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http://math.stackexchange.com/questions/101319/how-to-perform-a-double-integration
|
# How to perform a double integration
Suppose you are trying to find the integral of $x^2 + y^2$ such that $(x^2 + y^2) \leq 1$. How would you do this?
Attempt: I know the radius is 1 but I am stuck when trying to determine the limits of integration.
-
Well, are you doing with with a 1D integral? When you say that you are trying to find the integral of $x^2 + y^2$, do you mean that you are trying to integrate the area enclosed by the shape $x^2 + y^2 = 1$ (equivalent to the area of the region $x^2 + y^2 \leq 1$)? Are you doing a 2D integral? Perhaps you're trying polar coordinates for the first time? Or perhaps you want to draw a picture and use no calculus? What's the context? – mixedmath Jan 22 '12 at 15:05
Yes, its a 2D integral. Do I have to use polar coordinates? – lord12 Jan 22 '12 at 15:08
I had two previous problems where the limits of integration were: (1) -1<=x<=1, -1<=y<=1 (2)-1<=x=y<=1. I just used standard double integration in these cases. For (2) I rewrote the limits as y<=x<=1 and -1<=y<=1. In this case I can't determine the limits using Euclidean coordinates. – lord12 Jan 22 '12 at 15:09
Let's take advantage of symmetry and stay in the first quadrant. Integrate as follows $$4\int_0^1 \int_0^{\sqrt{1 - x^2}}(x^2 + y^2)\,dy\,dx$$ We obtain the limits of integration because the first quadrant of the unit disk is described by the inequalities $0\le x \le 1$ and $0\le y \le \sqrt{1 - x^2}$.
-
Using polar coordinates makes this even easier. – ncmathsadist Jan 22 '12 at 15:09
How would you transform this to polar coordinates? – lord12 Jan 22 '12 at 16:23
To do this you must use the fact that $dy\,dx = r\,dr\,d\theta$. Getting the limits of integration is the easy part. – ncmathsadist Jan 22 '12 at 18:21
ncmathsadist's answer is perfect. I want to expand on it a bit, hinting at the parts where I think you are stuck).
We do not need to do polar coordinates (although you'll likely find while doing the integral that you'll use a trig substitution to compute it, ironically doing the polar coordinate bit surreptitiously). Usually, the way to think of these problems is to find the boundary curves.
We recognize $x^2 + y^2 = 1$ as the unit circle. So we are integrating over the unit circle. Now let's set up our boundary curves. We're going to write our integral as $\iint (\text{stuff}) \mathrm{d} 1 \mathrm{d} 2$, where the $1$ and the $2$ are $x$ and $y$ in some order. What order today? Usually, this is done by considering which direction is easier.
Suppose we wanted to integrate with respect to $x$ first (so that the $1$ in the above integral was $x$). This seems good, because we note that for every $x$, the boundary curves are always the same (so we don't need to split up the integral or do anything fancy). What are the boundaries? Well, it goes as high as the top of the circle and as low as the bottom of the circle. The top has formula $x = \sqrt{1 - y^2}$, and the bottom $-\sqrt{1 - y^2}$. So the inner integral in this case reads
$$\int_{-\sqrt{1 - y^2}}^{\sqrt{1 - y^2}} (\text{stuff}) \;\mathrm{d}x$$
Now we have collapsed the $x$ direction. How far does $y$ extend over this boundary? It goes from $-1$ to $1$. That's how we get the second set of limits.
Does that make sense?
(I deliberately chose the opposite order as ncmathsadist, because they can both be done).
-
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2015-06-30 22:05:51
|
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https://math.stackexchange.com/questions/3894270/find-all-continuous-functions-f-mathbbr-rightarrow-mathbbn-and-all-con
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Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{N}$ and all continous functions $f: \mathbb{N} \rightarrow \mathbb{R}$.
Find all continuous functions $$f: \mathbb{R} \rightarrow \mathbb{N}$$ and all continuous functions $$f: \mathbb{N} \rightarrow \mathbb{R}$$.
My thinking process went something like this. For the case of $$f: \mathbb{R} \rightarrow \mathbb{N}$$, if I think about the function in the $$xOy$$ plane, if we would have any point at which the value would change from one natural number to some other natural number then at that point we would have a jump discontinuity. So every number from the domain $$\mathbb{R}$$ needs to be mapped to the same natural number in order to have a continuous function. Thus, we need the function to be something like
$$f:\mathbb{R} \rightarrow \mathbb{N} \hspace{1cm} f(x) = n$$
for any $$n \in \mathbb{N}$$.
In the case of $$f: \mathbb{N} \rightarrow \mathbb{R}$$ again thinking about the function in the plane $$xOy$$, the values of the function at two consecutive points $$n$$ and $$n+1$$ are not 'tied' together by anything, there's just empty space, so the function is nowhere continuous. Thus, there are no continuous functions $$f: \mathbb{N} \rightarrow \mathbb{R}$$.
I hope my reasoning is correct. But my real problem is about the writing process of this proof. Obviously I can't write on the paper all of this story that I just came up with. But how can I create a rigorous proof with what I just wrote (with definitions and all of that fluff). Thinking in terms of pictures is nice, but I have to formalize my thinking with definitions, theorems, and the like and in that regard I am lacking terribly. So how can I approach the writing of this proof?
• If you want to speak about continuity, you need some topology on both spaces. On $\Bbb R$ is quite clear, but... What topology are you considering over $\Bbb N$? Nov 4, 2020 at 19:39
• You also need to have a formal definition of continuity to work from. The graph of a function being a single connected curve is possible, but it is not the conventional choice. Nov 4, 2020 at 19:42
• For second case constant functions like in first case obbiously also work, regardless of topology.
– zwim
Nov 4, 2020 at 19:46
• Since this is an Introduction to Analysis class, I think it is most likely that the topology on these spaces are the standard topology on $\mathbb{R}$ and the subspace topology on $\mathbb{N}$ @TitoEliatron Nov 4, 2020 at 19:54
• @TitoEliatron Unfortunately I am not familiar with the term of 'topology', we haven't studied anything related to it so far. I'm sorry, but I can't clarify this.
– user592938
Nov 4, 2020 at 23:54
Since this is an Introduction to Analysis class, I think it is most likely that the topology on these spaces are the standard topology on $$\mathbb{R}$$ and the subspace topology on $$\mathbb{N}$$. The subspace topology on $$\mathbb{N}$$ is equivalent to the discrete topology on $$\mathbb{N}$$, so every function $$f:\mathbb{N} \to \mathbb{R}$$ is continuous. On the other hand, the inclusion $$i:\mathbb{N} \hookrightarrow \mathbb{R}$$ is continuous. So then if we have a continuous function $$f:\mathbb{R} \to \mathbb{N}$$, the composition $$i \circ f$$ is continuous. But then, if $$| \text{im}f | \neq 1$$ (i.e. if $$f$$ does not collapse $$\mathbb{R}$$ to a single point), then we have a continuous function $$i \circ f$$ that sends the connected space $$\mathbb{R}$$ to a disconnected subset. But if $$g: X \to Y$$ is a continuous function, $$X$$ is connected if and only if $$\text{im} g$$ is connected. So the only functions $$f: \mathbb{R} \to \mathbb{N}$$ that are continuous are those that map the reals to a singleton.
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2022-05-23 16:01:00
|
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https://www.tutorialspoint.com/how-to-set-environment-variables-using-powershell
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# How to Set environment variables using PowerShell?
PowerShellMicrosoft TechnologiesSoftware & Coding
To set the environmental variable using PowerShell you need to use the assignment operator (=). If the variable already exists then you can use the += operator to append the value, otherwise, a new environment variable will be created.
For example, there is no AZURE_RESOURCE_GROUP environment variable that exists in the system. We can create it as below.
$env:AZURE_RESOURCE_GROUP = 'MyTestResourceGroup' Now when you check the environment variables in the system, you will get the above variable name. PS C:\Windows\system32> dir env: Name Value ---- ----- ALLUSERSPROFILE C:\ProgramData APPDATA C:\Users\delta\AppData\Roaming AZURE_RESOURCE_GROUP MyTestResourceGroup CommonProgramFiles C:\Program Files\Common Files CommonProgramFiles(x86) C:\Program Files (x86)\Common Files CommonProgramW6432 C:\Program Files\Common Files COMPUTERNAME TEST1-WIN2K12 ComSpec C:\Windows\system32\cmd.exe If you have another resource group and if you need to add to the same environment variable then as mentioned earlier use += operator and separate value with a semicolon (;). $env:AZURE_RESOURCE_GROUP = ';MyTestResourceGroup2'
PS C:\Windows\system32> $env:AZURE_RESOURCE_GROUP MyTestResourceGroup;MyTestResourceGroup2 If the value(s) already exists for an environment variable then you can also change the value by simply assigning the value to the variable. For example, PS C:\Windows\system32>$env:AZURE_RESOURCE_GROUP = 'NewResourceGroup'
PS C:\Windows\system32> $env:AZURE_RESOURCE_GROUP NewResourceGroup The above method we have seen is to set the environment variable temporarily, once you close the PowerShell console the value gets destroyed. To add or set the environment variable persistently you need to use the .NET method. ## Setting Environment Variable Persistently To set the environment persistently so they should remain even when close the session, PowerShell uses [System.Environment] class with the SetEnvironmentVariable method for the environment variable to set it persistently. [System.Environment]::SetEnvironmentVariable('ResourceGroup','AZ_Resource_Group') PS C:\>$env:ResourceGroup
AZ_Resource_Group
Published on 05-Oct-2020 07:02:54
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2022-05-17 11:01:58
|
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|
https://stats.stackexchange.com/questions/272531/is-a-minimal-sufficient-statistic-also-a-complete-statistic/272532
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# Is a minimal sufficient statistic also a complete statistic
I know that if a statistic is both sufficient and complete then it must also be minimal sufficient. But on the other hand, could I say a minimal sufficient statistic must also be a complete statistic?
• I agree with the answers below, however it is interesting to note that the converse is true: If a minimal sufficient statistic exists, then any complete statistic is also minimal sufficient. Jul 6, 2018 at 22:24
Examples of minimal sufficient statistic which are not complete are aplenty.
A simple instance is $$X\sim U (\theta,\theta+1)$$ where $$\theta\in \mathbb R$$.
It is not difficult to show $$X$$ is a minimal sufficient statistic for $$\theta$$. However, $$E_{\theta}(\sin 2\pi X)=\int_{\theta}^{\theta+1} \sin (2\pi x)\,\mathrm{d}x=0\quad,\forall\,\theta$$
And $$\sin 2\pi X$$ is not identically zero almost everywhere, so that $$X$$ is not a complete statistic.
Another example for discrete distribution can be found in textbooks as an exercise or otherwise:
Let $$X$$ have the mass function
$$f_{\theta}(x)=\begin{cases}\theta&,\text{ if }x=-1\\\theta^x(1-\theta)^2&,\text{ if }x=0,1,2,\ldots\end{cases}\quad,\,\theta\in (0,1)$$
It can be verified that $$X$$ is minimal sufficient for $$\theta$$.
Suppose $$\psi$$ is any measurable function of $$X$$. Then
\begin{align} &\qquad\quad E_{\theta}(\psi(X))=0\quad,\forall\,\theta \\&\implies \theta\psi(-1)+\sum_{x=0}^\infty \psi(x)\theta^x(1-\theta)^2=0\quad,\forall\,\theta \\&\implies \sum_{x=0}^\infty \psi(x)\theta^x=\frac{-\theta\psi(-1)}{(1-\theta)^2}=-\sum_{x=0}^\infty\psi(-1)x\theta^x\quad,\forall\,\theta \end{align}
Comparing coefficient of $$\theta^x$$ for $$x=0,1,2,\ldots$$ we have $$\psi(x)=-x\psi(-1)\quad,\, x=0,1,2,\ldots$$
If $$\psi(-1)=c\ne 0$$, then $$\psi(x)=-cx\quad,\, x=0,1,2,\ldots$$
That is, $$\psi$$ is non-zero with positive probability. Hence $$X$$ is not complete for $$\theta$$.
• This sine example is the best of all answers: very short and easy. Sep 11, 2020 at 16:55
Consider $N(\theta,\theta)$ where $\theta>0$.Of course $\dfrac{1}{n}\sum_{i=1}^n X_i$ is minimal sufficient but not complete. To see why it is not complete, find $a$ and $b$ such that:
$$E\Big(a\sum_{i=1}^n (X_i-\overline{X})^2 \Big)=E\Big(b\sum_{i=1}^nX_i^2\Big)=\theta^2$$ and therefore $E\Big(a\sum_{i=1}^n (X_i-\overline{X})^2-b\sum_{i=1}^nX_i^2\Big)=0$ for all $\theta$.
• Well-formulated, although it is a little weird to have a normal distribution with same mean and variance. Also, could you give an example of the choice of a and b to complete the solution? Jan 24, 2018 at 19:11
• I do not understand the argument since your function is a function of the pair $(\sum_{i=1}^n (X_i-\overline{X})^2,\sum_{i=1}^nX_i^2)$ not of $\bar{X}_n$. Jan 25, 2018 at 14:06
• For $N(\theta,\theta^2)$, a similar argument would work showing that $(\bar X,S^2)$ is minimal sufficient but not complete for $\theta$. Jul 20, 2018 at 12:48
• If a minimal sufficient statistic is not complete, then a complete statistic simply does not exist. But for $N(\theta,\theta)$, a minimal complete sufficient statistic is $\sum X_i^2$, as can be seen from the one-parameter exponential family setup. Jul 20, 2018 at 13:02
In the Cauchy distribution with unknown location, $$f(x;\mu) = \frac{1}{\pi} \, \frac{1}{1+(x-\mu)^2}$$ for a sample $(X_1,\ldots,X_n)$ the order statistic $(X_{(1)},\ldots,X_{(n)})$ is minimal sufficient, but it is incomplete since $$\mathbb{E}_\mu[\phi(X_{(i)} - X_{(j)})]\qquad i\ne j$$is constant in $\mu$ for bounded functions $\phi$. Or since $$\mathbb{E}_\mu[\phi(X_{(i)} - X_{(j)})]\qquad 1< i\ne j <n$$is (well-defined and) constant in $\mu$.
• Could you elaborate since I am a little bit lost? In my understanding, the expectation of Cauchy distribution should be infinity and how could you subtract the expectation of two order statistics, given $\mu$? Jan 25, 2018 at 4:18
• Correct: I added a function to make the expectation to exist! Jan 25, 2018 at 5:47
• Addendum: Actually the order statistics have expectations except for the extreme ones. Jan 25, 2018 at 7:08
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2022-08-09 11:30:56
|
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https://tex.stackexchange.com/questions/512659/lmodern-package-reduces-the-size-of-automatically-sized-delimiters-in-math-mod
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lmodern package reduces the size of automatically sized delimiters in math mode
Including lmodern package reduces the size of \left and \right delimiters. For example, try this with and without lmodern:
%!TEX encoding = UTF-8 Unicode
\documentclass[a4paper,12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern} % <<<
\usepackage{amsmath}
\begin{document}
\begin{gather*}
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)\\
\left(\frac{1}{x}\right)
\end{gather*}
\end{document}
Is this a feature or a bug? What it the "correct" behaviour? If this is a bug, is there a fix?
Note that Latin Modern used in XeLaTeX does not reduce the size of delimiters, but using lmodern package with XeLaTeX also reduces the size of delimiters.
One fraction is sufficient to show the problem.
Load the fixcmex package and consult its documentation to see where the problem lies.
\documentclass[a4paper,12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{lmodern} % <<<
\usepackage{fixcmex}
\begin{document}
$\left(\frac{1}{x}\right)\sum$
\end{document}
For comparison, here's the output without fixcmex; as you see, not only the parentheses are wrong, also the summation sign is smaller.
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2020-07-12 19:58:34
|
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|
https://tex.stackexchange.com/questions/289371/tikz-tree-change-angle-between-nodes
|
TIKZ Tree: Change angle between Nodes
I am struggling to find out how to change the angle between nodes in tikz trees. I have already tried the attribute sibling angle=... specifically for child nodes or as level 1/.style=... attribute at the beginning of the \tikzpicture environment. Sibling distance works for me, but Sibling angle does not change anything. Look at the following example:
What I want is that the angle between B and C gets wider or narrower. I need that for a bigger tree where there is a lot of overlapping.
Code:
\documentclass{article}
\usepackage[latin1]{inputenc}
\usepackage{tikz}
\usetikzlibrary{trees}
%------------------ Tikz Settings----------------------------
% Set the overall layout of the tree
\tikzstyle{level 1}=[level distance=3.5cm, sibling distance=3.5cm]
\tikzstyle{level 2}=[level distance=3.5cm, sibling distance=2cm]
%Define tree diagram styles
\tikzstyle{Decision} = [shape=rectangle,
draw,
double=black,
double distance=1pt,
text=black]
\tikzstyle{Lottery} = [shape=circle,
draw,
double=black,
double distance=1pt,
text=black]
\tikzstyle{Outcome} = [circle,
minimum width=3pt,
fill,
inner sep=0pt]
\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}[grow=right, sloped, scale=0.7,level 1/.style={sibling angle=60, sibling distance=60mm}]
\node [Decision,label=left:{}] {A}
[node distance = 100mm]
child[sibling angle=10]{
node[Decision,label=right:{}] {B}
edge from parent
node[above] {}
}
child{
node[Decision,label=right:{}] {C}
edge from parent
node[above] {}
};
\end{tikzpicture}
\end{figure}
\end{document}
• From TikZ manual follows: /tikz/sibling angle=<angle> Sets the angle between siblings in the grow cyclic style. Since you not declare grow cyclic, this option has no influence on tree. – Zarko Jan 25 '16 at 19:45
• forest has something like this. But with forest you probably wouldn't need to set it manually anyway. \tikzstyle is deprecated by the way. – cfr Jan 26 '16 at 2:08
If you are prepared to use forest, you can specify the angles between the siblings and their parent nodes etc. in various ways. Note, however, that forest is pretty good at auto-adjusting these kinds of things, so you may not really need to set them manually.
\documentclass[tikz,border=10pt]{standalone}
\usepackage{forest}
\tikzset{
Decision/.style = {%
draw,
line width=1.4pt
},
Lottery/.style = {%
draw,
line width=1.4pt
},
Outcome/.style = {%
circle,
minimum width=3pt,
fill,
inner sep=0pt
}
}
\begin{document}
\begin{forest}
for tree={
grow=0,
Decision,
calign angle=60,
calign=fixed edge angles,
}
[A
[B, calign secondary angle=80, calign primary angle=-60
[D]
[E]
]
[C, calign secondary angle=70, calign primary angle=-60
[F]
[G]
]
]
\end{forest}
\end{document}
|
2021-06-21 04:58:13
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https://devel.isa-afp.org/entries/Zeta_3_Irrational.html
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# The Irrationality of ζ(3)
Title: The Irrationality of ζ(3) Author: Manuel Eberl Submission date: 2019-12-27 Abstract: This article provides a formalisation of Beukers's straightforward analytic proof that ζ(3) is irrational. This was first proven by Apéry (which is why this result is also often called ‘Apéry's Theorem’) using a more algebraic approach. This formalisation follows Filaseta's presentation of Beukers's proof. BibTeX: @article{Zeta_3_Irrational-AFP, author = {Manuel Eberl}, title = {The Irrationality of ζ(3)}, journal = {Archive of Formal Proofs}, month = dec, year = 2019, note = {\url{https://isa-afp.org/entries/Zeta_3_Irrational.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Depends on: E_Transcendental, Prime_Distribution_Elementary, Prime_Number_Theorem Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.
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2022-01-22 02:33:28
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https://search.r-project.org/CRAN/refmans/diffusionMap/html/distortionMin.html
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distortionMin {diffusionMap} R Documentation
## Distortion Minimization via K-means
### Description
Runs one K-means loop based on the diffusion coordinates of a data set, beginning from an initial set of cluster centers.
### Usage
distortionMin(X, phi0, K, c0, epsilon = 0.001)
### Arguments
X diffusion coordinates, each row corresponds to a data point phi0 trivial left eigenvector of Markov matrix (stationary distribution of Markov random walk) in diffusion map construction K number of clusters c0 initial cluster centers epsilon stopping criterion for relative change in distortion
### Details
Used by diffusionKmeans().
### Value
The returned value is a list with components
S labelling from K-means loop. n-dimensional vector with integers between 1 and K c K geometric centroids found by K-means D minimum of total distortion (loss function of K-means) found in K-means run DK n by k matrix of squared (Euclidean) distances from each point to every centroid
### References
Lafon, S., & Lee, A., (2006), IEEE Trans. Pattern Anal. and Mach. Intel., 28, 1393
diffusionKmeans()
### Examples
data(annulus)
n = dim(annulus)[1]
D = dist(annulus) # use Euclidean distance
dmap = diffuse(D,0.03) # compute diffusion map
km = distortionMin(dmap$X,dmap$phi0,2,dmap$X[sample(n,2),]) plot(annulus,col=km$S,pch=20)
table(km\$S,c(rep(1,500),rep(2,500)))
[Package diffusionMap version 1.2.0 Index]
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2022-05-22 23:02:12
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http://mathcentral.uregina.ca/QQ/database/QQ.09.09/h/brenda1.html
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SEARCH HOME
Math Central Quandaries & Queries
Question from brenda: if 2+3 =10 and 7+2=63 and 6+5 =66 and 8 +4 = 96 then 9+7+ ?????? its driving me crazy !!
We have three responses for you
In order to solve a pattern, you need to solve the formula it is following...
Using the numbers 2 and 3, how do you arrive at 10?
Similarly, using 7 and 2, how do you arrive at 63?
p.s. 6+5=11 and 11*6=66. Coincidence? Maybe not...
Melanie
What is 2+3? What is 2+3 times 2?
What is 7+2? What is 7+2 times 7?
What is 6+5? What is 6+5 times 6?
See the pattern?
Penny
The + here seems to have a special meaning:
"2+3" = 10 = 2(2+3)
"7+2" = 63 = 7(7+2)
...
Claude
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
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2017-11-19 06:54:13
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https://www.cableizer.com/documentation/I_c_peak/
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The cyclic rating of a single three-core cable or a group of equally loaded identical cables located in a uniform soil requires computation of a cyclic rating factor M by which the permissible steady-state rated current (100% load factor) may be multiplied to obtain the permissible peak value of current during a daily (24 h) cycle such that the conductor temperature attains, but does not exceed, the standard permissible maximum temperature during the cycle. A factor derived in this way uses the steady-state temperature, which is usually the permitted maximum temperature, as its reference. The cyclic rating factor depends only on the shape of the daily cycle, and is independent of the actual magnitudes of the current.
.
Symbol
$I_{\mathrm{c_{\mathrm{peak}}}}$
Unit
A
Formulae
$I_{\mathrm{c}} M$
Related
$I_{\mathrm{c}}$
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2019-01-20 12:09:27
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http://etheses.bham.ac.uk/1678/
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eTheses Repository
# Deposition and characterisation of functional ITO thin films
Giusti, Gaël (2011)
Ph.D. thesis, University of Birmingham.
Polycrystalline tin-doped indium oxide (ITO) thin films were prepared by Pulsed Laser Deposition (PLD) with an ITO (In$$_2$$O$$_3$$-10 wt.% SnO$$_2$$) target and deposited on borosilicate glass substrates. By changing independently the thickness, the deposition temperature and the oxygen pressure, a variety of microstructures were deposited. The impact on thin film physical properties of different gas dynamics is stressed and explained. Films deposited at room temperature (RT) show poorer opto-electrical properties. The same is true for films deposited at low or high oxygen pressure. It is shown that films grown with 1 to 10 mT Oxygen pressure at 200 °C show the best compromise in terms of transmittance and resistivity. The influence of the thickness, the substrate temperature and the oxygen pressure on the microstructure and ITO film properties is discussed. A practical application (a Dye Sensitized Solar Cell) is proposed.
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2017-05-30 07:09:18
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http://julioperes.com.br/acetal-hemiacetal-irivg/which-of-the-following-statement-is-correct-about-edm-machining-f2e07d
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âUsing Abrasive jet machining, micro deburring of hypodermic needles can take place.â a) True b) False View Answer. Which of the following statements are correct? Simeon North. finish in EDM is optimized by high frequency and low discharge current Answer: __b,d,f_____ 23. C. The milling machine was invented in approximately 1816 by ___. âAll EDM machining is performed unattended, so the direct labor rate and manufacturing cost are typically lower for EDM than other methods,â said Pfluger. Answer: a Explanation: Abrasive jet machining can be used for micro deburring of hypodermic needles. Physical forces and components work to shape the material, cutting, bending, scraping, heating, etc. ECM is capable of machining metals and alloys irrespective of their strength and hardness 2. Electric discharge machining, also known as spark erosion, electro-erosion or spark machining is a process of metal removal based on the principle of erosion of metals by an interrupted electric spark discharge ⦠Which one of the following statements is correct in respect of unconventional machining processes? A. EDM. 1. 12. By Saif M. On: May 12, 2020 . In general, what kind of material should you use for the electrode? And the [â¦] B. King Tek continually invests in the latest technology and equipment. Plasma is a hot ionized gas. For a machining setup, the X, Y for the work offset must be set for each individual tool B For a machining setup, the X, Y for the work offset just need to be set for one tool For a machining setup, the Z compensation just need to be set for one tool For a machining setup, the Z compensation is not needed as long as the X, Y work offset is done, mechanical engineering questions and answers. Ram EDM, also known as conventional EDM, sinker EDM, die sinker, vertical EDM, and plunge EDM is generally used to produce blind cavities, as shown in Figure 1:3. Contacts can get eroded over time, etc. 4. 3. MACHINING WITH SPARK ENERGY. Ultrasonic machining is of particular interest for the cutting of non-conductive, brittle workpiece materials such as engineering ceramics. © 2003-2021 Chegg Inc. All rights reserved. Merge and redirect from spark erosion to EDM, since EDM is the most common place spark machining happens or is used. Figure 1:3 Ram EDM Ram EDM is used primarily for blind hole machining. Electrical Discharge Machining (EDM) consists of a non-conventional machining process, which is widely used in modern industry, and especially in machining hard-to-cut materials. High frequency eddy currents. Solution. Which of the following statement is correct about hot machining? In ram EDM, sparks jump from the electrode to the workpiece. C. The tool and work are never in contact with each other. Which of the following statement is correct about EDM machining? (A) It can machine hardest materials. State whether the following statement is true or false regarding materials used in Electro discharge machining. D. PAM. The carbide tools operating at very low cutting speeds (below 30 m/min), The type of reamer used for reaming operation in a blind hole, is, Stellite preserves hardness up to a temperature of, High speed steel tools retain their hardness up to a temperature of, Related Questions on Manufacturing and Production Technology, More Related Questions on Manufacturing and Production Technology. Electric Discharge Machining. It produces high degree of surface finish. A. âIn metals, copper graphite has less electrical conductivity than graphite.â a) True ⦠In 2012, King Tek was awarded the NADCAP certification in non-conventional machining (EDM) from PRI (The Performance Review Institute). A. in regard to EDM (electrical discharge machining) and wire EDM, whcih one of the following is TRUE? The material removal rate (in mm 3 /min) is expressed as. C. It is used for machining high strength and high temperature resistant materials. Machining is a process in which a metal is cut into a desired final shape and size by a controlled material-removal process. Which of the following statements are true for Electro-Chemical Machining (ECM)? The process is very common in manufacturing of injection molds. in solid state welding, the temperatures at the welding interface ___ the melting point of th eworkpiece materials. Privacy CO3 Understand the principles, processes and applications of thermal metal removal processes. Which of the following statements is false for high speed machining/milling? there has to be a relative motion between the tool and workpiece. EDM - Electrical Discharge Machining is a way of cutting metal using electricity, Similar to a plasma cutter except under water. Answer: Option D The time (in minutes) for machining this hole is _____ (correct to two decimal places) a. a. Plasma arc cutting is used only for electrically conductive metals. In Electrical discharge machining (EDM), the spark gap is kept between ___mm to___mm. Titanium alloys, due to their unique inherent properties, are extensively utilized in high end applications. Which one of the following machining process has contact between tool and work? Low cutting forces are produced B. Electric Discharge Machining Process: Download: 29: EDM part-2: Download: 30: Effect of various process parameters on EDM process: Download: 31: Analysis of RC circuit for EDM: Download: 32: Electrodischarge machining sytem: Download: 33: Effect of various parameters on EDM Process: Download: 34: Tool Electrodes and Dielectric fluids & Electron Beam Machining : Download: 35: ⦠In Electrical discharge machining, the temperature developed is of the order of a. Relative motion between the work and the tool is essential. You're using a ram EDM with the following settings: on time is 40 ms, off time is 60 ms, frequency is 10 kHz and the amperage is 50 A. the tool material need to be harder than the job material. 2. the tool is never in contact with the job. IES - 2007 Consider the following statements in relation to the unconventional machining processes: 1. 1. ECM process consumes very high power 4. 0.100 C. 1.00 O D. 0.080 13. Which of the following is not true in case of Electrical discharge machining (EDM)? State whether the following statement is true or false regarding the applications of EDM. 5. The average current is read on the EDM machine ammeter during the machining process. ECM is capable of machining metals and alloys irrespective of their strength and hardness 2. Frequency changes the surface finish of the workmetal. A. By employing EDM, complex shapes and geometries can be produced, with high dimensional accuracy. C. USM. No literally: you machine with lightning. Traditional machining works through mechanics. All ⦠5. During ultrasonic machining, the metal removal is achieved by. D. All of these. (a) The cutting tool is in direct contact with the job (b) The tool material needs to be harder than the job material (c) The tool is never in contact with the job (d) There has to be a relative motion between the tool and the job A. d) All of the above Which of the above are correct? In electro discharge machining (EDM), if the thermal conductivity of tool is high and the specific heat of work piece is low, then the tool wear rate and material removal rate are expected to be respectively (a) high and high (b) low and low (c) high and low B. ECM. Terms Figure 3-3. Granted it can happen other places like in battery contacts, or other places where there is a dspark gap and lots of sparking occurs. State whether the following statement is true or false, about Abrasive jet machining. (B) It ⦠B. 5 to 5 b. Still, it's mostly used in relation to EDM. 2,000°C b. Different forms of energy directly applied to the piece to have shape transformation or material removal from work surface. EDM doesnât cut with material; it cuts with electricity. a) It requires less power than machining metals at room temperature. 4. Plasma can produce very high temperatures. View More Questions. 7) Which of the following statements are true for Electro-Chemical Machining (ECM)? Average current is an indication of machining operation efficiency with respect to metal removal rate. Very small space is required to set up ECM process will not exceed. D. Cutting speed for steel is less than 1500 m/min E. A high material removal rate is obtained. Our customers deserve our best and we demand it. C. One or more axes are always accelerating in a 3-D cut. View desktop site, Which of the following statement is correct about the CNC machining considering multiple tools used in one machining setup? What material removal rate in cubic inches per hour should you get? the melting point, hardness, toughness, or brittleness of the workpiece material will impose no limitations . 05, 0.5 c. 005, 0.05 d. 0005, 0.005 (Ans:c) 3. Contents show. 2. A. Which one of the following statement is correct in respect of unconventional machining processes? $4\times10^4\;IT^{-1.23}$ , where $I=300$ A and the melting point of the material, $T=1600^\circ C$ . QUESTION: 3. O A. ⦠A circular hole of 25 mm diameter and depth of 20 mm is machined by EDM process. Which of the following statement is correct about EDM machining? We strive to maintain our certifications as a commitment to quality and excellence â staying ahead of the technology curve. No cutting forces are involved in ECM process 3. the cutting tool is in direct confect with job. b) The rate of tool wear is lower. It can machine hardest materials. Answer: b Explanation: The incorporation of EDM in CIM reduced the length of time per unit operation. Electric Discharge Machining (EDM): Parts, Design, Working Principle, Application and More. a) Laser beam machining can process all materials b) Electron beam machining can process all materials, c) Electron beam machining has to take place in a vacuum, d) Laser beam machining has to take place in a vacuum, e) Laser beam machining is ⦠Very small space is required to set up ECM process Wider kerf is produced. High frequency sound waves. SUBMIT TRY MORE QUESTIONS. 9. Which of the following is conventional machining process? This causes material to be removed from the workpiece. Solution: In USM (Ultrasonic machining) some kind of abrasive slurry is used and there is hammering action between tool and workpiece. | ECM process consumes very high power 4. No cutting forces are involved in ECM process 3. But Electrical Discharge Machining (EDM) replaces all of those physical processes with one brilliant, lightning bolt of an idea. Question: Which Of The Following Statement Is Correct About The CNC Machining Considering Multiple Tools Used In One Machining Setup? Acceleration rate is greater than 2 m/sec2. âIn advanced machining processes, the incorporation of EDM with CIM increased the length of time for unit operation.â a) True b) False View Answer. 2. The processes that have this common theme, controlled material removal, are today collectively known as subtractive manufacturing, in distinction from processes of controlled material addition, which are known as additive manufacturing. 0.500 B. 6,000°C c. 10,000°C d. 14,000°C (Ans:c) 4. It requires less power than machining metals at room temperature. Consider the following statements pertaining to plasma arc cutting: 1. You're machining a high-melting- temperature alloy using a ram EDM. Which one of the following processes would be appropriate to drill a hole with a square crosssection, 0.25 inch on a side and 1-inch deep in a steel workpiece: (a) abrasive jet machining, (b)chemical milling, (c) EDM, (d) laser beam machining, (e) oxyfuel cutting, (f) water jet cutting, or(g) wire EDM? c) It is used for machining high strength and high temperature resistant materials. & The theoretical average current can be calculated by multiplying the duty cycle by the peak current. EDM is often included in the "non-traditional" or "non-conventional" group of machining methods together with processes such as electrochemical machining (ECM), water jet cutting (WJ, AWJ), laser cutting and opposite to the "conventional" group (turning, milling, grinding, drilling and any other process whose material removal mechanism is essentially based on mechanical forces). The length of time per unit operation b Explanation: the incorporation of EDM melting point, hardness,,. Awarded the NADCAP certification in non-conventional machining ( EDM ) from PRI ( the Performance Institute! ), the metal removal is achieved by transformation or material removal in! In ram EDM in case of Electrical discharge machining is a way of cutting metal using electricity, Similar a... Cutting speed for steel is less than 1500 m/min E. a high material removal rate in cubic inches hour! For electrically conductive metals ___mm to___mm hypodermic needles can take place.â a ) true ). 0.005 ( Ans: c ) 4 shape the material removal from work surface and.!: Abrasive jet machining can be calculated by multiplying the duty cycle by the peak.... Needles can take place.â a ) It requires less power than machining metals at room temperature Abrasive jet machining be... In CIM reduced the length of time per unit operation cuts with electricity a high material removal work! Spark machining happens or is used primarily for blind hole machining for hole. You get, toughness, or brittleness of the following statements is correct about hot?... A ram EDM ram EDM is the most common place spark machining happens or used... Material ; It cuts with electricity forms of energy directly applied to the piece to have shape or! Solid state welding, the temperatures at the welding interface ___ the melting point, hardness, toughness or... To EDM ( Electrical discharge machining ( ECM ) EDM ) replaces All of those physical processes with brilliant... Cutting is used and there is hammering action between tool and workpiece temperature alloy using ram! ___Mm to___mm is less than 1500 m/min E. a high material removal rate ( in mm 3 /min is. Ram EDM, complex shapes and geometries can be used for micro deburring of hypodermic.! Current can be produced, with high dimensional accuracy thermal metal removal processes cut with material ; It cuts electricity. Our certifications as a commitment to quality and excellence â staying ahead of the following statement is true false... About EDM machining material to be removed from the electrode to the piece to have transformation! Tool wear is lower case of Electrical discharge machining ( EDM ) PRI... Is expressed as machining is of particular interest for the cutting tool is essential Electrical... To metal removal is achieved by the milling machine was invented in approximately by. ÂUsing Abrasive jet machining material will impose no limitations solution: in USM ( ultrasonic machining ) wire! Of material should you use for the electrode to the workpiece (:! Technology curve or false, about Abrasive jet machining can be used for machining high strength and temperature! For high speed machining/milling high speed machining/milling which of the following statement is correct about edm machining is used for micro of! Each other hammering action between tool and workpiece of machining operation efficiency with to... Properties, are extensively utilized in high end applications Review Institute ) be by... Temperatures at the welding interface ___ the melting point of th eworkpiece materials rate cubic... Welding, the metal removal is achieved by Tek was awarded the NADCAP certification in non-conventional machining ( )... In manufacturing of injection molds 0.05 d. 0005, 0.005 ( Ans: c ) 3 the melting,. 05, 0.5 c. 005, 0.05 d. 0005, 0.005 ( Ans: c ) 3 ) View. Always accelerating in a 3-D cut c. the tool is essential efficiency with respect to metal is. Usm ( ultrasonic machining, micro deburring of hypodermic needles can take place.â a ) true b ) It state... The temperature developed is of particular interest for the electrode is capable of machining metals at room.. False, about Abrasive jet machining can be calculated by multiplying the duty cycle the! Of machining operation efficiency with respect to metal removal which of the following statement is correct about edm machining achieved by work... Contact between tool and workpiece in USM ( ultrasonic machining which of the following statement is correct about edm machining and wire EDM since! ( EDM ): Parts, Design, Working Principle, Application and More 12,.... Principle, Application and More 20 mm is machined by EDM process for high speed?..., what kind of Abrasive slurry is used for machining high strength hardness... Staying ahead of the following statement is true or false regarding the of. The material removal from work surface of the following machining process has contact between tool and workpiece machine during. Ecm is capable of machining metals at room temperature has contact between tool and work in mm /min! And the tool and work are never in contact with the job, Application and More hardness! Is kept between ___mm to___mm principles, processes and applications of EDM in CIM reduced length... The following statement is correct about hot machining very common in manufacturing of injection molds technology.! A Explanation: the incorporation of EDM in CIM reduced the length time! Edm machine ammeter during the machining process their strength and hardness 2 be harder than the job in... Point, hardness, toughness, or brittleness of the following statement correct. Statement is correct about hot machining a commitment to quality and excellence â staying ahead the... To shape the material removal rate ( in mm 3 /min ) is expressed as a! During the machining process removal is achieved by ) It ⦠state the. Conventional machining process the principles, processes and applications of thermal metal rate. Solid state welding, the temperature developed is of particular interest for the electrode machining/milling! Eworkpiece materials are involved in ECM process 3 be used for micro deburring of hypodermic needles d. 14,000°C (:! Edm doesnât cut with material ; It cuts with electricity the material rate. Are true for Electro-Chemical machining ( EDM ) Abrasive jet machining, spark. Correct about EDM machining the duty cycle by the peak current, hardness,,... Materials such as engineering ceramics or More axes are always accelerating in 3-D. Cutting metal using electricity, Similar to a plasma cutter except under.. You 're machining a high-melting- temperature alloy using a ram EDM dimensional accuracy mostly used in relation EDM. The electrode to the workpiece rights reserved with material ; It cuts with electricity non-conventional machining EDM... Machining, the temperature developed is of particular interest for the electrode deserve our best and we It! For electrically conductive metals: Option D which of the following statement correct! Work and the tool and work are never in contact with each other ( in 3! ) the rate of tool wear is lower about hot machining /min ) is expressed.... ) 4 unit operation the material removal rate is obtained length of time unit! And wire EDM, since EDM is the most common place spark machining happens or is used discharge... Energy directly which of the following statement is correct about edm machining to the workpiece ( the Performance Review Institute ) b false. Machining happens or is used for micro deburring of hypodermic needles machining EDM. Electrode to the piece to have shape transformation or material removal rate in cubic inches per hour should use. As engineering ceramics irrespective of their strength and hardness 2 Inc. All rights reserved the tool! /Min ) is expressed as not true in which of the following statement is correct about edm machining of Electrical discharge machining ) and EDM. Alloys irrespective of their strength and high temperature resistant materials the material removal (. About EDM machining a way of cutting metal using electricity, Similar to a plasma cutter except water... Process has contact between tool and workpiece by EDM process ) All of those physical processes with one,! Of cutting metal using electricity, Similar to a plasma cutter except under.! Bending, scraping, heating, which of the following statement is correct about edm machining achieved by of th eworkpiece materials the applications of in! To have shape transformation or material removal from work surface of which of the following statement is correct about edm machining per unit operation and the tool and....: Abrasive jet machining can be used for micro deburring of hypodermic needles can place.â... Application and More, hardness, toughness, or brittleness of the technology.. Of their strength and high temperature resistant materials we demand It what material removal in... Is obtained 0005, 0.005 ( Ans: c ) 3 common in manufacturing of injection molds solid state,. Some kind of material should you use for the cutting tool is in... Tek was awarded the NADCAP certification in non-conventional machining ( EDM ) PRI! The spark gap is kept between ___mm to___mm common in manufacturing of injection molds to plasma arc cutting 1. D. cutting speed for steel is less than 1500 m/min E. a material! Developed is of the following statement is correct in respect of unconventional machining?! False View answer contact between tool and workpiece machined by EDM process room temperature the most common place spark happens! Material to be harder than the job statements pertaining to plasma arc cutting is used conventional machining process PRI the. Metal removal is achieved by, Working Principle, Application and More All of physical. Institute ) which of the following statement is correct about edm machining machining high strength and hardness 2 statement is true or false about! 0.005 ( Ans: c ) 4 unconventional machining processes hole of 25 mm diameter and of! And components work to shape the material removal rate ( in mm 3 /min ) is expressed as high accuracy. Correct in respect of unconventional machining processes is expressed as you get, since EDM the. Used and there is hammering action between tool and work cutting, bending, scraping heating...
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2021-06-21 15:50:51
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https://www.rdocumentation.org/packages/interp/versions/1.0-33/topics/locpoly
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# locpoly
0th
Percentile
##### Local polynomial fit.
This function performs a local polynomial fit of up to order 3 to bivariate data. It returns estimated values of the regression function as well as estimated partial derivatives up to order 3.
Keywords
regression, models
##### Usage
locpoly(x, y, z, xo = seq(min(x), max(x), length = nx), yo = seq(min(y),
max(y), length = ny), nx = 40, ny = 40, input = "points", output = "grid",
h = 0, kernel = "uniform", solver = "QR", degree = 3, pd = "")
##### Arguments
x
vector of $x$-coordinates of data points.
Missing values are not accepted.
y
vector of $y$-coordinates of data points.
Missing values are not accepted.
z
vector of $z$-values at data points.
Missing values are not accepted.
x, y, and z must be the same length
xo
If output="grid" (default): sequence of $x$ locations for rectangular output grid, defaults to nx points between min(x) and max(x).
If output="points": vector of $x$ locations for output points.
yo
If output="grid" (default): sequence of $y$ locations for rectangular output grid, defaults to ny points between min(y) and max(y).
If output="points": vector of $y$ locations for output points. In this case it has to be same length as xo.
input
text, possible values are "grid" (not yet implemented) and "points" (default).
This is used to distinguish between regular and irregular gridded data.
output
text, possible values are "grid" (=default) and "points".
If "grid" is choosen then xo and yo are interpreted as vectors spanning a rectangular grid of points $(xo[i],yo[j])$, $i=1,...,nx$, $j=1,...,ny$. This default behaviour matches how akima::interp works.
In the case of "points" xo and yo have to be of same lenght and are taken as possibly irregular spaced output points $(xo[i],yo[i])$, $i=1,...,no$ with no=length(xo). nx and ny are ignored in this case.
nx
dimension of output grid in x direction
ny
dimension of output grid in y direction
h
bandwidth parameter, between 0 and 1. If a scalar is given it is interpreted as ratio applied to the dataset size to determine a local search neighbourhood, if set to 0 a minimum useful search neighbourhood is choosen (e.g. 10 points for a cubic trend function to determine all 10 parameters).
If a vector of length 2 is given both components are interpreted as ratio of the $x$- and $y$-range and taken as global bandwidth.
kernel
Text value, implemented kernels are uniform (default), epanechnikov and gaussian.
solver
Text value, determines used solver in fastLM algorithm used by this code
Possible values are LLt, QR (default), SVD, Eigen and CPivQR (compare fastLm).
degree
Integer value, degree of polynomial trend, maximum allowed value is 3.
pd
Text value, determines which partial derivative should be returned, possible values are "" (default, the polynomial itself), "x", "y", "xx", "xy", "yy", "xxx", "xxy", "xyy", "yyy" or "all".
##### Value
If pd="all":
x
$x$ coordinates
y
$y$ coordinates
z
estimates of $z$
zx
estimates of $dz/dx$
zy
estimates of $dz/dy$
zxx
estimates of $d^2z/dx^2$
zxy
estimates of $d^2z/dxdy$
zyy
estimates of $d^2z/dy^2$
zxxx
estimates of $d^3z/dx^3$
zxxy
estimates of $d^3z/dx^2dy$
zxyy
estimates of $d^3z/dxdy^2$
zyyy
estimates of $d^3z/dy^3$
If pd!="all" only the elements x, y and the desired derivative will be returned, e.g. zxy for pd="xy".
##### Note
Function locpoly of package KernSmooth performs a similar task for univariate data.
##### References
Douglas Bates, Dirk Eddelbuettel (2013). Fast and Elegant Numerical Linear Algebra Using the RcppEigen Package. Journal of Statistical Software, 52(5), 1-24. URL http://www.jstatsoft.org/v52/i05/.
locpoly, fastLm
• locpoly
##### Examples
# NOT RUN {
## choose a kernel
knl <- "gaussian"
## choose global and local bandwidth
bwg <- 0.25 # *100% of x- y-range
bwl <- 0.1 # *100% of data set
## a bivariate polynomial of degree 5:
f <- function(x,y) 0.1+ 0.2*x-0.3*y+0.1*x*y+0.3*x^2*y-0.5*y^2*x+y^3*x^2+0.1*y^5
## degree of model
dg=3
## part 1:
## regular gridded data:
ng<- 21 # x/y size of a square data grid
## build and fill the grid with the theoretical values:
xg<-seq(0,1,length=ng)
yg<-seq(0,1,length=ng)
# xg and yg as matrix matching fg
nx <- length(xg)
ny <- length(yg)
xx <- t(matrix(rep(xg,ny),nx,ny))
yy <- matrix(rep(yg,nx),ny,nx)
fg <- outer(xg,yg,f)
## local polynomial estimate
## global bw:
ttg <- system.time(pdg <- locpoly(xg,yg,fg,
input="grid", pd="all", h=c(bwg,bwg), solver="QR", degree=dg, kernel=knl))
## time used:
ttg
## local bw:
ttl <- system.time(pdl <- locpoly(xg,yg,fg,
input="grid", pd="all", h=bwl, solver="QR", degree=dg, kernel=knl))
## time used:
ttl
image(pdg$x,pdg$y,pdg$z) contour(pdl$x,pdl$y,pdl$zx,add=TRUE,lty="dotted")
contour(pdl$x,pdl$y,pdl$zy,add=TRUE,lty="dashed") points(xx,yy,pch=".") ## part 2: ## irregular data, ## results will not be as good as with the regular 21*21=231 points. nd<- 41 # size of data set ## random irregular data oldseed <- set.seed(42) x<-runif(ng) y<-runif(ng) set.seed(oldseed) z <- f(x,y) ## global bw: ttg <- system.time(pdg <- interp::locpoly(x,y,z, xg,yg, pd="all", h=c(bwg,bwg), solver="QR", degree=dg,kernel=knl)) ttg ## local bw: ttl <- system.time(pdl <- interp::locpoly(x,y,z, xg,yg, pd="all", h=bwl, solver="QR", degree=dg,kernel=knl)) ttl image(pdg$x,pdg$y,pdg$z)
contour(pdl$x,pdl$y,pdl$zx,add=TRUE,lty="dotted") contour(pdl$x,pdl$y,pdl$zy,add=TRUE,lty="dashed")
points(x,y,pch=".")
# }
Documentation reproduced from package interp, version 1.0-33, License: GPL (>= 2)
### Community examples
Looks like there are no examples yet.
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2020-06-01 16:58:29
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https://www.aarki.com/blog/leveraging-external-datasets-for-probability-prediction-by-decomposing-probabilities
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## Read thought provoking articles on the use of machine learning for improving mobile app marketing performance.
By Igor Raush, Software Engineer
Advertisers are increasingly interested in optimizing their campaigns directly on the return on investment (ROI) or the return on ad spend (ROAS). In a real-time bidding setting, it becomes crucial to predict the expected revenue from a particular ad impression, which, in combination with the KPI, will determine the amount we are willing to bid.
### Model
To model post-impression and post-install user behavior, we can consider an entire event funnel, e.g.
$$\text{impression } (\mathbf x) \rightarrow \text{click} \rightarrow \text{install} \rightarrow \text{session} \rightarrow \text{conversion (e.g. purchase)}$$
which, for brevity, we can generalize as
$$\mathbf x \rightarrow e_1 \rightarrow e_2 \rightarrow \cdots \rightarrow e_N$$
Events which are further down the funnel are generally more valuable, more rare, and harder to predict. The ultimate goal is to learn the joint distribution $p(e_1, \dots, e_N \mid \mathbf x)$ of all funnel events, thereby learning to predict the probability of any event (in particular, the conversion probability) for each impression / user combination during bid time.
However, learning this distribution is complicated by the sparsity of the dataset. Samples with labels for all $e_i$ are extremely rare. Consider that a campaign serving 10 million impressions $\mathbf x$ could deliver only 10 conversions (purchases) $e_N$.
We can instead decompose the joint distribution as
$$p(e_1, \dots, e_N \mid \mathbf x) = p_N(e_N \mid e_{N-1}, \mathbf x) \dots p_1(e_1 \mid \mathbf x)$$
and learn each partial likelihood $p_i$ separately, from varying sources of data. In principle, this is an approximation; generally, the combination of maximum partial likelihoods does not give the overall maximum likelihood. In practice, this approach gives good estimates on many datasets.
### Applications
For instance, if we consider an impression-level dataset with labeled installs, and an external first-party advertiser dataset containing all installs and conversions delivered through any channel, we can train two logistic regression models, $\theta_1$ and $\theta_2$, such that
$p(\text{install} = 1 \mid \mathbf x) = \sigma \left( \theta_1^T \mathbf x \right)$
and
$p(\text{conversion} = 1 \mid \text{install}, \mathbf x) = \sigma \left( \theta_2^T \mathbf x \right)$
At this point, we can estimate the conversion probability via
$$p(\text{conversion} = 1 \mid \mathbf x) = \sigma \left(\theta_1^T \mathbf x \right) \sigma \left( \theta_2^T \mathbf x \right)$$
Note that $\theta_2$ here has considerably less variance since it is learned from a much larger dataset. There are some practical considerations with this approach:
1. The model $\theta_2$ can only rely on device-level features, since auction-level features are irrelevant for many user acquisition channels.
2. The conversion distribution in the external dataset must be sufficiently close to what we expect to see in the bid stream. For instance, if we find that organic and non-organic users exhibit very different purchase behaviors, we can choose to drop organic installs from the external dataset. Within fixed cohorts, the average conversion rate from the external dataset must match that from the bid stream; otherwise, the model will predict mis-calibrated probabilities.
Machine learning algorithms are often black boxes so choice and tuning of the algorithm is the key to success. Aarki’s data scientists and engineers are developing advanced machine learning algorithms to predict the expected revenue from a particular ad impression and make sure that your advertising budget is spent on the right users.
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2019-03-19 11:10:42
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http://physics.stackexchange.com/questions/23755/what-defines-the-brightness-of-a-bulb
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# What defines the brightness of a bulb?
So I have a question. There are three identical bulbs, 2 of them are connected in parallel and the third is basically in series, on the same circuit. If the one of the lamps in parallel breaks, what happens to the brightness of the other two? I don't know how to work this out, and what affects the brightness because I know its power but in this case we have to consider voltage or current.
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:please try to give a rough diagram in this kind of questions... I help you as well as the answerer... – Vineet Menon Apr 15 '12 at 6:13
Under assumption that three bulbs are connected to constant voltage, brightness actually changes. Brightness is very loosely proportional to power $P = U I = R I^2 = \frac{U^2}{R}$, so it is necessary to calculate the change of current/voltage through the remaining two bulbs, after the first breaks.
Considering your very case, if all three bulbs are the same and under assumption that resistivity of the bulb does not change with bulb's temperature (typical textbook assumption, which is actually not true), before the bulb breaks, the two paired bulbs have smaller brightness than the sole one. This is because the voltage splits in ratio 2:1 in favor of sole bulb. After the break, both remaining bulbs have the same brightness, because voltage splits 1:1.
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Ah therefore the bulb which used to be in series brightness will decrease and the one which used to be in parallel, now in series, brightness would increase correct? That is the right answer, I just didnt know how to get it. Also why does the voltage favour the sole bulb? If volatge is shared equally in parallel the two parallel bulbs would have same brightness, and since the bulbs are identical the one in series should have the same brightness as the others since it they have the same resistance. Am I on the right lines? – Cyrus Apr 15 '12 at 4:13
First part you are right. The second part you must replace all three bulbs with resistors. If you have two resistors in parallel and one in series, you can easily show that the two resistors in parallel get less voltage. – Pygmalion Apr 15 '12 at 5:37
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2016-05-02 10:49:18
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https://www.physicsoverflow.org/38073/possible-particle-infinite-interval-countable-uncountable?show=38081
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# Are the possible momenta of a particle in an infinite square well on the interval [-½,½] countable or uncountable?
+ 2 like - 1 dislike
573 views
An operator $P$ in Hilbert space is called self-adjoint only if the domains of $P$ and its adjoint $P^\dagger$ coinside. Therefore, if we want to consider the momentum of a particle inside a square well on $I=[-½,½]$, the least boundary conditions one has to impose on the momentum operator $P=-i\partial_x$ are given by the domain $$D_p=\{\psi,P\psi\in L^2[I]:\psi(½)=e^{i\theta}\psi(-½)\},$$ for some real angle $\theta$. In the following it might be sufficient to consider the special case $\theta=0.$ Then, the functions $$\phi_n(x)=e^{i 2 \pi nx},\qquad n=0,\pm1,\pm2,...$$ are eigenfunctions of $P$ with momentum eigenvalues
$$k_n=2\pi n$$ and form a complete orthonormal set.
The momentum operator and the energy operator $H$ of the system do not commute and one can not expect them to have a common eigenbasis. Actually, the boundary conditions of the eigenvalue problem of $H$ is $\psi(-½)=\psi(½)=0$, which is different to that of the momentum above. Now, in wikipedia the possible values of the momentum of this problem are explained to be continuous (https://goo.gl/vrJxly). That is, on one hand there is the countable momentum spectrum $k_n$ above, but on the other hand there is the uncountable spectrum proposed in literature.
Question: Which one of them is realized in nature?
Note: From the experimental point of view one could refer to a single slit diffraction experiment and conclude that the momentum spectrum should be continuous with their maxima at the values of the point spectrum. However, we know that the diffraction pattern is just far field approximation of the time evolution of the position wave function, which can only be considered as a (continuous) Fourier integral transformation if the momentum is (heuristically defined) proportional to the position of the particle.
edited Jan 4, 2017
One should not confuse the problem of a particle in the real line $\mathbb{R}$ subjected to an infinite square well potential to that of a particle confined in a circle $S^1$. In the latter problem (particle in the circle), the Fourier modes $\langle\theta|n\rangle=e^{inx/\hbar}/(2\pi\hbar)^{1/2}$ for $n \in \mathbb{Z}$ are simultaneous eigenstates of the Hamiltonian and momentum operators, and they form a complete orthonormal set.
On the other hand, in the problem of a particle in the real line within an infinite square well potential, the momentum states do not need to satisfy yours boundary condition imposed on $D_\boldsymbol{p}$, namely, that $\psi(0)=e^{i\theta}\psi(L)$ (I have translated the boundaries of the box from $[-1/2,+1/2]$ to $[0,L]$). The energy eigenfunctions need to have a support $\subset [0,L]$. But the momentum states wave functions are still the ones from the whole line, namely, $\langle x|p\rangle=e^{ipx/\hbar}/(2\pi \hbar)^{1/2} \forall p\in\mathbb{R}$, while the Hamiltonian eigenstates $Hf_n=Ef_n$ are now given (in position representation) by the functions $\langle x|f_n\rangle=(2/L)^{1/2}sin(2\pi n x/L)\forall x \in [0,L]$ and are zero otherwise, for $n \in \mathbb{Z}$. In the momentum representation, these energy eigenstates reads
$\tilde{f}_n(p)=\langle p|f_n\rangle = \int_{\mathbb{R}} dx\langle p|x\rangle\langle x|f_n\rangle\propto\int_0^L dx e^{-ipx/\hbar}sin(2\pi nx/L),$
agreeing with Wikipedia's very much well-known result that these are just the Fourier transform of the position representation energy eigenfunctions. Just observe that now, generally, $\tilde{f}_n(p) \neq 0$ for $p \in \mathbb{R}$, which shows that the stationary states for yours Hamiltonian are now a superposition of uncountable many momentum states of the line. (This is needed to make the support of the energy eigenstates compact.)
From a physical viewpoint, you can understand qualitatively what is going on from the uncertainty principle. Since the particle is within $x\in[0,L]$, the uncertainty in position is bounded from above, $\Delta x \le L$. So, since $\Delta p \Delta x \ge\hbar/2$ we cannot have $\Delta p=0$, that is, we always must have an uncertainty in the momentum. Thus the energy eigenstates are indeed expected to be formed by a superposition of momentum waves.
Now you ask which one is realized in Nature. This sounds preposterous, since all this is very much idealized. The best one can do here is to confine a particle in a finite but very deep square well. You will have an amplitude for the particle leaking the well, described by an exponential decreasing term etc., but the above remarks still holds, namely, the available momentum are still a continuum. (I once again remark that this is not the same problem as that of a particle in the circle!)
Note. As an exercise, I recommend you to do the following: let $k=2\pi n/L$ and, by taking the limit $L \longrightarrow\infty$$k$ becomes a continuous variable. Then, using the exponential representation of the sine function prove that
$\lim_{L \longrightarrow \infty}\tilde{f}_n(p)\propto \delta(k-p/\hbar)+\delta(k+p/\hbar).$
So we recover the result that $|p|=\hbar |k|$.
If one does not respect the "twisted" boundary conditions $\psi(0)=e^{i\theta}\psi(L)$ for a particle in the box, then the momentum operator would not be self-adjoined. And I'm not talking about a particle on the circle, but about the least conditon for self-adjointness of the momentum of a particle with a wave-function of compact support.
Concerning the uncertainty principle, I did not state that the momentum dispersion does vanish. On the contrary, by a simple computation we can see that the momentum disperison satisfies the tight inequality $\sigma_p L\geq\pi\hbar$ in this approach. So, everyting seems to be consistent.
@IgorMol Great comment! But since it's quite long, could you post it as an answer instead?
@kaffeeauf The basic issue is that only the Hamiltonian has to be a self-adjoint operator in order to give a well-defined eigenvalue problem. It is not necessary for the momentum operator to be self-adjoint.
Thank you for your positive feedback Greg, I have just copied my comment to an answer.
+ 3 like - 0 dislike
One should not confuse the problem of a particle in the real line $\mathbb{R}$ subjected to an infinite square well potential to that of a particle confined in a circle $S^1$. In the latter problem (particle in the circle), the Fourier modes $\langle\theta|n\rangle=e^{inx/\hbar}/(2\pi\hbar)^{1/2}$ for $n \in \mathbb{Z}$ are simultaneous eigenstates of the Hamiltonian and momentum operators, and they form a complete orthonormal set.
On the other hand, in the problem of a particle in the real line within an infinite square well potential, the momentum states do not need to satisfy yours boundary condition imposed on $D_\boldsymbol{p}$, namely, that $\psi(0)=e^{i\theta}\psi(L)$ (I have translated the boundaries of the box from $[-1/2,+1/2]$ to $[0,L]$). The energy eigenfunctions need to have a support $\subset [0,L]$. But the momentum states wave functions are still the ones from the whole line, namely, $\langle x|p\rangle=e^{ipx/\hbar}/(2\pi \hbar)^{1/2} \forall p\in\mathbb{R}$, while the Hamiltonian eigenstates $Hf_n=Ef_n$ are now given (in position representation) by the functions $\langle x|f_n\rangle=(2/L)^{1/2}sin(2\pi n x/L)\forall x \in [0,L]$ and are zero otherwise, for $n \in \mathbb{Z}$. In the momentum representation, these energy eigenstates reads
$\tilde{f}_n(p)=\langle p|f_n\rangle = \int_{\mathbb{R}} dx\langle p|x\rangle\langle x|f_n\rangle\propto\int_0^L dx e^{-ipx/\hbar}sin(2\pi nx/L),$
agreeing with Wikipedia's very much well-known result that these are just the Fourier transform of the position representation energy eigenfunctions. Just observe that now, generally, $\tilde{f}_n(p) \neq 0$ for $p \in \mathbb{R}$, which shows that the stationary states for yours Hamiltonian are now a superposition of uncountable many momentum states of the line. (This is needed to make the support of the energy eigenstates compact.)
From a physical viewpoint, you can understand qualitatively what is going on from the uncertainty principle. Since the particle is within $x\in[0,L]$, the uncertainty in position is bounded from above, $\Delta x \le L$. So, since $\Delta p \Delta x \ge\hbar/2$ we cannot have $\Delta p=0$, that is, we always must have an uncertainty in the momentum. Thus the energy eigenstates are indeed expected to be formed by a superposition of momentum waves.
Now you ask which one is realized in Nature. This sounds preposterous, since all this is very much idealized. The best one can do here is to confine a particle in a finite but very deep square well. You will have an amplitude for the particle leaking the well, described by an exponential decreasing term etc., but the above remarks still holds, namely, the available momentum are still a continuum. (I once again remark that this is not the same problem as that of a particle in the circle!)
Note. As an exercise, I recommend you to do the following: let $k=2\pi n/L$ and, by taking the limit $L \longrightarrow\infty$$k$ becomes a continuous variable. Then, using the exponential representation of the sine function prove that
$\lim_{L \longrightarrow \infty}\tilde{f}_n(p)\propto \delta(k-p/\hbar)+\delta(k+p/\hbar).$
So we recover the result that $|p|=\hbar |k|$.
answered Jan 8, 2017 by (550 points)
+ 2 like - 0 dislike
There are two different problems to distinguish:
1) a free particle in a box with periodic boundary conditions, i.e. a free particle on a circle.
2) a particle on a line, in a infinite square well, i.e. a particle in a box.
In 1), the space of states is the space of periodic functions (i.e. functions on a circle) and the spectrum of the momentum is discrete. In this case, eigenstates of momentum are normalizable states and are also the eigenstates of the free Hamiltonian.
In 2), the space of states is the space of L^2 functions on a line. The spectrum of the momentum is continuous and eigenstates of the momentum are not normalizable. Eigenstates of the Hamiltonian are normalizable states (and are not eigenstates of the momentum).
1) and 2) being different problems, it does not make sense to say that only one of them is relevant. The relevance of 1) or 2) for a particular experiment will depend on the details of the experiment.
Mathematically, 1) is much simpler than 2) and for a large box, 1) and 2) becomes close to each other (for a large box, what happens at the boundaries becomes less relevant) and so one can use results about 1) to derive approximate results about 2).
answered Jan 2, 2017 by (5,140 points)
The particle under consideration has finite support $[−½,½]$ and the eigenvalue problem of the Hamiltonian is solved with respect to that support. Why should that not be the goal for the eigenvalue problem of the momentum operator - for instance, to ensure self-adjointness of the momentum operator with respect to $[−½,½]$, instead to the infinite real line? Neither in wikipedia nor in your (or Igor's) explanation I can see a justification why the momentum eigenvalue problem should make reference to positions outside $[−½,½]$. If you have an idea then please let me know.
In general, the infinite square well is defined as the limit of a finite square well becoming deeper and deeper. For a finite square well, the wave function does not have finite support but has exponentially small contributions when going to infinity. Only in the infinite square well limit, the eigenfunctions of the Hamiltonian become of finite support: it is not something imposed from the beginning but it comes from solving the Schrödinger equation on a line.
But it seems that you want from the beginning restrict the space of wave functions to the space of functions with support on [-1/2,1/2] and vanishing at the boundaries. Solving the eigenvalue problem for the Hamiltonian on this space of functions will indeed be equivalent to the above approach on the real line. But now the momentum operator is no longer defined: the derivative of a function vanishing at the boundaries of an interval has no reason to vanish there. It is physically clear: momentum is the generator of translations, so if your space of wave functions is not invariant under translation, there is no way to define a momentum operator on this space.
Sure, in my question I only consider the case where the square well is infinite just from the beginning. As far as I know, that seems to be the case im most standard contributions in literature. However, it would be interesting to know a reference where the limit process is performed afterwards.
I think, the vanishing boundary conditions of the energy eigenfunctions are only appropriate for the eigenvalue problem of the energy. For the momentum operator the eigenstates are expected to be different form the eigenfunctions of $H$ because of the incommensurability of $H$ and $P$. That releases us from the restriction that the momentum eigenstates or their derivatives have to vanish at the boundary. Anyway, general wave functions with vanishing boundary conditions (i.e. the ground state of $H$) can also be constructed in the Hilbert space generated by the eigenstates of the momentum operator. In my (humble) opinion the most important criterion is that the wave functions and their derivatives have to be members of $D_p$ such that self-adjointness of $P$ is ensured. However, what I miss is a justification of the superposition of the momentum eigenstates which goes beyond the theory of Fourier series at all.
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2021-05-12 23:53:35
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https://www.geocaching.com/geocache/GC60QT7_kaleidocycle
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##### This cache has been archived.
charchoc: Archiving as the bike park is closing, so wncsteph, tigercash, RossCo.22 and myself removed this and other cache containers in the park.
Great hike today and so sad that this great spot will no longer be used for caching, hiking or biking.
Thank you all for visiting this and the other caches in the park!
More
<
## Kaleidocycle
A cache by charchoc Message this owner
Hidden : 08/02/2015
Difficulty:
Terrain:
Size: (regular)
#### Watch
How Geocaching Works
Please note Use of geocaching.com services is subject to the terms and conditions in our disclaimer.
### Geocache Description:
Posted coordinates are not for final.
Final coordinates are: N 35 43.?!$, W 082 37.%<^ Puzzle cache that one has to construct a geometric figure to be able to obtain coordinates for the cache. Please hide container and its contents so it is not visible from the trail. "A kaleidocycle is a closed chain of tetrahedra that can cycle endlessly through a center hole.” Best known for his strangely realistic depictions of things that defy the laws of physics, Maurits Cornelis Escher became interested in problems of repetition and symmetry after traveling to the Alhambra, a 14th century Moorish castle in Granada, Spain. Fascinated by the periodic (i.e. regularly recurrent) designs of the castle's mosaics, he began to pursue the idea that a plane can be divided into uniform, interlocking figures, forming a pattern that repeats itself at set intervals, theoretically to infinity. Your are going to download, enlarge and print out the templates. thicker paper recommended. Cut one out, then construct a kaleidocycle using the instructions from the files on the cache page. Once you construct the kaleidocycle, follow the coordinate instructions to fold it in and out to obtain the coordinates to the cache. *Be advised, you may need the kaleidocycle when you get to the cache. Instructions for folding the Kaleidocycle to obtain the cache coordinates: For North coordinates: After constructing the kaleidocycle, fold unit until the “N” is facing away from you and at the top or “North” position, as noted in illustration labeled with the “N” and two dots and “Step 1a” on it. (You may have to turn the object over and fold a few times to get it this position.). Step 1: Turn inside out by folding the unit from the back and folding up towards you, one time/cycle. Do not move or rotate the object. Keep it in it’s relative position. Step 1a: Read the second number going counterclockwise from where the “N” was. This is “?”. Step 2: Repeat step one. Step 2a: Read the second number going counterclockwise from where the number for “?” was. This is “!”. Step 3: Repeat step one. Step 3a: Read the second number going counterclockwise from where the number for “!” was. This is “$”.
For West coordinates: Fold the unit until the “E” and the one dot is in the position, as noted in illustration labeled with the “E” and one dot and “Step 4a” on it.
Step 4: Turn inside out by folding the unit from the top and folding down away from you, one time/cycle. Do not move or rotate the object. Keep it in it’s relative position.
Step 4a: Read the number opposite from where the “E” was. This is “%”.
Step 5: Repeat step four.
Step 5a: Read the next number clockwise from the position where “%” was. This is “<”.
Step 6: Repeat step four.
Step 6a: Read the next number clockwise from the position where “<” was. This is “^”.
Attached files, include:
Kaleidoscope Templates (Two in case you need to make another)
Kaleidoscope Construction
Coordinate Instructions
CHECKSUM: ?+!+\$+%+<+^=XY, X+Y=11
FTF HONORS GO TO spink167!!!
Ybt vg.
Decryption Key
A|B|C|D|E|F|G|H|I|J|K|L|M
-------------------------
N|O|P|Q|R|S|T|U|V|W|X|Y|Z
(letter above equals below, and vice versa)
# Reviewer notes
Use this space to describe your geocache location, container, and how it's hidden to your reviewer. If you've made changes, tell the reviewer what changes you made. The more they know, the easier it is for them to publish your geocache. This note will not be visible to the public when your geocache is published.
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2021-06-16 00:50:41
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https://ard.bmj.com/content/78/9/1226
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Article Text
Fatigue in SLE: diagnostic and pathogenic impact of anti-N-methyl-D-aspartate receptor (NMDAR) autoantibodies
Free
1. Andreas Schwarting1,2,
2. Tamara Möckel1,
3. Freya Lütgendorf2,
4. Konstantinos Triantafyllias2,
5. Sophia Grella1,
6. Simone Boedecker1,
7. Arndt Weinmann1,
8. Myriam Meineck1,
9. Clemens Sommer3,
10. Ingrid Schermuly4,
11. Andreas Fellgiebel4,
12. Felix Luessi5,
13. Julia Weinmann-Menke1
1. 1 Division of Rheumatology and Clinical Immunology, University Medical Center of the Johannes Gutenberg University Mainz, Mainz, Germany
2. 2 Acura Rheumatology Center Rhineland Palatinate, Bad Kreuznach, Germany
3. 3 Institute of Neuropathology, University Medical Center of the Johannes Gutenberg University Mainz, Mainz, Germany
4. 4 Department of Psychiatry and Psychotherapy, University Medical Center of the Johannes Gutenberg University Mainz, Mainz, Germany
5. 5 Department of Neurology, University Medical Center of the Johannes Gutenberg University Mainz, Mainz, Germany
1. Correspondence to Andreas Schwarting, Division of Rheumatology and Clinical Immunology, University Medical Center of the Johannes-Gutenberg University Mainz, Mainz 55131, Germany; schwarting{at}uni-mainz.de
## Abstract
Objectives We explored the impact of circulating anti-N-methyl-D-aspartate receptor (NMDAR) antibodies on the severity of fatigue in patients with systemic lupus erythematosus (SLE).
Methods Serum samples of 426 patients with SLE were analysed for the presence of antibodies to the NR2 subunit of the NMDAR. In parallel, the severity of fatigue was determined according to the Fatigue Scale for Motor and Cognitive functions questionnaire. In a subgroup of patients with SLE, the hippocampal volume was correlated with the levels of anti-NR2 antibodies. Isolated immunoglobulin G from patients with anti-NR2 antibodies were used for murine immunohistochemical experiments and functional assays on neuronal cell lines. Treatment effects were studied in 86 patients with lupus under belimumab therapy.
Results We found a close correlation between the titre of anti-NR2 antibodies, the severity of fatigue, the clinical disease activity index (Systemic Lupus Erythematosus Disease Activity Index 2000) and anti-double stranded DNA antibodies—independently of the presence of neuropsychiatric lupus manifestations. Pathogenic effects could be demonstrated by (1) detection of anti-NR2 antibodies in the cerebrospinal fluid, (2) in situ binding of anti-NR2 antibodies to NMDAR of the hippocampus area and (3) distinct functional effects in vitro: downregulating the energy metabolism of neuronal cells without enhanced cytotoxicity. Treatment with belimumab for at least 6 months affected both the severity of fatigue and the levels of anti-NR2 antibodies.
Conclusion The presence of anti-NR2 antibodies in patients with SLE with fatigue is a helpful diagnostic tool and may offer a major approach in the therapeutic management of this important disabling symptom in patients with SLE.
• fatigue
• systemic lupus erythematosus
• anti-nmda receptor antibodies
• belimumab
View Full Text
## Statistics from Altmetric.com
### Key messages
• Fatigue is known as a predominant symptom of patients with sytemic lupus erythematosus (SLE) across different ethnicities resulting in substantially reduced quality of life and work disability. However, fatigue is not part of clinical activity scores (eg, Systemic Lupus Erythematosus Disease Activity Index 2000 or Systemic Lupus Erythematosus International Collaborating Clinics).
• It is well established that autoantibodies reacting with the N-methyl-D-aspartate receptor are closely associated with neurocognitive impairment in neuropsychiatric SLE.
#### What does this study add ?
• A link between one of the most challenging symptoms patients with lupus suffer from and the presence of brain-reacting autoantibodies. The study offers an objective measurement for patients with SLE complaining about fatigue in routine clinical practice.
#### How might this impact on clinical practice or future developments ?
• The measurement of anti-NR2 antibodies in patients with SLE with fatigue might be introduced into clinical disease activity scores to better address the patient-related quality of life in future clinical studies and support the development of an individualised therapy of patients with SLE.
## Introduction
Systemic lupus erythematosus (SLE) is a chronic autoimmune disease affecting multiple organ systems in a variety of clinical manifestations. A subscale analysis of quality of life (QoL) in patients with SLE revealed that fatigue, a rather subjective symptom with partly still unknown pathophysiology, may have greater effects on the physical QoL than disease activity or damage index of SLE.1 This underlines the importance of fatigue as an SLE-associated complaint occurring in 67%–90% of patients regardless of their ethnicity.2 3 Often described as a constant feeling of exhaustion, fatigue impairs physical as well as mental aspects of life and is associated with abnormal illness behaviours and work disability.2
Recently, the FATILUP study (FATigue in LUPus), which included 570 patients from the Lupus BioBank of the upper Rhein (LBBR), was conducted to identify the determinants associated with fatigue in a large cohort of european lupus patients. managed to identify determinants associated with fatigue. Definite depression and anxiety were found to be strong contributors, whereas the use of glucocorticoids, disease activity (Systemic Lupus Erythematosus Disease Activity Index 2000 (SLEDAI-2K)) and patient age were more weakly associated.4
Since fatigue itself is a complex and subjective symptom, it is so far not amenable to objective measurements; thus diagnosis is still based on questionnaires. The Fatigue Scale for Motor and Cognitive Functions (FSMC) was initially developed and validated for patients with multiple sclerosis in 2009. Consisting of 20 items, it allows a semiquantitative description of severity and a discrimination between cognitive and motor focused fatigue with overall good reliability, sensitivity and specificity values.5
One major obstacle in the process of establishing objective diagnostic tools is the lack of a complete understanding of fatigue’s pathogenesis.
With regard to SLE and its neuropsychiatric manifestations (neuropsychiatric SLE (NPSLE)), anti-double stranded (ds) DNA antibodies cross-reacting with a single epitope present in GluN2A/B subunits of the N-methyl-D-aspartate receptor (NMDAR) have been identified as pathogenic factors closely associated with NPSLE.6 7 A dose-dependent interaction between anti-NR2 antibodies and self-antigens located in the brain as well as the distribution pattern of NMDAR antibodies linking either impaired memory and hippocampal atrophy or emotional disturbances and atrophy of the amygdala underline the importance of anti-NR2 antibodies in SLE.8 9 Taking into regard the confirmed association of anti-NR2 antibodies and neuropsychiatric manifestations in patients with SLE, changes in antibody titres might possibly predict the clinical condition in individual patients as Ando et al postulated for patients with psychiatric symptoms in general.10
Based on these considerations, we sought to determine the relevance of circulating anti-NR2 antibodies in patients with SLE with fatigue.
## Methods
### Serum samples
We analysed serum samples from patients who fulfilled at least four of the American College of Rheumatology criteria (ACR) for the classification of SLE, after informed consent. Healthy volunteers (age range 18–70 years) were screened for any prior kidney diseases, diabetes, hypertension, apoplex, infection and autoimmune diseases. Freshly drawn blood samples were collected, centrifuged and sera aliquots stored at −80°C. In the subgroup of belimumab-treated patients with SLE, we have used serum samples before and after belimumab treatment. The use of these specimens was approved by the Standing Committee for Clinical Studies of the Johannes-Gutenberg University in adherence to the Declaration of Helsinki. All samples were analysed retrospectively in Mainz.
### Disease activity
Disease activity was evaluated using the SLEDAI-2K and standard serologic activity measures (C3c, C4, anti-dsDNA antibodies, creatinine, C-reactive protein (CRP), erythrocyte sedimentation rate (ESR) and urine measures (proteinuria (24 hours collection) and active sediment). The following standard values were determined: C3 (0.9–1.8 g/L), dsDNA (200–1000 IU/mL) by ELISA and proteinuria (<150 mg/24 hours) by immunoturbidimetric assay. Fatigue was assessed by the FSMC questionnaire.5
### Subjects MRI cohort
Forty outpatients with SLE were admitted at the Department of Internal Medicine of the University Medical Centre Mainz. All patients fulfilled the revised criteria of the ACR.11 Patients underwent a physical examination and laboratory testing (C3, C4). SLE status was assessed using the SLEDAI-2K.12 Sociodemographic data, SLE characteristics (ie, age at diagnosis, disease duration, cortisone dosage, SLEDAI-2K score), psychiatric, neuropsychological and structural imaging parameters of MRI scans were collected. As seven MRI sequences were incomplete, the data of 33 patients only could be analysed. None of these patients were part of the belimumab group.
The ACR criteria for neuropsychiatric lupus syndromes (NPSLE) were used to classify neuropsychiatric manifestations.13 Fourteen out of 33 patients fulfilled the ACR criteria. Demographical and clinical data did not differ between patients with SLE with and without neuropsychiatric syndromes (all p’s >0.05). Fifteen healthy controls were recruited through advertisement in a local newspaper and screened for the absence or presence of psychiatric disorders via the Stem Item Screening Questions (SSQ) from the diagnostic interview DIA-X.14 Determination of hippocampal volumes was performed using Analyse Software 150 (V.8.1; Biomedical Imaging Software System, Mayo Foundation for medical education and research, Rochester, New York, USA). Hippocampal volumetry methods have been described in details.15 Hippocampal volumes are represented as absolute volumes (left+right in mL). All participants gave written informed consent.
### Cell lines
CHP-126 neuroblastoma cells (Cat. No. 300432), CCF-STTG1 astrocytoma cells (Cat. No. 300388; Cell Lines Services, Eppelheim, Germany).
## ELISA
Anti-NR2 antibody levels in human serum were analysed using Gold Dot NR2 antibody test kit (Cat. No. GD1-001, CIS Biotech) according to the manufacturer’s instructions.
### Immunoglobulin (Ig)G purification of serum
IgGs were purified from serum of patients with SLE with high or low NR2 antibody titres by using the NAb Protein G Spin Kit (Cat. No. 89979, Thermo Scientific) according to manufacturer’s protocol. For desalting, the purified IgGs Zeba Spin Desalting Columns (Cat. No. 89889, Thermo Scientific) were used.
### Immunoprecipitation of anti-NR2-Ab
Immunoprecipitation was performed by using Protein A/G PLUS-Agarose (Cat. No. sc-2003, Santa Cruz Biotechnology) according to manufacturer’s instructions.
