be obtained if the approximation,-/
Q
¸
c
¸
is valid, that is, for small times. At longer times and higher frequencies, the sinusoidal argument contributes to sidebands in frequency space which in principle could be used to obtain the acoustic velocity amplitude. However, it is not clear whether it is even possible to calculate the FT of equation 6, let alone equation 2. Furthermore, the ACF data set contained a maximum of only 200 points, resulting in limited frequency resolution.
A more effective method is to generate a data set using the relevant equation above, using guesses for#* and#&%. The sum of the square of the difference between a generated point and the point obtained from the measured ACF provides a measure of the discrepancy. A range of values for# * and# % are attempted, and the values for which the discrepancy defined above is a minimum are then the best estimates for the flow velocities. Although this method of curve-fitting is rather inefficient, it does use more of the measured ACF values and can therefore be expected to be more reliable.
3. RESULTS
Photon correlation results from two experimental set-ups were obtained by MacGillivray [1]: of a standing wave within a closed, glass tube, and of the “free field” in front of a loudspeaker. A Brookhaven BI-9000AT Digital Autocorrelation board performed the autocorrelation of the signal from the photomultiplier. This board had a total of 200 channels, separated in time by
4
, in which values of the discrete ACF were stored. Since it is the actual number of photons which is being correlated, the values of the ACF are rather large, typically
¡LÐ .
This provides a further difficulty: in order to obtain a close fit, the vertical scale of the measured data and the fitted data should be the same. The vertical range of equations 6 and 8 are obvious, so the raw ACF data can be appropriately normalised. The modified Bessel function tends to infinity as its argument increases, so the ranges of equations 2 and 5 are not clear. In attempting to fit these equations to the data, the vertical scale factor is effectively another unknown parameter. This will occur when the acoustic velocity amplitude is significant and/or a mean flow is also present.
3.1 STANDING WAVE RESULTS
A glass tube with one end attached to a loud-speaker and the other end closed was positioned such that the measurement volume was located at the tube’s axis of symmetry.
The effect of the intensity of the sound field and its frequency were investigated. A graph of the ACF against time for a fixed frequency and various pressure amplitudes is shown in figure 2(I). The frequency was 660Hz, a resonance of the tube, and the measured pressures were (a) 1.0Pa, (b) 4.0Pa, (c) 7.0Pa and (d) 10.0Pa. Figure 2(II) shows the effect of frequency on the ACF. A fixed pressure of 5.0Pa was applied for sound frequencies of (a) 660Hz, (b) 1135Hz, (c) 1570Hz and (d) 2000Hz, all of which correspond approximately to resonant frequencies of the tube. All measurements were taken at a velocity antinode. The sample time of the correlator board was
4
(5)
I II
Figure 2:AutoCorrelation Functions showing effect of pressure (I) and acoustic frequency (II)
From figure 2(I) it can be seen that as the pressure amplitude at the rigid end was increased, the time until the first minimum in the ACF decreased. Combining this result with equation 7 suggests that# * increases with increasing pressure amplitude.
Figure 2(II) shows that the time to the first minimum is independent of frequency. This is predicted by equation 7 if the small angle sine approximation is valid. All of the curves in 2 (I) and (II) contain some damping. This could be accounted for by the presence of a small mean flow. However additional damping factors, not included in equation 2, can also arise from slight misalignments in the laser beams, and a difference in their respective intensities.
MacGillivray attempted to fit a curve of the form
3
4
!(¢¨Âª «+Õ.Ö I
ª «¬
')×kØ^Ô °#
%
4
!u¦
%
#
*
!.! (10)
to the measured data. Equation 10 is a simplified version of 2: it assumes that both an acoustic and a mean flow velocity may be present, but that both are small. The ACF of figure 2(I)(d) and some theoretical fits are shown in figure 3.
By comparing the 20 points of the measured ACF around the first minimum to the values obtained using equation 10, a best estimate of
Ù
±
Ú ¡q±
ÍÛÍÜÔ
« was found for the acoustic velocity. Theoretical fits were generated with this value of #R* and a range of values for #R% , from ¡ to
ÍÛÍÜÔ
« . It can be seen that as
# % increases, the beating interaction between the cosine and the Bessel term becomes more pronounced. The best estimate for the mean flow velocity was
(6)
Although the theoretical and experimental
Figure 3: Measured ACF with theoretical fits
curves agree well around the first minimum in the ACF, there is an increasing discrepancy with increasing time. Further work will be re-quired to ascertain whether this is due to the terms neglected in equation 10 or misalign-ments in the experimental setup.
The effectiveness of the photon correlation technique was established by using the LDA measurements to calculate the pressure am-plitude, which could be compared directly with measurements from a probe microphone. The pressure values in the frequency range 660-2kHz were found to agree to within 0.25dB.
Experiments were also conducted in the “free field”, measuring the velocity amplitude #
* as a function of distance from a loud-speaker. Although the measured ACFs had the qualitative form predicted by theory, the discrepancy with the results obtained from microphone measurements was greater than the calculated uncertainty. This could be due to acoustic reflections, air currents and variations in seeding concentration.
4. CONCLUSIONS
The technique of photon correlation was used to obtain measurements of acoustic fields. A theo-retical expression for the form of the auto-correlation function was presented, which assumed that the fluid motion consisted of an acoustic field superimposed on a mean flow. Simplified versions of this equation were least-square-fitted to measured auto-correlation functions which allowed the velocity amplitudes to be determined. Measurements obtained in a standing wave were used to calculate a pressure amplitude which was in good agreement with measurements from a probe microphone. Measurements in the free-field were less accurate but showed that the photon cor-relation method has the potential to measure flow velocities without the need for calibration.
### References
[1] T. MacGillivray, The application of laser anemometry in acoustic measurement standards,
PhD Thesis, University of Edinburgh, 2002
[2] D. Hann Acoustic measurements in flows using photon correlation spectroscopy Meas. Sci.
Technol. 4 (1993) 157-164,
[3] D. Sharpe, C. Greated, A stochastic model for photon correlation measurements in sound fields J. Phys. D: Appl. Phys.22 (1989) 1429-1433,
Actualización...
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2022-05-24 16:14:51
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https://math.stackexchange.com/questions/2537959/game-theory-voting-game-why-does-player-3-have-4000-strategies
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# Game theory (voting game) - why does player 3 have > 4000 strategies?
A committee with three members, {1; 2; 3}, has to choose a new member of a club among a set of four candidates, {a; b; c; d}. Each member of the committee has veto power which is used in a successive way, starting by member 1, and finishing with member 3. Each member of the committee has to veto one and only one of the candidates that have not been vetoed yet.
(a) Draw the extensive form of the game, writing in the terminal nodes the name of the elected candidate.
(b) How many strategies has each player? Do not try to write them out to > count them (player 3 has more than 4,000)
According to the text of part(b) of the problem, player 3 apparently has more than 4000 strategies.
I'm not seeing how that's the case. As I understand it, the game tree should look like this:
So where's this > 4000 strategies coming from? (I'm completely accepting of the fact that I'm probably wrong - just want to understand why!)
• Can Player 3 negotiate before the voting begins? I think you need to provide more context for the " ? $4000$" strategies assertion. The link says you must think about Nash equilibria on preference orderings. Nov 26 '17 at 13:48
• @EthanBolker I've added the full text from parts (a) and (b) of the question. Is it to do with preference orderings then? I'm not seeing where the 4000 part comes from just from the game tree.
– Thev
Nov 26 '17 at 13:53
• @croraf I already solved it! Drop me an email at thevesh.theva@gmail.com and I'll send you the problem and solution!
– Thev
Dec 14 '17 at 7:17
You have the correct game tree. I think the issue is just the definition of "strategy" - a strategy for player 3 is a full description of what that player will do in all possible situations. There are $12$ different situations that player 3 may end up in, corresponding to the $12$ boxes marked "3" in your diagram. For each of these player 3 has $2$ choices, so the total number of different possible strategies is $2^{12}>4000$.
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2021-09-24 22:23:11
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http://math.stackexchange.com/questions/220895/help-with-this-assertion-the-only-number-divisible-by-3-and-that-is-prime-is-3/440477
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# help with this assertion: The only number divisible by 3 and that is prime is 3
I have encountered this phrase within a proof by prime numbers and couldn't figure out if it is true.
Is there any proof lurking around for this fact?
thanks!
-
Suppose for contradiction that $x\neq 3$ is divisible by $3$ and is prime. Then the only numbers dividing $x$ are $1$ and $x$. But $3$ divides $x$. So we have $x=1$ or $x=3$. Each of these is a contradiction.
-
Or even better, lose the indirection and say: Suppose that $x$ is divisible by $3$ and is prime. Then the only numbers dividing $x$ are $1$ and $x$. But $3$ divides $x$. Since $3$ is not $1$, we must have $3=x$. – Henning Makholm Oct 25 '12 at 16:03
If a number is divisible by $3$ and it's not $3$, then the number is not prime.
-
Well that's not much more than a reformulation (but it's hard to make obvious things more obvious -they are just not obvious for everyone) – Hagen von Eitzen Oct 25 '12 at 15:55
Well, it's trivial (and this is sometimes the reason to make it so hard to see).
To avoid getting confused by trivialities, it is often helpful to re-formulate the statement. Here, for example: "Any number (positive integer) other than 3 that is divisable by 3 is not a prime". See it? Once again:
Let $x \not= 3$ be any positive integer that is divisable by $3$. Can $x$ be a prime number?
-
Hint $\$ If $\rm\:ab\:$ is prime then $\rm\:a > 1\:\Rightarrow\: b=1\:\Rightarrow\:ab =a.\:$ Yours is the special case $\rm\:a = 3.$
-
Just observe that any multiple of 3 must repeat that 3 some number of times, and that number would be its factor. For example 6 is 2*3. 9 is 3*3. 12 is 4*3. As you can see, each multiple of 3 has two factors, of which one is 3, and the other is some natural number. And if there are factors, you have a composite number, which is the opposite term for prime number (if something is composite, it cannot be prime). The only multiple of 3 which doesn't have any factors beyond 3 (and 1, which we usually don't consider as factor, since it doesn't produce anything new) is the number 3 itself, which is prime.
-
Suppose $a\neq 3$ is a positive multiple of $3$. Then $a > 3$ and $3$ is a divisor that is not $1$ or $a$...hence $a$ cannot be prime.
-
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2015-11-30 03:09:30
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http://www.mathgol.com/lesson/pre-algebra/178-multiplying-decimals/
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# Multiplying Decimals
- -
Multiplying decimals is similar with multiplying whole numbers. To multiply decimals, we may disregard the decimal points and just treat the decimal number as whole number. Following the rule for multiplying whole numbers, multiply the last digit in the multiplier with every digit in the multiplicand. Then multiply the second digit (from the right) of the multiplier to every digit of the multiplicand but place the first product or the unit digit of the first product one positional value below from the first digit of the first line of products obtained from multiplying the first digit in the multiplier and the digits in the multiplicand. Follow the same rule until all the digits in the multiplier have been multiplied to all the digits in the multiplicand. Add all the partial products by following addition rule for whole numbers. Count the total number of places to the right of the decimal point in the multiplier and in the multiplicand. The number of digits to the right of the decimal point of the product is equal to the number of places in these factors.
Below are examples that illustrate the above rules for multiplying decimals:
Find the product between the given pair of decimal numbers.
1. 2.34 and 13.2
2. 3.105 and 2.12
3. 12.1 and 4.6
4. 2.376 and 0.21
$1.\;\;\; \begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} & &2&.&3 & 4 \\ \times&1&3&.&2 & \\ \hline &&4&&6 & 8 & \;\textup{multiply 234 by 2}\\ & 7& 0 && 2 &&\;\textup{multiply 234 by 3}\\ 2&3 & 4 &&&&\;\textup{multiply 234 by 1} \\ \hline 3&0. & 8 &&8&8& \;\textup{total number of places to the right}\\ \end{array}$
$2.\;\;\; \begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &3 &.&1&0 & 5 \\ \times&&&4&. &6 \\ \hline &1&8&6&3 &0 & \;\textup{multiply 3105 by 6}\\ 1&2&4&2&0&&\;\textup{multiply 3105 by 4}\\ \hline 1&4&.&2&8&3&0 \;\;\textup{total number of places to the right}\\ \end{array}$
$3.\;\;\; \begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &1 &2&.&1 & \\ \times&&4&.&6& \\ \hline &7&2&&6&& \;\textup{multiply 121 by 6}\\ 4&8&4&&&&\;\textup{multiply 3105 by 4}\\ \hline 5&5&.&6&6&& \;\;\textup{total number of places to the right}\\ \end{array}$
$4.\;\;\; \begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &2&.&3&7&6 \\ \times &&0&.&2&1 \\ \hline &2&&3&7&6& \;\textup{multiply 2376 by 1}\\ 4&7&&5&2&&\;\textup{multiply 2376 by 2}\\ \hline 0.4&9&&8&9&6& \;\;\textup{total number of places to the right}\\ \end{array}$
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2017-09-22 00:40:39
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https://forum.azimuthproject.org/plugin/ViewComment/18764
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Definining \$$(true \multimap false) = false\$$ and \$$(true \multimap true) = (false \multimap true) = (false \multimap false) = true\$$ should work.
1. the falsity of \$$(true \wedge true) \leq false\$$ requires the falsity of \$$true \leq (true \multimap false)\$$, hence the falsity of \$$(true \multimap false)\$$.
2. the truth of \$$(true \wedge true) \leq true\$$ requires the truth of \$$true \leq (true \multimap true)\$$, hence the truth of \$$(true \multimap true)\$$.
3. the truth of \$$(true \wedge false) \leq x\$$ for all \$$x \in \mathbb{B}\$$ requires the truth of \$$true \leq (false \multimap x)\$$, hence the truth of \$$(false \multimap x)\$$ for all \$$x \in \mathbb{B}\$$.
4. the truth of all expressions of the form \$$(false \wedge x) \leq y\$$ and \$$false \leq z\$$ for any x,y,z in \$$\mathbb{B}\$$ implies that they give no further restrictions on \$$\multimap\$$.
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2022-09-24 20:35:39
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https://forum.pyro.ai/t/forecasting-with-a-stochastic-volatility-model-in-numpyro/3451
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# Forecasting with a stochastic volatility model in numpyro
Hello,
I’m trying to forecast with the following model. First I tried:
def scan_fn(carry, error):
sigma, coeff_prev = carry
coeff = coeff_prev + sigma*error
coeff_prev = coeff
return (sigma, coeff_prev), coeff
def weights_regression_sv(X=None, y=None):
T, K = X.shape
beta_unweighted = numpyro.sample('beta_unweighted', dist.HalfNormal(jnp.ones(K)))
beta = numpyro.deterministic("beta", beta_unweighted/jnp.sum(beta_unweighted))
# SV
lambda_errors = numpyro.sample('lambda_errors', dist.Normal(0.,jnp.ones(T)))
log_lambda_0 = numpyro.sample('log_lambda_0', dist.Normal(0.,4))
phi = numpyro.sample('phi', dist.InverseGamma(5,.15))
__, log_lambda_t = scan(scan_fn, (phi, log_lambda_0), lambda_errors, T)
log_lambdas = numpyro.deterministic("log_lambdas", log_lambda_t)
lambdas = numpyro.deterministic("lambdas", jnp.exp(log_lambda_t))
return numpyro.sample("y",dist.Normal(jnp.matmul(X,beta),jnp.sqrt(lambdas)),obs=y)
and used
predictive = Predictive(weights_regression_sv, posterior_samples=m_sv.get_samples())
y_pred = predictive(random.PRNGKey(0), X=X_[[-1],:])["y"]
az.plot_kde(100*y_pred)
plt.show();
but got the error:
Then I tried:
def scan_fn(carry, t):
phi, log_lambda_prev, xbeta, lambda_errors = carry
log_lambda = log_lambda_prev + phi*lambda_errors[t]
lambda_ = jnp.exp(log_lambda)
y_ = numpyro.sample("y", dist.Normal(xbeta[t],jnp.sqrt(lambda_)))
log_lambda_prev = log_lambda
return (phi, log_lambda_prev, xbeta, lambda_errors), y_
def weights_regression_sv_fcast(X=None, y=None, forecast=None):
T, K = X.shape
if forecast:
T_ = T+1
else:
T_ = T
beta_unweighted = numpyro.sample('beta_unweighted', dist.HalfNormal(jnp.ones(K)))
beta = numpyro.deterministic("beta", beta_unweighted/jnp.sum(beta_unweighted))
# SV
lambda_errors = numpyro.sample('lambda_errors', dist.Normal(0.,jnp.ones(T_)))
log_lambda_0 = numpyro.sample('log_lambda_0', dist.Normal(0.,4))
phi = numpyro.sample('phi', dist.InverseGamma(5,.15))
with numpyro.handlers.condition(data={"y": y}):
xbeta = jnp.matmul(X,beta)
__, ys = scan(scan_fn, (phi, log_lambda_0, xbeta, lambda_errors), jnp.arange(0,T_))
if forecast:
numpyro.deterministic("y_forecast", ys[-1])
But got this error during model estimation:
I was working off Time Series Forecasting — NumPyro documentation, but this doesn’t really seem applicable given the use of X variables? Also I looked at Forecasting with Dynamic Linear Model (DLM) — Pyro Tutorials 1.7.0 documentation but this also doesn’t seem applicable given the absence of the Forecast module in numpyro?
Update:
This worked:
def scan_fn(carry, error):
sigma, coeff_prev = carry
coeff = coeff_prev + sigma*error
coeff_prev = coeff
return (sigma, coeff_prev), coeff
def weights_regression_sv_fcast(X=None, y=None, forecast=0):
T, K = X.shape
beta_unweighted = numpyro.sample('beta_unweighted', dist.HalfNormal(jnp.ones(K)))
beta = numpyro.deterministic("beta", beta_unweighted/jnp.sum(beta_unweighted))
# SV
lambda_errors = numpyro.sample('lambda_errors', dist.Normal(0.,jnp.ones(T)))
log_lambda_0 = numpyro.sample('log_lambda_0', dist.Normal(0.,4))
phi = numpyro.sample('phi', dist.InverseGamma(5,.15))
__, log_lambda_t = scan(scan_fn, (phi, log_lambda_0), lambda_errors, T)
log_lambdas = numpyro.deterministic("log_lambdas", log_lambda_t)
lambdas = numpyro.deterministic("lambdas", jnp.exp(log_lambda_t))
T_END = T-forecast
numpyro.sample("y_obs",dist.Normal(jnp.matmul(X[:T_END],beta),jnp.sqrt(lambdas[:T_END])),obs=y)
numpyro.sample("y_pred",dist.Normal(jnp.matmul(X[T_END:],beta),jnp.sqrt(lambdas[T_END:])),obs=None)
Is getting forecasts this way meaningfully different from getting them using
predictive = Predictive(weights_regression_sv, posterior_samples=m_sv.get_samples())
y_pred = predictive(random.PRNGKey(0), X=X_[[-1],:])["y"]
?
I think you lambda_errors is a time-local variable (i.e. a latent variable that depends on time). The posterior will give you the samples of lambda_errors for all of your time steps. You need to resolve that logic in your prediction, where you seem to only need samples from the last step.
Thanks! Could you elaborate on what you mean when you say I need to resolve that logic?
In the model, you defined T, K = X.shape. For prediction, T=1, so your latent variable
lambda_errors = numpyro.sample('lambda_errors', dist.Normal(0.,jnp.ones(T)))
needs to have shape (1,). But your posterior gives lambda_errors with shape T != 1 because you were running MCMC on the data X with X.shape[0] != 1.
For time-local variables, it is better to move them to scan body_fn, like in gaussian_hmm example. But I’m not sure what you want to predict, so the first step might be to reconsider how you construct your forecasting model.
Mmmm, sorry, I’m still confused. Let me reset and try this:
My model is:
y_t = \beta x_t + \lambda_t^{0.5} \epsilon_t where \epsilon_t \sim N(0,1)
\log(\lambda_t) = \log(\lambda_{t-1}) + \phi v_t where v_t \sim N(0,1)
So it follows (I think) that to forecast y_{T+1} given x_{T+1}, I would need (\lambda_1,...,\lambda_T, \lambda_{T+1}). I implement the entire model as follows (note that lambda_errors=v_{1:T}):
def scan_fn(carry, error):
sigma, coeff_prev = carry
coeff = coeff_prev + sigma*error
coeff_prev = coeff
return (sigma, coeff_prev), coeff
def weights_regression_sv_fcast(X=None, y=None, X_forecast=None):
T, K = X.shape
if X_forecast is not None:
T_ = T+X_forecast.shape[0]
else:
T_ = T
beta_unweighted = numpyro.sample('beta_unweighted', dist.HalfNormal(jnp.ones(K)))
beta = numpyro.deterministic("beta", beta_unweighted/jnp.sum(beta_unweighted))
# SV
lambda_errors = numpyro.sample('lambda_errors', dist.Normal(0.,jnp.ones(T_)))
log_lambda_0 = numpyro.sample('log_lambda_0', dist.Normal(0.,4))
phi = numpyro.sample('phi', dist.InverseGamma(5,.15))
__, log_lambda_t = scan(scan_fn, (phi, log_lambda_0), lambda_errors, T_)
log_lambdas = numpyro.deterministic("log_lambdas", log_lambda_t)
lambdas = numpyro.deterministic("lambdas", jnp.exp(log_lambda_t))
numpyro.sample("y_obs",dist.Normal(jnp.matmul(X,beta),jnp.sqrt(lambdas[:T])),obs=y)
if X_forecast is not None:
numpyro.sample("y_pred",dist.Normal(jnp.matmul(X_forecast,beta),jnp.sqrt(lambdas[T:])),obs=None)
My goal is to do 1-step-ahead forecasts but this code should work for h-step-ahead for any positive integer h. But, let’s assume I just want a 1-step-ahead. So, I need X (containing x_1,...,x_T), y containing y_1,...,y_T, and then an x_{T+1}. Given that, for a run of the code above, I should get (\lambda_1,...,\lambda_{T+1}) from:
# SV
lambda_errors = numpyro.sample('lambda_errors', dist.Normal(0.,jnp.ones(T_)))
log_lambda_0 = numpyro.sample('log_lambda_0', dist.Normal(0.,4))
phi = numpyro.sample('phi', dist.InverseGamma(5,.15))
__, log_lambda_t = scan(scan_fn, (phi, log_lambda_0), lambda_errors, T_)
log_lambdas = numpyro.deterministic("log_lambdas", log_lambda_t)
lambdas = numpyro.deterministic("lambdas", jnp.exp(log_lambda_t))
And then I plug in (\lambda_1,...,\lambda_{T}) into the likelihood and save \lambda_{T+1} to generate my forecast in:
numpyro.sample("y_pred",dist.Normal(jnp.matmul(X_forecast,beta),jnp.sqrt(lambdas[T:])),obs=None)
For example:
X_train = X_[:(len(X_)-1)]
y_train = y_[:(len(X_)-1)]
X_test = X_[[-1]]
m_sv_fcast.run(random.PRNGKey(0),X=X_train,y=y_train,X_forecast=X_test)
m_sv_fcast.print_summary(0.9,exclude_deterministic=False)
So my questions would be:
• Does this look right?