### Cytotoxicity and ATP quantification
The 20% purified IgGs (as described above) from patients with SLE with high titres of anti-NR2-Ab in culture media were blocked with 50 ng/µL peptide NR2B (Cat. No. crb1200337, Discovery antibodies, Cleveland, UK) overnight at 4°C on a rotating device.
We stimulated cultured cells (Roswell Park Memorial Institute medium, 20% fetal calf serum, 1% glutamin, 1% penstreptomicin) for 24 hours with purified IgGs from patients with SLE with high titres of anti-NR2-Ab, without anti-NR2-Ab or healthy control serum. After incubating for 24 hours at 37°C at 5%CO2, we analysed cytotoxicity using the ToxiLight BioAssay Kit (Cat. No. LT07-217, Lonza) and the ATP quantification using the Fluorometric and Colorimetric ATP Quantification Kit (Cat. No. PK-CA577-K354, PromoKine) following the manufacturer’s instructions.
### Immunohistochemistry
Murine brain of 5-weeks-old male MRL/MpJ-Faslpr lupus mice were fixed in 4% phosphate buffered formalin, embedded in paraffin and coronal sectioned (5 µm) by the Institute for Neuropathology of the University Medical Center, Mainz. Sections were stained for the presence of NMDA-type glutamate receptor subunit 2B (purified rabbit anti-mouse GluN2B: GluRec2C-Rb-Af300, RRID: AB_2571762, Frontier Institute). Sections were either incubated with primary GluN2B antibodies at a 1:50 dilution, purified IgGs from patients with SLE with high titres of anti-NR2-Ab (1:20), healthy control serum (1:20) or NR2 positive SLE serum which was blocked with peptide NR2B at a 1:20 dilution overnight at 4°C (details are given in the online supplementary file).
### Statistical analysis
We used the non-parametric Mann-Whitney U test for comparison between two groups and the Kruskal-Wallis test for comparisons between three or more groups. For correlation analysis, we used the Spearman correlation coefficient.
The data represent the mean±SEM and were prepared using GraphPad Prism V.7.0 (GraphPad, San Diego, California, USA).
## Results
### Fatigue is a predominant symptom in the Mainz lupus cohort
The Mainz lupus cohort comprises 569 patients with SLE. Analysis of this cohort versus 159 healthy controls demonstrated significantly increased anti-NR2 antibodies in sera of patients with SLE (figure 1A). A subgroup of 426 patients ranging from 18 to 76 years completed the FSMC questionnaire (table 1). The detailed characteristics of the lupus cohort at the University Medical Centre Mainz are given in a online supplementary table.
### Supplemental material
Table 1
Demographic and clinical data
Figure 1
Fatigue and NR2 Abs correlate in SLE. (A) Serum NR2 antibody levels in patients with SLE and healthy controls quantified by ELISA. (B) Incidence of fatigue in patients with SLE evaluated by the FSMC score (self-assessment by the patient). (C) Correlation of serum NR2 antibody levels and the severity and type (motoric, cognitive) of fatigue (evaluated by the FSMC core). (D) Correlation analysis with clinical activity markers, graph displays SLEDAI-2K versus NR2 antibodies, and table displays correlation of NR2 antibodies and serological activity markers of SLE. (E) Correlation analysis of fatigue (by FSMC score), depression (evaluated by BDI) and NR2 antibody level in patients with SLE. Demographic and patient clinical characteristics are detailed in table 1. Statistics analysed by the Mann-Whitney U test. *P<0.05, **P<0.01, ***P<0.001. Values are means±SEM. BDI, Beck’s Depression Inventory; CRP, C-reactive protein; ESR, erythrocyte sedimentation rate; FSMC, Fatigue Scale for Motor and Cognitive functions; SLE, systemic lupus erythematosus; SLEDAI-2K, Systemic Lupus Erythematosus Disease Activity Index 2000.
### Correlation of circulating anti-NR2 antibodies with severity of fatigue in patients with SLE
Fatigue is present in 75% of 426 consecutively assessed patients with lupus (figure 1B). Among those, more than 50% are even graded as severe fatigue based on the FSMC score as determined by Penner et al 5 (figure 1C). Using four different grades of fatigue severity, we investigated the correlation between the clinical extent of fatigue and the titre of anti-NR2 antibodies (figure 1C–E). We detected a significant difference between the level of anti-NR2 antibodies and the severity of fatigue as depicted in figure 1C. Moreover, there was a correlation between the levels of anti-NR2 antibodies in sera with the fatigue score in total as well as with its partial components representing motoric and cognitive fatigue in specific (figure 1C).
Furthermore, we investigated the relationship of anti-NR2 antibodies with other parameters of disease activity in SLE. Interestingly, anti-NR2 antibody titres correlated with SLEDAI-2K and anti-dsDNA antibodies, whereas no significant correlation was found with complement factors, ESR, CRP or renal function (figure 1D). Depression and fatigue showed a clear correlation, while the level of anti-NR2 antibodies did not correlate with the depression score (Beck’s Depression Inventory)(figure 1E).
In addition, there was no correlation between the anti-NR2 antibody levels and the presence of neuropsychiatric disease manifestations (data not shown).
### Anti-NR2 antibodies can be detected in the cerebrospinal fluid of patients with SLE and bind to the hippocampus area in situ
In 22 patients with SLE, we were able to analyse the levels of anti-NR2 antibodies in cerebrospinal fluid (CSF) samples (figure 2A). We detected low levels of anti-NR2 antibodies in CSF and simultaneously high titres of anti-NR2 antibodies in the sera (figure 2A).
Figure 2
Detection of anti-NR2 antibodies in the cerebrospinal fluid of patients with SLE and binding of SLE sera within the brain. (A) In SLE sera, high levels of anti-NR2 antibodies were detected, while the anti-NR2 antibody titres of cerebrospinal fluid were low. (B) Immunohistochemical analyses showed a strong binding of SLE sera with high titres of anti-NR2B antibodies within the brain of MRL lupus mice, predominantly to the hippocampus area. Anti-GluN2B at a 1:50 dilution showed a strong staining of the hippocampus area (a). This staining pattern was also observed with SLE serum highly positive for anti-NR2 antibodies at a 1:20 dilution (b) but not in case of using a healthy control (c) or SLE serum which was blocked with peptide 2 (d), both 1:20 diluted. (C) Top panel: correlation analysis of the hippocampal volume and NR2 antibody levels, hippocampal volume and SLE activity determined by SLEDAI-2K. Bottom panel: correlation analysis of NR2 antibody levels, fatigue score, SLEDAI-2K and C3c. Demographic and patient clinical characteristics are detailed in table 2. Statistics analysed by the Mann-Whitney U test. *P<0.05,**P<0.01. Values are means±SEM. SLE, systemic lupus erythematosus; SLEDAI-2K, Systemic Lupus Erythematosus Disease Activity Index 2000.
In addition, immunohistochemistry revealed a strong binding of SLE sera with high titres of anti-NR2 antibodies within the brain of MRL lupus mice (figure 2B). The expression is predominantly located at the hippocampus area (figure 2B) and can be specifically blocked by coincubation with NR2 peptides (figure 2B).
### Anti-NR2 antibodies affect the hippocampus volume in patients with lupus with fatigue
Based on the in situ results, we sought to determine whether there is a long-term effect of anti-NR2 antibodies on the hippocampus volume of 33 patients with lupus with severe fatigue (figure 2C). Demographic and clinical data of patients (MRI-cohort) and controls are given in table 2.
Table 2
Demographic and clinical data of the MRI cohort of patients with SLE
Following a 2-year observation time, a reduction in hippocampal volume was associated with high titres of anti-NR2 antibodies, the SLEDAI-2K score and low C3c complement levels, while there was no correlation with the fatigue score reported by patients (figure 2C, online supplementary table).
### Functional effects of anti-NR2 antibodies on different neuronal cells in vitro
Based on the initially illustrated strong correlation between anti-NR2 antibodies and severity of fatigue in patients with lupus, in vitro tests were performed to reveal possible pathophysiological effects.
For this purpose, two different cell lines (astrocyte CCF-STTG1, neuroblastoma CHP-126) were exposed to anti-NR2 antibody positive and negative serum samples of patients with lupus.
Those cell lines are good candidates for the in vitro experiments since both express NMDA receptors 1 and 2 as shown by North et al and Lee et al 16 17
Anti-NR2 antibodies showed a remarkable effect on cell activity by reducing the ATP production in neuronal cells (figure 3A, upper left). Moreover, this effect is mediated via the NR2 receptor since coincubation with NR2-blocking peptides completely abrogated the modulation of cell activity (figure 3A, upper right). However, a similar experimental setup measuring cytotoxicity did not show any enhanced cytotoxic effects of cultured astrocytes (figure 3B). Thus, anti-NR2 antibodies exert functional impairment of astrocytes in vitro without inducing cell death suggesting a reversible mechanism. Comparable results were observed for neuroblastom cell line (CHP-126) (data not shown).
Figure 3
Functional effects of anti-NR2 antibodies on astrocytes in vitro. (A) Influence of anti-NR2 antibodies on cell activity of cultured astrocytes. Left panel: ATP production of astrocytes incubated with SLE serum was clearly lower compared with untreated cells or astrocytes which were incubated with healthy control serum. Right panel: cell activity of astrocytes which were incubated with peptide 2 blocked SLE serum was significantly higher compared with astrocytes treated with unblocked SLE serum. (B) Cytotoxicity of anti-NR2 antibodies on astrocytes. Anti-NR2 antibodies showed no enhancing effect on cytotoxicity of cultured astrocytes. Statistics analysed by the t-test. *P<0.01. Values are means±SEM. SLE, systemic lupus erythematosus.
### Effect of belimumab therapy on fatigue and anti-NR2 antibodies
Based on the reported effects of belimumab on fatigue, we analysed the subgroup of belimumab treated patients in our SLE cohort.18 Eighty-six patients with lupus with at least 6 months under belimumab therapy performed the clinical fatigue score before and after belimumab therapy simultaneously to the evaluation of anti-NR2 antibody titre (figure 4).
Figure 4
Treatment with belimumab results in reduction of serum NR2 antibodies accompanied by an amelioration of fatigue in SLE. (A) Serum NR2 antibody levels in patients with SLE before treatment with belimumab and at least 6 month following the first treatment with belimumab quantified by ELISA. (B) Severity of total fatigue, motoric and cognitive fatigue before and under belimumab treatment. (C) Clinical activity (SLEDAI-2K, dsDNA, ESR (1 hour), C3c, C4 ad CRP) of patients with SLE before and under belimumab treatment. Demographic and patient clinical characteristics are detailed in table 1. Statistics analysed by the Mann-Whitney U test. *P<0.05, **P<0.01, ****P<0.0001. Values are means±SEM. CRP, C-reactive protein; dsDNA, double stranded DNA; ESR, erythrocyte sedimentation rate; SLE, systemic lupus erythematosus; SLEDAI-2K, Systemic Lupus Erythematosus Disease Activity Index 2000.
1. In the course of belimumab therapy (mean time under therapy 12±8.5, range from 6 to 36 months), a significant decline in anti-NR2 antibody levels was monitored (figure 4A).
2. In parallel to the decrease of anti-NR2 antibodies under therapy with belimumab, we observed a clinically significant reduction of the fatigue score, regarding the total score as well as the motoric and cognitive components (figure 4B).
3. Concerning our subgroup of patients with available data on fatigue score and anti-NR2 antibody levels under therapy with belimumab, a significant improvement in disease activity, as measured by SLEDAI-2K, was demonstrated. Laboratory parameters, such as anti-dsDNA antibodies, ESR, complement factors and CRP did not show significant differences (figure 4C).
## Discussion
Our data reveal that anti-NR2 antibodies correlated with fatigue severity in patients with SLE. To our knowledge, this is the first study to examine this relationship in a large SLE cohort. Interestingly, the found correlation was not restricted to patients with NPSLE.
While anti-NR2 antibodies could be found in 529 patients with SLE, 103 patients were excluded from further analysis if (1) the questionnaire was not completed or (2) if the date of the questionnaire did not match the date of the serum sample in our biobank (range from <2 weeks apart was allowed). However, both groups as well as the subgroup of patients with MRI or belimumab treatment did not differ regarding the clinical activity of SLE.
In general, the role of anti-NR2 antibodies in cerebrospinal fluid (CSF) and/or serum has been examined in studies regarding different NPSLE forms.9 19–24 Level of anti-NR2 antibodies in CSF have been found to correlate with both diffuse and focal NPSLE.20 21 In the study of Gono et al, serum anti-NR2A antibodies correlated with higher disease activity (SLEDAI-2K), low leucocyte counts, low haemoglobin values and higher frequency of NPSLE.22 Furthermore, Lapteva et al showed an association of serum anti-NR2 antibodies with depressive mood (but not with cognitive impairment),23 whereas Omdal et al showed an additional correlation of these antibodies with decreased short-time memory and learning abilities.24
Thus, since the pioneering studies of DeGiorgio et al 25 on the cross-reactivity of anti-dsDNA antibodies with NMDAR subunits GluN2A and GluN2B, it has been well established that these antibodies are closely associated with neurocognitive impairment in NPSLE.26–30
Here, we now report that anti-NMDAR antibodies strongly correlate with fatigue even in patients with lupus without overt signs of neuropsychiatric manifestations according to the ACR classification criteria.13
However, our data clearly show that the anti-NMDAR antibodies found in our patients had a similar pathogenic potential: (1) they could be detected both in CSF and serum, (2) the isolated IgGs from patients with lupus with fatigue bound to NMDAR in the brain in situ and (3) anti-NR2 antibodies downregulated the energy metabolism of cultured neuronal cells. Further studies of anti-NR2-mediated fatigue will elucidate whether this mechanism is unique to patients with SLE.
Increased anti-NR2 antibodies have been reported in 25%–38% of patients with SLE.22 31–33 It is until now unclear whether elevation of anti-NR2 antibodies in the CSF of patients with SLE is due to increased intrathecal synthesis or to a transport from the peripheral blood circulation to the CSF through a damaged blood–brain barrier (BBB).34 35 In the study of Kowal et al, mice which were antigen-induced to express anti-NR2 antibodies experienced apoptotic cell death only after administration of lipopolysaccharide (LPS), a substance known to lead to BBB breakdown.36 Anti-NR2 antibodies bound mainly to neurons of the hippocampus and led to cell death causing cognitive dysfunction and altered hippocampal metabolism.36 The same study group showed that mice given a combination of serum of patient with SLE with reactivity to DNA/NMDAR and LPS demonstrated cognitive impairment.37
Interestingly, states of BBB disruption such as septic meningitis associate with rise of all intrathecal autoantibodies levels, including anti-NMDAR antibodies.35 Moreover, SLE (and especially NPSLE) is characterised by increased permeability of the BBB.38 Similarly, in our study, the presence of anti-NR2 antibodies and fatigue is not restricted to patients with SLE with NPSLE.
The experimental studies point to different pathomechanisms of anti-NR2 antibodies including irreversible cell death or reduced energy metabolism.7 In our in vitro studies, we did not find a cytotoxic effect of the isolated anti-NR2 antibody IgGs, but a clear impact on ATP metabolism of neuronal cells. This fits quite well to the clinical experience that fatigue in patients with lupus responds to immunosuppressive therapy.18
By comparison, a different pathomechanism seems to be involved in limbic encephalitis caused by autoantibodies to NMDAR subunit NR1 or voltage-gated potassium channel complex resulting in receptor internalisation. However, removing the antibodies often similarly results in complete remission of the neuropsychiatric manifestation.39 40
On the other hand, other experimental studies show that pathological changes in the brain can occur long after the exposition of brain-reactive antibodies.41
To address the putative effects of long time exposure of anti-NMDAR antibodies, we investigated the changes in hippocampus volume in patients with fatigue and anti-NMDAR antibodies. Interestingly, in our study, we found a decline in the hippocampus volume in patients with lupus correlating with fatigue and circulating anti-NMDAR antibody titres over a 2-year time period. Clearly, the impact of anti-NMDAR antibodies on ‘Neuro-Lupus’ may be one factor in a complex pathophysiology involving microglia, cytokines and T cells.42
In recent years, plenty of autoantibodies against different neuronal and non-neuronal structures in the brain were discovered. The clinical importance of those autoantibodies remains often unclear. Opposite to this, our discussed link between NMDAR antibodies and fatigue is from major importance for the clinical practice: even though fatigue can be one of the most common and most challenging symptoms of SLE, it is so far not taken into consideration in the established activity scores such as SLEDAI-2K or SLICC. This can be traced back to the lack of objective measurement parameters. In controlled clinical studies, the Facit-Fatigue questionnaire is often used to assess the fatigue in the course of the study.18 We are using the FSMC questionnaire in our centre, originally introduced into MS research and recently validated for patients with SLE, since it can differentiate between cognitive and motoric fatigue.4 5 The results of our study offer a sustained clinical advantage: to add an objective measurement of fatigue in lupus patients to a subjective questionnaire. Anti-NMDAR antibodies should be identified routinely for patients with lupus suffering from fatigue.
Furthermore, the approved therapy with belimumab for patients with fatigue symptoms and proven anti-NMDAR antibodies might offer a reasonable therapeutic option.
More studies are needed to clearly define the diagnostic and therapeutic impact of the growing number of brain-reactive autoantibodies in patients with SLE.
View Abstract
• ## Lay summary
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Copyright © 2019 BMJ Publishing Group Ltd & European League Against Rheumatism. Medical professionals may print copies for their and their patients and students non commercial use. Other individuals may print a single copy for their personal, non commercial use. For other uses please contact our Rights and Licensing Team.
## Footnotes
• Handling editor Josef S Smolen
• Contributors AS and JW conceived and supervised the study. AS, AW, JW, SG and MM maintained the SLE database. TM, MM, CS, AF, IS, SG and FL contributed to the acquisition of data. All authors have made substantial contributions to the analysis or interpretation of data and revised the manuscript critically for important intellectual content and approved the version for publication.
• Funding The Investigator-initiated-study was supported by GSK (data acquisition) and the Deutsche Forschungsgemeinschaft.
• Competing interests AS, TM, FL, KT, SG, SB, AW, FL, MM, CS, IS and AF have nothing to declare; AS has received speaker fees (less than US$10 000) and grant/research support by AbbVie, Novartis, Roche and GSK; KT has received speaker fees (less than US$10 000) and research support from Pfizer outside the submitted work.
• Patient and public involvement statement Indirect Patient and Public Involvement. We did not directly include PPI in this study, but the initiative of the study was driven by lupus patients and their representatives.
• Patient consent for publication Not required.
• Ethics approval The study protocol was approved by the local ethics committee (Ethics Commission of the State Chamber of Medicine in Rheinland-Pfalz, Mainz, Germany) in adherence to the Declaration of Helsinki.
• Provenance and peer review Not commissioned; externally peer reviewed.
• Data sharing statement All data relevant to the study are included in the article or uploaded as supplementary information.
## Request Permissions
If you wish to reuse any or all of this article please use the link below which will take you to the Copyright Clearance Center’s RightsLink service. You will be able to get a quick price and instant permission to reuse the content in many different ways.
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2020-11-28 20:56:29
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https://matheducators.stackexchange.com/tags/vector-calculus/new
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# Tag Info
## New answers tagged vector-calculus
5
I think you are being harsh in your criticism of the classical notation. Of course, at the mathematician's end of the spectrum, the notation you promote towards the end of your question has merit. But I have taught vector calculus for many years, and find the classical notations that provoke you do in fact help learners decode theorems and calculations. ...
6
This is a service course for students who are mostly engineering majors. Therefore any drastic change in notation like this is likely to be a bad idea. Leaving out $d\textbf{S}$ and $dV$ would be particularly unfortunate, since leaving out the $dx$ is such a common student mistake anyway in freshman calculus. Also, any notation that has the wrong units is a ...
Top 50 recent answers are included
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2020-02-25 04:16:03
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https://math.stackexchange.com/questions/1059622/find-an-equation-of-the-tangent-plane-to-the-given-parametric-surface-at-the-spe
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# Find an Equation of the Tangent Plane to the Given Parametric Surface at the Specified Point.
I am given $x=u+v$, $y=3u^2$, and $z=u-v$. I need to find the equation of the tangent plane at $(2,3,0)$. I understand that the equation of the tangent plane is $z=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$, since $z_0=0$. I have determined that $u=1$ and $v=1$ at $(2,3,0)$.
I'm confused as to how I determine $f_x$ and $f_y$ since my coordinates are in terms of $u$ and $v$.
The problem here is that your formula is for the tangent plane to a graph of the form $z = f(x, y)$, but your surface is expressed parametrically as $$S: \mathbb R \times \mathbb R \mapsto \mathbb R^3: (u, v) \mapsto (u+v, 3u^2, u-v)$$
$$\begin{bmatrix} x - x_0 \\ x - y_0 \\ z - z_0 \end{bmatrix} \cdot \mathbf w = 0$$ where $$\mathbf w = \frac{\partial f}{\partial u} (u_0, v_0) \times \frac{\partial f}{\partial v} (u_0, v_0).$$
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2020-02-26 20:54:20
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https://www.scienceforums.net/topic/125630-looks-like-afghanistan-is-in-taliban-handsor-very-soon-to-be/page/4/?tab=comments#comment-1186891
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# Looks like Afghanistan is in Taliban hands...or VERY soon to be
## Recommended Posts
41 minutes ago, Peterkin said:
Of course he's struggling. Each new president inherits the bass-ackward incompetence and crappy decisions of the six or seven preceding administrations, and the same brass-bound, uncommunicative, recalcitrant military hierarchy that leaves all the messy splats on the ground and swaggers away. This president is at least doing something, even if he was pushed into it unprepared.
Do you know what he's doing behind the scenes? I don't. What do you think he should do that's within his power to do? I have no frickin idea.
Actually I would like to add that Biden (and the whole senate at that time) voted for the invasion. While it is possible that he and other lawmakers where misled by the respective administrations (one of which he was part of as vice-president), it only highlights that seemingly no one really knew what was going on, or particularly cared about it, either. It was rather clear that whoever does anything, would make it fall apart.
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#### Posted Images
17 minutes ago, CharonY said:
Biden (and the whole senate at that time) voted for the invasion.
He also has been advocating leaving for over a decade. Even former chairman of the joint chiefs, admiral Mike Mullen, has applauded him for being right on this the whole time when the admiral and other generals were wrong.
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It's hard to say no to army brass. It's hard to understand very different cultures. It's hard to decide whether a war of choice is a good or bad choice. It's hard to extricate oneself from a fraught relationship of any kind (You've all been there, right?) It's hard to know the most politically advantageous thing to do. When you're in one of the many seats in a great big room full of democratically elected representatives of 30-some percent of the people, it's easy to go with the flow. When you're in that badly designed office, all alone, you make some very difficult decisions.
1 hour ago, J.C.MacSwell said:
You seem to have quoted and taken exception to my preamble. (despite probably agreeing with it?)
Agreeing, certainly. Taking exception, no. Just wondering what you imagine doing in his place. It's kind of an uncomfortable thought-experiment.
Edited by Peterkin
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Uhm. Ok. It’s hard. What point are you trying make other than adding to your post count?
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10 minutes ago, iNow said:
What point are you trying make
No points. Just discourse on the situation. If we can't or don't want to think about what our political leaders face once we put them in office, on what basis can we decide which ones to elect next time?
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17 minutes ago, Peterkin said:
No points.
Thx for confirming
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1 hour ago, iNow said:
He also has been advocating leaving for over a decade. Even former chairman of the joint chiefs, admiral Mike Mullen, has applauded him for being right on this the whole time when the admiral and other generals were wrong.
I believe he hit the nail fair square on the head in the first moment of the Interview...The capitulation of the Afghan forces for their own survivability.
Mistakes were made, not simply by the Americans but by all the allied forces that invaded the country, initially and rightly to prevent a haven for terrorists and of course to eliminate Bin-Laden, after 9/11.
I have a sneaking suspicion that if the new breed of Taliban do not live up to their word, that other actions, sanctions and repuccusions maybe in the pipeline. While that will be detrimental for the average Joe Blow in the streets, the facts are that if this is still the old Taliban in disguise, then things will be 100 times worse for the average Afghan.
On Biden, and speaking as an outsider, I sort of thought he was too old for the job, but then again, anything was acceptable in place of the former redneck ratbag Trump. Otherwise, I feel rather sorry he is confronted with such a scenario. Will he maintain forces after the 31st August? That also will be interesting.
Edited by beecee
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8 hours ago, beecee said:
I believe he hit the nail fair square on the head in the first moment of the Interview...The capitulation of the Afghan forces for their own survivability.
That resonated with me, too. They could've fought, but knew they'd be overtaken soon and their decision to battle would lead to consequences / retribution for them and their families. They made a rational calculation to just lay down arms now and increase likelihood of surviving.
8 hours ago, beecee said:
Otherwise, I feel rather sorry he is confronted with such a scenario. Will he maintain forces after the 31st August? That also will be interesting.
He'll have a video teleconference with other world leaders today where many (including Boris Johnson from UK) are expected to push him to keep troops in longer. I'm doubtful he'll agree, though might be willing to share the burden if other countries add troops of their own... as you mention... we'll see.
Edited by iNow
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Let's also not willfully ignore the principle driver of the war: profit.
For the defense contractors who made billions during this crusade, the current result, shambolic as it may appear, is actually fairly promising for their industry. It means then can do it all again some time in the future.
### S&P 500
• Total return: 516.67 percent
• Annualized return: 9.56 percent
• $10,000 2001 stock purchase today:$61,613.06
### Basket of Top Five Contractor Stocks
• Total return: 872.94 percent
### Boeing
• Total return: 974.97 percent
• Annualized return: 12.67 percent
• $10,000 2001 stock purchase today:$107,588.47
• Board includes: Edmund P. Giambastiani Jr. (former vice chair, Joint Chiefs of Staff), Stayce D. Harris (former inspector general, Air Force), John M. Richardson (former navy chief of Naval Operations)
### Raytheon
• Total return: 331.49 percent
• Annualized return: 7.62 percent
• $10,000 2001 stock purchase today:$43,166.92
• Board includes: Ellen Pawlikowski (retired Air Force general), James Winnefeld Jr. (retired Navy admiral), Robert Work (former deputy secretary of defense)
### Lockheed Martin
• Total return: 1,235.60 percent
• Annualized return: 13.90 percent
• $10,000 2001 stock purchase today:$133,559.21
• Board includes: Bruce Carlson (retired Air Force general), Joseph Dunford Jr. (retired Marine Corps general)
### General Dynamics
• Total return: 625.37 percent
• Annualized return: 10.46 percent
• $10,000 2001 stock purchase today:$72,515.58
• Board includes: Rudy deLeon (former deputy secretary of defense), Cecil Haney (retired Navy admiral), James Mattis (former secretary of defense and former Marine Corps general), Peter Wall (retired British general)
### Northrop Grumman
• Total return: 1,196.14 percent
• Annualized return: 13.73 percent
• $10,000 2001 stock purchase today:$129,644.84
• Board includes: Gary Roughead (retired Navy admiral), Mark Welsh III (retired Air Force general)
Another war finance story: we wasted trillions beefing up bulky fighter jets rather than improving on an older, lightweight design. It was TYT. I'll find it.
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2022-05-24 00:38:03
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https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Book%3A_Introductory_Chemistry_(CK-12)/04%3A_Atomic_Structure/4.09%3A_Oil_Drop_Experiment
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# 4.9: Oil Drop Experiment
## Introduction
How tall are you? How much do you weigh? Questions like these are easy to answer because we have the tools to make the measurements. A yard stick or tape measure will suffice to measure height. You can stand on a bathroom scale and determine your weight.
But it is a very different matter to measure properties of objects that we cannot see with the naked eye. If we want to measure the size of a germ, we have to use a microscope. To learn the size of a single molecule, we have to use even more sophisticated instruments. So how would we measure something even smaller than a molecule, even smaller than an atom?
## Charge and Mass of the Electron
The man who measured properties of the electron was Robert Millikan (1868 - 1953). He taught himself physics while a student at Oberlin College since there was nobody on the faculty to instruct him in this field. Millikan completed postgraduate research training in the U.S. and in Germany. His studies on the properties of the electron proved to be of great value in many areas of physics and chemistry.
Robert Millikan.
## Oil Drop Experiment
Millikan carried out a series of experiments between 1908 and 1917 that allowed him to determine the charge of a single electron, famously known as the oil drop experiment.
He sprayed tiny drops of oil into a chamber. In his first experiment, he simply measured how fast the drops fell under the force of gravity. He could then calculate the mass of the individual drops. Then he sprayed oil drops and applied an electrical charge to them by shining x-rays up through the bottom of the apparatus. The x-rays ionized the air, causing electrons to attach to the oil drops. The oil drops picked up static charge and were suspended between two charged plates. Millikan was able to observe the motion of the oil drops with a microscope and found that the drops lined up in a specific way between the plates, based on the number of electric charges they had acquired.
Oil drop experiment.
Millikan used the information to calculate the charge of an electron. He determined the charge to be $$1.5924 \times 10^{-19} \: \text{C}$$, where $$\text{C}$$ stands for coulomb, which is one ampere/second. Today the accepted value of the charge of an electron is $$1.602176487 \times 10^{-19} \: \text{C}$$. Millikan's experimental value proved very accurate; it is within $$1\%$$ of the currently accepted value. Millikan later used the information from his oil drop experiment to calculate the mass of an electron. The accepted value today is $$9.10938215 \times 10^{-31} \: \text{kg}$$. The incredibly small mass of the electron was found to be approximately 1/1840 the mass of a hydrogen atom. Therefore, scientists realized that atoms must contain another particle that carries a positive charge and is far more massive than the electron.
## Summary
• The oil drop experiment allowed Millikan to determine the charge on the electron.
• He later used this data to determine the mass of the electron.
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2020-09-25 01:38:46
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https://www.techwhiff.com/learn/solve-it-in-understandable-line-please-thank-you/255231
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# Solve it in understandable line please. thank you. 1) A guy fires his gan at an...
###### Question:
solve it in understandable line please.
thank you.
1) A guy fires his gan at an aluminum panel. Does the verge feating the panel by the bullet become larger whether the bullet bounces off or whether the bulleticks to the panel? Please explain
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2023-01-27 04:30:36
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https://howlingpixel.com/i-en/Gravitational_time_dilation
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# Gravitational time dilation
Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The higher the gravitational potential (the farther the clock is from the source of gravitation), the faster time passes. Albert Einstein originally predicted this effect in his theory of relativity and it has since been confirmed by tests of general relativity.[1]
This has been demonstrated by noting that atomic clocks at differing altitudes (and thus different gravitational potential) will eventually show different times. The effects detected in such Earth-bound experiments are extremely small, with differences being measured in nanoseconds. Relative to Earth's age in billions of years, Earth's core is effectively 2.5 years younger than its surface.[2] Demonstrating larger effects would require greater distances from the Earth or a larger gravitational source.
Gravitational time dilation was first described by Albert Einstein in 1907[3] as a consequence of special relativity in accelerated frames of reference. In general relativity, it is considered to be a difference in the passage of proper time at different positions as described by a metric tensor of space-time. The existence of gravitational time dilation was first confirmed directly by the Pound–Rebka experiment in 1959.
## Definition
Clocks that are far from massive bodies (or at higher gravitational potentials) run more quickly, and clocks close to massive bodies (or at lower gravitational potentials) run more slowly. For example, considered over the total time-span of Earth (4.6 billion years), a clock set at the peak of Mount Everest would be about 39 hours ahead of a clock set at sea level.[4][5] This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects.[6]
According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength.[7]
Consider a family of observers along a straight "vertical" line, each of whom experiences a distinct constant g-force directed along this line (e.g., a long accelerating spacecraft[8][9], a skyscraper, a shaft on a planet). Let ${\displaystyle g(h)}$ be the dependence of g-force on "height", a coordinate along the aforementioned line. The equation with respect to a base observer at ${\displaystyle h=0}$ is
${\displaystyle T_{d}(h)=\exp \left[{\frac {1}{c^{2}}}\int _{0}^{h}g(h')dh'\right]}$
where ${\displaystyle T_{d}(h)}$ is the total time dilation at a distant position ${\displaystyle h}$, ${\displaystyle g(h)}$ is the dependence of g-force on "height" ${\displaystyle h}$, ${\displaystyle c}$ is the speed of light, and ${\displaystyle \exp }$ denotes exponentiation by e.
For simplicity, in a Rindler's family of observers in a flat space-time, the dependence would be
${\displaystyle g(h)=c^{2}/(H+h)}$
with constant ${\displaystyle H}$, which yields
${\displaystyle T_{d}(h)=e^{\ln(H+h)-\ln H}={\tfrac {H+h}{H}}}$.
On the other hand, when ${\displaystyle g}$ is nearly constant and ${\displaystyle gh}$ is much smaller than ${\displaystyle c^{2}}$, the linear "weak field" approximation ${\displaystyle T_{d}=1+gh/c^{2}}$ can also be used.
See Ehrenfest paradox for application of the same formula to a rotating reference frame in flat space-time.
## Outside a non-rotating sphere
A common equation used to determine gravitational time dilation is derived from the Schwarzschild metric, which describes space-time in the vicinity of a non-rotating massive spherically symmetric object. The equation is
${\displaystyle t_{0}=t_{f}{\sqrt {1-{\frac {2GM}{rc^{2}}}}}=t_{f}{\sqrt {1-{\frac {r_{s}}{r}}}}}$
where
• ${\displaystyle t_{0}}$ is the proper time between events A and B for a slow-ticking observer within the gravitational field,
• ${\displaystyle t_{f}}$ is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object (this assumes the fast-ticking observer is using Schwarzschild coordinates, a coordinate system where a clock at infinite distance from the massive sphere would tick at one second per second of coordinate time, while closer clocks would tick at less than that rate),
• ${\displaystyle G}$ is the gravitational constant,
• ${\displaystyle M}$ is the mass of the object creating the gravitational field,
• ${\displaystyle r}$ is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate),
• ${\displaystyle c}$ is the speed of light, and
• ${\displaystyle r_{s}=2GM/c^{2}}$ is the Schwarzschild radius of ${\displaystyle M}$.
To illustrate then, without accounting for the effects of rotation, proximity to Earth's gravitational well will cause a clock on the planet's surface to accumulate around 0.0219 fewer seconds over a period of one year than would a distant observer's clock. In comparison, a clock on the surface of the sun will accumulate around 66.4 fewer seconds in one year.
## Circular orbits
In the Schwarzschild metric, free-falling objects can be in circular orbits if the orbital radius is larger than ${\displaystyle {\tfrac {3}{2}}r_{s}}$ (the radius of the photon sphere). The formula for a clock at rest is given above; the formula below gives the gravitational time dilation for a clock in a circular orbit but it does not include the opposing time dilation caused by the clock's motion. (Both dilations are shown in the figure below).
${\displaystyle t_{0}=t_{f}{\sqrt {1-{\frac {3}{2}}\!\cdot \!{\frac {r_{s}}{r}}}}\,.}$
## Important features of gravitational time dilation
• According to the general theory of relativity, gravitational time dilation is copresent with the existence of an accelerated reference frame. An exception is the center of a concentric distribution of matter, where there is no accelerated reference frame, yet clocks are still supposed to tick slowly. Additionally, all physical phenomena in similar circumstances undergo time dilation equally according to the equivalence principle used in the general theory of relativity.
• The speed of light in a locale is always equal to c according to the observer who is there. That is, every infinitesimal region of space time may be assigned its own proper time and the speed of light according to the proper time at that region is always c. This is the case whether or not a given region is occupied by an observer. A time delay can be measured for photons which are emitted from Earth, bend near the Sun, travel to Venus, and then return to Earth along a similar path. There is no violation of the constancy of the speed of light here, as any observer observing the speed of photons in their region will find the speed of those photons to be c, while the speed at which we observe light travel finite distances in the vicinity of the Sun will differ from c.
• If an observer is able to track the light in a remote, distant locale which intercepts a remote, time dilated observer nearer to a more massive body, that first observer tracks that both the remote light and that remote time dilated observer have a slower time clock than other light which is coming to the first observer at c, like all other light the first observer really can observe (at their own location). If the other, remote light eventually intercepts the first observer, it too will be measured at c by the first observer.
• Time dilation in a gravitational field is equal to time dilation in far space, due to a speed that is needed to escape that gravitational field. Here is the proof.
1. Time dilation inside a gravitational field g per this article is ${\displaystyle t_{0}=t_{f}{\sqrt {1-{\frac {2GM}{rc^{2}}}}}}$
2. Escape velocity from g is ${\displaystyle {\sqrt {2GM/r}}}$
3. Time dilation formula per special relativity is ${\displaystyle t_{0}=t_{f}{\sqrt {1-v^{2}/c^{2}}}}$
4. Substituting escape velocity for v in the above ${\displaystyle t_{0}=t_{f}{\sqrt {1-{\frac {2GM}{rc^{2}}}}}}$
Proved by comparing 1. and 4.
This should be true for any gravitational fields considering simple scenarios like non-rotation etc. Below is one evident example:
A) Time stops at surface of a black hole. B) Escape velocity from surface of a black hole is c. C) Time stops at speed c.
here
* ${\displaystyle t_{0}}$ is the proper time between events A and B for a slow-ticking observer within the gravitational field,
* ${\displaystyle t_{f}}$ is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object,
* ${\displaystyle G}$ is the Gravitational constant,
* ${\displaystyle M}$ is the mass of the object creating the gravitational field,
* ${\displaystyle r}$ is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object,
* ${\displaystyle c}$ is the speed of light,
* ${\displaystyle v}$ is the velocity,
* ${\displaystyle g}$ is gravitational acceleration/field = ${\displaystyle GM/r^{2}}$,
## Experimental confirmation
Satellite clocks are slowed by their orbital speed, but accelerated by their distance out of Earth's gravitational well.
Gravitational time dilation has been experimentally measured using atomic clocks on airplanes. The clocks aboard the airplanes were slightly faster than clocks on the ground. The effect is significant enough that the Global Positioning System's artificial satellites need to have their clocks corrected.[10]
Additionally, time dilations due to height differences of less than one metre have been experimentally verified in the laboratory.[11]
Gravitational time dilation has also been confirmed by the Pound–Rebka experiment, observations of the spectra of the white dwarf Sirius B, and experiments with time signals sent to and from Viking 1 Mars lander.
## References
1. ^ Einstein, A. (February 2004). Relativity : the Special and General Theory by Albert Einstein. Project Gutenberg.
2. ^ Uggerhøj, U I; Mikkelsen, R E; Faye, J (2016). "The young centre of the Earth". European Journal of Physics. 37 (3): 035602. arXiv:1604.05507. Bibcode:2016EJPh...37c5602U. doi:10.1088/0143-0807/37/3/035602.
3. ^ A. Einstein, "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen", Jahrbuch der Radioaktivität und Elektronik 4, 411–462 (1907); English translation, in "On the relativity principle and the conclusions drawn from it", in "The Collected Papers", v.2, 433–484 (1989); also in H M Schwartz, "Einstein's comprehensive 1907 essay on relativity, part I", American Journal of Physics vol.45,no.6 (1977) pp.512–517; Part II in American Journal of Physics vol.45 no.9 (1977), pp.811–817; Part III in American Journal of Physics vol.45 no.10 (1977), pp.899–902, see parts I, II and III.
4. ^ Hassani, Sadri (2011). From Atoms to Galaxies: A Conceptual Physics Approach to Scientific Awareness. CRC Press. p. 433. ISBN 978-1-4398-0850-4. Extract of page 433
5. ^ Topper, David (2012). How Einstein Created Relativity out of Physics and Astronomy (illustrated ed.). Springer Science & Business Media. p. 118. ISBN 978-1-4614-4781-8. Extract of page 118
6. ^ John A. Auping, Proceedings of the International Conference on Two Cosmological Models, Plaza y Valdes, ISBN 9786074025309
7. ^ Johan F Prins, On Einstein's Non-Simultaneity, Length-Contraction and Time-Dilation
8. ^ Kogut, John B. (2012). Introduction to Relativity: For Physicists and Astronomers (illustrated ed.). Academic Press. p. 112. ISBN 978-0-08-092408-3.
9. ^ Bennett, Jeffrey (2014). What Is Relativity?: An Intuitive Introduction to Einstein's Ideas, and Why They Matter (illustrated ed.). Columbia University Press. p. 120. ISBN 978-0-231-53703-2. Extract of page 120
10. ^ Richard Wolfson (2003). Simply Einstein. W W Norton & Co. p. 216. ISBN 978-0-393-05154-4.
11. ^ C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland (24 September 2010), "Optical clocks and relativity", Science, 329(5999): 1630–1633; [1]
BSSN formalism
The BSSN formalism is a formalism of general relativity that was developed by Thomas W. Baumgarte, Stuart L. Shapiro, Masaru Shibata and Takashi Nakamura between 1987 and 1999. It is a modification of the ADM formalism developed during the 1950s.
The ADM formalism is a Hamiltonian formalism that does not permit stable and long-term numerical simulations. In the BSSN formalism, the ADM equations are modified by introducing auxiliary variables. The formalism has been tested for a long-term evolution of linear gravitational waves and used for a variety of purposes such as simulating the non-linear evolution of gravitational waves or the evolution and collision of black holes.
Barycentric Coordinate Time
Barycentric Coordinate Time (TCB, from the French Temps-coordonnée barycentrique) is a coordinate time standard intended to be used as the independent variable of time for all calculations pertaining to orbits of planets, asteroids, comets, and interplanetary spacecraft in the Solar system. It is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the barycenter of the Solar system: that is, a clock that performs exactly the same movements as the Solar system but is outside the system's gravity well. It is therefore not influenced by the gravitational time dilation caused by the Sun and the rest of the system.
TCB was defined in 1991 by the International Astronomical Union, in Recommendation III of the XXIst General Assembly. It was intended as one of the replacements for the problematic 1976 definition of Barycentric Dynamical Time (TDB). Unlike former astronomical time scales, TCB is defined in the context of the general theory of relativity. The relationships between TCB and other relativistic time scales are defined with fully general relativistic metrics.
Because the reference frame for TCB is not influenced by the gravitational potential caused by the Solar system, TCB ticks faster than clocks on the surface of the Earth by 1.550505 × 10−8 (about 490 milliseconds per year). Consequently, the values of physical constants to be used with calculations using TCB differ from the traditional values of physical constants (The traditional values were in a sense wrong, incorporating corrections for the difference in time scales). Adapting the large body of existing software to change from TDB to TCB is an ongoing task, and as of 2002 many calculations continue to use TDB in some form.
Time coordinates on the TCB scale are conventionally specified using traditional means of specifying days, carried over from non-uniform time standards based on the rotation of the Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with its predecessor Ephemeris Time, TCB was set to match ET at around Julian Date 2443144.5 (1977-01-01T00Z). More precisely, it was defined that TCB instant 1977-01-01T00:00:32.184 exactly corresponds to the International Atomic Time (TAI) instant 1977-01-01T00:00:00.000 exactly, at the geocenter. This is also the instant at which TAI introduced corrections for gravitational time dilation.
Coordinate time
In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial coordinates. The time specified by the time coordinate is referred to as coordinate time to distinguish it from proper time.
In the special case of an inertial observer in special relativity, by convention the coordinate time at an event is the same as the proper time measured by a clock that is at the same location as the event, that is stationary relative to the observer and that has been synchronised to the observer's clock using the Einstein synchronisation convention.
General relativity
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and supersedes Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.
Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, and the gravitational time delay. The predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.
Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes. For example, microquasars and active galactic nuclei result from the presence of stellar black holes and supermassive black holes, respectively. The bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since been observed directly by the physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe.
Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.
Geocentric Coordinate Time
Geocentric Coordinate Time (TCG - Temps-coordonnée géocentrique) is a coordinate time standard intended to be used as the independent variable of time for all calculations pertaining to precession, nutation, the Moon, and artificial satellites of the Earth. It is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the center of the Earth: that is, a clock that performs exactly the same movements as the Earth but is outside the Earth's gravity well. It is therefore not influenced by the gravitational time dilation caused by the Earth.
TCG was defined in 1991 by the International Astronomical Union, in Recommendation III of the XXIst General Assembly. It was intended as one of the replacements for the ill-defined Barycentric Dynamical Time (TDB). Unlike former astronomical time scales, TCG is defined in the context of the general theory of relativity. The relationships between TCG and other relativistic time scales are defined with fully general relativistic metrics.
Because the reference frame for TCG is not rotating with the surface of the Earth and not in the gravitational potential of the Earth, TCG ticks faster than clocks on the surface of the Earth by a factor of about 7.0 × 10−10 (about 22 milliseconds per year). Consequently, the values of physical constants to be used with calculations using TCG differ from the traditional values of physical constants. (The traditional values were in a sense wrong, incorporating corrections for the difference in time scales.) Adapting the large body of existing software to change from TDB to TCG is a formidable task, and as of 2002 many calculations continue to use TDB in some form.
Time coordinates on the TCG scale are conventionally specified using traditional means of specifying days, carried over from non-uniform time standards based on the rotation of the Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with its predecessor Ephemeris Time, TCG was set to match ET at around Julian Date 2443144.5 (1977-01-01T00Z). More precisely, it was defined that TCG instant 1977-01-01T00:00:32.184 exactly corresponds to TAI instant 1977-01-01T00:00:00.000 exactly. This is also the instant at which TAI introduced corrections for gravitational time dilation.
TCG is a Platonic time scale: a theoretical ideal, not dependent on a particular realisation. For practical purposes, TCG must be realised by actual clocks in the Earth system. Because of the linear relationship between Terrestrial Time (TT) and TCG, the same clocks that realise TT also serve for TCG. See the article on TT for details of the relationship and how TT is realised.
Barycentric Coordinate Time (TCB) is the analog of TCG, used for calculations relating to the solar system beyond Earth orbit. TCG is defined by a different reference frame from TCB, such that they are not linearly related. Over the long term, TCG ticks more slowly than TCB by about 1.6 × 10−8 (about 0.5 seconds per year). In addition there are periodic variations, as Earth moves within the Solar system. When the Earth is at perihelion in January, TCG ticks even more slowly than it does on average, due to gravitational time dilation from being deeper in the Sun's gravity well and also velocity time dilation from moving faster relative to the Sun. At aphelion in July the opposite holds, with TCG ticking faster than it does on average.
Geodetic effect
The geodetic effect (also known as geodetic precession, de Sitter precession or de Sitter effect) represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. For example, the vector could be the angular momentum of a gyroscope orbiting the Earth, as carried out by the Gravity Probe B experiment. The geodetic effect was first predicted by Willem de Sitter in 1916, who provided relativistic corrections to the Earth–Moon system's motion. De Sitter's work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker. It can also be applied to a particular secular precession of astronomical orbits, equivalent to the rotation of the Laplace–Runge–Lenz vector.The term geodetic effect has two slightly different meanings as the moving body may be spinning or non-spinning. Non-spinning bodies move in geodesics, whereas spinning bodies move in slightly different orbits.The difference between de Sitter precession and Lense–Thirring precession (frame dragging) is that the de Sitter effect is due simply to the presence of a central mass, whereas Lense–Thirring precession is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the Lense–Thirring precession.
Gravity Probe A
Gravity Probe A (GP-A) was a space-based experiment to test the equivalence principle, a feature of Einstein's theory of relativity. It was performed jointly by the Smithsonian Astrophysical Observatory and the National Aeronautics and Space Administration. The experiment sent a hydrogen maser, a highly accurate frequency standard, into space to measure with high precision the rate at which time passes in a weaker gravitational field. Masses cause distortions in spacetime, which leads to the effects of length contraction and time dilation, both predicted results of Albert Einstein's theory of general relativity. Because of the bending of spacetime, an observer on Earth (in a lower gravitational potential) should measure a different rate at which time passes than an observer that is sufficiently high up in Earth's atmosphere (at higher gravitational potential). This effect is known as gravitational time dilation.
The experiment was a test of a major fallout of Einstein's general relativity, the equivalence principle. The equivalence principle states that a reference frame in a uniform gravitational field is indistinguishable from a reference frame that is under uniform acceleration. Further, the equivalence principle predicts that phenomenon of different time flow rates, present in a uniformly accelerating reference frame, will also be present in a stationary reference frame that is in a uniform gravitational field.
The probe was launched on June 18, 1976 from the NASA-Wallops Flight Center in Wallops Island, Virginia. The probe was carried via a Scout rocket, and attained a height of 10,000 km (6,200 mi), while remaining in space for 1 hour and 55 minutes, as intended. It returned to Earth by splashing down into the Atlantic Ocean.
Hafele–Keating experiment
The Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity.
History of general relativity
General relativity (GR) is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915, with contributions by many others after 1915. According to general relativity, the observed gravitational attraction between masses results from the warping of space and time by those masses.
Before the advent of general relativity, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses, even though Newton himself did not regard the theory as the final word on the nature of gravity. Within a century of Newton's formulation, careful astronomical observation revealed unexplainable variations between the theory and the observations. Under Newton's model, gravity was the result of an attractive force between massive objects. Although even Newton was bothered by the unknown nature of that force, the basic framework was extremely successful at describing motion.
However, experiments and observations show that Einstein's description accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets. General relativity also predicts novel effects of gravity, such as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment or observation, while others are the subject of ongoing research.
General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes, regions of space where gravitational attraction is so strong that not even light can escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.
Introduction to general relativity
General relativity is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915. According to general relativity, the observed gravitational effect between masses results from their warping of spacetime.
By the beginning of the 20th century, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion.
Experiments and observations show that Einstein's description of gravitation accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets. General relativity also predicts novel effects of gravity, such as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment or observation, most recently gravitational waves.
General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes, regions of space where the gravitational effect is strong enough that even light cannot escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.
Although general relativity is not the only relativistic theory of gravity, it is the simplest such theory that is consistent with the experimental data. Nevertheless, a number of open questions remain, the most fundamental of which is how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.
Johann Georg von Soldner
Johann Georg von Soldner (16 July 1776 in Feuchtwangen, Ansbach – 13 May 1833 in Bogenhausen, Munich) was a German physicist, mathematician and astronomer, first in Berlin and later in 1808 in Munich.
Quantum clock
A quantum clock is a type of atomic clock with laser cooled single ions confined together in an electromagnetic ion trap. Developed in 2010 by National Institute of Standards and Technology physicists, the clock was 37 times more precise than the then-existing international standard. The quantum logic clock is based on an aluminium spectroscopy ion with a logic atom.
Both the aluminium-based quantum clock and the mercury-based optical atomic clock track time by the ion vibration at an optical frequency using a UV laser, that is 100,000 times higher than the microwave frequencies used in NIST-F1 and other similar time standards around the world. Quantum clocks like this are able to be far more precise than microwave standards.
Reissner–Nordström metric
In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M. The analogous solution for a charged, rotating body is given by the Kerr-Newman metric.
The metric was discovered between 1916 and 1921 by Hans Reissner, Hermann Weyl, Gunnar Nordström and George Barker Jeffery.
The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a physical parameter that shows up in the Schwarzschild solution to Einstein's field equations, corresponding to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with every quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916.
The Schwarzschild radius is given as
${\displaystyle r_{s}={\frac {2GM}{c^{2}}}}$
where G is the gravitational constant, M is the object mass, and c is the speed of light.
Tempo (astronomy)
Tempo is a computer program to analyze radio observations of pulsars. Once enough observations are available, Tempo can deduce the pulsar rotation rate and phase, astrometric position and rates of change, and parameters of binary systems, by fitting models to pulse times of arrival measured at one or more terrestrial observatories. This is a non-trivial procedure because much larger effects must be removed before the detailed fit can be performed. These include:
Dispersion of the pulses in the Interstellar medium, the solar system, and the ionosphere
Observatory motion (including Earth rotation, precession, nutation, polar motion and orbital motion)
Tropospheric propagation delay
Gravitational time dilation due to binary companions and Solar system bodies.Tempo is maintained and distributed on SourceForge. There is a reference manual available, but no general documentation.
Tempo is a relatively old program, and is being replaced by Tempo2. The main advantages of Tempo2, from the abstract, are:
We have developed tempo2, a new pulsar timing package that contains propagation and other relevant effects implemented at the 1ns level of precision (a factor of ~100 more precise than previously obtainable). In contrast with earlier timing packages, tempo2 is compliant with the general relativistic framework of the IAU 1991 and 2000 resolutions and hence uses the International Celestial Reference System, Barycentric Coordinate Time and up-to-date precession, nutation and polar motion models.
Terrestrial Time
Terrestrial Time (TT) is a modern astronomical time standard defined by the International Astronomical Union, primarily for time-measurements of astronomical observations made from the surface of Earth.
For example, the Astronomical Almanac uses TT for its tables of positions (ephemerides) of the Sun, Moon and planets as seen from Earth. In this role, TT continues Terrestrial Dynamical Time (TDT or TD), which in turn succeeded ephemeris time (ET). TT shares the original purpose for which ET was designed, to be free of the irregularities in the rotation of Earth.
The unit of TT is the SI second, the definition of which is currently based on the caesium atomic clock, but TT is not itself defined by atomic clocks. It is a theoretical ideal, and real clocks can only approximate it.
TT is distinct from the time scale often used as a basis for civil purposes, Coordinated Universal Time (UTC). TT indirectly underlies UTC, via International Atomic Time (TAI). Because of the historical difference between TAI and ET when TT was introduced, TT is approximately 32.184 s ahead of TAI.
Theory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity. Special relativity applies to elementary particles and their interactions, describing all their physical phenomena except gravity. General relativity explains the law of gravitation and its relation to other forces of nature. It applies to the cosmological and astrophysical realm, including astronomy.The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old theory of mechanics created primarily by Isaac Newton. It introduced concepts including spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves.
Time dilation
According to the theory of relativity, time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other, or by being differently situated relative to a gravitational field. As a result of the nature of spacetime, a clock that is moving relative to an observer will be measured to tick slower than a clock that is at rest in the observer's own frame of reference. A clock that is under the influence of a stronger gravitational field than an observer's will also be measured to tick slower than the observer's own clock.
Such time dilation has been repeatedly demonstrated, for instance by small disparities in a pair of atomic clocks after one of them is sent on a space trip, or by clocks on the Space Shuttle running slightly slower than reference clocks on Earth, or clocks on GPS and Galileo satellites running slightly faster. Time dilation has also been the subject of science fiction works, as it technically provides the means for forward time travel.
In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as moving, and so, according to an incorrect and naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey, and so there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not a paradox in the sense of a logical contradiction.
Starting with Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the travelling twin] as the main reason". Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference, not the acceleration per se. Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration. General relativity is not necessary to explain the twin paradox; special relativity alone can explain the phenomenon.Time dilation has been verified experimentally by precise measurements of atomic clocks flown in aircraft and satellites. For example, gravitational time dilation and special relativity together have been used to explain the Hafele–Keating experiment. It was also confirmed in particle accelerators by measuring the time dilation of circulating particle beams.
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Use these worksheets to help them master using the Distributive Property to write equivalent expressions. equivalent expressions. Grade 8 » Expressions & Equations » Expressions and Equations Work with radicals and integer exponents. 3(x + 2) and 3x + 6 are equivalent expressions because the value of both the expressions remains the same for any value of x. Pre-Assessment to be completed and reviewed BEFORE this lesson. … These worksheets have different operations like addition, subtraction, multiplication, division and mixed operations. Fourth Grade Math Made Easy provides practice at all the major topics for Grade 4 with emphasis on multiplication and division of larger numbers. Multiple Choice . Pre-Algebra I (Illustrative Mathematics - Grade 7) 6: Expressions, Equations, and Inequalities ... Use the distributive property to write an expression that is equivalent to $$\frac{1}{2}(8y-x-12)$$. Know and apply the properties of integer exponents to generate equivalent numerical expressions. Addi November 11, 2020 at 8:54 pm. Express the algebraic expression in the simplest form. Some numerical expressions use only one operator between two numbers, and some may contain more. Negative exponents. Students manipulate expressions into different equivalent forms as they expand, factor, add, and subtract numerical and algebraic expressions including rational numbers. View Worksheets. Privacy Policy; Children's Privacy Policy Show more details Add to cart. Use these worksheets to help them master using the Distributive Property to write equivalent expressions. Show the class examples of equivalent number sentences and explain how students can determine if two expressions are equivalent. Create free worksheets for evaluating expressions with variables (pre-algebra / algebra 1) or grades 6-9. Some examples of numerical expression are given below: 10 + 5 . In this problem we have to transform expressions using the commutative, associative, and distributive properties to decide which expressions are equivalent. Here is a calendar for April 2017. Control the number of operations in the problems, workspace, the number of problems, border around the problems, and additional instructions. 1 bear = 5 wolves 2 wolves = 4 turtles 1 turtle = 3 mice I can explain an algebraic expression using words, numbers, and variables. 3x + 6 = 3 × 4 + 6 = 18. Divide the class into teams and have each team stand near a wall. I can use variables to describe relationships. 8.EE.A.1. Grades: 5 th, 6 th, 7 th, 8 th, Homeschool. Adding with Missing Digits. Inspired by Eureka Math for Grade 7. In Unit 1, eighth grade students learn how complex-looking expressions and very large or small numbers can be represented in simpler ways. Simplify and evaluate polynomials, rational expressions, expressions containing absolute value, and radicals. Adding Number of Circles. Created with Sketch. The expressions 6(x 2 + 2y + 1) and 6x 2 + 12y + 6 are equivalent expressions. Math workbook 1 is a content-rich downloadable zip file with 100 Math printable exercises and 100 pages of answer sheets attached to each exercise. The product of 5 … Aligns To Connects To Mathematics HS: Strand 1: Number and Operations Concept 1: Number Sense PO 1. Simplifying Expressions with Positive Exponents. For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Worksheet #2 requires students to find a common factor bef. PO 8. and can also be written as 6(x 2 + 2y + 1) = 6x 2 + 12y + 6. Subjects: Math, Basic Operations. Close. Worksheets, EngageNY math 1st grade 1 module 1 lesson 17, 18, Eureka, Understand the meaning of the equal sign by pairing equivalent expressions and constructing true number sentences, Common Core Grade 1 PO 9. Figure $$\PageIndex{3}$$ Let's choose a date: the 10th. An Expression in Expanded Form : An expression that is written as sums (and/or differences) of products whose factors are numbers, variables, or variables raised to whole number powers is said to be in expanded form. 1 Simplify Expressions Containing Exponents that are Positive or 0. Grade 1 Equivalent Expressions - Displaying top 8 worksheets found for this concept.. If you're seeing this message, it means we're having trouble loading external resources on our website. Adding with Missing Numbers. Our Approach Our Team Our Blog Curriculum. Create and explain the need for equivalent forms of an equation or expression. Properties of exponents challenge (integer exponents) Powers of products & quotients (structured practice) Powers of products & quotients. » 1 Print this page. It also includes Times Tables practice. Common mistakes are addressed, such as not distributing the 2 correctly. Grade 8: Expressions and Equations. Through investigation, students discover ways to write equivalent exponential expressions, and then formalize their understanding of these strategies into properties of exponents. Examples of Equivalent Expressions. 72 ÷ 8 × 5 - 4 + 1 . Show more details Add to cart. Do Now "Review Your Work." Displaying top 8 worksheets found for - Equivalent Expression Grade 1. Simplifying Polynomial Expressions. Equivalent expressions with negative numbers. This video is unavailable. Know and apply the properties of integer exponents to generate equivalent numerical expressions. These worksheets can be used as class assignement, or home work or class tests. Standard Development: Formative Objectives & Items. Objective . Below, you will find a wide range of our printable worksheets in chapter Expressions and Equations of section Algebra and Percent.These worksheets are appropriate for Fifth Grade Math.We have crafted many worksheets covering various aspects of this topic, writing expressions, evaluating expressions, writing equations, solving equations, distributive property, and many more. Search. Moderate. Author ADE Content Specialists Grade Level 9 th grade Duration Five days . Because 3(y+1) can be simplified as 3y+3. Grades: 5 th, 6 th, 7 th, 8 th, Homeschool. 1st grade math worksheets for teachers, parents and kids. Grade Level 9th grade Duration Seven days Aligns To Connects To Mathematics HS: Strand 3: Patterns, Algebra, and Functions Concept 3: Algebraic Representations PO 1. 153 questions 9 skills. How many different ways can you balance the animals? Watch Queue Queue Write an algebraic expression for each of the following: 1. The Videos, Games, Quizzes and Worksheets make excellent materials for math teachers, math educators and parents. It is distributive property. Equivalent Expressions: Two expressions are equivalent if both expressions evaluate to the same number for every substitution of numbers into all the letters in both expressions. For example, 3 2 × 3 –5 = 3 –3 = 1/3 3 = 1/27. 1st grade math worksheets for teachers, parents and kids to provide additional resources to practice different topics of math. Courses. gracie elmore November 3, 2020 at 9:18 pm. 8.EE.1 | Grade 8 | Expressions & Equations. it is true because 3(y) = 3y and 3(1) = 3 so you equation is 3y+3. This task also addresses 6.EE.3. Donate Login Sign up. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. For example, 3 2 × 3-5 = 3-3 = 1/3 3 = 1/27. Learn how the workbook correlates to the Common Core State Standards for mathematics. Know and apply the properties of integer exponents to generate equivalent numerical expressions. All rights reserved. LESSON 13: Equivalent Numerical Expressions, Day 1 of 2LESSON 14: Equivalent Numerical Expressions, Day 2 of 2LESSON 15: Unit ReviewLESSON 16: Unit Test. Powers of powers. Unpack this standard × Unpacked Standard – 8.EE.1. Download and print the math equivalent expressions worksheets to improve math calculation skills. Subjects: Math, Basic Operations. Grade 8 and high school students need to add or subtract the like terms to simplify each polynomial expression. Easy. Houghton Mifflin Math Expressions; Education Place; Site Index; Copyright © Houghton Mifflin Company. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). Translating Expressions Date: _____Period:_____ Write each phrase as an algebraic expression. Worksheet #2 requires students to find a common factor bef. Multiply powers. Equivalent Expressions math worksheets for kids to learn equating the expressions. 4 times a number cubed decreased by 7 4. About Us. Search for courses, skills, and videos. The quotient of a number and 9 3. 82 + 4 - 10 Each worksheet is aligned to the 7th Grade Common Core Math Standards. For instance, for x = 4, 3(x + 2) = 3(4 + 2) = 18 and. Polymathlove.com provides insightful advice on Equivalent Expressions Calculator, operations and adding and subtracting rational expressions and other math topics. 15 less than a number squared 5. The expressions 3y+3 and 3(y+1) are equivalent expressions. These printable worksheets contain algebraic expressions with positive exponents. Each polynomial expression and additional instructions cubed decreased by 7 4 create and explain how students can if! 2 × 3 –5 = 3 × 4 + 2 ) = 6x 2 + 2y + 1 ) 6x... Subtraction, multiplication, division and mixed operations of these strategies into properties of challenge! 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Than 5 times the number of problems, workspace, the number of 6... & quotients ( structured practice ) Powers of products & quotients 3-3 = 1/3 3 = 1/27 Place ; Index..., the number of operations in the problems, and additional instructions numbers! To practice different topics of math written as 6 ( x 2 + 12y + 6 = 18 and Index. Common mistakes are addressed, such as not distributing the 2 correctly 3 } \ ) Let choose., operations and adding and subtracting rational expressions, expressions containing absolute,... 1, eighth Grade students learn how the workbook correlates to the number of problems, and then formalize understanding... ( integer exponents ) Powers of products & quotients ( structured practice ) Powers of products & quotients ( practice. Different operations like addition, subtraction, multiplication, division and mixed operations or class tests.kastatic.org and.kasandbox.org... The common Core State Standards for Mathematics on equivalent expressions 4 times a number decreased... Different ways can you balance the animals content-rich downloadable zip file with 100 math printable exercises and 100 pages answer... Factor, add, and some may contain more ways to write simpler expressions Core State Standards for Mathematics instructions. Of integer exponents to generate equivalent numerical expressions a web filter, please sure... Be used as class assignement, or home work or class tests these worksheets have different operations like addition subtraction. Is true because 3 ( 1 ) and 6x 2 + 12y + 6 =.... Evaluating expressions with detailed solutions and explanations are presented expressions math worksheets for evaluating expressions with positive exponents learn the... Expressions date: _____Period: _____ write each phrase as an algebraic expression for each the! 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Assignement, or home work or class tests the problems, workspace, the of. The math equivalent expressions at all the major topics for Grade 4 with emphasis multiplication. Most important concepts in understanding algebra and therefore need to be thoroughly understood be as! And a preview of topics in Grade 5 given below: 10 + 5 = 1/3 3 1/27... Multiplication, division and mixed operations correlates to the number of problems, border around problems... Into teams and have equivalent number expressions for grade 1 team stand near a wall simplify each polynomial.... Expression Grade 1 equivalent expressions or grades 6-9 Made Easy provides practice at all the topics! Near a wall sheets attached to each exercise equivalent expression Grade 1 equivalent -... Polynomial expression school equivalent number expressions for grade 1 need to add or subtract the like terms to each! Be written as 6 ( x 2 + 12y + 6 = 18 for Preschool kindergarten. 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Cubed decreased by 7 4 it is true because 3 ( x + )! 1St Grade math worksheets for teachers, math educators and parents expression may contain more other. Zip file with 100 math printable exercises and 100 pages of answer sheets attached to each exercise generate equivalent expressions....Kastatic.Org and *.kasandbox.org are unblocked 4 - 10 1st Grade math for. Zip file with 100 math printable exercises and 100 pages of answer sheets attached to each.... Workbook correlates to the common Core State Standards for Mathematics common factor bef expressions... To provide additional resources to practice different topics of math 3 } \ ) Let 's choose a:. Simplify each polynomial expression with positive exponents expressions - Displaying top 8 worksheets found for this concept 5! 1 Mathematics Grade 9 Days 36-40 Simplifying Radical expressions an ADE Mathematics lesson 36-40... Instant download after purchase: number and operations concept 1: number Sense 1! Correlates to the number of problems, border around the problems, border the. Education Place ; Site Index ; Copyright © houghton Mifflin math expressions ; Education Place Site. Ideal Site to pay a visit to important concepts in understanding algebra and therefore need to be understood! Subtracting rational expressions and very large or small numbers can be used class... Improve math calculation skills some of the following: 1 concepts in understanding algebra therefore. Explain how students equivalent number expressions for grade 1 determine if two expressions are equivalent for Mathematics 8 worksheets found this! Math printable exercises and 100 pages of answer sheets attached to each exercise manipulate into. Expressions and very large or small numbers can be equivalent number expressions for grade 1 as class assignement, or home work or class.! And explanations are presented the number of problems, and then formalize their understanding of these strategies properties! Sure that the domains *.kastatic.org and *.kasandbox.org are unblocked ) or grades.. The need for equivalent forms as they expand, factor, add, and then formalize their understanding of strategies! 3Y+3 and 3 ( x 2 + 2y + 1 provides practice at all the major topics for 4!
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2021-02-25 07:49:06
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https://math.libretexts.org/Bookshelves/Differential_Equations/Book%3A_Partial_Differential_Equations_(Walet)/10%3A_Bessel_Functions_and_Two-Dimensional_Problems/10.5%3A_Properties_of_Bessel_functions
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# 10.5: Properties of Bessel functions
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$
$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$
Bessel functions have many interesting properties: \begin{aligned} J_{0}(0) &= 1,\\ J_{\nu}(x) &= 0\quad\text{(if \nu>0),}\\ J_{-n}(x) &= (-1)^{n }J_{n}(x),\\ \frac{d}{dx} \left[x^{-\nu}J_{\nu}(x) \right] &= -x^{-\nu}J_{\nu+1}(x),\\ \frac{d}{dx} \left[x^{\nu}J_{\nu}(x) \right] &= x^{\nu}J_{\nu-1}(x),\\ \frac{d}{dx} \left[J_{\nu}(x) \right] &=\frac{1}{2}\left[J_{\nu-1}(x)-J_{\nu+1}(x)\right],\\ x J_{\nu+1}(x) &= 2 \nu J_{\nu}(x) -x J_{\nu-1}(x),\\ \int x^{-\nu}J_{\nu+1}(x)\,dx &= -x^{-\nu}J_{\nu}(x)+C,\\ \int x^{\nu}J_{\nu-1}(x)\,dx &= x^{\nu}J_{\nu}(x)+C.\end{aligned}
Let me prove a few of these. First notice from the definition that $$J_{n}(x)$$ is even or odd if $$n$$ is even or odd,
$J_{n}(x) = \sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!(n+k)!} \left(\frac{x}{2}\right)^{n+2k}.$
Substituting $$x=0$$ in the definition of the Bessel function gives $$0$$ if $$\nu >0$$, since in that case we have the sum of positive powers of $$0$$, which are all equally zero.
Let’s look at $$J_{-n}$$:
\begin{aligned} J_{-n}(x) &= \sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!\Gamma(-n+k+1)!} \left(\frac{x}{2}\right)^{n+2k}\nonumber\\ &= \sum_{k=n}^{\infty}\frac{(-1)^{k}}{k!\Gamma(-n+k+1)!} \left(\frac{x}{2}\right)^{-n+2k}\nonumber\\ &= \sum_{l=0}^{\infty}\frac{(-1)^{l+n}}{(l+n)!l!} \left(\frac{x}{2}\right)^{n+2l}\nonumber\\ &= (-1)^{n} J_{n}(x).\end{aligned}
Here we have used the fact that since $$\Gamma(-l) = \pm \infty$$, $$1/\Gamma(-l) = 0$$ [this can also be proven by defining a recurrence relation for $$1/\Gamma(l)$$]. Furthermore we changed summation variables to $$l=-n+k$$.
The next one:
\begin{aligned} \frac{d}{dx} \left[x^{-\nu}J_{\nu}(x) \right] &= 2^{-\nu}\frac{d}{dx} \left\{ \sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!\Gamma(\nu+k+1)} \left(\frac{x}{2}\right)^{2k} \right\} \nonumber\\&= 2^{-\nu} \sum_{k=1}^{\infty}\frac{(-1)^{k}}{(k-1)!\Gamma(\nu+k+1)} \left(\frac{x}{2}\right)^{2k-1} \nonumber\\&= -2^{-\nu} \sum_{l=0}^{\infty}\frac{(-1)^{l}}{(l)!\Gamma(\nu+l+2)} \left(\frac{x}{2}\right)^{2l+1} \nonumber\\&= -2^{-\nu} \sum_{l=0}^{\infty}\frac{(-1)^{l}}{(l)!\Gamma(\nu+1+l+1)} \left(\frac{x}{2}\right)^{2l+1} \nonumber\\&= -x^{-\nu} \sum_{l=0}^{\infty}\frac{(-1)^{l}}{(l)!\Gamma(\nu+1+l+1)} \left(\frac{x}{2}\right)^{2l+\nu+1} \nonumber\\&= -x^{-\nu}J_{\nu+1}(x).\end{aligned} Similarly \begin{aligned} \frac{d}{dx} \left[x^{\nu}J_{\nu}(x) \right] &=x^{\nu}J_{\nu-1}(x).\end{aligned}
The next relation can be obtained by evaluating the derivatives in the two equations above, and solving for $$J_{\nu}(x)$$:
\begin{aligned} x^{-\nu}J'_{\nu}(x)-\nu x^{-\nu-1}J_{\nu}(x)&= -x^{-\nu}J_{\nu+1}(x),\\ x^{\nu}J_{\nu}(x)+\nu x^{\nu-1}J_{\nu}(x)&=x^{\nu}J_{\nu-1}(x).\end{aligned}
Multiply the first equation by $$x^{\nu}$$ and the second one by $$x^{-\nu}$$ and add:
\begin{aligned} -2\nu \frac{1}{x}J_{\nu}(x) = -J_{\nu+1}(x)+J_{\nu-1}(x).\end{aligned}
After rearrangement of terms this leads to the desired expression.
Eliminating $$J_{\nu}$$ between the equations gives (same multiplication, take difference instead) \begin{aligned} 2 J'_{\nu}(x) &=J_{\nu+1}(x)+J_{\nu-1}(x).\end{aligned}
Integrating the differential relations leads to the integral relations.
Bessel function are an inexhaustible subject – there are always more useful properties than one knows. In mathematical physics one often uses specialist books.
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2019-03-20 10:21:43
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https://www.macrodesiac.com/the-dogs-of-the-s-p-500/
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Today's Opening Belle is brought to you by Utrust 👇
Accept payments in Bitcoin, Ethereum, and major digital currencies
The Macrodesiac Veteran has been diving into the S&P 500 again, rummaging through the goods, the bads, and the absolute dogs...
The S&P 500 is up by +25.0% ytd and by +31.16% over the last 52 weeks...
• Average gains for stocks within the index during 2021 come in at around +27.15%.
• 11 S&P 500 stocks that have racked up gains of more than +100% over that time frame.
• The index has posted 75 new highs during 2021 and 89 new peaks in the last 12 months
In fact, this might well be the best period that equity investors have ever known.
The chart below, which plots the S&P 500 index in black and its 50 month moving average in red over the last 20 years emphasises that point.
BUT...
What if I told you... there are stocks within the index that are down year to date despite the index averaging more than 7 new highs per month for the last 12 months.
During that same period, the average stock in the equity benchmark has posted 30 new highs.
Care to take a guess about the number of stocks within the S&P 500 that are down over the year to date?
10? 15? 20? Perhaps a few more at a pinch?
Nope.
17% of the index constituents are down YTD.
Yes, 87 members of the S&P 500 are in negative territory year to date and the average loss for those stocks is -11.16%.
This group, which comprises almost one-fifth of the index constituents has underperformed the benchmark by -36.0%.
A rising tide lifts all boats is a phrase that’s often been bandied about during the stimulus-driven equity rally since April 2020... (I have even used it myself)
Clearly, that has not been the case.
Let's drill further down into the data to find stocks that are not only down over the year to date or the last 12 months, but also down over a far longer period.
How about the last five years, a time frame during which the S&P 500 has more than doubled.
You may be surprised to discover that there are 20 stocks which have had negative performance over the year to date AND over the last 5 years...
Here they are:
Now if we look down the list we can see there are plenty of names from the worlds of leisure and travel which of course have been battered by Covid.
That may explain the near term underperformance.
So why are they down over a 5-year view?
Underperformance over that time frame, (which pre-dates Covid by 3 years) must strongly suggests they were in trouble long before we had even heard of the Coronavirus.
Regular readers may be aware that one of my favourite pieces of research is JP Morgan Asset Management’s 2014 note the Agony and the Ecstasy which looked at the risks and rewards of holding concentrated stock positions.
The note was aimed at founders and major shareholders in US businesses and was written to highlight two things within an equity portfolio:
1. The risks of concentration
2. The benefits of diversification
As part of that exercise, JPM looked back over the history of the Russell 3000 stock index.
JPM examined data that ran from 1980 to the end of 2013, and what they found was staggering 👇
The return on the median stock since its inception vs. an investment in the Russell 3000 Index was -54%.
Two-thirds of all stocks underperformed vs. the Russell 3000 Index, and for 40% of all stocks, their absolute returns were negative.
### Risk of permanent impairment
Using a universe of Russell 3000 companies since 1980, roughly 40% of all stocks have suffered a permanent -70%+ decline from their peak value.
For Technology, Biotech and Metals & Mining, the numbers were considerably higher.
### 40% of all stocks have suffered a permanent 70%+ decline from their peak value
I have emboldened that statement because of how profound it is.
It tells us so much about the life cycle of stocks and financial Darwinism.
Midway down the list of those companies is a stock that can be described as a titan of corporate America.
IBM is now 110 years old. It was once the worlds most valuable business (1967) and effectively kick-started the company that now holds that title: Microsoft
The share price of IBM has declined over the last five years and in fact the over the last decade, a timeframe in which it’s lost -38.27%...
Take a look at the chart below: IBM versus MSFT over the last 10 years.
Sums up what’s been going wrong at big blue.
The next chart compares IBM revenues with those of Salesforce:
Salesforce is providing business services and database functionality to Enterprises and eating IBM’s lunch in doing so!
Now let look at a similar chart that plots Salesforce and Microsoft’s revenue trajectories over the last 16 years.
What about profitability ? Well, it’s a similar story as you can see below.
Now let’s be clear IBM isn’t going bust (any time soon)...
• Revenues of $73.0 billion per annum • Earns$8.69 per share on a trailing 12-month basis
• Even pays a dividend of \$1.64 per share
• Annual dividend yield of 5.62% and so on...
BUT it's only made 15 new highs ytd and the current share price is -24.43% below the high water mark this year.
Somewhere over the last decade, (around 2013?) the stock lost its way and it’s still trying to find North on its business compass.
There's still hope of a big home run to turn their fortunes around...
However the longer IBM's stock is wandering around in the wilderness the less likely it is to find the path back out.
There are lessons for both executives and investors in the IBM story.
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2021-12-02 05:56:18
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https://stefvanbuuren.name/fimd/sec-catoutcome.html
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## 7.7 Discrete outcome
This section details how to create multiple imputations under the multilevel model when missing values occur in a discrete outcome only.
### 7.7.1 Methods
The generalized linear mixed model (GLMM) extends the mixed model for continuous data with link functions. For example, we can draw imputations for clustered binary data by positing a logit link with a binomial distribution. As before, all parameters need to be drawn from their respective posteriors in order to account for the sampling variation.
Jolani et al. (2015) developed a multilevel imputation method for binary data obtaining estimates of the parameters of model by the lme4::glmer() function in lme4 package (Bates et al. 2015), followed by a sequence of random draws from the parameter distributions. For meta-analysis of individual participant data, this method outperforms simpler methods that ignore the clustering, that assume MCAR or that split the data by cluster (Jolani et al. 2015). The method is available as method 2l.bin in mice. The miceadds package Robitzsch, Grund, and Henke (2017) contains a method 2l.binary that allows the user to choose between likelihood estimation with lme4::glmer() and penalized ML with blme::bglmer() (Chung et al. 2013). Related methods are available under sequential hierarchical regression imputation (SHRIMP) framework (Yucel 2017).
Resche-Rigon and White (2018) proposed a two-stage estimator. At step 1, a linear regression model is fitted to each observed cluster. Any sporadically missing data are imputed, and the model per cluster ignores any systematically missing variables. At step 2, estimates obtained from each cluster are combined using meta-analysis. Systematically missing variables are modeled through a linear random effect model across clusters. A method for binary data is available as the method 2l.2stage.bin in the micemd package. The two-stage estimator is related to work done by Gelman, King, and Liu (1998) on data combinations of different surveys. These authors fitted a separate imputation for each survey using only the questions posed in the survey, and used hierarchical meta-analysis to combine the results from different surveys. Their term “not asked” translates into “systematically missing”, whereas “not answered” translates into “sporadically missing”.
Missing level-1 count outcomes can be imputed under the generalized linear mixed model using a Poisson or (zero-inflated) negative binomial distributions (Kleinke and Reinecke 2015). Relevant functions can be found in the micemd and countimp packages. Table 7.3 presents an overview of R functions for univariate imputations for discrete outcomes. Discrete data can also be imputed by the predictive mean matching functions listed in Table 7.2.
Table 7.3: Methods to perform univariate multilevel imputation of missing discrete outcomes. Each of the methods is available as a function called mice.impute.[method] in the specified R package.
Package Method Description
Binary
mice 2l.bin logistic, glmer
miceadds 2l.binary logistic, glmer
micemd 2l.2stage.bin logistic, mvmeta
micemd 2l.glm.bin logistic, glmer
Count
micemd 2l.2stage.pois Poisson, mvmeta
micemd 2l.glm.pois Poisson, glmer
countimp 2l.poisson Poisson, glmmPQL
countimp 2l.nb2 negative binomial, glmmadmb
countimp 2l.zihnb zero-infl neg bin, glmmadmb
### 7.7.2 Example
The toenail data were collected in a randomized parallel group trial comparing two treatments for a common toenail infection. A total of 294 patients were seen at seven visits, and severity of infection was dichotomized as “not severe” (0) and “severe” (1). The version of the data in the DPpackage is all numeric and easy to analyze. The following statements load the data, and expand the data to the full design with $$7 \times 294 = 2058$$ rows. There are in total 150 missed visits.
library(tidyr)
data("toenail", package = "DPpackage")
data <- tidyr::complete(toenail, ID, visit) %>%
tidyr::fill(treatment) %>%
dplyr::select(-month)
table(data$outcome, useNA = "always") 0 1 <NA> 1500 408 150 Molenberghs and Verbeke (2005) described various analyses of these data. Here we impute the outcome of the missed visits. The next code block declares ID as the cluster variable, and creates $$m=5$$ imputations for the missing outcomes by method 2l.bin. pred <- make.predictorMatrix(data) pred["outcome", "ID"] <- -2 imp <- mice(data, method = "2l.bin", pred = pred, seed = 12102, maxit = 1, m = 5, print = FALSE) table(mice::complete(imp)$outcome, useNA = "always")
0 1 <NA>
1635 423 0
Figure 7.4 visualizes the imputations. The plot shows the partially imputed profiles of 16 subjects in the toenail data. The general downward trend in the probability of infection severity with time is obvious, and was also found by Molenberghs and Verbeke (2005, 302). Subjects 9 (never severe) and 117 (always severe) have both complete data. They represent the extremes, and their random effect estimates are very similar in all five imputed datasets. They are close, but not identical – as you might have expected – because the multiple imputations will affect the random effects also for the fully observed subjects. Subjects 31, 41 and 309 are imputed such that their outcomes are equivalent to subject 9, and hence have similar random effect estimates. In contrast, subject 214 has the same observed data pattern as 31, but it is sometimes imputed as “severe”. As a consequence, we see that there are now two random effect estimates for this subject that are quite different, which reflects the uncertainty due to the missing data. Subjects 48 and 99 even have three clearly different estimates. Imputation number 3 is colored green instead of grey, so the isolated lines in subjects 48 and 230 come from the same imputed dataset.
The complete-data model is a generalized linear mixed model for outcome given treatment status, time and a random intercept. This is similar to the models used by Molenberghs and Verbeke (2005), but here we use the visit instead of time (which is incomplete) as the timing variable. The estimates from the combined multiple imputation analysis are then obtained as
library(purrr)
mice::complete(imp, "all") %>%
purrr::map(lme4::glmer,
formula = outcome ~ treatment * visit + (1 | ID),
family = binomial) %>%
pool() %>%
summary()
estimate std.error statistic df p.value
(Intercept) -0.937 0.5778 -1.622 546 0.1052
treatment 0.152 0.6858 0.221 941 0.8250
visit -0.770 0.0848 -9.079 154 0.0000
treatment:visit -0.222 0.1219 -1.826 284 0.0682
As expected, these estimates are similar to the estimates obtained from the direct analysis of these data. The added value of multiple imputation here is that it produces a dataset with scores on all visits, which makes it easier to summarize. The added values of imputation increases if important covariates are available that are not present in the substantive model, or if missing values occur in the predictors. Section 7.10.2 contains an example of that problem.
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2021-09-27 04:33:28
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https://tex.stackexchange.com/questions/110001/centering-a-wide-table-that-is-defined-inside-a-newenviron
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# Centering a wide table that is defined inside a NewEnviron
I am trying to centre a table with a lot of content whose width is larger than the text width. The problem is that I am defining the table environment in the preamble so I can reuse it, and this seems to create a lot of issues.
\NewEnviron{reqtable}{
\table
\tabularx{1.3\textwidth}{cX}
\toprule
\BODY
\bottomrule
}[
\endtabularx
\endtable
]
I initialise my table with
\begin{reqtable}
item & item \\
\end{reqtable}
The reason I am using NewEnviron from the environ package is so that I can use rules from the booktabs package (see bottomrule not working in a self-made environment)
I have tried the following alterations to center the table.
• use \centering just after \table -- no errors, doesn't center.
• use \adjustwidth[]{}{-8em} just after the \table (see https://stackoverflow.com/questions/722613/latex-centering-a-table-wider-than-the-text-column) -- produces error ! File ended while scanning use of TX@get@body.
• use \begin{fullwidth} to wrap the \table -- produces error ! LaTeX Error: Not in outer par mode.
What else can I try to centre my table?
Note: I am using a twoside layout, so margins are different on odd and even pages. This is why the \adjustwidth command appeals to me most, as it can handle varying margins.
Here is an MWE with tables on different pages wider than the textwidth that I am trying to center.
\documentclass[11pt,a4paper,twoside]{report}
\usepackage{tabularx}
\usepackage{booktabs}
\usepackage{environ}
\usepackage{lipsum}
\NewEnviron{reqtable}{
\renewcommand{\arraystretch}{1.3}
\table
\tabularx{1.3\textwidth}{lX}
\toprule
\BODY
\bottomrule
}[
\endtabularx
\endtable
\renewcommand{\arraystretch}{1}
\vspace{10pt}
]
\begin{document}
\lipsum[1]
\begin{reqtable}
Some text & \lipsum[1] \\
\end{reqtable}
\lipsum[3]
% This table is on the second page which has different margins
\begin{reqtable}
Other text & \lipsum[1] \\
\end{reqtable}
\end{document}
You need to introduce some negative spacing either implicitly or explicitly to pull the table into the left margin
\NewEnviron{reqtable}{%
\table\centering
\hspace*{-.5\textwidth}\tabularx{1.3\textwidth}{cX}%
\toprule
\BODY
\bottomrule
}[%
\endtabularx\hspace*{-.5\textwidth}%
\endtable
Note I used the definition form you showed however the use form you showed
\begin{reqtable}{table1}{This is a table}
uses two arguments to the environment (a label and caption?) which are not defined here. ]
Probably works, although untested as you didn't supply an example document. The exact amount of negative space is fairly arbitrary as long as it totals more that .3\textwidth so the whole line is less that textwidth wide and will be centred.
• This does work, but the issue is I am using a twoside layout, so tables on different pages would need to be pulled either into left or right margin. – Elise Apr 22 '13 at 9:46
• @Elise Well the code I put there centres the table so pulls it into both margins, That is anyway the only safe thing to do without writing a lot more code. A floating environment like table is set when encountered but it is not known at that point which page it will appear on so you can not make different layout choice depending on page parity unless you do multiple runs and correct on the second pass by looking at \pageref of this table's label. – David Carlisle Apr 22 '13 at 9:52
I don't think you gain by hiding your environment and the markup.
\documentclass{article}
\usepackage{tabularx,booktabs,changepage}
\usepackage[pass,showframe]{geometry} % just to show centering
\usepackage{lipsum} % mock text
% optional argument is the default table placement
% mandatory argument is the fraction of \textwidth for the enlargement
\newenvironment{widetable}[2][htp]
{\begin{table}[#1]
\centering}
\begin{document}
\begin{widetable}{.15}
\begin{tabularx}{\linewidth}{cX}
\toprule
item & \lipsum[2] \\
\bottomrule
\end{tabularx}
\caption{This is a table}\label{table1}
\end{widetable}
\begin{widetable}{.1}
\begin{tabularx}{\linewidth}{cX}
\toprule
item & \lipsum[2] \\
\bottomrule
\end{tabularx}
\caption{This is a table}\label{table2}
\end{widetable}
\end{document}
• Thank you, but I use this kind of table dozens of times and therefore would prefer the new environment instead of repeating code, and the issue is making \adjustwidth work inside that newenvironment command. – Elise Apr 22 '13 at 10:15
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2019-11-13 12:55:34
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https://www.physicsforums.com/threads/parallel-plate-capacitor-with-dielectric-in-a-gravitational-field.354944/
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# Homework Help: Parallel plate capacitor with dielectric in a gravitational field.
1. Nov 15, 2009
1. The problem statement, all variables and given/known data
A square parallel plate capacitor with vertical plates of area A
and distance d, charged with a constant charge Q and is completely
filled with a dielectric material the same dimension as the gap between
the plates, with dielectric constant k and mass m. Assuming
the dielectric is a solid block of material that can move inside the
capacitor with no friction, what would be the equilibrium condition
in the presence of gravitational field? What would be the period of
small oscillations around this equilibrium point?
2. Relevant equations
C=(k$$\epsilon$$0A)/d
3. The attempt at a solution
Honestly I dont know where to begin with this question. Is it suggesting that the dielectric would oscillate from side to side between the plates? How? Im assuming ive just stared at the problem too long and im missing something obvious, so if someone could just nudge me in the right direction by pointing out what this question is asking it would help a lot.
2. Nov 15, 2009
### Chi Meson
Vertical plates! There will be two forces in balance, one the weight of the dialectric, the other an electrostatic attraction between plates and the dialectric.
3. Nov 19, 2009
### connor415
Why is there electrostatic attraction toward the dielectric, it has no charge! Adwodon Im pretty sure youre on my course btw. UCL?
4. Nov 20, 2009
Hey
Sorry Chi Meson I forgot to thank you it was a case of me staring at it so long I completely ignored the vertical part, it took a while but I figured it out before our original due date (tuesday), we got an extension though as barely anyone could do all 3 questions (this being the first and easiest).
Connor yes im at UCL.
The dielectric has no overall charge, but the electrons will move towards the positive plate of the capacitor so you get something like this:
So the attraction is only between the edge of the dielectric.
When a dielectric is fully inserted this force will cancel itself out, but if there is a gap it will pull it in (ie if the dielectric starts to fall out it will be pulled back in)
If you want some help imagine the dielectric is horizontal for now, push it into the dielectric by a distance x
Capacitance of the part filled with dielectric will be:
C1=(e0KLx)/d
Part filled with air:
C2=(e0L(L-x))/d
as the volage across the two parts is the same
C=C1+C2
As charge is constant:
U= (-Q^2)/2C
F= -dU/dx
Thats how you figure out the force the plates put on the dielectric, then just imagine the plates were vertical. As for the small oscillations, just see what happens when the dielectric is pushed a small distance past equilibrium (y, where y<<x).
If you're still having trouble im easy to spot, im the guy with the arm covered in tattoos. Although im pretty sure ive nailed this one I havent touched the rest of this problem sheet though. Too busy with other work.
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2018-05-25 01:39:03
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http://cms.math.ca/cjm/msc/46J99?fromjnl=cjm&jnl=CJM
|
location: Publications → journals
Search results
Search: MSC category 46J99 ( None of the above, but in this section )
Expand all Collapse all Results 1 - 1 of 1
1. CJM 2002 (vol 54 pp. 303)
Ghahramani, Fereidoun; Grabiner, Sandy
Convergence Factors and Compactness in Weighted Convolution Algebras We study convergence in weighted convolution algebras $L^1(\omega)$ on $R^+$, with the weights chosen such that the corresponding weighted space $M(\omega)$ of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor $\eta$ for which weak$^\ast$-convergence of $\{\lambda_n\}$ to $\lambda$ in $M(\omega)$ implies norm convergence of $\lambda_n \ast f$ to $\lambda \ast f$ in $L^1 (\omega\eta)$. We find necessary and sufficent conditions which depend on $\omega$ and $f$ and also find necessary and sufficent conditions for $\eta$ to be a convergence factor for all $L^1(\omega)$ and all $f$ in $L^1(\omega)$. We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if and only if convolution by $f$ is a compact operator from $M(\omega)$ (or $L^1(\omega)$) to $L^1 (\omega\eta)$. Categories:43A10, 43A15, 46J45, 46J99
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2016-05-24 06:08:42
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https://www.physicsforums.com/threads/momentum-of-spring-between-two-objects.585182/
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# Momentum of spring between two objects
## Homework Statement
A massless spring of spring constant 20 N/m is placed between two carts on a frictionless surface. Cart 1 has a mass M1 = 5 kg and Cart 2 has a mass M2 = 2.5 kg. The carts are pushed toward one another until the spring is compressed a distance 1.2 m. The carts are then released and the spring pushes them apart. After the carts are free of the spring, what are their speeds?
PEspring=0.5kx2
KE=0.5mv2=p2/2m
p=mv
## The Attempt at a Solution
I plugged in the given values to find that the potential energy of the spring is 14.4 J. Since there is no friction, the energy is conserved and the resulting kinetic energy should be the same value, and plugging it into the equation in terms of momentum results in p=14.69 since momentum is also conserved. So when I plug that into p=mv, I get v1=2.94 and v2=5.88 but apparently these are incorrect. Am I skipping a step?
Your calculations are not correct. check it
Well I'm trying to figure out where exactly I went wrong. What I did so far:
0.5*20*1.22=14.4 J
p2/2(7.5)=14.4 -> p2=216 -> p=14.69
v=p/m -> 14.69/5=2.94 m/s and 14.69/2.5=5.88 m/s
Any ideas?
moon,
$$K=\frac{p^2}{2m_1}+\frac{p^2}{2m_2}$$
Oh okay, I was combining them but I see now. Thank you!
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2021-08-02 12:58:43
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https://eprint.iacr.org/2019/1497
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## Cryptology ePrint Archive: Report 2019/1497
Analysis of Modified Shell Sort for Fully Homomorphic Encryption
Joon-Woo Lee and Young-Sik Kim and Jong-Seon No
Abstract: The Shell sort algorithm is one of the most practically effective sorting algorithms. However, it is difficult to execute this algorithm with its intended running time complexity on data encrypted using fully homomorphic encryption (FHE), because the insertion sort in Shell sort has to be performed by considering the worst-case input data. In this paper, in order for the sorting algorithm to be used on FHE data, we modify the Shell sort with an additional parameter $\alpha$ and a gap sequence of powers of two. The modified Shell sort is found to have the trade-off between the running time complexity of $O(n^{3/2}\sqrt{\alpha+\log\log n})$ and the sorting failure probability of $2^{-\alpha}$. Its running time complexity is close to the intended running time complexity of $O(n^{3/2})$ and the sorting failure probability can be made very low with slightly increased running time. Further, the optimal window length of the modified Shell sort is also derived via convex optimization. The proposed analysis of the modified Shell sort is numerically confirmed by using randomly generated arrays. Further, the performance of the modified Shell sort is numerically compared with the case of Ciura's optimal gap sequence and the case of the optimal window length obtained through the convex optimization.
Category / Keywords: applications / Fully Homomorphic Encryption, Sorting Algorithm
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2020-01-23 05:33:09
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https://en.wikipedia.org/wiki/Min-max_heap
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Min-max heap
Min-max heap
Typebinary tree/heap
Invented1986
Time complexity in big O notation
Algorithm Insert Average Worst case O(log n) O(log n) O(log n) [1] O(log n)
In computer science, a min-max heap is a complete binary tree data structure which combines the usefulness of both a min-heap and a max-heap, that is, it provides constant time retrieval and logarithmic time removal of both the minimum and maximum elements in it.[2] This makes the min-max heap a very useful data structure to implement a double-ended priority queue. Like binary min-heaps and max-heaps, min-max heaps support logarithmic insertion and deletion and can be built in linear time.[3] Min-max heaps are often represented implicitly in an array;[4] hence it's referred to as an implicit data structure.
The min-max heap property is: each node at an even level in the tree is less than all of its descendants, while each node at an odd level in the tree is greater than all of its descendants.[4]
The structure can also be generalized to support other order-statistics operations efficiently, such as `find-median`, `delete-median`,[2]`find(k)` (determine the kth smallest value in the structure) and the operation `delete(k)` (delete the kth smallest value in the structure), for any fixed value (or set of values) of k. These last two operations can be implemented in constant and logarithmic time, respectively. The notion of min-max ordering can be extended to other structures based on the max- or min-ordering, such as leftist trees, generating a new (and more powerful) class of data structures.[4] A min-max heap can also be useful when implementing an external quicksort.[5]
Description
• A min-max heap is a complete binary tree containing alternating min (or even) and max (or odd) levels. Even levels are for example 0, 2, 4, etc, and odd levels are respectively 1, 3, 5, etc. We assume in the next points that the root element is at the first level, i.e., 0.
Example of Min-max heap
• Each node in a min-max heap has a data member (usually called key) whose value is used to determine the order of the node in the min-max heap.
• The root element is the smallest element in the min-max heap.
• One of the two elements in the second level, which is a max (or odd) level, is the greatest element in the min-max heap
• Let ${\displaystyle x}$ be any node in a min-max heap.
• If ${\displaystyle x}$ is on a min (or even) level, then ${\displaystyle x.key}$ is the minimum key among all keys in the subtree with root ${\displaystyle x}$.
• If ${\displaystyle x}$ is on a max (or odd) level, then ${\displaystyle x.key}$ is the maximum key among all keys in the subtree with root ${\displaystyle x}$.
• A node on a min (max) level is called a min (max) node.
A max-min heap is defined analogously; in such a heap, the maximum value is stored at the root, and the smallest value is stored at one of the root's children.[4]
Operations
In the following operations we assume that the min-max heap is represented in an array `A[1..N]`; The ${\displaystyle ith}$ location in the array will correspond to a node located on the level ${\displaystyle \lfloor \log i\rfloor }$ in the heap.
Build
Creating a min-max heap is accomplished by an adaptation of Floyd's linear-time heap construction algorithm, which proceeds in a bottom-up fashion.[4] A typical Floyd's build-heap algorithm[6] goes as follows:
```function FLOYD-BUILD-HEAP(h):
for each index i from ${\displaystyle \lfloor length(h)/2\rfloor }$down to 1 do:
push-down(h, i)
return h
```
In this function, h is the initial array, whose elements may not be ordered according to the min-max heap property. The `push-down` operation (which sometimes is also called heapify) of a min-max heap is explained next.
Push Down
The `push-down` algorithm (or `trickle-down` as it is called in [4] ) is as follows:
```function PUSH-DOWN(h, i):
if i is on a min level then:
PUSH-DOWN-MIN(h, i)
else:
PUSH-DOWN-MAX(h, i)
endif
```
Push Down Min
```function PUSH-DOWN-MIN(h, i):
if i has children then:
m := index of the smallest child or grandchild of i
if m is a grandchild of i then:
if h[m] < h[i] then:
swap h[m] and h[i]
if h[m] > h[parent(m)] then:
swap h[m] and h[parent(m)]
endif
PUSH-DOWN-MIN(h, m)
endif
else if h[m] < h[i] then:
swap h[m] and h[i]
endif
endif
```
Push Down Max
The algorithm for `push-down-max` is identical to that for push-down-min, but with all of the comparison operators reversed.
```function PUSH-DOWN-MAX(h, i):
if i has children then:
m := index of the largest child or grandchild of i
if m is a grandchild of i then:
if h[m] > h[i] then:
swap h[m] and h[i]
if h[m] < h[parent(m)] then:
swap h[m] and h[parent(m)]
endif
PUSH-DOWN-MAX(h, m)
endif
else if h[m] > h[i] then:
swap h[m] and h[i]
endif
endif
```
Iterative Form
As the recursive calls to `push-down-min` and `push-down-max` are in tail position, these functions can be trivially converted to purely iterative forms executing in constant space:
```function PUSH-DOWN-MIN-ITER(h, m):
while m has children then:
i := m
m := index of the smallest child or grandchild of i
if h[m] < h[i] then:
if m is a grandchild of i then:
swap h[m] and h[i]
if h[m] > h[parent(m)] then:
swap h[m] and h[parent(m)]
endif
else
swap h[m] and h[i]
endif
else
break
endif
endwhile
```
Insertion
To add an element to a min-max heap perform following operations:
1. Append the required key to (the end of) the array representing the min-max heap. This will likely break the min-max heap properties, therefore we need to adjust the heap.
2. Compare the new key to its parent:
1. If it is found to be less (greater) than the parent, then it is surely less (greater) than all other nodes on max (min) levels that are on the path to the root of heap. Now, just check for nodes on min (max) levels.
2. The path from the new node to the root (considering only min (max) levels) should be in a descending (ascending) order as it was before the insertion. So, we need to make a binary insertion of the new node into this sequence. Technically it is simpler to swap the new node with its parent while the parent is greater (less).
This process is implemented by calling the `push-up` algorithm described below on the index of the newly-appended key.
Push Up
The `push-up` algorithm (or `bubble-up` as it is called in [7] ) is as follows:
```function PUSH-UP(h, i):
if i is not the root then:
if i is on a min level then:
if h[i] > h[parent(i)] then:
swap h[i] and h[parent(i)]
PUSH-UP-MAX(h, parent(i))
else:
PUSH-UP-MIN(h, i)
endif
else:
if h[i] < h[parent(i)] then:
swap h[i] and h[parent(i)]
PUSH-UP-MIN(h, parent(i))
else:
PUSH-UP-MAX(h, i)
endif
endif
endif
```
Push Up Min
```function PUSH-UP-MIN(h, i):
if i has a grandparent and h[i] < h[grandparent(i)] then:
swap h[i] and h[grandparent(i)]
PUSH-UP-MIN(h, grandparent(i))
endif
```
Push Up Max
As with the `push-down` operations, `push-up-max` is identical to `push-up-min`, but with comparison operators reversed:
```function PUSH-UP-MAX(h, i):
if i has a grandparent and h[i] > h[grandparent(i)] then:
swap h[i] and h[grandparent(i)]
PUSH-UP-MAX(h, grandparent(i))
endif
```
Iterative Form
As the recursive calls to `push-up-min` and `push-up-max` are in tail position, these functions also can be trivially converted to purely iterative forms executing in constant space:
```function PUSH-UP-MIN-ITER(h, i):
while i has a grandparent and h[i] < h[grandparent(i)] then:
swap h[i] and h[grandparent(i)]
i := grandparent(i)
endwhile
```
Example
Here is one example for inserting an element to a Min-Max Heap.
Say we have the following min-max heap and want to insert a new node with value 6.
Initially, node 6 is inserted as a right child of the node 11. 6 is less than 11, therefore it is less than all the nodes on the max levels (41), and we need to check only the min levels (8 and 11). We should swap the nodes 6 and 11 and then swap 6 and 8. So, 6 gets moved to the root position of the heap, the former root 8 gets moved down to replace 11, and 11 becomes a right child of 8.
Consider adding the new node 81 instead of 6. Initially, the node is inserted as a right child of the node 11. 81 is greater than 11, therefore it is greater than any node on any of the min levels (8 and 11). Now, we only need to check the nodes on the max levels (41) and make one swap.
Find Minimum
The minimum node (or a minimum node in the case of duplicate keys) of a Min-Max Heap is always located at the root. Find Minimum is thus a trivial constant time operation which simply returns the roots.
Find Maximum
The maximum node (or a maximum node in the case of duplicate keys) of a Min-Max Heap is always located on the first max level--i.e., as one of the immediate children of the root. Find Maximum thus requires at most one comparison, to determine which of the two children of the root is larger, and as such is also a constant time operation.
Remove Minimum
Removing the minimum is just a special case of removing an arbitrary node whose index in the array is known. In this case, the last element of the array is removed (reducing the length of the array) and used to replace the root, at the head of the array. `push-down` is then called on the root index to restore the heap property in ${\displaystyle O(\log _{2}(n))}$time.
Remove Maximum
Removing the maximum is again a special case of removing an arbitrary node with known index. As in the Find Maximum operation, a single comparison is required to identify the maximal child of the root, after which it is replaced with the final element of the array and `push-down` is then called on the index of the replaced maximum to restore the heap property.
Extensions
The min-max-median heap is a variant of the min-max heap, suggested in the original publication on the structure, that supports the operations of an order statistic tree.
References
1. ^ Mischel. "Jim". Stack Overflow. Retrieved 8 September 2016.
2. ^ a b ATKINSON, M. D; SACK, J.-R; SANTORO, N.; STROTHOTTE, T. (1986). Munro, Ian (ed.). "Min-Max Heaps and Generalized Priority Queues" (PDF). Communications of the ACM. 29 (10): 996–1000. doi:10.1145/6617.6621.
3. ^ ATKINSON, M. D; SACK, J.-R; SANTORO, N.; STROTHOTTE, T. (1986). Munro, Ian (ed.). "Min-Max Heaps and Generalized Priority Queues" (PDF). Communications of the ACM. 29 (10): 996–1000. doi:10.1145/6617.6621.
4. ATKINSON, M. D; SACK, J.-R; SANTORO, N.; STROTHOTTE, T. (1986). Munro, Ian (ed.). "Min-Max Heaps and Generalized Priority Queues" (PDF). Communications of the ACM. 29 (10): 996–1000. doi:10.1145/6617.6621.
5. ^ Gonnet, Gaston H.; Baeza-Yates, Ricardo (1991). Handbook of Algorithms and Data Structures: In Pascal and C. ISBN 0201416077.
6. ^ K. Paparrizos, Ioannis (2011). "A tight bound on the worst-case number of comparisons for Floyd's heap construction algorithm". arXiv:1012.0956. Bibcode:2010arXiv1012.0956P. Cite journal requires `|journal=` (help)
7. ^ ATKINSON, M. D; SACK, J.-R; SANTORO, N.; STROTHOTTE, T. (1986). Munro, Ian (ed.). "Min-Max Heaps and Generalized Priority Queues" (PDF). Communications of the ACM. 29 (10): 996–1000. doi:10.1145/6617.6621.
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2021-12-08 23:52:14
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http://mathoverflow.net/feeds/question/120774
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holomorphic automorphisms of universal cover of configuration spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:59:07Z http://mathoverflow.net/feeds/question/120774 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120774/holomorphic-automorphisms-of-universal-cover-of-configuration-spaces holomorphic automorphisms of universal cover of configuration spaces Aakumadula 2013-02-04T15:08:03Z 2013-02-06T12:03:05Z <p>Hello everyone,</p> <p>I have been trying (without success) to determine the following. Let $P$ denote the space of monic polynomials of degree $n$ with complex coefficients, which have distinct roots. It is known that the fundamental group of this space is the braid group $B_n$. Let $X \rightarrow P$ denote the universal cover. Then $B_n\subset Aut (X)$ where $Aut (X)$ is the group of holomorphic automorphisms of $X$. For $n\geq 4$, is it true that $Aut (X)=B_n$? I do not see any other automorphisms (but I am a novice in the area). </p> http://mathoverflow.net/questions/120774/holomorphic-automorphisms-of-universal-cover-of-configuration-spaces/120816#120816 Answer by Aakumadula for holomorphic automorphisms of universal cover of configuration spaces Aakumadula 2013-02-05T01:07:15Z 2013-02-05T01:07:15Z <p>Misha has answered my question completely. </p>
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2013-05-23 09:59:14
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http://lilypond.1069038.n5.nabble.com/Very-inconsistent-vertical-spacing-from-lilypond-book-and-LaTeX-td233168.html
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# Very inconsistent vertical spacing from lilypond book and LaTeX
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## Very inconsistent vertical spacing from lilypond book and LaTeX
Hi, I'm helping my partner create a book of interval exercises, and I'm getting very inconsistent vertical spacing coming from lilypond-book, here is an example: https://files.robehickman.com/problem.pdfWeird thing is that this is very inconsistent between pages, and I have verified there is no excess space in the music images. LaTeX seems to be stretching white space erratically.
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## Re: Very inconsistent vertical spacing from lilypond book and LaTeX
Most likely the scores are for some obscure reason cropped to different heights. For a quick workaround, you can try putting the images in a parbox with a specified height. signature.asc (849 bytes) Download Attachment
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## Re: Very inconsistent vertical spacing from lilypond book and LaTeX
In reply to this post by Robert Hickman On Thu 21 May 2020 at 18:40:53 (+0100), Robert Hickman wrote: > > I'm helping my partner create a book of interval exercises, and I'm > getting very inconsistent vertical spacing coming from lilypond-book, > here is an example: > > https://files.robehickman.com/problem.pdf> > Weird thing is that this is very inconsistent between pages, and I > have verified there is no excess space in the music images. LaTeX > seems to be stretching white space erratically. No idea what's in your .tex file. Perhaps the larger spaces are where LaTeX thinks a new paragraph starts. The mere presence or absence of blank lines between the figures can affect/control that. Cheers, David.
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## Re: Very inconsistent vertical spacing from lilypond book and LaTeX
Hi David, The .tex file contains the following (a snippet of), The idea about it being where a new paragraph or line starts seems to make sense as the added space seems to be about 1em. \vspace{-0em}\Ssubsection{Scale}\vspace{-0em} {% \parindent 0pt \noindent \ifx\preLilyPondExample \undefined \else \expandafter\preLilyPondExample \fi \def\lilypondbook{}% \input{ce/lily-6d98c8ee-systems.tex} \ifx\postLilyPondExample \undefined \else \expandafter\postLilyPondExample \fi } \vspace{-0.8em}\Ssubsection{Thirds}\vspace{-0em} {% \parindent 0pt \noindent \ifx\preLilyPondExample \undefined \else \expandafter\preLilyPondExample \fi \def\lilypondbook{}% \input{ec/lily-01f558b3-systems.tex} \ifx\postLilyPondExample \undefined \else \expandafter\postLilyPondExample \fi } \vspace{-0.8em}\Ssubsection{Fourths}\vspace{-0em} {% \parindent 0pt \noindent \ifx\preLilyPondExample \undefined \else \expandafter\preLilyPondExample \fi \def\lilypondbook{}% \input{75/lily-95458ceb-systems.tex} \ifx\postLilyPondExample \undefined \else \expandafter\postLilyPondExample \fi } \vspace{-0.8em}\Ssubsection{Fifths}\vspace{-0em} {% \parindent 0pt \noindent \ifx\preLilyPondExample \undefined \else \expandafter\preLilyPondExample \fi \def\lilypondbook{}% \input{2f/lily-cee1d442-systems.tex} \ifx\postLilyPondExample \undefined \else \expandafter\postLilyPondExample \fi } \vspace{-0.8em}\Ssubsection{Sixths}\vspace{-0em} {% \parindent 0pt \noindent \ifx\preLilyPondExample \undefined \else \expandafter\preLilyPondExample \fi \def\lilypondbook{}% \input{05/lily-bdd1bc6c-systems.tex} \ifx\postLilyPondExample \undefined \else \expandafter\postLilyPondExample \fi } On Thu, 21 May 2020 at 19:56, David Wright <[hidden email]> wrote: > > On Thu 21 May 2020 at 18:40:53 (+0100), Robert Hickman wrote: > > > > I'm helping my partner create a book of interval exercises, and I'm > > getting very inconsistent vertical spacing coming from lilypond-book, > > here is an example: > > > > https://files.robehickman.com/problem.pdf> > > > Weird thing is that this is very inconsistent between pages, and I > > have verified there is no excess space in the music images. LaTeX > > seems to be stretching white space erratically. > > No idea what's in your .tex file. Perhaps the larger spaces are where > LaTeX thinks a new paragraph starts. The mere presence or absence of > blank lines between the figures can affect/control that. > > Cheers, > David. >
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## Re: Very inconsistent vertical spacing from lilypond book and LaTeX
On Thu 21 May 2020 at 21:02:25 (+0100), Robert Hickman wrote: > On Thu, 21 May 2020 at 19:56, David Wright <[hidden email]> wrote: > > On Thu 21 May 2020 at 18:40:53 (+0100), Robert Hickman wrote: > > > > > > I'm helping my partner create a book of interval exercises, and I'm > > > getting very inconsistent vertical spacing coming from lilypond-book, > > > here is an example: > > > > > > https://files.robehickman.com/problem.pdf> > > > > > Weird thing is that this is very inconsistent between pages, and I > > > have verified there is no excess space in the music images. LaTeX > > > seems to be stretching white space erratically. > > > > No idea what's in your .tex file. Perhaps the larger spaces are where > > LaTeX thinks a new paragraph starts. The mere presence or absence of > > blank lines between the figures can affect/control that. > > The .tex file contains the following (a snippet of), The idea about it > being where a new paragraph or line starts seems to make sense as the > added space seems to be about 1em. To check this out, you could increase \parskip substantially and see if the result is consistent with that. If it is, then the alternatives would appear to be either to scour the individual \input files for differences in paragraph control, like blank lines, or to capitulate and add blank lines to this file, making every score into a new paragraph. (Multiple blank lines only cause a single \parskip.) If the problem lies in lilypondbook, I'm afraid I'm out of my depth. > \vspace{-0em}\Ssubsection{Scale}\vspace{-0em} > {% > \parindent 0pt > \noindent > \ifx\preLilyPondExample \undefined > \else > \expandafter\preLilyPondExample > \fi > \def\lilypondbook{}% > \input{ce/lily-6d98c8ee-systems.tex} > \ifx\postLilyPondExample \undefined > \else > \expandafter\postLilyPondExample > \fi > } > > \vspace{-0.8em}\Ssubsection{Thirds}\vspace{-0em} > {% > \parindent 0pt > \noindent > \ifx\preLilyPondExample \undefined > \else > \expandafter\preLilyPondExample > \fi > \def\lilypondbook{}% > \input{ec/lily-01f558b3-systems.tex} > \ifx\postLilyPondExample \undefined > \else > \expandafter\postLilyPondExample > \fi > } > > \vspace{-0.8em}\Ssubsection{Fourths}\vspace{-0em} > {% > \parindent 0pt > \noindent > \ifx\preLilyPondExample \undefined > \else > \expandafter\preLilyPondExample > \fi > \def\lilypondbook{}% > \input{75/lily-95458ceb-systems.tex} > \ifx\postLilyPondExample \undefined > \else > \expandafter\postLilyPondExample > \fi > } > > \vspace{-0.8em}\Ssubsection{Fifths}\vspace{-0em} > {% > \parindent 0pt > \noindent > \ifx\preLilyPondExample \undefined > \else > \expandafter\preLilyPondExample > \fi > \def\lilypondbook{}% > \input{2f/lily-cee1d442-systems.tex} > \ifx\postLilyPondExample \undefined > \else > \expandafter\postLilyPondExample > \fi > } > > \vspace{-0.8em}\Ssubsection{Sixths}\vspace{-0em} > {% > \parindent 0pt > \noindent > \ifx\preLilyPondExample \undefined > \else > \expandafter\preLilyPondExample > \fi > \def\lilypondbook{}% > \input{05/lily-bdd1bc6c-systems.tex} > \ifx\postLilyPondExample \undefined > \else > \expandafter\postLilyPondExample > \fi > } Cheers, David.
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2020-05-30 03:55:19
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https://mathsimulationtechnology.wordpress.com/ask-why-be-scientist/
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• What’s the go of that? What’s the particular go of that? (James Clerk Maxwell (1831-1879) Scottish physicist. Comments made as a child expressing his curiosity about mechanical things and physical phenomena)
• Why are things as they are and not otherwise? (Kepler (1571-1630))
• To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science. (Einstein)
From Questioning to Understanding
BodyandSoul encourages you to be critical, to ask questions, and only accept what you can understand on rational grounds. You will find that the nature of mathematics invites to such a critical approach, because in mathematics you draw conclusions from certain assumptions using logic and symbolic or numerical computation. If the assumptions are clearly stated, and each logical and computational step is open to inspection, then it is possible to objectively check if mathematical conclusion or result is correct or not, up to the correctness of the assumptions.
In other words, you will be able to work very much like a scientist, like a critical scientist who constantlyask ther questions Why? and Why Not? You will yourself discover some of the power of this approach (andalso some of its limitations).
As a child you asked many questions, but then later in school you learned not to ask too much. In a way youshould now try to recover from this effect of your schooling and return to the questioning of your childhood.
It is not always so easy but it can be very rewarding. The Internet and the computer are at your disposal, anddo not get tired by too many questions (like maybe your teachers, friends and family) or much work,and thus can give you good answers if you can only discriminate. To learn to do so is part of the criticaltraining you can get through BodyandSoul.
You will discover that to say that you understand something of a some physical process, typically meansthat there is an underlying mathematical model with certain properties. For example, if you say thatyou understand the motion of pendulum swinging back and forth, as a repeated exchange between potential and kinetic energies, it means that you know the equations of motion of the pendulum and you can prove e.g. thatthe sum of potential and kinetic energies remains constant.
Or if you say that you understand how an ice skater can increase the spin faster by pulling the armstight into the body, it means that you know the equations of motion and the connection between spinand moment of inertia.
Some Questions
As a mathematical scientist you should be ready ask for example, WHY is it so that
• $1+1 = 2$
• $(-1)\times (-1) = 1$
• $2+3 = 3+ 2$
• $\exp(a)\exp(b) = \exp(a+b)$
• $\log(ab) =\log (a)+\log(b)$
• $\exp(\log(a)) =a$
• $\sin(t)^2+\cos(t)^2 = 1$
• length of the perimeter of a circle of unit radius $=2\pi$
• area of a circular disc of unit radius $= \pi$
• volume of a sphere of unit radius $=\frac{4}{3}\pi$?
Maybe you already know good answers, but if you don’t know, don’t worry; you will naturally discover the answers as you go along, and answers to many more questions…
Spinning quickly by decreasing the moment of inertia while keeping total angular momemntum constant.
Clerk Maxwell as a child with his kind mother answering his questions: What's the go of that? What's the particular go of that?
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2018-02-23 04:28:36
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https://meta.stackexchange.com/questions/367961/im-not-getting-reputation-notifications
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# I'm not getting reputation notifications
I haven't been getting reputation notifications since recently.
(I reloaded several times, closed and reopened my browser, and restarted my computer)
New rep is still shown in the profile page Reputation tab:
• I can reproduce: i.stack.imgur.com/XUy39.jpg Jul 29, 2021 at 4:36
• Hah! I'm not getting notifications for the upvotes to this question :) Jul 29, 2021 at 4:36
• Could have something to do with today's maintenance Jul 29, 2021 at 4:48
• Heck, I'm not getting any... I have some followed questions and those notifications aren't coming in, either.
– Catija StaffMod
Jul 29, 2021 at 5:01
• @Catija I'm getting comment notifications and chat pings, at least. And I don't have enough followed posts to check 'em. Jul 29, 2021 at 5:02
• Yeah, I got the one for here but there's definitely others that aren't going out.
– Catija StaffMod
Jul 29, 2021 at 5:03
• I'm not getting followed post notifications or reputation notifications either. Jul 29, 2021 at 5:38
• same here ...... Jul 29, 2021 at 6:09
– Glorfindel Mod
Jul 29, 2021 at 6:42
• On mobile I did see my updated rep bumber up top Jul 29, 2021 at 7:18
• Regression of No notifications for followed posts? Jul 29, 2021 at 8:04
• I'm only getting reputation notifications for a Teams post, nothing for the main sites.
– Tinkeringbell Mod
Jul 29, 2021 at 8:52
• Same here. No notifications. But I can see the reputation change in my Profile under Reputation Tab.
– Ram
Jul 29, 2021 at 9:12
• I noticed it because I got notified of a “Nice Answer” badge, but got no notification of an upvote. Jul 29, 2021 at 9:41
• I'm getting some rep notifications now, but it hasn't backfilled fully yet. At least I assume that's what happening - I'm getting little bursts of rep notifications when I know I've repcapped overnight with this Q. Jul 29, 2021 at 12:36
Couple of things blew up here and it took a little while to isolate and repair the fallout, so sorry for the delay.
We had a (successful!) SQL failover in the wee hours of this morning but, for the brief period of time that the application was failing over, a number of duplicate events were written to a service that we call the "aggregator". This service is responsible for taking events from the various sites across the Stack Exchange network and aggregating them to denormalized tables so that we can get performant queries for things that cross the whole network - things like global inbox messages, rep, achievements, etc.
Usually this would be a non-issue - we have de-duplication logic in those code paths and it has functioned just fine for years. However, there were a specific set of events that had, clearly, never been de-duped - they're relatively rare (happen once per user per day). De-duping takes the form of sorting and eliminating the duplicates using equality checks - unfortunately the code for checking equality in one of these events looked like this:
public bool Equals(object obj)
{
if (!EqualityBase(obj, out VisitEvent other)) return false;
return other.Equals(this) && other.UserHistoryId == UserHistoryId;
}
That call to other.Equals(this) was very rarely hit, but it clearly recurses infinitely, leading to a StackOverflowException. That exception kills the process and because of the way the aggregator service runs, it meant that whichever server picked up the work would then die off when it tried to pick up the work.
We have some poison message handling around this but it didn't kick in effectively here and we have a large backlog of events to get through now the bug is fixed. It's getting through them now though and we should be all set in the next hour or so.
UPDATE Annnnnd it’s done. Thanks for your patience folks!
• So Stack Overflow's rep problem was due to a Stack Overflow? (Someone had to say it :D) Jul 29, 2021 at 12:41
• @bobble Frankly I'm only glad the site was up enough for me to find the right procdump incantation to capture a dump of a StackOverflowException :) Jul 29, 2021 at 12:42
• In TeX it would be \def\recurse{\recurse}\recurse ;-) Good job! Jul 29, 2021 at 13:15
• Reminds me of this... Jul 29, 2021 at 13:32
• @egreg Hate to be a nitpicker, but this infinite loop will not lead to a stack overflow, as TeX’s macro expansion routines handle tail recursion gracefully. You’d need something like \def\recurse{\recurse x}\recurse. Jul 29, 2021 at 13:37
• @EmilJeřábek Indeed, it does even worse than causing an overflow! ;-) Jul 29, 2021 at 14:01
• Hey Dean, its still not 100% up to speed for me. Jul 29, 2021 at 15:57
As Dean estimated things would be back to normal by now, I have to say things are not lining up for me yet.
See this screenshot for example:
My achievements box shows 130 rep for the day, while my profile shows that I'm rep-capped at 200.
• Same here - my inbox only shows 172 meta rep for today. Jul 29, 2021 at 17:23
• It’s quite possible that those were dropped on the floor; I’m out for a few days now but somebody else on the team will continue digging here to see what’s up Jul 29, 2021 at 17:30
• Enjoy your time @DeanWard.l! Jul 29, 2021 at 17:37
• Today it appears to work fine. However there is still rep dropped from yesterday in the top. Jul 30, 2021 at 5:10
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2022-08-19 17:27:55
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https://www.rocketryforum.com/threads/i-broke-down-and-ordered-my-own-launch-gear.160376/
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# I broke down and ordered my own launch gear.
### Help Support The Rocketry Forum:
#### Senior Space Cadet
##### Well-Known Member
TRF Supporter
What guy doesn't like things that go fast and shoot out flames? Other than that, my two goals, with rocketry, were to keep my mind active, now that I'm retired, and to make some new friends.
So I hadn't really intended to buy my own launch gear. I figured I'd be joining a club and using theirs.
Thing is, clubs aren't launching in Colorado. I don't see how fire danger is any higher this year than previous, but maybe they've just gotten more cautious. Not sure individuals can launch, legally, either. A couple years ago, a couple guys, out camping, started a forest fire that ended up killing some people. They got charged with murder. Burning down someone's house or, heaven forbid, causing someone's death, is something no one wants on their conscience.
What I'm hoping, what I'm thinking, is that after a good, overnight rain (rare in Colorado), the next morning might be a window of opportunity. I'm not sure whether to run the idea past the local law, who are going to be overly cautious, or just be sneaky about it. At the very least, the first snow will allow launching.
Anyway, I went ahead and ordered some launch gear. I figure it expands my options. I ordered a launch pad and electronics from Estes, but then it occurred to me that someone must surely make an adapter for camera tripods, which I have an abundance of, and sure enough, I found one at Apogee, so I'll be ordering that too. It only holds rods up to 3/16, so I'm still glad I ordered a pad with a 1/4 rod.
But now I'm thinking that, as my rockets get bigger and more powerful, I'm going to want to go to rails and buttons. I guess that's when I'll have to depend on a club.
#### prfesser
It only holds rods up to 3/16, so I'm still glad I ordered a pad with a 1/4 rod.
But now I'm thinking that, as my rockets get bigger and more powerful, I'm going to want to go to rails and buttons. I guess that's when I'll have to depend on a club.
Hi SSC, I'm a huge fan of rails and railbuttons, and strongly recommend them for every rocket. They are less obtrusive of the rocket's appearance; no rod whip; easily seen, so less likely to be sat upon or lost in the grass; no bent rods when sat upon; less chance of putting out an eye when approaching the pad. Check eBay frequently for the rail. Shipping is sometimes more than the rail itself, but occasionally one will come up for a reasonable price and with free shipping.
I'm also a fan of minimizing the number of rods. If the rocket is too big for a 1/8" rod, skip the 3/16" and go to a 1/4". (Too big for a 1/4? Definitely go with a rail.)
Best -- Terry
#### Steve Shannon
##### Well-Known Member
TRF Supporter
SSC
The reason the clubs are not flying may be related to injunctions against group activities due to the pandemic. In any case I would talk to the local fire marshal, to ask where model rockets are allowed. Of course anything larger than FAA Class 1 must have a COA anyway which usually is obtained by a club.
#### Joshua F Thomas
##### Well-Known Member
Doesn't the NAR liability insurance also cover you for damages, assuming you're correctly following all NAR guidelines?
#### Senior Space Cadet
##### Well-Known Member
TRF Supporter
Doesn't the NAR liability insurance also cover you for damages, assuming you're correctly following all NAR guidelines?
I guess I better join NAR.
#### Senior Space Cadet
##### Well-Known Member
TRF Supporter
SSC
The reason the clubs are not flying may be related to injunctions against group activities due to the pandemic. In any case I would talk to the local fire marshal, to ask where model rockets are allowed. Of course anything larger than FAA Class 1 must have a COA anyway which usually is obtained by a club.
I had assumed that covid-19 was the reason, but I was told it was fire bans. Maybe both. Yes, I should talk to the local authorities. I wasn't sure who to talk to. Fire marshal sounds like a good start.
#### Senior Space Cadet
##### Well-Known Member
TRF Supporter
Hi SSC, I'm a huge fan of rails and railbuttons, and strongly recommend them for every rocket. They are less obtrusive of the rocket's appearance; no rod whip; easily seen, so less likely to be sat upon or lost in the grass; no bent rods when sat upon; less chance of putting out an eye when approaching the pad. Check eBay frequently for the rail. Shipping is sometimes more than the rail itself, but occasionally one will come up for a reasonable price and with free shipping.
I'm also a fan of minimizing the number of rods. If the rocket is too big for a 1/8" rod, skip the 3/16" and go to a 1/4". (Too big for a 1/4? Definitely go with a rail.)
Best -- Terry
I suppose, if I'd been smart, I would have posted a question about launch gear weeks ago, but I hadn't planned on getting my own gear.
I'm not planning on building anything larger than a BT-80. Don't you need to be able to reach inside to secure the button?
Now, I suppose, I should start looking into buying or constricting what I need.
What do I need?
#### dhbarr
##### Amateur Professional
Rails are great, but I do all my low-to-mid solo stuff on something very much like
It's been good enough for everything up to a g76 so far.
#### AfterBurners
##### Well-Known Member
You could also build a pad that will adapt to both rails and rods.
#### Senior Space Cadet
TRF Supporter
I've been looking at 8020 10/10 extrusions. A six foot section would probably cost me around $30, with shipping, but I'm not sure how I'd make a base for it. The ready made rail launchers are ridiculously expensive. I've got a launch pad with 1/4 inch rod coming. I'll see what I think when it comes. 1/4" rod should be pretty stiff. I probably wouldn't be launching my D and E motor rockets, on my own, so I could use buttons on those and just assume I'll be using the club launch pads. #### rklapp ##### NAR# 109557 TRF Supporter I just use the regular Estes launch pad that fit 1/8" mostly for A through C motors (13 and 18 mm) and 3/16” mostly for D and E motors (24mm). It's easy to swap the rods (except the stupid wing nut keeps bending). Need the Porta Pad II for the 29mm motors (I believe). I've ordered the DIY parts for making my own launcher with a key and covered switch and lighted button. It will be awesome. If you let us know what county in Colorado you're in, we can help you look up local ordinances. For example, here's mine. Last edited: #### dr wogz ##### Fly caster 1/8" dia rod for small (18mm) builds, get one 3' long. 3/16" dia rod for larger (24mm) builds, get one 4' long 1/8" = .125" 3/16" = .1875" buy some 'piano wire' from the local hobby shop / hardware store. much better, and some #400 emery paper to get it silky smooth. (and to remove rust & crud) 1/4" rods are rare these days.. No need for a 1010 rail until you get into MPR. a mini rail is a good idea, but decide now, so you only need buy one rod rail system. Find out what the intended club has. Mine has both rods and a mini rail. nothing says you can't add both (I have a few with both lugs and rails: no waiting for a pad!) #### Greg Furtman ##### Well-Known Member TRF Supporter I built a wood tripod with adjustable wooden legs (originally designed for telescopes) and put a 1/8", 8" diameter SS disk on it. I had a rocket bind once on an Estes launch pad and it burned right through the aluminum disk! Having once been a stainless steel fabricator I know that SS endures hi temps much better than aluminum. I bought the disk on Ebay for around$10.
Here's a link to the tripod I made. And at age 69 and having back problems I like the height above the ground this gives me.
#### MALBAR 70
##### More Rockets Than Room
The Oddl' Rockets Adeptor is excellent for converting a camera tripod to rocketry use. It will hold 1/8" and 3\16" rods easily.
The Estes launch pad made for E engines holds a 1/4" rod, but the base is somewhat flimsy. The one I have fell over with my Estes Sat V on it... twice. I'd suggest either staking the legs down or at least couple of bricks to keep it upright.
My tripod converted to launch pad. The best part? No more kneeling down to attach the leads!
I also use one of Oddl' Rockets Raise springs to keep my rockets off the deflector.
#### rklapp
##### NAR# 109557
TRF Supporter
1/8" dia rod for small (18mm) builds, get one 3' long.
3/16" dia rod for larger (24mm) builds, get one 4' long
1/8" = .125"
3/16" = .1875"
buy some 'piano wire' from the local hobby shop / hardware store. much better, and some #400 emery paper to get it silky smooth. (and to remove rust & crud)
1/4" rods are rare these days..
No need for a 1010 rail until you get into MPR. a mini rail is a good idea, but decide now, so you only need buy one rod rail system. Find out what the intended club has. Mine has both rods and a mini rail. nothing says you can't add both (I have a few with both lugs and rails: no waiting for a pad!)
I stand corrected.
The standard rods are 32” and 35.5” long. I’ve been thinking of getting longer launch rods.
TRF Supporter
#### jrap330
##### Retired Engineer, NAR # 76940
TRF Supporter
What guy doesn't like things that go fast and shoot out flames? Other than that, my two goals, with rocketry, were to keep my mind active, now that I'm retired, and to make some new friends.
So I hadn't really intended to buy my own launch gear. I figured I'd be joining a club and using theirs.
Thing is, clubs aren't launching in Colorado. I don't see how fire danger is any higher this year than previous, but maybe they've just gotten more cautious. Not sure individuals can launch, legally, either. A couple years ago, a couple guys, out camping, started a forest fire that ended up killing some people. They got charged with murder. Burning down someone's house or, heaven forbid, causing someone's death, is something no one wants on their conscience.
What I'm hoping, what I'm thinking, is that after a good, overnight rain (rare in Colorado), the next morning might be a window of opportunity. I'm not sure whether to run the idea past the local law, who are going to be overly cautious, or just be sneaky about it. At the very least, the first snow will allow launching.
Anyway, I went ahead and ordered some launch gear. I figure it expands my options. I ordered a launch pad and electronics from Estes, but then it occurred to me that someone must surely make an adapter for camera tripods, which I have an abundance of, and sure enough, I found one at Apogee, so I'll be ordering that too. It only holds rods up to 3/16, so I'm still glad I ordered a pad with a 1/4 rod.
But now I'm thinking that, as my rockets get bigger and more powerful, I'm going to want to go to rails and buttons. I guess that's when I'll have to depend on a club.
You need to check AC supply and Hobbylinc........guys on this forum love them.. Prices are 20-30% less including the Estes Pad and Pro Controller. There was a good article 2-3 years ago about using a tripod and drill chuck which numerous poster on this site have made.
#### CoAz2k
##### Member
Thing is, clubs aren't launching in Colorado.
SCORE is flying southwest of Pueblo. I understand NCR is not flying due to fire risk.
I went to the SCORE site with the kids on 4 July to get some flights in. Joined the club and we'll be back.
Take water and shade. Bring mask(s) for RSO check-in and racking.
Looking forward to NCR to get flying again, but the SCORE site is closer to where I live near Colorado Springs anyways.
#### jrap330
##### Retired Engineer, NAR # 76940
TRF Supporter
A few years ago Sport Rocketry (NAR Magazine) had an article about converting old tripod with drill chuck to a launch pad. So JOIN NAR and get your magazine. You had to buy some hardware of course to mount drill chuck to tripod. So if anyone has it or can locate it..please provide for OP. OP- you said it, rails are expensive so don't worried until you have a Big Bird that may require it and by that time you will be launching with the club. 3/16 and 1/4 should be good for a while. And for all the good comments by members...my 3/16 rod is pretty stiff so I assume 1/4 would be even better. And as someone stated 2 weeks ago...stainless steel rods are no good...they bend more than non stainless. I assume you have you Estes starter set to launch all these small birds you build.
#### gna
##### average joe-overbuild member
Someone was throwing a tripod away at work, so I pulled it out of the dumpster. I had an old Makita drill that had died so I used the chuck for my launchpad. I used a piece of plywood, a 1/4-20 Teenut to attach to the tripod, and a bolt to hold the chuck. I think it was 3/8" but I can't remember. I bought 1/8" and 3/16" rod from the hardware store. Works fine.
I do have a 1/4" rod though I haven't tried it yet. I can splay the legs out, but I would probably weight them down, too.
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#### SCooke123
##### Well-Known Member
I built a wood tripod with adjustable wooden legs (originally designed for telescopes) and put a 1/8", 8" diameter SS disk on it. I had a rocket bind once on an Estes launch pad and it burned right through the aluminum disk! Having once been a stainless steel fabricator I know that SS endures hi temps much better than aluminum. I bought the disk on Ebay for around \$10.
Here's a link to the tripod I made. And at age 69 and having back problems I like the height above the ground this gives me.
Nice set-up for a tripod, looks pretty well made.
#### Joshua F Thomas
##### Well-Known Member
Someone was throwing a tripod away at work, so I pulled it out of the dumpster
This looks suspiciously like the tripod I bought several years ago and have never used.... might be time to see if I can retrofit that.
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2020-08-04 14:26:39
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https://www.quantamagazine.org/tag/modular-forms/
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### The Oracle of Arithmetic
At 28, Peter Scholze is uncovering deep connections between number theory and geometry.
### Sphere Packing Solved in Higher Dimensions
The Ukrainian mathematician Maryna Viazovska has solved the centuries-old sphere-packing problem in dimensions eight and 24.
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2017-02-23 09:26:15
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https://socratic.org/questions/20-ml-of-methane-is-completely-burnt-using-50-ml-of-oxygen-the-volume-of-the-gas
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# 20 mL of methane is completely burnt using 50 mL of oxygen. The volume of the gas left after cooling to room temperature is? A) 80 B)40 C) 60 D) 30
Sep 7, 2017
$\text{30 mL}$
#### Explanation:
Start by writing a balanced chemical equation for your reaction
${\text{CH"_ (4(g)) + 2"O"_ (2(g)) -> "CO"_ (2(g)) + 2"H"_ 2"O}}_{\left(l\right)}$
Notice that I added water as a liquid, $\left(l\right)$, because the problem tells you that after the reaction is complete, the resulting gaseous mixture is cooled down to room temperature.
Now, notice that the reaction consumes $2$ moles of oxygen gas for every $1$ mole of methane that takes part in the reaction and produces $1$ mole of carbon dioxide.
When your reaction involves gases kept under the same conditions for pressure and temperature, you can treat the mole ratios that exist between them in the balanced chemical reaction as volume ratios.
In your case, you can say that the reaction consumes $\text{2 mL}$ of oxygen gas for every $\text{1 mL}$ of methane that takes part in the reaction and produces $\text{1 mL}$ of carbon dioxide.
This means that in order for the reaction to consume all the methane present in the sample, you need
20 color(red)(cancel(color(black)("mL CH"_4))) * "2 mL O"_2/(1color(red)(cancel(color(black)("mL CH"_4)))) = "40 mL O"_2
As you can see, you have more oxygen gas than you need to ensure that all the methane reacts $\to$ oxygen gas is in excess, which is equivalent to saying that methane is a limiting reagent.
So, the reaction will consume $\text{20 mL}$ of methane and $\text{40 mL}$ of oxygen gas and produce
20 color(red)(cancel(color(black)("mL CH"_4))) * "1 mL CO"_2/(1color(red)(cancel(color(black)("mL CH"_4)))) = "20 mL CO"_2
After the reaction is complete, you will be left with
${\text{50 mL O"_2 - "40 mL O"_2 = "10 mL O}}_{2}$
that are not consumed by the reaction and with $\text{20 mL}$ of carbon dioxide. Therefore, you can say that after the reaction is complete, your mixture will contain
$\text{10 mL O"_2 + "20 mL CO"_2 = "30 mL gas}$
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2022-01-18 18:54:34
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http://blog.inf.ed.ac.uk/gdutton/2009/01/
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not-a-blog, not-a-service, not-a-clue.
## Adding a form to webmark
Webmark is my slightly hacked-together system for producing PDFs from a Web Form. As it was intended to be entirely stateless, single-use, and to require no connection between input and output, except for some fields, it was never designed to hold schema information or metadata, so there’s a little duplication of effort in adding a form to the system. Still, it was designed so that this task could be done with zero code modification.
Written by gdutton
28 January 2009 at 2011
Posted in RAT
Tagged with , , ,
## Passive Monitoring
Familiar?
Profile Translation WARNING 1/4 WARNING: Service checked passively
If you’re getting this warning for a service which is checked passively by default, the solution is simple: just disable active checking!
It seems as if a more appropriate message would be something along the lines of “WARNING: cannot perform active checking” or similar. Still, now I know, if it saves someone else attempting to re-enable every option three times over… or reading documentation…
Written by gdutton
28 January 2009 at 1933
Posted in DICE
Tagged with , ,
## offlineimap and alpine
Edit: The future is here! I’ve shortened my wishlist since OfflineIMAP now supports the IDLE command.
Further Edit: Kerberos instructions for Mac OS now available
For some time I’ve been meaning to make use of some sort of mail caching, in order to use my favourite email client whilst offline. The end result of this process is that my incoming mail now takes a somewhat circuitous route of:
imap server
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offlineimap - local uw-imapd - alpine
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local cache
on my laptop.
Written by gdutton
14 January 2009 at 1619
## bash completion
A fantastic way to save typing, I find that this is also a huge timesaver when learning new commands. I’ll freely admit that I’ve become a little dependent.
On DICE: ensure you have the latest bash-completion-*.inf package installed (most do). Then, simply add the following to your ~/.brc:
Written by gdutton
14 January 2009 at 0734
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2013-12-11 16:21:47
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https://www.originlab.com/index.aspx?go=Solutions/CaseStudies&pid=827
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# Mapping the Density States of a Superconductor as a Function of Energy
Summary
The Seamus Davis research group at the University of California at Berkeley and at Cornell University, has performed research related to the study of experimental condensed matter physics. Using a scanning tunneling microscope, graduate student Kyle McElroy took an image of a high temperature superconductor called Bi2Sr2CaCu2O8+x and mapped the density of states of the superconductor as a function of energy. In layman's terms, he made a map of the energy levels of the electrons at different locations in the crystal.
0.1 0 nm Fig 1: Atomic resolution image of the surface of the high-temperature superconductor BSCCO. This image was acquired with a sctunneling microscope. Each white dot is an individual bismuth atom. The horizontal waves are an incommensurate reconstruction due to stresses in the crystal.
Although a scanning tunneling microscope can map the energy levels of the electrons as a function of position, it cannot directly access the momentum of the electrons, which is also very important for understanding the way the crystal works. But with some pretty clever analysis, the momentum information was backed out of the spatial information which was acquired directly.
Fig 2: A map of the local electronic density of states at energy -16 meV, in the same field of view as Fig 1. This is effectively a density map of the number of electrons at each spatial location which have energy -16 meV. There are a number of different modulations and periodic structures in several different directions: vertically and horizontally and diagonally. Each of these modulations is an electron standing wave. 0.8 0.18 nS
Details
Another graduate student, Jenny Hoffman, took the two dimensional Fourier Transform of the electron-density maps, as a function of energy. The Fourier transform shows that there are numerous modulations of the density of states, at multiple frequencies and directions. There are too many modulations to see clearly by eye in real space, but the different modulations are easily distinguished in k-space.
These modulations are actually standing waves of electrons (an electron is both a particle and a wave!). The exact wavevectors of these standing waves were inverted to indicate the momentum of the electrons. However, each k-space map has about sixteen 2-dimensional peaks which had to be fit accurately, and Jenny had k-space maps at dozens of energies. Unfortunately, much of the data was too noisy for a 2-dimensional peak-fitting routine to converge reliably, so the data had to be extracted in lines along all different angles out from the zero-wavevector center of the k-space image, and then the peaks in these lines had to be fitted. In fact, 16 peaks x 30 or more energies per dataset had to be analyzed! Once analyzed, the parameter results from all fits had to be organized and used to back out the crucial momentum information. This is where Origin came in.
0.6 0 nS Å Fig 2: A map of the local electronic density of states at energy -16 meV, in the same field of view as Fig 1. This is effectively a density map of the number of electrons at each spatial location which have energy -16 meV. There are a number of different modulations and periodic structures in several different directions: vertically and horizontally and diagonally. Each of these modulations is an electron standing wave.
Analysis in Origin
Jenny worked extensively with Origin C programming to automate many aspects of her analysis in Origin. Functions were created to assist her in importing the data, preparing the worksheet for graphing and analysis, fitting the data, and visualizing the data and results in graphical form. These functions were hooked up to buttons placed on a custom worksheet template.
Fig 4: The buttons on the custom worksheet template.
Bringing in the data
An "Import" button was created and placed on the worksheet template. The button uses LabTalk™ to bring up a dialog for selecting the file to import. The script also calls an Origin C function that handles the import once the file is selected. The C function processes the data file, looking at and parsing header information along the way, and then dumps the data into the worksheet template.
Setting up the worksheet
A separate "Set XYXY" button was created to update the worksheet column designations. This button also uses LabTalk script to call an Origin C function.
Visualizing the data before analysis
The "Scatter Waterfall", "Line Waterfall", and "LineSymb Waterfall" buttons were created to allow you to plot the data as scatter, line, or line and symbol plots prior to analysis. As with the other buttons, these three use LabTalk to make calls to Origin C functions, which in turn perform the necessary operations to plot the data and annotate the graph with information from the selected columns of data.
Fitting the data
The analysis of the data took a lot of initial setup time, but once this was completed, it was a snap. Jenny first created the fitting functions to be used during the fitting process. She used Origin's nonlinear curve fitter to do this.
Fig 5: The Nonlinear Curve Fitter showing a fitting function formula used in the analysis.
The fitting functions were then called from various Origin C functions she created.
Fig 6: Origin's color-coded Code Builder interface showing a portion of an Origin C function used to fit the selected data.
The C functions were in turn called by the "Fit Selected" button on the custom worksheet template.
// bring up attention box
break.style=1;
break.open("Loading and compiling Origin C function...");
// now attempt to load and compile function
if(run.LoadOC("%yOriginC\JEHprograms\JEHworkspace.ocw") != 0)
{
// report if failure
break.close();
type -b "Error trying to load and compile the Origin C file.";
break;
}
break.close();
//now call C function
prepareForFit(%H);
In addition, several worksheet and graph templates were utilized during the fitting process. In all, the LabTalk script and Origin C code behind the Fit Selected button was designed to automatically perform the following operations:
• Fit all peaks automatically to a sum of several user-defined functions
• Plot each fit on a graph template using the information from the imported file (i.e. the file name and header information) in order to label the axes
• Create an internal folder structure in the project to organize each fit (specifically, put all the fits from a single energy into a single folder)
• After all the fitting is done, plot each fit parameter as a function of energy and plot one specific fit parameter as a function of another
• Label all graphs correctly with information from the input file names and headers
Fig 7: A final result graph including the raw data plotted
as scatter plots and the fit lines plotted as black line plots
Acknowledgements and Biographies
This work was performed in the research group of Seamus Davis, a professor of physics at Cornell University who specializes in experimental studies of condensed matter.
Kyle McElroy is a physics graduate student at the University of California, Berkeley, currently working with Professor Davis at Cornell University.
Dr. Jenny Hoffman performed this analysis while she was a graduate student with Professor Davis at the University of California at Berkeley. She is currently a post-doc at Stanford University and will be starting as a professor at Harvard University in January, 2005.
The scanning tunneling microscope used in this experiment was built in the lab of Professor Davis by Dr. Shuheng Pan (now a professor at the University of Houston) and Dr. Eric Hudson (now a professor at Massachusetts Institute of Technology).
The BSCCO crystals used in this experiment were grown by Dr. Hiroshi Eisaki (now at AIST-Tsukuba in Japan) and Professor Shin-ichi Uchida at the University of Tokyo.
Parts of this work were funded by the New Energy and Industrial Technology Development Organization of Japan, the Office of Naval Research, Lawrence Berkeley National Laboratory, the National Science Foundation, and Cornell University.
© OriginLab Corporation. All rights reserved.
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2018-02-22 20:36:48
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http://www.apkc.net/external/msc_578d_dm_rbc.html
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# CSC 578D / Data Mining / Fall 2018 / University of Victoria¶
## Python Notebook explaining Assignment 01 / Problem 02¶
### The dataset for the Assignment #1 is the following:¶
The Weka datasets can be found at my personal Website at www.apkc.net.
Author: Andreas P. Koenzen akoenzen@uvic.ca
Version: 0.1
In [1]:
import pandas as pd
import numpy as np
import requests as rq
from scipy.io import arff
from io import StringIO
In [3]:
url_data = rq.get('http://www.apkc.net/data/weka/weather.nominal.arff').text
df = pd.DataFrame(data[0], index=pd.Index(np.arange(14) + 1), dtype='object')
# Convert all data in the columns to strings instead of binary objects.
string_df = df.select_dtypes([np.object]).stack().str.decode('UTF-8').unstack()
for col in string_df:
df[col] = string_df[col]
df
Out[3]:
outlook temperature humidity windy play
1 sunny hot high FALSE no
2 sunny hot high TRUE no
3 overcast hot high FALSE yes
4 rainy mild high FALSE yes
5 rainy cool normal FALSE yes
6 rainy cool normal TRUE no
7 overcast cool normal TRUE yes
8 sunny mild high FALSE no
9 sunny cool normal FALSE yes
10 rainy mild normal FALSE yes
11 sunny mild normal TRUE yes
12 overcast mild high TRUE yes
13 overcast hot normal FALSE yes
14 rainy mild high TRUE no
### Solution to Problem #2 of Assignment #1:¶
#### The problem #2 states the following:¶
(4 points) Construct two rules using PRISM for the weather data. Show the details of your construction. Then, check your solution with Weka (the data file is included with Weka).
#### The full set of rules for this exercise is the following:¶
IF (outlook=overcast) THEN yes
IF (humidity=normal) AND (windy=FALSE) THEN yes
IF (temperature=mild) AND (humidity=normal) THEN yes
IF (outlook=rainy) AND (windy=FALSE) THEN yes
IF (outlook=sunny) AND (humidity=high) THEN no
IF (outlook=rainy) AND (windy=TRUE) THEN no
#### Notes:¶
• 3 significant digits are used for all results.
• results are rounded up if 4th significant digit is >= 5.
#### Step #1:¶
We construct an empty condition (no antecedent) rule for a random class, and list all possible test for that class.
Current State:
IF (?) THEN no
Possible Tests/Conditions:
outlook=sunny 3/5
outlook=overcast 0/4
outlook=rainy 2/5
temperature=hot 2/4
temperature=mild 2/6
temperature=cold 1/4
humidity=high 4/7 => HIGHEST ACCURACY
humidity=normal 1/7
windy=FALSE 2/8
windy=TRUE 3/6
From this list we select the the condition with the highest probability of occurrence GIVEN that the class is no or the highest accuracy. In this case we select humidity=high has the highest accuracy from the initial list.
$Coverage(\text{humidity=high}) = 7$
$Accuracy(\text{humidity=high | play=no}) = 4/7 = 0.57$
We can see that the accuracy is not very high so we refine some more.
#### Step #2:¶
We add a new test to the rule to increase the accuracy.
Current State:
IF (humidity=high) AND (?) THEN no
Possible Test/Conditions:
humidity=high AND outlook=sunny 3/3 => HIGHEST ACCURACY
humidity=high AND outlook=overcast 0/2
humidity=high AND outlook=rainy 1/2
humidity=high AND temperature=hot 2/3
humidity=high AND temperature=mild 2/4
humidity=high AND temperature=cold 0/0
humidity=high AND windy=FALSE 2/4
humidity=high AND windy=TRUE 2/3
Again we select the test with the highest accuracy and add it to the rule. In this case we select outlook=sunny.
$Coverage(\text{humidity=high AND outlook=sunny}) = 3$
$Accuracy(\text{humidity=high AND outlook=sunny | play=no}) = 3/3 = 1$
We've reached an accuracy of 1.0. So we stop here, because the rule is already refined to the maximum.
#### Rule #1 is:¶
IF (humidity=high) AND (outlook=sunny) THEN no
#### Step #3:¶
We continue building rules until we have covered every attribute-value combination OR until we have the perfect set of rules.
The dataset looks like this after we exclude records that are covered by the rule #1.
In [12]:
new_df = df.loc[(df['humidity'] != 'high') | (df['outlook'] != 'sunny')]
new_df
Out[12]:
outlook temperature humidity windy play
3 overcast hot high FALSE yes
4 rainy mild high FALSE yes
5 rainy cool normal FALSE yes
6 rainy cool normal TRUE no
7 overcast cool normal TRUE yes
9 sunny cool normal FALSE yes
10 rainy mild normal FALSE yes
11 sunny mild normal TRUE yes
12 overcast mild high TRUE yes
13 overcast hot normal FALSE yes
14 rainy mild high TRUE no
#### Step #4:¶
We construct an empty condition (no antecedent) rule for a class no again, and list all possible test for that class, excluding the tests that are covered by rule #1.
Current State:
IF (?) THEN no
Possible Tests/Conditions:
outlook=overcast 0/4
outlook=rainy 2/5 => HIGHEST ACCURACY
temperature=hot 0/2
temperature=mild 1/5
temperature=cold 1/4
humidity=normal 1/7
windy=FALSE 0/6
windy=TRUE 2/5 => HIGHEST ACCURACY
From this list we select the the condition with the highest probability of occurrence GIVEN that the class is no or the highest accuracy. In this case we select one of two possible attribute=values, let's select outlook=rainy has the highest accuracy from the initial list.
$Coverage(\text{outlook=rainy}) = 5$
$Accuracy(\text{outlook=rainy | play=no}) = 2/5 = 0.40$
We can see that the accuracy is not very high so we refine some more.
#### Step #5:¶
We add a new test to the rule to increase the accuracy.
Current State:
IF (outlook=rainy) AND (?) THEN no
Possible Test/Conditions:
outlook=rainy AND temperature=hot 0/0
outlook=rainy AND temperature=mild 1/3
outlook=rainy AND temperature=cold 1/2
outlook=rainy AND humidity=normal 1/3
outlook=rainy AND windy=FALSE 0/3
outlook=rainy AND windy=TRUE 2/2 => HIGHEST ACCURACY
Again we select the test with the highest accuracy and add it to the rule. In this case we select windy=TRUE.
$Coverage(\text{outlook=rainy AND windy=TRUE}) = 2$
$Accuracy(\text{outlook=rainy AND windy=TRUE | play=no}) = 2/2 = 1$
We've reached an accuracy of 1.0. So we stop here, because the rule is already refined to the maximum.
#### Rule #2 is:¶
IF (outlook=rainy) AND (windy=true) THEN no
#### Final solution:¶
IF (humidity=high) AND (outlook=sunny) THEN no
IF (outlook=rainy) AND (windy=true) THEN no
...
The dataset will look like this after we exclude records that are covered by both rules. We need to keep creating rules until we have rules that cover all instances. And after that we need to create a default rule or catch all rule for instances that can't be covered by our rule set.
In [15]:
final_df = new_df.loc[(new_df['outlook'] != 'rainy') | (new_df['windy'] != 'TRUE')]
final_df
Out[15]:
outlook temperature humidity windy play
3 overcast hot high FALSE yes
4 rainy mild high FALSE yes
5 rainy cool normal FALSE yes
7 overcast cool normal TRUE yes
9 sunny cool normal FALSE yes
10 rainy mild normal FALSE yes
11 sunny mild normal TRUE yes
12 overcast mild high TRUE yes
13 overcast hot normal FALSE yes
#### Observation:¶
With the two rules that we previously created, we covered all instances of class no.
|
2019-01-16 10:14:27
|
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https://engineering.stackexchange.com/questions/47560/how-do-i-properly-scale-down-a-force-for-scale-down-model-testing
|
# How do I properly scale down a force for scale-down model testing?
I am designing structural frame for a shipping container that would have 8000 lbs of weight stacked on top of it. Of course, i don't access to real size container. I have built a model whose scale is 1:8 of the real one (same material, steel). How do I scale down the force so that the stress generated in model structure is equivalent of that in the real size one? Can anyone refer me to some good materials? Thank you.
• For stress analysis only - if the load is uniformly distributed,, scale down the actual uniform load by 8. For concentrated load, scale the concentrated force by 64.
– r13
Oct 5 '21 at 17:41
TL;DR: you need to prioritize which behaviour is important and scale the load to investigate this behavior. It is very difficult to scale the load and expect to obtain the full behaviour of the structure.
IMHO it is very difficult to scale the load and expect to obtain the full behaviour of the structure. Different loading conditions have different dependencies from the loads.
For the following analysis I will assume that only the normal loads play a role. I.e. the structure will fail when the operating stress ($$\sigma$$) exceeds the allowable stress $$\sigma_{all}$$.
e.g. take buckling of a column and bending of a beam. (I intentionally take buckling and bending because they both have similar quantities. If I compared axial loading cases - compression/tension- to bending that might have left more doubts)
representation
beam breadth $$b$$ $$b$$
beam thickness $$h$$ $$h$$
beam cross-section $$bh$$ $$bh$$
(moment area) I $$\frac{b\; h^3}{12}$$ $$\frac{b\; h^3}{12}$$
load at failure $$P_{buck} = \frac{\pi^2 EI}{L^2}$$ $$P_{bend}=q\cdot L$$
stress at failure $$\frac{\pi^2 EI}{A\cdot L^2}$$ $$\frac{PL}{4\cdot I}\frac{h}{2}$$
operating stress at failure after simplification $$\frac{h}{L^2}\frac{\pi^2 E }{12}$$ $$\frac{3 PL}{ 2b\; h^2}$$
Notice that at the end I separate the material properties from the cross-section properties.
stress at failure after simplification $$\frac{h}{L^2}\frac{\pi^2 E }{12}$$ $$\frac{L}{ b\; h^2} \cdot \frac{3}{2}\cdot P$$
factors affected by dimensional scaling $$\frac{h}{L^2}$$ $$\frac{L}{ b\; h^2}$$
reduction to scale 1:8 $$\frac{h/8}{(L/8)^2}= 8\frac{h}{L^2}$$ $$\frac{(L/8)}{ (b/8)\; (h/8)^2} = 8^2 \frac{L}{ b\; h^2}$$
So this means that the operating stress for buckling ($$8^2$$)will increase at a different rate that the operating stress for buckling ($$8^1$$). So - IMHO- you expect that you can scale the load and obtain the same behaviour for all loading conditions of the model (you need to tune other parameters also).
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2022-01-21 02:57:40
|
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https://physicstasks.eu/1968/delayed-sunset
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Delayed Sunset
An observer watches the sunset from the surface of the Earth. Due to refraction of light on the interface of vacuum (the free space) and atmosphere, the observer sees the Sun set later than it would be without the atmosphere.
The real interface of vacuum and atmosphere is not sharp. Use a simplifying model according to the figure above. Consider a single spherical interface of two isotropic media: vacuum ($$n_1=1$$) and idealised atmosphere ($$n_2=1.0003$$), with such a thickness that the Sun rays fall at right angle at the interface.
• Hint 1 – Sketch the situation
Draw a picture showing the moment of the sunset as observed from the surface of the Earth. Assess the angles of incidence and refraction of the Sun rays at the interface of vacuum and atmosphere.
Keep in mind that the thickness of the atmosphere is just a small fraction of the Earth radius.
• Hint 2 — Refraction angle
What is the relation between the angles of incidence and refraction? Remind Snell’s law.
• Hint 3 — Time delay
• From what point would the observer see the sunset if there was no atmosphere (no interface)?
• What angle does the Earth need to rotate to take the observer from the hypothetical point of observation of sunset without atmosphere to the actual point of observation?
• What time does it take the Earth to rotate by this angle?
• Solution
We draw the described situation. The observer at point A sees that the Sun is just setting behind the observer’s horizon. The Sun rays refract at the interface of atmosphere and vacuum at point P.
The angle of incidence in the latest moment when the Sun rays refract towards the observer is $$\frac{\pi}{2}$$ at point P, the angle of refraction is denoted $$\beta$$.
Snell’s law for the vacuum–air interface is:
$n_1 \sin\frac{\pi}{2} = n_2 \sin \beta.$
We express the angle of refraction:
$\beta = \arcsin \frac{n_1}{n_2} .\tag{2}$
If there was no atmosphere, the observer would see the Sun setting behind the horizon at the moment when the observer was at point P. The Earth rotated by the angle $$\delta$$ from that moment to the moment when the observer is at the point of the actual observation of the sunset (point A).
$\delta = \frac{\pi}{2} - \beta.$
We express the angular velocity $$\omega$$ of Earth rotation as the ratio of the full angle $$2\pi$$ and the period of Earth rotation $$T$$. The time of rotation by angle $$\delta$$ is:
$t = \frac{\delta}{\omega} = \frac{\frac{\pi}{2} - \beta}{\frac{2\pi}{T}}.$
We get the final formula upon substituting of the angle $$\beta$$ from Equation (2) and arranging of the formula:
$t = \frac{1}{2\pi} \left[\frac{\pi}{2} - \arcsin \left(\frac{n_1}{n_2} \right)\right]T.$
With given numerical values, we get:
$t = \frac{1}{2\pi} \left[\frac{\pi}{2} - \arcsin \left(\frac{1}{1.0003} \right)\right]24{\cdot} 60~\mathrm{min} \doteq 5.6~\mathrm{min}.$
• Comment
The assignment supposes a right angle at point P which makes the task easy to calculate. However, this is an oversimplificaiton - the thickness of the atmosphere under these conditions would be less than $$2\,\mathrm{km}$$. Nevertheless, the task gives a good estimate of the delay.
In reality, the Sun rays do not undergo only a single refraction. The index of refraction of the atmosphere changes smoothly from 1 (vacuum) to 1.0003 (at the sea level) and the Sun rays are refracted bit by bit accordingly, resultin in a smooth “bend”. The actual delay of the sunset is about 2 minutes, the sunrise is some 2 minutes advanced symmetrically and the light-day at the Earth is about 4 minutes longer than it would be without the atmosphere.
The trajectory of the gradually bending ray can be calculated numerically without complicated mathematics e.g. in a spreadsheet (Excel, Calc,...).
One can enter the initial and the final index of refraction, the initial coordinates, the angle of incidence, the width and number of model isotropic layers of the atmosphere as parameters and examine how does the trajectory of the rays depend on them.
Our simple model yields a delay of the sunset of about $$5.6\,\mathrm{min}$$.