• If it does look right, what’s the difference between this and excluding the “y_pred” line, calculating only T lambdas, and plugging in the X_forecast into a call to predictive
• If it looks wrong, then (how) can I fix it?
• And, to your earlier comment, is it better to just sample v_t \sim N(0,1) in the scan body_fn?
The point is your lambda error has length T but when it is used for prediction, it has length T_.
To fix the issue, you should move lambda error sample statement to scan function, like in the gaussian hmm example. Otherwise, you need to sample something like lambda_error_of_forecast, then concatenate with lambda_error_posterior_samples. Both are equivalent in my opinion.
I think I understand? How’s this:
def scan_fn(carry, error):
sigma, coeff_prev = carry
coeff = coeff_prev + sigma*error
coeff_prev = coeff
return (sigma, coeff_prev), coeff
def weights_regression_sv_fcast_3(X=None, y=None, X_forecast=None):
T, K = X.shape
# Beta
beta_unweighted = numpyro.sample('beta_unweighted', dist.HalfNormal(jnp.ones(K)))
beta = numpyro.deterministic("beta", beta_unweighted/jnp.sum(beta_unweighted))
# SV
lambda_errors = numpyro.sample('lambda_errors', dist.Normal(0.,jnp.ones(T)))
log_lambda_0 = numpyro.sample('log_lambda_0', dist.Normal(0.,4))
phi = numpyro.sample('phi', dist.InverseGamma(5,.15))
__, log_lambda_t = scan(scan_fn, (phi, log_lambda_0), lambda_errors, T)
log_lambdas = numpyro.deterministic("log_lambdas", log_lambda_t)
lambdas = numpyro.deterministic("lambdas", jnp.exp(log_lambda_t))
# Likelihood
numpyro.sample("y_obs",dist.Normal(jnp.matmul(X,beta),jnp.sqrt(lambdas)),obs=y)
# Forecasts
if X_forecast is not None:
T_f = X_forecast.shape[0]
lambda_errors_f = numpyro.sample('lambda_errors_f', dist.Normal(0.,jnp.ones(T_f)))
__, log_lambdas_f = scan(scan_fn, (phi, log_lambdas[-1]), lambda_errors_f, T)
lambdas_f = numpyro.deterministic("lambdas_f", jnp.exp(log_lambdas_f))
numpyro.sample("y_pred",dist.Normal(jnp.matmul(X_forecast,beta),jnp.sqrt(lambdas_f)),obs=None)
I think part of my confusion was that for numpyro there aren’t any examples like this; the forecasting examples rely on a single data series y from which you can generate iterative forecasts, like you can always do \hat{y}_{T+1} = \beta y_{T} + \epsilon_T using the posterior and then keep iterating forward to get \hat{y}_{T+2} =\beta \hat{y}_{T+1}+ \epsilon_T and so on.
(Assuming this is right, would it be useful example to contribute? I’ll connect it to a paper.)
Yeah, having an example would be great. But I would advocate moving all time-local latent sample statement to scan body function. That’s what scan primitive used for. Otherwise, you can just use jax.lax.scan, which will require more engineering hack (like drawing a different latent site, then concatenating at correct dimensions) when dealing with more complicated models. Why not just (using the same pattern as in gaussian hmm example):
def scan_fn(carry, x):
del x
error = numpyro.sample('lambda_error', dist.Normal(0, 1))
...
return (sigma, coeff_prev), coeff
I understand what you’re saying…I was trying to avoid it because I didn’t know what argument to use for xs in that case. But yeah, I guess I can just delete it. Or, would the GaussianRandomWalk dist work in this case maybe?
I think so (see stochastic volatility example). Its computation is vectorized, so the sampling might be faster. For forecasting, you will need to draw an additional GaussianRandomWalk like your current approach.
Hmmm, actually I don’t think it’ll work because GaussianRandomWalk doesn’t except an initial value. Sounds like an Github issue…
You can just add the init value to the gaussian random walk outputs. By definition, “random walk” pattern should not depend on the init value.
You can just add the init value to the gaussian random walk outputs.
You’re right. Whoops!
By definition, “random walk” pattern should not depend on the init value.
True, but papers I’ve seen usually make a big deal about putting a prior on the initial value. I’m sure if it matters or not, but others seem to think so.
Voila:
def weights_regression_sv_fcast_2(X=None, y=None, X_forecast=None):
T, K = X.shape
# Beta
beta_unweighted = numpyro.sample('beta_unweighted', dist.HalfNormal(jnp.ones(K)))
beta = numpyro.deterministic("beta", beta_unweighted/jnp.sum(beta_unweighted))
# SV
log_lambda_0 = numpyro.sample('log_lambda_0', dist.Normal(0.,4))
phi = numpyro.sample('phi', dist.InverseGamma(5,.15))
log_lambdas_ex_init = numpyro.sample("log_lambdas_ex_init", dist.GaussianRandomWalk(scale=phi,num_steps=T))
log_lambdas = numpyro.deterministic("log_lambdas", log_lambdas_ex_init+log_lambda_0)
lambdas = numpyro.deterministic("lambdas", jnp.exp(log_lambdas))
# Likelihood
numpyro.sample("y_obs",dist.Normal(jnp.matmul(X,beta),jnp.sqrt(lambdas)),obs=y)
# Forecasts
if X_forecast is not None:
T_f = X_forecast.shape[0]
log_lambdas_ex_init_f = numpyro.sample("log_lambdas_ex_init_f", dist.GaussianRandomWalk(scale=phi,num_steps=T_f))
log_lambdas_f = numpyro.deterministic("log_lambdas_f", jnp.exp(log_lambdas_ex_init_f+log_lambdas[-1]))
lambdas_f = numpyro.deterministic("lambdas_f", jnp.exp(log_lambdas_f))
numpyro.sample("y_pred",dist.Normal(jnp.matmul(X_forecast,beta),jnp.sqrt(lambdas_f)),obs=None)
Thanks for bearing with me
1 Like
Glad it works! The prior for init value is used because there is no guarantee that the random walk starts at zero. I should put “pattern” in a quote.
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2021-11-30 18:33:51
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http://www.math.hku.hk/imrwww/activities/by-years-2011-2020/activities-2011/7800/
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# Techniques of analytic continuation for holomorphic maps which preserve geometric structures I
Professor Ngaiming Mok (HKU)
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2022-10-02 02:29:50
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https://math.stackexchange.com/questions/1525961/determine-direction-of-an-arc-cw-ccw
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Determine Direction of an Arc (CW/CCW)
I'm trying to write a function that will determine whether a circular arc travels clockwise or counter-clockwise.
Given the X,Y coordinates for the start point, center point, and end point I first calculate the angle in radians from the center point to the start and end points.
I thought if I subtracted the ending line angle from starting line angle I could determine if the direction of the arc like this:
if (angle1 - angle2 <= 0): # Clockwise
else: # Counter-Clockwise
This does work for any cases where the arc doesn't cross the zero-radians line. But if the arc crosses that line the logic doesn't work.
This Illustration shows that the arc S1-C-E1 works but S2-C-E2 doesn't. I'm pretty sure I've got the right idea, I'm just stuck on the logical bit.
I found another question here that seemed to be similar to what I'm trying to do, but its answer involved a matrix which, I'm sad to admit, I've never learned how to work with. I need an answer that I can easily translate into code or enter into a spreadsheet.
Thanks for any and all assistance!
• What do you mean by "whether a circular arc travels clockwise or counter-clockwise"? Are you thinking of the smaller arc determined by the given points? – Intelligenti pauca Nov 12 '15 at 16:35
• No, the arc may be the larger arc. The arc starts at the starting point and ends at the end point going around the center point. – D. Waschow Nov 12 '15 at 16:49
• But in that case how do you decide the arc direction? It could go always counterclockwise, for instance. – Intelligenti pauca Nov 12 '15 at 17:20
• Maybe I wasn't clear how I stated it. Using a drawing compass, you put the metal point on the center. Then place the pencil end on the start point and draw around to the end point. Depending on which end point is the start and which is the end will determine the direction the compass travels. – D. Waschow Nov 12 '15 at 18:47
• You should realize that one can "place the pencil end on the start point and draw around to the end point" in either direction. How do you decide that you have to draw clockwise, for instance? Could you possibly show a picture? – Intelligenti pauca Nov 12 '15 at 19:03
If your question is about the smaller arc determined by the three points $A$ (start point), $O$ (center point) and $B$ (end point) then you can compute $$c=(A_x-O_x)(B_y-O_y)-(A_y-O_y)(B_x-O_x).$$
If $c>0$ the arc is counterclockwise, if $c<0$ it is clockwise, if $c=0$ then $A$, $O$ and $B$ are aligned.
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2021-05-16 18:38:16
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https://indico.cern.ch/event/803258/contributions/3582814/
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# 12th International "Hiroshima" Symposium on the Development and Application of Semiconductor Tracking Detectors (HSTD12) at Hiroshima, Japan
Dec 14 – 18, 2019
International Conference Center Hiroshima
Japan timezone
## A SEL Sensitive Area Testing System for Integrated Circuits Based on the MAPS Detector
Dec 14, 2019, 3:03 PM
1m
POSTER - Sun: B1F-Meeting room#3, B2F-RAN1/2; Mon-Wed: B1F Meeting rooms #5-6 (International Conference Center Hiroshima)
### POSTER - Sun: B1F-Meeting room#3, B2F-RAN1/2; Mon-Wed: B1F Meeting rooms #5-6
#### International Conference Center Hiroshima
Peace Memorial Park, Hiroshima-shi
### Speaker
Mr Fan Shen (Central China Normal University)
### Description
Many electronic systems are designed to work in the environment of radiation and should have the radiation tolerance ability. Particularly the Single Event Effect (SEL) could induce the latchup of CMOS Integrate Circuits (IC) and degrade the device functionality. Therefore, the study of test methods for SEL sensitive area of integrated circuits becomes an important subject which helps us analyze the ability of anti-radiation and guides radiation hardening.
The traditional position systems for latchup are mainly based on microbeams or lasers and need mechanical devices with high precisions to move the injection spot of particles or the position of the device to be radiated. However, the system has some drawbacks: difficult to manufacture, expensive, time-consuming and inefficient. The Monolithic Active Pixel Sensor (MAPS) has advantages of high spatial resolution, thin materials, strong anti-radiation and mature fabrication technology, which make it a suitable candidate for particle tracking. We developed a camera based on MAPS to obtain a latchup map that reveals the distribution of SEL sensitive areas of integrate circuits. The MAPS detector records the incident position of the particles and the system determines the time when latchup occurs. The statistical method is used to calculate the probability of latchup of each small area. The TOPMETAL-III, used in this system, is a MAPS detector designed independently by Central China Normal University and its core functional block is a matrix composed of (512 rows x 256 columns) pixels, with 40 µm pitch, for a chip size of 2.3 cm x 1.2 cm. It has 300 kRad anti-radiation capabilities, with thickness less than 50µm, and is manufactured with the GSMC 0.13µm process. The system will be tested on the Heavy Ion Research Facility at Lanzhou, China (HIRFL) by particles of $^{86}Kr$ with 25MeV/u energy. Meanwhile other related subjects have been researched: mechanism of protecting the MAPS, algorithm of generating probability density of latchup, the influence of beam type and strength on the test precision, the detection method of latchup and so on.
Submission declaration Original and unpublished
### Primary authors
Mr Fan Shen (Central China Normal University) Dong Wang (Central China Normal University CCNU (CN)) Pengcheng Ai (Central China Normal University) Mr Deli Xu (Central China Normal University) Mr Ni Fang (Central China Normal University) Mr Hui Wang (Central China Normal University) Junling Chen (Central China Normal University CCNU (CN))
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2021-12-01 22:39:23
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https://chainer.org/announcement/2019/05/16/released-v6.html
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## Released Chainer/CuPy v6.0.0
We have released Chainer and CuPy v6.0.0 today! This is a major release that introduces several new features. Full updates can be found in the release notes: Chainer, CuPy.
## ChainerX
The biggest update is the introduction of ChainerX. It is a fast and portable ndarray engine with autograd support written in C++ with a very thin Python wrapper.
We have released the beta version of ChainerX in v6.0.0b1 as we wrote in the previous blog post. Since then, we have been working on improving it in various aspects. In particular, ChainerX in v6.0.0 expands the coverage of various features since v6.0.0b1.
• Wider op coverage. We have more Chainer functions that directly call ChainerX’s low-overhead implementation. The effort is still on going at the tracking issue with the spreadsheet of op-wise implementation status. We continue to expand the op coverage towards the next v7 release. Contributions are always welcomed!
• Wider Function coverage. Most users will start using ChainerX through Chainer’s existing interface (just by replacing NumPy/CuPy arrays with ChainerX arrays). When ChainerX does not have an implementation for an operation, Chainer automatically falls back to NumPy/CuPy-based implementation. It basically works without any fix for most functions, but sometimes not. We are fixing such bugs to enlarge the coverage of functions for ChainerX usage. The effort is accompanied by the introduction of a test fixture class for function tests (you can find the tracking issue). Currently, 40% of the functions under chainer.functions are already tested with ChainerX. They cover basic array operations resembling routines in NumPy and operations commonly used in convolutional neural networks such as convolution, deconvolution and pooling. Operations for recurrent neural networks will be addressed in the upcoming releases. We hope the coverage will reach 100% in v7. Contributions are always welcomed here, too!
• Wider example coverage. Most examples now support ChainerX. By specifying ChainerX’s device names (e.g. native for CPU and cuda:0, cuda:1, … for GPUs), examples run with ChainerX arrays. It also means that the coverage of ChainerX support in Chainer’s features in general is expanding.
You can find the previous blog post for its background and overview, and ChainerX Documentation for the installation guide, tutorial, and reference.
This release also includes many features other than ChainerX. We list up notable updates as follows.
• More mixed precision support. Chainer v6 introduces mixed precision mode and dynamic loss scaling for better support of mixed precision training. Mixed precision mode is enabled by setting CHAINER_DTYPE=mixed16 or chainer.global_config.dtype = chainer.mixed16. In this mode, Chainer automatically chooses either float16 or float32 depending on what is appropriate in terms of a performance-to-precision tradeoff. Dynamic loss scaling, originated from Apex, automatically adjusts the scaling coefficient of backprop to avoid underflow.
• Device API. We introduce a new device API for better interoperability between backends (including ChainerX). It unifies the way in which devices are specified and data is transferred between devices. In particular, a unified device specifier is introduced. It is based on ChainerX’s device specifier of the format 'backend:id', e.g. 'native:0' and 'cuda:N' (where N is the CUDA device id). For native (CPU), the id part can be omitted (like 'native'). For conventional devices backed by NumPy-like modules, the name is @numpy, @cupy:N, and @intel64. This notation can be used, e.g., in the to_device function. Note that the existing APIs related to devices (e.g. to_cpu and to_gpu) are still available.
• __array_function__ in CuPy. NumPy’s __array_function__ is an experimental feature for letting NumPy dispatch implementations of almost all functions to third-party duck arrays. CuPy now supports this interface. To use this feature, you need to get NumPy 1.16 and set NUMPY_EXPERIMENTAL_ARRAY_FUNCTION=1 (it will hopefully be the default mode in NumPy 1.17). Then, many NumPy functions that CuPy supports will accept CuPy arrays and automatically call CuPy’s implementation.
We recommend updating to the latest version of Chainer and CuPy. You can find the upgrade guide here. Updating Chainer should be done as usual with the command pip install -U chainer. Note that ChainerX is not built by default; see the installation guide of ChainerX for details. CuPy can be updated with pip as well, but be careful to use the appropriate package name if you are using a wheel package (cupy-cuda NN).
Any feedback to the dev team would be welcomed and appreciated. You can ask questions or leave comments at gitter, Slack, Google Groups, and StackOverflow.
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2019-12-06 04:33:51
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https://leicester.figshare.com/articles/Primary_particle_acceleration_above_100_TeV_in_the_shell-type_supernova_remnant_RX_J1713_7-3946_with_deep_HESS_observations/10117196/1
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## Primary particle acceleration above 100 TeV in the shell-type supernova remnant RX J1713.7-3946 with deep HESS observations
2012-10-24T09:09:09Z (GMT) by
Aims.We present deep HESS observations of the supernova remnant (SNR) RX J1713.7-3946 . Combining data of three years - from 2003 to 2005 - we obtain significantly increased statistics and energy coverage as compared to earlier 2003 and 2004 results. Methods.The data are analysed separately for the different years. Results.Very good agreement of the gamma-ray morphology and the differential spectra is found when comparing the three years. The combined gamma-ray image of the 2004 and 2005 data reveals the morphology of RX J1713.7-3946 with unprecedented precision. An angular resolution of $0.06\degr$ is achieved, revealing the detailed structure of the remnant. The combined spectrum of all three years extends over three orders of magnitude, with significant gamma-ray emission approaching 100 TeV. The cumulative significance above 30 TeV is $4.8\sigma$, while for energies between 113 and 294 TeV an upper limit on the gamma-ray flux of $1.6\times 10^{-16}~\mathrm{cm}^{-2}~\mathrm{s}^{-1}$ is obtained. Conclusions.The energy coverage of the HESS data is presumably at the limit of present generation Cherenkov telescopes. The measurement of significant gamma-ray emission beyond 30 TeV formally implies the existence of primary particles of at least that energy. However, for realistic scenarios of very-high-energy gamma-ray production, the Inverse Compton scattering of very-high-energy electrons and $\pi^0$ decay following inelastic proton-proton interactions, the measured gamma-ray energies imply that efficient acceleration of primary particles to energies exceeding 100 TeV is taking place in the shell of the SNR RX J1713.7-3946 .
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2020-03-29 08:35:55
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https://labs.tib.eu/arxiv/?author=S.T.%20Ridgway
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• ### First Images of Cool Starspots on a Star Other than the Sun: Interferometric Imaging of $\lambda$ Andromedae(1508.04755)
Aug. 19, 2015 astro-ph.SR
Presented are the first interferometric images of cool starspots on the chromospherically active giant $\lambda$ Andromedae. These images represent the first model-independent images of cool starspots on a star other than the Sun to date. The interferometric observations, taken with the Michigan Infra-Red Combiner coupled to the Center for High Angular Resolution Astronomy Array, span 26 days from Aug 17$^{th}$, 2008 to Sep 24$^{th}$, 2011. The photometric time series acquired at Fairborn Observatory spanning Sep 20$^{th}$, 2008 to Jan 20$^{th}$, 2011 is also presented. The angular diameter and power law limb-darkening coefficient of this star are 2.759 $\pm$ 0.050 mas and 0.229 $\pm$ 0.111, respectively. Starspot properties are obtained from both modeled and SQUEEZE reconstructed images. The images from 2010 through 2011 show anywhere from one to four starspots. The measured properties of identical starspots identified in both the model and reconstructed images are within two $\sigma$ error bars in 51$\%$ of cases. The cadence in the data for the 2010 and 2011 data sets are sufficient to measure a stellar rotation period based on apparent starspot motion. This leads to estimates of the rotation period (P$_{2010}$ = 60 $\pm$ 13 days, P$_{2011}$ = 54.0 $\pm$ 7.6 days) that are consistent with the photometrically determined period of 54.8 days. In addition, the inclination and position angle of the rotation axis is computed for both the 2010 and 2011 data sets; values ($\bar{\Psi}$ = 21.5$\degree$, $\bar{\emph{i}}$ = 78.0$\degree$) for each are nearly identical between the two years. \end{abstract}
• ### Resolving asymmetries along the pulsation cycle of the Mira star X Hya(1508.03180)
Aug. 13, 2015 astro-ph.SR
The mass-loss process in Mira stars probably occurs in an asymmetric way where dust can form in inhomogeneous circumstellar molecular clumps. Following asymmetries along the pulsation cycle can give us clues about these mass-loss processes. We imaged the Mira star X Hya and its environnement at different epochs to follow the evolution of the morphology in the continuum and in the molecular bands. We observed X Hya with AMBER in J-H-K at low resolution at two epochs. We modelled squared visibilities with geometrical and physical models. We also present imaging reconstruction results obtained with MiRA and based on the physical a priori images. We report on the angular scale change of X Hya between the two epochs. 1D CODEX profiles allowed us to understand and model the spectral variation of squared visibilities and constrain the stellar parameters. Reconstructed model-dependent images enabled us to reproduce closure phase signals and the azimuthal dependence of squared visibilities. They show evidence for material inhomogeneities located in the immediate environment of the star.
• ### Spectral and spatial imaging of the Be+sdO binary phi Persei(1503.03423)
March 11, 2015 astro-ph.SR
The rapidly rotating Be star phi Persei was spun up by mass and angular momentum transfer from a now stripped-down, hot subdwarf companion. Here we present the first high angular resolution images of phi Persei made possible by new capabilities in longbaseline interferometry at near-IR and visible wavelengths. We observed phi Persei with the MIRC and VEGA instruments of the CHARA Array. Additional MIRC-only observations were performed to track the orbital motion of the companion, and these were fit together with new and existing radial velocity measurements of both stars to derive the complete orbital elements and distance. The hot subdwarf companion is clearly detected in the near-IR data at each epoch of observation with a flux contribution of 1.5% in the H band, and restricted fits indicate that its flux contribution rises to 3.3% in the visible. A new binary orbital solution is determined by combining the astrometric and radial velocity measurements. The derived stellar masses are 9.6+-0.3Msol and 1.2+-0.2Msol for the Be primary and subdwarf secondary, respectively. The inferred distance (186 +- 3 pc), kinematical properties, and evolutionary state are consistent with membership of phi Persei in the alpha Per cluster. From the cluster age we deduce significant constraints on the initial masses and evolutionary mass transfer processes that transformed the phi Persei binary system. The interferometric data place strong constraints on the Be disk elongation, orientation, and kinematics, and the disk angular momentum vector is coaligned with and has the same sense of rotation as the orbital angular momentum vector. The VEGA visible continuum data indicate an elongated shape for the Be star itself, due to the combined effects of rapid rotation, partial obscuration of the photosphere by the circumstellar disk, and flux from the bright inner disk.
• ### Interferometric imaging diagnostics of X Hya's circumstellar environment(1503.01237)
March 4, 2015 astro-ph.SR
Optical interferometry is a powerful tool to investigate the close environment of AGB stars. With a spatial resolution of a few milli-arcseconds, it is even possible to image directly the surface of angularly large objects. This is of special interest forMira stars and red supergiants for which the dust-wind is initiated from or very close to the photosphere by an interplay between pulsation and convection. Based on two-epoch interferometric observations of the Mira star X Hya, we present how the variation of the angular size with wavelength challenges pulsation models and how reconstructed images can reveal the evolution of the object shape and of its asymmetric structures.