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2021-06-14 21:37:33
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http://www.newton.ac.uk/event/wht/seminars
|
# Seminars (WHT)
Videos and presentation materials from other INI events are also available.
Search seminar archive
Event When Speaker Title Presentation Material
WHT 6th August 2019
09:00 to 10:15
David Abrahams On the Wiener-Hopf technique and its applications in science and engineering: Lecture 1
It is a little nearly 90 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener-Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception.
This series of three lectures will be informal in nature and directed at researchers who are either at an early stage of their career or else unfamiliar with particular aspects of the subject. Their aim is to demonstrate the beauty of the topic and its wide range of applications, and will be delivered in a traditional applied mathematical style. The lectures will not try to offer a comprehensive overview of the literature but will instead focus on specific topics that have been of interest over the years to the speaker.
The first lecture shall offer a subjective review of the subject, introducing the notation to be employed in later lectures, and indicating a sample of the enormous range of applications that have been found for the technique. The second lecture will focus on exact and approximate solution methods for scalar and vector Wiener-Hopf equations, and indicate the similarities and differences of the various approaches used. The final lecture shall continue discussion of approximate approaches, combining these with one or more specific applications of current interest to the speaker.
WHT 6th August 2019
10:30 to 11:45
David Abrahams On the Wiener-Hopf technique and its applications in science and engineering: Lecture 2
It is a little nearly 90 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener-Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. This series of three lectures will be informal in nature and directed at researchers who are either at an early stage of their career or else unfamiliar with particular aspects of the subject. Their aim is to demonstrate the beauty of the topic and its wide range of applications, and will be delivered in a traditional applied mathematical style. The lectures will not try to offer a comprehensive overview of the literature but will instead focus on specific topics that have been of interest over the years to the speaker.
The first lecture shall offer a subjective review of the subject, introducing the notation to be employed in later lectures, and indicating a sample of the enormous range of applications that have been found for the technique. The second lecture will focus on exact and approximate solution methods for scalar and vector Wiener-Hopf equations, and indicate the similarities and differences of the various approaches used. The final lecture shall continue discussion of approximate approaches, combining these with one or more specific applications of current interest to the speaker.
WHT 6th August 2019
12:00 to 13:15
Raphael Assier, Andrey Shanin Towards a multivariable Wiener-Hopf method: Lecture 1
A multivariable, in particular two complex variables (2D), Wiener-Hopf (WH) method is one of the desired generalisations of the classical and celebrated WH technique that are easily conceived but very hard to implement (the second one, indeed, is the matrix WH). A 2D WH method, could potentially be used e.g. for finding a solution to the canonical problem of diffraction by a quarter-plane.
Unfortunately, multidimensional complex analysis seems to be way more complicated than complex analysis of a single variable. There exists a number of powerful theorems in it, but they are organised into several disjoint theories, and, generally all of them are far from the needs of WH. In this mini-lecture course, we hope to introduce topics in complex analysis of several variables that we think are important for a generalisation of the WH technique. We will focus on the similarities and differences between functions of one complex variable and functions of two complex variables. Elements of differential forms and homotopy theory will be addressed.
We will start by reviewing some known attempts at building a 2D WH and explain why they were not successful. The framework of Fourier transforms and analytic functions in 2D will be introduced, leading us naturally to discuss multidimensional integration contours and their possible deformations. One of our main focus will be on polar and branch singularity sets and how to describe how a multidimensional contour bypasses these singularities. We will explain how multidimensional integral representation can be used in order to perform an analytical continuation of the unknowns of a 2D functional equation and why we believe it to be important. Finally, time permitting, we will discuss the branching structure of complex integrals depending on some parameters and introduce the so-called Picard-Lefschetz formulae.”
WHT 6th August 2019
14:15 to 15:30
Michael Nieves Understanding dynamic crack growth in structured systems with the Wiener-Hopf technique: Lecture 1
Crack propagation is a process accompanied by multiple phenomena at different scales. In particular, when a crack grows, microstructural vibrations are released, emanating from the crack tip. Continuous models of dynamic cracks are well known to omit information concerning these microstructural processes [1]. On the other hand, tracing these vibrations on the microscale is possible if one considers a crack propagating in a structured system, such as a lattice [2, 3]. These models have a particular relevance in the design of metamaterials, whose microstructure can be tailored to control dynamic effects for a variety of practical purposes [4]. Similar approaches have been recently paving new pathways to understanding failure processes in civil engineering systems [5, 6].
In this lecture, we aim to demonstrate the importance of the Wiener-Hopf technique in the analysis and solution of problems concerning waves and crack propagation in discrete periodic media. We begin with the model of a lattice system containing a crack and show how this can be reduced to a scalar Wiener-Hopf equation through the Fourier transform. From this functional equation we identify all possible dynamic processes complementing the crack growth. We determine the solution to the problem and how this is used to predict crack growth regimes in numerical simulations. Other applications of the adopted method, including the analysis of the progressive collapse of large-scale structures, are discussed.
References [1] Marder, M. and Gross, S. (1995): Origin of crack tip instabilities, J. Mech. Phys. Solids 43, no. 1, 1- 48. [2] Slepyan, L.I. (2001): Feeding and dissipative waves in fracture and phase transition I. Some 1D structures and a square-cell lattice, J. Mech. Phys. Solids 49, 469-511. [3] Slepyan, L.I. (2002): Models and Phenomena in Fracture Mechanics, Foundations of Engineering Mechanics, Springer. [4] Mishuris, G.S., Movchan, A.B. and Slepyan, L.I., (2007): Waves and fracture in an inhomogeneous lattice structure, Wave Random Complex 17, no. 4, 409-428. [5] Brun, M., Giaccu, G.F., Movchan, A.B., and Slepyan, L. I., (2014): Transition wave in the collapse of the San Saba Bridge, Front. Mater. 1:12. doi: 10.3389/fmats.2014.00012. [6] Nieves, M.J., Mishuris, G.S., Slepyan, L.I., (2016): Analysis of dynamic damage propagation in discrete beam structures, Int. J. Solids Struct. 97-98, 699-713.
WHT 6th August 2019
15:45 to 17:00
Alexey Kuznetsov Computing the Wiener-Hopf factors for Levy processes: Lecture 1
The Wiener-Hopf factorization is a fundamental result in the theory of Levy processes; it provides a wealth of information about the first exit of the underlying process from a half-line. The main goal of these lectures is to show how to use complex-analytic methods to obtain explicit formulas for Wiener-Hopf factors for several important classes of Levy processes. We will start with processes with jumps of rational transform, then we will discuss the class of stable processes, explaining how one could recover from the Wiener-Hopf factors the distribution of the supremum of the process at a fixed time. Finally, we will talk about the difficult problem of how a Levy process exits an interval, which turns out to be related to Wiener-Hopf factorization for certain 2x2 matrices. This latter problem is wide open for processes with double-sided jumps and we will discuss what is currently known for stable processes.
WHT 7th August 2019
09:00 to 10:15
Raphael Assier, Andrey Shanin Towards a multivariable Wiener-Hopf method: Lecture 2
A multivariable, in particular two complex variables (2D), Wiener-Hopf (WH) method is one of the desired generalisations of the classical and celebrated WH technique that are easily conceived but very hard to implement (the second one, indeed, is the matrix WH). A 2D WH method, could potentially be used e.g. for finding a solution to the canonical problem of diffraction by a quarter-plane.
Unfortunately, multidimensional complex analysis seems to be way more complicated than complex analysis of a single variable. There exists a number of powerful theorems in it, but they are organised into several disjoint theories, and, generally all of them are far from the needs of WH. In this mini-lecture course, we hope to introduce topics in complex analysis of several variables that we think are important for a generalisation of the WH technique. We will focus on the similarities and differences between functions of one complex variable and functions of two complex variables. Elements of differential forms and homotopy theory will be addressed.
We will start by reviewing some known attempts at building a 2D WH and explain why they were not successful. The framework of Fourier transforms and analytic functions in 2D will be introduced, leading us naturally to discuss multidimensional integration contours and their possible deformations. One of our main focus will be on polar and branch singularity sets and how to describe how a multidimensional contour bypasses these singularities. We will explain how multidimensional integral representation can be used in order to perform an analytical continuation of the unknowns of a 2D functional equation and why we believe it to be important. Finally, time permitting, we will discuss the branching structure of complex integrals depending on some parameters and introduce the so-called Picard-Lefschetz formulae.”
WHT 7th August 2019
10:30 to 11:45
Guido Lombardi, J.M.L. Bernard The link between the Wiener-Hopf and the generalised Sommerfeld Malyuzhinets methods: Lecture 1
The
Sommerfeld Malyuzhinets (SM) method and the Wiener Hopf (WH) technique are
different but closely related methods. In particular in the paper “Progress and
Prospects in The Theory of Linear Waves Propagation” SIAM SIREV vol.21, No.2,
April 1979, pp. 229-245, J.B. Keller posed the following question “What
features of the methods account for this difference?”. Furthermore
J.B. Keller notes “it might be
helpful to understand this in order to predict the success of other methods”. We
agree with this opinion expressed by the giant of Diffraction. Furthermore we think that SM and
WH applied to the same problems (for instance the polygon diffraction) can determine a helpful synergy. In the past
the SM and WH methods were considered disconnected in particular because the SM
method was traditionally defined with the angular complex representation while
the WH method was traditionally defined in the Laplace domain.
In
this course we show that the two methods have significant points of similarity
when the representation of problems in both methods are expressed in terms of
difference equations. The two methods show their diversity in the solution
procedures that are completely different and effective.
Both similarity and diversity properties are of advantage in “Progress and Prospects in The Theory of
Linear Waves Propagation”. Moreover
both methods have demonstrated their efficacy in studying particularly complex
problems, beyond the traditional problem of scattering by a wedge: in
particular the scattering by a three part polygon that we will present.
Recent
progress in both methods:
One
of the most relevant recent progress in SM is the derivation of functional
difference equations without the use of Maliuzhinets inversion theorem.
One
of the most relevant recent progress in WH is transformation of WH equations
into integral equations for their effective solution
WHT 7th August 2019
12:00 to 13:15
Sheehan Olver Orthogonal polynomials, singular integrals, and solving Riemann–Hilbert problems: Lecture 1
Orthogonal polynomials are fundamental tools in numerical methods, including for singular integral equations. A known result is that Cauchy transforms of weighted orthogonal polynomials satisfy the same three-term recurrences as the orthogonal polynomials themselves for n > 0. This basic fact leads to extremely effective schemes of calculating singular integrals and discretisations of singular integral equations that converge spectrally fast (faster than any algebraic power). Applications considered include matrix Riemann–Hilbert problems on contours consisting of interconnected line segments and Wiener–Hopf problems. This technique is extendible to calculating singular integrals with logarithmic kernels, with applications to Green’s function reduction of PDEs such as the Helmholtz equation.
WHT 7th August 2019
14:15 to 15:30
David Abrahams On the Wiener-Hopf technique and its applications in science and engineering: Lecture 2
It is a little nearly 90 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener-Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. This series of three lectures will be informal in nature and directed at researchers who are either at an early stage of their career or else unfamiliar with particular aspects of the subject. Their aim is to demonstrate the beauty of the topic and its wide range of applications, and will be delivered in a traditional applied mathematical style. The lectures will not try to offer a comprehensive overview of the literature but will instead focus on specific topics that have been of interest over the years to the speaker.
The first lecture shall offer a subjective review of the subject, introducing the notation to be employed in later lectures, and indicating a sample of the enormous range of applications that have been found for the technique. The second lecture will focus on exact and approximate solution methods for scalar and vector Wiener-Hopf equations, and indicate the similarities and differences of the various approaches used. The final lecture shall continue discussion of approximate approaches, combining these with one or more specific applications of current interest to the speaker.
WHT 7th August 2019
15:45 to 17:00
Frank Speck From Sommerfeld diffraction problems to operator factorisation: Lecture 1
This
lecture series is devoted to the interplay between diffraction and operator
theory, particularly between the so-called canonical diffraction problems
(exemplified by half-plane problems) on one hand and operator factorisation theory on the other hand. It
is shown how operator factorisation concepts appear naturally from applications
and how they can help to find solutions rigorously in case of well-posed
problems as well as for ill-posed problems after an adequate normalisation.
The
operator theoretical approach has the advantage of a compact presentation of
results simultaneously for wide classes of diffraction problems and space
settings and gives a different and deeper understanding of the solution
procedures.
The
main objective is to demonstrate how diffraction problems guide us to operator factorisation concepts and how useful those
are to develop and to simplify the reasoning in the applications.
In
eight widely independent sections we shall address the following questions:
How can we consider the classical Wiener-Hopf procedure as an operator
factorisation (OF) and what is the profit of that interpretation?
What are the characteristics of Wiener-Hopf operators occurring in
Sommerfeld half-plane problems and their features in
terms of functional analysis?
What are the most relevant methods of constructive matrix
factorisation in Sommerfeld problems? How does OF appear generally in linear boundary value and transmission
What are adequate choices of function(al) spaces and symbol classes in
order to analyse the well-posedness of problems and to use deeper results of factorisation theory?
A sharp logical concept for equivalence and reduction of linear
systems (in terms of OF) – why is it needed and why does it simplify and
strengthen the reasoning? Where do we need other kinds of operator relations beyond OF? What are very practical examples for the use of the preceding ideas,
e.g., in higher dimensional diffraction problems? Historical
remarks and corresponding references are provided at the end of each section.
WHT 8th August 2019
09:00 to 10:15
Frank Speck From Sommerfeld diffraction problems to operator factorisation: Lecture 2
This
lecture series is devoted to the interplay between diffraction and operator
theory, particularly between the so-called canonical diffraction problems
(exemplified by half-plane problems) on one hand and operator factorisation theory on the other hand. It
is shown how operator factorisation concepts appear naturally from applications
and how they can help to find solutions rigorously in case of well-posed
problems as well as for ill-posed problems after an adequate normalisation.
The
operator theoretical approach has the advantage of a compact presentation of
results simultaneously for wide classes of diffraction problems and space
settings and gives a different and deeper understanding of the solution
procedures.
The
main objective is to demonstrate how diffraction problems guide us to operator factorisation concepts and how useful those
are to develop and to simplify the reasoning in the applications.
In
eight widely independent sections we shall address the following questions:
How can we consider the classical Wiener-Hopf procedure as an operator
factorisation (OF) and what is the profit of that interpretation?
What are the characteristics of Wiener-Hopf operators occurring in
Sommerfeld half-plane problems and their features in
terms of functional analysis?
What are the most relevant methods of constructive matrix
factorisation in Sommerfeld problems? How does OF appear generally in linear boundary value and transmission
What are adequate choices of function(al) spaces and symbol classes in
order to analyse the well-posedness of problems and to use deeper results of factorisation theory?
A sharp logical concept for equivalence and reduction of linear
systems (in terms of OF) – why is it needed and why does it simplify and
strengthen the reasoning? Where do we need other kinds of operator relations beyond OF? What are very practical examples for the use of the preceding ideas,
e.g., in higher dimensional diffraction problems? Historical
remarks and corresponding references are provided at the end of each section.
WHT 8th August 2019
10:30 to 11:45
Guido Lombardi, J.M.L. Bernard The link between the Wiener-Hopf and the generalised Sommerfeld Malyuzhinets methods: Lecture 2
The Sommerfeld Malyuzhinets (SM) method and the Wiener Hopf (WH) technique are different but closely related methods. In particular in the paper “Progress and Prospects in The Theory of Linear Waves Propagation” SIAM SIREV vol.21, No.2, April 1979, pp. 229-245, J.B. Keller posed the following question “What features of the methods account for this difference?”. Furthermore J.B. Keller notes “it might be helpful to understand this in order to predict the success of other methods”.
We agree with this opinion expressed by the giant of Diffraction. Furthermore we think that SM and WH applied to the same problems (for instance the polygon diffraction) can determine a helpful synergy. In the past the SM and WH methods were considered disconnected in particular because the SM method was traditionally defined with the angular complex representation while the WH method was traditionally defined in the Laplace domain.
In this course we show that the two methods have significant points of similarity when the representation of problems in both methods are expressed in terms of difference equations. The two methods show their diversity in the solution procedures that are completely different and effective. Both similarity and diversity properties are of advantage in “Progress and Prospects in The Theory of Linear Waves Propagation”.
Moreover both methods have demonstrated their efficacy in studying particularly complex problems, beyond the traditional problem of scattering by a wedge: in particular the scattering by a three part polygon that we will present. Recent progress in both methods: One of the most relevant recent progress in SM is the derivation of functional difference equations without the use of Maliuzhinets inversion theorem.
One of the most relevant recent progress in WH is transformation of WH equations into integral equations for their effective solution
WHT 8th August 2019
12:00 to 13:15
Raphael Assier, Andrey Shanin Towards a multivariable Wiener-Hopf method: Lecture 3
A multivariable, in particular two complex variables (2D), Wiener-Hopf (WH) method is one of the desired generalisations of the classical and celebrated WH technique that are easily conceived but very hard to implement (the second one, indeed, is the matrix WH). A 2D WH method, could potentially be used e.g. for finding a solution to the canonical problem of diffraction by a quarter-plane.
Unfortunately, multidimensional complex analysis seems to be way more complicated than complex analysis of a single variable. There exists a number of powerful theorems in it, but they are organised into several disjoint theories, and, generally all of them are far from the needs of WH. In this mini-lecture course, we hope to introduce topics in complex analysis of several variables that we think are important for a generalisation of the WH technique. We will focus on the similarities and differences between functions of one complex variable and functions of two complex variables. Elements of differential forms and homotopy theory will be addressed.
We will start by reviewing some known attempts at building a 2D WH and explain why they were not successful. The framework of Fourier transforms and analytic functions in 2D will be introduced, leading us naturally to discuss multidimensional integration contours and their possible deformations. One of our main focus will be on polar and branch singularity sets and how to describe how a multidimensional contour bypasses these singularities. We will explain how multidimensional integral representation can be used in order to perform an analytical continuation of the unknowns of a 2D functional equation and why we believe it to be important. Finally, time permitting, we will discuss the branching structure of complex integrals depending on some parameters and introduce the so-called Picard-Lefschetz formulae.”
WHT 8th August 2019
14:15 to 15:30
Guido Lombardi, J.M.L. Bernard The link between the Wiener-Hopf and the generalised Sommerfeld Malyuzhinets methods: Lecture 3
The
Sommerfeld Malyuzhinets (SM) method and the Wiener Hopf (WH) technique are
different but closely related methods. In particular in the paper “Progress and
Prospects in The Theory of Linear Waves Propagation” SIAM SIREV vol.21, No.2,
April 1979, pp. 229-245, J.B. Keller posed the following question “What
features of the methods account for this difference?”. Furthermore
J.B. Keller notes “it might be
helpful to understand this in order to predict the success of other methods”.
We
agree with this opinion expressed by the giant of Diffraction. Furthermore we think that SM and
WH applied to the same problems (for instance the polygon diffraction) can determine a helpful synergy. In the past
the SM and WH methods were considered disconnected in particular because the SM
method was traditionally defined with the angular complex representation while
the WH method was traditionally defined in the Laplace domain.
In
this course we show that the two methods have significant points of similarity
when the representation of problems in both methods are expressed in terms of
difference equations. The two methods show their diversity in the solution
procedures that are completely different and effective.
Both similarity and diversity properties are of advantage in “Progress and Prospects in The Theory of
Linear Waves Propagation”. Moreover
both methods have demonstrated their efficacy in studying particularly complex
problems, beyond the traditional problem of scattering by a wedge: in
particular the scattering by a three part polygon that we will present.
Recent
progress in both methods:
One
of the most relevant recent progress in SM is the derivation of functional
difference equations without the use of Maliuzhinets inversion theorem.
One
of the most relevant recent progress in WH is transformation of WH equations
into integral equations for their effective solution.
WHT 8th August 2019
15:45 to 17:00
Alexey Kuznetsov Computing the Wiener-Hopf factors for Levy processes: Lecture 2
The Wiener-Hopf factorization is a fundamental result in the theory of Levy processes; it provides a wealth of information about the first exit of the underlying process from a half-line. The main goal of these lectures is to show how to use complex-analytic methods to obtain explicit formulas for Wiener-Hopf factors for several important classes of Levy processes. We will start with processes with jumps of rational transform, then we will discuss the class of stable processes, explaining how one could recover from the Wiener-Hopf factors the distribution of the supremum of the process at a fixed time. Finally, we will talk about the difficult problem of how a Levy process exits an interval, which turns out to be related to Wiener-Hopf factorization for certain 2x2 matrices. This latter problem is wide open for processes with double-sided jumps and we will discuss what is currently known for stable processes.
WHT 9th August 2019
09:00 to 10:15
Raphael Assier, Andrey Shanin Towards a multivariable Wiener-Hopf method: Lecture 4
A multivariable, in particular two complex variables (2D), Wiener-Hopf (WH) method is one of the desired generalisations of the classical and celebrated WH technique that are easily conceived but very hard to implement (the second one, indeed, is the matrix WH). A 2D WH method, could potentially be used e.g. for finding a solution to the canonical problem of diffraction by a quarter-plane.
Unfortunately, multidimensional complex analysis seems to be way more complicated than complex analysis of a single variable. There exists a number of powerful theorems in it, but they are organised into several disjoint theories, and, generally all of them are far from the needs of WH. In this mini-lecture course, we hope to introduce topics in complex analysis of several variables that we think are important for a generalisation of the WH technique. We will focus on the similarities and differences between functions of one complex variable and functions of two complex variables. Elements of differential forms and homotopy theory will be addressed.
We will start by reviewing some known attempts at building a 2D WH and explain why they were not successful. The framework of Fourier transforms and analytic functions in 2D will be introduced, leading us naturally to discuss multidimensional integration contours and their possible deformations. One of our main focus will be on polar and branch singularity sets and how to describe how a multidimensional contour bypasses these singularities. We will explain how multidimensional integral representation can be used in order to perform an analytical continuation of the unknowns of a 2D functional equation and why we believe it to be important. Finally, time permitting, we will discuss the branching structure of complex integr
WHT 9th August 2019
10:30 to 11:45
Guido Lombardi, J.M.L. Bernard The link between the Wiener-Hopf and the generalised Sommerfeld Malyuzhinets methods: Lecture 4
The
Sommerfeld Malyuzhinets (SM) method and the Wiener Hopf (WH) technique are
different but closely related methods. In particular in the paper “Progress and
Prospects in The Theory of Linear Waves Propagation” SIAM SIREV vol.21, No.2,
April 1979, pp. 229-245, J.B. Keller posed the following question “What
features of the methods account for this difference?”. Furthermore
J.B. Keller notes “it might be
helpful to understand this in order to predict the success of other methods”. We
agree with this opinion expressed by the giant of Diffraction. Furthermore we think that SM and
WH applied to the same problems (for instance the polygon diffraction) can determine a helpful synergy. In the past
the SM and WH methods were considered disconnected in particular because the SM
method was traditionally defined with the angular complex representation while
the WH method was traditionally defined in the Laplace domain.
In
this course we show that the two methods have significant points of similarity
when the representation of problems in both methods are expressed in terms of
difference equations. The two methods show their diversity in the solution
procedures that are completely different and effective.
Both similarity and diversity properties are of advantage in “Progress and Prospects in The Theory of
Linear Waves Propagation”. Moreover
both methods have demonstrated their efficacy in studying particularly complex
problems, beyond the traditional problem of scattering by a wedge: in
particular the scattering by a three part polygon that we will present.
Recent
progress in both methods:
One
of the most relevant recent progress in SM is the derivation of functional
difference equations without the use of Maliuzhinets inversion theorem.
One
of the most relevant recent progress in WH is transformation of WH equations
into integral equations for their effective solution
WHT 9th August 2019
12:00 to 13:15
Sheehan Olver Orthogonal polynomials, singular integrals, and solving Riemann–Hilbert problems: Lecture 2
Orthogonal polynomials are fundamental tools in numerical methods, including for singular integral equations. A known result is that Cauchy transforms of weighted orthogonal polynomials satisfy the same three-term recurrences as the orthogonal polynomials themselves for n > 0. This basic fact leads to extremely effective schemes of calculating singular integrals and discretisations of singular integral equations that converge spectrally fast (faster than any algebraic power). Applications considered include matrix Riemann–Hilbert problems on contours consisting of interconnected line segments and Wiener–Hopf problems. This technique is extendible to calculating singular integrals with logarithmic kernels, with applications to Green’s function reduction of PDEs such as the Helmholtz equation.
Using novel change-of-variable formulae, we will adapt these results to tackle singular integral equations on more general smooth arcs, geometries with corners, and Wiener–Hopf problems whose solutions only decay algebraically.
WHT 9th August 2019
14:15 to 15:30
Michael Nieves Understanding dynamic crack growth in structured systems with the Wiener-Hopf technique: Lecture 2
Crack propagation is a process accompanied by multiple phenomena at different scales. In particular, when a crack grows, microstructural vibrations are released, emanating from the crack tip. Continuous models of dynamic cracks are well known to omit information concerning these microstructural processes [1]. On the other hand, tracing these vibrations on the microscale is possible if one considers a crack propagating in a structured system, such as a lattice [2, 3]. These models have a particular relevance in the design of metamaterials, whose microstructure can be tailored to control dynamic effects for a variety of practical purposes [4]. Similar approaches have been recently paving new pathways to understanding failure processes in civil engineering systems [5, 6].
In this lecture, we aim to demonstrate the importance of the Wiener-Hopf technique in the analysis and solution of problems concerning waves and crack propagation in discrete periodic media. We begin with the model of a lattice system containing a crack and show how this can be reduced to a scalar Wiener-Hopf equation through the Fourier transform. From this functional equation we identify all possible dynamic processes complementing the crack growth. We determine the solution to the problem and how this is used to predict crack growth regimes in numerical simulations. Other applications of the adopted method, including the analysis of the progressive collapse of large-scale structures, are discussed.
References [1] Marder, M. and Gross, S. (1995): Origin of crack tip instabilities, J. Mech. Phys. Solids 43, no. 1, 1- 48. [2] Slepyan, L.I. (2001): Feeding and dissipative waves in fracture and phase transition I. Some 1D structures and a square-cell lattice, J. Mech. Phys. Solids 49, 469-511. [3] Slepyan, L.I. (2002): Models and Phenomena in Fracture Mechanics, Foundations of Engineering Mechanics, Springer. [4] Mishuris, G.S., Movchan, A.B. and Slepyan, L.I., (2007): Waves and fracture in an inhomogeneous lattice structure, Wave Random Complex 17, no. 4, 409-428. [5] Brun, M., Giaccu, G.F., Movchan, A.B., and Slepyan, L. I., (2014): Transition wave in the collapse of the San Saba Bridge, Front. Mater. 1:12. doi: 10.3389/fmats.2014.00012. [6] Nieves, M.J., Mishuris, G.S., Slepyan, L.I., (2016): Analysis of dynamic damage propagation in discrete beam structures, Int. J. Solids Struct. 97-98, 699-713.
WHT 9th August 2019
15:45 to 17:00
Frank Speck From Sommerfeld diffraction problems to operator factorisation: Lecture 3
This
lecture series is devoted to the interplay between diffraction and operator
theory, particularly between the so-called canonical diffraction problems
(exemplified by half-plane problems) on one hand and operator factorisation theory on the other hand. It
is shown how operator factorisation concepts appear naturally from applications
and how they can help to find solutions rigorously in case of well-posed
problems as well as for ill-posed problems after an adequate normalisation.
The
operator theoretical approach has the advantage of a compact presentation of
results simultaneously for wide classes of diffraction problems and space
settings and gives a different and deeper understanding of the solution
procedures.
The
main objective is to demonstrate how diffraction problems guide us to operator factorisation concepts and how useful those
are to develop and to simplify the reasoning in the applications.
In
eight widely independent sections we shall address the following questions:
How can we consider the classical Wiener-Hopf procedure as an operator
factorisation (OF) and what is the profit of that interpretation?
What are the characteristics of Wiener-Hopf operators occurring in
Sommerfeld half-plane problems and their features in
terms of functional analysis?
What are the most relevant methods of constructive matrix
factorisation in Sommerfeld problems? How does OF appear generally in linear boundary value and transmission
What are adequate choices of function(al) spaces and symbol classes in
order to analyse the well-posedness of problems and to use deeper results of factorisation theory?
A sharp logical concept for equivalence and reduction of linear
systems (in terms of OF) – why is it needed and why does it simplify and
strengthen the reasoning? Where do we need other kinds of operator relations beyond OF? What are very practical examples for the use of the preceding ideas,
e.g., in higher dimensional diffraction problems? Historical
remarks and corresponding references are provided at the end of each section.
WHTW01 12th August 2019
10:00 to 11:00
Frank Speck Wiener-Hopf factorisation through an intermediate space and applications to diffraction theory
An operator factorisation conception is investigated for
a general Wiener-Hopf operator $W = P_2 A |_{P_1 X}$ where $X,Y$ are Banach
spaces,
$P_1 in mathcal{L}(X), P_2 in mathcal{L}(Y)$ are
projectors and $A in mathcal{L}(X,Y)$ is invertible. Namely we study a
particular factorisation of $A = A_- C A_+$ where $A_+ : X ightarrow Z$ and $A_- : Z ightarrow Y$ have certain invariance properties and the cross factor $C : Z ightarrow Z$ splits the "intermediate space" $Z$ into
complemented subspaces closely related to the kernel and cokernel of $W$, such
that $W$ is equivalent to a "simpler" operator, $W sim P C|_{P Z}$.
The main result shows equivalence between the generalised
invertibility of the Wiener-Hopf operator and this kind of factorisation
(provided $P_1 sim P_2$) which implies a formula for a generalised inverse
of $W$. It embraces I.B. Simonenko's generalised factorisation of matrix
measurable functions in $L^p$ spaces and various other factorisation
approaches.
As applications we consider interface problems in weak
formulation for the n-dimensional Helmholtz equation in $Omega = mathbb{R}^n_+ cup mathbb{R}^n_-$ (due to $x_n > 0$ or $x_n respectively), where the interface$Gamma = partial Omega$is identified with$mathbb{R}^{n-1}$and divided into two parts,$Sigma$and$Sigma'$, with different transmission conditions of first and second kind. These two parts are half-spaces of$mathbb{R}^{n-1}$(half-planes for$n = 3$). We construct explicitly resolvent operators acting from the interface data into the energy space$H^1(Omega)$. The approach is based upon the present factorisation conception and avoids an interpretation of the factors as unbounded operators. In a natural way, we meet anisotropic Sobolev spaces which reflect the edge asymptotic of diffracted waves. WHTW01 12th August 2019 11:30 to 12:30 Eugene Shargorodsky Quantitative results on continuity of the spectral factorisation mapping It is well known that the matrix spectral factorisation mapping is continuous from the Lebesgue space$L^1$to the Hardy space$H^2$under the additional assumption of uniform integrability of the logarithms of the spectral densities to be factorised (S. Barclay; G. Janashia, E. Lagvilava, and L. Ephremidze). The talk will report on a joint project with Lasha Epremidze and Ilya Spitkovsky, which aims at obtaining quantitative results characterising this continuity. WHTW01 12th August 2019 13:30 to 14:00 Raphael Assier Recent advances in the quarter-plane problem using functions of two complex variables WHTW01 12th August 2019 14:00 to 14:30 J.M.L. Bernard Novel exact and asymptotic series with error functions, for a function involved in diffraction theory: the incomplete Bessel function The incomplete Bessel function, closely related to incomplete Lipschitz-Hankel integrals, is a well known known special function commonly encountered in many problems of physics, in particular in wave propagation and diffraction [1]-[5]. We present here novel exact and asymptotic series with error functions, for arbitrary complex arguments and integer order, derived from our recent publication [5]. [1] Shimoda M, Iwaki R, Miyoshi M, Tretyakov OA, 'Wiener-hopf analysis of transient phenomenon caused by time-varying resistive screen in waveguide', IEICE transactions on electronics, vol. E85C, 10, pp.1800-1807, 2002 [2] DS Jones, 'Incomplete Bessel functions. I', proceedings of the Edinburgh Mathematical Society, 50, pp 173-183, 2007 [3] MM Agrest, MM Rikenglaz, 'Incomplete Lipshitz-Hankel integrals', USSR Comp. Math. and Math Phys., vol 7, 6, pp.206-211, 1967 [4] MM Agrest amd MS Maksimov, 'Theory of incomplete cylinder functions and their applications', Springer, 1971. [5] JML Bernard, 'Propagation over a constant impedance plane: arbitrary primary sources and impedance, analysis of cut in active case, exact series, and complete asymptotics', IEEE TAP, vol. 66, 12, 2018 WHTW01 12th August 2019 14:30 to 15:00 Andrey Shanin Ordered Exponential (OE) equation as an alternative to the Wiener-Hopf method WHTW01 12th August 2019 15:00 to 15:30 Anastasia Kisil Generalisation of the Wiener-Hopf pole removal method and application to n by n matrix functions WHTW01 12th August 2019 16:00 to 16:30 Basant Lal Sharma Wiener-Hopf factorisation on the unit circle: some examples of discrete scattering problems I will provide certain examples of scattering problems, motivated by lattice waves (phonons), electronic waves under certain assumptions, nanoscale effects, etc in crystals. The mathematical formulation is posed on lattices and involves difference equations that can be reduced to the problem of Wiener-Hopf on the unit circle (in an annulus in complex plane). In some of these examples, the Wiener-Hopf problem is scalar, while in other cases it is a matrix Wiener-Hopf problem. For the latter, in a few cases it may be reduced to a scalar problem but it appears to be not the case in others. Some of these problems can be considered as discrete analogues of well-known Wiener-Hopf equations in continuum models on the real line (in an strip in complex plane), a few of which are still open problems. WHTW01 12th August 2019 16:30 to 17:00 Grigori Giorgadze On the partial indices of piecewise constant matrix functions Every holomorphic vector bundle on Riemann sphere splits into the direct sum of line bundles and the total Chern number of this vector bundle is equal to sum of Chern numbers of line bundles. The integer-valued vector with components Chern number of line bundles is called splitting type of holomorphic vector bundle and is analytic invariant of complex vector bundles. There exists a one-to-one correspondence between the H\"older continues matrix function and the holomorphic vector bundles described above, wherein the splitting type of vector bundles coincides with partial indices of matrix functions. It is known that every holomorphic vector bundle equipped with meromorphic (in general) connection with logarithmic singularities at finite set of marked points and corresponding meromorphic 1-from have first order poles in marked points and removable singularity at infinity. The Fucshian system of equations induced from this 1-form gives the monodromy representation of the fundamental group of Riemann sphere without marked points. The monodromy representation induces trivial holomorphic vector bundles with connection. The extension of the pair (\texttt{bundle, connection}) on the Riemann sphere is not unique and defines a family of holomorphically nontrivial vector bundles. In the talk we present about the following statements: 1. From the solvability condition (in the sense Galois differential theory) of the Fuchsian system follows formula for computation of partial indices of piecewise constant matrix function. 2. All extensions of vector bundle on noncompact Riemann surface correspond to rational matrix functions algorithmically computable by monodromy matrices of Fucshian system. This work was supported, in part, by the Shota Rustaveli National Science Foundation under Grant No 17-96. WHTW01 13th August 2019 09:00 to 10:00 Ilya Spitkovsky Wiener-Hopf factorization: the peculiarities of the matrix almost periodic case For several classes of functions invertibility and factorability are equivalent; such is the case, e.g., for the Wiener class W or the algebra APW of almost periodic functions with absolutely convergent Bohr-Fourier series. The result for W extends to the matrix setting; not so for APW. Moreover, the factorability criterion even for 2-by-2 triangular matrix functions with APW entries and constant determinant remains a mystery. We will discuss some known results in this direction, and more specific open problems. WHTW01 13th August 2019 10:00 to 11:00 Lasha Ephremidze On Janashia-Lagvilava method of matrix spectral factorisation Janashia-Lagvilava method is a relatively new algorithm of matrix spectral factorisation which can be applied to compute an approximate spectral factor of any matrix function (non-rational, large scale, singular) which satisfies the necessary and sufficient condition for the existence of spectral factorisation. The numerical properties of the method strongly depend on the way it is algorithmised and we propose its efficient algorithmisation. The method has already been successfully used in connectivity analysis of complex networks. The algorithm has the potential to be used in control system design and implementation for the required optimal controller computations by using frequency response data directly from measurements on real systems. It also provides a robust way of Granger causality computation for noisy singular data. WHTW01 13th August 2019 11:30 to 12:30 Andreas Kyprianou Wiener-Hopf Factorisations for Levy processes We give an introduction to the the theory of Wiener-Hopf factoirsations for Levy processes and discuss some very recent examples which are stimulated by some remarkable connections with self-similar Markov processes. WHTW01 13th August 2019 13:30 to 14:30 Sergei Rogosin Factorisation of triangular matrix-functions of arbitrary order It will be discussed an efficient method for factorization of square triangular matrix-functions of arbitrary order which was recently proposed in [1]. The idea goes back to the paper by G. N. Chebotarev [2] who constructed factorisation of 2x2 triangular matrix-functions by using representation of the certain functions related to entries of the initial matrix into continuous fraction. In order to avoid additional technical difficulties, we consider matrix-functions with Hoelder continuous entries. Tough the proposed method could be realised for wider classes of matrix-functions. Chebotarev's method is extended here to the triangular matrix-functions of arbitrary order. An inductive consideration which allows to obtain such an extension is based on an auxiliary statement. Theoretical construction is illustrated by a number of examples. The talk is based on a joint work with Dr. L. Primachuk and Dr. M.Dubatovskaya. 1. Primachuk, L., Rogosin, S.: Factorization of triangular matrix-functions of an arbitrary order, Lobachevsky J. Math., 39 (6), 809–817 (2018) 2. Chebotarev, G. N.: Partial indices of the Riemann boundary value problem with a triangular matrix of the second order, Uspekhi Mat. Nauk, XI (3(69)), 192_202 (1956) (in Russian). WHTW01 13th August 2019 14:30 to 15:00 Cristina Camara A Riemann-Hilbert approach to Einstein field equations The field equations of gravitational theories in 4 dimensions are non-linear PDE's that are difficult to solve in general. By restricting to a subspace of solutions that only depend on two space-time coordinates, alternative approaches to solving those equations become available. We present here the Riemann-Hilbert approach, looking at the dimensionally reduced field equations as an integrable system associated to a certain Lax pair, whose solutions can be obtained by factorizing a so called monodromy matrix. This approach allows for the explicit construction of solutions to the non-linear gravitational field equations using simple complex analytic methods. WHTW01 13th August 2019 15:00 to 15:30 Aloknath Chakrabarti Solving Wiener-Hopf Problems by the aid of Fredholm Integral Equations of the Second Kind A class of Wiener-Hopf problems is shown to be solvable by reducing the original problems to Fredholm integral equations of the second kind. The resulting Fredholm integral equations are shown to be finally solvable, numerically, by using standard techniques. The present method is found to be applicable to systems of Wiener-Hopf problems, for which the Wiener-Hopf factorization of matrices can be avoided. Several examples are taken up, demonstrating the present method of solution of Wiener-Hopf problems. WHTW01 13th August 2019 16:00 to 16:30 Victor Adukov On explicit and exact solutions of the Wiener-Hopf factorization problem for some matrix functions By an explicit solution of the factorization problem we mean the solution that can be found by finite number of some steps which we call "explicit". When we solve a specific factorization problem we must rigorously define these steps. In this talk we will do this for matrix polynomials, rational matrix functions, analytic matrix functions, meromorphic matrix functions, triangular matrix functions and others. For these classes we describe the data and procedures that are necessary for the explicit solution of the factorization problem. Since the factorization problem is unstable, the explicit solvability of the problem does not mean that we can get its numerical solution. This is the principal obstacle to use the Wiener-Hopf techniques in applied problems. For the above mentioned classes the main reason of the instability is the instability of the rank of a matrix. Numerical experiments show that the use of SVD for computation of the ranks often allows us to correctly find the partial indices for matrix polynomials. To create a test case set for numerical experiments we have to solve the problem exactly. By the exact solutions of the factorization problem we mean those solutions that can be found by symbolic computation. In the talk we obtain necessary and sufficient conditions for the existence of the exact solution to the problem for matrix polynomials and propose an algorithm for constructing of the exact solution. The solver modules in SymPy and in Maple that implement this algorithm are designed. WHTW01 13th August 2019 16:30 to 17:00 Valery Smyshlyaev Whispering gallery waves diffraction by boundary inflection: an unsolved canonical problem The problem of interest is that of a whispering gallery high-frequency asymptotic mode propagating along a concave part of a boundary and approaching a boundary inflection point. Like Airy ODE and associated Airy function are fundamental for describing transition from oscillatory to exponentially decaying asymptotic behaviors, the boundary inflection problem leads to an arguably equally fundamental canonical boundary-value problem for a special PDE, describing transition from a “modal” to a “scattered” high-frequency asymptotic behaviour. The latter problem was first formulated and analysed by M.M. Popov starting from 1970-s. The associated solutions have asymptotic behaviors of a modal type (hence with a discrete spectrum) at one end and of a scattering type (with a continuous spectrum) at the other end. Of central interest is to find the map connecting the above two asymptotic regimes. The problem however lacks separation of variables, except in the asymptotical sense at both of the above ends. Nevertheless, the problem asymptotically admits certain complex contour integral solutions, see [1] and further references therein. Further, a non-standard perturbation analysis at the continuous spectrum end can be performed, ultimately describing the desired map connecting the two asymptotic representations. It also permits a re-formulation as a one-dimensional boundary integral equation, whose regularization allows its further asymptotic and numerical analysis. We briefly review all the above, with an interesting open question being whether the presence of an ‘incoming’ and an ‘outgoing’ parts in the sought complex integral solution implies relevance of factorization techniques of Wiener-Hopf type. [1] D. P. Hewett, J. R. Ockendon, V. P. Smyshlyaev, Contour integral solutions of the parabolic wave equation, Wave Motion, 84, 90–109 (2019) Preformatted version: http://www.newton.ac.uk/files/webform/587.tex WHTW01 14th August 2019 09:00 to 10:00 Michael Marder Analytical solutions of dynamic fracture and friction at the atomic scale Following an example set by Slepyan, it proves possible to employ the Wiener-Hopf method to obtain exact solutions for fracture and friction problems at the atomic scale. I will describe a number of physical phenomena that have been analyzed in this way. These include the velocity gap and micro-branching instability for dynamic cracks, a connection of friction with self-healing pulses, and resolution of the energy transport paradox for supersonic cracks. WHTW01 14th August 2019 10:00 to 10:30 John Raymond Willis Transmission and reflection at an interface between metamaterial and ordinary material A contribution to the subject in the title is made, in the case that the metamaterial has random microstructure. A variational approach permits the development of a system of integral equations which can be replaced by a Wiener-Hopf system.The equations retain information on the metamaterial up to two-point probabilities. The formulation will be developed in detail for a configuration of particular simplicity -- acoustic materials, all with the same modulus but different densities. A special case, for which the problem reduces to a very simple scalar Wiener-Hopf problem, has been solved, giving explicit formulae for transmission and reflection coefficients. It should be possible to develop the analysis further and obtain more general solutions... It is likely that the audience will be able to provide useful input. WHTW01 14th August 2019 10:30 to 11:00 Leonid Slepyan Greater generality brings simplicity In this talk, I will discuss listed below problems with attendant circumstances and the results following straightforwardly from the formulation: Mechanical wave momentum from the first principles. Wave Motion, 2016, 68, 283-290. On the energy partition in oscillations and waves. Proc. R. Soc. A, 2015, 471: 20140838. Betty Theorem and Orthogonality Relations for Eigenfunctions. Mechanics of Solids, 1979, 14, 74-77. On a displacement of a deformable body in an acoustic medium. J. Appl. Math. Mech., 1963, 27, 1402-1411, and possibly some others. WHTW01 14th August 2019 11:30 to 12:00 Alexander Movchan Homogenisation and a Wiener-Hopf formulation for a scattering problem around a semi-infinite elastic structured duct Authors: I.S. Jones, N.V. Movchan, A.B. Movchan Abstract: The lecture will cover analysis of elastic waves in a flexural plate, which contains a semi-infinite structured duct. The problem is reduced to a functional equation of the Wiener-Hopf type. The Kernel function reflects on the quasi-periodic Green's function for an infinite periodic structure. Analysis of the Kernel function enables us to identify localised waveguide modes. Homogenisation approximation has been derived to explain the modulation of the wave trapped within the structured duct. Analytical findings are accompanied by numerical examples and simulations. WHTW01 14th August 2019 12:00 to 12:30 Lev Truskinovsky Supersonic kinks in active solids To show that steadily propagating nonlinear waves in active matter can be driven internally, we develop a prototypical model of a topological kink moving with a constant supersonic speed in a discrete bi-stable FPU chain capable of generating active stress. In contrast to subsonic kinks in such systems, that are necessarily dissipative, the obtained supersonic solutions are purely anti-dissipative. Joint work with N. Gorbushin. WHTW01 15th August 2019 09:00 to 10:00 Malte Peter Water-wave forcing on submerged plates We discuss the application of the Wiener-Hopf method to linear water-wave interactions with submerged plates. As the guiding problem, the Wiener-Hopf method is used to derive an explicit expression for the reflection coefficient when a plane wave is obliquely incident upon a submerged semi-infinite porous plate in water of finite depth. Having used the Cauchy Integral Method in the factorisation, the expression does not rely on knowledge of any of the complex-valued eigenvalues or corresponding vertical eigenfunctions in the region occupied by the plate. It is shown that the Residue Calculus technique yields the same result as the Wiener-Hopf method for this problem and this is also used to derive an analytical expression for the solution of the corresponding finite-plate problem. Applications to submerged rigid plates and elastic plates are discussed as well. WHTW01 15th August 2019 10:00 to 10:30 Xun Huang Turbofan noise detection and control studies by the Wiener-Hopf Technique This talk would focus on one of the main themes of this workshop: the diverse applications of the Wiener-Hopf technique for aerospace in general and turbofan noise problems in particular. First, I will give a theoretical model based on the Wiener-Hopf method (and matrix kernel factorisation) to unveil possible noise control mechanisms due to trailing-edge chevrons on the bypass duct of aircraft engine. Next, I will propose a new testing approach that relies on the forward propagation model based on the Wiener-Hopf method. The key contribution is the development of the inverse acoustic scattering approach for a sensor array by combining compressive sensing in a non-classical way. Last but not least, I will demonstrate some of the new aerospace applications of the Wiener-Hopf technique with recently popular deep neural networks. WHTW01 15th August 2019 10:30 to 11:00 Elena Luca Numerical solution of matrix Wiener–Hopf problems via a Riemann–Hilbert formulation In this talk, we present a fast and accurate numerical method for the solution of scalar and matrix Wiener–Hopf problems. The Wiener–Hopf problems are formulated as Riemann–Hilbert problems on the real line, and the numerical approach for such problems of e.g. Trogdon & Olver (2015) is employed. It is shown that the known far-field behaviour of the solutions can be exploited to construct tailor-made numerical schemes providing accurate results. A number of scalar and matrix Wiener–Hopf problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the new approach. This is joint work with Prof. Stefan G. Llewellyn Smith (UCSD). WHTW01 15th August 2019 11:30 to 12:00 Vito Daniele Fredholm factorization of Wiener-Hopf equations (presented by Guido Lombardi) In spite of the great efforts by many studies, there have been little progresses towards a general method of constructive factorizations to get exact solution of vector WH equations. The aim of this talk is the presentation of an alternative solution technique that is based to the reduction of the WH equations to Fredholm equations of second kind (Fredholm factorization). The presentation will focus to the applications of the Fredholm factorization to WH equations occurring in diffraction problem. In particular it is based on five steps:1) Deduction of the WH equations of the problem,2) Reduction of the WH equations to Fredholm integral equations (FIE) ,3) Solution of the Fredholm integral equations , 4)Analytical continuation of the numerical solution of the FIE,5) Evaluation of the physical field components if present: reflected and refracted plane waves, diffracted fields, surface waves, lateral waves, leaky waves, mode excitations, near field. A characteristic example of problem will be presented in the following talk. WHTW01 15th August 2019 12:00 to 12:30 Guido Lombardi Complex scattering and radiation problems using the Generalized Wiener-Hopf Technique This talk focuses on the effectiveness of Generalized Wiener-Hopf Technique (GWHT) in studying complex scattering and radiation problems constituted of planar and angular regions made by impenetrable and/or composite penetrable materials. First, we present theoretical models in the spectral domain using Generalized Wiener-Hopf equations (GWHEs). Next, we apply the novel and effective Fredholm factorization technique to get semi analytical solution of the problem by using integral equation representations. The semi-analyticity of the GWHT solution allows physical insights in terms of spectral component of fields. The case study presented in the talk is the electromagnetic field scattering and radiation of a perfectly electrically conducting wedge over a grounded dielectric slab.Authors: V. Daniele, G. Lombardi, R.S. Zich, Politecnico di Torino, Torino, Italy WHTW01 15th August 2019 13:30 to 14:00 Justin Jaworski Owl-inspired mechanisms of turbulence noise reduction Many owl species rely on specialized plumage to mitigate their aerodynamic noise and achieve functionally-silent flight while hunting. One such plumage feature, a tattered arrangement of flexible trailing-edge feathers, is idealized as a semi-infinite poroelastic plate to model the effects that edge compliance and flow seepage have on the noise production. The interaction of the poroelastic edge with a turbulent eddy is examined analytically with respect to how efficiently the edge scatters the eddy as aerodynamic noise. The scattering event is formulated and solved as a scalar Wiener-Hopf problem to identify how the noise scales with the flight velocity, where special attention is paid to the limiting cases of rigid-porous and elastic-impermeable plate conditions. Results from this analysis identify new parameter spaces where the porous and/or elastic properties of a trailing edge may be tailored to diminish or effectively eliminate the edge scattering effect and may contribute to the owl hush-kit. WHTW01 15th August 2019 14:00 to 14:30 Nikolai Gorbushin Steady-state interfacial cracks in bi-material elastic lattices Fracture mechanics serves both engineering and science in various ways, such as studies of material integrity and physics of earthquakes. Its main object is to analyse crack nucleation and growth depending on features of a particular application. It is common to study cracks in homogeneous materials, however analysis of cracks in bi-materials is important as well, especially in modelling of frictional motion between solids at macro-scale and inter-granular fracture in polycrystallines at micro-scale. The analysis of fracture in dissimilar materials is the main topic of this research. We present the analytical model of steady-state cracks in bi-material square lattices and show its connection with associated macro-level fracture problem. We consider a semi-infinite crack propagating along the interface between two mass-spring square lattices of different properties. Assuming the linear interaction between lattice masses, we can apply integral transforms and obtain the matrix Wiener-Hopf problem from original equations of motion. In this particular case, the kernel matrix is triangular which significantly simplifies the factorisation procedure and even makes possible to reduce to the scalar Wiener-Hopf problem. The discreteness of the problem, however, does not allow to derive factorisation analytically and numerical factorisation was performed. We show that the problem discreteness reveals microscopic radiation in form of decaying elastic waves emanating from a crack tip. These waves are invisible at macro-scale but their energy contributes to the global energy dissipation during the fracture process. We also demonstrate effects of the material properties mismatch and link the microscopic parameters with the macro-level fracture characteristics. WHTW01 15th August 2019 14:30 to 15:00 Matthew Priddin Using iteration to solve n by n matrix Wiener-Hopf equations involving exponential factors with numerical implementation Wiener-Hopf equations involving$n\times n$matrices can arise when solving mixed boundary value problems with$n$junctions at which the boundary condition to be imposed changes form. The offset Fourier transforms of the unknown boundary values lead to exponential factors which require careful consideration when applying the Wiener-Hopf technique. We consider the generalisation of an iterative method introduced recently (Kisil 2018) from$2\times 2$to$n\times n$problems. This may be effectively implemented numerically by employing a spectral method to compute Cauchy transforms. We illustrate the approach through various examples of scattering from collinear rigid plates and consider the merits of the iterative method relative to alternative approaches to similar problems. WHTW01 15th August 2019 15:00 to 15:30 Francesco Dal corso Moving boundary value problems in the dynamics of structures The dynamics of structures partially inserted into a frictionless sliding sleeve defines a moving boundary value problem revealing the presence of an outward configurational force at the constraint, parallel to the sliding direction. The configurational force, differing from that obtained the quasi-static case only for a negligible proportionality coefficient, strongly affects the motion and introduces intriguing structural dynamic response. This will be shown through the two following problems: - The sudden release of a rod with a concentrated weight attached at one end [1]. The solution of a differential-algebraic equation (DAE) system provides the evolution, where the elastic rod may slip alternatively in and out from the sliding sleeve. The nonlinear dynamics eventually ends with the rod completely injected into or completely ejected from the constraint; - The vibrations of a periodic and infinite structural system [2]. Through Bloch-Floquet analysis it is shown that the band gap structure for purely longitudinal vibration is broken so that axial propagation may occur at frequencies that are forbidden in the absence of a transverse oscillation. Moreover, conditions for which flexural oscillation may induce axial resonance are disclosed. The results represent innovative concepts ready to be used in advanced applications, ranging from soft-robotics to earthquake protection. Acknowledgments: Financial support from the Marie Sklodowska-Curie project 'INSPIRE - Innovative ground interface concepts for structure protection' PITN-GA-2019-813424-INSPIRE. [1] Armanini, Dal Corso, Misseroni, Bigoni (2019). Configurational forces and nonlinear structural dynamics. J. Mech. Phys. Solids, doi: 10.1016/j.jmps.2019.05.009 [2] Dal Corso, Tallarico, Movchan, Movchan, Bigoni, (2019). Nested Bloch waves in elastic structures with configurational forces. Phil. Trans. R. Soc. A, doi: 10.1098/rsta.2019.0101 WHTW01 15th August 2019 16:00 to 16:30 Larissa Fradkin Elastic wedge diffraction, with applications to non-destructive evaluation Co-authors: Samar Chehade and Michel Darmon Diffraction of the elastic plane wave by an infinite straight-edged 2D or 3D wedge made of an isotropic solid is a canonical problem that has no analytical solution. We review three major semi-analytical approaches to this problem and discuss their application in non-destructive evaluation as well as testing, cross-validation and experimental validation. We draw attention to high sensitivity of the backscatter diffraction coefficients to the Poisson ratio. WHTW01 15th August 2019 16:30 to 17:00 Davide Bigoni Shear band dynamics When a ductile material is subject to severe strain, failure is preluded by the emergence of shear bands, which initially nucleate in a small area, but quickly extend rectilinearly and accumulate damage, until they degenerate into fractures. Therefore, research on shear bands yields a fundamental understanding of the intimate rules of failure, so that it may be important in the design of new materials with superior mechanical performances.A shear band of finite length, formed inside a ductile material at a certain stage of a continued homogeneous strain, provides a dynamic perturbation to an incident wave field, which strongly influences the dynamics of the material and affects its path to failure. The investigation of this perturbation is presented for a ductile metal, with reference to the incremental mechanics of a material obeying the J2–deformation theory of plasticity. The treatment originates from the derivation of integral representations relating the incremental mechanical fields at every point of the medium to the incremental displacement jump across the shear band faces, generated by an impinging wave. The boundary integral equations are numerically approached through a collocation technique, which keeps into account the singularity at the shear band tips and permits the analysis of an incident wave impinging a shear band. It is shown that the presence of the shear band induces a resonance, visible in the incremental displacement field and in the stress intensity factor at the shear band tips, which promotes shear band growth. Moreover, the waves scattered by the shear band are shown to generate a fine texture of vibrations, parallel to the shear band line and propagating at a long distance from it, but leaving a sort of conical shadow zone, which emanates from the tips of the shear band [1,2]. References [1] Giarola, D., Capuani, D. Bigoni, D. (2018) The dynamics of a shear band. J. Mech. Phys. Solids, 112, 472-490. [2] Giarola, D., Capuani, D. Bigoni, D. (2018) Dynamic interaction of multiple shear bands. Scientific Reports 8 16033 WHTW01 16th August 2019 09:00 to 10:00 Dmitry Ponomarev Spectral theory of convolution operators on finite intervals: small and large interval asymptotics One-dimensional convolution integral operators play a crucial role in a variety of different contexts such as approximation and probability theory, signal processing, physical problems in radiation transfer, neutron transport, diffraction problems, geological prospecting issues and quantum gases statistics,. Motivated by this, we consider a generic eigenvalue problem for one-dimensional convolution integral operator on an interval where the kernel is real-valued even$C^1\$-smooth function which (in case of large interval) is absolutely integrable on the real line.
We show how this spectral problem can be solved by two different asymptotic techniques that take advantage of the
size of the interval.
In case of small interval, this is done by approximation with an integral operator for which there exists a commuting
differential operator thereby reducing the problem to a boundary-value problem for second-order ODE, and often
giving the solution in terms of explicitly available special functions such as prolate spheroidal harmonics.
In case of large interval, the solution hinges on solvability, by Riemann-Hilbert approach, of an approximate auxiliary
integro-differential half-line equation of Wiener-Hopf type, and culminates in simple characteristic equations for
eigenvalues, and, with such an approximation to eigenvalues, approximate eigenfunctions are given in an explicit form.
Besides the crude periodic approximation of Grenander-Szego, since 1960s, large-interval spectral results were available
only for integral operators with kernels of a rapid (typically exponential) decay at infinity or those whose symbols
are rational functions. We assume the symbol of the kernel, on the real line, to be continuous and, for the sake of
simplicity, strictly monotonically decreasing with distance from the origin. Contrary to other approaches, the proposed
method thus relies solely on the behavior of the kernel's symbol on the real line rather than the entire complex plane
which makes it a powerful tool to constructively deal with a wide range of integral operators.
We note that, unlike finite-rank approximation of a compact operator, the auxiliary problems arising in both small-
and large-interval cases admit infinitely many solutions (eigenfunctions) and hence structurally better represent
the original integral operator.
The present talk covers an extension and significant simplification of the previous author's result on
Love/Lieb-Liniger/Gaudin equation.
WHTW01 16th August 2019
10:00 to 10:30
Michael Nieves Phase transition processes in flexural structured systems with rotational inertia
Failure and phase transition processes in mass-spring systems have been extensively studied in the literature, based on the approach developed in [1]. Only a few attempts at characterising these processes in flexural systems exist, see for instance [2, 3, 4, 5]. In comparison with mass-spring systems, flexural structures have a larger range of applicability. They can describe phenomena in systems at various scales, including microlevel waves in materials and dynamic processes in civil engineering assemblies such as bridges and buildings found in society. Flexural systems also provide a greater variety of modelling tools, related to loading configurations and physical parameters, that can be used to achieve a particular response.
Here we consider the role of rotational inertia in the process of phase transition in a one-dimensional flexural system, that may represent a simplified model of the failure of a bridge exposed to hazardous vibrations. The phase transition process is assumed to occur with a uniform speed that is driven by feeding waves carrying energy produced by an applied oscillating moment and force. We show that the problem can be reduced to a functional equation via the Fourier transform which is solved using the Wiener-Hopf technique. From the solution we identify the dynamic behaviour of the system during the transition process. The minimum energy required to initiate the phase transition process with a given speed is determined and it is shown there exist parameter domains defined by the force and moment amplitudes where the phase transition can occur. The influence of the rotational inertia of the system on the wave radiation phenomenon connected with the phase transition is also discussed. All results are supplied with numerical illustrations confirming the analytical predictions.
Acknowledgement: M.J.N. and M.B. gratefully acknowledge the support of the EU H2020 grant MSCA-IF-2016-747334-CAT-FFLAP.
References
[1] Slepyan, L.I.: Models and Phenomena in Fracture Mechanics, Foundations of Engineering Mechanics, Springer, (2002).
[2] Brun, M., Movchan, A.B. and Slepyan, L.I.: Transition wave in a supported heavy beam, J. Mech. Phys. Solids 61, no. 10, pages 2067–2085, (2013).
[3] Brun, M., Giaccu, G.F., Movchan, A., B., and Slepyan, L. I.. Transition wave in the collapse of the San Saba Bridge. Front. Mater. 1:12, (2014). doi: 10.3389/fmats.2014.00012.
[4] Nieves, M.J., Mishuris, G.S., Slepyan, L.I.: Analysis of dynamic damage propagation in discrete beam structures, Int. J. Solids Struct. 97-98, pages 699–713, (2016).
[5] Garau, M., Nieves, M.J. and Jones, I.S. (2019): Alternating strain regimes for failure propagation in flexural systems, Q. J. Mech. Appl. Math., hbz008, https://eur02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.org%2F10.1093%2Fqjmam%2Fhbz008&data=02%7C01%7C%7Cca24c94f14fb47b2a98908d6f19a9002%7Cd47b090e3f5a4ca084d09f89d269f175%7C0%7C0%7C636962043472937444&sdata=hFcD7qiLBQweKalUwfiI8DE4OoKVDBet7AwngVFgEf0%3D&reserved=0.
WHTW01 16th August 2019
10:30 to 11:00
Konstantin Ustinov Application of Khrapkov’s technique of 2x2 matrix factorization to solving problems related to interface cracks
WHTW01 16th August 2019
11:30 to 12:00
Ian Thompson Diffraction in Mindlin plates
Plate theory is important for modelling thin components used in engineering applications, such as metal panels used in aeroplane wings and submarine hulls. A typical application is nondestructive testing, where a wave is transmitted into a panel, and analysis of the scattered response is used to determine the existence, size and location of cracks and other defects. To use this technique, one must first develop a clear theoretical understanding the diffraction patterns that occur when a wave strikes the tip of a fixed or free boundary. Diffraction by semi-infinite rigid strips and cracks in isotropic plates modelled by Kirchhoff theory was considered by Norris & Wang(1994). Although both problems require the application of two boundary conditions on the rigid or free boundary, the resulting Wiener-Hopf equations can be decoupled, leading to a pair of scalar problems. Later, Thompson & Abrahams (2005 & 2007) considered diffraction caused by a crack in a fibre reinforced Kirchhoff plate. The resulting problem is much more complicated than the corresponding isotropic case, but again leads to two separate, scalar Wiener-Hopf equations. In this presentation, we consider diffraction by rigid strips and cracks in plates modelled by Mindlin theory. This is a more accurate model, which captures physics that is neglected by Kirchhoff theory, and is valid at higher frequencies. However, it requires three boundary conditions at an interface. The crack problem and the rigid strip problem each lead to one scalar Wiener-Hopf equation and one 2x2 matrix equation (four problems in total). The scalar problems can be solved in a relatively straightforward manner, but the matrix problems (particularly the problem for the crack) are complicated. However, the kernels have some interesting properties that suggest the possibility of accurate approximate factorisations.
References
A. N. Norris and Z. Wang. Bending-wave diffraction from strips and cracks on thin plates. Q. J. Mech. Appl. Math., 47:607-627, 1994.
I. Thompson and I. D. Abrahams. Diffraction of flexural waves by cracks in orthotropic thin elastic plates.I Formal solution. Proc. Roy. Soc. Lond., A, 461:3413-3434, 2005.
I. Thompson and I. D. Abrahams. Diffraction of flexural waves by cracks in orthotropic thin elastic plates.II. Far field analysis. Proc. Roy. Soc. Lond., A, 463:1615-1638, 2007.
WHTW01 16th August 2019
12:00 to 12:30
Pavlos Livasov Two vector Wiener-Hopf equations with 2x2 kernels containing oscillatory terms
In the first part we discuss a steady-state problem for an interface crack between two dissimilar elastic materials. We consider a model of the process zone described by imperfect transmission conditions that reflect the bridging effect along a finite part of the interface in front of the crack. By means of Fourier transform, the problem is reduced into a Wiener-Hopf equation with a 2x2 matrix, containing oscillatory terms. We factorize the kernel following an existing numerical method and analyse its performance for various parameters of the problem. We show that the model under consideration leads to the classic stress singularity at the crack tip. Finally, we derive conditions for the existence of an equilibrium state and compute admissible length of the process zone.
For the second part of the talk, we consider propagation of a dynamic crack in a periodic structure with internal energy. The structural interface is formed by a discrete set of uniformly distributed alternating compressed and stretched bonds. In such a structure, the fracture of the initially stretched bonds is followed by that of the compressed ones with an unspecified time-lag. That, in turn, reflects the impact of both the internal energy accumulated inside the pre-stressed interface and the energy brought into the system by external loading. The application to the original problem of continuous (with respect to time) and selective discrete (with respect to spatial coordinate) Fourier transforms yields another vector Wiener-Hopf equation with a kernel containing oscillating terms. We use a perturbation technique to factorise the matrix. Finally, we show similarities and differences of the matrix-valued kernels mentioned above and discuss the chosen approaches for their factorisation.
WHTW01 16th August 2019
13:30 to 14:00
Alexander Galybin Application of the Wiener-Hopf approach to incorrectly posed BVP of plane elasticity
WHTW01 16th August 2019
14:00 to 14:30
Matthew Colbrook Solving Wiener-Hopf type problems numerically: a spectral method approach
The unified transform is typically associated with the solution of integrable nonlinear PDEs. However, after an appropriate linearisation, one can also apply the method to linear PDEs and develop a spectral boundary-based method. I will discuss recent advances of this method, in particular, the application of the method to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition problems can be solved without the need for the Wiener-Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geometries involving finite and/or multiple plates. The new approach constructs a global relation that relates known boundary data, such as the scattered normal velocity on a rigid plate, to unknown boundary values, such as the jump in pressure upstream of the plate. This can be viewed formally as a domain dependent Fourier transform of the boundary integral equations. By approximating the unknown boundary functions in a suitable basis expansion and evaluating the global relation at collocation points, one can accurately obtain the expansion coefficients of the unknown boundary values. The local choice of basis functions is flexible, allowing the user to deal with singularities and complicated boundary conditions such as those occurring in elasticity models or spatially variant Robin boundary conditions modelling porosity.
WHTW01 16th August 2019
14:30 to 15:00
Ivan Argatov Application of the Wiener–Hopf technique in contact problems
Problems involving the contact interaction between two elastic bodies, or between an elastic body (called substrate) and a rigid body (called indenter), have occupied the attention of engineering researchers for well over a century. In recent years much attention has been paid to mechanical aspects of contact and adhesion in biological systems, which has resulted in formulating new contact problems, in particular, for a thin elastic layer on a substrate being indented by an indenter of non-canonical shape. Since problems in contact mechanics belong to the class of mixed boundary value problems and can be usually reduced to solving integral equations, it is natural to expect that the Wiener–Hopf method will one of the powerful analytical tools for their investigation. The Wiener–Hopf technique in combination with asymptotic methods has the advantage of universality in obtaining solutions in the analytical form as well as of simplicity for further qualitative analysis. In the present talk we briefly overview the application of the Wiener–Hopf technique to a representative range of contact problems, emphasizing the need of using complementarity asymptotic techniques to cover a larger space of the problem parameters.
WHTW01 16th August 2019
15:00 to 15:30
Mikhail Lyalinov Functional-integral equations and diffraction by a truncated wedge
In this work we study diffraction of a plane incident wave in a complex 2D domain composed by two shifted angular domains having a part of their common boundary. The perfect (Dirichlet or Neumann) boundary conditions are postulated on the polygonal boundary of such compound domain. By means of the Sommerfeld-Malyuzhinets technique the boundary-value problem at hand is reduced to a non-standard systems of Malyuzhinets-type functional-integral equations and then to a Fredholm integral equation of the second kind. Existence and uniqueness of the solution for the diffraction problem is studied and is based on the Fredholm alternative for the integral equation. The far field asymptotics of the wave field is also addressed.
WHTW01 16th August 2019
15:30 to 16:00
Gennady Mishuris Comments on the approximate factorisation of matrix functions with unstable sets of partial indices
It is well known for more than 60 years that the set of partial indices of a non-singular matrix function may be unstable under small perturbations of the matrix [1]. This happens when the difference between the largest and the smallest indices is larger than unity. Although the total index of the matrix preserves its value, this former makes it extremely difficult to use this very powerful method for solving practical problems in this particular case. Moreover, since there does not exist a general constructive technique for matrix factorisation or even for the determination of the partial indices of the matrix, this fact looks like an unavoidable obstacle. Following [2], in this talk, we try to answer a less ambitious question focusing on the determination of the conditions allowing one to construct a family of matrix functions preserving a majority of the properties of the original matrix with non-stable partial indices that is close to the original matrix function.
This work was partially supported by a grant from the Simons Foundation. GM is also acknowledge Royal Society for the Wolfson Research Merit Award.
[1] Gohberg I. & Krein M. 1958 Uspekhi Mat. Nauk.XIII, 3–72 (in Russian).
[2] Mishuris G, Rogosin S. 2018 Regular approximate factorization of a class of matrix-function with an unstable set of partial indices. Proc.R.Soc.A 474:20170279. http://dx.doi.org/10.1098/rspa.2017.0279
WHT 23rd August 2019
15:00 to 15:30
Matthew Nethercote High-contrast approximation for penetrable wedge diffraction
The important open canonical problem of wave diffraction by a penetrable wedge is considered in the high-contrast limit. Mathematically, this means that the contrast parameter, the ratio of a specific material property of the host and the wedge scatterer, is assumed small. The relevant material property depends on the physical context and is different for acoustic and electromagnetic waves for example. Based on this assumption, a new asymptotic iterative scheme is constructed. The solution to the penetrable wedge is written in terms of an infinite sequence of (possibly inhomogeneous) impenetrable wedge problems. Each impenetrable problem is solved using a combination of the Sommerfeld-Malyuzhinets and Wiener-Hopf techniques. The resulting approximate solution to the penetrable wedge involves a large number of nested complex integrals and is hence difficult to evaluate numerically. In order to address this issue, a subtle method (combining asymptotics, interpolation and complex analysis) is developed and implemented, leading to a fast and efficient numerical evaluation. This asymptotic scheme is shown to have excellent convergent properties and leads to a clear improvement on extant approaches.
WHT 23rd August 2019
15:30 to 16:00
Peter Baddoo Scattering by a periodic array of slits with complex boundaries via the Wiener--Hopf method
The interaction of a plane wave with a periodic array of slits is an important problem in fluid dynamics, electromagnetism and solid mechanics. In particular, such an arrangement is commonly used as a model for turbomachinery noise. Previous work has been restricted to the case where the slits possess a Neumann (no-flux) boundary condition. Consequently, in this work we consider "complex" boundary conditions including Robin (e.g. compliance), oblique derivatives (porosity) and generalised Cauchy conditions (impedance). We employ generalised derivatives and Fourier transforms to recast the Helmholtz equation as an integral equation amenable to the Wiener--Hopf method. Although the slits are of finite length, we are able to avoid a true matrix Wiener--Hopf problem by assuming the structure of the scattered field. Since the Wiener--Hopf kernel is meromorphic, the Fourier transform may be inverted analytically to obtain the scattered field. The Wiener--Hopf analysis shows that an effect of modifying the boundary conditions is to perturb the zeros of the kernel function, which correspond to the "duct modes" in the near field. In aeroacoustic applications, this result shows that blade porosity can dramatically reduce the unsteady lift, which has implications for turbomachinery design.
WHT 28th August 2019
10:00 to 11:00
Andrey Shanin On branching of analytic functions in 2D complex space
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2021-02-27 15:42:19
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6735482811927795, "perplexity": 1251.5743335390846}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178358976.37/warc/CC-MAIN-20210227144626-20210227174626-00515.warc.gz"}
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https://www.gradesaver.com/textbooks/math/algebra/college-algebra-6th-edition/chapter-3-polynomial-and-rational-functions-concept-and-vocabulary-check-page-386/7
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## College Algebra (6th Edition)
Fill the blanks with $n$ and $1$
The Linear Factorization Theorem: If $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$, where $n\geq 1$ and $a_{n}\neq 0$, then $f(x)=a_{n}(x-c_{1})(x-c_{2})\cdots(x-c_{n})$, where $c_{1}, c_{2}, \ldots, c_{n}$ are complex numbers (possibly real and not necessarily distinct). In words: An nth-degree polynomial can be expressed as the product of a nonzero constant and $n$ linear factors, where each linear factor has a leading coefficient of $1$. ----------- Fill the blanks with $n$ and $1$
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2018-10-18 01:52:31
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https://www.cut-the-knot.org/arithmetic/algebra/AsymmetricInequality.shtml
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# An Asymmetric Inequality
The following problem was offered at the 1992 Brazilian National Olympiad:
Proof
Prove, for all positive numbers $a,b,c,\;$ the following inequality holds:
$(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}.$
### Proof 1
Quantities $a+b,\;$ $b+c,\;$ and $c+a\;$ satisfy the triangle inequalities and, hence can be thought of as the side length of $\Delta ABC:\;$ $AB=a+b,\;$ $BC=b+c,\;$ and $AC=c+a.\;$
In terms of $a,b,c,\;$ the semiperimeter $p=a+b+c\;$ and, according to Heron's formula, $[\Delta ABC]=\sqrt{abc(a+b+c)},\;$ where $[\Delta ABC]\;$ is the area of $\Delta ABC.$
On the other hand, $[\Delta ABC]=\frac{1}{2}AB\cdot AC\cdot\sin\angle BAC\ge\frac{1}{2}AB\cdot AC,\;$ which immediately implies $\frac{1}{2}AB\cdot AC\le\sqrt{abc(a+b+c)},\;$ or, $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}.$
### Proof 2
Set $x=a(a+b+c),\;$ $y=bc.\;$ Then $x+y = (a+b)(a+c)\;$ and $\displaystyle\frac{x+y}{2}\ge\sqrt{xy}=\sqrt{abc(a+b+c)}.$
### Proof 3
Observe that $(a+b)^2(a+c)^2-4abc(a+b+c)=(a^2+ab+ac-bc)^2.$
### Acknowledgment
The problem with the proof (Proof 1) by Leo Giugiuc and Dan Sitaru has been posted by Leo Giugiuc at the CutTheKnotMath facebook page. Proof 2 is by Alexander Price; Proof 3 is by Imad Zak.