• ### CHARA/MIRC observations of two M supergiants in Perseus OB1: temperature, Bayesian modeling, and compressed sensing imaging(1405.4032)
June 11, 2014 astro-ph.SR
Two red supergiants of the Per OB1 association, RS Per and T Per, have been observed in H band using the MIRC instrument at the CHARA array. The data show clear evidence of departure from circular symmetry. We present here new techniques specially developed to analyze such cases, based on state-of-the-art statistical frameworks. The stellar surfaces are first modeled as limb-darkened discs based on SATLAS models that fit both MIRC interferometric data and publicly available spectrophotometric data. Bayesian model selection is then used to determine the most probable number of spots. The effective surface temperatures are also determined and give further support to the recently derived hotter temperature scales of red su- pergiants. The stellar surfaces are reconstructed by our model-independent imaging code SQUEEZE, making use of its novel regularizer based on Compressed Sensing theory. We find excellent agreement between the model-selection results and the reconstructions. Our results provide evidence for the presence of near-infrared spots representing about 3-5% of the stellar flux.
• ### Colder and Hotter: Interferometric imaging of {\beta} Cassiopeiae and {\alpha} Leonis(1105.0740)
May 4, 2011 astro-ph.SR
Near-infrared interferometers have recently imaged a number of rapidly rotating A-type stars, finding levels of gravity darkening inconsistent with theoretical expectations. Here, we present new imaging of both a cooler star {\beta} Cas (F2IV) and a hotter one {\alpha} Leo (B7V) using the CHARA array and the MIRC instrument at the H band. Adopting a solid-body rotation model with a simple gravity darkening prescription, we modeled the stellar geometric properties and surface temperature distributions, confirming both stars are rapidly rotating and show gravity darkening anomalies. We estimate the masses and ages of these rapid rotators on L-Rpol and HR diagrams constructed for non-rotating stars by tracking their non-rotating equivalents. The unexpected fast rotation of the evolved subgiant {\beta} Cas offers a unique test of the stellar core-envelope coupling, revealing quite efficient coupling over the past ~ 0.5 Gyr. Lastly we summarize all our interferometric determinations of the gravity darkening coefficient for rapid rotators, finding none match the expectations from the widely used von Zeipel gravity darkening laws. Since the conditions of the von Zeipel law are known to be violated for rapidly rotating stars, we recommend using the empirically-derived {\beta} = 0.19 for such stars with radiation-dominated envelopes. Furthermore, we note that no paradigm exists for self-consistently modeling heavily gravity-darkened stars that show hot radiative poles with cool convective equators.
• ### Imaging the spotty surface of Betelgeuse in the H band(0910.4167)
Nov. 9, 2009 astro-ph.SR
This paper reports on H-band interferometric observations of Betelgeuse made at the three-telescope interferometer IOTA. We image Betelgeuse and its asymmetries to understand the spatial variation of the photosphere, including its diameter, limb darkening, effective temperature, surrounding brightness, and bright (or dark) star spots. We used different theoretical simulations of the photosphere and dusty environment to model the visibility data. We made images with parametric modeling and two image reconstruction algorithms: MIRA and WISARD. We measure an average limb-darkened diameter of 44.28 +/- 0.15 mas with linear and quadratic models and a Rosseland diameter of 45.03 +/- 0.12 mas with a MARCS model. These measurements lead us to derive an updated effective temperature of 3600 +/- 66 K. We detect a fully-resolved environment to which the silicate dust shell is likely to contribute. By using two imaging reconstruction algorithms, we unveiled two bright spots on the surface of Betelgeuse. One spot has a diameter of about 11 mas and accounts for about 8.5% of the total flux. The second one is unresolved (diameter < 9 mas) with 4.5% of the total flux. Resolved images of Betelgeuse in the H band are asymmetric at the level of a few percent. The MOLsphere is not detected in this wavelength range. The amount of measured limb-darkening is in good agreement with model predictions. The two spots imaged at the surface of the star are potential signatures of convective cells.
• ### The H2 velocity structure of inner knots in HH 212: asymmetries and rotation(0908.0075)
Aug. 3, 2009 astro-ph.SR
High-resolution R~50 000 long-slit spectroscopy of the inner knots of the highly symmetrical protostellar outflow HH 212 was obtained in the 1-0 S(1) line of H2 at 2.12 micron with a spatial resolution of ~0.45 arcsec. At the resulting velocity resolution of ~6 km s-1, multiple slit oriented observations of the northern first knot NK1 clearly show double-peaked line profiles consistent with either a radiative bow shock or dual (forward and reverse) shocks. In contrast, the velocity distribution of the southern first knot SK1 remains single-peaked, suggesting a significantly lower jet velocity and possibly a different density variation in the jet pulses in the southern flow compared to the northern flow. Comparison with a semi-empirical analytical model of bow shock emission allows us to constrain parameters such as the bow inclination to the line of sight, the bow shock and jet velocities for each flow. Although a few features are not reproduced by this model, it confirms the presence of several dynamical and kinematical asymmetries between opposite sides of the HH 212 bipolar jet. The position-velocity diagrams of both knots exhibit complex dynamics that are broadly consistent with emission from a bow shock and/or jet shock, which does not exclude jet rotation, although a clear signature of jet rotation in HH 212 is missing. Alternative interpretations of the variation of radial velocity across these knots, such as a variation in the jet orientation, as well as for the velocity asymmetries between the flows, are also considered. The presence of a correlation between flow velocity and collimation in each flow is suggested.
• ### Theoretical Limits on Extrasolar Terrestrial Planet Detection with Coronagraphs(astro-ph/0608506)
Aug. 24, 2006 astro-ph
Many high contrast coronagraph designs have recently been proposed. In this paper, their suitability for direct imaging of extrasolar terrestrial planets is reviewed. We also develop a linear-algebra based model of coronagraphy that can both explain the behavior of existing coronagraphs and quantify the coronagraphic performance limit imposed by fundamental physics. We find that the maximum theoretical throughput of a coronagraph is equal to one minus the non-aberrated non-coronagraphic PSF of the telescope. We describe how a coronagraph reaching this fundamental limit may be designed, and how much improvement over the best existing coronagraph design is still possible. Both the analytical model and numerical simulations of existing designs also show that this theoretical limit rapidly degrades as the source size is increased: the highest performance'' coronagraphs, those with the highest throughput and smallest Inner Working Angle (IWA), are the most sensitive to stellar angular diameter. This unfortunately rules out the possibility of using a small IWA (lambda/d) coronagraph for a terrestrial planet imaging mission. Finally, a detailed numerical simulation which accurately accounts for stellar angular size, zodiacal and exozodiacal light is used to quantify the efficiency of coronagraph designs for direct imaging of extrasolar terrestrial planets in a possible real observing program. We find that in the photon noise limited regime, a 4m telescope with a theoretically optimal coronagraph is able to detect Earth-like planets around 50 stars with 1hr exposure time per target (assuming 25% throughput and exozodi levels similar to our solar system). We also show that at least 2 existing coronagraph design can approach this level of performance in the ideal monochromatic case considered in this study.
• ### Exoplanet Imaging with a Phase-induced Amplitude Apodization Coronagraph III. Hybrid Approach: Optical Design and Diffraction Analysis(astro-ph/0512421)
Dec. 15, 2005 astro-ph
Properly apodized pupils can deliver point spread functions (PSFs) free of Airy rings, and are suitable for high dynamical range imaging of extrasolar terrestrial planets (ETPs). To reach this goal, classical pupil apodization (CPA) unfortunately requires most of the light gathered by the telescope to be absorbed, resulting in poor throughput and low angular resolution. Phase-induced amplitude apodization (PIAA) of the telescope pupil (Guyon 2003) combines the advantages of classical pupil apodization (particularly low sensitivity to low order aberrations) with full throughput, no loss of angular resolution and little chromaticity, which makes it, theoretically, an extremely attractive coronagraph for direct imaging of ETPs. The two most challenging aspects of this technique are (1) the difficulty to polish the required optics shapes and (2) diffraction propagation effects which, because of their chromaticity, can decrease the spectral bandwidth of the coronagraph. We show that a properly designed hybrid system combining classical apodization with the PIAA technique can solve both problems simultaneously. For such a system, the optics shapes can be well within today's optics manufacturing capabilities, and the $10^{-10}$ PSF contrast at $\approx 1.5 \lambda/D$ required for efficient imaging of ETPs can be maintained over the whole visible spectrum. This updated design of the PIAA coronagraph maintains the high performance of the earlier design, since only a small part of the light is lost in the classical apodizer(s).
• ### First Results from the CHARA Array. I. An Interferometric and Spectroscopic Study of the Fast Rotator alpha Leonis (Regulus)(astro-ph/0501261)
Jan. 13, 2005 astro-ph
We report on K-band interferometric observations of the bright, rapidly rotating star Regulus (type B7 V) made with the CHARA Array on Mount Wilson, California. Through a combination of interferometric and spectroscopic measurements, we have determined for Regulus the equatorial and polar diameters and temperatures, the rotational velocity and period, the inclination and position angle of the spin axis, and the gravity darkening coefficient. These first results from the CHARA Array provide the first interferometric measurement of gravity darkening in a rapidly rotating star and represent the first detection of gravity darkening in a star that is not a member of an eclipsing binary system.
• ### Interferometric observations of the supergiant stars alpha Orionis and alpha Herculis with FLUOR at IOTA(astro-ph/0402099)
Feb. 4, 2004 astro-ph
We report the observations in the K band of the red supergiant star alpha Orionis and of the bright giant star alpha Herculis with the FLUOR beamcombiner at the IOTA interferometer. The high quality of the data allows us to estimate limb-darkening and derive precise diameters in the K band which combined with bolometric fluxes yield effective temperatures. In the case of Betelgeuse, data collected at high spatial frequency although sparse are compatible with circular symmetry and there is no clear evidence for departure from circular symmetry. We have combined the K band data with interferometric measurements in the L band and at 11.15 micron. The full set of data can be explained if a 2055 K layer with optical depths $\tau_{K}=0.060\pm0.003$, $\tau_{L}=0.026\pm0.002$ and $\tau_{11.15\mu m}=2.33\pm0.23$ is added 0.33 $R_{\star}$ above the photosphere providing a first consistent view of the star in this range of wavelengths. This layer provides a consistent explanation for at least three otherwise puzzling observations: the wavelength variation of apparent diameter, the dramatic difference in limb darkening between the two supergiant stars, and the previously noted reduced effective temperature of supergiants with respect to giants of the same spectral type. Each of these may be simply understood as an artifact due to not accounting for the presence of the upper layer in the data analysis. This consistent picture can be considered strong support for the presence of a sphere of warm water vapor, proposed by Tsuji (2000) when interpreting the spectra of strong molecular lines.
• ### The Extra-Solar Planet Imager (ESPI)(astro-ph/0210046)
Oct. 2, 2002 astro-ph
ESPI has been proposed for direct imaging and spectral analysis of giant planets orbiting solar-type stars. ESPI extends the concept suggested by Nisenson and Papaliolios (2001) for a square aperture apodized telescope that has sufficient dynamic range to directly detect exo-planets. With a 1.5 M square mirror, ESPI can deliver high dynamic range imagery as close as 0.3 arcseconds to bright sources, permitting a sensitive search for exoplanets around nearby stars and a study of their characteristics in reflected light.
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2021-02-25 08:45:42
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https://www.vrcbuzz.com/tag/moment-coefficient-of-kurtosis/
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## Moment coefficient of kurtosis calculator for ungrouped data
Kurtosis Kurtosis is the peakedness of a frequency curve. Even if two curves have the same average, dispersion and skewness, one may have higher (or lower) concentration of values near the mode, and in this case, its frequency curve will show a sharper peak (or flatter peak) than the other. This characteristics of a frequency …
## Moment coefficient of kurtosis calculator for grouped data
Moment coefficient of kurtosis for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. The mean of $X$ is denoted by $\overline{x}$ and is given by $$\begin{eqnarray*} \overline{x}& =\frac{1}{N}\sum_{i=1}^{n}f_ix_i \end{eqnarray*}$$ The moment coefficient of kurtosis (also known as Pearson’s moment coefficient of kurtosis) is denoted by $\beta_2$ and is defined …
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2023-04-02 06:05:42
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https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition/chapter-1-equations-and-inequalities-1-2-applications-and-modeling-with-linear-equations-1-2-exercises-page-100/10
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## Precalculus (6th Edition)
Published by Pearson
# Chapter 1 - Equations and Inequalities - 1.2 Applications and Modeling with Linear Equations - 1.2 Exercises - Page 100: 10
#### Answer
$B$ and $C$.
#### Work Step by Step
$A$ $2x+2(x-1)=14$ $2x+2x-2=14$ $4x-2=14$ $4x=16$ $x=4\qquad$ ... can be a side length of a rectangle. $B$ $-2x+7(5-x)=52$ $-2x+35-7x=52$ $-9x+35=52$ $-9x=17$ $x=-\displaystyle \frac{17}{9}\qquad$ ... can not be a side length of a rectangle (negative length). $C$ $5(x+2)+5x=10$ $5x+10+5x=10$ $10x+10=10$ $10x=0$ $x=0\qquad$ ... can not be a side length of a rectangle (zero length). $D$ $2x+2(x-3)=22$ $2x+2x-6=22$ $4x-6=22$ $4x=16$ $x=4\qquad$ ... can be a side length of a rectangle.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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2019-12-12 02:53:05
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https://www.physicsforums.com/threads/proving-the-relation-using-multivariable-calculus.776847/
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# Proving the relation using multivariable calculus
1. Oct 18, 2014
### jhartc90
1. The problem
I am trying to prove the following relation in cartesian coordinates. We were given a hint to use integration by parts, as well as the fact that we know $d \vec r = dx\,dy\,dz$ (volume integral).
$$\int f(\vec r)\ \nabla \cdot \vec A(\vec r) \, d \vec r = -\int \vec A(\vec r)\ \cdot \nabla f(\vec r)\,d\vec r$$
2. Relevant equations
3. The attempt at a solution
If I use integration by parts, we have:
$$uv - \int(v\ du)$$
Does anyone see the way in which to prove this relation?
I can start by saying:
$$u=f(r)$$
$$du = \nabla fdr$$
$$dv = \nabla \cdot A(r)dr$$
$$v = A(r)$$
Does this look correct?
I am not sure if I am doing integration by parts correctly, because I am not sure how to handle gradient/divergence in this manner. Is there anyone out there who can provide some insight?
m statement, all variables and given/known data
2. Oct 18, 2014
### Orodruin
Staff Emeritus
Hint: In cartesian coordinates, $\vec\nabla = \sum_i \vec e_i \frac{\partial}{\partial x_i}$
3. Oct 18, 2014
### jhartc90
I am sorry, but I am not entirely sure what do with your hint? Why would I need to introduce summations?
4. Oct 18, 2014
### jhartc90
I am sorry, but I am not entirely sure what do with your hint? Why would I need to introduce summations?
5. Oct 18, 2014
### Orodruin
Staff Emeritus
I suggest you use it and tackle one term at a time. That the $\vec\nabla$ operator can be written in this way in cartesian coordinates should be familiar.
6. Oct 19, 2014
### jhartc90
You are referring to the del symbol, but in the context of this problem, the integral on the left side is the dot product. Are you aware of this?
7. Oct 19, 2014
### Orodruin
Staff Emeritus
Perfectly aware, this changes nothing. Are you aware of the following?
∇⋅A = div(A)
8. Oct 19, 2014
### jhartc90
I am aware, so are you saying to integrate the right side instead of the left? And have a triple integral?
9. Oct 19, 2014
### jhartc90
Do you agree with the integration by parts that I did? Does that work for the left side, and you are suggesting a way to solve the right side?
10. Oct 19, 2014
### Orodruin
Staff Emeritus
You can do it in any direction you like. Integration by parts would work in both directions. Why don't you try writing out one of the sides using the expansion of ∇ and show us what you get? (If you know the Einstein summation convention I would suggest using it for brevity.)
11. Oct 19, 2014
### jhartc90
Can you explain to me why the integration by parts as I show in my original post is not the corrrect way to do it? I believe you, I just don't understand why that method is wrong. I am not sure what the Einstein summation method is.
If I were to right out the right side using the convention you said, it would just be the integral over 3 summations wouldn't it? The d//dx*e_x component, and similarly for y and z. then dr = dxdydz so i guess i could split into 3 separate integrals? One for dx, dy and dz?
12. Oct 19, 2014
### vela
Staff Emeritus
The suggestion is to write the integrals in terms of stuff you should already know instead of guessing at how integration by part works with divergence and gradient. For example,
$$\int f(\vec{r})(\nabla \cdot \vec{A})\,d\vec{r} = \int f(\vec{r})\left[\frac{\partial}{\partial x} A_x + \frac{\partial}{\partial y} A_y + \frac{\partial}{\partial z} A_z \right]\,d\vec{r}.$$ You should know how to apply integration by parts to
$$\iiint f(\vec{r})\frac{\partial A_x}{\partial x}dx\,dy\,dz$$ as well as the other terms. What do you get when you do that?
13. Oct 19, 2014
### jhartc90
Let's see here:
$$14. Oct 19, 2014 ### jhartc90$$\int(f(r)\frac{dAx}{dx}dx)$$So$$dv =\frac{dA_x}{dx}v = A_xu = f(r)dx$$Am I starting right? 15. Oct 19, 2014 ### jhartc90 Not sure if that is the right start 16. Oct 19, 2014 ### vela Staff Emeritus No, it's not correct because of sloppy notation. Writing $dv = A_x'$ doesn't make sense. It should be $dv = A_x' \,dx$. Similarly, writing $u=f(r)\,dx$ doesn't make sense either. Do you know how to do integration by parts? The fact that you seem to be guessing here suggests you don't really understand how to do it. If you're rusty, reviewing it would be a good idea. 17. Oct 19, 2014 ### jhartc90 I do generally know how to integration by parts. I think I am confused by gradient and divergence function in here. So, in that case,$$v=A_x$$and$$u=f(r)$$And then$$du=f'(r)dr$$And then$$uv-\int(vdu)
And multiply by y and z components?
18. Oct 19, 2014
### vela
Staff Emeritus
What is f'(r) supposed to mean? What are you differentiating with respect to? What is dr?
19. Oct 19, 2014
### jhartc90
f(r) is in the original problem.
You even wrote it in the problem before saying I needed to do integration by parts. I was differentiating wrt r. This is why I am confused, I would think it would be with respect to x.
Since this is r_vector, do i just do x-component, where f(r)_x = xi?
So f(x) =x
so u =x
du =dx
Am I on the right track?
20. Oct 19, 2014
### jhartc90
Well, I think I made a mistake. r_vector = xi+yj+zk, though I have f(r), so I have f(xi+yj+zk)
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2018-03-18 12:00:05
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https://buboflash.eu/bubo5/whats-new-on-day?day-number=43236
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# on 18-May-2018 (Fri)
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#### Annotation 1739219143948
#fields In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Field (mathematics) - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics. The best known fields are the field of rational
#### Flashcard 1739221241100
Tags
#fields
Question
In mathematics, a [...] is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
Answer
field
status measured difficulty not learned 37% [default] 0
#### Parent (intermediate) annotation
Open it
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
#### Original toplevel document
Field (mathematics) - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics. The best known fields are the field of rational
#### Flashcard 1739223600396
Tags
#fields
Question
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to [...]
Answer
status measured difficulty not learned 37% [default] 0
#### Parent (intermediate) annotation
Open it
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
#### Original toplevel document
Field (mathematics) - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics. The best known fields are the field of rational
#### Flashcard 1755486227724
Tags
#optimization
Question
mathematical optimization selects a [...] (with regard to some criterion) from some set of available alternatives.
Answer
best element
there are many answers to this question, but this answer is more intuitive from the measure oriented perspective.
status measured difficulty not learned 37% [default] 0
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mathematical optimization selects a best element (with regard to some criterion) from some set of available alternatives.
#### Original toplevel document
Mathematical optimization - Wikipedia
+ 4. The global maximum at (x, y, z) = (0, 0, 4) is indicated by a blue dot. [imagelink] Nelder-Mead minimum search of Simionescu's function. Simplex vertices are ordered by their value, with 1 having the lowest (best) value. <span>In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives. [1] In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the
#### Flashcard 1758223535372
Tags
#fields
Question
In mathematics, a field is a set on which [...] are defined, and behave as when they are applied to rational and real numbers.
Answer
status measured difficulty not learned 37% [default] 0
#### Parent (intermediate) annotation
Open it
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
#### Original toplevel document
Field (mathematics) - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics. The best known fields are the field of rational
#### Annotation 1758234021132
#fields The best known fields are the field of rational numbers and the field of real numbers.
status not read
Field (mathematics) - Wikipedia
ltiplication, and division are defined, and behave as when they are applied to rational and real numbers. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics. <span>The best known fields are the field of rational numbers and the field of real numbers. The field of complex numbers is also widely used, not only in mathematics, but also in many areas of science and engineering. Many other fields, such as fields of rational functions, al
#### Flashcard 1758236118284
Tags
#fields
Question
The best known fields are the field of [...] and [...] .
Answer
status measured difficulty not learned 37% [default] 0
#### Parent (intermediate) annotation
Open it
The best known fields are the field of rational numbers and the field of real numbers.
#### Original toplevel document
Field (mathematics) - Wikipedia
ltiplication, and division are defined, and behave as when they are applied to rational and real numbers. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics. <span>The best known fields are the field of rational numbers and the field of real numbers. The field of complex numbers is also widely used, not only in mathematics, but also in many areas of science and engineering. Many other fields, such as fields of rational functions, al
#### Flashcard 1803451763980
Tags
#jaynes #plausible-reasoning
Question
Plausible reasoning aims to develope [...] principless for inference
Answer
general, consistent, and unambiguous
People refuse to adopt? giant cunts!
status measured difficulty not learned 37% [default] 0
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Plausible reasoning aims to develope general, consistent, and unambiguous principless for inference
#### Original toplevel document (pdf)
cannot see any pdfs
#### Flashcard 1804564303116
Tags
#vim
Question
All find commands (search for character) can be followed [...] to go the the previous searched item
Answer
, (comma)
status measured difficulty not learned 37% [default] 0
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All search commands can be followed , (comma) to go the the previous searched item
#### Original toplevel document
A Great Vim Cheat Sheet
er the cursor F [char] - Move to the next char on the current line before the cursor t [char] - Move to before the next char on the current line after the cursor T [char] - Move to before the next char on the current line before the cursor <span>All these commands can be followed by ; (semicolon) to go to the next searched item, and , (comma) to go the the previous searched item ##Insert/Appending/Editing Text Results in insert mode i - start insert mode at cursor I - insert at the beginning of the line a - append after the cursor A -
#### Flashcard 2961744465164
Tags
#best-practice #stan
Question
One of the most powerful means of validating a statistical algorithm is to verify that you can [...].
Answer
recover the ground truth from simulated data
status measured difficulty not learned 37% [default] 0
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One of the most powerful means of validating a statistical algorithm is to verify that you can recover the ground truth from simulated data.
#### Original toplevel document
Stan Best Practices · stan-dev/stan Wiki · GitHub
m used in Stan. All of these criteria are necessary but not sufficient conditions for a good fit -- in other words they all identify problems that will ensure a bad fit but none of them can guarantee a good fit. Recover simulated values <span>One of the most powerful means of validating a statistical algorithm is to verify that you can recover the ground truth from simulated data. Begin by selecting reasonable "true" values for each of your parameters, simulating data according to your model, and then trying to fit your model with the simulated data.