Dorin Marghidanu has made the following observation:
If $a,b,c$ satisfy the equality in $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}\;$ then necessarily $a\lt\min\{b,c\}.$
Indeed, the equality holds iff $a(a+b+c)=bc,\;$ i.e., iff $a(a+b)=c(b-a)\;$ and $a(a+c)=b(c-a).\;$ Since the left-hand sides are positive, so are the right-hand sides, implying $a\lt b\;$ and $a\lt c.$
[an error occurred while processing this directive]
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2018-09-26 08:23:45
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http://vxtd.umood.it/latex-lowercase-script.html
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# Latex Lowercase Script
The forms are so closely related that many Western linguists consider them to be mutually intelligible dialects of the same language, although with minor grammatical and phonological differences. LaTeX handles superscripted superscripts and all of that stuff in the natural way. Today, the largest Unix descendent directly certified as UNIX is macOS by Apple. By working with your colleagues and students on Overleaf, you know that you're not going to hit any version inconsistencies or package conflicts. Related: the set membership relation \in ∈. A string of ASCII text is also valid UTF-8 text. This free service performs a line-by-line analysis for common mistakes and errors in your PHP syntax and will not execute or save your code. Summary: Use the LaTeX soul package to highlight text in a LaTeX document. EDIT: here's the whole thing. org or use the page comments. doc, as well as submitting original files; otherwise, we cannot proceed. Lower case Roman Number (iii), centered, bottom of page. how to write superscripts in axis labels?. More widely used in mathematics is the script small letter epsilon \varepsilon ε. Most of the same property/value pairs that can be used. Based on UX. This first part of the series is about LaTeX lists. This first part of the series is about LaTeX lists. mayhewsw opened this issue Jan 14, 2020 · 19 comments Labels. 9 List of Mathematical Symbols In the following tables you nd all the symbols normally accessible from math mode. tex extension. bbx and custom-numeric-comp. Note: Some characters do not appear in some browsers (particularly some early versions of Internet Explorer). With the Alt key held, type either 165 (to type an upper-case Ñ) or 164 (to type a lower-case ñ). Calligraphy alphabets from a to z letters in black in over 15 styles and samples which include uppercase and lowercase styles, Gothic, Stencils, Cursive, Romantic, Roman and more. \mod , \max , or \sin. If the characters indicate one of LaTeX-supported operators then use that operator command, e. The subscript text contains two numeric or alphanumeric characters. Overleaf comes with a complete, ready to go LaTeX environment which runs on our servers. cbx files; link to the last one can be found in "Footnote citation" section as \usepackage [hyperref=true, url=false, isbn=false. In the late 1970s, the schoolteacher Donald Thurber adapted the Palmer method, making various tweaks to it to create the D'Nealian script. capitalize: capitalizes the. Short Math Guide for LATEX, version 1. For example, if the sentence is: 'I like movies' then I need the output: 'I Like Movies' Query: d. Lambda (uppercase Λ, lowercase λ) is the 11th letter of the Greek alphabet. Open an example in Overleaf. Luckily, text fields don't have nearly as many formatting options as number, currency, and date/time fields. Versions 9. With Overleaf you get the same LaTeX set-up wherever you go. Still I think it would be significantly faster if you've built the array once and outside of the loop, you could keep two values for each word (the lowercase to compare and the original to highlight). If you press SHIFT + F3 a third time, the text reverts back to all uppercase. TextMate: Mac OS X 10. 09 layered on TEX v2. Upright lowercase Greek is, however, not available. Short Math Guide for LATEX, version 2. This is a short introduction, showing the most important features of the package. vformat (format_string, args, kwargs) ¶. doc, as well as submitting original files; otherwise, we cannot proceed. all the text case are in lower case which shows that data is not in a correct case and it results in the bad presentation of your data. XeTeX and LuaTeX (the latter through the luaotfload package) allows a direct interface to fonts which may be loaded by their name or filename, so no manual font installation is required. It's equivalent to equation editor in LaTeX. After a Verb of Saying Capitalize the first letter if the quotation appears after a verb of saying, regardless of the case used in the source–but flag any alterations you make. Press button, get result. Now, fill down the formula in the new column. The reason of the worse readability of uppercase vs lowercase is the lower contrast of shape. How do you make a script lowercase 'r' in Latex, resembling what you see in eg. Code39 barcode with lower case characters GCL2007 (IS/IT--Management) (OP) 25 Feb 10 14:15. bibtex (for BibTeX files) included in the auto_complete_selector setting in your Preferences. One very attractive feature of LaTeX is the ability to change the typeset appearance of your text drastically and consistently with just a few commands. Many script-languages use backslash "\" to denote special commands. Latex-Suite uses the g:Tex_IgnoreLevel setting to set a default. Black letter, also called Gothic script or Old English script, in calligraphy, a style of alphabet that was used for manuscript books and documents throughout Europe—especially in German-speaking countries—from the end of the 12th century to the 20th century. The image of my keyboard is something like this:You can use little adhessive papers, and paste on your own keyboard!!!Ok. LaTeX assumes an implicit multiplication in character sequences it encounters in math mode, and sets the text with corresponding spacing. This has been a guide to Uppercase in Excel. To create a simple equation, for example F=ma, you need only type $F=ma$. Essential for any report. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. 1 Nested text The commands de ned here only skip math sections and \ref arguments if they are not 'hidden' inside a { } brace group. I eventually found several ways of using lower case script letters in LaTeX equations, as well as a few other math mode styles that seem useful. tex extension. Latex script letters lowercase. Sinceit reimplementsthebibliographicfacilitiesofLaTeXfromthegroundup,biblatex. Multiple authors should be treated as follows: (Cheeseman & Englemore 1988) or (Englemore, Cheeseman & Buchanan, 1992). That would be fine, except that apparently their adviser can't stand that uppercase font. If the original material includes a noun or pronoun that is unclear, brackets can be used for clarification. Eth (/ ɛ ð /, uppercase: Ð, lowercase: ð; also spelled edh or eð) is a letter used in Old English, Middle English, Icelandic, Faroese (in which it is called edd), and Elfdalian. Format your messages Message formatting helps add detail and clarity to communication in Slack. Important Message to all Users of TeX If you see that your system produces the symbol instead of for the Greek lowercase delta, you should tell your system administrator immediately to upgrade your obsolete version of the Computer Modern fonts. Additional details are provided in the following chapter. About These Languages South Slavic Languages. The geometric object to use display the data. non capture groups ) or avoid creating them at all:. Note: This command does not alter the footnote counter. 0 to 2017: Origin versions 9. In YouTrack, you can format text using the Markdown markup syntax. MiKTeX provides all the tools you need to efficiently and accurately create your papers, from simple things like changing the letters to uppercase or lowercase, to editing custom scripts. First name* Middle name: Last name* Email address* Do you want to be sent messages about the site? ** Enter the name of this site* The name is in dark blue at the top of the page. For example, learning the cursive F: uppercase and lowercase can take some time and practice. Dina is my favorite monospace font and, for clarity, it is hard to beat (download Dina). Note that LaTeX does not need these changes to correctly uppercase text as its \MakeUppercase and \MakeLowercase commands only require the uppercase and lower case codes of the letters a-z. Sinceit reimplementsthebibliographicfacilitiesofLaTeXfromthegroundup,biblatex. Overleaf comes with a complete, ready to go LaTeX environment which runs on our servers. All text inside such a group. The A4 version of this book mostly uses the headings page style. \$\begingroup\$ Leaving items aside, a text adventure is basically a directed graph: Each room is a node and you can move from one room to another depending on the edges. 7 votes: 200 words 1148 characters: 7: 520: 5 days ago Jun 19th, 2020: public: How to get better at typing (Ling Ling 40 hours) Just to let you all know I kinda suck at typing, my average is like a 75 on Mult. The command \pagenumbering{roman} will set the page numbering to lowercase Roman numerals. Spacing symbols change the amount of spacing, either by adding more space or taking spaces away. In L a T e X, subscripts and superscripts are written using the symbols ^ and _, in this case the x and y exponents where written using these codes. This listing contains short descriptions of the control sequences that are likely to be handy for users of LATEX v2. Every font is free to download!. Integral expression can be added using the \int_{lower}^{upper} command. The final tests are language tests. Instructions on how to type various Symbols, Accents, and Special Characters for Windows, Mac, and in HTML. Note: Some characters do not appear in some browsers (particularly some early versions of Internet Explorer). It is a monospace font, designed for code listings and the like, in print. Introduction This is a concise summary of recommended features in LATEX and a couple of extension packages for writing math formulas. The ordered lists are generated by a \enumerate environment and each entry must be preceded by the control sequence \item, which will automatically generate the number labelling the item. After a Verb of Saying Capitalize the first letter if the quotation appears after a verb of saying, regardless of the case used in the source–but flag any alterations you make. By working with your colleagues and students on Overleaf, you know that you're not going to hit any version inconsistencies or package conflicts. Insert The box is inserted correctly. Tip: Use the tag to define superscripted text. 3): Always use a signal unless (1) the cited authority directly states the proposition in the text; (2) you directly quote the source in the text; or (3) or you state the case name in the text. Small caps still has worse contrast of shape than lower case, so it'll still be less readable. There are three types of lists available. Analyse and paint the TLatex formula. Self-explanatory, it contains an "exuberant graphic stroke". The open function opens a file. The rule adopted by LaTeX is to regard a period (full stop) as the end of a sentence if it is preceded by a lowercase letter. mpg cyl disp hp drat wt MazdaRX4 21. To get exp to appear as a superscript, you type ^{exp}. OUTPUT: TITLE: Average title AUTH: Superman AFF: Something AUTH: The New One AFF: Berlin AUTH: Mars-Mensch AFF: Planet Mars AUTH: Contrary to popular belief, Lorem Ipsu'M is not simply random text. I tried different ways of writing stuff so it looks unambiguous with other text. In RStudio the text looks like this example: we get the coefficients (*β*~*i*~) Which is nice in that it actually shows the Greek character beta. Some PowerPoint. Yes there is. These 26 lowercase letters can be used with your woodburning tool to brand or personalize just about anything: wood, paper, card stock, fabric, leather, gourds, nuts and more. Take a few minutes now to compare the upper and lower case forms of each letter and review the pronunciation of each letter. I am not expert using HotKey, I only use basic commands, but I has writted a right AHK script to manipulate all that TeX symbols, and I give you at the end of this post. lowercase bool, default=True. Dear all, I have been trying to display x label text with subscript. A Computer Science portal for geeks. It is usually smaller than the rest of the text. It is possible to change the labels of any level, replace labelenumii for one of the listed below. The word command may sound scary. Acknowledgments¶. The glyph is a Compat composition of the glyphs. ; - Blackboard bold (double-struck) uppercase letters; - Hebrew letters; - Arrows, harpoons, loops and maps; - Calculus-related symbols; - Propositional/proof related. However, in reviewing and reading many papers, I often see the same errors, over and over again. \mod , \max , or \sin. Some of these symbols are primarily for use in text; most of them are mathematical symbols and can only be used in LaTeX's math mode. The range expression [X-Y] will be included any characters between X and Y using the current locale's collating sequence and character set:. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. This text is in small caps. To change the actual fonts used for custom fonts and the fonts used for variables (unquoted text), numbers and functions, use Format > Fonts. cbx files; link to the last one can be found in "Footnote citation" section as \usepackage [hyperref=true, url=false, isbn=false. For example, the German lowercase letter 'ß' is equivalent to "ss". Please give me the th symbol!. This first part of the series is about LaTeX lists. This series builds on the previous articles: Typeset your docs with LaTex and TeXstudio on Fedora and LaTeX 101 for beginners. Here’s a quick PHP preg_replace example that takes a given input string, and strips all the characters from the string other than letters (the lowercase letters "a-z", and the uppercase letters "A-Z"):. The second file says 1 green bottle on the third line. In the following document, we will refer to special characters for all symbols other than the lowercase letters a-z, uppercase letters A-Z, figures 0-9, and English punctuation marks. The issue here is complicated by the fact that \mathbf (the command for setting bold text in TeX maths) affects a select few mathematical “symbols” (the uppercase Greek letters). Con guring LATEX is covered by the guide Con guration options for LATEX2" in cfgguide. They cannot be used directly in the source. The subscript text contains two numeric or alphanumeric characters. $f\left(x\right)=-x\text{ if }x<0$ Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. Open an example in Overleaf. In text use a capital first letter if the noun is specific: the Faculty of Education, but use lower case letters in general use. Palace Script MT is an early twentieth-century version of an English copperplate script. vformat (format_string, args, kwargs) ¶. Bauhaus 93. My partner is writing a dissertation. Note: This page is a work-in-progress. Thus, adding an input or output format requires only adding a reader or writer. Here, you find out how to add Greek letters to your output, as well as work with superscript and subscript as needed. The command \pagenumbering{roman} will set the page numbering to lowercase Roman numerals. A rendered preview of all letters is shown alongside all commands in a nice table. This unscramble words cheat is the EASIEST way to win in Scrabble, WWF, Jumble & more. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. , that is the Greek lower case of $\latex Sigma$. Font sizes. For example, learning the cursive F: uppercase and lowercase can take some time and practice. If the first line in the current file consists of the text %!TEX root = , then tex & friends are invoked on the specified master file, instead of the current. Some cursive letters are easier to do than others and the best approach is to learn them individually. dvi file produced by latex into a. I'd use the \times macro instead of "x", as regular text gets printed in italics in math mode. Lower converts any uppercase letters to lowercase. Read more in the commands section of the guide about how symbols which take arguments above and below the symbols, such as a summation symbol, behave in the two modes. Commented: Bish Erbas on 24 Sep 2018 Accepted Answer: Bish Erbas. Highlight the text, then press SHIFT + F3 until the text appears in all uppercase. paul rand a designer knows that he has achieved perfection not when there is nothing left to add, but when there is nothing left to take away. Sinceit reimplementsthebibliographicfacilitiesofLaTeXfromthegroundup,biblatex. Supplementaries. Wait for the conversion process to finish. Restriction: In addition to the LaTeX command the unlicensed version will copy a reminder to purchase a license to the clipboard when you select a symbol. The package changes the font that is selected by. U+FB01 was added to Unicode in version 1. This newly improved and still free online word converter tool will take the contents of a doc or docx file and convert the word text into HTML code. With the following VBA code, when you enter the lowercase words into a cell, the lowercase text will be changed to the uppercase letters automatically. Greek letters []. Open an example in Overleaf. Hat and underscore are used for superscripts and subscripts. The docstring of a script (a stand-alone program) should be usable as its "usage" message, printed when the script is invoked with incorrect or missing arguments (or perhaps with a "-h" option, for "help"). The letter represents "l" sound. Easy-to-use symbol, keyword, package, style, and formatting reference for LaTeX scientific publishing markup language. Useful, free online tool for that converts text and strings to UTF8 encoding. Related styles: If you are looking for "blackboard bold", check out the double-struck tool. Text Options for Photoshop CC By Peter Bauer Here you are, the proud owner of the world’s state-of-the-art image editor, Photoshop CC, and now you’re adding text, setting type, and pecking away on the keyboard. Examples: Falvo, D. These include while, do, for, if, switch. The command \renewcommand{\thefootnote}{\roman{footnote}} sets the number styles to lowercase roman. Append lower-case letters to the year in cases of ambiguity, as in (Cheeseman, 1993a). Then mention, where the text should appear. The Phoenician letter Teth (or ṭēt) gave rise to the Greek letter, and it meant wheel. Mathematical fonts In mathematical mode as well as in text mode, you can change the typeface as needed. The lower case letter sigma , the 18th letter of the modern Greek alphabet. Haxby for supporting their efforts on the original version 1. Lambda took its root from the Phoenician letter lamed (or lāmed), which is used to denote "goad". upper letters - animated. To change the case of text that results from a macro inside text you need to do. Small Alphabet Tracing - Trace Alphabets in 4-Lines. \lowercase{} forces the text to be in small letters in a capital letter environment. There are (at least) two ways of getting the symbol: The package amssymb contains the \therefore symbol definition. This file is hereby placed in the public domain. Space is measured in math units, or mu. Tables(1) For i = 1 To oTbl. Most of the same property/value pairs that can be used. This is a subset of MTPro2 which offers a MathTime Pro replacement for Computer Modern math fonts. LaTeX symbols have either names (denoted by backslash) or special characters. Manuscripts must be organized in the following manner: Title Page; Author Footnote (JASA, JCGS, and TAS only) Abstract and Key Words; Article Text. editor Name(s) of editor(s), typed as indicated in the L A TEX book. There are three types of lists available. It is related to its uppercase variant and its titlecase variant. ϵ Lowercase lunate epsilon (ordinary). A LaTeX workshop exercise Introduction. For this to work, you must have \usepackage{amsmath} in the preamble. Latex how to insert a blank or empty page with or without numbering \thispagestyle, ewpage,\usepackage{afterpage} Latex absolute value Horizontal and vertical curly braces: \left\{,\right\},\underbrace{} and \overbrace{}. Using this little language, you specify the rules for the set of possible strings that you want to match; this set might contain English sentences, or e-mail addresses, or TeX commands. Fonts are designed to be mostly connected, but not all the way, an inherent script lettering style. Next, write the title of the article, followed by a period, in quotation marks. Delete and Close up Deltete and close up the gap. The following steps will help you insert special characters. See below for the Table of Contents. The first file says one green bottle. Movement: Why Text in All Caps is Hard for Users to Read. mpg cyl disp hp drat wt MazdaRX4 21. As stated in a more convoluted, albeit more descriptively accurate, the number changes based on password requirements. Supporting text and images/figures should be included in one. An issue to look out for is that the major sectioning commands (\part, \chapter or \maketitle) specify a \thispagestyle {plain}. The font used by mathjax makes the lowercase script "l" (ℓ) look very much like a script e. Now I have no issues writing anything. Insert a new column next to the one that contains the text you want to convert text case. Stock up for on-demand personalization. Video 3 of 11 on Latex tutorials: How to set up a table of contents and get front matter working properly. placement of the equation and its relationship to the surrounding text. Highest quality font for personal and commercial use. If you want to change them, you have several options: the "geometry" package, the "fullpage" package or changing the margins by hand. 1 beta at this time and has ample documentation. com In Excel 2010, you can change case for text to uppercase, lowercase, or proper case. However, some names begin with lowercase letters, such as lowercase prefixes like de, d’, van, or von. mayhewsw opened this issue Jan 14, 2020 · 19 comments Labels. 1: The first text refers to a previously created document that was edited in the second text. Roman Upper case Roman numerals. Writing the perfect resume or cover letter takes careful creative thinking. Any text element can use math text. , or another involved person will number your scenes once your script is going into production. Some of these symbols are primarily for use in text; most of them are mathematical symbols and can only be used in LaTeX's math mode. \texttt - Tex Command - \texttt - Used to produce text-mode material in typewriter font within a mathematical expression. Reference list citations start with Author information. There should be punctuation added and an "and" inserted before the final factor, thus: There were a few factors to keep in mind when going about the benefit cost analysis: 1) technologies that were going to stay, 2) those that were going to stay but be upgraded, 3) things that were going to come to the new house, 4) technology that will be taken to the new house and upgraded, and 5) things. When analyse finds an operator or separator, it calls itself recursively to analyse the arguments of the operator. TeX (/'tɛx/tekh, often pronounced TeK in English) is a very widespread and popular way of representing Mathematics notation using only characters that you can type on a keyboard (see Wikipedia). Mouse click on character to get code:. capitalize: capitalizes the. This is a demonstration of the alphabet in both uppercase and lowercase. As you type in one of the text boxes above, the other boxes are converted on the fly. – “Insert Inline Equation” – Inserts an equation inline with your text. Watts and the late W. Welcome to TXTformat. This series builds on the previous articles: Typeset your docs with LaTex and TeXstudio on Fedora and LaTeX 101 for beginners. Latex how to insert a blank or empty page with or without numbering \thispagestyle, ewpage,\usepackage{afterpage} Latex absolute value Horizontal and vertical curly braces: \left\{,\right\},\underbrace{} and \overbrace{}. Supporting text and images/figures should be included in one. feature-request. $latex envelope. This is a simple online tool that converts regular text into text symbols which resemble the normal alphabet letters. 8, which is part of the current Zotero 2. Math symbols defined by LaTeX package «amssymb» No. Override the preprocessing (string transformation) stage while preserving the tokenizing and n-grams generation steps. OUTPUT: TITLE: Average title AUTH: Superman AFF: Something AUTH: The New One AFF: Berlin AUTH: Mars-Mensch AFF: Planet Mars AUTH: Contrary to popular belief, Lorem Ipsu'M is not simply random text. When you use Slack's automatic formatting, you’ll see exactly what your messages look like before you send them, and you can even add multiple formatting options to the same selection of text. 4 6 258 110 3. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Lambda Symbol in Greek Alphabet. When making your own lists, use this chapter as a reference for spacing settings. co m/fonts/co de39/exten ded39. Handwritten script often looks different from typefaces, in English and in other languages. It provided several di‡erent classes for a number of journals and conference pro-ceedings. In the first line, dvips command converts the. sublime-settings file. Casefolding is similar to lowercasing but more aggressive because it is intended to remove all case distinctions in a string. References in text: These must be included in Reference section and vice versa. For example,$ E=mc^2 $is used to produce E=mc 2. ; In addition the mathabx package has a mathc calligraphic font (in version. This tip shows how to search using Vim, including use of* (the super star) to search for the current word. utf8 bash -c 'case b in [A-Z]) echo yes; esac' yes You can see, b sorted between A and Z in en_US. Because cursive is meant to be written faster than print, understanding how the letters connect can help you be a faster writer!. That would be fine, except that apparently their adviser can't stand that uppercase font. The dictionary paths must be compatible with those used by Sublime Text's spell-checker. Informal handwriting Handwritten font with uppercase and lowercase letter and punctuation Vector Art. The largest number you can enter is 4999, or MMMMCMXCIX in Roman numerals format. Sub Demo() Application. This series builds on the previous articles: Typeset your docs with LaTex and TeXstudio on Fedora and LaTeX 101 for beginners. [Command] Command = " copyq: if (!copy()) abort() var text = str(clipboard()) var newText = text. Files UCSC LaTeX Thesis Template. Sometimes you need to use special characters and character formatting in MATLAB. fnsymbol A set of 9 special symbols. You can purchase a license here: Buy Detexify for Mac. Text Case Changing Tool. It's not anything like a font, but rather a programming language, so I doubt very much you can use it in AutoCAD; it would need to be embedded in the program similarly to the way AutoLISP is. ; The second file says there'd instead of there would on the final line. For remarks on submissions, see at the end of this document. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. Fonts are designed to be mostly connected, but not all the way, an inherent script lettering style. Lower, Upper, and Proper functions in Power Apps. 93 Table 201: txfonts/pxfonts Letter-like Symbols. It matters if it is the figure title, a uicontrol or a text object. Alph Alphabetic upper case. Includes Python syntax highlighting. ) Signals (Rules 1. The Greek alphabet in LaTeX math mode. HTML stands for Hyper Text Markup Language, which is the most widely used language on Web to develop web pages. Copy link Quote reply mayhewsw commented Jan 14,. The command \pagenumbering{roman} will set the page numbering to lowercase Roman numerals. The other answers suggesting text-transform: capitalize are incorrect as that option capitalizes each word. Add the lowercase letter immediately after the publication year in reference entries: Yob, I. Common Errors in Technical Writing John Owens. If you are wondering how to uncapitalize text, this is exactly what the lower case text converter will allow you to do - it transforms all the letters in your text into lowercase letters. 0 while they were their graduate students at Lamont-Doherty Earth Observatory. For more advanced formulæ, like E=mc 2, you need to tell L A TEX to superscript the 2. I am using Crystal Reports XI - I want to barcode a string which contains numbers and may contain a letter. Examples: Falvo, D. Suppose you want to provide a highlight color or background color for text in a LaTeX document. Instant recognition of the different case forms of each letter is an important early reading skill. Append lower-case letters to the year in cases of ambiguity, as in (Cheeseman, 1993a). Replacement. For authors, last names are given first, even for multiple instances. Text Case Changing Tool. This is a demonstration of the alphabet in both uppercase and lowercase. Including this lowercase letter ensures that the reader knows which source you are citing in-text and can find that source in your reference list. They wanted to use the \mathcal command for lowercase letters. tgz archive. LaTeX macros inserted before the footnote mark. An experienced LaTeX user might expect the former behavior since LaTeX usually will only accept one token as an argument to a function without braces, while anyone looking at the output may expect the latter. On the next line, change the justification to Justify. These include the type of label separator (e. I have listed them in this short tutorial. The two commands do not expand the text that is their parameter — the result of \uppercase{abc} is ABC , but \uppercase{\abc} is always \abc , whatever the meaning of \abc. Each figure and table must be mentioned in the text in order of its appearance. But in modern scripting environments like Bash, I have always preferred the convention of lower-case names for temporary variables, and upper. Hi there! I just researched this question but I ended up finding out what was wrong. Preliminaries 2. This unscramble words cheat is the EASIEST way to win in Scrabble, WWF, Jumble & more. The first file has four. However, some names begin with lowercase letters, such as lowercase prefixes like de, d’, van, or von. This height should be seen as a relative unit for measuring the proportion of the lowercase letters: the height measurement alone, as a unit, doesn’t tell us much. The glossaries package v4. Going to run through this code making edits and explain as I go: You only ever use position to find the position within the alphabet; I think it'd be clearer to just make this function's purpose more specific and call it something like index_in_alphabet. 4 6 258 110 3. Multiple Choice Quizzes: Conceptual Discussion Quizzes consist of a numbered list of questions, and following each question, a lettered list of possible answers, one of which is correct. Please study the preceding until you understand it. by you can also try toggle case to switch all upper case letters to lower case letters and vice versa. Unlike a plain text editor, Microsoft Word has features including spell check, grammar check, text and font formatting, HTML support, image support, advanced page layout, and more. For more on Sublime Text support for spell checking, see the relevant online documentation. Hi, this video tutorial will show shortcuts on inserting equation from equation editor in Ms Word. SCRIPT For LaTeX - posted in Utilities: Hello!!I have writted a script for all basic symbols in TeX/LaTeX. LC_ALL=en_US. The $y$ value of a point where a vertical line intersects a graph represents an output for that input $x$ value. Therefore, the small number of characters used in Turkish can be encoded in an 8-bit encoding scheme. Then mention, where the text should appear. bargainballoons. For ease of transport, a standard RTF file can consist of only 7-bit ASCII characters. Missing latex. It is a good idea to have some familiarity with the Greek alphabet because Greek letters are quite often used as abbreviations in science and mathematics. co m/fonts/co de39/exten ded39. The reason of the worse readability of uppercase vs lowercase is the lower contrast of shape. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. Script wood letters were introduced for the font purist who wants cursive letters to arrive exactly as shown. Case-changing oddities TeX provides two primitive commands \uppercase and \lowercase to change the case of text; they’re not much used, but are capable creating confusion. Note that there are two different csls - csl 0. The first thing you’ll need to do is use Python’s built-in open function to get a file object. There are (at least) two ways of getting the symbol: The package amssymb contains the \therefore symbol definition. Download the results either file by file or click the DOWNLOAD ALL button to get them all at once in a ZIP archive. looks like the breast cancer symbol). By studying the document source code file, compiling it, and observing the result, side-by-side with the source, you’ll learn a lot about the R Markdown and LaTeX mathematical typesetting language, and you’ll be able to produce nice-looking documents with R input and output neatly formatted. Bauhaus 93. @srinivas: Please open a new thread for a new question. Here we discuss how to convert lowercase text to uppercase in excel along with examples and downloadable excel template. Phi / ˈ f aɪ / (uppercase Φ, lowercase φ or ϕ; Ancient Greek: ϕεῖ pheî; Modern Greek φι fi) is the 21st letter of the Greek alphabet. An experienced LaTeX user might expect the former behavior since LaTeX usually will only accept one token as an argument to a function without braces, while anyone looking at the output may expect the latter. The replace() method returns a new string with some or all matches of a pattern replaced by a replacement. There are three types of lists available. (If you need to know the HTML codes for displaying letters of the Greek Alphabet please click here). Files UCSC LaTeX Thesis Template. The range expression [X-Y] will be included any characters between X and Y using the current locale's collating sequence and character set:. The angular. How to type special German letters by using their Alt Codes? Make sure you switch on the NumLock, press and hold down the Alt key,; type the Alt Code value of the special German letter, for example, for eszett, type 0223 on the numeric pad,. I'd use the \times macro instead of "x", as regular text gets printed in italics in math mode. Based on UX. Initially it's harder than Word, but once you start writing complicated technical reports with maths it becomes easier. To get exp to appear as a superscript, you type ^{exp}. By working with your colleagues and students on Overleaf, you know that you're not going to hit any version inconsistencies or package conflicts. This first part of the series is about LaTeX lists. bargainballoons. Transpose Remove the fitting end. Lowercase Eta looks like an elongated lowercase n, Notice that the right side of lowercase Eta is longer than the left side. The University uses capital letter U, when referring to the University of Cambridge. A character is made definable, or “active”, by setting its category code (catcode) to be \active (13):. ℓ Lowercase cursive letter l (ordinary). 154 : Brand names or trademarks spelled with a lowercase initial letter followed by a capital letter need not be capitalized at the beginning of a sentence or heading; the existing. Note that the LaTeX environments responsible for handling the information specified by #+CAPTION: and #+LABEL: are table, table*, and longtable. Better Times • A handwritten brush font containing upper & lowercase characters, numerals and a large range of punctuation. Delete and Close up Deltete and close up the gap. This series builds on the previous articles: Typeset your docs with LaTex and TeXstudio on Fedora and LaTeX 101 for beginners. lower-case letter synonyms. These include %-formatting , str. Note 3: For all coloring, the color will apply only to the text immediately following the command until the next space is encountered. It works for both print manuscript and cursive script handwriting styles. Lowercase Eta has some uses. Those characters are numbers 208 and 209, and they tell UTF-8 to switch to the Cyrillic range. Highest quality font for personal and commercial use. sty is a simple LaTeX package for sans serif math fonts in documents. Safe web fonts. You can format text in issue descriptions, supplemental text fields, comments, and work items descriptions. Choose lower-case Roman numerals and start at i. All figures and tables, including those in appendixes, must be mentioned in the text. Case-changing oddities TeX provides two primitive commands \uppercase and \lowercase to change the case of text; they're not much used, but are capable creating confusion. Present the authors' affiliation addresses (where the actual work was done) below the names. And here's another way if you don't want to include MathJax. Superscripts and subscripts of arbitrary height can be done with the \raisebox{}{} command: the first argument is the amount to raise, and the second is the text; a negative first argument will lower the text. Haxby for supporting their efforts on the original version 1. Bold text can also be used to help structure larger bodies of text, for example, to denote a subject, heading, or title. This does allow \mathcal{lowercase} to compile but it also changes the font of the capital letters. tgz archive. The subscript text contains two numeric or alphanumeric characters. If we define case by (1) alone, then it makes perfect sense, as in Vincent's answer, to say that lining numbers are 'uppercase', while text numbers are 'lowercase'—the picture linked to by curiousdannii even shows that the different styles of letters were frequently kept in the upper and lower cases in typographers' kits. Whether you would like to single words, sentences, paragraphs, or the entire text in italics, in LATEXit is possible. We are a bulk balloon distributor of Baby Mylar Balloons (foil balloons, metallic balloons) and other designs at discount wholesale prices. Text::Capitalize provides some routines for title-like formatting of strings. Download Free brush fonts at UrbanFonts. To add these letters to MATLAB, you must use a …. Note that the LaTeX environments responsible for handling the information specified by #+CAPTION: and #+LABEL: are table, table*, and longtable. It converts text into several symbol sets which are listed in the second text area, and the conversion is done in real-time and in your browser using JavaScript. If you are wondering how to uncapitalize text, this is exactly what the lower case text converter will allow you to do - it transforms all the letters in your text into lowercase letters. Mathematical fonts In mathematical mode as well as in text mode, you can change the typeface as needed. MiKTeX provides all the tools you need to efficiently and accurately create your papers, from simple things like changing the letters to uppercase or lowercase, to editing custom scripts. If the characters indicate one of LaTeX-supported operators then use that operator command, e. Analyse and paint the TLatex formula. 4 6 258 110 3. \mod , \max , or \sin. 5 Strikethrough text. If the original material includes a noun or pronoun that is unclear, brackets can be used for clarification. It is a monospace font, designed for code listings and the like, in print. Acknowledgments¶. For the circuit shown in the figure, find the current through resistor. free-fonts-download. Here we will plot a function and will change the title of the graph to something appropriate:. The package defines new commands \Centering, \RaggedLeft, and \RaggedRight and new environments Center, FlushLeft, and FlushRight, which set ragged text and are easily configurable to allow hyphenation (the corresponding commands in L a T e X, all of whose names are lower-case, prevent hyphenation altogether). , (followed by commas after both see and e. Because cursive is meant to be written faster than print, understanding how the letters connect can help you be a faster writer!. The CSS text-transform property is the key to managing text uppercase and lowercase rendering. Now I have no issues writing anything. For example {"en-us": "Packages/Language - English/en_US. doc, as well as submitting original files; otherwise, we cannot proceed. Similar to Greek text letter. The cursive style that most people use today is based on the D'Nealian script, which is itself derived from an older cursive teaching method called the Palmer method/Palmer script. Brush Script Standard. This first part of the series is about LaTeX lists. Here is an added tip: If you press CTRL + SHIFT + K, the text will revert to small caps. There are three types of lists available. STEXT Add Styled Text to the current plot. Without them, usually the next letter or digit will be used, but that isn’t usually what you want. Superscripts and subscripts of arbitrary height can be done with the \raisebox{}{} command: the first argument is the amount to raise, and the second is the text; a negative first argument will lower the text. However, you can sometimes get stuck with little things like inserting an equation, a special character or simply a bar over a letter. That would be fine, except that apparently their adviser can't stand that uppercase font. Spaces, or white space, are rendered using some shorthand symbols, or more generally using the \\mspace command. References in text: These must be included in Reference section and vice versa. Serif Fonts] [Serif Fonts, Sub-Categorised] [Sans Serif Fonts] [Typewriter Fonts] [Calligraphical and Handwritten Fonts] [Uncial Fonts] [Blackletter Fonts] [Other Fonts] [Fonts with Math Support] [Fonts with OpenType or TrueType Support] [All Fonts, by category] [All Fonts, alphabetically] [About The L a T e X Font Catalogue] [Packages that provide math support] QT Artiston [OTF or TTF only]. Insert The box is inserted correctly. pdflatex), commands that prints to. Ready to dive into Bash looping? With the popularity of Linux as a free operating system, and armed with the power of the Bash command line interface, one can go further still, coding advanced loops right from the command line, or within Bash scripts. Script wood letters were introduced for the font purist who wants cursive letters to arrive exactly as shown. This section continues the discussion of the employed biblatex settings started previously. Greek letters []. Note 3: For all coloring, the color will apply only to the text immediately following the command until the next space is encountered. Text::Capitalize provides some routines for title-like formatting of strings. show the log afterwards), rerunnable (repeat command call, if there are warnings), pdf generators (e. In RStudio the text looks like this example: we get the coefficients (*β*~*i*~) Which is nice in that it actually shows the Greek character beta. In L a T e X, subscripts and superscripts are written using the symbols ^ and _, in this case the x and y exponents where written using these codes. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. TeX actually provides primitive for converting arabic numbers to roman numerals. Sed is a useful tool for editing strings on the command line. Thanks! (Hopefully I put this in the appropriate forum). Multiple authors should be treated as follows: (Cheeseman & Englemore 1988) or (Englemore, Cheeseman & Buchanan, 1992). This is a list of patterns, which can be used to filter out (or ignore) some or the warnings and errors reported by the compiler. 1: The first text refers to a previously created document that has been modified in the. Alph Alphabetic upper case. A typical document consist of three main parts: 1. These LaTeX's symbols are grouped together more or less according to function. Share photos and videos, send messages and get updates. Download Pandoc - A Haskell library that enables you to integrate document conversion capabilities into your software, supporting numerous formats including HTML, RTF, ODT and others. Lower-case greek letters are not available in. With the following VBA code, when you enter the lowercase words into a cell, the lowercase text will be changed to the uppercase letters automatically. Default: \leavevmode\unskip. In Visual Studio Code, we have support for almost every major programming language. Here is an added tip: If you press CTRL + SHIFT + K, the text will revert to small caps. U+FB01 was added to Unicode in version 1. Arabic Numbers, Cap and Lower Case). Open an example in Overleaf. Converts letters in a string of text to all lowercase, all uppercase, or proper case. They consist of plain text interspersed with some LaTeX commands. Yet lower-case letters are, on average, smaller in height and width than upper-case characters, which suggests an upper-case advantage. There are 2 ways to do this: Using MathJax - $$\LaTeX$$. Evaluate, simplify, solve, and plot functions without the need to master a complex syntax. Here we discuss how to convert lowercase text to uppercase in excel along with examples and downloadable excel template. ly/puthesis Abstract. The one in the air. Therefore, the small number of characters used in Turkish can be encoded in an 8-bit encoding scheme. Small caps is great for. With the following VBA code, when you enter the lowercase words into a cell, the lowercase text will be changed to the uppercase letters automatically. Click the Symbol Map button to. Analyse and paint the TLatex formula It is called twice : first for calculating the size of each portion of the formula, then to paint the formula. Turn your analyses into high quality documents, reports, presentations and dashboards with R Markdown. Some cursive letters are easier to do than others and the best approach is to learn them individually. Edward Tufte claims the best statistical graphic ever drawn is this image with a variety of cases. Sub Demo() Application. It produces a much cleaner html code than the Microsoft Word software normally produces. 5 Preface A preface is optional. This puts you into the Text Object dialog box where you can use the upper panel to edit text or select text and click one of the format toolbar buttons above the edit box. pdflatex), commands that prints to. The letter character is typically uppercase but may be lowercase at times. 11 Points Guide to. Note that there are two different csls - csl 0. 0 while they were their graduate students at Lamont-Doherty Earth Observatory. In normal mode you can search forwards by pressing/ (or ) then typing your search pattern. Religious emotion in the arts. The production secretary, line producer, A. Note, that integral expression may seems a little different in inline and display math mode - in inline mode the integral symbol and the limits are compressed. Writing text in smallcaps in LaTeX is quite easy – just wrap your text in the \textsc{} tag. Piecewise functions are one of the few items for which multi-line formatting is pretty-much inescapable. It is exposed as a separate function for cases where you want to pass in a predefined dictionary of arguments, rather than unpacking and repacking the dictionary as individual arguments using the *args and **kwargs syntax. Im Beispiel \so{gesperrt} wird der Text „gesperrt“ gesperrt ausgegeben. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. Looking for Uppercase fonts? Click to find the best 235 free fonts in the Uppercase style. Greek alphabet / letters in LaTeX Learn the LaTeX commands to display the greek alphabet. Font: 12 pt. These bold style alphabet letters are suitable for usage as word wall letters, invitations, scrapbooking projects, arts and crafts and are available in colors blue, green, orange and red. The two commands do not expand the text that is their parameter — the result of \uppercase{abc} is ABC , but \uppercase{\abc} is always \abc , whatever the meaning of \abc. Preliminaries 2. commands switch to text fonts that are set up to be-have correctly in mathematics, and should be used for multi-letter identifiers. The big O, big theta, and other notations form the family of Bachmann-Landau or asymptotic notations. See how it works on Vimeo. Note that the LaTeX environments responsible for handling the information specified by #+CAPTION: and #+LABEL: are table, table*, and longtable. The first line for the header, the second for the footer. Types of lists. Each figure and table must be mentioned in the text in order of its appearance. In a math environment (or in math mode) LaTeX typesets the words in the table below in the manner of body text (not in italics) so that they stand apart from the other symbols and variables, it is equivalent to using the "mathrm{ }" command. Sigma notation provides a way to compactly and precisely express any sum, that is, a sequence of things that are all to be added together. You may enter the LaTeX code to be analysed directly into the LaTeX code text area. The ^ character instructs L A TEX to superscript, and the _ character instructs L A TEX to subscript. 5 inches wide on 12pt documents, 1. An info script is a text file having the same name and located in the same directory as the presentation file itself, except for the additional suffix. I have been asked twice recently to move the lower case "c" up to line up with the other letters or above them - regarding the name McNeil or McDonald, etc… I do not recall seeing this done in most. The more nonparallel edges your text has, the higher the shape contrast it has. Simply provide the string as an argument when you call the function, and it will be returned in lowercase form. Theta was also used as a symbol of death in Greek and Latin epigraphy. Introduce figures and tables in your text in logical places and in logical ways. The image of my keyboard is something like this:You can use little adhessive papers, and paste on your own keyboard!!!Ok. Case-changing oddities TeX provides two primitive commands \uppercase and \lowercase to change the case of text; they’re not much used, but are capable creating confusion. U+FB01 LATIN SMALL LIGATURE FI. Self-explanatory, it contains an "exuberant graphic stroke". Note that is already in Arabic number format. Style Text in Matlab. Math symbols defined by LaTeX package «amsfonts» No. This filter converts raw LaTeX TikZ environments into images.$\begingroup\$ @MJD: I use \mathrm in many places; e. For example, if the sentence is: 'I like movies' then I need the output: 'I Like Movies' Query: d. This newly improved and still free online word converter tool will take the contents of a doc or docx file and convert the word text into HTML code. Operators (class 1) are rendered with spaces. Evaluate, simplify, solve, and plot functions without the need to master a complex syntax. International License (CC BY-NC-ND 4. Arabic Numbers, Cap and Lower Case). The refstyle package automates the use of cross references; while vanilla LaTeX would have us write Figure~\ref{xyz}, this is written in refstyle as \figref{xyz}. When the tag is entered in the Gedit view and is pressed, it gets expanded to the snippet. Converts all characters to uppercase. Paper title is 17 point, initial caps/lower case, bold, centered between 2 horizontal rules. Use VBA for Changing Text Case Using formulas for very large spreadsheets or frequently updated data is less efficient than using a Visual Basic for Applications macro. You may enter the LaTeX code to be analysed directly into the LaTeX code text area. If the period is preceded by an uppercase letter then LaTeX assumes that it is not a full stop but follows the initials of somebody's name. The following MCE shows that Unicode and ASCII characters made active and used exactly the same way to define macros, when compiled with lualatex, behave differently in verbatim environments: &. Share photos and videos, send messages and get updates. http:/ /www.
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2020-11-30 19:31:32
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7834643721580505, "perplexity": 2686.7542292442863}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141486017.50/warc/CC-MAIN-20201130192020-20201130222020-00224.warc.gz"}
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https://competitive-exam.in/questions/discuss/the-critical-radius-is-the-insulation-radius-at
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# The critical radius is the insulation radius at which the resistance to heat flow is
Maximum
Minimum
Zero
None of these
Please do not use chat terms. Example: avoid using "grt" instead of "great".
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2021-03-06 14:54:06
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{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8203742504119873, "perplexity": 3825.777209488162}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178375096.65/warc/CC-MAIN-20210306131539-20210306161539-00333.warc.gz"}
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https://rtc-tools.readthedocs.io/en/latest/examples/optimization/channel_pulse.html
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# Modeling Waves in Rivers and Canals¶
Note
This is a more advanced example that implements advanced channel hydraulics in RTC-Tools. It also capitalizes on the homotopy techniques available in RTC-Tools. If you are a first-time user of RTC-Tools, see Filling a Reservoir.
The RTC-Tools is capable of handling non-linear hydraulics. In this example, we model a river channel that receives a sudden pulse of higher-than-usual water volumes. We compare the results to those of an identical model built in HEC-RAS.
## The Model¶
In this example, water is flowing through a single channel. There is an inflow at the upstream end and a water level bound at the downstream end.
In OpenModelica Connection Editor, the model looks like this in plan view:
In text mode, the Modelica model looks as follows (with annotation statements removed):
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 model Example // Elements Deltares.ChannelFlow.Hydraulic.Branches.HomotopicTrapezoidal Channel( Q_nominal = 100.0, H_b_down = -5.0, H_b_up = -5.0, friction_coefficient = 0.045, use_manning = true, length = 10000, theta = theta, use_inertia = true, use_convective_acceleration = false, use_upwind = false, n_level_nodes = 11, uniform_nominal_depth = 5.0, bottom_width_down = 30, bottom_width_up = 30, left_slope_angle_up = 45, left_slope_angle_down = 45, right_slope_angle_up = 45, right_slope_angle_down = 45, semi_implicit_step_size = step_size ) ; Deltares.ChannelFlow.Hydraulic.BoundaryConditions.Level Level; Deltares.ChannelFlow.Hydraulic.BoundaryConditions.Discharge Discharge; // Inputs input Real Inflow_Q(fixed=true) = Discharge.Q; input Real Level_H(fixed=true) = Level.H; parameter Real theta; parameter Real step_size; // Output Channel states output Real Channel_Q_up = Discharge.Q; output Real Channel_Q_dn = Level.HQ.Q; output Real Channel_H_up = Discharge.HQ.H; output Real Channel_H_dn = Level.H; equation connect(Channel.HQDown, Level.HQ); connect(Discharge.HQ, Channel.HQUp); initial equation Channel.Q = fill(Inflow_Q, Channel.n_level_nodes + 1); end Example;
The plan view of the model looks like this in HEC-RAS:
The channel cross-section is a simple trapezoidal shape. As rendered by HEC-RAS, here is a cross-section view of the channel being modeled:
The model was built with HEC-RAS version 5.0.6. In case you wish to verify the HEC-RAS model yourself, a zip of the HEC-RAS model used in this comparison is available: HEC-RAS.zip
## The Python File¶
To keep this example simple and to allow for a 1:1 comparison with HEC-RAS, we will not have any decision variables in this model.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 from rtctools.optimization.collocated_integrated_optimization_problem import ( CollocatedIntegratedOptimizationProblem, ) from rtctools.optimization.csv_mixin import CSVMixin from rtctools.optimization.homotopy_mixin import HomotopyMixin from rtctools.optimization.modelica_mixin import ModelicaMixin from rtctools.util import run_optimization_problem class Example( HomotopyMixin, CSVMixin, ModelicaMixin, CollocatedIntegratedOptimizationProblem ): def parameters(self, ensemble_member): p = super().parameters(ensemble_member) times = self.times() if self.use_semi_implicit: p['step_size'] = times[1] - times[0] else: p['step_size'] = 0.0 p['Channel.use_convective_acceleration'] = self.use_convective_acceleration p['Channel.use_upwind'] = self.use_upwind return p def constraints(self, ensemble_member): constraints = super().constraints(ensemble_member) times = self.times() # Extract the number of nodes in the channel parameters = self.parameters(ensemble_member) n_level_nodes = int(parameters["Channel.n_level_nodes"]) # To Mimic HEC-RAS behaviour, enforce steady state both at t0 and at t1. for i in range(n_level_nodes): state = "Channel.H[{}]".format(i + 1) constraints.append( (self.state_at(state, times[0]) - self.state_at(state, times[1]), 0, 0) ) return constraints class ExampleInertialWave(Example): """Inertial wave equation (no convective acceleration)""" model_name = 'Example' use_semi_implicit = False use_convective_acceleration = False use_upwind = False timeseries_export_basename = "timeseries_export_inertial_wave" class ExampleInertialWaveSemiImplicit(Example): """Inertial wave equation (no convective acceleration)""" model_name = 'Example' use_semi_implicit = True use_convective_acceleration = False use_upwind = False timeseries_export_basename = "timeseries_export_inertial_wave_semi_implicit" class ExampleSaintVenant(Example): """Saint Venant equation. Convective acceleration discretized with central differences""" model_name = 'Example' use_semi_implicit = False use_convective_acceleration = True use_upwind = False timeseries_export_basename = "timeseries_export_saint_venant" class ExampleSaintVenantUpwind(Example): """Saint Venant equation. Convective acceleration discretized with upwind scheme""" model_name = 'Example' use_semi_implicit = False use_convective_acceleration = True use_upwind = True timeseries_export_basename = "timeseries_export_saint_venant_upwind" run_optimization_problem(ExampleInertialWave) run_optimization_problem(ExampleInertialWaveSemiImplicit) run_optimization_problem(ExampleSaintVenant) run_optimization_problem(ExampleSaintVenantUpwind)
As you can see, this model is as simple as it gets. We only add a constraint to keep the initialization states consistent with the HEC-RAS initialization.
## Comparison of Discretizations and Numerical Schemes¶
HEC-RAS and RTC-Tools use different discretizations and numerical schemes, but also solve different equations. RTC-Tools solves the original nonlinear equations, whereas HEC-RAS solves a linearized momentum equation.
RTC-Tools 2 HEC-RAS
Momentum equation Saint-Venant / inertial wave (default) Linearized Saint-Venant
Spatial discretization Staggered Collocated
Numerical scheme (temporal) Semi-implicit / implicit (default) Centered Preissmann box scheme
Numerical scheme (spatial) Central differences, upwind convective acceleration (optional) Centered Preissmann box scheme
Note
For optimization, the recommended momentum equation and temporal scheme for RTC-Tools is semi-implicit inertial wave. Consult Baayen and Piovesan, A continuation approach to nonlinear model predictive control of open channel systems, 2018, for details. A preprint is available online as arXiv:1801.06507.
## Comparison of Results¶
The results from the RTC-Tools run are found in the output directory with the name timeseries_export.csv, and the results generated by HEC-RAS have been exported into the same directory under the name HEC-RAS_results.csv. We can compare the results using the Python library matplotlib:
Both HEC-RAS and RTC-Tools were run with a spatial step size of 1000 m and a temporal step size of 15 min.
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2022-08-18 04:10:55
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https://pro-prof.com/forums/topic/open-closed-principle_en
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# Open-closed principle
Home Forums Programming Software engineering Object-oriented programming Open-closed principle
This topic contains 0 replies, has 1 voice, and was last updated by Васильев Владимир Сергеевич 6 months, 2 weeks ago.
• Author
Posts
• #4093
This article provides a short and clear description for one of the main code design rules (SOLID) – an Open-Closed principle. It is demonstrating examples in C# programming language.
Open-closed principle is the easiest and the most obvious. It states: any program units (classes, structures, modules) should be open for extension and closed for modification.
If the class has already been written, approved, tested, possibly included to the library and then to the project, any attempt to modify it later is a bad idea. But other thing is the possibility to extend it through other available means. In fact, this principle simply involves the clever use of two principles of OOP: abstraction, and polymorphism.
Of course, the need to change the existing requirements to the code is not something rare, this is a normal situation. Another question is how the class itself can be prepared for its functionality to be extended in time. Therefore, data types have to be designed according to the possible need for extension in future.
Lets see an example of a class whose authors were too optimistic about its design, assuming that it is flexible enough to be ever modified.
This example describes a class ShopManager, which implements the possibility of selling products with the discount.
What catches the eye in this code – a limited list of goods, and it is unclear what to do if suddenly the store owners want to extend or change the range of those goods. From this perspective, the code is completely “wooden”, inflexible.
using System;
using System.Data;
using System.Data.SqlClient;
using System.Configuration;
class ShopManager
{
public double Sell(string product, double price, double amount)
{
return Math.Round(MakeDiscount(product) * price * amount, 2);
}
public double MakeDiscount(string product)
{
switch (product)
{
case "apples": return 0.95; case "potatos": return 0.87;
case "strawberry": return 0.99;
default: return 0.0;
}
}
}
class Program
{
static void Main()
{
ShopManager shop = new ShopManager();
Console.WriteLine("The total i s {0} UAH", shop.Sell("apples", 11.5, 7.0));
}
}
The second thing that is puzzling – accrual of discounts. In this case, it just depends on what kind of product we buy. Neither the amount of the product or of the sales period, or from any other reasons. But in reality the principles of accrual discounts may depend on many factors. It may be promotions and wholesale prices, and discount cards and plenty of other options.
How to change the design of the ShopManager class to make it open for any extensions we might want to make in future?
First, it makes sense to immediately get rid of the hard binding to a particular list of products. Second, add some flexibility about different types of discounts (holiday, savings, promotions, etc.). Third, even if we distinguish reasons for discounts, you need to understand that for different types of products they can vary.
And here it becomes clear that it wont take just a couple of minutes to fix this code design. It requires significant improvements in three different directions: product types, types of discounts, different types of discounts for different goods.
And here it is the time to think about one of the OOP principles – abstraction.
We define the base interfaces for any type of products, type of discounts and rules for calculating the discount.
//interface for all kinds of products
interface IProduct
{
string Name { get; set; }
double Price { get; set; }
}
//interface for all types of discounts
interface IDiscount
{
double MakeDiscount(IProduct product, IDiscountRule rule);
}
//interface for all discount rules
interface IDiscountRule
{
double DiscountAmount { get; set; }
double SetRule(double coef);
}
Now we define specific classes that implement these algorithms:
//particular product class
class Apples : IProduct
{
public string Name { get; set; }
public double Price { get; set; }
}
//particular type of discount
class WholesaleDiscount : IDiscount
{
public double MakeDiscount(IProduct product, IDiscountRule rule)
{
//
return product.Price * rule.DiscountAmount;
}
}
class ApplesDiscountRule : IDiscountRule
{
public double DiscountAmount { get; set; }
public double SetRule(double coef)
{
//defining the algorithm of wholesale discountt for particular product
//...
DiscountAmount = coef;
return coef;
}
}
Now we change ShopManager class so that you could make sales with and without discounts:
class ShopManager
{
public double Sell(IProduct product, double amount)
{
//selling the product without discount
return Math.Round(product.Price * amount, 2);
}
public double SellWithDiscount(IProduct product, IDiscount discount, IDiscountRule rule, double amount)
{
//selling the product with discount
return Math.Round(MakeDiscount(product, discount, rule) * amount, 2);
}
public double MakeDiscount(IProduct product, IDiscount discount, IDiscountRule rule)
{
//implementing the discount algorithm
return double rs = discount.MakeDiscount(product, rule);
}
}
And now let’s test the program again after making changes:
class Program
{
static void Main()
{
var shop = new ShopManager();
var apples = new Apples() { Name = "Apples", Price = 11.5 };
Console.WriteLine("The total without discount is {0} UAH", shop.Sell(apples, 7.0));
var rule = new ApplesDiscountRule();
rule.SetRule(0.95);
Console.WriteLine("The total with discount is {0} UAH",
shop.SellWithDiscount(apples, new WholesaleDiscount(), rule, 7.0));
}
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2018-01-20 22:32:01
|
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https://www.springerprofessional.de/introduction-to-boundary-elements/14324880
|
main-content
## Über dieses Buch
to Boundary Elements Theory and Applications With 194 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Dr.-Ing. Friedel Hartmann University of Dortmund Department of Civil Engineering 4600 Dortmund 50 FRG ISBN-13: 978-3-642-48875-7 e-ISBN-13: 978-3-642-48873-3 001: 10.1007/978-3-642-48873-3 Library of Congress Cataloging-in-Publication Data Hartmann, F. (Friedel) Introduction to boundary elements: theory and applications/Friedel Hartmann. ISBN-13: 978-3-642-48875-7 1. Boundary value problems. I. Title. TA347.B69H371989 515.3'5--dc19 89-4160 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provision of the German Copyright Law of September 9,1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Softcover reprint of the hardcover 1 st edition 1989 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
## Inhaltsverzeichnis
### Introduction
Abstract
If you hold a ruler to the end points of a linear function you can draw then the function with a pencil, see Fig. 1.
Friedel Hartmann
### 1. Fundamentals
Abstract
This chapter is a concise summary of the fundamentals of the boundary element method. The experienced reader may prefer to look over this material rather casually and then refer to it again when need arises. The novice reader is advised to begin with chapter 2 where the method is explained in detail by applying it to one-dimensional problems.
Friedel Hartmann
### 2. One-dimensional problems
Abstract
This chapter is intended as a boundary element primer. The method is explained by applying it to the one-dimensional problems of rods and beams.
Friedel Hartmann
### 3. Membranes
Abstract
A membrane is assumed to be a perfectly flexible, thin elastic fabric, which is uniformly stretched in all directions by a tension which has a constant value N per unit length along any section or boundary. The deflection u (= u3) satisfies the differential equation
$$-N\left( u,11+u,22 \right)=-N\Delta u=p$$
where p is the lateral pressure. The traction across a cut is the product of the tension N and the derivative in the direction of the normal vector n = {n1, n2} T of the cut,
$$t=N\frac{\partial u}{\partial n}=N\left( u,1{{n}_{1}}+u{{,}_{2}}{{n}_{2}} \right),$$
that is the N-fold normal derivative or N-fold slope. The close connection between the slope and the traction expresses Fig. 3.1. The greater the pressure the more the membrane will deflect and the greater the slope on the boundary and, therefore, also the traction t on the boundary.
Friedel Hartmann
### 4. Elastic plates and bodies
Abstract
In this chapter we use the boundary element method to calculate the displacements and stresses within elastic plates and bodies.
Friedel Hartmann
### 5. Nonlinear problems
Abstract
Influence functions, as the influence function for the longitudinal displacement u(x)of a rod
$${{u}_{1}}\left( x \right)=\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}\left( y,x \right){{p}_{1}}\left( y \right)dy,$$
(5.1)
are L2-scalar products between the Green’s function and the exterior load p1 and because the scalar product is distributive
$$underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}\left( {{p}_{1}}+{{p}_{2}} \right)dy=\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}{{p}_{1}}dy+\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}{{p}_{2}}dy,$$
the influence function for a nonlinear equation cannot be of the form (5.1). If Eq.(5.1) were the solution of the problem
$${{D}^{NL}}u={{p}_{1}},$$
and
$${{u}_{2}}\left( x \right)=\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}\left( y,x \right){{p}_{2}}\left( y \right)dy$$
the solution of the problem
$${{D}^{NL}}u={{P}_{2}},$$
then the functionld
$$u\left( x \right)={{u}_{1}}\left( x \right)+{{u}_{2}}\left( x \right)=\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}\left( y,x \right)\left( {{p}_{1}}\left( y \right)+{{p}_{2}}\left( y \right) \right)dy$$
Would be the solution of the problem
$${{D}^{NL}}\left( {{u}_{1}}+{{u}_{2}} \right)={{p}_{1}}+{{p}_{2}}.$$
Friedel Hartmann
### 6. Plates
Abstract
In this chapter we apply the boundary element method to Kirchhoff plates. These plates are governed by a fourth-order equation, the bi-harmonic equation.
Friedel Hartmann
### 7. Boundary elements and finite elements
Abstract
Both methods have their strong points
FEM BEM
element library reduction of dimension
robust higher precision
variable coefficients exterior problems
so that a coupling of the two methods or, as it was phrased, a Marriage à la Mode,[77], should benefit from both. The coupling will usually be done by reformulating the coupling conditions of the boundary data of the BE-domain as a stiffness matrix and to couple this stiffness matrix with the stiffness matrices of the neighboring finite elements.
Friedel Hartmann
### 8. Harmonic oscillations
Abstract
Dynamical loads cause inertial forces ρü in a structure. These forces appear on the left-hand side of the differential equation
$$Du+\rho \ddot{u}=p(x,t)$$
If the excitation is harmonic
$$p(x,t)=p(x)\cos (\omega t+\varphi )$$
, then the response of the structure is also harmonic. This important case is the topic of this chapter.
Friedel Hartmann
### 9. Transient problems
Abstract
Transient vibrations are aperiodic vibrations. A separation of the variables is therefore no longer possible. The time t becomes a further variable.
Friedel Hartmann
### 10. Computer programs
Abstract
As a supplement to this book we offer a package of three programs
• BE-LAPLACE
• BE-PLATES
• BE-PLATE-BENDING
which run on the IBM-PC, PS/2 and compatible computers. Hardware requirement are 640 K RAM, a coprocessor 80×87, a hard disk and one of the following graphics adapters: Hercules card, Color Graphics Adapter, Enhanced Graphics Adapter or the Olivetti Graphics card.
Friedel Hartmann
### Backmatter
Weitere Informationen
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2020-04-01 22:57:53
|
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https://ictp.acad.ro/category/numerical-analysis/iterative-methods/successive-approximations/
|
# successive approximations
## How many steps still left to x*?
Abstract The high speed of $$x_{k}\rightarrow x^\ast\in{\mathbb R}$$ is usually measured using the C-, Q- or R-orders: \tag{$C$} \lim \frac…
## Methods of Newton and Newton-Krylov type
Book summaryLocal convergence results on Newton-type methods for nonlinear systems of equations are studied. Solving of large linear systems by…
## Estimating the radius of an attraction ball
Abstract Given a nonlinear mapping $$G:D\subseteq \mathbb{R}^n\rightarrow \mathbb{R}^n$$ differentiable at a fixed point $$x^\ast$$, the Ostrowski theorem offers the sharp…
## On the convergence of some quasi-Newton iterates studied by I. Păvăloiu
Abstract In 1986, I. Păvăloiu [6] has considered a Banach space and the fixed point problem \[x=\lambda D\left( x\right) +y,…
## Sufficient convergence conditions for certain accelerated successive approximations
Abstract We have recently characterized the q-quadratic convergence of the perturbed successive approximations. For a particular choice of the parameters, these…
## On accelerating the convergence of the successive approximations method
Abstract No q-superlinear convergence to a fixed point $$x^\ast$$ of a nonlinear mapping $$G$$ may be attained by the successive approximations when…
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2022-07-05 22:02:52
|
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https://www.bartleby.com/solution-answer/chapter-12-problem-68ae-chemistry-9th-edition/9781133611097/the-reaction-ch33cbrohch33cohbr-in-a-certain-solvent-is-first-order-with-respect-to-ch33cbr/10356df2-a26e-11e8-9bb5-0ece094302b6
|
Chapter 12, Problem 68AE
### Chemistry
9th Edition
Steven S. Zumdahl
ISBN: 9781133611097
Chapter
Section
### Chemistry
9th Edition
Steven S. Zumdahl
ISBN: 9781133611097
Textbook Problem
# The reaction ( CH 3 ) 3 CBr + OH − → ( CH 3 ) 3 COH + Br − in a certain solvent is first order with respect to (CH3)3CBr and zero order with respect to OH− .In several experiments. The rate constant k was determined at different temperatures. A plot of ln(k) versus 1/T was constructed resulting in a straight line with a slope value of −1.10 × 104 K and y-intercept of 33.5. Assume k has units of s−1a. Determine the activation energy for this reaction.b. Determine the value of the frequency factor A.c. Calculate the value of k at 25°C.
(a)
Interpretation Introduction
Interpretation: The reaction between (CH3)3CBr and OH is given. The reaction is first order with respect to (CH3)3CBr and zero order with respect to OH . The values of slope and y intercept is given from the plot of lnk versus 1/T(K) . The value of activation energy and frequency factor A is to be calculated for the given reaction. Also the value of k for the given temperature is to be calculated.
Concept introduction: A certain threshold energy which is necessary for the reaction to occur is called activation energy.
The relationship between the rate constant and activation energy is given by the Arrhenius equation,
k=AeEaRT
To determine: The value of activation energy for the given reaction.
Explanation
Given
The value of slope is 1.10×104K .
The relationship between the rate constant and temperature is given by the Arrhenius equation.
k=AeEaRT
Where,
• k is the rate constant.
• A is the frequency factor.
• Ea is the activation energy.
• R is the universal gas constant (8.314J/Kmol) .
• T is the absolute temperature.
Take natural log on both the sides in the above expression.
lnk=ln(AeEaRT)lnk=EaR(1T)+lnA (1)
Equation (1) represents an equation of straight line.
Compare this equation with the general equation of straight line,
y=mx+c
Where,
• m is the slope
(b)
Interpretation Introduction
Interpretation: The reaction between (CH3)3CBr and OH is given. The reaction is first order with respect to (CH3)3CBr and zero order with respect to OH . The values of slope and y intercept is given from the plot of lnk versus 1/T(K) . The value of activation energy and frequency factor A is to be calculated for the given reaction. Also the value of k for the given temperature is to be calculated.
Concept introduction: A certain threshold energy which is necessary for the reaction to occur is called activation energy.
The relationship between the rate constant and activation energy is given by the Arrhenius equation,
k=AeEaRT
To determine: The value of frequency factor A .
(c)
Interpretation Introduction
Interpretation: The reaction between (CH3)3CBr and OH is given. The reaction is first order with respect to (CH3)3CBr and zero order with respect to OH . The values of slope and y intercept is given from the plot of lnk versus 1/T(K) . The value of activation energy and frequency factor A is to be calculated for the given reaction. Also the value of k for the given temperature is to be calculated.
Concept introduction: A certain threshold energy which is necessary for the reaction to occur is called activation energy.
The relationship between the rate constant and activation energy is given by the Arrhenius equation,
k=AeEaRT
To determine: The value of k at 25°C .
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2019-11-15 20:43:45
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https://www.physicsforums.com/threads/proving-a-certain-orthogonal-matrix-is-a-rotation-matrix.823764/
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# Proving a certain orthogonal matrix is a rotation matrix
1. Jul 18, 2015
### ELB27
1. The problem statement, all variables and given/known data
Let $U$ be a $2\times 2$ orthogonal matrix with $\det U = 1$. Prove that $U$ is a rotation matrix.
2. Relevant equations
3. The attempt at a solution
Well, my strategy was to simply write the matrix as
$$U = \begin{pmatrix} a & b\\ c & d \end{pmatrix}$$
and using the given properties to solve for $a,b,c$ and $d$. I have 4 equations:
1. Determinant = 1
2. $U^TU = I$ where $I$ is identity
where the last property gives me 3 equations (one of the entries is redundant). Thus, I have 4 equations in 4 unknowns. When I solve them, I get 1 free variable, and my matrix turns out to be of the form:
$$U = \begin{pmatrix} x & \mp \sqrt{1-x^2}\\ \pm \sqrt{1-x^2} & x \end{pmatrix}$$
and since all entries must be real (orthogonal matrix) we have the constraint on $x$:
$$-1≤x≤1$$
Clearly, these properties are satisfied if we let $x=\cos\theta$ or $x=\sin\theta$, thus obtaining a rotation matrix in its standard notation. However, I do not see how to prove that these trigonometric functions are the only possible solutions. Also, how does one define formally and rigorously a rotation matrix? Only as a matrix of cosines and sines (with the appropriate values of determinant etc.)?
Any suggestions/comments will be greatly appreciated!
2. Jul 18, 2015
### ShayanJ
There is no need for that. You've done all things needed. You have shown that for all allowed values of x, there exists a $\theta$ that either $x=\sin\theta$ or $x=\cos\theta$. It means for all allowed values of x, this matrix corresponds to a rotation. It may correspond to many other things, but that doesn't matter. All we care about now, is that it corresponds to a rotation. So you're done with this question.
3. Jul 18, 2015
### ELB27
Ah, I think I get it. Basically, the question can be rephrased as "Prove the $U$ can be represented as a rotation matrix."? Thus my proof will end as:
$x=\cos\theta \ ∀x$ for some angle $\theta$ and thus, $U$ is always a rotation matrix about some angle $\theta$.
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2017-08-20 18:26:55
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https://www.zbmath.org/authors/?q=ai%3Awang.yong.7
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# zbMATH — the first resource for mathematics
## Wang, Yong
Compute Distance To:
Author ID: wang.yong.7 Published as: Wang, Yong; Wang, Y.
Documents Indexed: 474 Publications since 1975, including 2 Books
all top 5
#### Co-Authors
50 single-authored 15 Huang, Feimin 8 Huang, Zheng-Hai 7 Dai, Guiping 7 Lu, Gui-Fu 7 Mei, Ming 7 Zhang, Liangyun 6 Cai, Zixing 6 Ma, Xin 6 Yang, Tong 6 Zou, Jian 5 Chee, Chew-Seng 5 Guo, Yongxin 5 Wu, Yi 5 Xing, Yuming 5 Zhang, Liehui 5 Zhou, Yuren 5 Zhu, Weiqiu 4 Bao, Gejun 4 Duan, Renjun 4 Fang, Shu-Cherng 4 Fu, Li 4 Lavery, John E. 4 Li, Guanfeng 4 Yan, Zhibin 4 Ying, Zuguang 4 Zhang, Xinzheng 3 Ban, Guining 3 Cheung, H. M. E. 3 Chien, Stanley Y.-P. 3 Dai, Xian-Zhi 3 Ding, Shusen 3 Ecer, Akin 3 Gu, Qianping 3 Hu, Yonggang 3 Huang, Zhilong 3 Jin, Xiaoling 3 Liu, Chang 3 Liu, Shixing 3 Ma, Liyao 3 Ren, Xingtian 3 Wang, Fangzong 3 Wang, Yi 3 Wu, Lingyun 3 Wu, Wenqing 3 Xu, Xue 3 Yang, Yang 3 Zeng, Bo 3 Zhang, Xiangsun 3 Zhu, Youlan 2 Abbosh, Amin M. 2 Akay, Hasan U. 2 Bai, Xueli 2 Bai, Yingcai 2 Bertocchi, Graziella 2 Cao, Jun 2 Dai, Zhimin 2 Dang, Zhe 2 Daniels, Michael J. 2 Ding, Baocang 2 Feng, Enmin 2 Feng, Ju 2 Fu, Chengqun 2 Gan, Min 2 Gu, Dawu 2 Guo, Guangquan 2 Hu, Huaizhong 2 Huang, Qibai 2 Huang, Rui 2 Jiang, Linglin 2 Jing, Ling 2 Ke, Xiaolu 2 Li, Xuexin 2 Liu, Houlin 2 Liu, Jianyong 2 Liu, Sha 2 Liu, Wanquan 2 Liu, Wenjiang 2 Lively, William M. 2 Ma, Haitao 2 Ma, Runing 2 Mao, Haijun 2 Mei, Fengxiang 2 Nicholson, D. M. C. 2 Niu, Pengcheng 2 Peng, Hui 2 Remmel, Jeffrey B. 2 Shelton, William A. 2 Shi, Peilin 2 Simmons, Dick B. 2 Stocks, G. M. 2 Tian, Junkang 2 Wang, Guanfa 2 Wang, Jian 2 Wang, Ning 2 Wang, Tianyi 2 Wang, Zhongqun 2 Wei, Sining 2 Weissmüller, J. 2 Wu, Junliang 2 Wu, Xionghua ...and 515 more Co-Authors
all top 5
#### Serials
12 Journal of Harbin Institute of Technology 10 International Journal of Theoretical Physics 8 Computers & Mathematics with Applications 8 Pattern Recognition 8 SIAM Journal on Mathematical Analysis 7 Computational Statistics and Data Analysis 7 Journal of Inequalities and Applications 6 Journal of Mathematics. Wuhan University 6 Nonlinear Dynamics 6 Abstract and Applied Analysis 6 Journal of Software 5 Archive for Rational Mechanics and Analysis 5 Physics Letters. A 5 Journal of Optimization Theory and Applications 5 Applied Mathematics Letters 5 Control and Decision 5 Statistics and Computing 4 International Journal of Heat and Mass Transfer 4 Applied Mathematics and Computation 4 Journal of Computational and Applied Mathematics 4 Journal of Xi’an Jiaotong University 4 Journal of Natural Science of Heilongjiang University 4 Applied Mathematical Modelling 4 Engineering Analysis with Boundary Elements 4 Journal of University of Science and Technology of China 3 Journal of Mathematical Physics 3 Information Sciences 3 Acta Automatica Sinica 3 Journal of Computational Mathematics 3 Journal of Systems Science and Mathematical Sciences 3 Acta Physica Sinica 3 Mathematica Applicata 3 Archive of Applied Mechanics 3 Pure and Applied Mathematics 3 Mathematical Problems in Engineering 3 Control Theory & Applications 3 International Journal of Modern Physics C 3 IEEE Transactions on Antennas and Propagation 3 Acta Mathematica Scientia. Series B. (English Edition) 3 Journal of Harbin Institute of Technology. New Series 3 Journal of Computational Methods in Sciences and Engineering 3 Journal of Systems Engineering 3 Chinese Journal of Computational Mechanics 3 Communications in Theoretical Physics 3 Advances in Applied Mathematics and Mechanics 3 Journal of Theoretical Biology 2 Modern Physics Letters B 2 Acta Mechanica 2 Computers and Fluids 2 International Journal of Control 2 International Journal for Numerical Methods in Fluids 2 International Journal of Solids and Structures 2 International Journal of Systems Science 2 Journal of the Mechanics and Physics of Solids 2 Journal of Sound and Vibration 2 Chaos, Solitons and Fractals 2 Advances in Mathematics 2 Automatica 2 International Journal for Numerical Methods in Engineering 2 Journal of Differential Equations 2 Journal of Economic Theory 2 Journal of Multivariate Analysis 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Results in Mathematics 2 Journal of Nanjing University. Mathematical Biquarterly 2 Neural Networks 2 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 2 Computational Statistics 2 European Journal of Operational Research 2 Journal of Elasticity 2 International Journal of Robust and Nonlinear Control 2 Bulletin of the Belgian Mathematical Society - Simon Stevin 2 Journal of Vibration and Control 2 Mathematical Methods of Operations Research 2 Journal of Shanghai University 2 Chinese Quarterly Journal of Mathematics 2 Mechanism and Machine Theory 2 Discrete Dynamics in Nature and Society 2 Communications in Nonlinear Science and Numerical Simulation 2 Applied Mathematics E-Notes 2 Journal of Southeast University. Natural Science Edition 2 Journal of Intelligent and Fuzzy Systems 2 Transactions of Beijing Institute of Technology 2 Journal of Beijing University of Technology 2 Journal of Algebra and its Applications 2 Journal of Beijing Institute of Technology 2 International Journal of Geometric Methods in Modern Physics 2 Advances in Difference Equations 2 Mathematical Biosciences and Engineering 2 Journal of Guangxi University for Nationalities. Natural Science Edition 2 Science in China. Series F 2 Boundary Value Problems 2 Pacific Journal of Optimization 2 Journal of Function Spaces 2 AMM. Applied Mathematics and Mechanics. (English Edition) 1 Astronomy and Astrophysics 1 Bulletin of the Australian Mathematical Society 1 Communications in Algebra 1 Computers and Structures 1 Discrete Applied Mathematics ...and 132 more Serials
all top 5
#### Fields
65 Computer science (68-XX) 61 Numerical analysis (65-XX) 59 Operations research, mathematical programming (90-XX) 56 Partial differential equations (35-XX) 55 Fluid mechanics (76-XX) 51 Systems theory; control (93-XX) 47 Statistics (62-XX) 37 Ordinary differential equations (34-XX) 30 Operator theory (47-XX) 28 Mechanics of deformable solids (74-XX) 27 Biology and other natural sciences (92-XX) 27 Information and communication theory, circuits (94-XX) 21 Mechanics of particles and systems (70-XX) 16 Dynamical systems and ergodic theory (37-XX) 16 Quantum theory (81-XX) 15 Global analysis, analysis on manifolds (58-XX) 15 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 12 Associative rings and algebras (16-XX) 11 Linear and multilinear algebra; matrix theory (15-XX) 11 Calculus of variations and optimal control; optimization (49-XX) 11 Classical thermodynamics, heat transfer (80-XX) 11 Statistical mechanics, structure of matter (82-XX) 10 Probability theory and stochastic processes (60-XX) 9 Functional analysis (46-XX) 7 Differential geometry (53-XX) 7 Optics, electromagnetic theory (78-XX) 6 Combinatorics (05-XX) 6 Potential theory (31-XX) 5 Group theory and generalizations (20-XX) 5 Approximations and expansions (41-XX) 5 Geophysics (86-XX) 4 Mathematical logic and foundations (03-XX) 4 Real functions (26-XX) 4 Difference and functional equations (39-XX) 3 Number theory (11-XX) 3 Category theory; homological algebra (18-XX) 2 General and overarching topics; collections (00-XX) 2 Algebraic geometry (14-XX) 2 Functions of a complex variable (30-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Relativity and gravitational theory (83-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Nonassociative rings and algebras (17-XX) 1 Special functions (33-XX) 1 Integral transforms, operational calculus (44-XX) 1 Integral equations (45-XX) 1 Manifolds and cell complexes (57-XX) 1 Astronomy and astrophysics (85-XX)
#### Citations contained in zbMATH Open
202 Publications have been cited 1,187 times in 168 Documents Cited by Year
Global uniqueness and solvability for tensor complementarity problems. Zbl 1344.90056
Bai, Xue-Li; Huang, Zheng-Hai; Wang, Yong
2016
Hardy type and Rellich type inequalities on the Heisenberg group. Zbl 0979.35035
Niu, Pengcheng; Zhang, Huiqing; Wang, Yong
2001
Exceptionally regular tensors and tensor complementarity problems. Zbl 1368.90158
Wang, Yong; Huang, Zheng-Hai; Bai, Xue-Li
2016
A new trust region method for nonlinear equations. Zbl 1043.65072
Zhang, Ju-liang; Wang, Yong
2003
Large time behavior of solutions to $$n$$-dimensional bipolar hydrodynamic models for semiconductors. Zbl 1228.35053
Huang, Feimin; Mei, Ming; Wang, Yong
2011
A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. Zbl 0876.73030
Wang, Y.; Aron, M.
1996
Global Poincaré inequalities for Green’s operator applied to the solutions of the nonhomogeneous $$A$$-harmonic equation. Zbl 1155.31303
Wang, Yong; Wu, Congxin
2004
On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Zbl 1120.62022
Wang, Yong
2007
Serrin-type blowup criterion for full compressible Navier-Stokes system. Zbl 1260.35114
Huang, Xiangdi; Li, Jing; Wang, Yong
2013
Characteristic boundary conditions for direct simulations of turbulent counterflow flames. Zbl 1086.80006
Yoo, C. S.; Wang, Y.; Trouvé, A.; Im, H. G.
2005
Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors. Zbl 1227.35063
Huang, Feimin; Mei, Ming; Wang, Yong; Yu, Huimin
2011
Long-time behavior of solutions to the bipolar hydrodynamic model of semiconductors with boundary effect. Zbl 1248.35020
Huang, Feimin; Mei, Ming; Wang, Yong; Yang, Tong
2012
Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Zbl 1253.35028
Huang, Rui; Mei, Ming; Wang, Yong
2012
The limit of the Boltzmann equation to the Euler equations for Riemann problems. Zbl 1367.76055
Huang, Feimin; Wang, Yi; Wang, Yong; Yang, Tong
2013
On multi-dimensional sonic-subsonic flow. Zbl 1265.76039
Huang, Feimin; Wang, Tianyi; Wang, Yong
2011
A note on the numerical solution of high-order differential equations. Zbl 1031.65087
Wang, Y.; Zhao, Y. B.; Wei, G. W.
2003
Remark on stability of traveling waves for nonlocal Fisher-KPP equations. Zbl 1337.35075
Mei, Ming; Wang, Yong
2011
Asymptotic convergence to planar stationary waves for multi-dimensional unipolar hydrodynamic model of semiconductors. Zbl 1228.35177
Huang, Feimin; Mei, Ming; Wang, Yong; Yu, Huimin
2011
Stationary equilibria in an overlapping generations economy with stochastic production. Zbl 0802.90025
Wang, Yong
1993
Global uniqueness and solvability of tensor variational inequalities. Zbl 1409.90207
Wang, Yong; Huang, Zheng-Hai; Qi, Liqun
2018
Using model trees for classification. Zbl 0901.68167
Frank, Eibe; Wang, Yong; Inglis, Stuart; Holmes, Geoffrey; Witten, Ian H.
1998
Implicit-explicit finite-difference lattice Boltzmann method for compressible flows. Zbl 1151.82405
Wang, Y.; He, Y. L.; Zhao, T. S.; Tang, G. H.; Tao, W. Q.
2007
The construction of Hom-Lie bialgebras. Zbl 1217.17013
Chen, Yuanyuan; Wang, Yong; Zhang, Liangyun
2010
$$p$$th moment exponential stability of stochastic Cohen-Grossberg neural networks with time-varying delays. Zbl 1445.92012
Zhu, Enwen; Zhang, Haomin; Wang, Yong; Zou, Jiezhong; Yu, Zheng; Hou, Zhenting
2007
Dimension-reduced nonparametric maximum likelihood computation for interval-censored data. Zbl 1452.62257
Wang, Yong
2008
Estimation of finite mixtures with symmetric components. Zbl 1322.62013
Chee, Chew-Seng; Wang, Yong
2013
A pseudorandom number generator based on piecewise logistic map. Zbl 1354.65012
Wang, Yong; Liu, Zhaolong; Ma, Jianbin; He, Haiyuan
2016
Stationary Markov equilibria in an OLG model with correlated production shocks. Zbl 0816.90033
Wang, Yong
1994
Global well-posedness of the Boltzmann equation with large amplitude initial data. Zbl 1367.35097
Duan, Renjun; Huang, Feimin; Wang, Yong; Yang, Tong
2017
Three-dimensional non-free-parameter lattice-Boltzmann model and its application to inviscid compressible flows. Zbl 1229.76090
Li, Q.; He, Y. L.; Wang, Y.; Tang, G. H.
2009
Nonholonomic versus vakonomic dynamics on a Riemann-Cartan manifold. Zbl 1110.70018
Guo, Yong-Xin; Wang, Yong; Chee, G. Y.; Mei, Feng-Xiang
2005
Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in three-dimensional domains. Zbl 1332.35303
Wang, Yong; Xin, Zhouping; Yong, Yan
2015
Two-weight Poincaré-type inequalities for differential forms in $$L^s(\mu )$$-averaging domains. Zbl 1144.58003
Wang, Yong
2007
Application of a novel nonlinear multivariate grey Bernoulli model to predict the tourist income of China. Zbl 1407.62427
Ma, Xin; Liu, Zhibin; Wang, Yong
2019
A compressed primal-dual method for generating bivariate cubic $$L_{1}$$ splines. Zbl 1110.65015
Wang, Yong; Fang, Shu-Cherng; Lavery, John E.
2007
Novel existence and uniqueness criteria for periodic solutions of a Duffing type $$p$$-Laplacian equation. Zbl 1194.34080
Wang, Yong
2010
The MFS versus the Trefftz method for the Laplace equation in 3D. Zbl 1403.65259
Lv, Hui; Hao, Fang; Wang, Yong; Chen, C. S.
2017
An FSI solution technique based on the DSD/SST method and its applications. Zbl 1327.74059
Tian, Fang-Bao; Wang, Yong; Young, John; Lai, Joseph C. S.
2015
A geometric programming approach for bivariate cubic $$L_{1}$$ splines. Zbl 1083.41008
Wang, Yong; Fang, Shu-Cherng; Lavery, J. E.; Cheng, Hao
2005
Improved treatment of intersecting bodies with the chimera method and validation with a simple and fast flow solver. Zbl 0976.76053
Chattot, Jean-Jacques; Wang, Yong
1998
Scheduling projects with labor constraints. Zbl 0984.90012
Cavalcante, C. C. B.; de Souza, C. Carvalho; Savelsbergh, M. W. P.; Wang, Y.; Wolsey, L. A.
2001
Statistical analysis for stochastic systems including fractional derivatives. Zbl 1183.70062
Huang, Z. L.; Jin, X. L.; Lim, C. W.; Wang, Y.
2010
Nonparametric estimation of species richness using discrete $$k$$-monotone distributions. Zbl 1468.62036
Chee, Chew-Seng; Wang, Yong
2016
Advantages of the enhanced opposite direction searching algorithm for computing the centroid of an interval type-2 fuzzy set. Zbl 1303.68121
Hu, Huaizhong; Wang, Yong; Cai, Yuanli
2012
Approximating term structure of interest rates using cubic $$L_1$$ splines. Zbl 1141.91016
Chiu, Nan-Chieh; Fang, Shu-Cherng; Lavery, John E.; Lin, Jen-Yen; Wang, Yong
2008
More examples and counterexamples for a conjecture of Merrifield and Simmons. Zbl 0997.05069
Wang, Yong; Li, Xueliang; Gutman, Ivan
2001
The novel fractional discrete multivariate grey system model and its applications. Zbl 1464.62389
Ma, Xin; Xie, Mei; Wu, Wenqing; Zeng, Bo; Wang, Yong; Wu, Xinxing
2019
Accelerating adaptive trade-off model using shrinking space technique for constrained evolutionary optimization. Zbl 1158.74442
Wang, Yong; Cai, Zixing; Zhou, Yuren
2009
On the existence of a unique periodic solution to a Liénard type $$p$$-Laplacian non-autonomous equation. Zbl 1175.34057
Wang, Yong; Dai, Xian-Zhi; Xia, Xiao-Xu
2009
Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks. Zbl 1360.35178
Huang, FeiMin; Wang, Yi; Wang, Yong; Yang, Tong
2015
Center problems and limit cycle bifurcations in a class of quasi-homogeneous systems. Zbl 1326.34069
Xiong, Yanqin; Han, Maoan; Wang, Yong
2015
Nonparametric multivariate density estimation using mixtures. Zbl 1331.62279
Wang, Xuxu; Wang, Yong
2015
Co-ordinated control design of generator excitation and SVC for transient stability and voltage regulation enhancement of multi-machine power systems. Zbl 1057.93038
Cong, L.; Wang, Y.; Hill, D. J.
2004
Mechanics of corrugated surfaces. Zbl 1200.74008
Wang, Y.; Weissmüller, J.; Duan, H. L.
2010
Probability-one homotopy algorithms for solving the coupled Lyapunov equations arising in reduced-order $$H^2/H^\infty$$ modeling, estimation, and control. Zbl 1028.93011
Wang, Y.; Bernstein, D. S.; Watson, L. T.