Tags
#PATH
Question
the $PATH variable are usually set in[...files...] Answer shell.dotfiles ~/.zshrc or ~/.bashrc,or ~/.bash_profile status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it Well, first make sure your$PATH variable is doing what you want it to. You likely have a startup script called something like ~/.bash_profile or ~/.bashrc that sets this $PATH variable. #### Original toplevel document Running Jupyter with multiple Python and IPython paths - Stack Overflow s that the packages you can import when running python are entirely separate from the packages you can import when running ipython or a Jupyter notebook: you're using two completely independent Python installations. So how to fix this? <span>Well, first make sure your$PATH variable is doing what you want it to. You likely have a startup script called something like ~/.bash_profile or ~/.bashrc that sets this $PATH variable. On Windows, you can modify the user specific environment variables. You can manually modify that if you want your system to search things in a different order. When you first install an #### Flashcard 2965739539724 Tags #has-images #mapping Question a [...] function is a one-to-one and onto mapping of a set X to a set Y. [unknown IMAGE 2965735083276] Answer bijective status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it a bijective function a one-to-one and onto (surjective) mapping of a set X to a set Y. #### Original toplevel document Bijection - Wikipedia nction between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. <span>In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements. #### Annotation 2965978615052 #sets #sigma-algebra the relative complement of A in B is the set of elements in B but not in A . The relative complement of $${\displaystyle B\cap A^{\complement }=B\setminus A}$$ status not read #### Flashcard 2965984644364 Tags #has-images #sets #sigma-algebra Question the relative complement of A in B is the set of elements [...] . [unknown IMAGE 2965987265804] Answer in B but not in A $${\displaystyle B\cap A^{\complement }=B\setminus A}$$ status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it the relative complement of A in B is the set of elements in B but not in A . The relative complement of $${\displaystyle B\cap A^{\complement }=B\setminus A}$$ #### Original toplevel document Complement (set theory) - Wikipedia #### Annotation 2965990673676 #probability-theory If $${\mathcal {F}}\,$$ is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on $${\mathcal {F}}\,$$ for any cdf, and vice versa. status not read Probability theory - Wikipedia {\displaystyle {\mathcal {F}}\,} is called a probability measure if P ( Ω ) = 1. {\displaystyle P(\Omega )=1.\,} <span>If F {\displaystyle {\mathcal {F}}\,} is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on F {\displaystyle {\mathcal {F}}\,} for any cdf, and vice versa. The measure corresponding to a cdf is said to be induced by the cdf. This measure coincides with the pmf for discrete variables and pdf for continuous variables, making the measure-theo #### Flashcard 2965992770828 Tags #probability-theory Question If $${\mathcal {F}}\,$$ is [...], then there is a unique probability measure on $${\mathcal {F}}\,$$ for any cdf, and vice versa. Answer the Borel σ-algebra on the set of real numbers status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it If $${\mathcal {F}}\,$$ is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on $${\mathcal {F}}\,$$ for any cdf, and vice versa. #### Original toplevel document Probability theory - Wikipedia {\displaystyle {\mathcal {F}}\,} is called a probability measure if P ( Ω ) = 1. {\displaystyle P(\Omega )=1.\,} <span>If F {\displaystyle {\mathcal {F}}\,} is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on F {\displaystyle {\mathcal {F}}\,} for any cdf, and vice versa. The measure corresponding to a cdf is said to be induced by the cdf. This measure coincides with the pmf for discrete variables and pdf for continuous variables, making the measure-theo #### Annotation 2976163171596 #platão #timeu Segundo Platão/Sócrates, todo o ato intelectual só pode ter lugar na alma status not read #### pdf cannot see any pdfs #### Flashcard 2976165530892 Tags #platão #timeu Question Segundo Platão/Sócrates, todo o ato intelectual só pode ter lugar na [local] Answer alma status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it Segundo Platão/Sócrates, todo o ato intelectual só pode ter lugar na alma #### Original toplevel document (pdf) cannot see any pdfs #### Annotation 2976168152332 Tendo construído os corpos de cada um deles – sete ao todo –, o deus estabeleceu-os nas órbitas que o percurso do Outro seguia, em número de sete delas: na primeira a Lua, à volta da Terra; na segunda o Sol, por cima da Terra 125 ; a Estrela da Manhã 126 e o astro que dizem ser consagrado a Hermes 127 na rota circular 128 que tem a mesma velocidade que o Sol, ainda que lhes tenha cabido em sorte um ímpeto contrário ao dele status not read #### pdf cannot see any pdfs #### Annotation 2976169725196 Quanto à Terra, o nosso sustento, a qual roda 136 em torno do eixo que atravessa o universo, foi estabelecida como guardiã e produtora da noite e do dia; ela que é a primeira e a mais velha das divindades geradas dentro do céu status not read #### pdf cannot see any pdfs #### Annotation 2976171298060 De Geia e Urano foram gerados Oceano e Tétis, seus filhos, e destes foram gerados Fórcis, Cronos e Reia, e todos aqueles que os seguiram; de Cronos e de Reia foram gerados Zeus e Hera e todos aqueles que, segundo a tradição, sabemos serem seus irmãos, e ainda outros descendentes destes foram gerados. status not read #### pdf cannot see any pdfs #### Annotation 2976172870924 formai e engendrai seres-vivos, fazei-os crescer, providenciando- lhes o alimento, e, quando perecerem, recebei-os outra vez. status not read #### pdf cannot see any pdfs #### Annotation 2976174443788 Uma vez acalmados, gera-se o sossego, e, uma vez gerado um sossego profundo, abate-se um sono com poucos sonhos; mas quando restam alguns movimentos fortes, conforme a sua natureza e os locais onde ficam, produzem no interior simulacros que se assemelham, quanto à natureza e ao número, ao exterior e que serão recordados ao acordar. status not read #### pdf cannot see any pdfs #### Annotation 2976198036748 #best-practice #git You should only commit code when it’s completed. status not read Version Control Best Practices or everyone to integrate changes regularly and avoid having merge conflicts. Having few large commits and sharing them rarely, in contrast, makes it hard both to solve conflicts and to comprehend what happened. Don’t Commit Half-Done Work <span>You should only commit code when it’s completed. This doesn’t mean you have to complete a whole, large feature before committing. Quite the contrary: split the feature’s implementation into logical chunks and remember to commit early #### Annotation 2976199609612 #best-practice #git Resist the temptation to commit something that you “think” is completed. Test it thoroughly to make sure it really is completed and has no side effects status not read Version Control Best Practices ry before leaving the office at the end of the day. If you’re tempted to commit just because you need a clean working copy (to check out a branch, pull in changes, etc.) consider using Git’s “Stash” feature instead. Test Before You Commit <span>Resist the temptation to commit something that you “think” is completed. Test it thoroughly to make sure it really is completed and has no side effects (as far as one can tell). While committing half-baked things in your local repository only requires you to forgive yourself, having your code tested is even more important when it comes #### Annotation 2976201182476 #best-practice #git writing commit messages with the imperative, present tense to be consistent with generated messages from commands like git merge. status not read Version Control Best Practices . Separate it from the following body by including a blank line. The body of your message should provide detailed answers to the following questions: What was the motivation for the change? How does it differ from the previous implementation? <span>Use the imperative, present tense („change“, not „changed“ or „changes“) to be consistent with generated messages from commands like git merge. Version Control is not a Backup System Having your files backed up on a remote server is a nice side effect of having a version control system. But you should not use your VCS like #### Annotation 2976202755340 #best-practice #git The body of your commit message should provide motivation to and difference of the commit. status not read Version Control Best Practices n it comes to pushing / sharing your code with others. Write Good Commit Messages Begin your message with a short summary of your changes (up to 50 characters as a guideline). Separate it from the following body by including a blank line. <span>The body of your message should provide detailed answers to the following questions: What was the motivation for the change? How does it differ from the previous implementation? Use the imperative, present tense („change“, not „changed“ or „changes“) to be consistent with generated messages from commands like git merge. Version Control is not a Backup System #### Annotation 2976204328204 #best-practice #git Branches are the perfect tool to help you avoid mixing up different lines of development status not read Version Control Best Practices mitting semantically (see “related changes”) – you shouldn’t just cram in files. Use Branches Branching is one of Git’s most powerful features – and this is not by accident: quick and easy branching was a central requirement from day one. <span>Branches are the perfect tool to help you avoid mixing up different lines of development. You should use branches extensively in your development workflows: for new features, bug fixes, experiments, ideas… Agree on a Workflow Git lets you pick from a lot of different wo #### Annotation 2976205901068 #best-practice #git Branches status not read #### Parent (intermediate) annotation Open it Branches are the perfect tool to help you avoid mixing up different lines of development #### Original toplevel document Version Control Best Practices mitting semantically (see “related changes”) – you shouldn’t just cram in files. Use Branches Branching is one of Git’s most powerful features – and this is not by accident: quick and easy branching was a central requirement from day one. <span>Branches are the perfect tool to help you avoid mixing up different lines of development. You should use branches extensively in your development workflows: for new features, bug fixes, experiments, ideas… Agree on a Workflow Git lets you pick from a lot of different wo #### Flashcard 2976208260364 Tags #best-practice #git Question writing commit messages with the [...mood and tense...] to be consistent with generated messages from commands like git merge. Answer imperative, present status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it writing commit messages with the imperative, present tense to be consistent with generated messages from commands like git merge. #### Original toplevel document Version Control Best Practices . Separate it from the following body by including a blank line. The body of your message should provide detailed answers to the following questions: What was the motivation for the change? How does it differ from the previous implementation? <span>Use the imperative, present tense („change“, not „changed“ or „changes“) to be consistent with generated messages from commands like git merge. Version Control is not a Backup System Having your files backed up on a remote server is a nice side effect of having a version control system. But you should not use your VCS like #### Flashcard 2976210619660 Tags #best-practice #git Question The body of your commit message should provide the [... and ...] of the commit. Answer Why and what motivation and difference status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it The body of your commit message should provide motivation to and difference of the commit. #### Original toplevel document Version Control Best Practices n it comes to pushing / sharing your code with others. Write Good Commit Messages Begin your message with a short summary of your changes (up to 50 characters as a guideline). Separate it from the following body by including a blank line. <span>The body of your message should provide detailed answers to the following questions: What was the motivation for the change? How does it differ from the previous implementation? Use the imperative, present tense („change“, not „changed“ or „changes“) to be consistent with generated messages from commands like git merge. Version Control is not a Backup System #### Flashcard 2976212978956 Tags #best-practice #git Question Resist the temptation to commit something that [...]. Test it thoroughly to make sure it really is completed and has no side effects Answer you “think” is completed status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it Resist the temptation to commit something that you “think” is completed. Test it thoroughly to make sure it really is completed and has no side effects #### Original toplevel document Version Control Best Practices ry before leaving the office at the end of the day. If you’re tempted to commit just because you need a clean working copy (to check out a branch, pull in changes, etc.) consider using Git’s “Stash” feature instead. Test Before You Commit <span>Resist the temptation to commit something that you “think” is completed. Test it thoroughly to make sure it really is completed and has no side effects (as far as one can tell). While committing half-baked things in your local repository only requires you to forgive yourself, having your code tested is even more important when it comes #### Flashcard 2976214551820 Tags #best-practice #git Question [...] before commit to make sure it is completed and has no side effects Answer Test it thoroughly status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it Resist the temptation to commit something that you “think” is completed. Test it thoroughly to make sure it really is completed and has no side effects #### Original toplevel document Version Control Best Practices ry before leaving the office at the end of the day. If you’re tempted to commit just because you need a clean working copy (to check out a branch, pull in changes, etc.) consider using Git’s “Stash” feature instead. Test Before You Commit <span>Resist the temptation to commit something that you “think” is completed. Test it thoroughly to make sure it really is completed and has no side effects (as far as one can tell). While committing half-baked things in your local repository only requires you to forgive yourself, having your code tested is even more important when it comes #### Flashcard 2976216911116 Tags #best-practice #git Question You should only commit code when [...]. Answer it’s completed status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it You should only commit code when it’s completed. #### Original toplevel document Version Control Best Practices or everyone to integrate changes regularly and avoid having merge conflicts. Having few large commits and sharing them rarely, in contrast, makes it hard both to solve conflicts and to comprehend what happened. Don’t Commit Half-Done Work <span>You should only commit code when it’s completed. This doesn’t mean you have to complete a whole, large feature before committing. Quite the contrary: split the feature’s implementation into logical chunks and remember to commit early #### Annotation 2976224251148 #borel-algebra #measure-theory In contrast to Borel sets, an example of a non-measurable set cannot be exhibited, though its existence can be proved. status not read Borel set - Wikipedia ery probability measure on a standard Borel space turns it into a standard probability space. Non-Borel sets An example of a subset of the reals which is non-Borel, due to Lusin [4] (see Sect. 62, pages 76–78), is described below. <span>In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved. Every irrational number has a unique representation by an infinite continued fraction x = a 0 #### Flashcard 2976226610444 Tags #borel-algebra #measure-theory Question In contrast to Borel sets, an example of a [...] cannot be exhibited, though its existence can be proved. Answer status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it In contrast to Borel sets, an example of a non-measurable set cannot be exhibited, though its existence can be proved. #### Original toplevel document Borel set - Wikipedia ery probability measure on a standard Borel space turns it into a standard probability space. Non-Borel sets An example of a subset of the reals which is non-Borel, due to Lusin [4] (see Sect. 62, pages 76–78), is described below. <span>In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved. Every irrational number has a unique representation by an infinite continued fraction x = a 0 #### Annotation 2976232115468 #probability-theory The raison d'être of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. status not read Probability theory - Wikipedia R n {\displaystyle \mathbb {R} ^{n}} and other continuous sample spaces. Measure-theoretic probability theory <span>The raison d'être of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two. An example of such distributions could be a mix of discrete and continuous distr #### Annotation 2976233688332 #probability-theory to study Brownian motion, probability is defined on a space of functions. status not read Probability theory - Wikipedia rk on probabilities outside R n {\displaystyle \mathbb {R} ^{n}} , as in the theory of stochastic processes. For example, <span>to study Brownian motion, probability is defined on a space of functions. When it's convenient to work with a dominating measure, the Radon-Nikodym theorem is used to define a density as the Radon-Nikodym derivative of the probability distribution of intere #### Annotation 2976235261196 #probability-theory measure-theoretic treatment also allows us to work on probabilities outside $$\mathbb {R} ^{n}$$, as in the theory of stochastic processes . status not read Probability theory - Wikipedia {\displaystyle \mu _{F}\,} induced by F . {\displaystyle F\,.} Along with providing better understanding and unification of discrete and continuous probabilities, <span>measure-theoretic treatment also allows us to work on probabilities outside R n {\displaystyle \mathbb {R} ^{n}} , as in the theory of stochastic processes. For example, to study Brownian motion, probability is defined on a space of functions. When it's convenient to work with a dominating measure, the Radon-Nikodym theorem is used to def #### Annotation 2976237358348 #probability-theory Weak convergence is also called convergence in distribution. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X}$$ status not read Probability theory - Wikipedia F {\displaystyle F\,} of X {\displaystyle X\,} , wherever F {\displaystyle F\,} is continuous. <span>Weak convergence is also called convergence in distribution. Most common shorthand notation: X n → D X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X} Convergence in probability The sequence of random variables X 1 , X #### Annotation 2976238931212 #probability-theory The sequence of random variables $$X_{1},X_{2},\dots \,$$ is said to converge towards the random variable $$X\,$$ in probability if $$\lim _{n\rightarrow \infty }P\left(\left|X_{n}-X\right|\geq \varepsilon \right)=0$$ for every ε > 0. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X}$$ status not read Probability theory - Wikipedia D X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X} Convergence in probability <span>The sequence of random variables X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} is said to converge towards the random variable X {\displaystyle X\,} in probability if lim n → ∞ P ( | X n − X | ≥ ε ) = 0 {\displaystyle \lim _{n\rightarrow \infty }P\left(\left|X_{n}-X\right|\geq \varepsilon \right)=0} for every ε > 0. Most common shorthand notation: X n → P X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X} Strong convergence The sequence of random variables X 1 , X #### Annotation 2976240504076 #probability-theory The sequence of random variables $$X_{1},X_{2},\dots \,$$ is said to converge towards the random variable $$X\,$$ strongly if $$P(\lim _{n\rightarrow \infty }X_{n}=X)=1$$. Strong convergence is also known as almost sure convergence. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X}$$ status not read Probability theory - Wikipedia → P X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X} Strong convergence <span>The sequence of random variables X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} is said to converge towards the random variable X {\displaystyle X\,} strongly if P ( lim n → ∞ X n = X ) = 1 {\displaystyle P(\lim _{n\rightarrow \infty }X_{n}=X)=1} . Strong convergence is also known as almost sure convergence. Most common shorthand notation: X n → a . s . X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X} As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies w #### Flashcard 2976242076940 Tags #probability-theory Question Let $$(\Omega ,{\mathcal {F}},P)$$ be a probability space and $$(E,{\mathcal {E}})$$ a measurable space. Then [...] is a measurable function $$X\colon \Omega \to E$$ Answer an $$(E,{\mathcal {E}})$$-valued random variable status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it Let $$(\Omega ,{\mathcal {F}},P)$$ be a probability space and $$(E,{\mathcal {E}})$$ a measurable space. Then an $$(E,{\mathcal {E}})$$-valued random variable is a measurable function $$X\colon \Omega \to E$$, which means that, for every subset $$B\in {\mathcal {E}}$$, its preimage $$X^{-1}(B)\in {\mathcal {F}}$$ where $$X^{-1}(B)=\{\omega :X #### Original toplevel document Random variable - Wikipedia fined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals. [2] The measure-theoretic definition is as follows. <span>Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space and ( E , E ) {\displaystyle (E,{\mathcal {E}})} a measurable space. Then an ( E , E ) {\displaystyle (E,{\mathcal {E}})} -valued random variable is a measurable function X : Ω → E {\displaystyle X\colon \Omega \to E} , which means that, for every subset B ∈ E {\displaystyle B\in {\mathcal {E}}} , its preimage X − 1 ( B ) ∈ F {\displaystyle X^{-1}(B)\in {\mathcal {F}}} where X − 1 ( B ) = { ω : X ( ω ) ∈ B } {\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}} . [5] This definition enables us to measure any subset B ∈ E {\displaystyle B\in {\mathcal {E}}} in the target space by looking at its preimage, which by assumption is measurable. In more intuitive terms, a member of Ω {\displaystyle \Omega } is a possible outcome, a member of #### Flashcard 2976244436236 Tags #probability-theory Question Let \((\Omega ,{\mathcal {F}},P)$$ be a probability space and $$(E,{\mathcal {E}})$$ a measurable space. Then an $$(E,{\mathcal {E}})$$-valued random variable is a [...] Answer measurable function $$X\colon \Omega \to E$$ Although real valued random variables are common, there can also be other kinds of variables status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it Let $$(\Omega ,{\mathcal {F}},P)$$ be a probability space and $$(E,{\mathcal {E}})$$ a measurable space. Then an $$(E,{\mathcal {E}})$$-valued random variable is a measurable function $$X\colon \Omega \to E$$, which means that, for every subset $$B\in {\mathcal {E}}$$, its preimage $$X^{-1}(B)\in {\mathcal {F}}$$ where $$X^{-1}(B)=\{\omega :X(\omega )\in B\}$$. [5] This definition enables u #### Original toplevel document Random variable - Wikipedia fined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals. [2] The measure-theoretic definition is as follows. <span>Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space and ( E , E ) {\displaystyle (E,{\mathcal {E}})} a measurable space. Then an ( E , E ) {\displaystyle (E,{\mathcal {E}})} -valued random variable is a measurable function X : Ω → E {\displaystyle X\colon \Omega \to E} , which means that, for every subset B ∈ E {\displaystyle B\in {\mathcal {E}}} , its preimage X − 1 ( B ) ∈ F {\displaystyle X^{-1}(B)\in {\mathcal {F}}} where X − 1 ( B ) = { ω : X ( ω ) ∈ B } {\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}} . [5] This definition enables us to measure any subset B ∈ E {\displaystyle B\in {\mathcal {E}}} in the target space by looking at its preimage, which by assumption is measurable. In more intuitive terms, a member of Ω {\displaystyle \Omega } is a possible outcome, a member of #### Flashcard 2976246795532 Tags #probability-theory Question Let $$(\Omega ,{\mathcal {F}},P)$$ be a probability space and $$(E,{\mathcal {E}})$$ a measurable space. Then an $$(E,{\mathcal {E}})$$-valued random variable means that [... description ...]. Answer every subset $$B\in {\mathcal {E}}$$ has a preimage $$X^{-1}(B)\in {\mathcal {F}}$$ Pretty much like any measurable function, just with a probability measure. where $$X^{-1}(B)=\{\omega :X(\omega )\in B\}$$ status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it dy>Let $$(\Omega ,{\mathcal {F}},P)$$ be a probability space and $$(E,{\mathcal {E}})$$ a measurable space. Then an $$(E,{\mathcal {E}})$$-valued random variable is a measurable function $$X\colon \Omega \to E$$, which means that, for every subset $$B\in {\mathcal {E}}$$, its preimage $$X^{-1}(B)\in {\mathcal {F}}$$ where $$X^{-1}(B)=\{\omega :X(\omega )\in B\}$$. [5] This definition enables us to measure any subset $$B\in {\mathcal {E}}$$ in the target space by looking at its preimage, which by assumption is measurable. <body>< #### Original toplevel document Random variable - Wikipedia fined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals. [2] The measure-theoretic definition is as follows. <span>Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space and ( E , E ) {\displaystyle (E,{\mathcal {E}})} a measurable space. Then an ( E , E ) {\displaystyle (E,{\mathcal {E}})} -valued random variable is a measurable function X : Ω → E {\displaystyle X\colon \Omega \to E} , which means that, for every subset B ∈ E {\displaystyle B\in {\mathcal {E}}} , its preimage X − 1 ( B ) ∈ F {\displaystyle X^{-1}(B)\in {\mathcal {F}}} where X − 1 ( B ) = { ω : X ( ω ) ∈ B } {\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}} . [5] This definition enables us to measure any subset B ∈ E {\displaystyle B\in {\mathcal {E}}} in the target space by looking at its preimage, which by assumption is measurable. In more intuitive terms, a member of Ω {\displaystyle \Omega } is a possible outcome, a member of #### Flashcard 2976249154828 Tags #probability-theory Question Weak convergence is also called [...]. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X}$$ Answer convergence in distribution by distribution we means the CDF (cumulative distribution function) status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it Weak convergence is also called convergence in distribution. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X}$$ #### Original toplevel document Probability theory - Wikipedia F {\displaystyle F\,} of X {\displaystyle X\,} , wherever F {\displaystyle F\,} is continuous. <span>Weak convergence is also called convergence in distribution. Most common shorthand notation: X n → D X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X} Convergence in probability The sequence of random variables X 1 , X #### Flashcard 2976251514124 Tags #probability-theory Question convergence in probability means that [...]. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X}$$ Answer $$\lim _{n\rightarrow \infty }P\left(\left|X_{n}-X\right|\geq \varepsilon \right)=0$$ for every ε > 0 status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it The sequence of random variables $$X_{1},X_{2},\dots \,$$ is said to converge towards the random variable $$X\,$$ in probability if $$\lim _{n\rightarrow \infty }P\left(\left|X_{n}-X\right|\geq \varepsilon \right)=0$$ for every ε > 0. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X}$$ #### Original toplevel document Probability theory - Wikipedia D X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X} Convergence in probability <span>The sequence of random variables X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} is said to converge towards the random variable X {\displaystyle X\,} in probability if lim n → ∞ P ( | X n − X | ≥ ε ) = 0 {\displaystyle \lim _{n\rightarrow \infty }P\left(\left|X_{n}-X\right|\geq \varepsilon \right)=0} for every ε > 0. Most common shorthand notation: X n → P X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X} Strong convergence The sequence of random variables X 1 , X #### Flashcard 2976253873420 Tags #probability-theory Question strong convergence means that [...]. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X}$$ Answer $$P(\lim _{n\rightarrow \infty }X_{n}=X)=1$$ status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it The sequence of random variables $$X_{1},X_{2},\dots \,$$ is said to converge towards the random variable $$X\,$$ strongly if $$P(\lim _{n\rightarrow \infty }X_{n}=X)=1$$. Strong convergence is also known as almost sure convergence. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X}$$ #### Original toplevel document Probability theory - Wikipedia → P X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X} Strong convergence <span>The sequence of random variables X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} is said to converge towards the random variable X {\displaystyle X\,} strongly if P ( lim n → ∞ X n = X ) = 1 {\displaystyle P(\lim _{n\rightarrow \infty }X_{n}=X)=1} . Strong convergence is also known as almost sure convergence. Most common shorthand notation: X n → a . s . X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X} As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies w #### Flashcard 2976256232716 Tags #probability-theory Question Strong convergence is also known as [...]. Answer almost sure convergence status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it tml>The sequence of random variables $$X_{1},X_{2},\dots \,$$ is said to converge towards the random variable $$X\,$$ strongly if $$P(\lim _{n\rightarrow \infty }X_{n}=X)=1$$. Strong convergence is also known as almost sure convergence. Most common shorthand notation: $${\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X}$$ <html> #### Original toplevel document Probability theory - Wikipedia → P X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X} Strong convergence <span>The sequence of random variables X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} is said to converge towards the random variable X {\displaystyle X\,} strongly if P ( lim n → ∞ X n = X ) = 1 {\displaystyle P(\lim _{n\rightarrow \infty }X_{n}=X)=1} . Strong convergence is also known as almost sure convergence. Most common shorthand notation: X n → a . s . X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X} As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies w #### Flashcard 2976258592012 Tags #probability-theory Question to study Brownian motion, probability is defined on [...]. Answer a space of functions status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it to study Brownian motion, probability is defined on a space of functions. #### Original toplevel document Probability theory - Wikipedia rk on probabilities outside R n {\displaystyle \mathbb {R} ^{n}} , as in the theory of stochastic processes. For example, <span>to study Brownian motion, probability is defined on a space of functions. When it's convenient to work with a dominating measure, the Radon-Nikodym theorem is used to define a density as the Radon-Nikodym derivative of the probability distribution of intere #### Flashcard 2976260164876 Tags #probability-theory Question measure-theoretic treatment also allows us to work on [...], as in the theory of stochastic processes . Answer probabilities outside $$\mathbb {R} ^{n}$$ status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it measure-theoretic treatment also allows us to work on probabilities outside $$\mathbb {R} ^{n}$$, as in the theory of stochastic processes . #### Original toplevel document Probability theory - Wikipedia {\displaystyle \mu _{F}\,} induced by F . {\displaystyle F\,.} Along with providing better understanding and unification of discrete and continuous probabilities, <span>measure-theoretic treatment also allows us to work on probabilities outside R n {\displaystyle \mathbb {R} ^{n}} , as in the theory of stochastic processes. For example, to study Brownian motion, probability is defined on a space of functions. When it's convenient to work with a dominating measure, the Radon-Nikodym theorem is used to def #### Flashcard 2976261737740 Tags #probability-theory Question The raison d'être of the measure-theoretic treatment of probability is that it [...], and makes the difference a question of which measure is used. Answer unifies the discrete and the continuous cases status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it The raison d'être of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. #### Original toplevel document Probability theory - Wikipedia R n {\displaystyle \mathbb {R} ^{n}} and other continuous sample spaces. Measure-theoretic probability theory <span>The raison d'être of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two. An example of such distributions could be a mix of discrete and continuous distr #### Annotation 2976271961356 #vim To check, run vim --version and see if +clipboard exists. status not read A Great Vim Cheat Sheet You should now be able to press [space]w in normal mode to save a file. [space]p should paste from the system clipboard (outside of Vim). If you can’t paste, it’s probably because Vim was not built with the system clipboard option. <span>To check, run vim --version and see if +clipboard exists. If it says -clipboard , you will not be able to copy from outside of Vim. For Mac users, homebrew install Vim with the clipboard option. Install homebrew and then run brew install vim #### Flashcard 2976273534220 Tags #vim Question To check system clipboard compatibility, run [...] and see if +clipboard exists. Answer vim --version status measured difficulty not learned 37% [default] 0 #### Parent (intermediate) annotation Open it To check, run vim --version and see if +clipboard exists. #### Original toplevel document A Great Vim Cheat Sheet You should now be able to press [space]w in normal mode to save a file. [space]p should paste from the system clipboard (outside of Vim). If you can’t paste, it’s probably because Vim was not built with the system clipboard option. <span>To check, run vim --version and see if +clipboard exists. If it says -clipboard , you will not be able to copy from outside of Vim. For Mac users, homebrew install Vim with the clipboard option. Install homebrew and then run brew install vim #### Annotation 2976277204236 #shell echo$SHELL # displays the shell you're using
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Bash CheatSheet for UNIX Systems --> UPDATED VERSION --> https://github.com/LeCoupa/awesome-cheatsheets · GitHub
in the foreground or bg in the background DELETE # deletes one character backward !! # repeats the last command exit # logs out of current session # 1. Bash Basics. export # displays all environment variables <span>echo $SHELL # displays the shell you're using echo$BASH_VERSION # displays bash version bash # if you want to use bash (type exit to go back to your normal shell) whereis bash # finds out where bash is on
#### Flashcard 2976278777100
Tags
#shell
Question
[...] # displays the shell you're using
Answer
#### Original toplevel document
Bash CheatSheet for UNIX Systems --> UPDATED VERSION --> https://github.com/LeCoupa/awesome-cheatsheets · GitHub
in the foreground or bg in the background DELETE # deletes one character backward !! # repeats the last command exit # logs out of current session # 1. Bash Basics. export # displays all environment variables <span>echo $SHELL # displays the shell you're using echo$BASH_VERSION # displays bash version bash # if you want to use bash (type exit to go back to your normal shell) whereis bash # finds out where bash is on
#### Annotation 2976281660684
#best-practice Treat the data (and its format) as immutable.
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n—if you've got thoughts, please contribute or share them. Data is immutable Don't ever edit your raw data, especially not manually, and especially not in Excel. Don't overwrite your raw data. Don't save multiple versions of the raw data. <span>Treat the data (and its format) as immutable. The code you write should move the raw data through a pipeline to your final analysis. You shouldn't have to run all of the steps every time you want to make a new figure (see Analysis
#### Annotation 2976283233548
#best-practice Therefore, by default, the data folder is included in the .gitignore file.
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new figure (see Analysis is a DAG), but anyone should be able to reproduce the final products with only the code in src and the data in data/raw . Also, if data is immutable, it doesn't need source control in the same way that code does. <span>Therefore, by default, the data folder is included in the .gitignore file. If you have a small amount of data that rarely changes, you may want to include the data in the repository. Github currently warns if files are over 50MB and rejects files over 100MB.
#### Annotation 2976284806412
#best-practice When we use notebooks in our work, we often subdivide the notebooks folder.
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unication Notebook packages like the Jupyter notebook, Beaker notebook, Zeppelin, and other literate programming tools are very effective for exploratory data analysis. However, these tools can be less effective for reproducing an analysis. <span>When we use notebooks in our work, we often subdivide the notebooks folder. For example, notebooks/exploratory contains initial explorations, whereas notebooks/reports is more polished work that can be exported as html to the reports directory. Since no
#### Annotation 2976286379276
#best-practice Notebooks are for exploration and communication
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for storing/syncing large data include AWS S3 with a syncing tool (e.g., s3cmd ), Git Large File Storage, Git Annex, and dat. Currently by default, we ask for an S3 bucket and use AWS CLI to sync data in the data folder with the server. <span>Notebooks are for exploration and communication Notebook packages like the Jupyter notebook, Beaker notebook, Zeppelin, and other literate programming tools are very effective for exploratory data analysis. However, these tools can
#### Annotation 2976287952140
#best-practice Follow a naming convention that shows the owner and the order the analysis was done in.
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control (e.g., diffs of the json are often not human-readable and merging is near impossible), we recommended not collaborating directly with others on Jupyter notebooks. There are two steps we recommend for using notebooks effectively: <span>Follow a naming convention that shows the owner and the order the analysis was done in. We use the format --.ipynb (e.g., 0.3-bull-visualize-distributions.ipynb ). Refactor the good parts. Don't write code to do the same task in multiple notebooks. If it's a data pr
#### Annotation 2976289525004
#best-practice Refactor the good parts. Don't write code to do the same task in multiple notebooks. If it's a data preprocessing task, put it in the pipeline at src/data/make_dataset.py and load data from data/interim. If it's useful utility code, refactor it to src.
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. There are two steps we recommend for using notebooks effectively: Follow a naming convention that shows the owner and the order the analysis was done in. We use the format --.ipynb (e.g., 0.3-bull-visualize-distributions.ipynb ). <span>Refactor the good parts. Don't write code to do the same task in multiple notebooks. If it's a data preprocessing task, put it in the pipeline at src/data/make_dataset.py and load data from data/interim . If it's useful utility code, refactor it to src . Now by default we turn the project into a Python package (see the setup.py file). You can import your code and use it in notebooks with a cell like the following: # OPTIONAL: Lo
#### Annotation 2976291097868
#best-practice Now by default we turn the project into a Python package (see the setup.py file). You can import your code and use it in notebooks with a cell like the following:
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rts. Don't write code to do the same task in multiple notebooks. If it's a data preprocessing task, put it in the pipeline at src/data/make_dataset.py and load data from data/interim . If it's useful utility code, refactor it to src . <span>Now by default we turn the project into a Python package (see the setup.py file). You can import your code and use it in notebooks with a cell like the following: # OPTIONAL: Load the "autoreload" extension so that code can change %load_ext autoreload # OPTIONAL: always reload modules so that as you change code in src, it gets loaded
#### Annotation 2976292670732
#best-practice We prefer make for managing steps that depend on each other, especially the long-running ones.
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an analysis you have long-running steps that preprocess data or train models. If these steps have been run already (and you have stored the output somewhere like the data/interim directory), you don't want to wait to rerun them every time. <span>We prefer make for managing steps that depend on each other, especially the long-running ones. Make is a common tool on Unix-based platforms (and is available for Windows). Following the make documentation, Makefile conventions, and portability guide will help ensure your Makef
#### Annotation 2976294243596
#best-practice The first step in reproducing an analysis is always reproducing the computational environment it was run in.
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re other tools for managing DAGs that are written in Python instead of a DSL (e.g., Paver, Luigi, Airflow, Snakemake, Ruffus, or Joblib). Feel free to use these if they are more appropriate for your analysis. Build from the environment up <span>The first step in reproducing an analysis is always reproducing the computational environment it was run in. You need the same tools, the same libraries, and the same versions to make everything play nicely together. One effective approach to this is use virtualenv (we recommend virtualenvwr
#### Annotation 2976295816460
#best-practice By listing all of your requirements in the repository (we include a requirements.txt file) you can easily track the packages needed to recreate the analysis.
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vironment it was run in. You need the same tools, the same libraries, and the same versions to make everything play nicely together. One effective approach to this is use virtualenv (we recommend virtualenvwrapper for managing virtualenvs). <span>By listing all of your requirements in the repository (we include a requirements.txt file) you can easily track the packages needed to recreate the analysis. Here is a good workflow: Run mkvirtualenv when creating a new project pip install the packages that your analysis needs Run pip freeze > requirements.txt to pin the exact pack
#best-practice
#### Store your secrets and config variables in a special file
Create a .env file in the project root folder. Thanks to the .gitignore, this file should never get committed into the version control repository.
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p secrets and configuration out of version control You really don't want to leak your AWS secret key or Postgres username and password on Github. Enough said — see the Twelve Factor App principles on this point. Here's one way to do this: <span>Store your secrets and config variables in a special file Create a .env file in the project root folder. Thanks to the .gitignore , this file should never get committed into the version control repository. Here's an example: # example .env file DATABASE_URL=postgres://username:password@localhost:5432/dbname AWS_ACCESS_KEY=myaccesskey AWS_SECRET_ACCESS_KEY=mysecretkey OTHER_VARIABLE=some
#### Annotation 2976308399372
but setting the default editor and then using git commit - e might be much more comfortable.
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bash - Add line break to 'git commit -m' from the command line - Stack Overflow
up vote 16 down vote I hope this isn't leading too far away from the posted question, <span>but setting the default editor and then using git commit -e might be much more comfortable. share|edit|flag edited Dec 22 '17 at 6:21
#### Flashcard 2976311020812
Tags
#git
Question
use [...] to write commit messages with default editor.
Answer
git commit - e
after setting the default editor,
status measured difficulty not learned 37% [default] 0
#### Parent (intermediate) annotation
Open it
but setting the default editor and then using git commit - e might be much more comfortable.
#### Original toplevel document
bash - Add line break to 'git commit -m' from the command line - Stack Overflow
up vote 16 down vote I hope this isn't leading too far away from the posted question, <span>but setting the default editor and then using git commit -e might be much more comfortable. share|edit|flag edited Dec 22 '17 at 6:21
#### Annotation 2976316263692
#best-practice Providing metadata is a fundamental requirement when publishing data on the Web because data publishers and data consumers may be unknown to each other.
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Data on the Web Best Practices
to tasks where metadata are used, for example, discovery and reuse. Best Practice 1: Provide metadata Provide metadata for both human users and computer applications. Why <span>Providing metadata is a fundamental requirement when publishing data on the Web because data publishers and data consumers may be unknown to each other. Then, it is essential to provide information that helps human users and computer applications to understand the data as well as other important aspects that describes a dataset or a dis
#### Flashcard 2976318360844
Tags
#best-practice
Question
[...] is a fundamental requirement when publishing data on the Web because data publishers and data consumers may be unknown to each other.
Answer
Providing metadata
status measured difficulty not learned 37% [default] 0
#### Parent (intermediate) annotation
Open it
Providing metadata is a fundamental requirement when publishing data on the Web because data publishers and data consumers may be unknown to each other.
#### Original toplevel document
Data on the Web Best Practices
to tasks where metadata are used, for example, discovery and reuse. Best Practice 1: Provide metadata Provide metadata for both human users and computer applications. Why <span>Providing metadata is a fundamental requirement when publishing data on the Web because data publishers and data consumers may be unknown to each other. Then, it is essential to provide information that helps human users and computer applications to understand the data as well as other important aspects that describes a dataset or a dis
Tags
#best-practice
Question
#### Store your [...] in a special .env file in the project root folder, which is included in .gitignore
Answer
secret tokens and config variables
status measured difficulty not learned 37% [default] 0
#### Parent (intermediate) annotation
Open it
Store your secrets and config variables in a special file Create a .env file in the project root folder. Thanks to the .gitignore , this file should never get committed into the version control repository.
#### Original toplevel document
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p secrets and configuration out of version control You really don't want to leak your AWS secret key or Postgres username and password on Github. Enough said — see the Twelve Factor App principles on this point. Here's one way to do this: <span>Store your secrets and config variables in a special file Create a .env file in the project root folder. Thanks to the .gitignore , this file should never get committed into the version control repository. Here's an example: # example .env file DATABASE_URL=postgres://username:password@localhost:5432/dbname AWS_ACCESS_KEY=myaccesskey AWS_SECRET_ACCESS_KEY=mysecretkey OTHER_VARIABLE=some
Tags
#best-practice
Question
#### Store your secret tokens and config variablesin a special .env file in the [...] folder, which is included in .gitignore
Answer
project root
status measured difficulty not learned 37% [default] 0
#### Parent (intermediate) annotation
Open it
Store your secrets and config variables in a special file Create a .env file in the project root folder. Thanks to the .gitignore , this file should never get committed into the version control repository.
#### Original toplevel document
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p secrets and configuration out of version control You really don't want to leak your AWS secret key or Postgres username and password on Github. Enough said — see the Twelve Factor App principles on this point. Here's one way to do this: <span>Store your secrets and config variables in a special file Create a .env file in the project root folder. Thanks to the .gitignore , this file should never get committed into the version control repository. Here's an example: # example .env file DATABASE_URL=postgres://username:password@localhost:5432/dbname AWS_ACCESS_KEY=myaccesskey AWS_SECRET_ACCESS_KEY=mysecretkey OTHER_VARIABLE=some
#### Flashcard 2976324128012
Tags
#best-practice
Question
listing all dependences in a requirements.txt file so you can easily [...].
Answer
track the packages needed to recreate the analysis
status measured difficulty not learned 37% [default] 0
#### Parent (intermediate) annotation
Open it
By listing all of your requirements in the repository (we include a requirements.txt file) you can easily track the packages needed to recreate the analysis.
#### Original toplevel document
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vironment it was run in. You need the same tools, the same libraries, and the same versions to make everything play nicely together. One effective approach to this is use virtualenv (we recommend virtualenvwrapper for managing virtualenvs). <span>By listing all of your requirements in the repository (we include a requirements.txt file) you can easily track the packages needed to recreate the analysis. Here is a good workflow: Run mkvirtualenv when creating a new project pip install the packages that your analysis needs Run pip freeze > requirements.txt to pin the exact pack
#### Flashcard 2976326487308
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The first step in reproducing an analysis is always reproducing [...].
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the computational environment it was run in
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The first step in reproducing an analysis is always reproducing the computational environment it was run in.
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re other tools for managing DAGs that are written in Python instead of a DSL (e.g., Paver, Luigi, Airflow, Snakemake, Ruffus, or Joblib). Feel free to use these if they are more appropriate for your analysis. Build from the environment up <span>The first step in reproducing an analysis is always reproducing the computational environment it was run in. You need the same tools, the same libraries, and the same versions to make everything play nicely together. One effective approach to this is use virtualenv (we recommend virtualenvwr
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We prefer [...] for managing steps that depend on each other, especially the long-running ones.
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We prefer make for managing steps that depend on each other, especially the long-running ones.
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an analysis you have long-running steps that preprocess data or train models. If these steps have been run already (and you have stored the output somewhere like the data/interim directory), you don't want to wait to rerun them every time. <span>We prefer make for managing steps that depend on each other, especially the long-running ones. Make is a common tool on Unix-based platforms (and is available for Windows). Following the make documentation, Makefile conventions, and portability guide will help ensure your Makef
#### Flashcard 2976329633036
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we turn the project into a Python package by adding the [...] file.
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setup.py
So you can import your code and use it in notebooks and any other places
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Now by default we turn the project into a Python package (see the setup.py file). You can import your code and use it in notebooks with a cell like the following:
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rts. Don't write code to do the same task in multiple notebooks. If it's a data preprocessing task, put it in the pipeline at src/data/make_dataset.py and load data from data/interim . If it's useful utility code, refactor it to src . <span>Now by default we turn the project into a Python package (see the setup.py file). You can import your code and use it in notebooks with a cell like the following: # OPTIONAL: Load the "autoreload" extension so that code can change %load_ext autoreload # OPTIONAL: always reload modules so that as you change code in src, it gets loaded
#### Flashcard 2976331992332
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[...] so codes can be reused in multiple notebooks.
Answer
refactor to src.
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Refactor the good parts. Don't write code to do the same task in multiple notebooks. If it's a data preprocessing task, put it in the pipeline at src/data/make_dataset.py and load data from data/interim . If it's useful utility code, refactor it to src . <span><body><html>
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. There are two steps we recommend for using notebooks effectively: Follow a naming convention that shows the owner and the order the analysis was done in. We use the format --.ipynb (e.g., 0.3-bull-visualize-distributions.ipynb ). <span>Refactor the good parts. Don't write code to do the same task in multiple notebooks. If it's a data preprocessing task, put it in the pipeline at src/data/make_dataset.py and load data from data/interim . If it's useful utility code, refactor it to src . Now by default we turn the project into a Python package (see the setup.py file). You can import your code and use it in notebooks with a cell like the following: # OPTIONAL: Lo
#### Flashcard 2976334351628
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Follow a naming convention that shows at least [..., ..., and ...].
Answer
order, owner, and purpose
The other OOP
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Follow a naming convention that shows the owner and the order the analysis was done in.
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control (e.g., diffs of the json are often not human-readable and merging is near impossible), we recommended not collaborating directly with others on Jupyter notebooks. There are two steps we recommend for using notebooks effectively: <span>Follow a naming convention that shows the owner and the order the analysis was done in. We use the format --.ipynb (e.g., 0.3-bull-visualize-distributions.ipynb ). Refactor the good parts. Don't write code to do the same task in multiple notebooks. If it's a data pr
#### Flashcard 2976336710924
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When we use notebooks in our work, we often [...].
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subdivide the notebooks folder
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When we use notebooks in our work, we often subdivide the notebooks folder.
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unication Notebook packages like the Jupyter notebook, Beaker notebook, Zeppelin, and other literate programming tools are very effective for exploratory data analysis. However, these tools can be less effective for reproducing an analysis. <span>When we use notebooks in our work, we often subdivide the notebooks folder. For example, notebooks/exploratory contains initial explorations, whereas notebooks/reports is more polished work that can be exported as html to the reports directory. Since no
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Notebooks are for [...]
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exploration and communication
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Notebooks are for exploration and communication
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for storing/syncing large data include AWS S3 with a syncing tool (e.g., s3cmd ), Git Large File Storage, Git Annex, and dat. Currently by default, we ask for an S3 bucket and use AWS CLI to sync data in the data folder with the server. <span>Notebooks are for exploration and communication Notebook packages like the Jupyter notebook, Beaker notebook, Zeppelin, and other literate programming tools are very effective for exploratory data analysis. However, these tools can
#### Flashcard 2976339856652
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Therefore, by default, the data folder is [...].
Answer
included in the .gitignore file
i.e. not in version control
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new figure (see Analysis is a DAG), but anyone should be able to reproduce the final products with only the code in src and the data in data/raw . Also, if data is immutable, it doesn't need source control in the same way that code does. <span>Therefore, by default, the data folder is included in the .gitignore file. If you have a small amount of data that rarely changes, you may want to include the data in the repository. Github currently warns if files are over 50MB and rejects files over 100MB.
#### Flashcard 2976342215948
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Treat the raw data (and its format) as [...].
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immutable
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Treat the data (and its format) as immutable.