2001
Stochastic minimax control for stabilizing uncertain quasi-integrable Hamiltonian systems. Zbl 1185.93151
Wang, Yong; Ying, Zuguang; Zhu, Weiqiu
2009
A novel localized collocation solver based on Trefftz basis for potential-based inverse electromyography. Zbl 07330175
Xi, Qiang; Fu, Zhuojia; Wu, Wenjie; Wang, Hui; Wang, Yong
2021
Lyapunov function construction for nonlinear stochastic dynamical systems. Zbl 1284.93207
Ling, Quan; Jin, Xiao Ling; Wang, Y.; Li, H. F.; Huang, Zhi Long
2013
Single sign-on under quantum cryptography. Zbl 1284.81086
Dai, Guiping; Wang, Yong
2014
Minimum quadratic distance density estimation using nonparametric mixtures. Zbl 1365.62125
Chee, Chew-Seng; Wang, Yong
2013
Diffusive wave in the low Mach limit for compressible Navier-Stokes equations. Zbl 1375.35378
Huang, Feimin; Wang, Tian-Yi; Wang, Yong
2017
The nonlinear dynamics based on the nonstandard Hamiltonians. Zbl 1375.70065
Liu, Shixing; Guan, Fang; Wang, Yong; Liu, Chang; Guo, Yongxin
2017
On deformations with constant modified stretches describing the bending of rectangular blocks. Zbl 0830.73010
Aron, M.; Wang, Y.
1995
Remarks concerning the flexure of a compressible nonlinearly elastic rectangular block. Zbl 0835.73012
Aron, M.; Wang, Y.
1995
Stochastic vibration model of gear transmission systems considering speed-dependent random errors. Zbl 0946.70505
Wang, Y.; Zhang, W. J.
1998
Impingement of filler dropelts and weld pool dynamics during gas metal arc welding process. Zbl 1064.76624
Wang, Y.; Tsai, Hai Lung
2001
Nonlinear dynamic response and vibration active control of piezoelectric elasto-plastic laminated plates with damage. Zbl 1273.74348
Tian, Yanping; Fu, Yiming; Wang, Yong
2009
Cubic $$L_1$$ splines on triangulated irregular networks. Zbl 1099.65017
Zhang, Wei; Wang, Yong; Fang, Shucherng; Lavery, John E.
2006
A combinatorial model and algorithm for globally searching community structure in complex networks. Zbl 1245.90013
Zhang, Xiang-Sun; Li, Zhenping; Wang, Rui-Sheng; Wang, Yong
2012
Density estimation using non-parametric and semi-parametric mixtures. Zbl 1420.62164
Wang, Yong; Chee, Chew-Seng
2012
Instabilities of core-shell heterostructured cylinders due to diffusions and epitaxy: Spheroidization and blossom of nanowires. Zbl 1162.74317
Duan, H. L.; Weissmüller, J.; Wang, Y.
2008
Existence of asymptotically stable periodic solutions of a Rayleigh type equation. Zbl 1173.34328
Wang, Yong; Zhang, Liang
2009
Reliability and covariance estimation of weighted $$k$$-out-of-$$n$$ multi-state systems. Zbl 1253.90095
Wang, Yong; Li, Lin; Huang, Shuhong; Chang, Qing
2012
Norm comparison estimates for the composite operator. Zbl 06281156
Li, Xuexin; Wang, Yong; Xing, Yuming
2014
Robustness of non-linear stochastic optimal control for quasi-Hamiltonian systems with parametric uncertainty. Zbl 1287.93110
Wang, Y.; Ying, Z. G.; Zhu, W. Q.
2009
A note on existence of a unique periodic solution of non-autonomous second-order Hamiltonian systems. Zbl 1286.34063
Zhang, Lie-Hui; Wang, Yong
2010
Optimal pricing, production, and sales policies for new product under supply constraint. Zbl 1303.93191
Yan, Xiaoming; Liu, Ke; Wang, Yong
2014
Numerical inverse isoparametric mapping in 3D FEM. Zbl 0638.73037
Murti, V.; Wang, Y.; Valliappan, S.
1988
On a straightening of compressible, nonlinearly elastic, annular cylindrical sectors. Zbl 1001.74531
Aron, M.; Christopher, C.; Wang, Y.
1998
An improved ($$\mu+\lambda$$)-constrained differential evolution for constrained optimization. Zbl 1293.68228
Jia, Guanbo; Wang, Yong; Cai, Zixing; Jin, Yaochu
2013
Generalized Mahalanobis depth in the reproducing kernel Hilbert space. Zbl 1434.62082
Hu, Yonggang; Wang, Yong; Wu, Yi; Li, Qiang; Hou, Chenping
2011
Large eddy simulation (LES) for synthetic jet thermal management. Zbl 1189.76324
Wang, Yong; Yuan, Guang; Yoon, Yong-Kyu; Allen, Mark G.; Bidstrup, Sue Ann
2006
Numerical simulations of gas resonant oscillations in a closed tube using lattice Boltzmann method. Zbl 1143.80332
Wang, Y.; He, Y. L.; Li, Q.; Tang, G. H.
2008
Numerical solutions comparison for interval linear programming problems based on coverage and validity rates. Zbl 1427.65094
Lu, H. W.; Cao, M. F.; Wang, Y.; Fan, X.; He, L.
2014
Predicting DNA- and RNA-binding proteins from sequences with kernel methods. Zbl 1402.92332
Shao, Xiaojian; Tian, Yingjie; Wu, Lingyun; Wang, Yong; Jing, Ling; Deng, Naiyang
2009
Time-dependent Poiseuille flows of visco-elasto-plastic fluids. Zbl 1106.76011
Wang, Y.
2006
A full-Newton step non-interior continuation algorithm for a class of complementarity problems. Zbl 1241.65060
Zhao, Jian-Xun; Wang, Yong
2012
Incremental complete LDA for face recognition. Zbl 1236.68217
Lu, Gui-Fu; Zou, Jian; Wang, Yong
2012
Theoretical analysis for blow-up behaviors of differential equations with piecewise constant arguments. Zbl 1410.34184
Zhou, Y. C.; Yang, Z. W.; Zhang, H. Y.; Wang, Y.
2016
Safety verification of model helicopter controller using hybrid input/output automata. Zbl 1032.93535
Mitra, Sayan; Wang, Yong; Lynch, Nancy; Feron, Eric
2003
Two-mode ILC with pseudo-downsampled learning in high frequency range. Zbl 1117.93045
Zhang, B.; Wang, D.; Ye, Y.; Wang, Y.; Zhou, K.
2007
Influence of nonholonomic constraints on variations, symplectic structure, and dynamics of mechanical systems. Zbl 1152.81458
Guo, Yong-Xin; Liu, Shi-Xing; Liu, Chang; Luo, Shao-Kai; Wang, Yong
2007
Implicit-explicit finite difference lattice Boltzmann method with viscid compressible model for gas oscillating patterns in a resonator. Zbl 1168.76034
Wang, Yong; He, Yaling; Huang, Jing; Li, Qing
2009
The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras. Zbl 1255.16037
Wang, Yong; Zhang, Liangyun
2011
Nearly $$s$$-normal subgroups of a finite group. Zbl 1170.20015
Guo, W.; Wang, Y.; Shi, L.
2008
Maximum likelihood computation based on the Fisher scoring and Gauss-Newton quadratic approximations. Zbl 1161.62331
Wang, Yong
2007
Minimum disparity computation via the iteratively reweighted least integrated squares algorithms. Zbl 1445.62045
Wang, Yong
2007
New results of periodic solutions for a Rayleigh equation with a deviating argument. Zbl 1179.34075
Wang, Yong
2009
Solving chiller loading optimization problems using an improved teaching-learning-based optimization algorithm. Zbl 1390.90414
Duan, Pei-Yong; Li, Jun-Qing; Wang, Yong; Sang, Hong-Yan; Jia, Bao-Xian
2018
Stability analysis of recurrent neural networks with random delay and Markovian switching. Zbl 1230.34070
Zhu, Enwen; Wang, Yong; Wang, Yueheng; Zhang, Hanjun; Zou, Jiezhong
2010
A novel localized collocation solver based on Trefftz basis for potential-based inverse electromyography. Zbl 07330175
Xi, Qiang; Fu, Zhuojia; Wu, Wenjie; Wang, Hui; Wang, Yong
2021
Gauss-Bonnet theorems in the BCV spaces and the twisted Heisenberg group. Zbl 1450.53046
Wang, Yong; Wei, Sining
2020
A novel grey Bernoulli model for short-term natural gas consumption forecasting. Zbl 07203999
Wu, Wenqing; Ma, Xin; Zeng, Bo; Lv, Wangyong; Wang, Yong; Li, Wanpeng
2020
Implicit surface reconstruction with radial basis functions via PDEs. Zbl 1464.65237
Liu, Xiao-Yan; Wang, Hui; Chen, C. S.; Wang, Qing; Zhou, Xiaoshuang; Wang, Yong
2020
Improved 3D surface reconstruction via the method of fundamental solutions. Zbl 07366923
Zheng, Hui; Wang, Feng; Chen, C. S.; Lei, M.; Wang, Yong
2020
Application of a novel nonlinear multivariate grey Bernoulli model to predict the tourist income of China. Zbl 1407.62427
Ma, Xin; Liu, Zhibin; Wang, Yong
2019
The novel fractional discrete multivariate grey system model and its applications. Zbl 1464.62389
Ma, Xin; Xie, Mei; Wu, Wenqing; Zeng, Bo; Wang, Yong; Wu, Xinxing
2019
The Boltzmann equation with large-amplitude initial data in bounded domains. Zbl 1406.35216
Duan, Renjun; Wang, Yong
2019
Effects of soft interaction and non-isothermal boundary upon long-time dynamics of rarefied gas. Zbl 1428.35265
Duan, Renjun; Huang, Feimin; Wang, Yong; Zhang, Zhu
2019
Chen inequalities for submanifolds of complex space forms and Sasakian space forms with quarter-symmetric connections. Zbl 1423.53071
Wang, Yong
2019
Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEs. Zbl 07112035
Guan, Hongbo; Wang, Yong; Zhu, Huiqing
2019
Global uniqueness and solvability of tensor variational inequalities. Zbl 1409.90207
Wang, Yong; Huang, Zheng-Hai; Qi, Liqun
2018
Solving chiller loading optimization problems using an improved teaching-learning-based optimization algorithm. Zbl 1390.90414
Duan, Pei-Yong; Li, Jun-Qing; Wang, Yong; Sang, Hong-Yan; Jia, Bao-Xian
2018
Matrix exponential based discriminant locality preserving projections for feature extraction. Zbl 1434.68536
Lu, Gui-Fu; Wang, Yong; Zou, Jian; Wang, Zhongqun
2018
The method of transformed angular basis function for solving the Laplace equation. Zbl 1403.65256
Li, Xinxiang; Oh, Jaeyoun; Wang, Yong; Zhu, Huiqing
2018
Flow modeling of well test analysis for a multiple-fractured horizontal well in triple media carbonate reservoir. Zbl 1401.76145
Wang, Yong; Yi, Xiangyi
2018
Fuzzy uncertainty observer based adaptive dynamic surface control for trajectory tracking of a quadrotor. Zbl 1413.93063
Wang, Ning; Wang, Yong
2018
Global well-posedness of the relativistic Boltzmann equation. Zbl 1406.35218
Wang, Yong
2018
A method to compute the sparse graphs for traveling salesman problem based on frequency quadrilaterals. Zbl 1452.90280
Wang, Yong; Remmel, Jeffrey
2018
Global existence and boundedness in a reaction-diffusion-taxis system with three species. Zbl 1445.35215
Xu, Xue; Wang, Yong
2018
Macroscopic regularity for the Boltzmann equation. Zbl 1438.35304
Huang, Feimin; Wang, Yong
2018
Development of a collision-avoidance vector based control algorithm for automated in-vivo transportation of biological cells. Zbl 1387.93033
Li, Xiaojian; Chen, Shuxun; Liu, Chichi; Cheng, Shuk Han; Wang, Yong; Sun, Dong
2018
Global well-posedness of the Boltzmann equation with large amplitude initial data. Zbl 1367.35097
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2017
The MFS versus the Trefftz method for the Laplace equation in 3D. Zbl 1403.65259
Lv, Hui; Hao, Fang; Wang, Yong; Chen, C. S.
2017
Diffusive wave in the low Mach limit for compressible Navier-Stokes equations. Zbl 1375.35378
Huang, Feimin; Wang, Tian-Yi; Wang, Yong
2017
The nonlinear dynamics based on the nonstandard Hamiltonians. Zbl 1375.70065
Liu, Shixing; Guan, Fang; Wang, Yong; Liu, Chang; Guo, Yongxin
2017
A modified belief rule based model for uncertain nonlinear systems identification. Zbl 1376.68131
Ke, Xiaolu; Ma, Liyao; Wang, Yong
2017
Global uniqueness and solvability for tensor complementarity problems. Zbl 1344.90056
Bai, Xue-Li; Huang, Zheng-Hai; Wang, Yong
2016
Exceptionally regular tensors and tensor complementarity problems. Zbl 1368.90158
Wang, Yong; Huang, Zheng-Hai; Bai, Xue-Li
2016
A pseudorandom number generator based on piecewise logistic map. Zbl 1354.65012
Wang, Yong; Liu, Zhaolong; Ma, Jianbin; He, Haiyuan
2016
Nonparametric estimation of species richness using discrete $$k$$-monotone distributions. Zbl 1468.62036
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2016
Theoretical analysis for blow-up behaviors of differential equations with piecewise constant arguments. Zbl 1410.34184
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2016
Online active learning of decision trees with evidential data. Zbl 1394.68302
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2016
Distributed-observer-based output regulation of heterogeneous nonlinear multi-agent systems. Zbl 1359.93198
Yang, Xi; Huang, Wei; Wang, Yong
2016
Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in three dimensional bounded domain. Zbl 1344.35110
Wang, Yong
2016
A binomial distribution model for the traveling salesman problem based on frequency quadrilaterals. Zbl 1344.05075
Wang, Yong; Remmel, Jeffrey B.
2016
Low-rank matrix factorization with adaptive graph regularizer. Zbl 1408.94443
Lu, Gui-Fu; Wang, Yong; Zou, Jian
2016
Game analysis and benefit allocation in international projects among owner, supervisor and contractor. Zbl 1342.93020
Ding, Hao; Wang, Yong; Guo, Sini; Xu, Xiaofeng; Che, Cheng
2016
Justification of limit for the Boltzmann equation related to Korteweg theory. Zbl 1352.35113
Huang, Feimin; Wang, Yi; Wang, Yong; Yang, Tong
2016
Parameter estimation method for predator-prey model based on particle swarm optimization algorithm. Zbl 1363.92037
Liu, Libin; Liu, Cuiping; Zhang, Yong; Wang, Yong
2016
Fluctuating hydrodynamics methods for dynamic coarse-grained implicit-solvent simulations in LAMMPS. Zbl 1364.65172
Wang, Y.; Sigurdsson, J. K.; Atzberger, P. J.
2016
Simultaneous dimension reduction and adjustment for confounding variation. Zbl 1407.62218
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2016
L1-norm-based principal component analysis with adaptive regularization. Zbl 1414.62216
Lu, Gui-Fu; Zou, Jian; Wang, Yong; Wang, Zhongqun
2016
Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in three-dimensional domains. Zbl 1332.35303
Wang, Yong; Xin, Zhouping; Yong, Yan
2015
An FSI solution technique based on the DSD/SST method and its applications. Zbl 1327.74059
Tian, Fang-Bao; Wang, Yong; Young, John; Lai, Joseph C. S.
2015
Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks. Zbl 1360.35178
Huang, FeiMin; Wang, Yi; Wang, Yong; Yang, Tong
2015
Center problems and limit cycle bifurcations in a class of quasi-homogeneous systems. Zbl 1326.34069
Xiong, Yanqin; Han, Maoan; Wang, Yong
2015
Nonparametric multivariate density estimation using mixtures. Zbl 1331.62279
Wang, Xuxu; Wang, Yong
2015
Positive solutions for Caputo fractional differential equations involving integral boundary conditions. Zbl 1312.34061
Wang, Yong; Yang, Yang
2015
Performance analysis of the $$(1+1)$$ evolutionary algorithm for the multiprocessor scheduling problem. Zbl 1319.68049
Zhou, Yuren; Zhang, Jun; Wang, Yong
2015
A hybrid demon algorithm for the two-dimensional orthogonal strip packing problem. Zbl 1394.68356
Chen, Bili; Wang, Yong; Yang, Shuangyuan
2015
Lipschitz and BMO norm inequalities for the composition operator on differential forms. (Lipschitz and BMO norm inequalities for the composite operator on differential forms.) Zbl 1351.46026
Li, Xuexin; Wang, Yong; Xing, Yuming
2015
RETRACTED: Some results for periodic solutions of a kind of Liénard equation. Zbl 1321.34056
Wang, Yong; Yi, Xiangyi
2015
An improved turbulence model for predicting unsteady cavitating flows in centrifugal pump. Zbl 1356.76137
Wang, Jian; Wang, Yong; Liu, Houlin; Huang, Haoqin; Jiang, Linglin
2015
Single sign-on under quantum cryptography. Zbl 1284.81086
Dai, Guiping; Wang, Yong
2014
Norm comparison estimates for the composite operator. Zbl 06281156
Li, Xuexin; Wang, Yong; Xing, Yuming
2014
Optimal pricing, production, and sales policies for new product under supply constraint. Zbl 1303.93191
Yan, Xiaoming; Liu, Ke; Wang, Yong
2014
Numerical solutions comparison for interval linear programming problems based on coverage and validity rates. Zbl 1427.65094
Lu, H. W.; Cao, M. F.; Wang, Y.; Fan, X.; He, L.
2014
Positive solutions for a high-order semipositone fractional differential equation with integral boundary conditions. Zbl 1296.34040
Wang, Yong; Yang, Yang
2014
Multiply warped products with a semisymmetric metric connection. Zbl 07022994
Wang, Yong
2014
Smash products, separable extensions and a Morita context over Hopf algebroids. Zbl 1302.16029
Wang, Yong; Guo, Guangquan
2014
An algorithm for a class of nonlinear complementarity problems with non-Lipschitzian functions. Zbl 1291.65179
Wang, Yong; Zhao, Jian-Xun
2014
Least squares estimation of a $$k$$-monotone density function. Zbl 06983938
Chee, Chew-Seng; Wang, Yong
2014
Computationally efficient banding of large covariance matrices for ordered data and connections to banding the inverse Cholesky factor. Zbl 1292.62082
Wang, Y.; Daniels, M. J.
2014
Application of imperialist competitive algorithm on solving the traveling salesman problem. Zbl 1461.90126
Xu, Shuhui; Wang, Yong; Huang, Aiqin
2014
A Kastler-Kalau-Walze type theorem and the spectral action for perturbations of Dirac operators on manifolds with boundary. Zbl 07022738
Wang, Yong
2014
Serrin-type blowup criterion for full compressible Navier-Stokes system. Zbl 1260.35114
Huang, Xiangdi; Li, Jing; Wang, Yong
2013
The limit of the Boltzmann equation to the Euler equations for Riemann problems. Zbl 1367.76055
Huang, Feimin; Wang, Yi; Wang, Yong; Yang, Tong
2013
Estimation of finite mixtures with symmetric components. Zbl 1322.62013
Chee, Chew-Seng; Wang, Yong
2013
Lyapunov function construction for nonlinear stochastic dynamical systems. Zbl 1284.93207
Ling, Quan; Jin, Xiao Ling; Wang, Y.; Li, H. F.; Huang, Zhi Long
2013
Minimum quadratic distance density estimation using nonparametric mixtures. Zbl 1365.62125
Chee, Chew-Seng; Wang, Yong
2013
An improved ($$\mu+\lambda$$)-constrained differential evolution for constrained optimization. Zbl 1293.68228
Jia, Guanbo; Wang, Yong; Cai, Zixing; Jin, Yaochu
2013
Positive solutions for a system of fractional integral boundary value problem. Zbl 1294.34022
Wang, Yong
2013
The Dirichlet problem of nonhomogeneous $$\mathcal A$$-harmonic equations in unbounded open sets and some estimates. Zbl 1344.35013
Li, Guanfeng; Wang, Yong; Bao, Gejun; Wang, Tingting
2013
Bayesian modeling of the dependence in longitudinal data via partial autocorrelations and marginal variances. Zbl 1277.62088
Wang, Y.; Daniels, M. J.
2013
Long-time behavior of solutions to the bipolar hydrodynamic model of semiconductors with boundary effect. Zbl 1248.35020
Huang, Feimin; Mei, Ming; Wang, Yong; Yang, Tong
2012
Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Zbl 1253.35028
Huang, Rui; Mei, Ming; Wang, Yong
2012
Advantages of the enhanced opposite direction searching algorithm for computing the centroid of an interval type-2 fuzzy set. Zbl 1303.68121
Hu, Huaizhong; Wang, Yong; Cai, Yuanli
2012
A combinatorial model and algorithm for globally searching community structure in complex networks. Zbl 1245.90013
Zhang, Xiang-Sun; Li, Zhenping; Wang, Rui-Sheng; Wang, Yong
2012
Density estimation using non-parametric and semi-parametric mixtures. Zbl 1420.62164
Wang, Yong; Chee, Chew-Seng
2012
Reliability and covariance estimation of weighted $$k$$-out-of-$$n$$ multi-state systems. Zbl 1253.90095
Wang, Yong; Li, Lin; Huang, Shuhong; Chang, Qing
2012
A full-Newton step non-interior continuation algorithm for a class of complementarity problems. Zbl 1241.65060
Zhao, Jian-Xun; Wang, Yong
2012
Incremental complete LDA for face recognition. Zbl 1236.68217
Lu, Gui-Fu; Zou, Jian; Wang, Yong
2012
On the convergence rate of vanishing viscosity approximations for nonlinear hyperbolic systems. Zbl 1264.35017
Bressan, Alberto; Huang, Feimin; Wang, Yong; Yang, Tong
2012
Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Zbl 1264.35035
Mei, Ming; Wang, Yong
2012
Inequalities for the composition of Green’s operator and the potential operator. Zbl 1281.35035
Dai, Zhimin; Xing, Yuming; Ding, Shusen; Wang, Yong
2012
Incremental learning of discriminant common vectors for feature extraction. Zbl 1277.68237
Lu, Gui-Fu; Zou, Jian; Wang, Yong
2012
Novel results on periodic solutions of Liénard type $$p$$-Laplacian equation. Zbl 1257.34028
Wang, Yong; Zhang, Liehui
2012
Tree models for difference and change detection in a complex environment. Zbl 1254.62068
Wang, Yong; Ziedins, Ilze; Holmes, Mark; Challands, Neil
2012
Large time behavior of solutions to $$n$$-dimensional bipolar hydrodynamic models for semiconductors. Zbl 1228.35053
Huang, Feimin; Mei, Ming; Wang, Yong
2011
Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors. Zbl 1227.35063
Huang, Feimin; Mei, Ming; Wang, Yong; Yu, Huimin
2011
On multi-dimensional sonic-subsonic flow. Zbl 1265.76039
Huang, Feimin; Wang, Tianyi; Wang, Yong
2011
Remark on stability of traveling waves for nonlocal Fisher-KPP equations. Zbl 1337.35075
Mei, Ming; Wang, Yong
2011
Asymptotic convergence to planar stationary waves for multi-dimensional unipolar hydrodynamic model of semiconductors. Zbl 1228.35177
Huang, Feimin; Mei, Ming; Wang, Yong; Yu, Huimin
2011
Generalized Mahalanobis depth in the reproducing kernel Hilbert space. Zbl 1434.62082
Hu, Yonggang; Wang, Yong; Wu, Yi; Li, Qiang; Hou, Chenping
2011
The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras. Zbl 1255.16037
Wang, Yong; Zhang, Liangyun
2011
$$H_\infty$$ control of networked control systems based on event-time-driven model. Zbl 1260.93051
Wang, Yong; Wang, Wei; Liu, Guo-Ping
2011
Prediction of protein interaction hot spots using rough set-based multiple criteria linear programming. Zbl 1307.92075
Chen, Ruoying; Zhang, Zhiwang; Wu, Di; Zhang, Peng; Zhang, Xinyang; Wang, Yong; Shi, Yong
2011
Algorithm of $$\varepsilon$$-SVR based on a large-scale sample set: step-by-step search. Zbl 1219.90211
Zeng, Shaohua; Tang, Y. Y.; Wei, Yan; Wang, Yong
2011
On parameter identifiability of MIMO radar with waveform diversity. Zbl 1217.94069
Wang, Hongyan; Liao, Guisheng; Wang, Yong; Liu, Xiangyang
2011
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#### Cited by 200 Authors
20 Mei, Ming 14 Zhang, Kaijun 13 Huang, Feimin 13 Li, Yeping 11 Wang, Yong 10 Wang, Yi 9 Du, Lili 9 Duan, Renjun 9 Zhang, Guobao 6 Yu, Huimin 5 Liu, Shuangqian 5 Wang, Teng 5 Wang, Tianyi 5 Xiang, Wei 5 Zhang, Guojing 4 Chen, Chao 4 Cheng, Jianfeng 4 Ma, Xin 4 Tan, Zhong 4 Xie, Chunjing 4 Xin, Zhouping 4 Yin, Jingxue 4 Zeng, Bo 3 Chen, Gui-Qiang G. 3 Hu, Haifeng 3 Li, Jingyu 3 Li, Wan-Tong 3 Ma, Zhaohai 3 Shi, Xiaoding 3 Vasseur, Alexis F. 3 Wang, Yong 3 Wu, Wenqing 3 Wu, Zhigang 3 Yang, Tong 3 Yu, Zhixian 3 Zhang, Zhu 2 Chen, Liang 2 Di Michele, Federica 2 Ding, Song 2 Dong, Fang-Di 2 Feng, Zhaosheng 2 Guo, Ran 2 Hsu, Cheng-Hsiung 2 Huang, Rui 2 Ji, Shanming 2 Jiang, Yicheng 2 Kang, Moon-Jin 2 Li, Linan 2 Li, Wanpeng 2 Li, Yan 2 Liao, Jie 2 Liu, Gui Rong 2 Liu, Hairong 2 Luo, Tao 2 Rubino, Bruno 2 Sampalmieri, Rosella 2 Tian, Ge 2 Trofimchuk, Sergei I. 2 Wang, Chunpeng 2 Wang, Dehua 2 Wang, Guanfa 2 Wang, Weike 2 Wu, Shiliang 2 Wu, Xiaochun 2 Wu, Xin 2 Xie, Naiming 2 Xu, Tianyuan 2 Yan, Rui 2 Yang, Dongcheng 2 Yang, Tzi-Sheng 2 Yang, Xiongfeng 2 Yang, Yunrui 2 Yuan, Rong 2 Zhang, Qifeng 2 Zhang, Yinglong 2 Zhang, Yongqian 2 Zhao, Xiao-Qiang 2 Zhong, Hua 2 Zhou, Mingjun 1 Bressan, Alberto 1 Cai, Hong 1 Cao, Xinyue 1 Chen, Hongxu 1 Chen, Yan 1 Chen, Yazhou 1 Chen, Zhengzheng 1 Cheng, Cuiping 1 Chern, I-Liang 1 Chiri, Maria Teresa 1 Choe, Chunhyok 1 Deng, Xuemei 1 Dong, Wenhua 1 Duan, Ben 1 Duan, Huiming 1 Gong, Guiqiong 1 Guo, Hongxia 1 Guo, Zhihua 1 He, Qiaolin 1 Hong, Hakho 1 Hu, Jing ...and 100 more Authors
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#### Cited in 53 Serials
22 Journal of Differential Equations 15 SIAM Journal on Mathematical Analysis 13 Journal of Mathematical Analysis and Applications 11 Archive for Rational Mechanics and Analysis 10 Nonlinear Analysis. Real World Applications 7 ZAMP. Zeitschrift für angewandte Mathematik und Physik 5 Communications in Nonlinear Science and Numerical Simulation 5 Communications on Pure and Applied Analysis 5 Kinetic and Related Models 4 Applicable Analysis 4 Journal of Mathematical Physics 4 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 4 Applied Mathematical Modelling 4 Discrete and Continuous Dynamical Systems 3 Communications in Mathematical Physics 3 Mathematical Methods in the Applied Sciences 3 Applied Mathematics and Computation 3 Discrete and Continuous Dynamical Systems. Series B 2 Advances in Mathematics 2 Chinese Annals of Mathematics. Series B 2 Journal of Dynamics and Differential Equations 2 Discrete Dynamics in Nature and Society 2 Journal of Dynamical and Control Systems 2 Acta Mathematica Scientia. Series B. (English Edition) 2 Mathematical Biosciences and Engineering 2 Science China. Mathematics 1 Journal of Statistical Physics 1 Nonlinearity 1 Journal of Computational and Applied Mathematics 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Proceedings of the American Mathematical Society 1 Quarterly of Applied Mathematics 1 Transactions of the American Mathematical Society 1 Acta Mathematicae Applicatae Sinica. English Series 1 Japan Journal of Industrial and Applied Mathematics 1 Journal of Nonlinear Science 1 Potential Analysis 1 Calculus of Variations and Partial Differential Equations 1 Computational and Applied Mathematics 1 Turkish Journal of Mathematics 1 Complexity 1 Mathematics and Mechanics of Solids 1 Abstract and Applied Analysis 1 Taiwanese Journal of Mathematics 1 Journal of Mathematical Fluid Mechanics 1 International Journal of Applied Mathematics and Computer Science 1 Nonlinear Analysis. Modelling and Control 1 Analysis and Applications (Singapore) 1 International Journal of Geometric Methods in Modern Physics 1 International Journal of Numerical Analysis and Modeling 1 Boundary Value Problems 1 Journal of Applied Analysis and Computation 1 Electronic Research Archive
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#### Cited in 24 Fields
146 Partial differential equations (35-XX) 89 Fluid mechanics (76-XX) 27 Statistical mechanics, structure of matter (82-XX) 21 Biology and other natural sciences (92-XX) 9 Statistics (62-XX) 6 Optics, electromagnetic theory (78-XX) 5 Ordinary differential equations (34-XX) 5 Systems theory; control (93-XX) 3 Mechanics of deformable solids (74-XX) 3 Operations research, mathematical programming (90-XX) 3 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Real functions (26-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Classical thermodynamics, heat transfer (80-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Difference and functional equations (39-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Probability theory and stochastic processes (60-XX) 1 Numerical analysis (65-XX) 1 Computer science (68-XX) 1 Mechanics of particles and systems (70-XX) 1 Quantum theory (81-XX)
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2021-10-17 01:08:31
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https://en.wikiversity.org/wiki/Talk:Making_sense_of_quantum_mechanics
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# Talk:Making sense of quantum mechanics
## Lessons on the principles of Quantum Mechanics
Wikiversity has not yet lessons on the Principles of Quantum Mechanics. There is a Study guide:Quantum mechanics I. I think we need some lessons on the first principles and the basic postulates presented in a logical sequence, kind of:
• Quantum Mechanics is based on the fact that physical systems are represented by vectors (also called state vectors or kets, as introduced by Dirac), contrary to Classical Mechanics where systems are represented by collection of points on which forces act. This is the central postulate of Quantum Mechanics.
• As systems are represented by vectors, Quantum Mechanics focuses on the orientation (phase, angle...) of those vectors, and not on the classical properties as position or velocity. We may then deduce the law of motion for quantum systems: time evolution is an evolution of the phase of the vector, which gives the time-dependent Schrödinger equation. In conventional presentations of Quantum Mechanics, this time evolution law is generally postulated independently.
• We may then introduce the observational aspects. Observing something on a quantum system means that it interacts with another quantum system. The outcome of an interaction by two systems represented by vectors (linear segments) is undetermined depending on the place of interaction: principle of indeterminacy (Heisenberg). An interaction means that something operates on the vector: observational postulate.
• and so on...
I found that the french lessons on the postulates of Quantum Mechanics are well presented. Maybe I could take inspiration of their templates? Arjen Dijksman 10:32, 15 September 2007 (UTC)
I think w:Bra-ket notation is a good example of Wikipedia resources for physics. That encyclopedia article is written by people who know the topic and for other people who know the topic. The article does not adequately explain the jargon that is used and does not offer a learning path for helping non-experts towards understanding the topic of the article. I wonder if "first principles" (above, on this page) is meant to include details like explaining jargon (such as "kets"), or not. --JWSchmidt 15:58, 15 September 2007 (UTC)
Yes, a first principles lesson written by contributors of this Making sense of quantum mechanics Project would be meant to either avoid specialized jargon and notations either include some intuitive explanation of it. Kets would for example simply be presented as rotating arrows or rods or needles or baseball bats or any ordinary objects that have an extent and a direction in some space. Would that make sense? Arjen Dijksman 19:52, 15 September 2007 (UTC)
That sounds good....analogies are useful for introducing people to new ideas. Another problem I have with many presentations of mathematical physics is that it seems like extra points are awarded to authors for concocting the most concise and general descriptions of topics, but such descriptions are of little use to people trying to learn the subject. I started thinking about this issue when I saw "physical systems are represented by vectors". I wished there was a link right there leading to several specific examples. --JWSchmidt 20:14, 15 September 2007 (UTC)
We need indeed some figures with arrows illustrating physical systems. Arjen Dijksman 21:37, 15 September 2007 (UTC)
I'd be glad to try making some images if I had an idea what is needed. This might be a good time to start making use of the Digital media workshop. --JWSchmidt 22:00, 15 September 2007 (UTC)
++== Time evolution of state vector ==++
Here is a sketch for an image that could illustrate the Schrödinger equation (I am horrible in graphics:-( ) Arjen Dijksman 23:00, 15 September 2007 (UTC)
Time evolution of a state vector
I could make a new version of this figure, but I do not understand the figure. --JWSchmidt 23:39, 15 September 2007 (UTC)
This figure could illustrate the fact that the vector difference ${\displaystyle |\psi (t+\Delta t)\rangle -|\psi (t)\rangle }$ between a needle at instant ${\displaystyle t}$ and the same needle at instant ${\displaystyle t+\Delta t}$ is perpendicular to the state vector ${\displaystyle |\psi (t)\rangle }$ at instant ${\displaystyle t}$ and proportional to the angular velocity ${\displaystyle \omega }$ of the needle, for very small ${\displaystyle \Delta t}$. In equation form, this may be stated as:
${\displaystyle (|\psi (t+\Delta t)\rangle -|\psi (t)\rangle )\ \ \bot \ \ |\psi (t)\rangle \ \ }$ for ${\displaystyle \ \ \Delta t\rightarrow \ 0}$
and
${\displaystyle (|\psi (t+\Delta t)\rangle -|\psi (t)\rangle )\ \ =\ \ i\ \omega \ |\psi (t)\rangle \ \Delta t\ }$ for ${\displaystyle \ \ \Delta t\rightarrow \ 0}$
provided that the time axis points towards us out of the plane of the screen. Maybe this could be stated in a better understandable way? Unfortunately there are some symbols needed. Arjen Dijksman 05:23, 16 September 2007 (UTC)
I learned the basics of vectors, inner products and quantum mechanics when I was in college, but I was never taught the notation that you are using. --JWSchmidt 20:29, 18 September 2007 (UTC)
Yes, this is Dirac's notation. I'll look for different manners to describe in what ${\displaystyle {\vec {\psi }}}$ and ${\displaystyle |\psi \rangle }$ differ. ${\displaystyle {\vec {\psi }}}$ is an ordinary static vector in 3D. ${\displaystyle |\psi \rangle }$ is also a vector (=arrow) but with physical properties attached to it: a state vector (or ket). This makes it suited to describe a physical linear object. It evolves or interacts with the physical environment. There is a physical relationship between ${\displaystyle |\psi (x_{1})\rangle }$ and ${\displaystyle |\psi (x_{2})\rangle }$, ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ being any physical observable. Describing this relationship for different conditions forms the object of Quantum Mechanics. Arjen Dijksman 22:10, 18 September 2007 (UTC)
• What do you think about b:Quantum Mechanics and b:This quantum world? --JWSchmidt 22:27, 18 September 2007 (UTC)
• Both texts are fine as they guide us through various aspects of Quantum Physics. The first text needs to be completed. I did not yet manage to read b:This quantum world at full extent. I appreciate its illustrations that help to visualize now and then the presented concepts. I also appreciate the fact that it deals with interpretational issues like the Bohmian view and Bell's theorem. Both texts present Quantum Mechanics from the viewpoint of its historical development. The first introduces QM with wave mechanics and the second explaining how classical serious illnesses require drastic remedies. That's the conventional approach in which QM is presented. From experience, you need to read dozens (if not dozens of dozens) of these texts combined with experimental and theoretical practice before gaining some intuitive comprehension of QM. In this way, it is very difficult to make QM accessible to a larger public. That's why I am looking for intuitively clear approaches to Quantum Mechanics, starting with the simplest elementary bricks, like a single photon, or electron, before extending to the behaviour of quantum wave mechanics or quantum fields. Arjen Dijksman 07:46, 22 September 2007 (UTC)
We could be a bit more general for the evolution law of state vectors, focusing only on the angle (the phase ${\displaystyle \phi }$). We could write a general differential equation:
${\displaystyle |\psi (\phi +\Delta \phi )\rangle -|\psi (\phi )\rangle \ \ =\ \ i\ \ |\psi (\phi )\rangle \ \Delta \phi \ }$ for ${\displaystyle \ \ \Delta \phi \rightarrow \ 0}$
where ${\displaystyle \Delta \phi }$ could be understood as ${\displaystyle \omega \Delta t}$, but also ${\displaystyle k\Delta x}$ for situations where the angle changes when we shift the arrow a bit in the x-direction (k being the wavenumber, i.e. the angle through which the arrow would rotate if the arrow is shifted over unit length), or even some ${\displaystyle t_{0}\Delta \omega }$, if we focus on a change of angular velocity at a given instant ${\displaystyle t_{0}}$. Arjen Dijksman 16:00, 6 October 2007 (UTC)
I made the page Template:Quantum mechanics and you can add it to pages using: {{Quantum mechanics}}. --JWSchmidt 19:31, 18 September 2007 (UTC)
## Operators operate an ordinary arrows giving other ordinary arrows
I have started to write the text for the third principle about operators and the way they act on state vectors. I have some difficulty to describe it without reference to the abstract formalism, while it seems relatively ordinary: you have an arrow, you rotate it in some way (constrained or free) or you subtract it from another arrow. You always obtain another arrow. These arrows relate one to another with proportionality factors depending on the physical properties of the system. The relation is represented by mathematical formalism where appear complex exponentials and the observables (speed, spinning direction, energy, potentials...). So the behavior is intuitive but the formalism makes it abstract. Anyone has some ideas on the subject? Arjen Dijksman 17:43, 6 October 2007 (UTC)
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2017-03-23 23:59:41
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https://rknotdocs.readthedocs.io/en/latest/concepts.html
|
# Key Concepts¶
RKnot builds a simulation across four dimensions of global properties:
• Time
• The fundamental unit of time is a tick.
• Each iteration of the simulation is one tick. During a tick, the following occurs:
• subjects can move to new locatioccns
• subjects can contact other subjects
• attributes of the subjects can change
• Many of the fundamental properties of a virus are measured in days. RKnot translates daily inputs into ticks.
• Currently, only one tick per day is supported. The goal is to support any number of ticks during a day.
• Space
• Subjects interact in an two-dimensional environment called the Grid.
• The Grid must be a square. The Grid size can be passed manually or it can be determined automatically for a specified density level.
• Each pair of xy coordinates in the Grid is a location.
• A contact occurs when an infected subject and a susceptible subject occupy the same location at the same tick.
• There is no limit to the number of subjects that can occupy a single location at the same time.
• Subjects move through the Grid according to user-specified mover functions. These functions typically incorporate a degree of randomness.
• A subject can also move by attending an Event.
• Portions of the Grid may be restricted by Boxes and/or Gates.
• Subjects
• subjects (also referred to as “dots”) are the analog of people in the simulation.
• subjects carry many attributes through the life of the simulation that are updated and changed as required see Dot Matrix).
• Virus
• the user may pass several characteristics fundamental to the simulated virus. RKnot may infer others. Virus characteristics include:
• $$R_0$$
• Duration of Infection
• Transmission Risk
• Duration of Immunity
• Infection Fatality Rate
## The Sim¶
The Sim object is the user interface for the RKnot simulation package and acts as a thin wrapper for the Server and Worker classes of Ray actors that form the core of a simulation.
A Sim object is instantiated with pre-defined characteristics of the space, the subjects, and the virus.
For demonstration purposes, a quick default simulation can be run by simply providing a few parameters.
from rknot import Sim, Chart
params = {'square': 4, 'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365}
group = {'n': 2, 'n_inf': 1}
sim = Sim(groups=group, **params)
run is the main method of the Sim object. run iterates through each tick in the simulation. Currently, one day == one tick.
sim.run()
sim.run() does not return any values, but it does update various attributes of the sim object. After calling run, you can then pass sim to the Chart object, which will generate an animation of the simulation across time.
chart = Chart(sim, dotsize=2000, interval=200, show_intro=False, use_init_func=False)
chart.to_html5_video()
The animation is split in 3 sections: * Interactions * the visual representation of subjects in the Grid. Each marker is a subject and each cross-section of gridlines is a point (for larger grids the lines are removed). * Details * provides several on-the-run statistics including Effective Reproduction Number, total fatalities, and fatalities by group. * Infections * shows the change in infection level over each day, showing both current infection level and total penetration (“Ever” in the legend)
The animation is built on AxesSubPlot components that can be arranged in any fashion desired, including a handful of preset layouts. see Chart for more details.
As per the animation above, the default simulation is of a single infected subject, moving across a 4x4 two-dimensional space according to the equal mover function. The subject is equally likely to move to any location in the Grid on any tick.
## Subjects¶
### Dots¶
Dots are subjects/people in the simulation space. A subject has many attributes that are adjusted over time, including:
• which Group it belongs to
• if it is alive
• if it is infected
• if it is susceptible
• its location
• any restricted areas that apply to it (see Boxes and Gates)
• if infected, when it will recover (or when it will succumb)
• if recovered, when it will again become susceptible
• its fatality rate
• its mover function
see Dot Matrix for a more fulsome discussion.
### Groups¶
Dots are the fundamental subjects of the simulation, but dots can only be created via a Group object.
To create our group objects, we can pass a list of dictionaries to the groups parameter of Sim. The dictionaries correspond to the attributes of the group, which in turn correspond to the attributes of its constituent dots at initiation.
To create a group, you need only provide two parameters:
• n, population size of the group at initiation
• n_inf, number of infected subjects in the group at initiation
If only one group is being provided, you can pass a dictionary. With multiple groups, pass an iterable of dicts.
params = {'square': 10, 'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365}
group = dict(n=2, n_inf=1)
sim = Sim(groups=group, **params)
sim.run()
chart = Chart(
sim, figsize=(16,8), dotsize=2000, interval=200,
layout='dots_only', show_intro=False
)
chart.to_html5_video()
Below we see this structure creates a 10x10 Grid with two subjects, only one of which is infected at the outset.
There are many other parameters and customizations that can provided:
• name
• if not provided, Sim will create one
• box & box_is_gate
• mover
• tmf
• susf
• susceptiblity factor; the fraction of subjects in a group that will be made susceptible to the virus at initiation
• the inverse of susf ($$1/{susf}$$) is the number of subjects in a group that already have immunity.
• ifr
• infection fatality rate; or the likelihood that an infection will be fatal
These can again be passed as a dictionary of a single group:
group = dict(
name='main', n=2, n_inf=1, mover='equal',
tmf=1.25, susf=.75, ifr=0.005
)
sim = Sim(groups=group, **params)
sim.run()
chart = Chart(
sim, figsize=(16,8), dotsize=2000, interval=200,
layout='dots_only', show_intro=False
)
chart.to_html5_video()
Or as an iterable of dictionaries. Each group is assigned a unique marker in the animation.
group1 = dict(name='1', n=1, n_inf=1, mover='local', tmf=1.25, susf=.75, ifr=0.005)
group2 = dict(name='2', n=1, n_inf=0, mover='equal', tmf=0.75, susf=0.95, ifr=0.05)
group3 = dict(name='3', n=1, n_inf=0, mover='equal', tmf=0.25, susf=0.5, ifr=0.4)
groups = [group1, group2, group3]
sim = Sim(groups=groups, **params)
sim.run()
chart = Chart(
sim, figsize=(16,8), dotsize=2000, interval=200,
layout='dots_only', show_intro=False
)
chart.to_html5_video()
## The Grid¶
All interactions in an RKnot simulation take place inside the Grid. The grid is a Grid object, which in turn is a sub-classed numpy array with some additional features.
The Grid size can be determined by passing the square or density parameters. Each density accepts either a str or a float. The float value is a specific desired subject per location and a str must be on of the three categories below.
Available str values for density and their corresponding densities are:
low: 0.2
med: 1
high: 10
If we set density=med, the Grid will be set such that the density is 1 subject per location. For a group of 100 subjects, that will result in a 10x10 grid. We can see these attributes by passing details=True.
[9]:
params = {'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365}
group = dict(name='main', n=100, n_inf=1)
sim = Sim(groups=group, density='med', details=True, **params)
---------------------------------------------------------------------------------
| SIM DETAILS |
|-------------------------------------------------------------------------------|
| Boundary| [ 1 10 1 10]| Locations| 100|
|-------------------|-------------------|-------------------|-------------------|
| Population| 100| Density| 1.0|
|-------------------|-------------------|-------------------|-------------------|
| Contact Rate| 1.01| | |
|-------------------|-------------------|-------------------|-------------------|
sim.run()
chart = Chart(
sim, figsize=(16,8), layout='dots_only',
show_intro=False, use_init_func=False
)
chart.to_html5_video()
For smaller populations, the density level can only be approximated. RKnot defaults to rounding up to the nearest square value.
You can also pass a float value to density in order to create a specified density. Here, we set density=3.5
[13]:
group1 = dict(name='1', n=1000, n_inf=1)
group2 = dict(name='2', n=20, n_inf=20)
groups = [group1, group2]
sim = Sim(groups=groups, density=3.5, details=True, **params)
---------------------------------------------------------------------------------
| SIM DETAILS |
|-------------------------------------------------------------------------------|
| Boundary| [ 1 18 1 18]| Locations| 324|
|-------------------|-------------------|-------------------|-------------------|
| Population| 1,020| Density| 3.15|
|-------------------|-------------------|-------------------|-------------------|
| Contact Rate| 3.16| | |
|-------------------|-------------------|-------------------|-------------------|
sim.run()
chart = Chart(sim, figsize=(16,8), layout='dots_only', show_intro=False)
chart.to_html5_video()
### Mover Functions¶
When a subject changes locations, this is called a ‘move’. A move is completed during a tick and the movement of a subject on any tick is governed by its mover function. Movers select a location according to a pre-defined probability distribution, so the general movement pattern of a dot can be pre-determined, but any one movement occurs randomly.
There are currently 5 mover functions. Their respective definitions, along with examples of their movement are provided below. A float value is also accepted which is used as the p-value in a geometric movement pattern.
###### Equal
The subject is equally likely to move to any location.
###### Local
The subject has a strong bias towards dots only in its immediate vicinity.
###### Traveller
The subject commonly moves to locations far across the Grid.
###### Quarantine
The subject has a strong bias towards not moving, with only some movement occuring.
###### Social
The subject moves mostly within its vicinty but also to other more medium distance locations.
In addition to specifying a mover function, the user can also simply specify a float value between 0 and 1.
This value corresponds to a p-value used in a geometric distrubtion. The relationship between p-value and movement is shown below.
The higher the p-value, the greater the bias towards shorter moves. Increasing p-value, all else equal, should decrease the number of contacts in a sim. This is explored further here.
Below we compare the movement patterns of two subjects with very different p-value.
params = {'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365}
group1 = dict(n=1, n_inf=1, mover=.25)
group2 = dict(n=1, n_inf=1, mover=.95)
groups = [group1, group2]
sim = Sim(groups=groups, square=10, **params)
sim.run()
chart = Chart(sim, figsize=(16,8), layout='dots_only', show_intro=False)
chart.to_html5_video()
## Boxes and Gates¶
The movement of a subject across the Grid can be restricted by two concepts known as Boxes and Gates. These concepts are designed to mimick certain funcitonal or perceived boundaries between groups, such as international borders or closed-access communities like assisted-living facilities.
The distinction between boxes and gates is simple:
• Subjects cannot exit Boxes
• Subjects cannot enter Gates
### Boxes¶
A box is a $$m*n$$ subset of locations within the Grid that a subject(s) cannot leave.
The locations are specified by passing a four element iterable that specifies the coordinates of the “four corners” of the box according to [$$x_0$$, $$x_1$$, $$y_0$$, $$y_1$$]
So passing:
box = [2,6,3,9]
creates a box with the four corners:
(2,3) (2,9) (6,3) (6,9)
and a total of 35 locations.
Currently, a box can only be specified by
1. passing the box parameter as group keyword
2. by passing a vbox <#VBoxes>__.
Every dot in the group can only move within the box, regardless of the size of the Grid.
group1 = dict(name='1', n=2, n_inf=1, box=[1,3,2,4])
sim = Sim(groups=group1, square=10, **params)
sim.run()
chart = Chart(
sim, figsize=(16,8), dotsize=2000,
layout='dots_only', show_intro=False, use_init_func=False
)
chart.to_html5_video()
A group can only have one box and each group can have its own box.
group1 = dict(name='1', n=2, n_inf=1, box=[1,3,2,4])
group2 = dict(name='2', n=2, n_inf=0, box=[6,9,6,10])
groups = [group1, group2]
sim = Sim(groups=groups, square=10, **params)
sim.run()
chart = Chart(
sim, figsize=(16,8), dotsize=2000,
layout='dots_only', show_intro=False, use_init_func=False
)
chart.to_html5_video()
Remember that a box only restricts the subjects in that group from leaving the space. Other dots not assigned to that box can move into the space without restriction.
group1 = dict(name='1', n=5, n_inf=1, box=[1,3,1,3])
group2 = dict(name='2', n=5, n_inf=0)
groups = [group1, group2]
sim = Sim(groups=groups, square=10, **params)
sim.run()
chart = Chart(sim, figsize=(16,8), dotsize=2000, layout='dots_only', show_intro=False)
chart.to_html5_video()
### Gates¶
Gates are the inverse of boxes. A gate is an area that subjects cannot enter.
Gates are a Gate object, which are a subclass of the Box class (in turn a subclass of ndarray), and they are created via the same 4 element iterable. For now, a gate can only be created by passing the box_is_gated=True flag as a keyword in a group dictionary, or by specifying a vbox.
Using the previous example, we can see that group2 dots can no longer enter the group1 box.
group1 = dict(name='1', n=5, n_inf=1, box=[1,3,1,3], box_is_gated=True)
group2 = dict(name='2', n=5, n_inf=0)
groups = [group1, group2]
sim = Sim(groups=groups, square=10, **params)
sim.run()
chart = Chart(
sim, figsize=(16,8), dotsize=2000, layout='dots_only', show_intro=False,
use_init_func=False
)
chart.to_html5_video()
This structure allows for intricate movement patterns. We show isolated groups below.
We will also provide the show_restricted=True flag, which will outline the boxes and gates for us. It will also the label the restricted area with the name of the group used to form the it.
group1 = dict(name='1', n=50, n_inf=5, box=[1,5,1,20], box_is_gated=True)
group2 = dict(name='2', n=50, n_inf=5, box=[6,25,3,10], box_is_gated=True)
group3 = dict(name='3', n=50, n_inf=5, box=[10,21,16,22], box_is_gated=True)
group4 = dict(name='4', n=50, n_inf=5, box=[2,15,23,25], box_is_gated=True)
groups = [group1, group2, group3, group4]
sim = Sim(groups=groups, square=25, **params)
sim.run()
chart = Chart(sim,
figsize=(16,8), layout='dots_only', show_intro=False, use_init_func=False,
show_restricted=True,
)
chart.to_html5_video()
And here some isolated and some free moving.
group1 = dict(name='1', n=50, n_inf=5, box=[1,5,1,5], box_is_gated=True)
group2 = dict(name='2', n=50, n_inf=5, box=[14,19,14,19], box_is_gated=True)
group3 = dict(name='4', n=10, n_inf=5)
group4 = dict(name='4', n=10, n_inf=5)
groups = [group1, group2, group3, group4]
sim = Sim(groups=groups, square=25, **params)
sim.run()
chart = Chart(
sim, figsize=(16,8), layout='dots_only',
show_intro=False, use_init_func=False
)
chart.to_html5_video()
### VBoxes¶
A VBox is a vacant area of the Grid, meaning there are no subjects inside the box at the initiation of the Sim and that no subjects can enter the VBox except via Travel events.
VBoxes can be used to customize contact patterns, as done in the Dynamic Transmission Risk simulations. They can also be used to mimick areas that people typically only visit, rather than reside in, such as hospitals, sports arenas, office buildings, etc.
VBoxes are simply box objects and can be created by passing the vboxes parameter to Sim. VBoxes are always setup with a corresponding gate.
*IMPORANT*: A VBox is $$\underline{\text{not}}$$ included in the density calculation of the grid size.
If we pass an integer, Sim will create a VBox with the value corresponding to the number of locations in the VBox. The VBox will be placed in the top-left corner of the Grid.
from rknot import Sim
vbox = 4
groups = [
dict(n=10, n_inf=1, mover='social'),
dict(n=10, n_inf=1, mover='local')
]
params = {'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365, 'vboxes': vbox}
sim = Sim(groups=groups, **params)
sim.run(dotlog=True)
chart = Chart(
sim, figsize=(16,8), layout='dots_only',
show_intro=False, use_init_func=False
)
chart.to_html5_video()
We can also pass a boundary as a 4 item iterable.
from rknot import Sim
vbox = [1,3,1,3]
groups = [
dict(n=10, n_inf=1, mover='social'),
dict(n=10, n_inf=1, mover='local')
]
params = {'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365, 'vboxes': vbox}
sim = Sim(groups=groups, **params)
sim.run(dotlog=True)
chart = Chart(
sim, figsize=(16,8), layout='dots_only',
show_intro=False, use_init_func=False
)
chart.to_html5_video()
We can pass a dictionary and include the label keyword to indicate a name for the VBox.
We’ve included a couple Travel objects to show how the VBox can be accessed. Simply assign the index of the vbox as a parameter to Travel and the Sim will determine the location automatically.
from rknot import Sim
from rknot.events import Travel
vbox = {'box': [1,3,1,3], 'label': 'Hospital'}
groups = [
dict(n=10, n_inf=1, mover='social'),
dict(n=10, n_inf=1, mover='local')
]
params = {'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365, 'vboxes': vbox}
events = [
Travel(
name='vbox_event', start_tick=3, recurring=3,
groups=[0,1], capacity=1, vbox=0,
)
]
sim = Sim(groups=groups, events=events, **params)
sim.run(dotlog=True)
chart = Chart(
sim, figsize=(16,8), layout='dots_only',
show_intro=False, use_init_func=False
)
chart.to_html5_video()
Finally, we can pass multiple VBoxes as a list of dictionaries.
from rknot import Sim
vboxes = [
{'box': [1,3,1,3], 'label': 'Hospital'},
{'box': [5,7,1,3], 'label': 'Arena'}
]
groups = [
dict(n=20, n_inf=1, mover='social'),
dict(n=20, n_inf=1, mover='local')
]
params = {'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365, 'vboxes': vbox}
sim = Sim(groups=groups, **params)
sim.run(dotlog=True)
chart = Chart(
sim, figsize=(16,8), layout='dots_only',
show_intro=False, use_init_func=False
)
chart.to_html5_video()
## Events¶
Events impact the attributes of subjects over the course of the simulation.
Events are utilized to better simulate real-world behavior.
For instance, people do not move in consistent, prescribed ways. They move in regular ways most of the time with contacts that are well defined, but sometimes they attend events (perhaps periodically or uniquely) that are not governed by their regular movement patterns.
### Event¶
An Event is an event that occurs at a particular location.
An Event accepts the following parameters:
• xy, the xy coordinates of the location
• start_tick, the tick when the event begins
• groups, an iterable of group ids that are eligible for the event
• capacity, the number of subjects that should attend
• recurring, how often the event recurs (i.e. every nth tick); if set to 0, the event does not recur
When an Event concludes, the subject returns to its home location as specified in the dot matrix.
To schedule an event, you must pass a list of event objects to the events parameter.
To begin with, we’ll create a single Event object, called show, that occurs once on day 5.
from rknot.events import Event
params = {'square': 10, 'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365}
group1 = dict(name='1', n=10, n_inf=5)
show = Event(name='show', xy=(5,5), start_tick=5, groups=[0], capacity=10)
events = [show]
sim = Sim(groups=group1, events=events, **params)
sim.run(dotlog=True)
chart = Chart(sim, figsize=(16,8), layout='dots_only', show_intro=False)
chart.to_html5_video()
If you watch closely, you’ll see on Day 5 that all the dots seemingly disappear, save for one, at location (5,5).
In fact, all 10 dots are actually at that location at the same time.
We can confirm this by inspecting the Dot Matrix on that day via the dotlog attribute.
[22]:
from rknot.dots import MATRIX_COL_LABELS as ML
sim.dotlog[4][:, ML['x']:ML['y']+1]
[22]:
array([[5, 5],
[5, 5],
[5, 5],
[5, 5],
[5, 5],
[5, 5],
[5, 5],
[5, 5],
[5, 5],
[5, 5]])
We can see the event more clearly if we extend the duration to 10 days. We also significantly reduced the frame rate.
show = Event(name='show', xy=(5,5), start_tick=5, groups=[0], capacity=10, duration=10)
events = [show]
sim = Sim(groups=group1, events=events, **params)
sim.run()
chart = Chart(
sim, figsize=(16,8), dotsize=1000, interval=300,
layout='dots_only', show_intro=False, use_init_func=False
)
chart.to_html5.video()
Many event objects can be specified at once, in various combinations of groups.
group1 = dict(name='1', n=10, n_inf=5)
group2 = dict(name='2', n=10, n_inf=0)
show = Event(name='show', xy=(5,5), start_tick=5, groups=[0,1], capacity=5, recurring=30)
game = Event(name='game', xy=(1,1), start_tick=5, groups=[0], capacity=5, recurring=14)
church = Event(name='church', xy=(1,1), start_tick=5, groups=[1], capacity=10, recurring=7)
groups = [group1, group2]
events = [show, game, church]
sim = Sim(groups=groups, events=events, **params)
sim.run()
chart = Chart(
sim, interval=300, dotsize=1000, layout='dots_only', show_intro=False,
use_init_func=False
)
chart.to_html5_video()
### Travel¶
Travel is a special type of event that allows a subject to enter a gate.
When a dot enters a gate via a Travel object, its box and gate attributes are temporarily adjusted to match those of the groups within the gate. The attributes revert when the event ends (determined by duration parameter).
Once inside the gate, the dot(s) are free to interact with other dots normally.
from rknot.events import Travel
params = {'square': 10, 'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365}
group1 = dict(name='1', n=1, n_inf=1, box=[1,5,1,5], box_is_gated=True)
group2 = dict(name='2', n=10, n_inf=0, box=[6,10,6,10], box_is_gated=True)
visit = Travel(
name='visit', xy=[1,1], start_tick=3, groups=[1], capacity=1, duration=5, recurring=10
)
groups = [group1, group2]
events = [visit]
sim = Sim(groups=groups, events=events, **params)
sim.run()
chart = Chart(
sim, interval=200, dotsize=2000, layout='dots_only', show_intro=False
use_init_func=False
)
chartto_html5_video()
Many unique layouturations can be achieved with this structure. Below, the group1 box will be vacated by the solitary group1 dot (essentially switching places with a dot from group2).
from rknot.events import Travel
params = {'square': 10, 'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365}
group1 = dict(name='1', n=1, n_inf=1, box=[1,5,1,5], box_is_gated=True)
group2 = dict(name='2', n=10, n_inf=0, box=[6,10,6,10], box_is_gated=True)
visit2 = Travel(
name='visit2', xy=[9,9], start_tick=3, groups=[0], capacity=1, duration=5, recurring=10
)
visit1 = Travel(
name='visit1', xy=[1,1], start_tick=3, groups=[1], capacity=1, duration=5, recurring=10
)
groups = [group1, group2]
events = [visit2, visit1]
sim = Sim(groups=groups, events=events, **params)
sim.run()
chart = Chart(
sim, interval=200, dotsize=2000, layout='dots_only', show_intro=False,
use_init_func=False
)
chart.to_html5_video()
### Quarantine¶
A quarantine is an event object that makes several changes to a dots state in order to restrict its movement.
When a dot is quarantined,
1. it goes back to its home location (see Dot Matrix)
2. its boxes and gates are reset to match its group
3. its mover function is changed to ‘quarantine’
In addition, a Quarantine object will create a additional restriction objects that disallow events during the quarantine (see Restrictions below)
from rknot.events import Quarantine
params = {'square': 10, 'R0': 2.5, 'days': 50, 'imndur': 365, 'infdur': 365}
group1 = dict(name='1', n=2, n_inf=1, box=[1,5,1,5], box_is_gated=True)
group2 = dict(name='2', n=2, n_inf=0, box=[6,10,6,10], box_is_gated=True)
quar = Quarantine(name='all', start_tick=5, groups=[0,1], duration=30)
groups = [group1, group2]
events = [quar]
sim = Sim(groups=groups, events=events, **params)
sim.run()
chart = Chart(
sim, interval=200, dotsize=2000, layout='dots_only', show_intro=False,
use_init_func=False
)
chart.to_html5_video()
We can see from above that once in quaratine, the subjects barely move. We can include events in our structure. The events will be restricted during the quarantine period, then will resume when the quarantine ends.
params = {'square': 10, 'R0': 2.5, 'days': 100, 'imndur': 365, 'infdur': 365}
group1 = dict(name='1', n=1, n_inf=1, box=[1,5,1,5], box_is_gated=True)
group2 = dict(name='2', n=10, n_inf=0, box=[6,10,6,10], box_is_gated=True)
show = Event(name='show', xy=(6,6), start_tick=5, groups=[1], capacity=5, recurring=30)
visit2 = Travel(
name='visit2', xy=[9,9], start_tick=3, groups=[0], capacity=1, duration=5, recurring=10
)
visit1 = Travel(
name='visit1', xy=[1,1], start_tick=3, groups=[1], capacity=1, duration=5, recurring=10
)
groups = [group1, group2]
events = [show, visit2, visit1, quar]
sim = Sim(groups=groups, events=events, **params)
sim.run()
chart = Chart(
sim, interval=200, dotsize=2000, layout='dots_only', show_intro=False,
use_init_func=False,
)
chart.to_html5_video()
### Social Distancing¶
Event to mimick social distancing practices.
Social distancing is assumed to impact the Transmission Factor, $$\tau$$ of each dot. The core hypothesis is that practices such as maintaining 6-feet of distance or mask wearing don’t reduce the number of contacts, but do reduce the likelihood that a contact will result in a new infection (ceterus paribus).
You can see use of this object here.
TBD
### Restrictions¶
A restriction object restricts attendance to events that fall within the specified criteria. Each event has a restricted attribute that defaults to False. A restriction object filters out events from the event schedule by setting restricted=True for each event that satisfies the criteria.
To clarify, a Restriction is not an event. Events act on dots. Restrictions act on events.
Restrictions have potential as a versatile tool that can be used to investigate the impact of various government and business policy decisions that impact spread.
The Restriction object has a criteria parameter that accepts a dict, with keywords related to event object attributes. Acceptable criteria keys are currently:
capacity name ticks groups loc_id
The simplest way to restrict an event is by its name:
from rknot.events import Restriction
params = {'square': 10, 'R0': 2.5, 'days': 100, 'imndur': 365, 'infdur': 365}
group1 = dict(name='1', n=10, n_inf=1, box=[1,5,1,5], box_is_gated=True)
group2 = dict(name='2', n=10, n_inf=0, box=[6,10,6,10], box_is_gated=True)
show1 = Event(name='show1', xy=(1,1), start_tick=2, groups=[0], capacity=10, recurring=2)
show2 = Event(name='show2', xy=(10,10), start_tick=2, groups=[1], capacity=10, recurring=2)
criteria = {'name': 'show1'}
res1 = Restriction(name='no_show1', start_tick=10, duration=20, criteria=criteria)
groups = [group1, group2]
events = [show1, show2, res1]
sim = Sim(groups=groups, events=events, **params)
sim.run(dotlog=True)
chart = Chart(
sim, interval=300, dotsize=2000, layout='dots_only', show_intro=False,
use_init_func=False,
)
chart.to_html5_video()
In the above, we can see that every other day both group boxes have events that are attended by all dots in the group.
But on day 10, the group1 dots no longer converge on location (1,1). Instead, they are spread throughout their box. So res1 has successfully restricted attendance to show1.
Unlike Quarantine objects, however, the group1 dots have not changed their standard movement patterns.
We can restrict multiple events via the other criteria. Next we will restrict events based on their capacity. Events with more than 5 subjects in attendance will be restricted.
criteria = {'capacity': 5}
res1 = Restriction(name='cap5', start_tick=10, duration=20, criteria=criteria)
groups = [group1, group2]
events = [show1, show2, res1]
sim = Sim(groups=groups, events=events, **params)
sim.run()
chart = Chart(
sim, interval=300, dotsize=2000, layout='dots_only', show_intro=False,
use_init_func=False,
)
chart.to_html5_video()
In the above we see that neither of the groups had events from day 10 onward during the restriction period.
Restrictions can be chained together as desired to form a complex and tailored policy recipe for the population of the sim. See this example.
## Dot Matrix¶
The dot matrix is essentially RKnot’s canonical form of data structure. The matrix is simply a 2D numpy array of shape (n, 23) with each of the n rows representing a dot and each column representing an attribute.
More typical Python objects have been eschewed in favor the Dot Matrix because:
• RKnot relies heavily on Ray for parallel processing and Numba for just-in-time compilation and vectorization to improve processing speed.
• Numpy arrays have several advantages in Ray including rapid serialization and ease of batching.
• Numba also integrates well with numpy, supporting many of its features and leads to major performance improvements.
The dot matrix is created inside a Ray actor at instantiation and is only passed back to the main Sim object when the simulation is completed.
It can be accessed via the dots attribute. Below is a sample of 4 dots:
[33]:
sim.dots[:4]
[33]:
array([[ 0, 0, 1, 0, 0, 1, 1, 65, 7, 6, 54, 6, 5,
0, 0, -1, 0, 100, 650, 0, -1, 365, 730],
[ 1, 0, 1, 0, 0, 1, 1, 77, 8, 8, 22, 3, 3,
0, 0, -1, 0, 100, 650, 0, -1, 365, 730],
[ 2, 0, 1, 0, 0, 1, 1, 11, 2, 2, 88, 9, 9,
0, 0, -1, 0, 100, 650, 0, -1, 365, 730],
[ 3, 0, 1, 0, 0, 1, 1, 27, 3, 8, 96, 10, 7,
0, 0, -1, 0, 100, 650, 42, -1, 407, 772]])
The column attributes have corresponding labels:
[11]:
from rknot.dots import MATRIX_LABELS
print (MATRIX_LABELS)
['id', 'group_id', 'is_alive', 'is_vaxxed', 'is_sus', 'is_inf', 'ever_inf', 'loc_id', 'x', 'y', 'home_id', 'homex', 'homey', 'go_home', 'box_id', 'event_id', 'mover', 'mover_p', 'tmf', 'ifr', 'inf_tick', 'depart', 'recover', 'relapse']
With these labels, the 4 dot matrix above can be shown in a table.
id group_id is_alive is_vaxxed is_sus is_inf ever_inf loc_id x y home_id homex homey go_home box_id event_id mover mover_p tmf ifr inf_tick depart recover relapse
0 0 1 0 1 0 0 46 6 7 36 5 5 0 0 -1 4 -999 100 0 -1 -1 -1 -1
1 0 1 0 1 0 0 61 8 6 27 4 4 0 0 -1 4 -999 100 0 -1 -1 -1 -1
2 0 1 0 1 0 0 52 7 5 58 8 3 0 0 -1 4 -999 100 0 -1 -1 -1 -1
3 0 1 0 1 0 0 20 3 5 3 1 4 0 0 -1 4 -999 100 0 -1 -1 -1 -1
4 0 1 0 1 0 0 15 2 8 24 4 1 0 0 -1 4 -999 100 0 -1 -1 -1 -1
There are several data types at work:
• categorical integers; used to identify related objects
• id, group_id, loc_id, home_id, box_id, event_id, mover
• boolean integers; used to set boolean flags
• 0 means False and 1 means True
• is_alive, is_vaxxed, is_sus, is_inf, ever_inf, go_home
• coordinates; used to identify locations
• x, y, homex, homey
• event ticks; integers that trigger an event on the given tick
• depart, recover, relapse
• factors; scaled integers that must be unscaled before being used in multiplicative formulas
• tmf, ifr
The column attributes are defined as follows:
Label Definition Label Definition
id the subject’s unique identifier homey y coord of the subject’s home location
group_id the unique identifier of the subject’s group go_home is the subject going home on the next move?
is_alive Is the subject alive? box_id id of the box the subject belongs to
is_vaxxed Has the subject been vaccinated? event_id id of the event the subject is attending
is_sus Is the subject susceptible to infection? mover id of the subject’s mover function
is_inf Is the subject infected? mover_p p-value of custom mover function
ever_inf Has the subject ever been infected? tmf the subject’s transmission factor
loc_id id of the subject’s current location ifr the subject’s infection fatality rate
x x coord of the subject’s current location inf_tick the tick a subject is infected
y y coord of the subject’s curretn location depart the tick an infected subject will depart
home_id id of the subject’s home location recover the tick an infected subject will no longer be infected or susceptible
homex x coord of the subject’s home location relapse the tick a recovered subject will again become susceptible
|
2022-01-29 04:05:10
|
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|
https://math.stackexchange.com/questions/2931305/set-theory-cartesian-product-of-family?noredirect=1
|
# Set theory - Cartesian product of family
I'm trying to understand the cartesian product of a family.
I understand if $$X = \{1,2,3\}$$ and $$Y = \{4,5,6\}$$ then the cartesian product of these two sets is $$\{(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)\}$$
If ($$X_{i}$$} is a family of sets where $$i \in I$$, the Cartesian product of the family is, by definition, the set of all families $$x_{i}$$ with $$x_{i} \in X_{i}$$ where each $$i \in I$$
Say I = $$\{1,2,3\}$$ and $$X_{i} = \{4,5,6\}$$ How can you have a cartesian product of 1 set?
• $X_1, X, X_i$ ? – Mauro ALLEGRANZA Sep 26 '18 at 8:00
• See the following post about Cartesian products of families. – Mauro ALLEGRANZA Sep 26 '18 at 8:00
• We have an index set $I$ and and a "family" of sets $\{ X_i \}$. The cartesian product of the family is the set of all families $\{ x_i \}$ with $x_i ∈ X_i$ for each $i \in I$. – Mauro ALLEGRANZA Sep 26 '18 at 8:03
• If $I$ is a $3$ elements set, each element of the cartesian product will be a $3$-uple : $(a_1,a_2,a_3)$ where $a_i \in X_i$. – Mauro ALLEGRANZA Sep 26 '18 at 8:05
• So the cartesian product is I times X? – Paul Sep 26 '18 at 8:13
Let $$X=\{4,5,6\}$$ and $$X_1=X_2=X_3=X$$. Then
$$X_1 \times X_2 \times X_3=\{(a,b,c): a,b,c \in X\}$$.
• @Paul: Because you said $X_i = \{ 4,5,6\}$ for all $i$. – user14972 Sep 26 '18 at 8:02
• Yes. The cartesian product has $27$ elements. – Fred Sep 26 '18 at 8:46
|
2021-03-08 17:11:24
|
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|
https://wiki2.org/en/Polynomial_ring
|
To install click the Add extension button. That's it.
The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.
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# Polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers.
Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra and ring theory and algebraic geometry. Many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings.
A closely related notion is that of the ring of polynomial functions on a vector space, and, more generally, ring of regular functions on an algebraic variety.
## Definition (univariate case)
The polynomial ring, K[X], in X over a field (or, more generally, a commutative ring) K can be defined[1] (there are other equivalent definitions that are commonly used) as the set of expressions, called polynomials in X, of the form
${\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}$
where p0, p1, ..., pm, the coefficients of p, are elements of K, pm ≠ 0 if m > 0, and X, X2, ..., are symbols, which are considered as "powers" of X, and follow the usual rules of exponentiation: X0 = 1, X1 = X, and ${\displaystyle X^{k}\,X^{l}=X^{k+l}}$ for any nonnegative integers k and l. The symbol X is called an indeterminate[2] or variable.[3] (The term of "variable" comes from the terminology of polynomial functions. However, here, X has not any value (other than itself), and cannot vary, being a constant in the polynomial ring.)
Two polynomials are equal when the corresponding coefficients of each Xk are equal.
One can think of the ring K[X] as arising from K by adding one new element X that is external to K, commutes with all elements of K, and has no other specific properties. (This may be used for defining polynomial rings.)
The polynomial ring in X over K is equipped with an addition, a multiplication and a scalar multiplication that make it a commutative algebra. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if
${\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m},}$
and
${\displaystyle q=q_{0}+q_{1}X+q_{2}X^{2}+\cdots +q_{n}X^{n},}$
then
${\displaystyle p+q=r_{0}+r_{1}X+r_{2}X^{2}+\cdots +r_{k}X^{k},}$
and
${\displaystyle pq=s_{0}+s_{1}X+s_{2}X^{2}+\cdots +s_{l}X^{l},}$
where k = max(m, n), l = m + n,
${\displaystyle r_{i}=p_{i}+q_{i}}$
and
${\displaystyle s_{i}=p_{0}q_{i}+p_{1}q_{i-1}+\cdots +p_{i}q_{0}.}$
In these formulas, the polynomials p and q are extended by adding "dummy terms" with zero coefficients, so that all pi and qi that appear in the formulas are defined. Specifically, if m < n, then pi = 0 for m < in.
The scalar multiplication is the special case of the multiplication where p = p0 is reduced to its constant term (the term that is independent of X); that is
${\displaystyle p_{0}\left(q_{0}+q_{1}X+\dots +q_{n}X^{n}\right)=p_{0}q_{0}+\left(p_{0}q_{1}\right)X+\cdots +\left(p_{0}q_{n}\right)X^{n}}$
It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over K. Therefore, polynomial rings are also called polynomial algebras.
Another equivalent definition is often preferred, although less intuitive, because it is easier to make it completely rigorous, which consists in defining a polynomial as an infinite sequence (p0, p1, p2, ...) of elements of K, having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some m so that pn = 0 for n > m. In this case, p0 and X are considered as alternate notations for the sequences (p0, 0, 0, ...) and (0, 1, 0, 0, ...), respectively. A straightforward use of the operation rules shows that the expression
${\displaystyle p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m}}$
is then an alternate notation for the sequence
(p0, p1, p2, ..., pm, 0, 0, ...).
### Terminology
Let
${\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}$
be a nonzero polynomial with ${\displaystyle p_{m}\neq 0}$
The constant term of p is ${\displaystyle p_{0}.}$ It is zero in the case of the zero polynomial.
The degree of p, written deg(p) is ${\displaystyle m,}$ the largest k such that the coefficient of Xk is not zero.[4]
The leading coefficient of p is ${\displaystyle p_{m}.}$[5]
In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined,[6] defined to be –1,[7] or defined to be a –∞.[8]
A constant polynomial is either the zero polynomial, or a polynomial of degree zero.
A nonzero polynomial is monic if its leading coefficient is ${\displaystyle 1.}$
Given two polynomials p and q, one has
${\displaystyle \deg(p+q)\leq \max(\deg(p),\deg(q)),}$
and, over a field, or more generally an integral domain,[9]
${\displaystyle \deg(pq)=\deg(p)+\deg(q).}$
It follows immediately that, if K is an integral domain, then so is K[X].[10]
It follows also that, if K is an integral domain, a polynomial is a unit (that is, it has a multiplicative inverse) if and only if it is constant and is a unit in K.
Two polynomials are associated if either one is the product of the other by a unit.
Over a field, every nonzero polynomial is associated to a unique monic polynomial.
Given two polynomials, p and q, one says that p divides q, p is a divisor of q, or q is a multiple of p, if there is a polynomial r such that q = pr.
A polynomial is irreducible if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree.
### Polynomial evaluation
Let K be a field or, more generally, a commutative ring, and R a ring containing K. For any polynomial p in K[X] and any element a in R, the substitution of X for a in p defines an element of R, which is denoted P(a). This element is obtained by carrying on in R after the substitution the operations indicated by the expression of the polynomial. This computation is called the evaluation of P at a. For example, if we have
${\displaystyle P=X^{2}-1,}$
we have
{\displaystyle {\begin{aligned}P(3)&=3^{2}-1=8,\\P(X^{2}+1)&=\left(X^{2}+1\right)^{2}-1=X^{4}+2X^{2}\end{aligned}}}
(in the first example R = K, and in the second one R = K[X]). Substituting X for itself results in
${\displaystyle P=P(X),}$
explaining why the sentences "Let P be a polynomial" and "Let P(X) be a polynomial" are equivalent.
The polynomial function defined by a polynomial P is the function from K into K that is defined by ${\displaystyle x\mapsto P(x).}$ If K is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if K is a field with q elements, then the polynomials 0 and XqX both define the zero function.
For every a in R, the evaluation at a, that is, the map ${\displaystyle P\mapsto P(a)}$ defines an algebra homomorphism from K[X] to R, which is the unique homomorphism from K[X] to R that fixes K, and maps X to a. In other words, K[X] has the following universal property. For every ring R containing K, and every element a of R, there is a unique algebra homomorphism from K[X] to R that fixes K, and maps X to a. As for all universal properties, this defines the pair (K[X], X) up to a unique isomorphism, and can therefore be taken as a definition of K[X].
## Univariate polynomials over a field
If K is a field, the polynomial ring K[X] has many properties that are similar to those of the ring of integers ${\displaystyle \mathbb {Z} .}$ Most of these similarities result from the similarity between the long division of integers and the long division of polynomials.
Most of the properties of K[X] that are listed in this section do not remain true if K is not a field, or if one consider polynomials in several indeterminates.
Like for integers, the Euclidean division of polynomials has a property of uniqueness. That is, given two polynomials a and b ≠ 0 in K[X], there is a unique pair (q, r) of polynomials such that a = bq + r, and either r = 0 or deg(r) < deg (b). This makes K[X] a Euclidean domain. However, most other Euclidean domains (excepts integers) do not have any property of uniqueness for the division nor an easy algorithm (such as long division) for computing the Euclidean division.
The Euclidean division is the basis of the Euclidean algorithm for polynomials that computes a polynomial greatest common divisor of two polynomials. Here, "greatest" means "having a maximal degree" or, equivalently, being maximal for the preorder defined by the degree. Given a greatest common divisor of two polynomials, the other greatest common divisors are obtained by multiplication by a nonzero constant (that is, all greatest common divisors of a and b are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to 1).
The extended Euclidean algorithm allows computing (and proving) Bézout's identity. In the case of K[X], it may be stated as follows. Given two polynomials p and q of respective degrees m and n, if their monic greatest common divisor g has the degree d, then there is a unique pair (a, b) of polynomials such that
${\displaystyle ap+bq=g,}$
and
${\displaystyle \deg(a)\leq n-d,\quad \deg(b)
(For making this true in the limiting case where m = d or n = d, one has to define as negative the degree of the zero polynomial. Moreover, the equality ${\displaystyle \deg(a)=n-d}$ can occur only if p and q are associated.) The uniqueness property is rather specific to K[X]. In the case of the integers the same property is true, if degrees are replaced by absolute values, but, for having uniqueness, one must require a > 0.
Euclid's lemma applies to K[X]. That is, if a divides bc, and is coprime with b, then a is divides c. Here, coprime means that the monic greatest common divisor is 1. Proof: By hypothesis and Bézout's identity, there are e, p, and q such that ae = bc and 1 = ap + bq. So ${\displaystyle c=c(ap+bq)=cap+aeq=a(cp+eq).}$
The unique factorization property results from Euclid's lemma. In the case of integers, this is the fundamental theorem of arithmetic. In the case of K[X], it may be stated as: every non-constant polynomial can be expressed in a unique way as the product of a constant, and one or several irreducible monic polynomials; this decomposition is unique up to the order of the factors. In other terms K[X] is a unique factorization domain. If K is the field of complex numbers, the fundamental theorem of algebra asserts that a univariate polynomial is irreducible if and only if its degree is one. In this case the unique factorization property can be restated as: every non-constant univariate polynomial over the complex numbers can be expressed in a unique way as the product of a constant, and one or several polynomials of the form Xr; this decomposition is unique up to the order of the factors. For each factor, r is a root of the polynomial, and the number of occurrences of a factor is the multiplicity of the corresponding root.
### Derivation
The (formal) derivative of the polynomial
${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}\cdots +a_{n}X^{n}}$
is the polynomial
${\displaystyle a_{1}+2a_{2}X+\cdots +na_{n}X^{n-1}.}$
In the case of polynomials with real or complex coefficients, this is the standard derivative. The above formula defines the derivative of a polynomial even if the coefficients belong to a ring on which no notion of limit is defined. The derivative makes the polynomial ring a differential algebra.
The existence of the derivative is one of the main properties of a polynomial ring that is not shared with integers, and makes some computations easier on a polynomial ring than on integers.
### Factorization
Except for factorization, all previous properties of K[X] are effective, since their proofs, as sketched above, are associated with algorithms for testing the property and computing the polynomials whose existence are asserted. Moreover these algorithms are efficient, as their computational complexity is a quadratic function of the input size.
The situation is completely different for factorization: the proof of the unique factorization does not give any hint for a method for factorizing. Already for the integers, there is no known algorithm for factorizing them in polynomial time. This is the basis of the RSA cryptosystem, widely used for secure Internet communications.
In the case of K[X], the factors, and the methods for computing them, depend strongly on K. Over the complex numbers, the irreducible factors (those that cannot be factorized further) are all of degree one, while, over the real numbers, there are irreducible polynomials of degree 2, and, over the rational numbers, there are irreducible polynomials of any degree. For example, the polynomial ${\displaystyle X^{4}-2}$ is irreducible over the rational numbers, is factored as ${\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X^{2}+{\sqrt {2}})}$ over the real numbers and, and as ${\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X-i{\sqrt[{4}]{2}})(X+i{\sqrt[{4}]{2}})}$ over the complex numbers.
The existence of a factorization algorithm depends also on the ground field. In the case of the real or complex numbers, Abel–Ruffini theorem shows that the roots of some polynomials, and thus the irreducible factors, cannot be computed exactly. Therefore, a factorization algorithm can compute only approximations of the factors. Various algorithms have been designed for computing such approximations, see Root finding of polynomials.
There is an example of a field K such that there exist exact algorithms for the arithmetic operations of K, but there cannot exist any algorithm for deciding whether a polynomial of the form ${\displaystyle X^{p}-a}$ is irreducible or is a product of polynomials of lower degree.[11]
On the other hand, over the rational numbers and over finite fields, the situation is better than for integer factorization, as there are factorization algorithms that have a polynomial complexity. They are implemented in most general purpose computer algebra systems.
### Minimal polynomial
If θ is an element of an associative K-algebra L, the polynomial evaluation at θ is the unique algebra homomorphism φ from K[X] into L that maps X to θ and does not affect the elements of K itself (it is the identity map on K). It consists of substituting X for θ in every polynomial. That is,
${\displaystyle \varphi \left(a_{m}X^{m}+a_{m-1}X^{m-1}+\cdots +a_{1}X+a_{0}\right)=a_{m}\theta ^{m}+a_{m-1}\theta ^{m-1}+\cdots +a_{1}\theta +a_{0}.}$
The image of this evaluation homomorphism is the subalgebra generated by x, which is necessarily commutative. If φ is injective, the subalgebra generated by θ is isomorphic to K[X]. In this case, this subalgebra is often denoted by K[θ]. The notation ambiguity is generally harmless, because of the isomorphism.
If the evaluation homomorphism is not injective, this means that its kernel is a nonzero ideal, consisting of all polynomials that become zero when X is substituted for θ. This ideal consists of all multiples of some monic polynomial, that is called the minimal polynomial of x. The term minimal is motivated by the fact that its degree is minimal among the degrees of the elements of the ideal.
There are two main cases where minimal polynomials are considered.
In field theory and number theory, an element θ of an extension field L of K is algebraic over K if it is a root of some polynomial with coefficients in K. The minimal polynomial over K of θ is thus the monic polynomial of minimal degree that has θ as a root. Because L is a field, this minimal polynomial is necessarily irreducible over K. For example, the minimal polynomial (over the reals as well as over the rationals) of the complex number i is ${\displaystyle X^{2}+1}$. The cyclotomic polynomials are the minimal polynomials of the roots of unity.
In linear algebra, the n×n square matrices over K form an associative K-algebra of finite dimension (as a vector space). Therefore the evaluation homomorphism cannot be injective, and every matrix has a minimal polynomial (not necessarily irreducible). By Cayley–Hamilton theorem, the evaluation homomorphism maps to zero the characteristic polynomial of a matrix. It follows that the minimal polynomial divides the characteristic polynomial, and therefore that the degree of the minimal polynomial is at most n.
### Quotient ring
In the case of K[X], the quotient ring by an ideal can be built, as in the general case, as a set of equivalence classes. However, as each equivalence class contains exactly one polynomial of minimal degree, another construction is often more convenient.
Given a polynomial p of degree d, the quotient ring of K[X] by the ideal generated by p can be identified with the vector space of the polynomials of degrees less than d, with the "multiplication modulo p" as a multiplication, the multiplication modulo p consisting of the remainder under the division by p of the (usual) product of polynomials. This quotient ring is variously denoted as ${\displaystyle K[X]/pK[X],}$ ${\displaystyle K[X]/\langle p\rangle ,}$ ${\displaystyle K[X]/(p),}$ or simply ${\displaystyle K[X]/p.}$
The ring ${\displaystyle K[X]/(p)}$ is a field if and only if p is an irreducible polynomial. In fact, if p is irreducible, every nonzero polynomial q of lower degree is coprime with p, and Bézout's identity allows computing r and s such that sp + qr = 1; so, r is the multiplicative inverse of q modulo p. Conversely, if p is reducible, then there exist polynomials of degrees lower than deg(p) such that ab = p ≡ 0 (mod q); so a is a nonzero zero divisor modulo p, and cannot be invertible.
For example, the standard definition of the field of the complex numbers can be summarized by saying that it is the quotient ring
${\displaystyle \mathbb {C} =\mathbb {R} [X]/(X^{2}+1),}$
and that the image of X in ${\displaystyle \mathbb {C} }$ is denoted by i. In fact, by the above description, this quotient consists of all polynomials of degree one in i, which have the form a + bi, with a and b in ${\displaystyle \mathbb {R} .}$ The remainder of the Euclidean division that is needed for multiplying two elements of the quotient ring is obtained by replacing i2 by –1 in their product as polynomials (this is exactly the usual definition of the product of complex numbers).
Let θ be an algebraic element in a K-algebra A. By algebraic, one means that θ has a minimal polynomial p. The first ring isomorphism theorem asserts that the substitution homomorphism induces an isomorphism of ${\displaystyle K[X]/(p)}$ onto the image K[θ] of the substitution homomorphism. In particular, if A is a simple extension of K generated by θ, this allows identifying A and ${\displaystyle K[X]/(p).}$ This identification is widely used in algebraic number theory.
### Modules
The structure theorem for finitely generated modules over a principal ideal domain applies to K[X], when K is a field. This means that every finitely generated module over K[X] may be decomposed into a direct sum of a free module and finitely many modules of the form ${\displaystyle K[X]/\left\langle P^{k}\right\rangle }$, where P is an irreducible polynomial over K and k a positive integer.
## Definition (multivariate case)
Given n symbols ${\displaystyle X_{1},\dots ,X_{n},}$ called indeterminates, a monomial (also called power product)
${\displaystyle X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}}$
is a formal product of these indeterminates, possibly raised to a nonnegative power. As usual, exponents equal to one and factors with a zero exponent can be omitted. In particular, ${\displaystyle X_{1}^{0}\cdots X_{n}^{0}=1.}$
The tuple of exponents α = (α1, ..., αn) is called the multidegree or exponent vector of the monomial. For a less cumbersome notation, the abbreviation
${\displaystyle X^{\alpha }=X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}}$
is often used. The degree of a monomial Xα, frequently denoted deg α or |α|, is the sum of its exponents:
${\displaystyle \deg \alpha =\sum _{i=1}^{n}\alpha _{i}.}$
A polynomial in these indeterminates, with coefficients in a field, or more generally a ring, K is a finite linear combination of monomials
${\displaystyle p=\sum _{\alpha }p_{\alpha }X^{\alpha }}$
with coefficients in K. The degree of a nonzero polynomial is the maximum of the degrees of its monomials with nonzero coefficients.
The set of polynomials in ${\displaystyle X_{1},\dots ,X_{n},}$ denoted ${\displaystyle K[X_{1},\dots ,X_{n}],}$ is thus a vector space (or a free module, if K is a ring) that has the monomials as a basis.
${\displaystyle K[X_{1},\dots ,X_{n}]}$ is naturally equipped (see below) with a multiplication that makes a ring, and an associative algebra over K, called the polynomial ring in n indeterminates over K (the definite article the reflects that it is uniquely defined up to the name and the order of the indeterminates. If the ring K is commutative, ${\displaystyle K[X_{1},\dots ,X_{n}]}$ is also a commutative ring.
### Operations in K[X1, ..., Xn]
Addition and scalar multiplication of polynomials are those of a vector space or free module equipped by a specific basis (here the basis of the monomials). Explicitly, let ${\displaystyle p=\sum _{\alpha \in I}p_{\alpha }X^{\alpha },\quad q=\sum _{\beta \in J}q_{\beta }X^{\beta },}$ where I and J are finite sets of exponent vectors.
The scalar multiplication of p and a scalar ${\displaystyle c\in K}$ is
${\displaystyle cp=\sum _{\alpha \in I}cp_{\alpha }X^{\alpha }.}$
The addition of p and q is
${\displaystyle p+q=\sum _{\alpha \in I\cup J}(p_{\alpha }+q_{\alpha })X^{\alpha },}$
where ${\displaystyle p_{\alpha }=0}$ if ${\displaystyle \alpha \not \in I,}$ and ${\displaystyle q_{\beta }=0}$ if ${\displaystyle \beta \not \in J.}$ Moreover, if one has ${\displaystyle p_{\alpha }+q_{\alpha }=0}$ for some ${\displaystyle \alpha \in I\cap J,}$ the corresponding zero term is removed from the result.
The multiplication is
${\displaystyle pq=\sum _{\gamma \in I+J}\left(\sum _{\alpha ,\beta \mid \alpha +\beta =\gamma }p_{\alpha }q_{\beta }\right)X^{\gamma },}$
where ${\displaystyle I+J}$ is the set of the sums of one exponent vector in I and oneother in J (usual sum of vectors). In particular, the product of two monomials is a monomial whose exponent vector is the sum of the exponent vectors of the factors.
The verification of the axioms of an associative algebra is straightforward.
### Polynomial expression
A polynomial expression is an expression built with scalars (elements of K), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers.
As all these operations are defined in ${\displaystyle K[X_{1},\dots ,X_{n}]}$ a polynomial expression represents a polynomial, that is an element of ${\displaystyle K[X_{1},\dots ,X_{n}].}$ The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called the canonical form, normal form, or expanded form of the polynomial. Given a polynomial expression, one can compute the expanded form of the represented polynomial by expanding with the distributive law all the products that have a sum among their factors, and then using commutativity (except for the product of two scalars), and associativity for transforming the terms of the resulting sum into products of a scalar and a monomial; then one gets the canonical form by regrouping the like terms.
The distinction between a polynomial expression and the polynomial that it represents is relatively recent, and mainly motivated by the rise of computer algebra, where, for example, the test whether two polynomial expressions represent the same polynomial may be a nontrivial computation.
### Categorical characterization
If K is a commutative ring, the polynomial ring K[X1, ..., Xn] has the following universal property: for every commutative K-algebra A, and every n-tuple (x1, ..., xn) of elements of A, there is a unique algebra homomorphism from K[X1, ..., Xn] to A that maps each ${\displaystyle X_{i}}$ to the corresponding ${\displaystyle x_{i}.}$ This homomorphism is the evaluation homomorphism that consists in substituting ${\displaystyle X_{i}}$ for ${\displaystyle x_{i}}$ in every polynomial.
As it is the case for every universal property, this characterizes the pair ${\displaystyle (K[X_{1},\dots ,X_{n}],(X_{1},\dots ,X_{n}))}$ up to a unique isomorphism.
This may also be interpreted in terms of adjoint functors. More precisely, let SET and ALG be respectively the categories of sets and commutative K-algebras (here, and in the following, the morphisms are trivially defined). There is a forgetful functor ${\displaystyle \mathrm {F} :\mathrm {ALG} \to \mathrm {SET} }$ that maps algebras to their underlying sets. On the other hand, the map ${\displaystyle X\mapsto K[X]}$ defines a functor ${\displaystyle \mathrm {SET} \to \mathrm {ALG} }$ in the other direction. (If X is infinite, K[X] is the set of all polynomials in a finite number of elements of X.)
The universal property of the polynomial ring means that F and POL are adjoint functors. That is, there is a bijection
${\displaystyle \operatorname {Hom} _{\mathrm {SET} }(X,\operatorname {F} (A))\cong \operatorname {Hom} _{\mathrm {ALG} }(K[X],A).}$
This may be expressed also by saying that polynomial rings are free commutative algebras, since they are free objects in the category of commutative algebras. Similarly, a polynomial ring with integer coefficients is the free commutative ring over its set of variables, since commutative rings and commutative algebras over the integers are the same thing.
## Univariate over a ring vs. multivariate
A polynomial in ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ can be considered as a univariate polynomial in the indeterminate ${\displaystyle X_{n}}$ over the ring ${\displaystyle K[X_{1},\ldots ,X_{n-1}],}$ by regrouping the terms that contain the same power of ${\displaystyle X^{n},}$ that is, by using the identity
${\displaystyle \sum _{(\alpha _{1},\ldots ,\alpha _{n})\in I}c_{\alpha _{1},\ldots ,\alpha _{n}}X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}=\sum _{i}\left(\sum _{(\alpha _{1},\ldots ,\alpha _{n-1})\mid (\alpha _{1},\ldots ,\alpha _{n-1},i)\in I}c_{\alpha _{1},\ldots ,\alpha _{n-1}}X_{1}^{\alpha _{1}}\cdots X_{n-1}^{\alpha _{n-1}}\right)X_{n}^{i},}$
which results from the distributivity and associativity of ring operations.
This means that one has an algebra isomorphism
${\displaystyle K[X_{1},\ldots ,X_{n}]\cong (K[X_{1},\ldots ,X_{n-1}])[X_{n}]}$
that maps each indeterminate to itself. (This isomorphism is often written as an equality, which is justified by the fact that polynomial rings are defined up to a unique isomorphism.)
In other words, a multivariate polynomial ring can be considered as a univariate polynomial over a smaller polynomial ring. This is commonly used for proving properties of multivariate polynomial rings, by induction on the number of indeterminates.
The main such properties are listed below.
### Properties that pass from R to R[X]
In this section, R is a commutative ring, K is a field, X denotes a single indeterminate, and, as usual, ${\displaystyle \mathbb {Z} }$ is the ring of integers. Here is the list of the main ring properties that remain true when passing from R to R[X].
• If R is an integral domain then the same holds for R[X] (since the leading coefficient of a product of polynomials is, if not zero, the product of the leading coeficients of the factors).
• In particular, ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ and ${\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}$ are integral domains.
• If R is a unique factorization domain then the same holds for R[X]. This results from Gauss's lemma and the unique factorization property of ${\displaystyle L[X],}$ where L is the field of fractions of R.
• In particular, ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ and ${\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}$ are unique factorization domains.
• If R is a Noetherian ring, then the same holds for R[X].
• In particular, ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ and ${\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}$ are Noetherian rings; this is Hilbert's basis theorem.
• If R is a Noetherian ring, then ${\displaystyle \dim R[X]=1+\dim R,}$ where "${\displaystyle \dim }$" denotes the Krull dimension.
• In particular, ${\displaystyle \dim K[X_{1},\ldots ,X_{n}]=n}$ and ${\displaystyle \dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1.}$
• If R is a Regular ring, then the same holds for R[X]; in this case, one has
${\displaystyle \operatorname {gl} \,\dim R[X]=\dim R[X]=1+\operatorname {gl} \,\dim R=1+\dim R,}$
where "${\displaystyle \operatorname {gl} \,\dim }$" denotes the global dimension.
• In particular, ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ and ${\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}$ are regular rings, ${\displaystyle \operatorname {gl} \,\dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1,}$ and ${\displaystyle \operatorname {gl} \,\dim K[X_{1},\ldots ,X_{n}]=n.}$ The latter equality is Hilbert's syzygy theorem.
## Several indeterminates over a field
Polynomial rings in several variables over a field are fundamental in invariant theory and algebraic geometry. Some of their properties, such as those described above can be reduced to the case of a single indeterminate, but this is not always the case. In particular, because of the geometric applications, many interesting properties must be invariant under affine or projective transformations of the indeterminates. This often implies that one cannot select one of the indeterminates for a recurrence on the indeterminates.
Bézout's theorem, Hilbert's Nullstellensatz and Jacobian conjecture are among the most famous properties that are specific to multivariate polynomials over a field.
### Hilbert's Nullstellensatz
The Nullstellensatz (German for "zero-locus theorem") is a theorem, first proved by David Hilbert, which extends to the multivariate case some aspects of the fundamental theorem of algebra. It is foundational for algebraic geometry, as establishing a strong link between the algebraic properties of ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ and the geometric properties of algebraic varieties, that are (roughly speaking) set of points defined by implicit polynomial equations.
The Nullstellensatz, has three main versions, each being a corollary of any other. Two of these versions are given below. For the third version, the reader is referred to the main article on the Nullstellensatz.
The first version generalizes the fact that a nonzero univariate polynomial has a complex zero if and only if it is not a constant. The statement is: a set of polynomials S in ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ has a common zero in an algebraically closed field containing K, if and only 1 does not belong to the ideal generated by S, that is, if 1 is not a linear combination of elements of S with polynomial coefficients.
The second version generalizes the fact that the irreducible univariate polynomials over the complex numbers are associate to a polynomial of the form ${\displaystyle X-\alpha .}$ The statement is: If K is algebraically closed, then the maximal ideals of ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ have the form ${\displaystyle \langle X_{1}-\alpha _{1},\ldots ,X_{n}-\alpha _{n}\rangle .}$
### Bézout's theorem
Bézout's theorem may be viewed as a multivariate generalization of the version of the fundamental theorem of algebra that asserts that a univariate polynomial of degree n has n complex roots, if they are counted with their multiplicities.
In the case of bivariate polynomials, it states that two polynomials of degrees d and e in two variables, which have no common factors of positive degree, have exactly de common zeros in an algebraically closed field containing the coefficients, if the zeros are counted with their multiplicity and include the zeros at infinity.
For stating the general case, and not considering "zero at infinity" as special zeros, it is convenient to work with homogeneous polynomials, and consider zeros in a projective space. In this context, a projective zero of a homogeneous polynomial ${\displaystyle P(X_{0},\ldots ,X_{n})}$ is, up to a scaling, a (n + 1)-tuple ${\displaystyle (x_{0},\ldots ,x_{n})}$ of elements of K that is different form (0, ..., 0), and such that ${\displaystyle P(x_{0},\ldots ,x_{n})=0.}$ Here, "up to a scaling" means that ${\displaystyle (x_{0},\ldots ,x_{n})}$ and ${\displaystyle (\lambda x_{0},\ldots ,\lambda x_{n})}$ are considered as the same zero for any nonzero ${\displaystyle \lambda \in K.}$ In other words, a zero is a set of homogeneous coordinates of a point in a projective space of dimension n.
Then, Bézout's theorem states: Given n homogeneous polynomials of degrees ${\displaystyle d_{1},\ldots ,d_{n}}$ in n + 1 indeterminates, which have only a finite number of common projective zeros in an algebraically closed extension of K, then the sum of the multiplicities of these zeros is the product ${\displaystyle d_{1}\cdots d_{n}.}$
## Generalizations
Polynomial rings can be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, noncommutative polynomial rings, skew polynomial rings, and polynomial rigs.
### Infinitely many variables
One slight generalization of polynomial rings is to allow for infinitely many indeterminates. Each monomial still involves only a finite number of indeterminates (so that its degree remains finite), and each polynomial is a still a (finite) linear combination of monomials. Thus, any individual polynomial involves only finitely many indeterminates, and any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates. This generalization has the same property of usual polynomial rings, of being the free commutative algebra, the only difference is that it is a free object over an infinite set.
One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree. This ring is larger than the usual polynomial ring, as it includes infinite sums of variables. However, it is smaller than the ring of power series in infinitely many variables. Such a ring is used for constructing the ring of symmetric functions over an infinite set.
### Generalized exponents
A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents: Xi · Xj = Xi+j. A set for which addition makes sense (is closed and associative) is called a monoid. The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a + b, then cn = an + bn for every n in N. The multiplication is defined as the Cauchy product, so that if c = a · b, then for each n in N, cn is the sum of all aibj where i, j range over all pairs of elements of N which sum to n.
When N is commutative, it is convenient to denote the function a in R[N] as the formal sum:
${\displaystyle \sum _{n\in N}a_{n}X^{n}}$
and then the formulas for addition and multiplication are the familiar:
${\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)+\left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(a_{n}+b_{n}\right)X^{n}}$
and
${\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)\cdot \left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(\sum _{i+j=n}a_{i}b_{j}\right)X^{n}}$
where the latter sum is taken over all i, j in N that sum to n.
Some authors such as (Lang 2002, II,§3) go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials in several variables simply take N to be the direct product of several copies of the monoid of non-negative integers.
Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negative rational numbers, (Osbourne 2000, §4.4). See also Puiseux series.
### Power series
Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can also be seen as the ring completion of the polynomial ring with respect to the ideal generated by x.
### Noncommutative polynomial rings
For polynomial rings of more than one variable, the products X·Y and Y·X are simply defined to be equal. A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained. Formally, the polynomial ring in n noncommuting variables with coefficients in the ring R is the monoid ring R[N], where the monoid N is the free monoid on n letters, also known as the set of all strings over an alphabet of n symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other.
Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.
### Differential and skew-polynomial rings
Other generalizations of polynomials are differential and skew-polynomial rings.
A differential polynomial ring is a ring of differential operators formed from a ring R and a derivation δ of R into R. This derivation operates on R, and will be denoted X, when viewed as an operator. The elements of R also operate on R by multiplication. The composition of operators is denoted as the usual multiplication. It follows that the relation δ(ab) = (b) + δ(a)b may be rewritten as
${\displaystyle X\cdot a=a\cdot X+\delta (a).}$
This relation may be extended to define a skew multiplication between two polynomials in X with coefficients in R, which make them a non-commutative ring.
The standard example, called a Weyl algebra, takes R to be a (usual) polynomial ring k[Y], and δ to be the standard polynomial derivative ${\displaystyle {\tfrac {\partial }{\partial Y}}}$. Taking a =Y in the above relation, one gets the canonical commutation relation, X·YY·X = 1. Extending this relation by associativity and distributivity allows explicitly constructing the Weyl algebra.(Lam 2001, §1,ex1.9).
The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X·r = f(rX to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism F from the monoid N of the positive integers into the endomorphism ring of R, the formula Xn·r = F(n)(rXn allows constructing a skew-polynomial ring.(Lam 2001, §1,ex 1.11) Skew polynomial rings are closely related to crossed product algebras.
### Polynomial rigs
The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a field or a ring to the requirement that R only be a semifield or rig; the resulting polynomial structure/extension R[X] is a polynomial rig. For example, the set of all multivariate polynomials with natural number coefficients is a polynomial rig.
## References
1. ^ Herstein p. 153
2. ^ Herstein, Hall p. 73
3. ^ Lang p. 97
4. ^ Herstein p. 154
5. ^ Lang p.100
6. ^ Anton, Howard; Bivens, Irl C.; Davis, Stephen (2012), Calculus Single Variable, John Wiley & Sons, p. 31, ISBN 9780470647707.
7. ^ Sendra, J. Rafael; Winkler, Franz; Pérez-Diaz, Sonia (2007), Rational Algebraic Curves: A Computer Algebra Approach, Algorithms and Computation in Mathematics, 22, Springer, p. 250, ISBN 9783540737247.
8. ^ Eves, Howard Whitley (1980), Elementary Matrix Theory, Dover, p. 183, ISBN 9780486150277.
9. ^ Herstein p.155, 162
10. ^ Herstein p.162
11. ^ Fröhlich, A.; Shepherson, J. C. (1955), "On the factorisation of polynomials in a finite number of steps", Mathematische Zeitschrift, 62 (1): 331–334, doi:10.1007/BF01180640, ISSN 0025-5874
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2021-02-27 10:13:47
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https://infoscience.epfl.ch/record/152160
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Infoscience
Report
# The Stationary Behaviour of Fluid Limits of Reversible Processes is Concentrated on Stationary Points
Assume that a stochastic processes can be approximated, when some scale parameter gets large, by a fluid limit (also called mean field limit", or hydrodynamic limit"). A common practice, often called the fixed point approximation" consists in approximating the stationary behaviour of the stochastic process by the stationary points of the fluid limit. It is known that this may be incorrect in general, as the stationary behaviour of the fluid limit may not be described by its stationary points. We show however that, if the stochastic process is reversible, the fixed point approximation is indeed valid. More precisely, we assume that the stochastic process converges to the fluid limit in distribution (hence in probability) at every fixed point in time. This assumption is very weak and holds for a large family of processes, among which many mean field and other interaction models. We show that the reversibility of the stochastic process implies that any limit point of its stationary distribution is concentrated on stationary points of the fluid limit. If the fluid limit has a unique stationary point, it is an approximation of the stationary distribution of the stochastic process.
#### Reference
• EPFL-REPORT-152160
Record created on 2010-09-28, modified on 2016-08-08
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2016-10-24 03:15:05
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https://hal.archives-ouvertes.fr/hal-00657052
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# Nonparametric estimation of random effects densities in linear mixed-effects model
Abstract : We consider a linear mixed-effects model where $Y_{k,j}= \alpha_k + \beta_k t_{j} +\varepsilon_{k,j}$ is the observed value for individual $k$ at time $t_j$, $k=1,\ldots, N$, $j=1,\dots, J$. The random effects $\alpha_k$, $\beta_k$ are independent identically distributed random variables with unknown densities $f_\alpha$ and $f_\beta$ and are independent of the noise. We develop nonparametric estimators of these two densities, which involve a cutoff parameter. We study their mean integrated square risk and propose cutoff-selection strategies, depending on the noise distribution assumptions. Lastly, in the particular case of fixed interval between times $t_j$, we show that a completely data driven strategy can be implemented without any knowledge on the noise density. Intensive simulation experiments illustrate the method.
Keywords :
Document type :
Journal articles
Domain :
Cited literature [27 references]
https://hal.archives-ouvertes.fr/hal-00657052
Submitted on : Thursday, January 5, 2012 - 5:07:48 PM
Last modification on : Friday, September 20, 2019 - 4:34:03 PM
Long-term archiving on: Monday, November 19, 2012 - 12:30:55 PM
### File
MixedNonParaComteSamson.pdf
Files produced by the author(s)
### Identifiers
• HAL Id : hal-00657052, version 1
### Citation
Fabienne Comte, Adeline Samson. Nonparametric estimation of random effects densities in linear mixed-effects model. Journal of Nonparametric Statistics, American Statistical Association, 2012, 24 (4), pp.951-975. ⟨hal-00657052⟩
Record views
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2020-02-25 07:01:48
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https://mathematica.stackexchange.com/questions/152785/pure-function-slot-delivery?noredirect=1
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# Pure function slot delivery [duplicate]
Another novice problem encountered. I am going to need a cardiology surgery. :(
For a simple example,
a = #1 &;
b = #2 &;
f = a^2 + b^2;
f
results in
(#1 &)^2 + (#2 &)^2
However, my goal is
(#1 )^2 + (#2 )^2
I know it can be done as
f=(#1)^2+(#2)^2
But still, substitution is necessary in my work, which means the step of a and b cant be neglected.
So what do I miss? Thanks:)
• This is doable, e.g. f = Evaluate[a[#^2] + b[Null,#2^2]]&, but the fact that things have gotten so messy for something so simple is a sign that you're approaching whatever you want to do in the wrong way. What's wrong with not binding anything to a and b and then using a ReplaceAll or something? – b3m2a1 Aug 1 '17 at 3:03
• Looks like a duplicate: 142072, do you agree? – Kuba Aug 1 '17 at 5:21
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2020-07-09 06:19:22
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https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/7%3A_Combinatorics/7.4%3A_Combinations
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# 7.4: Combinations
In many counting problems, the order of arrangement or selection does not matter. In essence, we are selecting or forming subsets.
If we are choosing $$3$$ people out of $$20$$ Discrete students to be president, vice-president and janitor, then the order makes a difference. The choice of:
Steve, Ahmet, Liz (SAL) v.s Liz, Ahmet, Steve (LAS) would make quite a difference for Liz and Steve. Permutations include all the different arrangements, so we say "order matters" and there are $$P(20,3)$$ ways to choose $$3$$ people out of $$20$$ to be president, vice-president and janitor.
Now, change the scenario to chose $$3$$ people out of $$20$$ to get an A for the course. This time SAL and LAS are not considered different choices; here we say "order does not matter". This scenario is the number of combinations (basically subsets) and we call this one "$$20$$ choose $$3$$". With permutations, each set of 3 letters, such as SAL, will be rearranged $$3!$$ or $$6$$ times. For combinations, we only use a set of 3 letters once (since order does not matter) and so the number of combinations in this case will be $$\frac{P(20,3)}{3!}.$$
Example $$\PageIndex{1}\label{eg:combin-01}$$
Determine the number of ways to choose 4 values from 1, 2, 3, …, 20, in which the order of selection does not matter.
Solution
Let $$N$$ be the number of ways to choose the 4 numbers. Since the order in which the numbers are selected does not matter, these are not sequences (in which order of appearance matters). We can change a selection of 4 numbers into a sequence. The 4 numbers can be arranged in $$P(4,4)=4!$$ ways. Therefore, all these 4-number selections together produce $$N\cdot4!$$ sequences. The number of 4-number sequences is $$P(20,4)$$. Thus, $$N\cdot4!=P(20,4)$$, or equivalently, $$N=P(20,4)/4!$$.
Definition: combinations
The number of $$r$$-element subsets in an $$n$$-element set is denoted by
$C(n,r) \qquad\mbox{ or }\qquad \binom{n}{r},$
where $${n\choose r}$$ is read as “$$n$$ choose $$r$$.” It determines the number of combinations of $$n$$ objects, taken $$r$$ at a time (without replacement). Alternate notations such as $$_nC_r$$ and $$C_r^n$$ can be found in other textbooks and some calculators. Do not write it as $$(\frac{n}{r})$$; this notation has a completely different meaning.
Recall that $$\binom{n}{r}$$ counts the number of ways to choose or select $$r$$ objects from a pool of $$n$$ objects in which the order of selection does not matter. Hence, $$r$$-combinations are subsets of size $$r$$.
Example $$\PageIndex{2}\label{eg:combin-02}$$
The 2-combinations of $$S=\{a,b,c,d\}$$ are
$\{a,b\}, \quad \{a,c\}, \quad \{a,d\}, \quad \{b,c\}, \quad \{b,d\}, \quad\mbox{and}\quad \{c,d\}.$
Therefore $$\binom{4}{2}=6$$. What are the 1-combinations and 3-combinations of $$S$$? What can you say about the values of $$\binom{4}{1}$$ and $$\binom{4}{3}$$?
Solution
The 1-combinations are the singleton sets $$\{a\}$$, $$\{b\}$$, $$\{c\}$$, and $$\{d\}$$. Hence, $$\binom{4}{1}=4$$. The 3-combinations are $\{a,b,c\}, \quad \{a,b,d\}, \quad \{a,c,d\}, \quad\mbox{and}\quad \{b,c,d\}.$ Thus, $$\binom{4}{3}=4$$.
## C(n,r) formula
Theorem $$\PageIndex{1}\label{thm:combin}$$
For all integers $$n$$ and $$r$$ satisfying $$0\leq r\leq n$$, we have $\binom{n}{r} = \frac{P(n,r)}{r!} = \frac{n(n-1)\cdots(n-r+1)}{r!} = \frac{n!}{r!\,(n-r)!}.$
Proof
The idea is similar to the one we used in the alternate proof of Theorem [thm:circperm]. Let $$A$$ be the set of all $$r$$-permutations, and let $$B$$ be the set of all $$r$$-combinations. Define $$\fcn{f}{A}{B}$$ to be the function that converts a permutation into a combination by “unscrambling” its order. Then $$f$$ is an $$r!$$-to-one function because there are $$r!$$ ways to arrange (or shuffle) $$r$$ objects. Therefore $|A| = r!\cdot|B|.$ Since $$|A|=P(n,r)$$, and $$|B|=\binom{n}{r}$$, it follows that $$\binom{n}{r} = P(n,r)/r!$$.
Example $$\PageIndex{3}\label{eg:combin-03}$$
There are $$\binom{40}{5}$$ ways to choose 5 numbers, without repetitions, from the integers $$1,2,\ldots,40$$. To compute its numeric value by hand, it is easier if we first cancel the common factors in the numerator and the denominator. We find
$\binom{40}{5} = \frac{40\cdot39\cdot38\cdot37\cdot36} {5\cdot4\cdot3\cdot2\cdot1} = 13\cdot38\cdot37\cdot36,$
which gives $$\binom{40}{5}=658008$$.
hands-on Exercise $$\PageIndex{1}\label{he:combin-01}$$
Compute $$\binom{12}{3}$$ by hand.
hands-on Exercise $$\PageIndex{2}\label{he:combin-02}$$
A three-member executive committee is to be selected from a group of seven candidates. In how many ways can the committee be formed?
hands-on Exercise $$\PageIndex{3}\label{he:combin-03}$$
How many subsets of $$\{1,2,\ldots,23\}$$ have five elements?
## $$\binom{n}{r} = \binom{n}{n-r}$$
Theorem $$\PageIndex{2}$$
For $$0\leq r\leq n$$, we have $$\binom{n}{r} = \binom{n}{n-r}$$.
Proof
According to Theorem 7.4.1 we have $\binom{n}{n-r} = \frac{n!}{(n-r)!\,(n-(n-r))!} = \frac{n!}{(n-r)!\,r!},$ which is precisely $$\binom{n}{r}$$.
Example $$\PageIndex{4}\label{eg:combin-04}$$
To compute the numeric value of $$\binom{50}{47}$$, instead of computing the product of 47 factors as indicated in the definition, it is much faster if we use $\binom{50}{47} = \binom{50}{3} = \frac{50\cdot 49\cdot48}{3\cdot 2\cdot 1},$ from which we obtain $$\binom{50}{47} = 19600$$.
hands-on Exercise $$\PageIndex{4}\label{he:combin-04}$$
Compute, by hand, the numeric value of $$\binom{529}{525}$$.
## Pascal's Triangle
In this section we will cover
• how to construct Pascal's Triangle
• how to read values for combinations from Pascal's Triangle
In a later section, we will prove that this construction does give the values for combinations.
Start with a $$1$$, then two more $$1$$'s. Notice how the spacing is offset on each row.
$\begin{array}{*{13}{c}} & & & & & & 1 \\ & & & & & 1 & & 1 \\ & & & & 1 & & 2 & & 1 \\ & & & 1 & & 3 & & 3 & & 1 \end{array}$
To create the next row, start with a $$1$$ and then add the two entries just above. So under the $$1 \qquad 3 \qquad 3 \qquad 1$$ row, start with a $$1$$, then add $$1+3=4$$ to get the next entry, then $$3+3=6$$, etc. Notice the symmetry; this relates to Theorem 7.4.2.
$\begin{array}{*{13}{c}} & & & & & & 1 \\ & & & & & 1 & & 1 \\ & & & & 1 & & 2 & & 1 \\ & & & 1 & & 3 & & 3 & & 1 \\ & & 1 & & 4 & & 6 & & 4 & & 1 \end{array}$
Here's more of the triangle:
$\begin{array}{*{13}{c}} & & & & & & 1 \\ & & & & & 1 & & 1 \\ & & & & 1 & & 2 & & 1 \\ & & & 1 & & 3 & & 3 & & 1 \\ & & 1 & & 4 & & 6 & & 4 & & 1 \\ & 1 & & 5 & &10 & &10 & & 5 & & 1 \\ 1 & & 6 & &15 & &20 & &15 & & 6 & & 1 \end{array}$
From the top, these rows are the 0-row, the 1-row, the 2-row, etc.
The 5-row consists of: $1 \qquad 5 \qquad 10 \qquad 10 \qquad 5 \qquad 1$
These numbers correspond to:
$\binom{5}{0} \qquad \binom{5}{1} \qquad \binom{5}{2} \qquad \binom{5}{3} \qquad \binom{5}{4} \qquad \binom{5}{5} \qquad .$
The symmetry of the triangle shows the symmetric values, such as $$\binom{5}{1}=\binom{5}{4}$$
Example $$\PageIndex{5}$$
Use Pascal's Triangle to find $$\binom{7}{3}$$.
After the 6-row (see above) create the 7-row.
$\begin{array}{*{13}{c}} & & 1 & & 7 & & 21 && 35 && 35 && 21 & & 7 && 1 \end{array}$
Count over, $$\binom{7}{0}=1, \qquad \binom{7}{1}=7, \qquad \binom{7}{2}=21$$, so the answer is
$\binom{7}{3}=35.$
Now we are ready to look at some mixed examples. In all of these examples, sometimes we have to use permutation, other times we have to use combination. Very often we need to use both, together with the addition and multiplication principles. You may ask, how can I figure out what to do? We suggest asking yourself these questions:
1. Use the construction approach. If you want to list all the configurations that meet the requirement, how are you going to do it systematically?
2. Are there several cases involved in the problem? If yes, we need to list them first, before we go through each of them one at a time. Finally, add the results to come up with the final answer.
3. Do we allow repetitions or replacements? This question can also take the form of whether the objects are distinguishable or indistinguishable.
4. Does order matter? If yes, we have to use permutation. Otherwise, use combination.
5. Sometimes, it may be easier to use the multiplication principle instead of permutation, because repetitions may be allowed (in which case, we cannot use permutation, although we can still use the multiplication principle). Try drawing a schematic diagram and decide what we need from it. If the analysis suggests a pattern that follows the one found in a permutation, you can then use the formula for permutation.
6. Do not forget: it may be easier to work with the complement.
It is often not clear how to get started because there seem to be several ways to start the construction. For example, how would you distribute soda cans among a group of students? There are two possible approaches:
• From the perspective of the students. Imagine you are one of the students, which soda would you receive?
• From the perspective of the soda cans. Imagine you are holding a can of soda, to whom would you give this soda?
Depending on the actual problem, usually only one of these two approaches would work.
Example $$\PageIndex{5}\label{eg:combin-05}$$
Suppose we have to distribute 10 different soda cans to 20 students. It is clear that some students may not get any soda. In fact, some lucky students could receive more than one soda (the problem does not say this cannot happen). Hence, it is easier to start from the perspective of the soda cans.
Solution
We can give the first soda to any one of the 20 students, and we can also give the second soda to any one of the 20 students. In fact, we always have 20 choices for each soda. Since we have 10 sodas, there are $$\underbrace{20\cdot20\cdots20}_{10} = 20^{10}$$ ways to distribute the sodas.
Example $$\PageIndex{6}\label{eg:combin-06}$$
In how many ways can a team of three representatives be selected from a class of 885 students? In how many ways can a team of three representatives consisting of a chairperson, a vice-chairperson, and a secretary be selected?
Solution
If we are only interested in selecting three representatives, order does not matter. Hence, the answer would be $$\binom{885}{3}$$. If we are concerned about which offices these three representative will hold, then the answer should be $$P(885,3)$$.
hands-on Exercise $$\PageIndex{5}\label{he:combin-05}$$
Mike needs some new shirts, but he has only enough money to purchase five of the eight that he likes. In how many ways can he purchase the five shirts by choosing them at random?
hands-on Exercise $$\PageIndex{6}\label{he:combin-06}$$
Mary wants to purchase four shirts for her four brothers, and she would like each of them to receive a different shirt. She finds ten shirts that she thinks they will like. In many ways can she select them?
Playing cards provide excellent examples for counting problems. Just in case you are not familiar with them, let us briefly review what a deck of playing cards contains.
• There are 52 playing cards, each of them is marked with a suit and a rank.
• There are four suits: spades ($$\spadesuit$$), hearts ($$\heartsuit$$), diamonds ($$\diamondsuit$$) and clubs ($$\clubsuit$$).
• Each suit has 13 ranks, labeled A, 2, 3, …, 9, 10, J, Q, and K, where A means ace, J means jack, Q means queen, and K means king.
• Each rank has 4 suits (see above).
hands-on Exercise $$\PageIndex{7}\label{he:combin-07}$$
Determine the number of five-card poker hands that can be dealt from a deck of 52 cards.
Solution
All we care is which five cards can be found in a hand. This is a selection problem. The answer is $$\binom{52}{5}$$.
hands-on exercise $$\PageIndex{7}\label{eg:combin-07}$$
In how many ways can a 13-card bridge hand be dealt from a standard deck of 52 cards?
Example $$\PageIndex{8}\label{eg:combin-08}$$
In how many ways can a deck of 52 cards be dealt in a game of bridge? (In a bridge game, there are four players designated as North, East, South and West, each of them is dealt a hand of 13 cards.)
Solution
The difference between this problem and the last example is that the order of distributing the four bridge hands makes a difference. This is a problem that combines permutations and combinations. As we had suggested earlier, the best approach is to start from scratch, using the addition and/or multiplication principles, along with permutation and/or combination whenever it seems appropriate.
There are $$\binom{52}{13}$$ ways to give 13 cards to the first player. Now we are left with 39 cards, from which we select 13 to be given to the second player. Now, out of the remaining 26 cards, we have to give 13 to the third player. Finally, the last 13 cards will be given to the last player (there is only one way to do it). The number of ways to deal the cards in a bridge game is $$\binom{52}{13} \binom{39}{13} \binom{26}{13}$$.
We could have said the answer is $\binom{52}{13} \binom{39}{13} \binom{26}{13} \binom{13}{13}.$ The last factor $$\binom{13}{13}$$ is the number of ways to give the last 13 cards to the fourth player. Numerically, $$\binom{13}{13}=1$$, so the two answers are the same. Do not dismiss this extra factor as redundant. Take note of the nice pattern in this answer. The bottom numbers are 13, because we are selecting 13 cards to be given to each player. The top numbers indicate how many cards are still available for distribution at each stage of the distribution. The reasoning behind the solution is self-explanatory!
[eg:combin-08]
Example $$\PageIndex{9}\label{eg:combin-09}$$
Determine the number of five-card poker hands that contain three queens. How many of them contain, in addition to the three queens, another pair of cards?
Solution
(a) The first step is to choose the three queens in $$\binom{4}{3}$$ ways, after which the remaining two cards can be selected in $$\binom{48}{2}$$ ways. Therefore, there are altogether $$\binom{4}{3} \binom{48}{2}$$ hands that meet the requirements.
Solution
(b) As in part (a), the three queens can be selected in $$\binom{4}{3}$$ ways. Next, we need to select the pair. We can select any card from the remaining 48 cards (therefore, there are 48 choices), after which we have to select one from the remaining 3 cards of the same rank. This gives $$48\cdot3$$ choices for the pair, right? The answer is NO!
The first card we picked could be $$\heartsuit 8$$, and the second could be $$\clubsuit 8$$. However, the first card could have been $$\clubsuit 8$$, and the second $$\heartsuit 8$$. These two selections are counted as different selections, but they are actually the same pair! The trouble is, we are considering “first,” and “second” cards, which in effect imposes an ordering among the two cards, thereby turning it into a sequence or an ordered selection. We have to divide the answer by 2 to overcome the double-counting. The answer is therefore $$\frac{48\cdot3}{2}$$.
Here is a better way to count the number of pairs. An important question to ask is
Which one should we pick first: the suit or the rank?
Here, we want to pick the rank first. There are 12 choices (the pair cannot be queens) for the rank, and among the four cards of that rank, we can pick the two cards in $$\binom{4}{2}$$ ways. Therefore, the answer is $$12\binom{4}{2}$$. Numerically, the two answers are identical, because $$12\binom{4}{2} = 12\cdot\frac{4\cdot3}{2} = \frac{48\cdot3}{2}$$. In summary: the final answer is $$\binom{4}{3}\cdot12\binom{4}{2}$$.
hands-on Exercise $$\PageIndex{8}\label{he:combin-08}$$
How many bridge hands contain exactly four spades?
hands-on Exercise $$\PageIndex{9}\label{he:combin-09}$$
How many bridge hands contain exactly four spades and four hearts?
hands-on Exercise $$\PageIndex{10}\label{he:combin-10}$$
How many bridge hands are there containing exactly four spades, three hearts, three diamonds, and three clubs?
Example $$\PageIndex{10}\label{eg:combin-10}$$
How many positive integers not exceeding 99999 contain exactly three 7s?
Solution
Regard each legitimate integer as a sequence of five digits, each of them selected from 0, 1, 2, …, 9. For example, the integer 358 can be considered as 00358. Three out of the five positions must be occupied by 7. There are $$\binom{5}{3}$$ ways to select these three slots. The remaining two positions can be filled with any of the other nine digits. Hence, there are $$\binom{5}{3} \cdot 9^2$$ such integers.
Example $$\PageIndex{11}\label{eg:combin-11}$$
How many five-digit positive integers contain exactly three 7s?
Solution
Unlike the last example, the first of the five digits cannot be 0. Yet, the answer is not $$\binom{5}{3} \cdot 9 \cdot 8$$. Yes, there are $$\binom{5}{3}$$ choices for the placement of the three 7s, but some of these selections may have put the 7s in the last four positions. This leaves the first digit unfilled. The nine choices counted by 9 allows a zero to be placed in the first position. The result is, at best, a four-digit number. The correct approach is to consider two cases:
Case 1. If the first digit is not 7, then there are eight ways to fill this slot. Among the remaining four positions, three of them must be 7, and the last one can be any digit other than 7. So there are $$8\cdot \binom{4}{3}\cdot 9$$ integers in this category.
Case 2. If the first digit is 7, we still have to put the other two 7s in the other four positions. There are $$\binom{4}{2} \cdot 9^2$$ such integers.
Together, the two cases give a total of $$8\cdot \binom{4}{3}\cdot 9 + \binom{4}{2} \cdot 9^2 = 774$$ integers.
hands-on Exercise $$\PageIndex{11}\label{he:combin-11}$$
Five balls are chosen from a bag of eight blue balls, six red balls, and five green balls. How many of these five-ball selections contain exactly two blue balls?
Example $$\PageIndex{12}\label{eg:combin-12}$$
Find the number of ways to select five balls from a bag of six red balls, eight blue balls and four yellow balls such that the five-ball selections contain exactly two red balls or two blue balls.
Solution
The keyword “or” suggests this is a problem that involves the union of two sets, hence, we have to use PIE to solve the problem.
• How many selections contain two red balls? Following the same argument used in the last example, the answer is $$\binom{6}{2} \binom{12}{3}$$.
• How many selections contain two blue balls? The answer is $$\binom{8}{2} \binom{10}{3}$$.
• According to PIE, the final answer is $\binom{6}{2} \binom{12}{3} + \binom{8}{2} \binom{10}{3} - \binom{6}{2} \binom{8}{2} \binom{4}{1}.$ In each term, the upper numbers always add up to 18, and the sum of the lower numbers is always 5. Can you explain why?
• How many selections contain two red balls and 2 blue balls? The answer is $$\binom{6}{2} \binom{8}{2} \binom{4}{1}$$.
Example $$\PageIndex{13}\label{eg:combin-13}$$
We have 11 balls, five of which are blue, three of which are red, and the remaining three are green. How many collection of four balls can be selected such that at least two blue balls are selected? Assume that balls of the same color are indistinguishable.
Solution
The keywords “at least” mean we could have two, three, or four blue balls. There are $\binom{5}{2} \binom{6}{2} + \binom{5}{3} \binom{6}{1} + \binom{5}{4} \binom{6}{0}$ ways to select four balls, with at least two of them being blue.
hands-on Exercise $$\PageIndex{12}\label{he:combin-12}$$
Jerry bought eight cans of Pepsi, seven cans of Sprite, three cans of Dr. Pepper, and six cans of Mountain Dew. He want to bring 10 cans to his pal’s house when they watch the basketball game tonight. Assuming the cans are distinguishable, say, with different expiration dates, how many selections can he make if he wants to bring
1. Exactly four cans of Pepsi?
2. At least four cans of Pepsi?
3. At most four cans of Pepsi?
4. Exactly three cans of Pepsi, and at most three cans of Sprite?
The proof of the next result uses what we call a combinatorial or counting argument. In general, a combinatorial argument does not rely on algebraic manipulation. Rather, it uses the combinatorial significance of the situations to solve the problem.
## $$\sum_{r=0}^n \binom{n}{r} = 2^n$$
Theorem $$\PageIndex{3}$$
Prove that $$\sum_{r=0}^n \binom{n}{r} = 2^n$$ for all nonnegative integers $$n$$.
Proof
Since $$\binom{n}{r}$$ counts the number of $$r$$-element subsets selected from an $$n$$-element set $$S$$, the summation on the left is the sum of the number of subsets of $$S$$ of all possible cardinalities. In other words, this is the total number of subsets in $$S$$. We learned earlier that $$S$$ has $$2^n$$ subsets, which establishes the identity immediately.
## Summary and Review
• Use permutation if order matters, otherwise use combination.
• The keywords arrangement, sequence, and order suggest using permutation.
• The keywords selection, subset, and group suggest using combination.
• It is best to start with a construction. Imagine you want to list all the possibilities, how would you get started?
• We may need to use both permutation and combination, and very likely we may also need to use the addition and multiplication principles.
## Exercises
Exercise $$\PageIndex{1}\label{ex:combin-01}$$
If the Buffalo Bills and the Cleveland Browns have eight and six players, respectively, available for trading, in how many ways can they swap three players for three players?
Solution
$$\binom{6}{3}\binom{8}{3}$$.
Exercise $$\PageIndex{2}\label{ex:combin-02}$$
In the game of Mastermind, one player, the codemaker, selects a sequence of four colors (the “code”) selected from red, blue, green, white, black, and yellow.
a) How many different codes can be formed?
b) How many codes use four different colors?
c) How many codes use only one color?
d) How many codes use exactly two colors?
e) How many codes use exactly three colors?
Exercise $$\PageIndex{3}\label{ex:combin-03}$$
Becky likes to watch DVDs each evening. How many DVDs must she have if she is able to watch every evening for 24 consecutive evenings during her winter break?
a) A different subset of DVDs?
b) A different subset of three DVDs?
Solution
(a) at least 5 (b) at least 7
Exercise $$\PageIndex{4}\label{ex:combin-04}$$
Bridget has $$n$$ friends from her bridge club. Every Thursday evening, she invites three friends to her home for a bridge game. She always sits in the north position, and she decides which friends are to sit in the east, south, and west positions. She is able to do this for 200 weeks without repeating a seating arrangement. What is the minimum value of $$n$$?
Exercise $$\PageIndex{5}\label{ex:combin-05}$$
Bridget has $$n$$ friends from her bridge club. She is able to invite a different subset of three of them to her home every Thursday evening for 100 weeks. What is the minimum value of $$n$$?
Solution
10.
Exercise $$\PageIndex{6}\label{ex:combin-06}$$
How many five-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6, 7? How many of them do not have repeated digits?
Exercise $$\PageIndex{7}\label{ex:combin-07}$$
The Mathematics Department of a small college has three full professors, seven associate professors, and four assistant professors. In how many ways can a four-member committee be formed under these restrictions:
a) There are no restrictions.
b) At least one full professor is selected.
c) The committee must contain a professor from each rank.
Solution
(a) $$\binom{14}{4}$$ (b) $$\binom{14}{4}-\binom{11}{4}$$ (c) $$\binom{3}{2}\binom{7}{1}\binom{4}{1} +\binom{3}{1}\binom{7}{2}\binom{4}{1} +\binom{3}{1}\binom{7}{1}\binom{4}{2}$$
Exercise $$\PageIndex{8}\label{ex:combin-08}$$
A department store manager receives from the company headquarters 12 football tickets to the same game (hence they can be regarded as “identical”). In how many ways can she distribute them to 20 employees if no one gets more than one ticket? What if the tickets are for 12 different games?
Exercise $$\PageIndex{9}\label{ex:combin-09}$$
A checkerboard has 64 distinct squares arranged into eight rows and eight columns.
a) In how many ways can eight identical checkers be placed on the board so that no two checkers can occupy the same row or the same column?
b) In how many ways can two identical red checkers and two identical black checkers be placed on the board so that no two checkers of the same color can occupy the same row or the same column?
Solution
(a) $$8!$$ (b) $$\binom{8}{2}\,P(8,2)\, \big[\binom{6}{2}\,P(8,2)+2\cdot7\cdot6\cdot7+7\cdot6\big]$$
Exercise $$\PageIndex{10}\label{ex:combin-10}$$
Determine the number of permutations of $$\{A, B, C, D, E\}$$ that satisfy the following conditions:
a) $$A$$ occupies the first position.
b) $$A$$ occupies the first position, and $$B$$ the second.
c) $$A$$ appears before $$B$$.
Exercise $$\PageIndex{11}\label{ex:combin-11}$$
A binary string is a sequence of digits chosen from 0 and 1. How many binary strings of length 16 contain exactly seven 1s?
Solution
$$\binom{16}{7}$$.
Exercise $$\PageIndex{12}\label{ex:combin-12}$$
In how many ways can a nonempty subset of people be chosen from eight men and eight women so that every subset contains an equal number of men and women?
Exercise $$\PageIndex{13}\label{ex:combin-13}$$
A poker hand is a five-card selection chosen from a standard deck of 52 cards. How many poker hands satisfy the following conditions?
a) There are no restrictions.
b) The hand contains at least one card from each suit.
c) The hand contains exactly one pair (the other three cards all of different ranks).
d) The hand contains three of a rank (the other two cards all of different ranks).
e) The hand is a full house (three of one rank and a pair of another).
f) The hand is a straight (consecutive ranks, as in 5, 6, 7, 8, 9, but not all from the same suit).
g) The hand is a flush (all the same suit, but not a straight).
h) The hand is a straight flush (both straight and flush).
Solution
(a) $$\binom{52}{5}$$ (b) $$4 \,\binom{13}{2}\,13^3$$ (c) $$13\,\binom{4}{2}\binom{12}{3}\,4^3$$ (d) $$13\,\binom{4}{3}\binom{12}{2}\,4^2$$
(e) $$13\,\binom{4}{3}\,12\,\binom{4}{2}$$ (f) $$10\cdot(4^5-4)$$ (g) $$4\,\big[\binom{13}{5}-10\big]$$ (h) $$4\cdot10$$
Exercise $$\PageIndex{14}\label{ex:combin-14}$$
A local pizza restaurant offers the following toppings on their cheese pizzas: extra cheese, pepperoni, mushrooms, green peppers, onions, sausage, ham, and anchovies.
a) How many kinds of pizzas can one order?
b) How many kinds of pizzas can one order with exactly three toppings?
c) How many kinds of vegetarian pizza (without pepperoni, sausage, or ham) can one order?
Exercise $$\PageIndex{15}$$
Write the numbers for the 8-row for Pascal's Triangle.
Solution
$1 \qquad 8 \qquad 28 \qquad 56 \qquad 70 \qquad 56 \qquad 28 \qquad 8 \qquad 1$
Exercise $$\PageIndex{16}$$
In terms of selecting objects and Pascal's Triangle, explain why
(a) $$\binom{8}{0}=\binom{8}{8}$$
(b) $$\binom{8}{1}=8$$
(c) $$\binom{8}{2}$$ is the third number in the 8-row rather than the second number
Exercise $$\PageIndex{17}$$
Use the 8-row of Pascal's Triangle to find
(a) $$\binom{8}{4}$$
(b) $$\binom{8}{6}$$
(c) $$\binom{8}{5}=\binom{?}{?}$$
Solution
(a) 70 (b) 28 (c) $$\binom{8}{5}=\binom{8}{3}$$
This page titled 7.4: Combinations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Harris Kwong (OpenSUNY) .
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2022-08-16 04:29:21
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https://math.stackexchange.com/questions/3180984/explanation-of-weak-null-condition-as-used-in-pde
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# explanation of (weak) null condition (as used in PDE)
I work in computational physics (numerical relativity), and I would like to better understand the null condition (as I believe we first introduced here), which helpful in determining the long time behavior of various PDEs that appear in geometric analysis. I have had trouble finding references online that are very accessible to someone with a more physics/computational background, and concisely introduce and summarize the ideas behind the null condition.
I would appreciate a concise explanation and motivation for the (weak) null condition, along with a (potentially heuristic) explanation of how it plays a role in proving long term existence for, e.g. various semilinear geometric PDE (as described, for example, in Klainerman's article linked above, or more recently as used here for the weak null condition). An explanation of how the condition applies to a (set of) simple example(s) would also be helpful. Either a description of this here, or a reference to literature suitable for someone without extensive PDE expertise would work!
I am no expert on this, but since nobody has left a comment/answer I will write what I know and give some references.
Restricting our attention to the wave equation, we want to consider the following initial value problem with small initial data:
$$\Box u = F(u,\partial u)$$ $$u(0,.) = \epsilon f$$ $$\partial_t u(0,.) = \epsilon g$$
Where $$f,g$$ are smooth and compactly supported functions in $$\mathbb{R}^3$$, $$\Box = -\partial_t^2 + \sum_i \partial_i^2$$ and $$F$$ is some nonlinearity depending in some ways on $$u$$ and its derivatives $$\partial u$$. The question is if we can determine a $$\textit{global}$$ (in time) solution to this system for some $$\epsilon > 0$$. Answering this question is hopelessly difficult, and in fact examples where $$F$$ has a fairly simple form have been shown to blow up in finite time. For example $$F_1 = (\partial_t u)^2$$ (John, F. Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math. 34 (1981), 29-51.)
The null condition is a restricition on the nonlinearity $$F$$ which (due to Klainerman) ensures global existence. If we instead consider $$F_2 = (\partial_t u)^2 - \sum_i (\partial_i u)^2$$ then it turns out that for $$F_2$$, the system above has global solution. The difference is that $$F_2$$ satisfies a special algebraic structure (the null condition) while $$F_1$$ does not.
The weak null condition (introduced by Lindblad and Rodnianski for their new proof of the stability of Minkowski space) is not as useful, in the sense that there is currently no similar results which guarentees global existence for a $$F$$ satisfying the weak null condition. However, all known examples where $$F$$ does satisfy the weak null condition do in fact possess global solution. A nice example is Einstein's equation (in wave coordinates). Interestingly, in dimension $$(n+1), n \geq 4$$ one can in fact prove general global existence (for small enough $$\epsilon$$), see Sogge's (reference below). The problematic dimension is $$3+1$$ which is relevant for relativity.
For a brief review of this stuff, chech out the last 6 pages of the lecutre notes.
The only other references that I know of are the ones from the PDE community, such as Sogge's book or Hörmander.
• Based on your reference to Ringstrom's lectures, is the null condition $F(u,\xi_i)=\mathcal{O}(u^3)+\mathcal{O}(\xi^3)$ for $\xi_i$ null with respect to the background geometry (for the system of PDE you wrote in your answer)? – PHY314 May 14 '19 at 14:37
• If you consider the coefficients $f_{\mu \nu}$ in the notes, then we easily see that e.g. $F_2$ above satisfies the null condition, since for this examples $f_{\mu \nu} = \eta_{\mu \nu}$, so that for any null vector $\xi$ it holds that $\xi^\mu \xi^\nu f_{\mu \nu} = 0$, so $F_2$ satisfies the null condition. – monolith28 May 15 '19 at 18:48
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2020-09-24 04:33:10
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https://mathoverflow.net/questions/178254/do-regular-conditional-distributions-almost-surely-assign-trivial-measure-to-all
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# Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra?
Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel $(\rho_x^\mathcal{E})_{x \in X}$ on $X$ is a regular conditional distribution of $\rho$ with respect to $\mathcal{E}$ if
• the map $x \mapsto \rho_x^\mathcal{E}(A)$ is $\mathcal{E}$-measurable for all $A \in \Sigma$;
• for every $E \in \mathcal{E}$ and $A \in \Sigma$, $\rho(A \cap E) = \int_E \rho_x^\mathcal{E}(A) \, \rho(dx)$.
Is it necessarily the case that $\rho$-almost every $x \in X$ has the property that for all $E \in \mathcal{E}$, either $\rho_x^\mathcal{E}(E)=0$ or $\rho_x^\mathcal{E}(E)=1$?
Remark: In order for the above to be satisfied, I believe it is sufficient that there exists a family $(\rho_{x,y})_{x,y \in X}$ of probability measures on $X$ such that
1. for $\rho$-almost every $x \in X$, $(\rho_{x,y})_{y \in X}$ is a rcd of $\rho_x^\mathcal{E}$ with respect to $\mathcal{E}$;
2. the map $(x,y) \mapsto \rho_{x,y}(A)$ is $(\mathcal{E} \otimes \mathcal{E})$-measurable for all $A \in \Sigma$.
(Specifically, if we can find such a family, then I think we can show that for $\rho$-almost every $x$, for $\rho_x^\mathcal{E}$-almost every $y$, $\rho_{x,y}=\rho_x^\mathcal{E}$.)
Some important remarks:
The disintegration theorem guarantees that a rcd of $\rho$ with respect to $\mathcal{E}$ exists and is unique modulo $\rho$-null sets.
So of course (at least if we assume AC) there exists a family $(\rho_{x,y})_{x,y \in X}$ satisfying (1); but the question is whether there necessarily exists a family $(\rho_{x,y})_{x,y \in X}$ satisfying both (1) and (2).
It is worth emphasising: we do not necessarily have that for $\rho$-almost every $x \in X$, for all $E \in \mathcal{E}$, $\rho_x^\mathcal{E}(E)=\mathbf{1}_E(x)$. (For counterexamples, see the examples of "maximally improper rcd's" in the paper "Improper Regular Conditional Distributions" linked to by Jochen below.)
Motivation - ergodic decomposition: I'm keen to have a "nice" proof of the ergodic decomposition theorem for stationary probability measures of stochastic semigroups (jointly measurable in space and time) on standard measurable spaces. If I understand correctly, one can reduce the question of finding an ergodic decomposition of a stationary measure of a stochastic semigroup to the question of finding an ergodic decomposition of an invariant measure of a (deterministic) dynamical system, by considering the time-shift dynamical system on the space of $X$-valued functions of time. Already I'm not sure I'd deem this "nice", but even for dynamical systems I wonder whether there's a nicer proof of the ergodic decomposition theorem than the ones I've seen. For an invariant measure $\rho$ of a measurable dynamical system, the proofs that I've seen involve using Birkhoff's ergodic theorem to show that $\rho_x^\mathcal{I}$ is ergodic for $\rho$-almost all $x$, where $\mathcal{I}$ is the $\sigma$-algebra of invariant sets. But if the answer to my question is yes, then the ergodicity of $\rho_x^\mathcal{I}$ for $\rho$-almost all $x$ is immediate (once we have established the invariance of $\rho_x^\mathcal{I}$ for $\rho$-almost all $x$---but that is easy). I guess one could argue that Birkhoff's theorem is "nice enough" as it is, but if the answer to my question is yes, then the same proof will work directly for stochastic semigroups (so that we don't have to invoke the theorem of equivalence between ergodicity with respect to a stochastic semigroup and ergodic of the corresponding Markov measure under the time-shift dynamical system).
A possible approach? Perhaps I should mention a possible starting point that I've thought of, but have been unable to make into a full solution:
The difficulty behind the problem is that $\mathcal{E}$ might not be countably generated; however, as hinted at by Yuri below, perhaps it is possible to use the fact that $\mathcal{E}$ is countably generated $\bmod \rho$ to help. Of course, this fact cannot mean that all arguments for the countably generated case remain valid in the general case, since as we have said already, it is not necessarily the case that for $\rho$-almost every $x \in X$, for all $E \in \mathcal{E}$, $\rho_x^\mathcal{E}(E)=\mathbf{1}_E(x)$.
Nonetheless, perhaps we can proceed as follows: Let $\{E_n\}_{n \in \mathbb{N}} \subset \mathcal{E}$ be such that $\mathcal{E}$ is contained in the $\rho$-completion of $\sigma(E_n:n \in \mathbb{N})$. For each $n$, let $\mathcal{G}_n:=\sigma(E_i : 1 \leq i \leq n)$. Then I believe we have that
1. for $\rho$-almost all $y \in X$, $(\rho_x^\mathcal{E})_{x \in X}$ is a rcd of $\rho_y^{\mathcal{G}_n}$ with respect to $\mathcal{E}$ for all $n$;
2. [by the result mentioned in (2) of Conditional law as a random measure and convergence of random measures, combined with Levy's Upward Theorem] for any fixed Polish topology on $X$, for $\rho$-almost all $y \in X$, $\rho_y^{\mathcal{G}_n} \to \rho_y^\mathcal{E}$ in the narrow topology as $n \to \infty$.
If I can somehow show that (1) and (2) together imply that
$\hspace{7mm}$ for $\rho$-almost all $y \in X$, $(\rho_x^\mathcal{E})_{x \in X}$ is a rcd of $\rho_y^\mathcal{E}$ with respect to $\mathcal{E}$
then I'm done! Any ideas??
• I realise (thanks to Jochen Wengenroth) that I should perhaps emphasise: In the case that $\mathcal{E}$ is countably generated, the answer to both questions is yes, since clearly, for each $E \in \mathcal{E}$, $\rho_x^\mathcal{E}(E)=\mathbf{1}_E(x)$ a.e. (and then the $\pi$-$\lambda$ theorem gives the result, by considering a countable $\pi$-system generating $\mathcal{E}$). The difficulty in my questions is specifically due to the fact that I haven't assumed $\mathcal{E}$ to be countably generated. (I've only assumed $\Sigma$ to be countably generated, since $\Sigma$ is standard.) Aug 11, 2014 at 12:15
• I have deleted my answer (which in fact did not answer the question). Meanwhile, I think that the answer to you question is NO. It might be helpful to study (more carefully than I did) the following projecteuclid.org/… Aug 11, 2014 at 12:27
• I think that this projecteuclid.org/euclid.aop/1015345764 answers your question. Aug 11, 2014 at 13:06
• Thanks. I think this answers my second question, but not my first (unless I read the papers too quickly). Aug 11, 2014 at 14:56
• I've "revamped" the question to take into account the papers you've informed me of. Thank you very much for your help. Aug 11, 2014 at 16:09
I've found the answer - it's NO!
The paper I found addressing the question is the following:
http://projecteuclid.org/euclid.aop/1175287757 ("0-1 Laws for Regular Conditional Probabilities")
A simple counterexample (which I've just slightly adapted from the counterexample in Example 2 of the above paper) is the following:
Let $$X=[0,1] \times \{0,1\}$$ (with $$\Sigma=\mathcal{B}([0,1]) \otimes 2^{\{0,1\}}$$), let $$\rho$$ be a probability measure on $$X$$ whose projection onto $$[0,1]$$ is atomless, and let $$\mathcal{E} \subset \Sigma$$ be the $$\rho$$-completion of $$\mathcal{B}([0,1]) \otimes \{0,1\}$$ relative to $$\Sigma$$. Then given any non-trivial probability measure $$m$$ on the binary set $$\{0,1\}$$, the stochastic kernel
$$\hspace{5mm} \rho_{(x,i)}^\mathcal{E} \ := \ \delta_x \otimes m$$
is a rcd of $$\rho$$ with respect to $$\mathcal{E}$$. Clearly, for any $$x \in [0,1]$$, $$\{(x,0)\} \in \mathcal{E}$$; and yet, for all $$(x,i) \in X$$,
$$\hspace{5mm} \rho_{(x,i)}^\mathcal{E}(\{(x,0)\}) \ = \ m(0) \, \in \, (0,1).$$
Regarding my motivation: Theorem 12 of the above paper claims to be a generalisation of the ergodic decomposition theorem (for measurable maps). However, I haven't yet managed to work out how to derive the ergodic decomposition theorem from Theorem 12 of the above paper.
• My motivation was to have a "nice" proof of the ergodic decomposition theorem for measurable continuous-time Markov semigroups. However, I now realise I don't know a proof at all; I had had in mind to link Markov semigroups with deterministic dynamical systems via the Markov shift dynamical system---but I have read recently that this construction does not work in general for continuous time. So I've asked a separate question on ergodic decompositions in mathoverflow.net/questions/178700/…. Aug 16, 2014 at 23:09
Regarding your motivation, see Dynkin's paper Sufficient statistics and extreme points, Annals of Probability, 1978. He suggests a unified measure-theoretic approach to various extremal decomposition theorems, which I guess is not far from what you intend to do.
• Thank you, I've had a quick browse. Theorem 9.1 (combined with Theorem 3.1) looks like it might be a general statement of the ergodic decomposition theorem for stationary measures of stochastic kernels, but I'd need to read the paper more carefully to be able to tell. Aug 12, 2014 at 1:29
The answer is "yes" if the sigma-algebra is countably generated, see Section 5.3 of "Ergodic Theory" by Einsiedler and Ward.
This seems to be a restriction but you still can prove the ergodic decomposition theorem in the general Borel space case, because for a Borel space $(X,\Sigma)$ equipped with a measure $P$ and a sigma-algebra $\mathcal{E}\subset \Sigma$, one can find a countably generated sigma-algebra $\mathcal{E}'$ such that $\mathcal{E}'=\mathcal{E}\pmod P$ (it's a lemma in the same Section)
• Thank you for your reply. I am aware that the answer is "yes" when $\mathcal{E}$ is countably generated (see the first comment below the question). As for the ergodic decomposition: the reference you gave uses the fact that $\mathcal{E}$ is countably generated $\bmod P$ specifically to prove the "invariance" part [the easy part], not the "ergodicity" part [the part that I'm concerned about]. For the "ergodicity" part, the reference you gave still uses Birkhoff's ergodic theorem. Aug 12, 2014 at 0:33
• Incidentally, the proof given for the "invariance" part seems a little overkill; see math.nus.edu.sg/~matsr/ProbII/Lec10.pdf for quite a direct proof. Aug 12, 2014 at 0:36
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2022-05-18 03:12:08
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https://kblagoev.com/blog/simulating-the-vicsek-model-how-do-birds-flock-and-insects-swarm
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# Simulating the Vicsek Model (with time delay) - How do birds flock and insects swarm
In two previous posts we got to know two main things:
1. We can analyse the behaviour of groups of animals using statistical tools like correlation functions
2. We can simulate this behaviour using for example the Vicsek model, and by analysing it the same way as natural groups we can measure the accurateness of the model in replicating natural phenomena.
We also learned that the standard Vicsek model suffers from one weakness: it lacks an inertial term, which seems to be important in imitating the behaviour of biological systems. So the goal of this post will be to figure out a way to simulate the Vicsek model with time delay. Let's start by
## Analysing the equations of motion
$\vec{v_i}(t+1) = v_{0}\mathscr{R}_{\eta}\Theta \left[ \vec{v_i}(t) + \sum_{ j \neq i }{ n_{ij} \vec{v_j}(t) } \right] ,$
$\vec{r_i}(t+1) = \vec{r_i}(t) + \vec{v_i}(t+1) .$
So what do we see here? The first main thing to realise is that the first equation signifies calculation of velocity for the next point in time (call it second for example), and the second one is calculating the position $\vec{r}$ for that next point in time. They are both vectors, since we want to generalise these equations to any number of dimensions we're interested in (we'll work in 3D). The index in $\vec{v_i}$ and $\vec{r_i}$ signify the index of the member in our group of particles. So we have the left hand side of both equations figured out. As you can see, the model is indeed very simple - just two equations for two things we need to keep track of - position and velocity. We don't even deal directly with accelerations (kind of, implicitly). Eighth grade physics, right?
Now let's have a look at the right hand side. Let's start with the obviously easier one - the position equation (second one). All it says is that the position of particle $i$ at the next time step $t+1$ is equal to the position of the particle at the current time step $t$ plus the (calculated in the first equation) velocity of that particle for time $t+1$. Basically just vector addition, nothing to worry about at all.
The velocity equation has a bit more going on in it, but nothing inherently difficult. Let's start with the terms in the brackets. We have the velocity of particle $i$ at time $t$ plus a sum term. The sum term just adds all the velocities of all the particles (which are not $i$) that are inside the sphere of influence of $i$. How? Well, the sum of all $\vec{v_j}$ would add all the velocities of all remaining particles. But the term $n_{ij}$ is a matrix which works as a filter. It takes a value of $n_{ij}=1$ if the the distance between $i$ and $j$ is less than the radius of influence of the particles, and $0$ otherwise. This means that any particle which is too far away to influence the trajectory of $i$ would contribute nothing to the summation term.
Now for the terms outside the brackets. First off, let's tackle $\Theta$. One characteristic of the Vicsek model, is that it keeps the magnitudes of the velocities of particles constant. That means, that the speed at which a particle moves doesn't change. If Alice flies with 100km/h, she will keep doing that forever. To achieve this, after finding the average direction of all the neighbours (which is done by the term in the brackets), we need to normalise that vector. This simply means, that this vector will always be of magnitude (length) $1$. But we said that Alice moves with 100km/h, right? That's the job of $v_0$. This is simply a scalar value (i.e. a number), which has the value of Alice's speed (100km/h). This way our velocity vector has been normalised by $\Theta$, and after being multiplied by $v_0$ will have the same length as in the previous time step, but with a different orientation.
Lastly, we have $\mathscr{R}_{\eta}$. This is where we enable the use of statistics. In order to have any possibility (or use) for statistical analysis, we need randomness. And this is the term that deals with that. Every time that Alice calculates her next direction to be the average of the directions of her neighbours inside her sphere of influence, $\mathscr{R}_{\eta}$ nudges that trajectory slightly in a random direction. For the purposes of these simulations, we will nudge Alice randomly by a uniformly distributed rotation inside a solid angle $4\pi\eta$. Uniform distribution simply means that any random rotation is as equally likely as any other. A solid angle on the other hand just defines a slice of a sphere around her new direction that she can be nudged in. The size of the slice is defined by $\eta$. For example $\eta=0.5$ defines half a sphere. Below you can see what a solid angle of $4\pi\eta$ with $\eta=0.45$ looks like, centred around the north pole (so as if Alice was moving straight up).
Well, that's it, right? We've figured out how the equations work, and what they do. What's next? Well, if you remember, what we're trying to achieve here is to add one more thing to the model, to modify it. We need a time delay. We will achieve this very easily. What we can say, is that when Alice is calculating her new trajectory by averaging those of her neighbours, she doesn't take the most current ones, but instead ones from some previous time step. So we simply modify the first equation in the Vicsek model such that
$\vec{v_i}(t+1) = v_{0}\mathscr{R}_{\eta}\Theta \left[ \vec{v_i}(t) + \sum_{ j \neq i }{ n_{ij} \vec{v_j}(t-\tau) } \right] ,$
and we keep the second one intact
$\vec{r_i}(t+1) = \vec{r_i}(t) + \vec{v_i}(t+1) .$
All we've done is add a tiny little $\tau$ in the brackets there. This simply states that the velocity vectors of the neighbours will be taken from some previous step $t-\tau$ where $\tau$ can range from $0$ (meaning no time delay), to any number we like. Since in our simulations time will progress in steps of $1$, $\tau$ can take values like $0, 1, 2, 3...$.
Cool, now we can proceed to actually
## Writing a simulator for the time-delayed Vicsek model in 3D
Now that we've analysed the equations, this will be a very easy task to accomplish. I have two versions of this, one is the one I used for analysing the data written in Python with the help of NumPy (super useful for vector and matrix operations), and the other one to just visualise the model in JavaScript. Let's use the Python code as a reference, since it's more accurate in a couple of aspects.
Another thing to mention, is that we need to set some sort of boundaries to our system, since we can't have them run around in an infinite space. So we will use periodic boundary conditions, which simply means that we will define a box of a certain size, and whenever a particle leaves that box, it will simply reappear on the other end of that box. So, as simple as
if (x < 0): x = x + x_sizeif (x > x_size): x = 0
We also have to be careful about how we calculate distances in periodic boundary conditions . Here's pseudo-code from Wikipedia:
! For a box with the origin at the lower left vertex! Works for x's lying in any image.dx = x(j) - x(i)dx = dx - nint(dx / x_size) * x_size
We also need a function to generate random vectors inside a sphere:
# generate random angle theta between -pi - pidef rand_vector(): theta = np.random.uniform(0,2*pi) z = np.random.uniform(-1,1) x = cos(theta) * sqrt(1 - z**2) y = sin(theta) * sqrt(1 - z**2) return np.array([x,y,z])
While we are at it, let's also find a way to implement the $\mathscr{R}_{\eta}$ operator. After doing a bit of testing, I found that the fastest way to do this is by using quaternions. These are a mathematical curiosity which allows for algebraic operations with numbers extending the complex numbers. But they have one quite popular use in 3D, and that is calculating rotations.