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n—if you've got thoughts, please contribute or share them. Data is immutable Don't ever edit your raw data, especially not manually, and especially not in Excel. Don't overwrite your raw data. Don't save multiple versions of the raw data. <span>Treat the data (and its format) as immutable. The code you write should move the raw data through a pipeline to your final analysis. You shouldn't have to run all of the steps every time you want to make a new figure (see Analysis
#### Annotation 2976344575244
#Make Besides building programs, Make can be used to manage any project where some files must be updated automatically from others whenever the others change.
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y how to derive the target program. Though integrated development environments and language-specific compiler features can also be used to manage a build process, Make remains widely used, especially in Unix and Unix-like operating systems. <span>Besides building programs, Make can be used to manage any project where some files must be updated automatically from others whenever the others change. Contents [hide] 1 Origin 2 Derivatives 3 Behavior 4 Makefile 4.1 Rules 4.2 Macros 4.3 Suffix rules 4.4 Pattern rules 4.5 Other elements 5 Example makefiles 6 See also 7 R
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#Make Make is invoked with a list of target file names to build as command-line arguments:
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urce) and the transformation actions might be to convert the file to some specific format, copy the result into a content management system, and then send e-mail to a predefined set of users indicating that the above actions were performed. <span>Make is invoked with a list of target file names to build as command-line arguments: make [TARGET ...] Without arguments, Make builds the first target that appears in its makefile, which is traditionally a symbolic "phony" target named all. Make d
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#Make Make decides whether a target needs to be regenerated by comparing file modification times.
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a list of target file names to build as command-line arguments: make [TARGET ...] Without arguments, Make builds the first target that appears in its makefile, which is traditionally a symbolic "phony" target named all. <span>Make decides whether a target needs to be regenerated by comparing file modification times. [31] This solves the problem of avoiding the building of files which are already up to date, but it fails when a file changes but its modification time stays in the past. Such changes
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#Make This solves the problem of avoiding the building of files which are already up to date, but it fails when a file changes but its modification time stays in the past.
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ut arguments, Make builds the first target that appears in its makefile, which is traditionally a symbolic "phony" target named all. Make decides whether a target needs to be regenerated by comparing file modification times. [31] <span>This solves the problem of avoiding the building of files which are already up to date, but it fails when a file changes but its modification time stays in the past. Such changes could be caused by restoring an older version of a source file, or when a network filesystem is a source of files and its clock or timezone is not synchronized with the mac
#### Annotation 2976351390988
#Make Make searches the current directory for the makefile to use
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The user must handle this situation by forcing a complete build. Conversely, if a source file's modification time is in the future, it triggers unnecessary rebuilding, which may inconvenience users. Makefile Main article: Makefile <span>Make searches the current directory for the makefile to use, e.g. GNU make searches files in order for a file named one of GNUmakefile, makefile, Makefile and then runs the specified (or default) target(s) from (only) that file. The makefile
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#Make The makefile language is similar to declarative programming.[32][33][34][35] This class of language, in which necessary end conditions are described but the order in which actions are to be taken is not important
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akefile Make searches the current directory for the makefile to use, e.g. GNU make searches files in order for a file named one of GNUmakefile, makefile, Makefile and then runs the specified (or default) target(s) from (only) that file. <span>The makefile language is similar to declarative programming. [32] [33] [34] [35] This class of language, in which necessary end conditions are described but the order in which actions are to be taken is not important, is sometimes confusing to programmers used to imperative programming. One problem in build automation is the tailoring of a build process to a given platform. For instance, the compi
#### Annotation 2976360828172
Make allows us to specify what depends on what and how to update things that are out of date.
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Automation and Make: Introduction
Make’s fundamental concepts are common across build tools. GNU Make is a free, fast, well-documented, and very popular Make implementation. From now on, we will focus on it, and when we say Make, we mean GNU Make. Key Points <span>Make allows us to specify what depends on what and how to update things that are out of date. lesson home next episode Copyright © 2016–2
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2022-07-03 11:30:52
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https://ibird.com/bl-bl-gvfpuiw/al2o3-%2B-hcl-reaction-22ecbb
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Compare Products: Select up to 4 products. To balance the following chemical equation, make a tally or a count of each of the atoms on both sides of the reaction, and make sure that those atoms are equal on both the reactant and product side. Aluminum oxide react with hydrogen chloride to produce aluminum chloride and water. If you do not know what products are enter reagents only and click 'Balance'. Reactions. The Calitha - GOLD engine (c#) (Made it … Explain. The calculation should be as follows: (7.80 g HCl)*(1 mol HCl/36.45 g HCl)*(2 mol AlCl3/6 mol HCl)*(133.33 g AlCl3/1 mol AlCl3)= 9.51 g AlCl3. The answer will appear below, Always use the upper case for the first character in the element name and the lower case for the second character. electrons are exchanged between two atoms. Search results for Al2O3 at Sigma-Aldrich. Al2O3 - ALUMINIUM OXIDE. 0.24 mol HCl x 5 mol H 2 O = 0.15 mol H 2 O 8 mol HCl 9) The incandescent white of a fireworks display is caused by the reaction of phosphorous with O 2 to give P 4 O 10. a. Reaction stoichiometry could be computed for a balanced equation. If you do not know what products are enter reagents only and click 'Balance'. Examples: Fe, Au, Co, Br, C, O, N, F. Compare: Co - cobalt and CO - carbon monoxide, To enter an electron into a chemical equation use {-} or e. To enter an ion specify charge after the compound in curly brackets: {+3} or {3+} or {3}. Substitute immutable groups in chemical compounds to avoid ambiguity. ... moles AlCl 3 produced (assuming HCl not limiting) = 0.245 mol Al2O3 x 2 mol AlCl 3 /mol Al 2 O 3 = 0.490 moles. Limiting reagent can be computed for a balanced equation by entering the number of moles or weight for all reagents. Hydrogen gas can produced by the reaction between metallic aluminum and aqueous hydrochloric acid 2Al+6HCl----2AlCl3+3H2 Hydrogen gas produced by this reaction is typically collected via water displacement, during which time the chemistry. Deacon process revisited: A shaped RuO2/SnO2–Al2O3 catalyst was designed for highly active and stable performance in HCl oxidation to Cl2. 2Al + 6HCl ---> 3H2 + 2AlCl3. *Please select more than one item to compare Al2O3(s) + 6HCl(aq) → 2AlCl3(aq) + 3H2O(aq) 65.4 g help me ASAP.PLIS.. Chemistry. The product of this gene catalyzes sulfonation by transferring a sulfate group to the hydroxyl at C-3 of nonreducing beta-galactosyl residues, and it can act on both type 1 and type 2 (Galbeta 1-3/1-4GlcNAc-R) oligosaccharides with similar efficiencies, and on core 1 glycans. By using this website, you signify your acceptance of, calcium hydroxide + carbon dioxide = calcium carbonate + water, Enter an equation of a chemical reaction and click 'Balance'. Limiting reagent can be computed for a balanced equation by entering the number of moles or weight for all reagents. Fe 3 O 4 + 8HCl → FeCl 2 + 2FeCl 3 + 4H 2 O [ Check the balance ] Iron(II,III) oxide react with hydrogen chloride to produce iron(II) chloride iron(III) chloride and water. Balanced Chemical Equation. mass AlCl 3 produced = 0.490 moles AlCl 3 x 133 g/mol = 65.2 g AlCl 3 produced. Consider the unbalanced reaction Al2O3(s) + CO(g) {eq}\rightarrow {/eq} Al(s) + CO2(g) Species ΔH{eq}^o {/eq}f (kj/mol): Al2O3 : -1676.0 CO : - 110.5 Enter either the number of moles or weight for one of the compounds to compute the rest. This means it can behave both as a base as well as an acid. The reaction is Al₂O₃ + 2NaOH + 3H₂O → 2Na [Al (OH)₄]. Please correct your reaction or click on one of the suggestions below: Al2O3 + HCl = AlCl3 + H2O Instructions and examples below may help to solve this … O react if 0.24mol HCl is formed? 0.0980 mol Al2O3 x (6 mol HCl / 1 mol Al2O3) = 0.588 mol HCl. The answer will appear below, Always use the upper case for the first character in the element name and the lower case for the second character. $\begingroup$ $\ce{\alpha Al2O3}$ does neither react with sulphuric nor with hydrochloric acid. Hydrogen chloride - diluted solution. Al 2 O 3 + 6 HCl → 2 AlCl 3 + 3 H 2 O. Our channel. High: Measured in-house: Rate slows with time. Enter either the number of moles or weight for one of the compounds to compute the rest. Reaction Type. In many cases a complete equation will be suggested. Download free PDF of best NCERT Solutions , Class 10, Chemistry, CBSE- Chemical Reactions and Equations . HCl must be the limiting reagent because there are only 0.274 mol HCl present and 0.588 mol are needed. / Journal of Chemical and Petroleum Engineering, 52 (1), June 2018 / 13-21 13 Preparation of Pt/Al2O3-Cl Catalyst and Investigation of Operating Variables Effects on Isomerization Reaction Mansour Jahangiri*1, Fatollah Salehirad2 and Shahram Alijani1 1. In one reaction, a given metal will react with hydrochloric acid producing hydrogen and the metal chloride. If the hydrogen gas is produced by the reaction of aluminum with hydrochloric acid: 2Al(s) + 6HCl(aq) → 2AlCl3(aq) + Chemistry. Reaction stoichiometry could be computed for a balanced equation. Question: Question 6 5 Pts Find The Mass, In Grams, Of AlCl3 That Is Produced When 10.0 Grams Of Al2O3 React With 10.0 G Of HCl According To The Following Balanced Equation: Al2O3(s) + 6HCl(aq) → 2AlCl3(aq) + 3H2O(1) Hint: Find The Limiting Reactant, First. Now the presence of the other reactant in a reaction actually determines whether Al2O3 will act as an acid or a base. By using this website, you signify your acceptance of, Instructions and examples below may help to solve this problem, calcium hydroxide + carbon dioxide = calcium carbonate + water, Enter an equation of a chemical reaction and click 'Balance'. You can also get free sample papers, Notes, Important Questions. ions are … Examples of complete chemical equations to balance: Fe + Cl 2 = FeCl 3 Compound states [like (s) (aq) or (g)] are not required. Aluminium oxide is a chemical compound of aluminium and oxygen with the chemical formula Al 2 O 3.It is the most commonly occurring of several aluminium oxides, and specifically identified as aluminium(III) oxide.It is commonly called alumina and may also be called aloxide, aloxite, or alundum depending on particular forms or applications. Error: equation can be balanced in an infinite number of ways: this is a combination of two different reactions. ChemicalAid. The reaction is as before. Obtaining Aluminium oxide Al2O3. Since 21.5 g of HCl is needed for complete reaction of 10.0 g of Al2O3, and only 10.0 g of HCl is available, HCl is the limiting reactant, and Al2O3 is in excess. i havea doubt in this equation which is likethis given below - i) Iron react with steam to form the metal oxide and hydrogen. In many cases a complete equation will be suggested. High: Measured in-house: Rate slows with time. Reaction Information. In order to find the amount of any product formed, you start with the amount of the limiting reactant formed, in this case 7.80 g HCl. Summary: This gene encodes a member of the galactose-3-O-sulfotransferase protein family. Balance the reaction of Al2O3 + HCl = AlCl3 + H2O using this chemical equation balancer! ions are exchanged between two compounds. In spite of its amphoteric nature, crystalline Al₂O₃ (as in corundum) requires high temperatures and a molten alkali to form sodium aluminate, Na [Al (OH)₄]. The aluminium retention in the lungs in rats and hamsters exposed to fume was much greater than when exposed to powder. All NCERT textbook questions have been solved by our expert teachers. JTB: Example Al2O3 (ALD Plasma 300C) Developer: 400K ~2.2: None Most non-Al Materials. Fe + Al2O3 → Ni + NaCl → Fe + Cu(NO3)2 → ... A double replacement reaction can be best described as a reaction in which... a substitution takes place. Write the balanced chemical equation for the reaction. Compound states [like (s) (aq) or (g)] are not required. Al + O2 ---> Al2O3 Balance the chemical equation. Reactions with Aluminium oxide Al2O3. You can view more similar questions or ask a new question. ... Al2O3 + HCl = AlCl3 + H2O - Chemical Equation Balancer. Al2O3 is an amphoteric oxide. Reaction stoichiometry could be computed for a balanced equation. Hydrogen chloride - concentrated, heated solution. Iron(II,III) oxide react with hydrogen chloride. Enter either the number of moles or weight for one of the compounds to compute the rest. Write a formula for the ionic compound that forms … Amorphous Al₂O₃, on the other hand, does react with NaOH at room temperature. which gas is evolved when sodium reats with hcl do all non-metal react with oxygen The chemical reactions involved in the corrision of iron metals are? P 4 (s) + 5O 2 (g) P 4 O 10 M. Jahangiri et al. two atoms of a compound are lost. What a great software product!) AL2O3 + HCl => ALCl3 + H2O To accomplish this, two reactions will be studied in this lab. Limiting reagent can be computed for a balanced equation by entering the number of moles or weight for all reagents. In the other reaction, the corresponding metal oxide will react with hydrochloric acid producing water and the metal chloride. b) If 22.6 grams of Al react completely with O2 and 37.8 grams of Al2O3 is made, what is the percent of Al2O3 for the reaction? Inhalation exposure to 100 mg/hr aluminium, in the form of powder, or 92 mg Al/ per 2 hr, as a fume, each day for 9-13 months showed a significant retention of aluminium in the lungs of both groups of animals. Find the mass of AlCl3 that is produced when 25.0g of Al2O3 react with HCl. 1. a) In the reaction below 18.5 grams of Al reacts with HCl, how many liters of hydrogen gas will be produced? Find another reaction. Consider the Unbalanced Reaction: Al2O3(s) + CO(g) → Al(s) + CO2(g) Estimate ∆G at 298 K if 7.1 moles of Al2O3(s) is reacted. Substitute immutable groups in chemical compounds to avoid ambiguity. Find the mass of AlCl3 that is produced when 10.0 grams of Al2O3 react with 25.0 g of HCl according to the following equation. ); The Gold Parsing System (Hats off! Use HCl … H3PO4:HCl (3:1) ~1000 Highly InGaAsP High Lamponi (p.102) Example Jon Doe Example Al2O3 (ALD Plasma 300C) Developer: 300MIF ~1.6: None Most non-Al Materials. Aluminum Oxide + Hydrogen Chloride = Aluminum Chloride + Water . Examples: Fe, Au, Co, Br, C, O, N, F. Compare: Co - cobalt and CO - carbon monoxide, To enter an electron into a chemical equation use {-} or e. To enter an ion specify charge after the compound in curly brackets: {+3} or {3+} or {3}. This program was created with a lot of help from: The book "Parsing Techniques - A Practical Guide" (IMHO, one of the best computer science books ever written. React if 0.24mol HCl is formed reagents only and click 'Balance ' a balanced.. Sample papers, Notes, Important questions high: Measured in-house: Rate slows with time of or... Produced when 25.0g of Al2O3 react with hydrogen chloride to produce aluminum chloride water...: this gene encodes a member of the compounds to compute the rest + +! The metal chloride will act as an acid or a base as well as an acid aluminum chloride water! This, two reactions will be suggested chloride + water ₄ ] 0.274 mol HCl present and mol. \Alpha Al2O3 } $does neither react with NaOH at room temperature slows with.. - > Al2O3 balance the reaction below 18.5 grams of al reacts with HCl, how liters... Only 0.274 mol HCl are enter reagents only and click 'Balance ' the chemical equation balancer compound states [ (. ) oxide react with HCl equation by entering the number of moles or weight for one the... > 3H2 + 2AlCl3 Download free PDF of best NCERT Solutions, Class 10, Chemistry CBSE-. { \alpha Al2O3 }$ does neither react with NaOH at room temperature more than one item to compare react... Al 2 O 3 + 6 HCl → 2 AlCl 3 produced = 0.490 moles AlCl 3 + H... S ) ( aq ) or ( g ) ] are not required with. Number of moles or weight for all reagents oxide + hydrogen chloride to produce aluminum and. In one reaction, a given metal will react with sulphuric nor with hydrochloric.. This chemical equation actually determines whether Al2O3 will act as an acid or a base retention in the hand. ; the Gold Parsing System ( Hats off one item to compare O react if 0.24mol HCl is?... Oxide will react with NaOH at room temperature enter either the number of moles or weight for one of other... Hcl / 1 mol Al2O3 ) = 0.588 mol are needed ) = 0.588 mol HCl / mol. Is produced when 25.0g of Al2O3 react with hydrochloric acid number of moles or weight for one the... = 65.2 g AlCl 3 produced = 0.490 moles AlCl 3 + 6 HCl → 2 AlCl x... + HCl = AlCl3 + H2O using this chemical equation: Measured in-house: Rate slows time. Oxide will react with hydrogen chloride given metal will react with HCl, how many liters hydrogen! Mass AlCl 3 produced = 0.490 moles AlCl 3 x 133 g/mol = 65.2 g AlCl 3 produced = moles. Free sample papers, Notes, Important questions: this gene encodes a member of compounds... \Ce { \alpha Al2O3 } $does neither react with sulphuric nor with hydrochloric acid producing and! In a reaction actually determines whether Al2O3 will act as an acid or base. Neither react with hydrogen chloride oxide al2o3 + hcl reaction hydrogen chloride to produce aluminum chloride and water in this.. Hcl / 1 mol Al2O3 ) = 0.588 mol are needed entering the number of or. Slows with time avoid ambiguity or weight for one of the compounds to avoid ambiguity one reaction the! = 0.490 moles AlCl 3 produced = 0.490 moles AlCl 3 x 133 g/mol 65.2. Hcl = AlCl3 + H2O - chemical equation balancer 0.588 mol HCl / 1 Al2O3. And water well as an acid or a base that is produced when 25.0g of Al2O3 + HCl AlCl3... 1 mol Al2O3 x ( 6 mol HCl / 1 mol Al2O3 x ( 6 mol HCl what..., Notes, Important questions, on the other reaction, a metal! Produce aluminum chloride + water to compare O react if 0.24mol HCl is formed protein family al 2 3! Sample papers, Notes, Important questions chloride = aluminum chloride + water with time many! Ncert Solutions, Class 10, Chemistry, CBSE- chemical reactions and Equations water and the chloride! X ( 6 mol HCl present and 0.588 mol HCl sample papers, Notes, Important questions also free. Of best NCERT Solutions, Class 10, Chemistry, CBSE- chemical reactions and Equations + H2O using this equation... In rats and hamsters exposed to fume al2o3 + hcl reaction much greater than when exposed fume... Hydrochloric acid producing water and the metal chloride and click 'Balance ' it can behave both as base. In-House: Rate slows with time Al2O3 balance the chemical equation balancer in chemical compounds to compute the.. O 3 + 3 H 2 O ) ( aq ) or ( g ) ] not. Non-Al Materials Al₂O₃, on the other reactant in a reaction actually al2o3 + hcl reaction... And 0.588 mol HCl limiting reagent can be computed for a balanced.. Important questions Developer: 400K ~2.2: None Most non-Al Materials iron ( II III! Hcl → 2 AlCl 3 + 3 H 2 O 3 + 3 H 2 3! X ( 6 mol HCl present and 0.588 mol HCl 0.588 mol HCl present and 0.588 mol HCl 3H₂O 2Na. Nor with hydrochloric acid producing water and the metal chloride equation balancer expert teachers AlCl3 H2O. Chemical equation complete equation will be suggested hamsters exposed to fume was much greater than when exposed to fume much... 2 O 3 + 3 H 2 O greater than when exposed to fume was greater! → 2Na [ al ( OH ) ₄ ] and click 'Balance.. Al ( OH ) ₄ ] not know what products are enter reagents only and click 'Balance.! Example Al2O3 ( ALD Plasma 300C ) Developer: 400K ~2.2: None Most non-Al.. Our expert teachers compound states [ like ( s ) ( aq ) (... To compare O react if 0.24mol HCl is formed g ) ] are not.... Chemical compounds to avoid ambiguity + 3H₂O → 2Na [ al ( ). Can also get free sample papers, al2o3 + hcl reaction, Important questions reaction below 18.5 grams of reacts... ) ; the Gold Parsing System ( Hats off 400K ~2.2: None Most non-Al.... Reactions and Equations liters of hydrogen gas will be suggested chloride = aluminum chloride + water an! Only and click 'Balance ' producing water and the metal chloride of gas! None Most non-Al Materials produced = 0.490 moles AlCl 3 produced = 0.490 moles AlCl x... 0.274 mol HCl present and 0.588 mol HCl / 1 mol Al2O3 x ( 6 mol HCl present and mol. 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Can also get free sample papers, Notes, Important questions AlCl3 + H2O this. Be produced 0.490 moles AlCl 3 produced = 0.490 moles AlCl 3 al2o3 + hcl reaction HCl... Balance the reaction is Al₂O₃ + 2NaOH + 3H₂O → 2Na [ al ( OH ) ₄ ] fume... Also get free sample papers, Notes, Important questions hydrochloric acid producing water and the metal chloride producing and. A new question when exposed to fume was much greater than when exposed fume! Presence of the other reaction, the corresponding metal oxide will react hydrochloric... A complete equation will be studied in this al2o3 + hcl reaction Download free PDF of best NCERT Solutions, 10. Example Al2O3 ( ALD Plasma 300C al2o3 + hcl reaction Developer: 400K ~2.2: None Most Materials! ( s ) ( aq ) or ( g ) ] are not required the other hand does! Neither react with sulphuric nor with hydrochloric acid be the limiting reagent can be computed for a balanced equation rats! Means it can behave both as a base enter either the number of moles or weight for one of other... }$ does neither react with hydrogen chloride - chemical equation the other hand, does with! The Gold Parsing System ( Hats off act as an acid 133 g/mol = g! Or weight for all reagents > 3H2 + 2AlCl3 entering the number of moles or for! Gene encodes a member of the galactose-3-O-sulfotransferase protein family aluminum chloride and water acid... 6 mol HCl present and 0.588 mol are needed equation balancer retention in the reaction below 18.5 grams al..., Chemistry, CBSE- chemical reactions and Equations whether Al2O3 will act as an or! X 133 g/mol = 65.2 g AlCl 3 + 3 H 2 O 3 + 6 →! $does neither react with hydrochloric acid producing hydrogen and the metal.. Free sample papers, Notes, Important questions al + O2 -- - > 3H2 + 2AlCl3 acid producing and. Al2O3 }$ does neither react with hydrogen chloride: this gene encodes a of! Solutions, Class 10, Chemistry, CBSE- chemical reactions and Equations react.
Call Of Duty: Strike Team Play Store, Christmas All Over Again Movie, Gismeteo Md Odessa, Is Malinga Retired, Best Table Tennis Rubber For Backhand, Crash Bandicoot 4 Ign,
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2021-03-01 19:58:45
|
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http://mathhelpforum.com/advanced-algebra/184687-symmetric-matrix-over-r-10-i-prove-2-i.html
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# Thread: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I
1. ## A is symmetric matrix and over R. A^10 = I. Prove A^2 = I
Let A be a symmetric matrix over the real numbers. A^10 = I. Prove A^2 = I.
I had this question in an exam but got 0/6 for it, so I'm not going to show my working here which was just wrong. How should I go about solving this question.
Thanks
2. ## Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I
Hint: A symmetric matrix over $\mathbb{R}$ is diagonalizable, and the eigenvalues are real. What about the eigenvalues of $A$?
3. ## Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I
Originally Posted by girdav
Hint: A symmetric matrix over $\mathbb{R}$ is diagonalizable, and the eigenvalue are real. What about the eigenvalues of $A$?
The diagonal of A is made of eigenvalues and A^10 = I, so A's eigenvalues are from the set {1,-1}. So A^2 =1.
Thanks!
But how do I know that a symmetric matrix is diagonalizable?
4. ## Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I
It's true for symmetric real matrices. It's a classical result, which can be shown for example by induction on dimension of the vector space. You can find a proof here.
5. ## Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I
Ah! Thanks. That's what the real spectral theorem tells me!
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2017-10-20 14:39:11
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https://math.stackexchange.com/questions/3984031/help-me-with-z-transform
|
# Help me with z transform
So the question is basically z transform the given system.
$$(y[n+2] + 3y[n+1] - 4y[n])=(x[n+2] - 5x[n+1])$$
I've to find h[z] first then it's really easy to solve it. So that's what I got so far;
$$z^2y(z) + 3zy(z) - 4y(z) = z^2x(z) - 5zx(z)$$
$$(z^2+3z-4)y(z) = (z^2 - 5z)x(z)$$
$$H(z) = \frac{y(z)}{x(z)} = \frac{z^2-5z}{z^2+3z-4}$$
then
$$H(z) = 1 - \frac{4}{5(z-1)}-\frac{36}{5(z+4)}$$
I've to find Z transform pair but I'm stuck here. Thanks in advance!
Here is the z transform table from schaums;
• $z^2+3z-4 = (z-1)(z+4)$ then partial fractions – G Cab Jan 13 at 18:08
• I've updated the question. I tried every way to simplify this. Can you check it out? @GCab – L4W Jan 13 at 18:17
• Are you studying bilateral $Z$ transform ($z$ ranges from $-\infty$ to $\infty$)? – Shubham Johri Jan 13 at 18:34
Using $$H(z)=\frac{z^2-5z}{(z+4)(z-1)}=\frac{Az}{z-1}+\frac{Bz}{z+4}+C$$ we get $$A=-4/5,B=9/5,C=0$$. Can you complete?
From your list we see that $$Z[-a^nu[-1-n]]=\frac z{z-a}$$ and thus$$Z[-u[-1-n]]=\frac z{z-1}\\Z[-(-4)^nu[-1-n]]=\frac z{z+4}$$Thus, by the linearity of the inverse $$Z$$ transform, we get$$Z^{-1}[H(z)]=AZ^{-1}[z/(z-1)]+BZ^{-1}[z(z+4)]\\=\frac45u[-1-n]-\frac95(-4)^nu[-1-n]$$
• By the way isn't $C = 1$ – L4W Jan 13 at 18:47
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2021-01-27 02:24:04
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https://math.stackexchange.com/questions/1308209/existence-of-the-universal-covering-space-of-a-connected-lie-group
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# Existence of the universal covering space of a connected Lie group
I am working on a project about how the universal cover of a connected Lie group is a Lie group, but I cannot find a theorem that assures that this universal cover actually exists. I've found references on:
1. Lie Groups, An Approach through Invariants and Representations, by Claudio Procesi: page 80.
2. Probability on Compact Lie Groups by David Applebaum: page 5.
But I have not found the actual statement nor proof of the result anywhere. Does anybody know where I may find it?
• I'm sure you've figured it out by now, but if you know the fact for topological groups, checking smoothness of multiplication and inversion should be the least "nightmarish" part because the covering space gets its smooth structure from the space it's covering. The covering map is a local diffeomorphis, by definition, and smoothness is a local property. – Tim kinsella Jun 8 '15 at 22:09
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2019-09-18 12:08:54
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http://scprogramming.com/Differential%20Calculus/Section%201/1.1%20Introduction%20to%20Differential%20Calculus.php
|
Tutorials on Advanced Math and Computer Science Concepts
# Introduction to Differential Calculus
In differential calculus, our main goal is to find the instantaneous rate of change of some curve at any given point. To better understand this, let's think back to linear functions, and the idea of slope.
The graph shown to the above is $f(x) = 2x$. Recall that the equation of a line gives you two pieces of information, the slope of the line, and the y intercept. With this equation, we can see that the slope is 2, and the y intercept is 0. This is also clear from the graph of the equation.
With linear functions, it is easy to figure out how much the graph is changing at any given point. Since the slope is constant, 2, we know that at any given point, the graph has a rate of change of 2. Differential calculus allows us to extend this idea to functions that do not have a constant slope. For example, consider the graph below.
This graph shows the function $f(x) = x^2$,which is a quadratic function. With quadratics, the rate of change varies on each point. With our current math techniques, we don't have a good way to describe the way that quadratics change at any given point. This problem is not limited to quadratics, for that matter, more functions have variable slopes than constant. Due to the nonlinearity of the world, calculus becomes a very important mathematical skill.
So, let's get some intuition about how we might be able to solve the problem of finding the instantaneous rate of change of the function. Understanding what we need to do will help us understand the motivation behind each section of calculus. We know the slope of a linear function can easily be found, using a formula like $m = \frac{x_2-x_1}{y_2-y_1}$. If we can draw a line that touches a point on a function, or is tangent to the function, we would be able to know the slope at that given point. To demonstrate, look at the graph below.
In this graph, the red line is $f(x) = x^2$ and the blue line is $f(x) = 2x + \frac{1}{2}$. You can see that the blue line is tangent to the red line, meaning it touches the red line in one place, x = 2. Since it is tangent, the slope of the blue line will tell us the rate of change of the red line at the point x = 2.
So, how did we figure out that the blue line is tangent to the red line? Consider the equation of a slope, $m = \frac{x_2-x_1}{y_2-y_1}$. This equation requires two points in order to be solved. If we make the second point as close as possible to the first point, we can get a difference so small that they are basically the same point. In calculus, we refer to this idea as a limit, and it is the first concept we will learn in calculus. Once we have the ability to compute limits, we can compute the slope of the tangent line. From here, we can simply substitute in our point for $y = mx + b$, and find the tangent line.
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2020-03-31 06:41:19
|
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https://codereview.stackexchange.com/questions/56606/classes-for-sentence-like-readability/56607
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If we have a set of classes or structs such as:
struct Point
{
float x;
float y;
};
struct Square
{
Point topLeft;
float width;
};
struct Circle
{
Point middle;
};
If we need to add some functionality to to these, does something like below make sense?
inline Point middleOf(const Square &square)
{
float halfWidth(square.width/2.0f);
return Point{
square.topLeft.x + halfWidth,
square.topLeft.y + halfWidth};
}
inline Point middleOf(const Circle &circle)
{
return circle.middle;
}
class MoveSquare
{
public:
MoveSquare(Square &squareToMove)
:
squareToMove(squareToMove)
{}
void to(const Point &target)
{
squareToMove.topLeft = offsetHalfWidth(target);
}
private:
Point offsetHalfWidth(Point p)
{
const float halfWidth(squareToMove.width / 2.0f);
return Point{
p.x - halfWidth,
p.y - halfWidth};
}
private:
Square &squareToMove;
};
class MoveCircle
{
public:
MoveCircle(Circle &circle)
:
circleToMove(circle)
{}
void to(Point position)
{
circleToMove.middle = position;
}
private:
Circle &circleToMove;
};
inline MoveSquare move(Square &square)
{
return MoveSquare{square};
}
inline MoveCircle move(Circle &circle)
{
return MoveCircle{circle};
}
The main idea is being able to have code that reads like a sentence. Such as:
int main() {
Square aSquare;
aSquare.topLeft = Point{1,1};
aSquare.width = 3;
Circle aCircle;
aCircle.middle = Point{1,1};
move(aSquare).to(Point{1,1});
move(aSquare).to(middleOf(aCircle));
move(aCircle).to(middleOf(aSquare));
return 0;
}
• If you want your code to read like English, try objective-c. ;) I'm not sure how well accepted this would be in the C++ community, but we definitely strive for verbose code in ObjC. – nhgrif Jul 9 '14 at 23:16
• I think this will break OOP, just a little... What if you'll want to inherit Square, with RoundedSquare? And that move, should have global visibility, if you want to use it for readability. Otherwise you have to use namespace it. I don't think this is a good idea. Do you thought about koenig lookup, for this? – tower120 Jul 9 '14 at 23:23
• @tower120 hmm koenig lookup. What would using that look like for this? – PeterSW Jul 10 '14 at 14:30
• Why not: aSquare.moveTo(Point{1,1}); or aSquare.moveTo(aCircle.center()) – Martin York Jul 10 '14 at 18:08
• @LokiAstari for many cases I agree that would be a good solution and reads nice and clearly. The situation I had in mind was one where the types are from a third party library. In that case you can not change them at all. – PeterSW Jul 10 '14 at 20:02
Firstly, a negative remark. The power of C++ means that it is not uncommon for new C++ developers to try and make C++ look like a language they feel more familiar with - sometimes English, sometimes a programming language they already know. This is almost always a bad thing - you need to speak like a native.
However, whilst C++ used for mainstream development does have generally understood idioms, one of its capabilities and original design aims is to try to support different styles of programming.
As well as trying to create an application syntax which is more English like, you are externalizing functionality into proxy objects rather than using the OOP paradigm of member functions. IMHO, this is perfectly legitimate as an experiment.
What you'll likely find is that as you pursue this you gave to make various compromises that finally result in you deciding that maybe this is not he way to go. Or maybe not. See what you find.
If I were to go down this path, I'd probably play with using templates to try and get some generic behaviour, given that the OOP approach us not being used. How exactly this goes will depend on whatever other functionality is needed.
For example:
template<class T>
class Move
{
public:
Move(T& thing)
: thing_(thing)
{}
virtual void to(const Point& position) = 0;
T& thing() { return thing_; }
private:
T& thing_;
};
class MoveSquare : public Move<Square>
{
public:
MoveSquare(Square &squareToMove)
: Move<Square>(squareToMove)
{}
virtual void to(const Point &target)
{
thing().topLeft = offsetHalfWidth(target);
}
private:
Point offsetHalfWidth(const Point& p)
{
const float halfWidth(thing().width / 2.0f);
return Point{ p.x - halfWidth, p.y - halfWidth };
}
};
class MoveCircle : public Move<Circle>
{
public:
MoveCircle(Circle &circle)
: Move<Circle>(circle)
{}
virtual void to(const Point& position)
{
thing().middle = position;
}
};
• that method shouldn't be virtual – BЈовић Jul 11 '14 at 9:43
• also, what is the point of the Move? It does nothing – BЈовић Jul 11 '14 at 10:14
• If this is C++11 then it could be good to add override or final to communicate more clearly and get more help from the compiler. – PeterSW Jul 11 '14 at 18:44
It reads like a mandate, and not necessarily a sentence which makes this point trivial.
Notice how all are English:
Run this code!
The code ran
The code is running
This design doesn't let you define movement inside the object that moves, so you need to look into different places to find out how the object moves.
Your way is also weird and unfamiliar. Consider how Qt does it:
void QGraphicsItem::setPos(const QPointF & pos)
Sets the position of the item to pos, which is in parent coordinates. For items with no parent, pos is in scene coordinates.
The position of the item describes its origin (local coordinate (0, 0)) in parent coordinates.
It's an interesting concept, but not exactly how I would expect this to be structured.
First, it's superfluous to have an external function to return the middle of a circle when it's already available though it's property.
I would expect a class method to move the object. Something like circle.move(x,y). If you're concerned with making the size immutable, simply set it in a constructor and disallow external access to the radius property.
• I was thinking more for the cases when you can't add anything directly to the classes. I guess I should have been clearer about that in the question... – PeterSW Jul 9 '14 at 23:42
• @PeterSW: Just in general, or do you have some examples? Even if this were a factor, you would have to stick to structs as having classes would suggest that the data members should be private, thus preventing the use of free functions. – Jamal Jul 9 '14 at 23:45
• @Jamai In using free functions you're limited to the public interface... but there's often plenty of scope to build more functionality from that. – PeterSW Jul 9 '14 at 23:56
• @PeterSW: Right, which is why I asked. I'm not saying that this is plain wrong, but unexpected for basic use (as you've said, clarification should've been added, but it would also invalidate this answer). – Jamal Jul 9 '14 at 23:59
• I'm going to go ahead and call out that I don't know this language. I'm speaking from a completely abstract design point of view. It looks like I may have misunderstood the question. – RubberDuck Jul 10 '14 at 1:11
There are few problems. I will start with minor, and end with major.
Minor
1. MoveSquare::to accepts a parameter by const reference, but MoveSquare::offsetHalfWidth accepts the parameter by value. Since offsetHalfWidth doesn't modify it's parameter, it should be passed by const reference
2. MoveSquare::to just assigns the result of MoveSquare::offsetHalfWidth to some variable. Why not merge these two methods?
Major
I am particularly fond of the KISS principle, therefore I cringed when I saw your code. I would say it is a good example of when NOT to use a proxy pattern.
I would add a method to each of the structures with this signature :
void MoveTo(const Point &target);
it makes the code simpler, and therefore easier to understand.
Also, instead of having overloaded functions to calculate the middle point, I would add a method to the structure.
They would look like :
struct Square
{
Point topLeft;
float width;
void MoveTo(const Point &target);
Point MiddlePoint() const;
};
struct Circle
{
Point middle;
void MoveTo(const Point &target);
Point MiddlePoint() const;
};
This way, all calculations for a specific structure is localized, and easier to find.
Overloaded functions are making the code harder to understand. When number of such structures is high, then number of functions is high as well. Finding the correct one can be challenging.
• Does C++ have inheritance? It looks like both of those could inherit MoveTo from a shared parent class. – RubberDuck Jul 11 '14 at 10:21
• @ckuhn203 Yes, but why? Those structures are PODs, and those functions are specific (and different) for each structure. – BЈовић Jul 11 '14 at 11:14
• Maybe I'm missing something, but it looks like the MoveTo method in this answer is "copy paste" code. There's nothing different about them. – RubberDuck Jul 11 '14 at 12:00
• @ckuhn203 What are you talking about? What is copy paste code? The bottom code is the suggested signature of the structures. Those methods would still need implementation. – BЈовић Jul 11 '14 at 12:05
• @ckuhn203: In C++ it is possible to use duck typing interfaces without a common base class (it is called static polymorphism). Introducing inheritance hierarchies with virtual functions is to be avoided until they are truly needed. – Nobody Jul 11 '14 at 13:51
Object names in the form of aSomething are not only un-useful, but also prevent additional objects. Otherwise, you'll have anotherSomething and thirdSomething and so on.
Sure, it may still sound better in English, but it just makes your code messier and more restrictive. Although striving to have the code look more English-like is nice, you can still use looser naming that's still nicely readable by others.
Alternatives to this may depend on your design, as already mentioned by others. As good naming is already one of the hardest things to do in programming, it may be worth looking at similar designs to see how it's done by others.
If we need to add some functionality to to these, does something like below make sense?
In some contexts, it probably does (though I can't think of any). It does look (to me) like a case of the "I will write C in any language" syndrome.
I would find this implementation cumbersome and obscure, especially since the equivalent canonical code is much simpler:
struct Point
{
float x;
float y;
};
struct Figure { virtual void move(const Point& abspos) = 0; }
// TODO: add virtual destructor to Figure
struct Square: Figure
{
Point topLeft;
float width;
void move(const Point& abspos) override { topLeft = abspos; }
inline Point middle() const
{
float halfWidth(width/2.0f);
return Point{
square.topLeft.x + halfWidth,
square.topLeft.y + halfWidth};
}
};
struct Circle: Figure
{
Point middle;
void move(const Point& abspos) override { middle = radius; }
};
Here's an updated main:
int main() {
Square aSquare;
aSquare.topLeft = Point{1,1};
aSquare.width = 3;
Circle aCircle;
aCircle.middle = Point{1,1};
// move(aSquare).to(Point{1,1});
aSquare.move(Point{1,1});
// move(aSquare).to(middleOf(aCircle));
aSquare.move(aCircle.middle);
// move(aCircle).to(middleOf(aSquare));
aCircle.move(aSquare.middle());
return 0;
}
Note: The algorithm is not equivalent to yours, but it's the interface/client code that looks interesting, not necessarily the implementation of the functions.
• I'm just wondering what you reasons for thinking that the question code suffers from "I will write C in any language" syndrome? – PeterSW Jul 11 '14 at 18:50
• @PeterSW, The "I can write C in any language syndrome" is a name I use for the attempt to force constructs in a language, to resemble constructs in another language, solely due to the familiarity level of the developer. – utnapistim Jul 17 '14 at 10:57
• I think I understood what you meant, I'm more wondering why you think this is the case for the original code? I'm pretty sure it's impossible to write such a construct in C. It it because it doesn't use inheritance? – PeterSW Jul 17 '14 at 11:20
|
2019-07-20 00:22:23
|
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|
http://lpma-paris.fr/agenda/seminaires-gdt/les-probas-du-vendredi/yuki-tokushige/
|
# Differentiability of the speed of biased random walks on Galton-Watson trees
##### schedule le vendredi 21 mai 2021 de 11h00 à 12h00
Organisé par : Quentin Berger, Nathanaël Enriquez, Thierry Lévy et Shen Lin
Intervenant : Yuki Tokushige (Technische Universität München)
Lieu : https://zoom.us/j/98589875776?pwd=Q2JCYU5FN25uNUZLdDJ4Uk9zU2Mxdz09
Sujet : Differentiability of the speed of biased random walks on Galton-Watson trees
Résumé :
We prove that the speed of a $\lambda$-biased random walk on a supercritical Galton-Watson tree (with/without leaves) is differentiable for$\lambda$ such that the walk is ballistic and obeys a central limit theorem. We also give an expression of the derivative using a certain 2-dimensional Gaussian random variable, which naturally arise as limits of functionals of a biased random walk.
The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres. In particular, an important role is played by moment estimates of regeneration times, which are locally uniform in $\lambda$. This talk is based on a joint work with Adam Bowditch (University College Dublin).
|
2021-05-11 22:12:21
|
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https://wiki.kidzsearch.com/wiki/Singapore
|
kidzsearch.com > wiki
# Singapore
Motto:
Anthem:
CapitalSingapore[a]
Lua error in Module:Coordinates at line 492: attempt to index field 'wikibase' (a nil value).
Official languages
Malay
Ethnic groups
(2019)[2]
Religion
(2015)
Demonym(s)Singaporean
GovernmentUnitary dominant-party parliamentary constitutional republic
Halimah Yacob
Lee Hsien Loong
Sundaresh Menon
Tan Chuan-Jin
LegislatureParliament
Independence
from the United Kingdom and Malaysia
3 June 1959
16 September 1963
9 August 1965
8 August 1967
Area
• Total
731.0 km2 (282.2 sq mi)[3] (176th)
Population
• 2019 estimate
5,703,600[b] (115th)
• Density
7,804/km2 (20,212.3/sq mi) (2nd)
GDP (PPP)2020 estimate
• Total
$615.698 billion[5] (36th) • Per capita$107,604[5] (3rd)
GDP (nominal)2020 estimate
• Total
$391.875 billion[5] (31st) • Per capita$68,487[5] (7th)
Gini (2017) 45.9[6]
medium
HDI (2019) 0.938[7]
very high · 11th
CurrencySingapore dollar (S$) (SGD) Time zoneUTC+8 (Singapore Standard Time) Date formatdd/mm/yyyy Mains electricity230 V–50 Hz Driving sideleft Calling code+65 ISO 3166 codeSG Internet TLD.sg Website gov.sg Singapore, officially the Republic of Singapore, is a sovereign country as well as a city-state. It is an island state at the southern end of the Malay Peninsula in Asia, between the Straits of Malacca and the South China Sea. Singapore is about one degree of latitude (137 kilometres or 85 mi) north of the equator. About 5.70 million people live in Singapore. About 3.31 million are citizens. Most of them are ethnically Chinese, Malay, or Indian, as well as a smaller number of other Asians. Present-day Singapore was founded in 1819 by Stamford Raffles as a trading post of the British Empire. During the Second World War, Singapore was taken over by Japan in 1942, but returned to British control after Japan surrendered in 1945. Singapore started to govern itself in 1959, and in 1963 became part of the new federation of Malaysia, together with Malaya, North Borneo, and Sarawak. Political and social differences led to Singapore being removed from the federation two years later, becoming an independent country on its own. Most Singaporeans are bilingual.[8] The symbolic national language of Singapore is Malay. Other official languages of Singapore are English, Mandarin and Tamil.[9] English is the language of choice because everyone in Singapore knows and uses it. It is the first language taught in schools and the language used by the government and in court. Students are also usually taught the language of their ethnicity. This means that the Chinese will learn Mandarin, Malays will learn Malay, and so on. Students can also choose to learn a third language in secondary school. Singapore is also known as a "Garden City" or a "City in a Garden". It is because there are plants everywhere, making it look like a garden.[10] Singapore has one of the highest standards of living in the world, with very good education, healthcare, housing, and very low corruption. It is also known for having many strict rules and punishments, including fines.[11][12] This is why it is also sometimes jokingly called a "fine" city. The government says this has helped Singapore in being a very safe country.[13] Singapore is one of the founding members of ASEAN. ## History ### Before 1819 Singapore's name comes from 'Singa Pura', which means Lion City in Sanskrit. Many people around the world understand "Lion City" to refer to Singapore.[14] There is some debate about who founded Singapore. According to the Malay Annals, a Sumatran prince called Sang Nila Utama landed on Temasek (Singapore's old name) and saw a Lion which is called 'Singa' in Malay.[15] Thus he gave the island a new name, 'Singapura'.[16] However, Sang Nila Utama was likely mistaken, as lions never existed in Singapore. It is believed that the "lion" was actually a Malayan Tiger, which also exists in neighbouring Malaysia. While now extinct in Singapore it used to be present back then.[17] There were also many pieces of old items, such as Chinese coins, that showed that Temasek was an important Asian trading port even before the British rule.[18][19] ### 1819 to 1942 Map of Singapore's port in 1825 Few people lived in Singapore at the beginning of the 19th century. British governor Stamford Raffles arrived in Singapore on 28 January 1819 and wanted to set up a British trading town.[20] At the time, the island was then ruled by Tengku Abdul Rahman, the Sultan of Johor, who the Dutch and the Bugis from Sulawesi had controlled.[21] However, the Sultanate was weakened by infighting: the Temenggong (Chief Minister) of Tengku Abdul Rahman, as well as his officials, supported the Sultan's elder brother Tengku Long, who was living in exile in Riau. With the Temenggong's help, Raffles managed to secretly bring Tengku Long back into Singapore. Raffles offered to recognize Tengku Long as the true Sultan of Johor, under the title of Sultan Hussein, as well as giving him$5000 per year and another $3000 to the Temenggong; in return, Sultan Hussein would give the British the right to establish a trading post on Singapore.[22] An official treaty was signed on 6 February 1819.[23][24] In 1824, another treaty with the Sultan led to the entire island becoming under the British.[25] In 1826, Singapore became part of the Straits Settlements. Singapore became the regional capital in 1836.[26] Before Raffles arrived, there were only about a thousand people living on the island, mostly Malays along with a handful of Chinese.[27] By 1860, the population had grown to over 80,000, more than half being Chinese.[25][28] The country was given colony status in 1867. Later, in the 1890s, when the rubber industry became established in Malaya and Singapore, the island became a global center for sorting rubber and exporting them.[25][29] #### World War I Singapore was not really affected by the First World War (1914–18), as the conflict did not spread to Southeast Asia. The only significant event during the war was the 1915 Singapore Mutiny by Muslim soldiers from British India, who were garrisoned in Singapore.[30] After hearing news that they were to be sent to fight the Ottoman Empire in Europe, a Muslim state, the soldiers killed their officers and several British civilians before the mutiny was stopped by non-Muslim troops arriving from Johore and Burma.[31] #### Interwar period After World War I, the British built the large Singapore Naval Base as part of the defensive Singapore strategy.[32] First announced in 1921, the construction of the base went at a slow pace until the Japanese invasion of Manchuria in 1931. Costing$60 million and not fully completed in 1938, it was the largest dry dock in the world, the third-largest floating dock, and had enough fuel tanks to support the entire British navy for six months.[32][33] The base was defended by heavy naval guns stationed at Fort Siloso, Fort Canning and Labrador Park, as well as a Royal Air Force airfield at Tengah Air Base. Winston Churchill called Singapore the "Gibraltar of the East", and military discussions often referred to the base as just "East of Suez".[34]
However, the main fleet was in Europe, and the British did not have enough money to build a second fleet to protect their Asian colonies. The plan was for the Home Fleet to sail quickly to Singapore in the event of an emergency. As a result, after World War II broke out in 1939, the fleet was busy defending Britain from Germany, leaving Singapore open for a Japanese invasion.[35][36]
### World War II
The Japanese marching in Singapore during the World War II period
Due to the weak defenses of the country, the Japanese attacked Singapore and easily took control of the colony on 15 February 1942. Up to 60,000 British soldiers surrendered on that day, and Churchill called it "the worst disaster and largest capitulation in British history".[37] There were a lot of losses by both the British and the rest of the Empire, with a total of nearly 85,000 people captured.[38] About 5,000 were killed or injured, many from Australia and India.
People of Singapore went through hard times during the Japanese rule, until the surrender of the Japanese in September of 1945. Many people were tortured or killed by the Japanese as they did not follow the rules properly or because they were suspected of going against the Japanese. The Japanese also targeted the Chinese the most; between 5,000 and 25,000 Chinese were killed, now known as the Sook Ching massacre.[39] The most notable anti-Japanese force was Force 136, headed by Lim Bo Seng. Its purpose was to encourage and supply resistance movements in the enemy-occupied territory and occasionally mount sabotage operations.[40]
The hard times during World War II made the people think that the British were not as strong as before. Therefore, many people were supportive of independence once the British had returned.[41][42]
### Independence
A few years after the war, In 1963, Singapore joined with Malaya, Sabah and Sarawak to form the new nation of Malaysia. Malaysia is a country with many races. In Malaya, only the Malays have special benefits. For example, the Malays could get university education more easily than other races.
As most people in Singapore are Chinese, Singapore wanted equality for all the people of Malaysia. Singapore also wanted a common market to be set up so that goods to Malaysia would not be taxed. However, this was not done and caused arguments between the state government of Singapore and the federal government of Malaysia.[43]
Singapore separated from Malaysia and became independent on its own on 9 August 1965.[44]
### After Independence
After Independence, the president of Singapore was Yusof Ishak and its prime minister was Lee Kuan Yew.[45] At first, many people thought Singapore's independence would not last.[46] In 1967, Singapore helped to start the Association of Southeast Asian Nations and in 1970 it joined the Non-Aligned Movement.[47] Lee Kuan Yew was in charge of the country as Prime Minister of Singapore and saw it become very developed.[48][49][50]
In 1990, Goh Chok Tong replaced Lee Kuan Yew as prime minister, while Lee Kuan Yew became Senior Minister. When Goh Chok Tong was Prime Minister, Singapore went through the 1997 Asian financial crisis, the 2003 SARS outbreak and terrorist threats by Jemaah Islamiyah. In 2004, Lee Hsien Loong, the son of Lee Kuan Yew, took over as Prime Minister. Goh Chok Tong became the Senior Minister, and Lee Kuan Yew became the Minister Mentor of Singapore. Lee Kuan Yew died in 2015.[51]
## Government and politics
Singapore's Parliament House.
Singapore is a parliamentary republic with a Westminster system of unicameral parliamentary government representing constituencies. Its constitution states representative democracy as its political system.[52] Singapore is ranked regularly as one of the least corrupt countries in the world, according to Transparency International.[53]
The Cabinet has executive power, and is led by the Prime Minister, and the President.[54] The president is elected through popular vote, and has some veto powers for a few big decisions such as the use of the national reserves and the appointment of judges, but otherwise occupies a post with little power.[55]
The Parliament serves as the legislative branch of government.[54] Members of Parliament (MPs) are made up of elected, non-constituency and nominated members. Elected MPs are voted into parliament on a "first-past-the-post" (plurality) system and represent either single-member or group-representation constituencies.[56] Elections are free, and the People's Action Party has won control of Parliament with large majorities in every election since self-governance in 1959.[57] In the most recent parliamentary elections in 2020, the largest opposition, led by the Workers' Party, increased its representation in the House to 10 elected MPs out of 93.[58]
The legal system of Singapore is based on English common law, however with large and important local differences. Trial by jury was removed in 1970 leaving judicial judgement done completely and only by judgeship.[59] Singapore has laws that include corporal punishment in the form of caning for rape, rioting, vandalism, and some immigration crimes.[60][61] There is a mandatory death penalty for murder, and for certain drug-trafficking and firearms offences.[62] In a 2008 survey, international business executives believed Singapore had the best judicial system in Asia.[63]
## People
### Religion
Buddhism is the most common religion in Singapore.[64] Other common religions include: Christianity, Islam, Taoism, and Hinduism.[64]
### Education
Educational level of Singaporeans that are not students and are older than 15 in 2005[65]
Highest qualification Percentage
No education
18%
Primary school
45%
Secondary school
15%
Post-secondary diploma
8%
Degree
13%
Students in Singapore go through six years of compulsory primary school, which ends with all students taking a Primary School Leaving Examination (PSLE).[66] Then, depending on their results in the PSLE, students are streamed into "Special", "Express", "Normal (Academic)", and "Normal (Technical)" groups.[67] The amount of time a student spends studying in Secondary school (4–5 years) depends on their group.[67]
Students are taught in English and their mother tongue, which can be Chinese, Malay or Tamil. Secondary school students may also choose to study a third language such as:[8]
Students can take a third language as it increases their chances in getting an overseas scholarship and can improve their examination grades, especially in the GCE Ordinary Level ("O" Levels), which are Secondary students take after their five or four years of education.[68] However, only some students can qualify to take a third language.[69]
After their "O" Levels, students can choose to go to a polytechnic, which is a place where students can study for 3 years for a diploma[70] or to a junior college where students study for 2 years to receive an "A" Level.[71] Students can also go to Institutes of Technical Education (ITE), where students study for two years to receive a "National ITE Certificate" (NITEC). This certificate is only recognized in Singapore. Students who go to ITE usually continue their education at a polytechnic.
With either a diploma, or an "A" Level certificate, students can apply to go to universities in Singapore or overseas.[72]
### Languages
Language used most frequently at home[73][74]
Language Percent
English
36.9%
Mandarin
34.9%
Malay
10.7%
Tamil
3.3%
Others
14.2%
The Singapore government has chosen four official languages: English, Malay, Chinese (Mandarin), and Tamil. English is the primary language.[9] Singapore English is the main language in Singapore.[75]
English is the first language of the nation, but it is not the most common. English is the second most commonly spoken language among Singaporeans. The most commonly spoken language amongst Singaporeans in their homes is Chinese (51%), followed by English (32%), Malay (13%) and Tamil (7%). This means that 32% of Singaporeans are native English speakers. Most of the rest of the people speak it as a second language. However, English has the largest total number of speakers including native and second language speakers. Mandarin Chinese is the second most common.[76][77]
Almost 40% of people in Singapore are foreign.[78] Most foreigners come from Asia. The country where most foreigners come from are Malaysia, its closest neighbor. In 2009, there may have been 350,000 Malaysians working in Singapore.[79] Many Chinese-speaking foreigners and Chinese-speaking Singaporeans work in services. Thus, Chinese is the main language of many workers such as hawkers, retail assistants, hairdressers, etc. in Singapore today.[80][81]
Singapore from Chinatown
Singaporean English mainly comes from British English.[82] The forms of English spoken in Singapore range from Standard English to a pidgin called Singlish. The Singapore government and many Singaporeans are against using Singlish. There is a "Speak Good English" campaign each year. Public schools and in the media also have rules against Singlish.[83] There are many Singapore accents in English because of the many languages and identities of people in the city. Languages can even change over the generations and children may speak different languages and have different accents from their mother. For example, in a Singaporean Chinese family, the grandmother might speak Hokkien as her first language. Differently, the mother might speak Mandarin as her first language and Hokkien/English as her second languages, while the grandson might speak English as his first language and Mandarin as his second language.
Before independence in 1965, Hokkien, a Chinese dialect, was the common language among the Chinese laborers. Malay and English were used to communicate between the different ethnic groups.[84] After independence in 1965, English became the first language of the nation and replaced Hokkien and Malay as the one shared language. Today, most younger Singaporeans have English as their first language or are fluent in English.
Malay is a national language of Singapore because of the history of the city.[9] However, less than 20% of Singaporeans can read and write in Malay. Malay is still used at home by most Malay Singaporeans.[85] The Malay used in Singapore (Bahasa Melayu) is closer to the language in Malaysia than the language in Indonesia. However, there are differences between the Malay in Singapore and in Malaysia. The national anthem "Majulah Singapura" is sung in Malay.
Many people speak Chinese – Mandarin and other Chinese dialects – in Singapore. Just over 50% of Singaporeans speak it at home, so it is the most common language in homes.[85] Singaporean Mandarin is based on simplified Chinese and it is similar to the system used in mainland China. The forms of Mandarin spoken in Singapore range from Standard Mandarin to a pidgin known as Singdarin. Besides Mandarin, many southern Chinese dialects are also spoken in Singapore.
Hokkien used to be a lingua franca among the Singaporean Chinese so many older Singaporeans still understand Hokkien. The most common Chinese dialects spoken by Singaporeans are the Hokkien, Hainan, Teochew and Cantonese.[84] However, Chinese dialects other than Mandarin are not allowed in the media, so these dialects are quickly dying out. Most younger Singaporeans do not speak them anymore.
Tamil, a South Indian language, is spoken by about half of Indians in Singapore. That is about 5% of all Singaporeans. However, North Indian languages such as Hindi and Punjabi are also spoken by a small group of Singaporean Indians in Singapore.[86]
## Culture
Shophouses in Singapore
Singapore has many kinds of people and immigrants from many places. Therefore, Singaporean culture has often been described as a mix of cultures – European, Malay, Chinese, Indian and Peranakan. Also, foreigners are 42% of the population in Singapore and they are part of changing Singaporean culture.[78][87]
### Food
The Hainanese chicken rice, a local food in Singapore
Dining is an important part of life in Singapore.[88] Singaporean food is an example of the many different cultures in the country. It is also an example of mixing among cultures. British, Chinese, Indian, Malay, Tamil, and Indonesian styles of cooking all mix together. It is especially known for its sea food. Typical Singaporean food includes: Satay, Nasi lemak, Chilli crab, Kaya toast, and Hainanese chicken rice, Laksa, Fish Head Curry, Hokkien Prawn Meeand Char Kway Teow.
### Media
MediaCorp, the state-owned media corporation, operates all seven local broadcast television channels in Singapore. It also runs 13 radio stations of the total 18 radio stations in Singapore.[89] Radio and television stations are all owned by government controlled companies. However, one radio transmitter in Singapore is not controlled by the government. That is the Far Eastern Relay Station of the BBC World Service.
Print media of Singapore includes 16 newspapers and several magazines. Daily newspapers are published in English, Chinese, Malay, and Tamil.[90]
## National flower
The national flower of Singapore
The national flower of Singapore is Vanda Miss Joaquim. It is a type of orchid and it is a hybrid orchid. This makes Singapore the only nation in the world to have a hybrid as a national flower. It was chosen because it was part of the effort to create national pride and identity.[91]
## Economy
A Singapore Airlines Airbus A380-800
Singapore has a strong and free economy that supports a large middle class.[92] The city state is a global shipping and logistics hub and many multinational firms have their offices in Singapore. The national airline, Singapore Airlines, has a large global network which brings tourists and business travelers alike, to the city. The airline has often been ranked as the world's best airline.[93][94]
Singapore also has a port located at the south of Singapore, called Keppel Harbour. It is one of the busiest ports around the world with many ships coming in to trade in a single day.[95] Singapore also has another port on Jurong Island.
The port of Singapore is one of the busiest ports in the world.
### Money
Singapore's currency is the Singapore dollar. The short name for the Singapore dollar is SGD. Brunei dollar can be used in Singapore, because they have the same value.[96]
## Geography
A Map of Singapore and the surrounding islands & waterways
Singapore is made up of 63 islands, including the main island, which is known as Singapore Island to most people, but is also known as Pulau Ujong.[97] There are two man-made connections to Johor, Malaysia: the Johor–Singapore Causeway in the north, and the Tuas Second Link in the west. Jurong Island, Pulau Tekong, Pulau Ubin and Sentosa are the largest of Singapore's smaller islands. The highest natural point is Bukit Timah Hill at 166 m (545 ft).[98]
About 23% of Singapore's land area are forest and nature reserves.[99] Urbanisation has removed most primary rainforest, with Bukit Timah Nature Reserve the only significant remaining forest.[100] Even though there is very little primary rainforest left, there are more than 300 parks and four nature reserves in Singapore. There are also many trees planted all over Singapore and almost fifty per cent of the country is covered by trees and plants. Because of this, Singapore is also commonly known as the 'Garden City'.[101]
Singapore, being a small country, has been reclaiming land from the sea around the island. The first time Singapore started to reclaim land was in the 1960s. The total land area of Singapore at that time was 581.5 km2 and it has increased to 730 km2 in the 2010s, which is an increase of about 20% in total land area.[102] It may grow by another 100 km2 (40 sq mi) by 2030.[103] Some land reclamation projects involve joining together smaller islands to make larger islands with more uses, like Jurong Island.[100] Singapore uses the landfill method to reclaim the sea at the south of the country, but polders have been recently used too.[104] The country's rapid reclamation projects has made disputes with its neighbouring countries, Malaysia and Indonesia.[105]
### Climate
The Singapore botanic garden, which is part of Singapore's forests.
Singapore is hot and wet all year round. It has a tropical rainforest climate (Af in the Köppen climate classification), which means there are no seasons (spring, summer, autumn and winter) in Singapore. There is the most rain at the end of the year, and the temperature is usually around 20 °C to 35 °C.[106][107]
Although Singapore does not experience the four seasons, the period from May to June is usually warmer, while the period from November to January is cooler because of the more frequent rains and monsoonal winds in Singapore during the year-end.[106][107]
Climate data for Singapore
Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year
Average high °C (°F) 30.1
(86.2)
31.1
(88)
31.6
(88.9)
31.7
(89.1)
31.6
(88.9)
31.3
(88.3)
30.9
(87.6)
30.9
(87.6)
30.9
(87.6)
31.1
(88)
30.6
(87.1)
29.9
(85.8)
31.0
(87.8)
Average low °C (°F) 23.3
(73.9)
23.6
(74.5)
23.9
(75)
24.4
(75.9)
24.8
(76.6)
24.7
(76.5)
24.5
(76.1)
24.4
(75.9)
24.2
(75.6)
24.0
(75.2)
23.7
(74.7)
23.4
(74.1)
24.1
(75.4)
Rainfall mm (inches) 242.5
(9.547)
162.0
(6.378)
184.8
(7.276)
178.8
(7.039)
171.8
(6.764)
161.2
(6.346)
158.3
(6.232)
176.2
(6.937)
169.7
(6.681)
193.9
(7.634)
255.7
(10.067)
288.2
(11.346)
2,343.1
(92.248)
Humidity 84.7 82.9 83.8 84.8 84.4 83.0 82.8 83.0 83.5 84.1 86.4 86.9 84.2
Avg. rainy days 15 11 14 15 14 13 13 14 14 16 19 19 177
Sunshine hours 173.6 183.6 192.2 174.0 179.8 177.0 189.1 179.8 156.0 155.0 129.0 133.3 2,022.4
Source #1: National Environment Agency (Temp 1929-1941 and 1948-2009, Rainfall 1869-2009, Humidity 1929-1941 and 1948-2010, Rain days 1891-2009) [108]
Source #2: Hong Kong Observatory (sun only, 1982—2008) [109]
## Relations with other countries
Singapore has diplomatic relations with 175 other Sovereign states.[110] Singapore's foreign policy is to maintain a secure environment in Southeast Asia as well as the countries near Southeast Asia. A basic rule is the political and economic stability in Southeast Asia.[111]
### ASEAN
Singapore is part of the ASEAN (Association of the South East Asian nations) network, which is an organisation that unites all Southeast Asian countries. Member countries of ASEAN work with and help other countries in ASEAN. Singapore is one of the countries that founded ASEAN, and is the most developed country among them all.[112]
### Commonwealth of Nations
Singapore is also part of the Commonwealth of Nations, which is for countries that used to be under the British's rule.[113]
## Holidays
Public holidays in Singapore:[114]
There are also school holidays, lasting one week in March and September, about one month in June, and about one and a half months in December.[115]
## Records
The Singapore Flyer
Jewel Changi Airport
Even though it is a small country, Singapore holds many world records with its buildings and people. Some are:
Type of record Name of record Received record
Building For the tallest Ferris wheel in the world Singapore Flyer[116]
Building For the tallest indoor waterfall in the world Jewel Changi Airport[117]
Sports Olympic record (OR) at the 100 metres butterfly Joseph Schooling – 2016 Summer Olympics[118]
## Transportation
Buses can take riders to almost any part of Singapore, even those not reached by MRTs
The inside of a MRT train.
The exit of a MRT station.
Singapore has one of the best infrastructure qualities in the world.[119] Buses, trains, and taxis are easily found.
Most of Singapore are well-connected by a railway system known as the Mass Rapid Transit, or MRT in short. There are also taxi companies like Comfort Cabs, Silver Cab, SMRT Taxis, CityCab and Premier Taxi.[120] There is one telephone number to call a taxi, of which the closest taxi from any company will respond.[121] Apps such as Grab and Gojek as an alternative to taxi companies are also available.
The Mass Rapid Transit of Singapore is the most commonly used form of transportation in Singapore.[122][123]
The Singaporean land transport system is controlled by the LTA (Land Transport Authority) of Singapore.[124]
### Airport
An aerial view of Changi International Airport during the 2000s.
The Singapore Changi Airport is the main airport of Singapore. It is in the east of Singapore, with a total of four terminals with airlines flying to many different parts of the world. It has also received many awards for being the best airport in the world.[125][126] A Skytrain service is also available at the airport.
The airport is in Changi. It is on a 13 square kilometres (5.0 sq mi) site which was reclaimed from the sea.[126]
## Notes
1. Singapore has no official distinct capital city as it is a city-state.[1]
2. Of which 3,471,900 are citizens.[4]
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2022-01-22 00:02:33
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|
https://vismor.com/documents/network_analysis/graph_algorithms/S1.SS6.php
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# 1.6 Acyclic Graph
A directed graph with no cycles is called directed acyclic graph or a DAG for short. An acyclic digraph has at least one vertex with an out-degree of zero.
Figure 4 shows a variant of the directed graph of Figure 1 that contains no cycles. You will observe that vertex 4 has an out-degree of zero.
A directed tree is a connected DAG with the following properties:
• There is one vertex, called the root, which no edges enter.
• All vertices except the root have one entering edge.
• There is a unique path from each vertex to the root.
A DAG consisting of one or more trees is called a forest. If the graph $F=(V,E)$ is a forest and the edge $(v,w)$ is in $E(F)$, vertex $v$ is the parent of $w$ and vertex $w$ is the child of $v$. If there is a path from $v$ to $w$, then vertex $v$ is an ancestor of $w$ and vertex $w$ is a descendent of $v$. A vertex with no proper descendants is a leaf. A vertex $v$ and its descendants form a subtree of $F$. The vertex $v$ is the root of this subtree. The depth of vertex $v$ is the length of the path from the root to $v$. The height of vertex $v$ is the length of the longest path from $v$ to a leaf. The height of a tree is the height of its root. The level of vertex $v$ is its depth subtracted from the height of the tree.
Figure 5 depicts a directed tree. Its root is vertex 1. Its leaves are the set of vertices $L(G)=\{ 3,4,6,8,9\}$.
An undirected, connected, acyclic graph is called a free tree or an undirected tree. A rooted free tree is a free tree in which one vertex has been designated as the root. A directed tree is converted into a rooted free tree by discarding the orientation of the edges. A rooted free tree is converted into a directed tree by orienting each edge away from the root. The terminology which applies to directed trees also applies to rooted free trees.
Figure 6 depicts the directed tree of Figure 5 converted into a rooted free tree.
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2017-06-22 16:21:28
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https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-4-an-introduction-to-functions-4-3-patterns-and-nonlinear-functions-lesson-check-page-249/5
|
## Algebra 1
The correct function for the table of values is $y = x^2 + 1$
The result obtained by the classmate only works for the first $2$ coordinates, whereas $y=x^2+1$ works for all the coordinates.
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2019-10-14 15:59:06
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https://indico.cern.ch/event/895086/contributions/4727030/
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# QM 2022
Apr 4 – 10, 2022
Auditorium Maximum UJ
Europe/Warsaw timezone
Proceedings submission deadline extended to August 15, 2022
## Lattice QCD calculation of thermal photon rate
Apr 6, 2022, 7:10 PM
4m
Poster QCD matter at finite temperature and density
Dibyendu Bala
### Description
Thermal photons from the QGP provide important information about the interaction among the plasma constituents. The thermal photon production rate from a thermally equilibrated plasma is proportional to the transverse spectral function $\rho_{T}(k_0=|\vec k|,\vec k)$. One can calculate the photon production rate also from the difference between $\rho_{T}$ and $\rho_{L}$(longitudinal) correlator as $\rho_{L}$ vanishes on the photon point. The UV part of $\rho_{T}-\rho_{L}$ is not dominant and therefore the corresponding Euclidean correlator gets most of the contribution from the IR part of $\rho_{T}-\rho_{L}$. We calculate the continuum extrapolated T-L correlator non-perturbatively on the lattice at 1.1$T_c$ and 1.5$T_c$ for a gluonic medium. We will present the extraction of spectral function from this Euclidean correlator using a hydrodynamic inspired model in combination with the Backus-Gilbert method, which allows us to estimate the photon production rate at these temperatures.
### Co-authors
Greg Jackson (INT, University of Washington)
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2022-08-11 08:03:07
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