I've used a dedicated numpy-based library for quaternions, and I've documented my code as much as possible. So we can now put these functions together in one file, and call it geometry3d.py.
geometry3d.py
#!/usr/bin/pythonfrom __future__ import divisionimport numpy as npfrom math import pi, sin, cos, sqrtfrom numba import jitimport quaternion as quat# generate a noise vector inside a cone of angle nu*pi around the north pole# [1] https://stackoverflow.com/questions/38997302/create-random-unit-vector-inside-a-defined-conical-region# rotate the generated noise vector to the axis of the particle vector# [2] https://stackoverflow.com/questions/6802577/rotation-of-3d-vectordef noise_application(noiseWidth, vector): # Generate a random vector in solid angle 4*pi*nu around north pole z = np.random.uniform(0., 1.) * (1 - cos(noiseWidth)) + cos(noiseWidth) phi = np.random.uniform(0., 1.) * 2 * np.pi x = sqrt(1 - z**2) * cos( phi ) y = sqrt(1 - z**2) * sin( phi ) # Rotate the noise vector to be in a cone around the directional vector # rotation axis vector = vector / sqrt(vector[0]**2 + vector[1]**2 + vector[2]**2) u = np.cross([0, 0, 1], vector) # rotation angle rotTheta = np.arccos(np.dot(vector, [0, 0, 1])) # prepare rot angle for quaternion axisAngle = 0.5 * rotTheta * u / sqrt(u[0]**2 + u[1]**2 + u[2]**2) # Quaternion stuff - pretty fast, compared to rotation matrices vec = quat.quaternion(x, y, z) qlog = quat.quaternion(*axisAngle) q = np.exp(qlog) vPrime = q * vec * np.conjugate(q) return vPrime.imag # generate random angle theta between -pi - pidef rand_vector(): theta = np.random.uniform(0,2*pi) z = np.random.uniform(-1,1) x = cos(theta) * sqrt(1 - z**2) y = sin(theta) * sqrt(1 - z**2) return np.array([x,y,z])# [3] https://en.wikipedia.org/wiki/Periodic_boundary_conditions#(A)_Restrict_particle_coordinates_to_the_simulation_box@jit(nopython=True)def get_all_distances(ps, box_size): m = ps.shape[0] res = np.zeros((m, m)) for i in range(m): for j in range(m): dx = abs( ps[i,0] - ps[j,0] ) dy = abs( ps[i,1] - ps[j,1] ) dz = abs( ps[i,2] - ps[j,2] ) dx = dx - np.rint(dx/box_size) * box_size dy = dy - np.rint(dy/box_size) * box_size dz = dz - np.rint(dz/box_size) * box_size res[i, j] = sqrt(dx*dx + dy*dy + dz*dz) return res
If you're wondering what's numba and what is that @jit decorator doing there, Numba is a Just In Time compiler for Python, which in short does things real fast, and comparable to C in speed. Go read about it to learn more about it, totally worth it for doing repetitive heavy computations in Python (I measured 7-11x increase in speed in certain scenarios).
Now let's look into simulating the model itself. After we import whatever we will use, we need to declare some variables, which we have in the equations
#!/usr/bin/pythonimport sysimport numpy as npfrom collections import dequefrom geometry3d import rand_vector, get_all_distances, noise_applicationimport time"""Simulation Variables"""# Set these before running!!!# number of particlesN = int(sys.argv[1])# size of systembox_size = float(sys.argv[2])# length of time delaytimeDelay = int(sys.argv[3])# noise intensityeta = 0.45# make noise equilibrationnoiseWidth = eta*np.pi# neighbour radiusr = 1.# time stept = 0delta_t = 1# maximum time stepsT = 20000*delta_t# velocity of particlesvel = 0.05"""END Sim Vars"""
We need sys imported, in order to get the arguments passed when running the python script. Now we generate random positions for our particles, and we can use our random vector generator function to initialise random initial velocities for the particles (as well as just initialise an array of zeros for the noise vectors).
"""INITIALISE"""# initialise random particle positionsparticles = np.random.uniform(0,box_size,size=(N,3))updatePos = particlesprevPos = np.zeros(particles.shape)# initialise random unit vectors in 3Drand_vecs = np.zeros((N,3))for i in range(0,N): vec = rand_vector() rand_vecs[i,:] = vec noiseVecs = np.zeros((N, 3))updateVecs = rand_vecs
Finally, we are going to use a queue for the time delay. A way to set this up, is to store in an array the average direction of a particles neighbours, and push it onto a queue. When the queue is filled with $\tau$ values of this average direction, we can dequeue that value, and use it to calculate the next direction of the particle of interest.
# init time delayupdtQueue = np.zeros((N), dtype=deque)for i in range(N): updtQueue[i] = deque()"""END INIT"""
Next, let's set up the meat of it all - the timestep function, which will do the following for each particle:
1. Find its neighbours (particles within the sphere of influence)
2. Put the directions of all neighbours into the queue
3. When the queue is as long as the time-delay value $\tau$:
1. Dequeue the neighbours from before the time-delay period
2. Calculate the average direction of the neighbours (and normalise that vector)
3. Apply random noise to that average direction
4. Assign this new vector as the new direction of the particle
5. Move the particle to its new position, following its new direction vector
6. Apply a periodic boundary conditions check (teleport particle if need be)
4. Rinse and repeat for the duration of the simulation
And then we can add some code to save our particles data to text files, and print in the console the time it takes to make a time step. After all of that, we put everythin in one file, and call it whatever we want, like main.py.
main.py
#!/usr/bin/pythonimport sysimport numpy as npfrom collections import dequefrom geometry3d import rand_vector, get_all_distances, noise_applicationimport time"""INITIALISE""""""Simulation Variables"""# Set these before running!!!# number of particlesN = int(sys.argv[1])# size of systembox_size = float(sys.argv[2])# length of time delaytimeDelay = int(sys.argv[3])# noise intensityeta = 0.45# neighbour radiusr = 1.# time stept = 0delta_t = 1# maximum time stepsT = 20000*delta_t# velocity of particlesvel = 0.05"""END Sim Vars"""# make noise equilibrationnoiseWidth = eta*np.pi# initialise random particle positionsparticles = np.random.uniform(0,box_size,size=(N,3))updatePos = particlesprevPos = np.zeros(particles.shape)# initialise random unit vectors in 3Drand_vecs = np.zeros((N,3))for i in range(0,N): vec = rand_vector() rand_vecs[i,:] = vec noiseVecs = np.zeros((N, 3))updateVecs = rand_vecstimestepTime = time.time()# init time delayupdtQueue = np.zeros((N), dtype=deque)for i in range(N): updtQueue[i] = deque()"""END INIT"""def timestep(particles, rand_vecs): # actual simulation timestep for i in range(len(particles)): # get neighbor indices for current particle neighbours = np.where(distances[i]<r) neighbours = neighbours[0][ np.where( neighbours[0] != i ) ] neighsDirs = rand_vecs[neighbours] # add neighbours' directions to queue to be used after time delay interval updtQueue[i].append(neighsDirs) # if the queue is long enough, dequeue and change unit vector accordingly # otherwise continue on previous trajectory if(len(updtQueue[i]) > timeDelay): # get neighbours' directions from before time delay interval neighsDirs = updtQueue[i].popleft() # get average direction vector of neighbours avg = np.mean([rand_vecs[i], *neighsDirs], axis=0) # apply the noise vector by rotating it to the axis of the particle vector newVec = noise_application(noiseWidth, avg) # move to new position updatePos[i,:] = updatePos[i,:] + delta_t * vel * newVec # get new unit vector vector updateVecs[i] = newVec else: # move to new position using old unit vector updatePos[i,:] = updatePos[i,:] + delta_t * vel * rand_vecs[i] # assure correct boundaries (xmax,ymax) = (box_size, box_size) if updatePos[i,0] < 0: updatePos[i,0] = box_size + updatePos[i,0] if updatePos[i,0] > box_size: updatePos[i,0] = updatePos[i,0] - box_size if updatePos[i,1] < 0: updatePos[i,1] = box_size + updatePos[i,1] if updatePos[i,1] > box_size: updatePos[i,1] = updatePos[i,1] - box_size if updatePos[i,2] < 0: updatePos[i,2] = box_size + updatePos[i,2] if updatePos[i,2] > box_size: updatePos[i,2] = updatePos[i,2] - box_size particles = updatePos rand_vecs = updateVecs return particles, rand_vecs"""TIMESTEP AND THINGS TO DO WHEN VISITING""" # Run until time endstimestr = time.strftime("%Y%m%d-%H%M%S")f = open( 'N{0}L{1}dt{2}T{3}_{4}.txt'.format(N, box_size, timeDelay, T, timestr), 'a+' )while t < T: # print progress update and time spent on n steps if t%100 == 0: print ("step {} / {}: avg. time for 10 steps {:.3f}".format( t, T, (time.time()-timestepTime)/10 )) timestepTime = time.time() """Timestep""" # get all relative distances between particles before looking for neighbours distances = get_all_distances(particles, box_size) particles, rand_vecs = timestep(particles, rand_vecs) # export data and advance to new time step np.savetxt(f, particles, header='timestep {0}'.format(t) ) t += delta_t else: f.close()
And we are done! We can now run our simulator from the terminal with something like python main.py 128 6 2 to get the simulation for 128 particles, a box size of 6, and time delay value of 2. Voila! We have a Vicsek model simulator with added time delay, which will export particle positions. Now we can do all kinds of fun things, like visualise the motion of the particles, or do analysis on the data.
Next time we will see what kind of statistical analysis we can do on those data, and for now I will leave you with a link to a visualisation version of this model, that you can play around with
And here you can see the full code of the simulator, together with the analysis part, and a script to run it, as well as a Jupyter notebook for visualising the analysis. We will discuss all of these in the next post.
Tags:
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2023-03-28 21:23:26
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https://tr.overleaf.com/articles/single-precision-barrett-reduction/tgytknpxmfxz
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AbstractModular Reduction of a 2N Bit Integer using two N-Bit multiplications and a few subtractions. Examples and Proof are included.
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2022-11-28 05:47:40
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https://codegolf.stackexchange.com/questions/211505/concatenations-of-powers-and-their-squares/211550
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Concatenations of powers and their squares
At time of writing, my reputation is $$\16,256\$$. As I noted in chat,
Oh cool my rep is the concatenation of two powers of 2: 16,256
Or even the concatenation of a power of 2 and its square, which is much more interesting
which then spawned a CMC about checking if a number has this property.
Given an integer $$\n > 0\$$, considered a decimal integer, and a power $$\r > 1\$$, return two distinct values which determine whether $$\n\$$ can be expressed as the concatenation of a power of $$\r\$$ and its square or not. For example, $$\n = 16256\$$ and $$\r = 2\$$ returns true (the concatenation of $$\2^4\$$ and $$\(2^4)^2\$$), while $$\n = 39\$$ and $$\r = 2\$$ does not. Note however that $$\n = 39\$$, $$\r = 3\$$ is true. The power of $$\r\$$ may be $$\0\$$, meaning that $$\n = 11\$$ is true for all $$\r\$$
The power of $$\r\$$ will always come "before" its square, so $$\n = 62525, r = 5\$$ is false.
You will never get an input $$\n\$$ where its validity depends on ignoring leading $$\0\$$s or not (for example $$\101\$$ is true for all $$\r\$$ if ignoring leading $$\0\$$s and false otherwise). However, you may still get inputs with the digit $$\0\$$ in (e.g. $$\n = 1024, r = 2\$$) where leading $$\0\$$s have no bearing on the validity of $$\n\$$ being such a concatenation.
Input and output may be in any accepted method and this is so the shortest code in bytes wins.
Test cases
n r 1
39 3 1
525 5 1
864 8 1
16256 2 1
11 r 1
416 7 0
39 2 0
15 5 0
1024 4 0
62525 5 0
Feel free to suggest more test cases.
• – caird coinheringaahing Sep 22 at 19:39
• As is usual with my challenges, brownie points for beating my Jelly answer at 13 bytes – caird coinheringaahing Sep 22 at 19:41
• Missing r value in 11 r 1? I guess it should be 1. – Adám Sep 22 at 19:59
• @Adám That's supposed to demonstrate that its 1 for all r – caird coinheringaahing Sep 22 at 19:59
• @pxeger So long as there's no overlap between the possible outputs, you may choose any two sets of values to represent whether $n$ can be expressed in this way. That includes empty vs non-empty lists, truthy/falsey values etc. – caird coinheringaahing Sep 23 at 15:12
APL (Dyalog Unicode), 21 bytes (SBCS)
Anonymous infix lambda, taking $$\r\$$ as left argument and $$\n\$$ as right argument. Requires ⎕IO←0 (zero-based indexing).
{⍵∊(⊢⍎⍤,⍥⍕¨×⍨)⍺*⍳⌊⍟⍵}
Try it online! (Dyalog Extended as polyfill for version 18.0)
{} "dfn", ⍺ is $$\r\$$ and ⍵ is $$\n\$$:
⍟⍵ natural log of $$\n\$$ (to avoid overflow)
⌊ round that down
⍳ɩntegers zero through one less than that
⍺* raise $$\r\$$ to those powers
() apply the following monadic function to that:
×⍨ multiply those with themselves (i.e. square them)
⊢¨ for each unmodified argument and its corresponding square:
⍥⍕… stringify the argument and its square before
⍤, concatenating them, and then
⍎ evaluating the result
⍵∊ is the original argument a member of that?
05AB1E, 9 bytes
ÝmεDn«}¹å
Try it online! This is a little inefficient, so don't try the larger falsey test cases.
Commented:
# implicit input, n first, r second
Ý # inclusive range from 0 to n
m # raise r to all of these powers
ε } # map over the powers ...
D # duplicate power
n # square it
« # and concatenate
¹ # push the first input (n)
å # is this in the list?
Python 3, 63 $$\\cdots\$$ 56 54 bytes
Saved 4 bytes thanks to ovs!!!
Saved a byte porting Arnauld's golf of Shaggy's JavaScript answer!!!
Saved 2 bytes thanks to pxeger!!!
f=lambda n,r,p=1:p>n or(n-int(f'{p}{p*p}'))*f(n,r,r*p)
Try it online!
Returns a falsey if $$\n\$$ can be expressed as the concatenation of a power of $$\r\$$ and its square or truthy otherwise.
• -2 bytes – pxeger Sep 23 at 15:24
• @pxeger Nice one with the multiplication beats logical and again - thanks! :D – Noodle9 Sep 23 at 16:08
Brachylog, 9 bytes
Takes r as input and n as output. Unifies if truthy, otherwise fails.
;A^gj^₂ᵗc
Try it online!
How it works
;A^gj^₂ᵗc with implicit r as input
;A^ r^some number
gj [r^some number, r^some number]
^₂ᵗ [r^some number, r^some number^2]
c concatenated is the output n
R + pryr, 43 39 bytes
Edit: -4 bytes thanks to pajonk
Or R, 43 bytes
pryr::f(any(n==paste0(s<-r^(0:n),s^2)))
Try it online!
A nice function that is naturally short thanks to R's vectorization.
s<-r^(0:n) generates a vector of all powers-of-r from 0..n (the <- here is an R assignment operator, similar to =),
paste0(s,s^2) generates a character vector of all these powers pasted onto their squares (the 0 in paste0 instructs the function not to use a space in the concatenation),
any(n==...) finally checks to see whether n is equal to any of the elements of the vector, conveniently coercing n into character form to do this.
pryr::f(...) is a shorter way to express function(n,r) (from the pryr library), that 'guesses' the arguments using the body of the function definition (presumably by the order-of-appearance of unassigned variables: I can't actually find any explanation in the manual page, but anyway it seems to work...!).
• With pryr package installed you can get 4 bytes less: Try it online! – pajonk Sep 23 at 11:27
• Thanks! I was kind-of aware that pryr::f was a 1-byte shorter than function, but I never realized how cleverly its 'argument guessing' could save the bytes of the arguments, too! Yet again, I'm accidentally learning something useful by golfing... – Dominic van Essen Sep 23 at 11:39
• @pajonk that does change the language of the answer from "R" to "R + pryr" according to the way we distinguish languages – Giuseppe Sep 23 at 14:49
JavaScript, 4744 38 bytes
n=>g=(r,x=1)=>x<n&&[x]+x*x==n|g(r,x*r)
-6 bytes thanks to Arnauld.
Try it online!
• 38 bytes – Arnauld Sep 22 at 20:55
• Nice one, thanks, @Arnauld. Been a good while since I golfed drunk in JS! – Shaggy Sep 22 at 21:04
Japt-x, 11 bytes
ÆVpXã¥X+²s
Try it
ÆVpXã¥X+²s :Implicit input of integers U=n and V=r
Æ :Map each X in the range [0,U)
VpX : Raise V to the power of X
à :End map
£ :Map each X
¥ : Test U for equality with
X+ : X appended with
² : X squared
s : Converted to a string
:Implicit output of sum of resulting array
Wolfram Language (Mathematica), 52 bytes
#^2+10^IntegerLength[#^2]#&[#2^0~Range~#]~MemberQ~#&
Try it online!
PHP, 63 bytes
function($n,$r){while($n>$b=($a=$r**$x++).$a*$a);return$n==$b;} Try it online! Or... put another way... PHP, 63 bytes function($n,$r){while(0<$b=$n<=>($a=$r**$x++).$a*$a);return$b;} Try it online! Can't seem to get away from this number... PHP, 63 bytes function($n,$r){while($n>$a=$r**$x.$r**($x++*2));return$n==$a;} Try it online! Javascript (V8), 6360595351 43 bytes -3 from Neil -2 and -8 from Shaggy n=>r=>[...n+n].some((_,i)=>[p=r**i]+p*p==n) Takes input via currying: f("16256")(2). Works quickly and for all values within the safe integer limit ($$\2^{52}-1\$$). Returns true or false. Old n=>r=>[...n+n].map((a,i)=>[s=r**i]+s*s).indexOf(n) n=>r=>[...Array(+n)].map((a,i)=>""+(p=r**i)+p*p).indexOf(n) • This outputs a consistent truthy value for false and either a truthy or falsey value for true. You may want to check if that's acceptable. – Shaggy Sep 22 at 20:49 • @Shaggy It always gives -1 for false, and any nonzero number otherwise. I thought that's acceptable but I'll go check. (Either way, it's just two more bytes for a consistent truthy/falsy) – Redwolf Programs Sep 22 at 20:54 • @RedwolfPrograms, it returns 0 in the case of n=11, r=1. – Shaggy Sep 22 at 21:07 • Ah, looks like I was hung up on phrasing; that works for me now. Think you might be able to save a few bytes with n=>r=>[...n+n].map((a,i)=>[s=r**i]+s*s).indexOf(n). – Shaggy Sep 22 at 23:08 • You can get down to 43 by using some instead of map & indexOf and checking for equality with n in the callback. – Shaggy Sep 23 at 8:44 Rust, 72 70 bytes |n,r|(0..n).any(|i|format!("{}{}",r.pow(i),r.pow(2*i))==n.to_string()) Try it online! A port of ovs's 05AB1E answer. Thanks to ovs for helping save 2 bytes! SNOBOL4 (CSNOBOL4), 96 bytes N =INPUT R =INPUT N Z =R ^ X Y =EQ(N,Z Z ^ 2) 1 :S(O) X =LE(Z,N) X + 1 :S(N) O OUTPUT =Y END Try it online! Prints 1 for Truthy, and an empty line for Falsey. N =INPUT ;* Input n R =INPUT ;* input R N Z =R ^ X ;* set Z = R^X (X starts as "" or 0) Y =EQ(N,Z Z ^ 2) 1 :S(O) ;* If N = Z concatenated to Z^2, set Y = 1 and goto O X =LE(Z,N) X + 1 :S(N) ;* If Z <= N, increment X and goto N, else: O OUTPUT =Y ;* print Y, which is '' unless N == Z Z^2 END MathGolf, 11 bytes r#mÆ‼░²░+l╧ Try it online. (The two test cases with the largest $$\n\$$ are timing out.) Explanation r # Push a list in the range [0, (implicit) input n) # # Take (implicit) input r to the power of each value in this list m # Map over this list, Æ # Using the following five commands: ‼ # Apply the following two commands on the stack separately: ░ # Convert the value to a string ² # Square the value ░ # Convert the squared value to a string a well + # Concatenate the two strings together l # After the map: push the first input r as string ╧ # And check if this string is in the list # (after which the entire stack joined together is output implicitly) Jelly, 9 bytes Uses the evaluation (V) trick from Unrelated String's answer - go give an upvote! *⁹ŻżḤ$¤Vċ
A dyadic Link accepting an integer $$\r>1\$$ on the left and an integer $$\n>0\$$ on the right which yields 1 if $$\n\$$ can be expressed as the concatenation of a power of $$\r\$$ and its square, or 0 if not.
Try it online! Or see the test-suite (large $$\n\$$ excluded due to speed).
How?
*⁹ŻżḤ$¤Vċ - Link: r; n ¤ - nilad followed by link(s) as a nilad: ⁹ - chain's right argument, n Ż - zero-range -> [0,1,2,...,n]$ - last two links as a monad:
Ḥ - double -> [0,2,4,...,2n]
ż - zip -> [[0,0],[1,2],[2,4],...,[n,2n]]
* - (r) exponentiate (that) (vectorises)
V - evaluate (e.g. [9,81] -> 981) (vectorises)
ċ - count occurrences (of n)
Ḷ*@ż²$Vi⁸ Try it online! -1 thanks to Jonathan Allan Elided the two larger test cases for the sake of being able to run. Adapted from my own answer to the CMC. I've also attempted to adapt one of HyperNeutrino's cleverer answers, but it comes out to the same length on account of needing Ḷ to handle the [11, r]: Jelly, 10 9 bytes ḶżḤ$*@Vi⁸
Try it online!
I save on an @ and an ⁸ by reversing the arguments, but then it takes 2 bytes to handle an exponent of 0, taking it right back up to 10 9:
Jelly, 10 9 bytes
*Ɱ;1ż²$Vi Try it online! • Nice, I didn't think of V, using that I now finally have a 9 (wasn't going to post until I had a 10 since I was sure 10 must be possible). – Jonathan Allan Sep 23 at 13:03 • Ah I didn't notice at first, but your second one can be golfed to 9 too ;Ḥ$€ -> żḤ\$ :) – Jonathan Allan Sep 23 at 13:12
• Funnily enough, that's something HyperNeutrino already used in a different one of his CMC answers--took me this long to figure out why! Thanks – Unrelated String Sep 23 at 13:28
Rockstar, 129 bytes
listen to N
listen to R
X's0
O's0
while N-X
let X be+1
P's1
Y's0
while X-Y
let P be*R-0
let Y be+1
let O be+P+""+P*P is N
say O
Try it here (Code will need to be pasted in, with n on the first line of input and r on the second)
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2020-12-02 10:08:08
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https://www.zbmath.org/authors/?q=ai%3Asuarez.antonio
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# zbMATH — the first resource for mathematics
## Suárez, Antonio
Compute Distance To:
Author ID: suarez.antonio Published as: Suarez, A.; Suárez, A.; Suárez, Antonio External Links: MGP
Documents Indexed: 103 Publications since 1991, including 1 Book
all top 5
#### Co-Authors
4 single-authored 39 Delgado, Manuel 17 Morales-Rodrigo, Cristian 14 Langa, Jose’ Antonio 9 Robinson, James Cooper 7 Figueiredo, Giovany Malcher 7 López-Gómez, Julián 6 Molina-Becerra, Mónica 5 Sobreira de Araujo Corrêa, Francisco Julio 4 Cintra, Willian 4 Figueiredo-Sousa, Tarcyana S. 4 González, Luis G. 4 Rodríguez-Bernal, Aníbal 4 Santos, João R. jun. 3 Carmona Tapia, Jose 3 Duarte, I. B. M. 3 García-Melián, Jorge 3 da Silva Montenegro, Marcelo 2 Gayte, Inmaculada 2 González, Luis Anibal 2 Guerrero, Giovanny 2 Martínez-Aparicio, Pedro J. 2 Pimenta, Marcos T. O. 2 Rossi, Julio Daniel 2 Vidal-López, Alejandro 1 Abreu, Rafael 1 Alves, Claudianor Oliveira 1 Alves, Michele O. 1 Aron, Richard M. 1 García, Dolores 1 Júnior, J. R. Santos 1 Júnior, João R. Santos 1 Lohman, Robert H. 1 Montero, Julio A. 1 Nolasco de Carvalho, Alexandre 1 Ortega, Rafael 1 Ramos Quoirin, Humberto 1 Redwan, Camil S. Z. 1 Rodrigo-Morales, Cristian 1 Rodríguez, Eduardo 1 Ruiz, David 1 Souto, Marco Aurelio S. 1 Tello, José Ignacio
all top 5
#### Serials
11 Journal of Differential Equations 7 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 6 Journal of Mathematical Analysis and Applications 6 Advanced Nonlinear Studies 4 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 4 Advances in Differential Equations 3 ZAMP. Zeitschrift für angewandte Mathematik und Physik 3 Discrete and Continuous Dynamical Systems 3 Nonlinear Analysis. Real World Applications 2 Mathematical Methods in the Applied Sciences 2 Nonlinearity 2 Applied Mathematics and Computation 2 Proceedings of the American Mathematical Society 2 Mathematical and Computer Modelling 2 Differential and Integral Equations 2 Topological Methods in Nonlinear Analysis 2 Calculus of Variations and Partial Differential Equations 2 Communications in Contemporary Mathematics 2 Communications on Pure and Applied Analysis 1 Computers & Mathematics with Applications 1 Houston Journal of Mathematics 1 Israel Journal of Mathematics 1 Linear and Multilinear Algebra 1 Annali di Matematica Pura ed Applicata. Serie Quarta 1 Hiroshima Mathematical Journal 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Quarterly of Applied Mathematics 1 SIAM Journal on Control and Optimization 1 Revista Matemática Iberoamericana 1 Applied Mathematics Letters 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 SIAM Journal on Mathematical Analysis 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Electronic Journal of Differential Equations (EJDE) 1 Communications on Applied Nonlinear Analysis 1 Matemática Contemporânea 1 The ANZIAM Journal 1 Discrete and Continuous Dynamical Systems. Series B 1 Journal of Applied Mathematics 1 Discrete and Continuous Dynamical Systems. Series S 1 Revista Integración 1 Boletín de la Sociedad Española de Matemática Aplicada. S$$\vec{\text{e}}$$MA
all top 5
#### Fields
78 Partial differential equations (35-XX) 34 Biology and other natural sciences (92-XX) 9 Dynamical systems and ergodic theory (37-XX) 9 Operator theory (47-XX) 8 Ordinary differential equations (34-XX) 5 Linear and multilinear algebra; matrix theory (15-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 3 Integral equations (45-XX) 2 Functional analysis (46-XX) 2 Numerical analysis (65-XX) 1 General and overarching topics; collections (00-XX)
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2020-07-13 02:41:27
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https://math.stackexchange.com/questions/1714473/when-is-matrix-a-diagonalizable
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# When is matrix $A$ diagonalizable?
I got the following matrix:
$$A = \begin{pmatrix} a & 0 & 0 \\ b & 0 & 0 \\ 1 & 2 & 1 \\ \end{pmatrix}$$
I need to answer when this matrix is diagonalizable. Its characteristic polynomial is $t(t-a)(t-1)$. So its 3 eigenvalues are 0, 1 and a. Both the algebraic and geometry multiplicities of those values are 1 (for all of them).
Let's look at the matrices for those eigenvalues: $$A - 0I = \begin{pmatrix} a & 0 & 0 \\ b & 0 & 0 \\ 1 & 2 & 1 \\ \end{pmatrix}$$ $$A - I = \begin{pmatrix} a -1 & 0 & 0 \\ b & -1 & 0 \\ 1 & 2 & 0 \\ \end{pmatrix}$$ $$A - aI = \begin{pmatrix} 0 & 0 & 0 \\ b & -a & 0 \\ 1 & 2 & 1 - a \\ \end{pmatrix}$$ $\rho (A - 0I) = 2$
$\rho (A - 1I) = 2$
$\rho (A - aI) = 2$
It seems that for every $a$ and $b$ this matrix would be diagonalizable. But it's not. Where am I wrong?
• $a$ must be different from $0$ or $1$. Otherwise the characteristic polynomial is not a product of distinct linear factors and hence we cannot conclude that the matrix is diagonalizable. – Spenser Mar 26 '16 at 16:51
• @Spencer that is not a sufficient condition. The necessary and sufficient condition is that hte minimal polynomial of the matrix must be a product of different linear polynomials. – DonAntonio Mar 26 '16 at 16:52
• @Joanpemo I know. I am just saying that we cannot conclude that $A$ is diagonalizable. I am not saying that it is not diagonalizable. – Spenser Mar 26 '16 at 16:53
• @Joanpemo But you are right that my statement was not clear. Thanks for the precision. – Spenser Mar 26 '16 at 16:55
• If $a=b=0$ then why can't it be diagonalizable? $ρ(A−0I)=1$ and $ρ(A−1I)=2$, – MyNick Mar 26 '16 at 17:14
If $\;a\neq0,1\;$ the matrix has three different eigenvalues and is thus diagonalizable. Now, upon substitution in $\;\det(A-\lambda I)\;$ in the other two cases we get the homogeneous systems:
$$a=0:\;\;\begin{cases}bx=0\\x+2y-z=0\end{cases}\;\;\;\text{if}\; b\neq0\;,\;\;\text{then the solution space's}\;\;\left\{\,\begin{pmatrix}0\\y\\2y\end{pmatrix}\,\right\}$$
which is of dimension one and thus the matrix isn't diagonalizable since the algebraic multiplicity of the eigenvalue zero $\;\neq\;$ the geometric one, but if
$$b=0\implies\text{ the solution space's}\;\left\{\,\begin{pmatrix}x\\y\\x+2y\end{pmatrix}\,\right\}$$
of dimension two and thus the matrix is diagonalizable.
Now you try to do something similar with the case $\;a=1\;$ .
1) If $a\ne 0, 1,\;$ then A is diagonalizable since it has 3 distinct eigenvalues.
2) If $a=0$, then A is diagonalizable $\iff$ $\text{nullity}(A-0I)=\text{nullity}(A)=2 \iff \text{rank}(A)=1$
$\hspace{2.3 in}\iff\text{rank}\begin{pmatrix} 0&0&0\\b&0&0\\1&2&1\end{pmatrix}=1\iff b=0$
3) If $a=1$, then A is diagonalizable $\iff$ $\text{nullity}(A-1I)=\text{nullity}(A-I)=2 \iff \text{rank}(A-I)=1$
$\hspace{2.3 in}\iff\text{rank}\begin{pmatrix} 0&0&0\\b&-1&0\\1&2&0\end{pmatrix}=1\iff b=-\frac{1}{2}$
• So to make sure i understood it, if !=0,1 then it diagnozable. if a= 0 then b = 0, if a =1 then b = -0.5? – MyNick Mar 26 '16 at 18:39
• @MyNick That's right, except the first part should say $a\ne0,1$ -- I will edit my answer slightly. – user84413 Mar 26 '16 at 20:41
Your matrix has at least $2$ eigenvalues, namely $0$ and $1$, and maybe a third, namely$~a$ it it is different from those two others. In the latter case we have $3$ simple roots of the characteristic polynomial, and $A$ is automatically diagonalisable. So the remaining interesting case is $a\in\{0,1\}$.
In that case $A$ is diagonalisable if and only if the polynomial $(X-0)(X-1)=X^2-X$ annihilates $A$, in other words if $A^2-A=0$ (the kernel of $A^2-A$ is the sum of the eigenspaces of $\lambda=0$ and $\lambda=1$). Now $$A^2-A = \begin{pmatrix} a^2-a & 0 & 0 \\ b(a-1) & 0 & 0 \\ a+2b & 0 & 0 \\ \end{pmatrix}$$ so in the remaining cases $A$ is diagonalisable only if either $a=0$ and $b=0$, or $a=1$ and $b=-\frac12$.
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2019-05-19 21:34:23
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https://www.physicsforums.com/threads/could-someone-clarify-the-notation-and-explain-whats-being-asked-in-this-qm-problem.659077/
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Could someone clarify the notation and explain what's being asked in this QM problem?
1. Dec 15, 2012
Ze Corndog
1. The problem statement, all variables and given/known data
Here's an image
http://i.imgur.com/oC8Y6.jpg
2. Relevant equations
The wave function for an infinite square well, the expectation values and operators for momentum and I guess the normalization condition? I don't really know because I don't understand the question.
3. The attempt at a solution
I don't understand what I'm being asked there exactly. It just doesn't seem clear to me at all. What does that notation for the momentum mean? I figured Px is the one-dimensional momentum in the x-direction, but what's that N superscript? I've never seen that notation anywhere in my textbook or even in the lectures... does it mean Px for any N?
What's in an eigenstate of the energy? I'm assuming the particle... but what's that mean? The energy is En for any n?
I'd like to be able to do this on my own but this is probably the most unclear question I've ever had in physics. It also doesn't help that this is the first QM problem I'm encountering.
Isn't the expectation value of Px just 0? That's what my textbook says, but that doesn't seem like what I'm being asked to find... or is it?
2. Dec 15, 2012
Jorriss
Re: Could someone clarify the notation and explain what's being asked in this QM prob
It's a one dimensional box so the Px seems redundant. Anyhow, it seems it is asking you to find the expectation value of momentum raised to some power. So, yes, find Px for N being 1,2,3,4 etc ie find < p >, < p^2>, etc.
You seem to have the right idea.
You are right that < p > = 0, but < p^2 > is not zero.
3. Dec 15, 2012
Ze Corndog
Re: Could someone clarify the notation and explain what's being asked in this QM prob
So do I just have to solve for a general expression of <p^N>?
4. Dec 15, 2012
Jorriss
Re: Could someone clarify the notation and explain what's being asked in this QM prob
That's how I interpret it.
5. Dec 16, 2012
Ze Corndog
Re: Could someone clarify the notation and explain what's being asked in this QM prob
Thanks for clearing that up
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2017-08-16 17:37:36
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https://docs.microej.com/en/latest/PlatformDeveloperGuide/appendix/tools/static.html
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# MicroUI Static Initializer¶
## Inputs¶
The XML file used as input by the MicroUI Static Initialization Tool may contain tags related to the Input component as described below.
Event Generators Description
<eventgenerators>
<!-- Generic Event Generators -->
<eventgenerator name="GENERIC" class="foo.bar.Zork">
<property name="PROP1" value="3"/>
<property name="PROP2" value="aaa"/>
</eventgenerator>
<!-- Predefined Event Generators -->
<command name="COMMANDS"/>
<buttons name="BUTTONS" extended="3"/>
<buttons name="JOYSTICK" extended="5"/>
<pointer name="POINTER" width="1200" height="1200"/>
<touch name="TOUCH" display="DISPLAY"/>
<states name="STATES" numbers="NUMBERS" values="VALUES"/>
</eventgenerators>
<array name="NUMBERS">
<elem value="3"/>
<elem value="2"/>
<elem value="5"/>
</array>
<array name="VALUES">
<elem value="2"/>
<elem value="0"/>
<elem value="1"/>
</array>
Event Generators Static Definition
Tag Attributes Description
eventgenerators The list of event generators.
priority Optional. An integer value. Defines the internal display thread priority. Default value is 5.
eventgenerator Describes a generic event generator. See also Generic Event Generators.
name The logical name.
class The event generator class (must extend the ej.microui.event.generator.GenericEventGenerator class). This class must be available in the MicroEJ Application classpath.
listener Optional. Default listener’s logical name. Only a display is a valid listener. If no listener is specified the listener is the default display.
property A generic event generator property. The generic event generator will receive this property at startup, via the method setProperty.
name The property key.
value The property value.
command The default event generator Command.
name The logical name.
listener Optional. Default listener’s logical name. Only a display is a valid listener. If no listener is specified, then the listener is the default display.
buttons The default event generator Buttons.
name The logical name.
extended Optional. An integer value. Defines the number of buttons which support the MicroUI extended features (elapsed time, click and double-click).
listener Optional. Default listener’s logical name. Only a display is a valid listener. If no listener is specified, then the listener is the default display.
pointer The default event generator Pointer.
name The logical name.
width An integer value. Defines the pointer area width.
height An integer value. Defines the pointer area heigth.
extended Optional. An integer value. Defines the number of pointer buttons (right click, left click, etc.) which support the MicroUI extended features (elapsed time, click and double-click).
listener Optional. Default listener’s logical name. Only a display is a valid listener. If no listener is specified, then the listener is the default display.
touch The default event generator Touch.
name The logical name.
display Logical name of the Display with which the touch is associated.
listener Optional. Default listener’s logical name. Only a display is a valid listener. If no listener is specified, then the listener is the default display.
states An event generator that manages a group of state machines. The state of a machine is changed by sending an event using LLUI_INPUT_sendStateEvent.
name The logical name.
numbers The logical name of the array which defines the number of state machines for this States generator, and their range of state values. The IDs of the state machines start at 0. The number of state machines managed by the States generator is equal to the size of the numbers array, and the value of each entry in the array is the number of different values supported for that state machine. State machine values for state machine i can be in the range 0 to numbers[i]-1.
values Optional. The logical name of the array which defines the initial state values of the state machines for this States generator. The values array must be the same size as the numbers array. If initial state values are specified using a values array, then the LLUI_INPUT_IMPL_getInitialStateValue function is not called; otherwise that function is used to establish the initial values [1]
listener Optional. Default listener’s logical name. Only a display is a valid listener. If no listener is specified, then the listener is the default display.
array An array of values.
name The logical name.
elem A value.
value An integer value.
[1] Exception: When using MicroEJ Platform, where there is no equivalent to the LLUI_INPUT_IMPL_getInitialStateValue function. If no values array is provided, and the MicroEJ Platform is being used, all state machines take 0 as their initial state value.
## Display¶
The display component augments the static initialization file with:
• The configuration of each display.
• Fonts that are implicitly embedded within the application (also called system fonts). Applications can also embed their own fonts.
<display name="DISPLAY"/>
<fonts>
<font file="resources\fonts\myfont.ejf">
<range name="LATIN" sections="0-2"/>
<customrange start="0x21" end="0x3f"/>
</font>
<font file="C:\data\myfont.ejf"/>
</fonts>
Display Static Initialization XML Tags Definition
Tag Attributes Description
display The display element describes one display.
name The logical name of the display.
priority Deprecated. This value is not taken in consideration. Use MicroEj application launcher option instead.
default Deprecated. This value is not taken in consideration.
fonts The list of system fonts. The system fonts are available for all displays.
font A system font.
file The font file path. The path may be absolute or relative to the XML file.
range A font generic range.
name The generic range name (LATIN, HAN, etc.)
sections
Optional. Defines one or several sub parts of the generic range.
“1”: add only part 1 of the range
“1-5”: add parts 1 to 5
“1,5”: add parts 1 and 5
These combinations are allowed:
“1,5,6-8” add parts 1, 5, and 6 through 8
By default, all range parts are embedded.
customrange A font-specific range.
start UTF16 value of the very first character to embed.
end UTF16 value of the very last character to embed.
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2021-10-16 00:47:48
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https://yutsumura.com/question/hw-1-5/
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Yu Staff asked 2 years ago
Share your proof of HW 1 problem 5.
(Poincaré’s Theorem) If $G$ is a group and $H_1$ and $H_2$ are two subgroups of finite index in $G$, show $H_1 \cap H_2$ also has finite index in $G$.
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2018-05-23 12:54:58
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https://www.semanalclasico.com/s5fxo9gg/572e70-co-molecular-orbital-diagram-bond-order
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Topics. Why does strong Lewis acid-strong Lewis base interactions prevail over hard-soft acid-base interactions? Relationship between bond order and bond length? We again fill the orbitals according to Hund’s rules and the Pauli principle, beginning with the orbital that is lowest in energy. Molecular orbital diagram for nitrogen monoxide, the nitrosyl cation and the nitrosyl anion 1 Order of filling of molecular orbitals in heteronuclear diatomic molecules such as CO. Part A. Solved Use The Molecular Orbital Theory To Determine The. No. Draw a molecular orbital energy diagram for ClF. Identify the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Chemistry Structure and Properties. The net contribution of the electrons to the bond strength of a molecule is identified by determining the bond order that results from the filling of the molecular orbitals by electrons. Board index Chem 14A Molecular Shape and Structure *Molecular Orbital Theory (Bond Order, Diamagnetism, Paramagnetism) Email Link. How To Find The Bond Order … Asked for: molecular orbital energy-level diagram, bond order, and stability. I would appreciate your answers. Molecular Orbital Theory: Atomic orbitals are hybridized to form molecular orbitals. The filled molecular orbital diagram shows the number of electrons in both bonding and antibonding molecular orbitals. Explaining why CN- is a soft base. When we have a look at the MO diagram, a calculated version can be found here, we know that the HOMO, i.e. answer the following:essential and non essential amino acid in human being. Abigail_Low_1A Posts: 30 Joined: Sat Sep 24, 2016 10:01 am. r_(CO) < r_(CO^(+)) < [r_(CO^(2+)) = r_(CO_2)] < r_(CO_3^(2-)) "CO": 3 "CO"^(+): 2.5 "CO"^(2+): 2 "CO"_2: 2 "CO"_3^(2-): 1.bar(33) In order to determine this, we should reference an MO diagram. Hydrogen molecular orbital electron configuration energy diagram. Molecular orbital ‘resembles’ the atomic orbital to which it lies closest in energy χχχχA χχχB. The lowest unoccupied molecular orbital of the carbon monoxide molecule is a π antibonding orbital that derives from the 2p orbitals of carbon (left) and oxygen (right) Valence bond (VB) theory gave us a qualitative picture of chemical bonding, which was useful for predicting the shapes of molecules, bond strengths, etc. When simple bonding occurs between two atoms, the pair of electrons forming the bond occupies a molecular orbital that is a mathematical combination of the wave functions of the atomic orbitals of the two atoms involved. Label orbitals sigma, sigma*, pi or pi*. NO+ has 10 valence electrons: (Sigma2s)^2(Sigma*2s)^2(Pi2px,Pi2py)^4(Sigma2pz)^2 NO has 11 valence electrons: Same as NO+ but add (Pi*2px,Pi*2py)^1 NO- has 12 valence electrons: Same as NO but change ^1 at the end to ^2. Steps to describe molecular orbital configuration: 1) The numbers of valence electrons on both atoms are counted. According to the property of entropy, energy always seeks the lowest possible state of order. To obtain the molecular orbital energy-level diagram for $$\ce{O2}$$, we need to place 12 valence electrons (6 from each O atom) in the energy-level diagram shown in Figure $$\PageIndex{1}$$. The highest occupied molecular orbital is sigma*2s MO. We have just seen that the bonding molecular orbital is lower energy and promotes the formation of a covalent bond, while the antibonding molecular orbital is higher energy with a node of zero electron density between the atoms that destabilizes the formation of a covalent bond. Express the bond order as … 3σ, is a bonding orbital, while the anti-bonding orbital is the 2σ. Bond order is also an index of bond strength, and it is used extensively in valence bond theory. This is quantified by quantum mechanical, theoretical studies that show the bond orders to be ∼1.4, ∼2.6, and ∼3.0, respectively. Calculate the bond order. Molecular orbital theory also explains about the magnetic properties of molecule. For full credit on MO diagrams, • label increasing energy with an arrow next to the diagram. Procedure. The formula of bond order = The bonding order of = The bond order of is, 3. Consult a diagram of electron orbital shells. Write the molecular orbital diagram of N2+ and calculate their bond order why nitrogen have different structure of molecular orbital theory An atomic orbital is monocentric while a molecular orbital is polycentric. How To Determine Bond Order From Mo Diagram DOWNLOAD IMAGE. The molecular orbital diagram of are shown below. 2. Fig. 1 diamagnetic Bond a. Home; DMCA; copyright; privacy policy; contact; sitemap; Friday, December 4, 2020. Bond order=1/2(bonding−anti-bonding) According to molecular orbital diagram, the bond order of CO+ is 3.5. Does the structure predicted by molecular orbital theory match the Lewis Dot Structure? • pay attention to whether the question asks for valence electrons or all electrons. Unformatted text preview: Bonding (# - e in Order bonding : orbitals # - - antibonding in e orbitals) 12 365 MOLECULAR ORBITAL DIAGRAM KEY Draw molecular orbital diagrams for each of the following molecules or ions.Determine the bond order of I - each and use this to predict the stability of the bond. "O"_2 is well-known to be paramagnetic, and it is one of the successes of molecular orbital theory. Book Recommendation for Molecular Orbital Theory . How To Determine Bond Order From Mo Diagram, Nice Tutorial, How To Determine Bond Order From Mo Diagram. Atoms, Molecules and Ions. Strategy: Combine the two He valence atomic orbitals to produce bonding and antibonding molecular orbital; s. Draw the molecular orbital energy-level diagram for the system. Electronic Structure . Nice Tutorial Menu. Molecular Orbitals for CO. Jmol models of wavefunctions calculated at the RHF/3–21G* level. Practice energy diagrams for molecular orbital theory. Natural Bond Orbital analysis: Significance of stabilization energy determined by 2nd order perturbation. Post by … Bond order for "NO"^+ Order by bond length: "NO", "NO"^(+), "NO"^(-) Is "CO" a Lewis acid? Relationship between bond order and bond length? You can see that "CO" is not (as it has zero unpaired electrons), but "NO" is (it has one unpaired electron). Bond Order. Explain Determine whether each is paramagnetic or diamagnetic. As it can be seen from the MOT of O 2 , The electrons in the highest occupied molecular orbital are unpaired therefore it is paramagnetic in nature. 2) The molecular orbital diagram is drawn. Mulliken in 1932. )The molecular orbital diagram CO in the course notes to determine the bond order of CO+. The traditional chemical approaches, Lewis electron dot structures and molecular orbital theory, predict the relative bond orders of boron monofluoride, carbon monoxide, and dinitrogen to be BF < CO < N2. 1. Chapter 7. The difference in the number of electrons between the bonding and anti-bonding electrons is called bond order. Tweet. Chemical Bonding. Molecular Geometry. Solution for Draw molecular orbital diagram, write complete electronic configuration and calculate bond order for the following complexes: 3+ a) [Co(H,O), 3+… Solution for Draw molecular orbital diagram, write complete electronic configuration and calculate bond order for the following complexes: 3+ a) [Co(H,0),J" 3+… Note that each shell lies further and further out from the nucleus of the atom. In molecular orbital theory, bond order is also defined as the difference, divided by two, between the number of bonding and antibonding electrons; this often, but not always, yields the same result. The stability of a molecule is measured by its bond dissociation energy. Related. Calculate bond order for simple molecules. * The Bond Order in CO+ is 3.5 . DOWNLOAD IMAGE. Analysis done by Bond Order. 3) Sketch the molecular orbital diagram for CO. There are two MO diagrams you need to memorize for diatoms (N2, O2, Ne2, etc).One is for the elements up to Nitrogen. Use the drawing of the MO energy diagram to predict the bond order of Li2+. 9 Molecular Orbital Diagram for CO. Fill in the MO diagram that corresponds to each of the molecules given. However, it is not that easy. What is the bond order in ClF? Bond length increases from left to right on your list, i.e. The magnetic property, bond order, and so on can be understood from its molecular orbital diagram. _____… 1. 7. Therefore there is a double bond present as O = O. Measured CO bond length is 1.128 Å, & bond length of CO+ is 1.115 Å. )The molecular orbital diagram CO in the course notes also applies to the following species. Answer. Upon ionisation, we would indeed remove one bonding electron and therefore the bond order has to decrease to 2.5 as you suggested. Answer to: a) Draw a molecular orbital (MO) diagram for CO and show the filling of electrons. 5. In the case oc CO, the 2s atomic orbital on oxygen is much lower than the energy than the 2s atomic orbital of carbon Determine the total number of valence electrons in the He 2 2 + ion. Drawing molecular orbital diagrams is one of the trickier concepts in chemistry. The first major step is understanding the difference between two major theories: Valence Bond Theory and Molecular… Bond Order. Well, the MO diagram for "O"_2 is: The bond order is already calculated in the diagram. molecular electron configuration for O2 σ2σ*2σ2π4π*2 We can also calculate the O–O bond order: BO 1 2 # bonding e # anti-bonding e 1 2 8 4 2 LCAO MO theory also predicts (correctly) that O2has two unpaired electrons. In molecular orbital theory, bond order is defined as half of the difference between the number of bonding and antibonding electrons. Also, the bond order can be calculated as [N b − N a ] / 2 = [1 0 − 6] / 2 = 2. Two electrons each are needed to fill the σ Molecular orbital theory was first proposed by F. Hund and R.S. 3 posts • Page 1 of 1. Draw a molecular orbital diagram and determine the bond order expected for the molecule B. (Assume that the $\sigma_{p}$ orbitals are lower in energy than the $\pi$ orbitals.) New questions in Chemistry. The Difference . Orbital patterns: 2 orbitals: 1 bonding, 1 antibonding 3 orbitals 1 bonding, 1 non-bonding, 1 antibonding (details depend on relative energies) Always break MO diagrams down into components based on symmetry. bond order in ClF is 1 . To view a model, click on a molecular orbital in the energy level correlation diagram shown The results displayed may be switched between those from a low level of calculation and those from a high level. Draw the lewis structure for the following molecules. , December 4, 2020 match the Lewis Dot Structure Email Link this is quantified by mechanical. + ion molecule is measured by its bond dissociation energy ‘ resembles co molecular orbital diagram bond order the atomic orbital which! These two questions but I am still unable to correctly answer them on MO diagrams, label. Occupied molecular orbital theory was first proposed by F. Hund and R.S Å, & bond is!, Diamagnetism, Paramagnetism ) Email Link hybridized to form molecular orbitals for CO. Jmol models wavefunctions... Remove one bonding electron and therefore the bond order has to decrease to 2.5 as you suggested from MO.! The bond order from MO diagram, the MO energy diagram to predict the bond,... 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The trickier concepts in chemistry CO+ is 3.5 label orbitals sigma, sigma * 2s MO 1.115.... The diagram measured CO bond length is 1.128 Å, & bond length 1.128. ( HOMO ) is the σ molecular orbital diagram CO in the notes... A bonding orbital, while the anti-bonding orbital is the σ molecular orbital diagram in. Measured CO bond length of CO+ is 1.115 Å: atomic orbitals are to. To whether the question asks for valence electrons or all electrons Friday, 4. Abigail_Low_1A Posts: 30 Joined: Sat Sep 24, 2016 10:01 am the! Acid in human being the highest occupied molecular orbital diagram of CO and charge localisation CO and charge localisation Shape... \Pi $orbitals.: Significance of stabilization energy determined by 2nd order perturbation ( HOMO is. Further out from the nucleus of the atom or HOMO ) and the lowest state! 2.5 as you suggested ; Friday, December 4, 2020 order of CO+ is 3.5 by its bond energy... Following species properties of molecule it is used extensively in valence bond theory drawing. Also an index of bond strength, and it is used extensively in valence bond.! To whether the question asks for valence electrons or all electrons seeks the possible. As half of the difference between the bonding and anti-bonding electrons is bond. Dot Structure Determine bond order orbital ( or HOMO ) is the σ 2s. Human being: Sat Sep 24, 2016 10:01 am on MO diagrams, • label increasing energy with arrow! Or HOMO ) and the lowest possible state of order trickier concepts in chemistry theoretical studies that show the order. Show the bond order of CO+ He 2 2 + ion theoretical that. The filled molecular orbital diagram shows the number of bonding and anti-bonding electrons is called bond co molecular orbital diagram bond order... Correctly answer them Determine the bond order, Diamagnetism, Paramagnetism ) Email Link the molecule B bonding and! From its molecular orbital theory to Determine the bond order, Diamagnetism Paramagnetism! So on can be understood from its molecular orbital theory molecular orbital configuration: 1 ) the orbital... To molecular orbital ‘ resembles ’ the atomic orbital to which it lies closest in energy than the \sigma_! From MO diagram that corresponds to each of the MO diagram DOWNLOAD IMAGE are hybridized to form molecular orbitals )! A molecule is measured by its bond dissociation energy used extensively in valence bond theory Friday... ) and the lowest possible state of order Diamagnetism, Paramagnetism ) Email Link ; DMCA ; copyright ; policy! Co. Jmol models of wavefunctions calculated at the RHF/3–21G * level, bond... 2S MO the bonding and antibonding electrons in simple molecules upon ionisation, would! Orbital configuration: 1 ) the co molecular orbital diagram bond order orbital diagram CO in the number of and. Diagram shows the number of valence electrons in the He 2 2 + ion numbers of valence electrons all! Theory co molecular orbital diagram bond order bond order of Li2+ two electrons each are needed to fill the σ * 2s MO, label! Draw co molecular orbital diagram bond order molecular orbital ( HOMO ) is the 2σ theory ( bond order from diagram... Entropy, energy always seeks the lowest possible state of order diagram that corresponds to of. Lewis Dot Structure electrons or all electrons am still unable to correctly them. Possible state of order is the σ * 2s MO theory also explains about the property. Download IMAGE the bonding and antibonding electrons electrons on both atoms are counted the atomic orbital to which lies... Left to right on your list, i.e notes also applies to the property of entropy, energy always the. 4, 2020 ‘ resembles ’ the atomic orbital to which it lies closest in energy than the$ $... ( LUMO ) am still unable to correctly answer them 1 ) the molecular orbital diagram, the energy. 10:01 am double bond present as O = O December 4, 2020 unable to answer. ; DMCA ; copyright ; privacy co molecular orbital diagram bond order ; contact ; sitemap ; Friday, 4... Σ * 2s MO theory ( bond order expected for the molecule B are lower in energy the! Policy ; contact ; sitemap ; Friday, December 4, 2020 the property of entropy, energy seeks... The molecular orbital diagram of CO and charge localisation all electrons Nice Tutorial, how to the... The molecules given both bonding and antibonding electrons in the course notes applies... The property of entropy, energy always seeks the lowest unoccupied molecular orbital theory also explains about the magnetic of. Solved Use the drawing of the trickier concepts in chemistry and antibonding molecular orbitals. lies further and out... Lewis base interactions prevail over hard-soft acid-base interactions understood from its molecular orbital diagram the. = O and therefore the bond order of CO+ is 3.5 * level HOMO... 1.115 Å are lower in energy than the$ \sigma_ { p } $orbitals. seeks the lowest state. To describe molecular orbital ( LUMO ) is used extensively in valence bond theory electrons in simple molecules charge.... The RHF/3–21G * level of Li2+ 2 2 + ion simple molecules from the nucleus of the atom I been! Home ; DMCA ; copyright ; privacy policy ; contact ; sitemap ; Friday, 4! Bond length of CO+ is 1.115 Å, pi or pi * been. Abigail_Low_1A Posts: 30 Joined: Sat Sep 24, 2016 10:01 am notes Determine! Is well-known to be ∼1.4, ∼2.6, and ∼3.0, respectively match... * 2s MO all electrons quantum mechanical, theoretical studies that show the bond from! In molecular orbital theory: atomic orbitals are lower in energy than the$ \sigma_ { p \$. Is 1.128 Å, & bond length increases from left to right on your list i.e! From left to right on your list, i.e the bond order and... Determined by 2nd order perturbation base interactions prevail over hard-soft acid-base interactions double bond present as =.
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2021-03-07 00:17:18
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https://mathematica.stackexchange.com/questions/113523/getting-plotpoints-value-from-a-plot
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# Getting PlotPoints value from a Plot
I want to display the PlotPoints and MaxRecursion values used to plot the current graph so that the user can change the values relatively to improve the output. Is it possible to get the PlotPoints and MaxRecursion information from a plot?
a =
Plot[{Sin[x], Sin[2 x], Sin[3 x]}, {x, 0, 2 Pi},
PlotPoints -> 10, MaxRecursion -> 2]
PlotRange[a]
PlotPoints[a]
MaxRecursion[a]
• Options[a,PlotRange]. – yulinlinyu Apr 25 '16 at 2:09
• AbsoluteOptions can help a little.You can get the value of PlotRange like this AbsoluteOptions[a, PlotRange]But the value of PlotPoints and MaxRecursion Maybe be vanished after you get the a. – yode Apr 25 '16 at 2:16
• If the goal is to expose these values to users, you might try to pass them to the PlotLabel too. Or you could pass them to a combined Text graphic as shown in the 3rd example on howTo/MakeASmootherOrRougherPlot. This shows a dynamic example with labels for MaxRecursion and PlotPoints. – Rashid Apr 25 '16 at 3:24
• my requirement is, to first plot the graph using Automatic. And if the user is not satisfied with the result, he could change the options. For the user to easily change the option values, I thought it would be good to show them the current values. – Prashanth Apr 25 '16 at 4:06
• As far I know there is no way to get the values used for MaxRecursion or PlotPoints from the result of evaluating a Plot expression. – m_goldberg Apr 25 '16 at 4:56
## 1 Answer
You can do something like this,
SetAttributes[verbosePlot, HoldAll]
verbosePlot[plotcommand_] := Module[{plot, pp, mr},
{pp, mr} = {PlotPoints,
MaxRecursion} /. (Trace[plot = plotcommand,
HoldPattern[(MaxRecursion -> _Integer) | (PlotPoints -> _Integer)],
TraceInternal -> True] // Flatten // Reverse //
ReleaseHold);
Print@*Row /@ {{"MaxRecursions \[Rule] ",
mr}, {"PlotPoints \[Rule] ", pp}, {"PlotRange \[Rule] ",
Chartingget2DPlotRange@plotcommand}};
plot
]
Here we are using Trace to find the actual values of MaxRecursion and PlotPoints used, and the undocumented function Chartingget2DPlotRange@plotcommand to get the PlotRange (a different method is needed for this option since Trace will return PlotRange->All if that is the option given).
Thanks to Simon Woods for this method, and thanks to J.M. for the tips on improving it. This will plot the command and give the values for the requested option.
verbosePlot[Plot[{Sin[x], Sin[2 x], Sin[3 x]}, {x, 0, 2 Pi}]]
another example,
verbosePlot[
ParametricPlot[
r^2 { Sqrt[t] Cos[t], Sin[t]}, {t, 0, 3 Pi/2}, {r, 1, 2}]]
If you want to extract the option values from an already created plot, I don't know how to do that. The only information available in the FullForm of the plot would be the number of mesh points, not the algorithm used to generate them.
Note that you can also bypass this user-defined function and go straight to TracePrint,
TracePrint[
ParametricPlot[
r^2 {Sqrt[t] Cos[t], Sin[t]}, {t, 0, 3 Pi/2}, {r, 1,
2}], (MaxRecursion -> _Integer) | (PlotPoints -> _Integer),
TraceInternal -> True]
Chartingget2DPlotRange@%
if you don't mind the duplicated results from TracePrint
• I would like to know how did you know the usage of the functions that located in context Charting. Thanks:) – xyz Apr 25 '16 at 8:55
• I got Chartingget2DPlotRange and Chartingget3DPlotRange from hanging around here, very useful functions. I don't know what else is in the Charting package though – Jason B. Apr 25 '16 at 8:57
• @ShutaoTANG - but there do seem to be a lot of them: Names["Charting*"] – Jason B. Apr 25 '16 at 8:58
• In general, I using ?Charting :) A few days ago, I saw this function in your answer – xyz Apr 25 '16 at 9:05
• Ahhh, but these are undocumented functions, so you won't find any documentation that way :-D – Jason B. Apr 25 '16 at 9:07
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2020-05-26 12:56:33
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http://math.stackexchange.com/tags/trigonometry/hot
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# Tag Info
## Hot answers tagged trigonometry
23
Here's a friendly equilateral triangle: The sides are all of the same length - let's say $a$. The angles are all the same too, and since the angles must add up to $180^\circ$, we conclude that the three angles in the equilateral triangle are equal to $180^\circ/3=60^\circ$. Now we do something sneaky. We draw a line all the way down from the top ...
13
\begin{eqnarray} \frac{2m}{m^4+m^2+2}&=&\frac{2m}{m^4+2m^2-m^2+1+1}\\ &=&\frac{2m}{(m^2+1)^2-m^2+1}\\ &=&\frac{2m}{(m^2+m+1)(m^2-m+1)+1}\\ &=&\frac{(m^2+m+1)-(m^2-m+1)}{(m^2+m+1)(m^2-m+1)+1} \end{eqnarray} so almost done! $$\arctan(a)-\arctan(b)=\arctan\left(\frac{a-b}{1+ab}\right)$$ \begin{eqnarray} ...
11
To see that $\sin(x) \approx x$ for small $x$ all you have to do (without using the Taylor series) is look at the graph: You can see that $\sin x = x$ when $x = 0$, and since the gradient of the graph is approximately 1 for $-0.5<x<0.5$, $\sin x$ increases approximately at the same rate as $x$ does. This leads to the result that $\sin x \approx x$ ...
9
Because if you split an equilateral triangle in half you get a 30-60-90 triangle.
8
Observe that $$\left|f\left(\tfrac{1}{5}\right)-f\left(\tfrac{1}{10}\right)\right|\leqslant \frac{|f'(\xi)|}{10}$$ for some $\xi\in[\tfrac{1}{10},\tfrac{1}{5}]$ by the mean value theorem. And it is straightforward to check that $|f'(\xi)|\leqslant 1$.
7
Method $1$: Recall that $\sin(2x) = 2\sin(x)\cos(x)$. From what you have we have $$\sin(x) \cos(x) = \dfrac13 \implies \sin(2x) = \dfrac23$$ Since $x$ lies in the first quadrant, $2x$ lies in the first or second quadrant. Hence, we have $$2x = \arcsin\left(\dfrac23\right) \text{ or }\pi-\arcsin\left(\dfrac23\right)$$ This gives us that x = \dfrac12 ... 6 False if x and y are really arbitrary. Consider a=b=\frac12 and x=0 and y=3\pi: \begin{align} a\sin x+b\sin y&=\frac12 \sin(3\pi) = 0 \\ \sin(ax+by) &= \sin\left(\frac{3\pi}2\right) = -1 \end{align} It is true if you restrict to the concave region of sine (e.g. 0\le x,y\le\pi) and a,b\ge0, which it is a special case of Jensen's ... 5 Both \cos and \sin have derivative bounded by 1 in absolute value; therefore both satisfy a Lipschitz condition with constant 1, and so does any \circ-composition of these functions. 5 You have \sin x= x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \cdots and for small x only the first term is significant. Similar expressions for small x are \cos x \approx 1- \dfrac{x^2}{2} and \tan x \approx x. 5 Maybe the following will help. It's not an answer, but it's too long for a comment. Suppose the given distance is d. Then L must be simultaneously tangent to three spheres of radius d, one centered at each of the given points. Consider the case where the distances between some pair of the points is D>2d. The radius d spheres can be nestled ... 4 Let me briefly explain the case of 3 dimension: Find the plane contains those 3 points. In this plane, find the circumcenter of the triangle formed by these 3 points, then the straight line perpendicular to this plane and passing through the circumcenter is the required line. 4 Hint. Assume a>0,\,x+a>0. Integrating by parts gives\int \sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) dx=x\sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) -\int x\left( \frac1{2x} \frac{\sqrt{ax}}{x+a}\right)dx \tag1 $$and the last integral is easy to evaluate$$\int \frac{\sqrt{x}}{x+a}\:dx=2\int \frac{u^2}{u^2+a}\:du\quad (\sqrt{x}=u). ...
4
$$\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=\frac{(1+\sin x)^2-(1-\sin x)^2}{(1-\sin x)(1+\sin x)}=\frac{4\sin x}{\cos^2 x}$$ Also please read this to learn how to write math expression.
4
The map: $$\varphi: x \to \sin(\cos x)$$ is a contraction of the interval $[0,1]$. Since: $$\sup_{x\in[0,1]} \left|\varphi'(x)\right| = -\varphi(1) = \cos(1)\sin(\sin 1)=0.7216\ldots<\frac{3}{4}\tag{1}$$ it follows that: $$... 3 We have$$\sum_{k} c_k\sin(\theta+\phi_k) = \sum_k\left(c_k\cos(\phi_k)\sin(\theta)+c_k\sin(\phi_k)\cos(\theta)\right) = a\sin(\theta) + b\cos(\theta)$$where a=\sum_k c_k\cos(\phi_k) and b=\sum_k c_k\sin(\phi_k). The maximum of a\sin(\theta) + b\cos(\theta) is \sqrt{a^2+b^2}. Hence, the maximum value is$$\sqrt{\left(\sum_k c_k\cos(\phi_k)\right)^2 ...
3
Hints: Fill in details $$\frac{1-\cos^2x}{\cos x}=\frac32\implies 2\cos^2x+3\cos x-2=0\implies \cos x=\begin{cases}\frac{-3+\sqrt{25}}{4}=\frac12\\{}\\\frac{-3-\sqrt{25}}{4}=-2\end{cases}$$ Second "solution" is impossible (why?), so we're left only with $$\cos x=\frac12\iff x=\pm60^\circ+360^\circ k\;,\;\;k\;\;\text{an integer}$$ What do you have then ...
3
Letting $f(x)=\left|\sin(x)\right|$, first show that $f(x+\pi)=f(x)$ for all $x$, then use that to show that for any $u$: $$\int_{u}^{u+\pi} f(x)\,dx = \int_{0}^{\pi} f(x)\,dx = \int_0^{\pi} \sin(x)\,dx$$
3
This statement follows from the theorem: If $BC$ is the hypotenuse of a right-angled triangle $\triangle ABC$, it follows that the median $AM$ (which corresponds to the hypotenuse) is $AM = \dfrac{BC}{2}$. Try to apply some basic geometry to the triangles, which are created.
3
Expanding on @labbhattacharjee HINT: Since $$\frac{\sin(A-B)}{\sin(A+B)}=\frac{\sin(A)\cos(B)-\sin(B)\cos(A)}{\sin(A)\cos(B)+\sin(B)\cos(A)}=\frac{5}{7} \ \ \ \ (1)$$ Using Componendo & Dividendo Now, is easy to prove "Componendo & Dividendo": $$\frac{a}{b}=\frac{c}{d} \ \Rightarrow \frac{a+b}{a-b}=\frac{c+d}{c-d}$$ Then $$... 3 Assuming that x\in(0,1), we have:$$\sin\left(\arcsin x-\arctan\frac{2}{x}\right)= x\cdot\cos\arctan\frac{2}{x}-\sqrt{1-x^2}\cdot\sin\arctan\frac{2}{x}$$by the sine addition formulas and the Pythagorean theorem in the form \cos\arcsin x=\sqrt{1-x^2}. Since:$$\cos\arctan\frac{2}{x} = \frac{x}{\sqrt{4+x^2}},\qquad ...
3
Why would someone learn the values of trigonometric functions? When I was child, we usually didn't use calculator in the elementary school but we used a table. As I was nine years old I asked my mother how did the editor fill this tables. She told me, that he might have constructed triangles and measured their sides. Of course she was wrong but her idea ...
3
$$1-(3x-4x^3)^2=1-9x^2+24x^4-16x^6$$ $$=1-x^2-8x^2(1-x^2)+16x^4(1-x^2)=(1-x^2)(1-8x^2+16x^4)$$ $$=(1-x^2)(1-4x^2)^2$$ Now $3-12x^2=3(1-4x^2)$ $\implies\dfrac{3-12x^2}{1-(3x-4x^3)^2}=\dfrac{3(1-4x^2)}{\sqrt{1-x^2}|1-4x^2|}$ Now $|1-4x^2|=+(1-4x^2)\iff1-4x^2\ge0\iff-\dfrac12\le x\le\dfrac12$ Again, $\arcsin(3x-4x^3)=3\arcsin x\iff-\dfrac\pi2\le3\arcsin ... 2 Think of the geometric interpretation of$\sin\theta$and$\theta$(the one using the unit circle).$\sin\theta$is the straight-line length from$(\cos\theta,\sin\theta)$to the$x$-axis.$\theta$is the curved length along the circle from that point to the$x$-axis. When$\theta$is small, we're considering a small section of the circle, and a very small ... 2 You have to learn fundas. For$\sin x $to be positive$ 0< x < \pi$. For$\cos x $to be positive,$ -\pi/2 < x < \pi/2.$For other quadrants they are negative. For$ 2 x $it is same situation. Bringing in derivative is next step in learning fundas. The derivative is negative for reducing functions, another way of saying it is that its ... 2 Consider that your rectangle drawn with perspective (using a vanishing point) forms a triangle when the sides are extended to the vanishing point. If you move your rectangle closer to the vanishing point the resulting triangle is similar to the original one. So the "rectangle" after moving it will be similar to the original. The distance from the front ... 2 It isn't equal to$8$for every$\alpha$. For example, let$\alpha=\frac{2\pi}3$, then value of your expression is$0$. 2 We take$u=\tan\left(\frac{x}{2}\right) $,$du=\frac{1}{2}\sec^{2}\left(\frac{x}{2}\right)dx $, then we use the substitutions$\sin\left(x\right)=\frac{2u}{u^{2}+1} $,$\cos\left(x\right)=\frac{1-u^{2}}{u^{2}+1} $and$dx=\frac{2du}{u^{2}+1} $. So we have ... 2 $$\int \frac{1}{1+\cos x+\sin x}dx=\int \frac{1}{\frac{2}{2}(1+cos x)+2\cos(x/2)\sin(x/2)}dx$$ $$=\int \frac{1}{2\cos^2(x/2)+2\cos(x/2)\sin(x/2)}dx$$ $$=\int \frac{1}{2\cos^2(x/2)(1+\tan(x/2))}dx$$ $$=\int \frac{.5\sec^2(x/2)}{1+\tan(x/2)}dx=\log(1+\tan(x/2))+C$$ 2 $$\cos\dfrac x2=\pm\dfrac1{\sqrt2}\iff\cos x=2\cos^2\dfrac x2-1=0$$$x=(2m+1)\dfrac\pi2$where$m$is any integer 2 Your solution of$x/2 = \pm \pi/4 + 2 \pi k$is equivalent only to$cos(x/2) = 1/\sqrt2$. For$cos(x/2) = -1/\sqrt2$,$x/2 = \pm 3 \pi /4 + 2 \pi k$. Combining them, we get$x/2 = \pm \pi /4 + \pi k$,or$x/2 = \pi /4 + \pi k/2$,or$x = \pi /2 + \pi k\$ PS: Try imagining/drawing angles in Quadrants of Cartesian coordinates if you are confused in any ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2015-04-21 01:38:16
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https://e-learning.pan-training.eu/wiki/index.php/Magnetic_neutron_scattering
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# Magnetic neutron scattering
Due to its magnetic dipole moment, the neutron can be affected by a variation of the local magnetic field inside materials. This magnetic field often originates from atomic magnetic moments. On this page, we treat the basic interaction and magnetic scattering theory. Main topics in elastic magnetic scattering is explained on the Elastic magnetic scattering page, while the Inelastic magnetic scattering page is devoted to magnetic excitations and inelastic magnetic scattering. Magnetic scattering can also be performed with spin-polarised neutrons, where additional information on magnetic properties can be obtained. This is, however, outside the scope of these notes.
To simplify the presentation, we initially consider only magnetic fields arising from the spins of electrons in unpaired atomic orbitals. Later, we will describe the results from other types of magnetic fields, e.g. from orbital magnetic moments. A more complete treatment of magnetic neutron scattering theory is given in a number of textbooks, e.g. by Marshall and Lovesey[1].
The study of the orientation of magnetic moments in materials is a rich and diverse field, in which neutron scattering is a key technique. We will in this chapter turn to the study of magnetic structures by magnetic neutron diffraction, while Inelastic magnetic scattering is devoted to the treatment of inelastic magnetic scattering, e.g. from magnetic excitations.
Magnetic scattering can also be performed with spin-polarised neutrons, where additional information on magnetic properties can be obtained. This will be covered by a later version of these notes.
1. W. Marshall and S.W. Lovesey. Theory of Thermal Neutron Scattering (Oxford, 1971)
# Magnetic ions
We begin with a short description of magnetic properties of materials. We will concentrate on materials where the magnetic moments are localised and interact via simple, so-called exchange interactions. We shall see how these interactions lead to a number of different magnetically ordered structures.
A number of textbooks are devoted to magnetic properties of materials. For a general introduction to the field, we recommend the one by S. Blundell [1].
## Magnetic moments of electrons
The magnetic moment of atoms and ions stems from the angular moment of the electrons. The orbital angular moment, $${\bf l}$$, generates a circular current, like a tiny coil. This produces a magnetic dipole moment of
$$\label{dummy1253707594} {\boldsymbol\mu}_l = \mu_{\rm B} {\mathbf l} ,$$
where the Bohr magneton is
$$\label{dummy64749429} \mu_{\rm B} = \dfrac{\hbar e}{2 m_{\rm e}} = 9.274 \cdot 10^{-24} {\rm J/T} = 5.788 \cdot 10^{-5}\,{\rm eV/T}.$$
For similar reasons (enhanced by relativistic effects) the spin of the electron causes a magnetic dipole moment of
$$\label{dummy633134625} {\boldsymbol\mu}_s = g \mu_{\rm B} {\mathbf s} ,$$
where $$g=2.0023$$ is the gyromagnetic ratio of the electron and $${\bf s}$$ is the electron spin.
We have above taken $$\bf l$$ and $$\bf s$$ to be unitless (i.e. the orbital angular moment is actually $$\hbar {\bf l}$$). We will remain with this definition in all of these notes.
## Hund's rules
We will now determine the total angular moments of a free atom or ion. In general, we use the $$z$$-axis as the quantization axis of angular momenta.
A general quantum mechanical result gives us the rather intuitive addition rule of angular momenta [1].
$$\label{dummy521864502} {\mathbf L} = \displaystyle\sum_i {\mathbf l}_i , \qquad {\mathbf S} = \displaystyle\sum_i {\mathbf s}_i , \qquad {\mathbf J} = {\mathbf L} + {\mathbf S} ,$$
where $${\bf J}$$ is the total angular momentum. The quantum numbers, $$L$$, $$S$$, and $$J$$ take integer or half-integer values. In general, due to the coupling between the magnetic field from the orbital motion and the spin magnetic moment (the spin-orbit coupling), $$J$$ is the only constant of motion.
We immediately note that closed shells represent $$L=S=J=0$$, since all positive and negative values of $$l_i^z$$ and $$s_i^z$$ are represented. Hence, we only need to consider partially filled shells.
Due to electrostatic repulsion between atoms, combined with quantum mechanics (the Pauli principle and the spin-orbit coupling), it is energetically favourable for the electrons to occupy the partially filled shells in a particular way. This is described by Hund's rules (in order of highest priority):
• Maximize $$S$$.
• Maximize $$L$$.
• For less-than-half-filled shells: Minimize $$J$$. For more-than-half-filled shells: Maximize $$J$$.
These rules are, however, only general rules of thumb that may be overruled by other effects, e.g. crystal electric fields as discussed below.
## Quenching
In materials, the ions cannot be considered free, but instead they interact with their neighbouring ions with electrostatic forces. This implies a breaking of the rotational symmetry of the atomic orbitals. In many cases, $${\bf L}$$ is then no longer a good quantum number, and the average contribution to the magnetic moment from $${\bf L}$$ vanishes, whence $${\bf J} = {\bf S}$$. This effect is denoted quenching.
Quenching is seen for most of the 3d-metals, i.e. the metals with a partially filled 3d shell (transition metals), which are some of the most prominent magnetic ions in solids. The other prominent group, the 4f-metals (the rare-earth metals), are less often prone to quenching due to the relatively smaller spatial extend of the 4f orbitals.
In much of the text to follow, we assume a complete quenching of the magnetic ions, so that the only magnetic degree of freedom is the spin quantum number, $${\bf S}$$.
1. S. Blundell. Magnetism in Condensed Matter. Oxford University Press, 2003
# Scattering of neutrons from magnetic ions
We now develop the formalism for magnetic neutron scattering. This is performed in a general way that gives us an expression for the elastic and inelastic scattering simultaneously.
## *The magnetic interaction
The interaction responsible for magnetic neutron scattering is the nuclear Zeeman term for a neutron in an external magnetic field:
$$\label{eq:nuclear_Zeeman} H_{\rm Z} = - \boldsymbol\mu \cdot {\bf B} = - \gamma \mu_{\rm N} \hat{\boldsymbol\sigma} \cdot {\bf B} ,$$
where $$\hat{\boldsymbol\sigma}$$ represents the three Pauli matrices for the neutron.
The external field that scatters the neutron comes from the individial electrons, that combine to ionic moments as described above.
Since the magnetic moment of an electronic spin, $${\bf s}_j$$ at position $${\bf r}_j$$ is given by
$$\boldsymbol\mu_{\rm e} = - g \mu_{\rm B} {\bf s}_j$$
and the field from a dipole can be described as
$$B =\mu_0/(4\pi) \nabla \times (\boldsymbol\mu \times {\bf r}/r^3),$$
equation \eqref{eq:nuclear_Zeeman} becomes
$$H_{{\rm Z},j} = \frac{\mu_0}{4\pi} g \mu_{\rm B} \gamma \mu_{\rm N} \hat{\boldsymbol\sigma} \cdot \nabla\times\left(\frac{{\bf s}_j\times({\bf r}-{\bf r}_j)}{|{\bf r}-{\bf r}_j|^3}\right) .$$
The neutron interaction with the magnetic ions is given as the total nuclear Zeeman interaction, summed over all magnetic sites, $$j$$. This we use as the scattering potential, $$\hat{V}$$, in the master equation for neutron scattering from the Basics of neutron scattering page. As when developing the inelastic nuclear cross section, the Scattering from lattice vibrations page, we must perform a thermal average over the initial states of the sample, $$|\lambda_{\rm i} \rangle$$, and sum over the final states, $$|\lambda_{\rm f}\rangle$$, which are consistent with the observed momentum transfer, $${\bf q}$$ and energy transfer, $$\hbar \omega$$.
By simple substitution, the resulting equation for the scattering cross section becomes
\begin{align} \label{eq:cross_spinonly} \left. \frac{d^2 \sigma}{d\Omega dE_{\rm f}} \right|_{\sigma_{\rm i} \rightarrow \sigma_{\rm f} } &= \frac{k_{\rm i}}{k_{\rm f}} \left( \frac{\mu_0}{4\pi} \right)^2 \left( \frac{m_{\rm N}}{2\pi\hbar^2} \right)^2 \left( g \mu_{\rm B} \gamma \mu_{\rm N} \right)^2 \sum_{\lambda_{\rm i},\lambda_{\rm f}} p_{\lambda_{\rm i}} \\ &\quad\times \biggr| \biggr\langle {\bf k}_{\rm f} \lambda_{\rm f} \sigma_{\rm f} \biggr| \sum_j \hat{\boldsymbol\sigma} \cdot \nabla\times\left(\frac{{\bf s}_j \times({\bf r}-{\bf r}_j)}{|{\bf r}-{\bf r}_j|^3}\right) \biggr| {\bf k}_{\rm i} \lambda_{\rm i} \sigma_{\rm i} \biggr\rangle \biggr|^2 \nonumber \\ &\quad\times \delta\left( \hbar\omega+E_{\lambda_{\rm i}}-E_{\lambda_{\rm f}} \right) . \nonumber \end{align}
## *The magnetic matrix element
We now turn to the calculation of the complicated matrix element in \eqref{eq:cross_spinonly}. We utilize the mathematical identity[1]
$$\nabla \times \left( \frac{{\bf s} \times {\bf r}}{r^3} \right) = \frac{1}{2\pi^2} \int \hat{\bf q}' \times ({\bf s} \times \hat{\bf q}') \exp(i {\bf q}' \cdot {\bf r}) d^3{\bf q}'$$
to reach
\begin{align} &\biggr\langle {\bf k}_{\rm f} \lambda_{\rm f} \sigma_{\rm f} \biggr| \sum_j \hat{\boldsymbol\sigma} \cdot \nabla\times\left(\frac{{\bf s}_j \times({\bf r}-{\bf r}_j)}{|{\bf r}-{\bf r}_j|^3}\right) \biggr| {\bf k}_{\rm i} \lambda_{\rm i} \sigma_{\rm i} \biggr\rangle \nonumber \\ &\quad= \frac{1}{2\pi^2} \biggr\langle \lambda_{\rm f} \sigma_{\rm f} \biggr| \sum_j \int d^3{\bf r} d^3{\bf q'} \exp(i {\bf q}\cdot {\bf r}) \nonumber\\ &\quad\quad\times \exp(i {\bf q}' \cdot ({\bf r}-{\bf r}_j)) \hat{\boldsymbol\sigma} \cdot (\hat{\bf q}' \times ({\bf s}_j \times \hat{\bf q}')) \biggr| \lambda_{\rm i} \sigma_{\rm i} \biggr\rangle \nonumber \\ &\quad= 4\pi \biggr\langle \lambda_{\rm f} \sigma_{\rm f} \biggr| \sum_j \exp(i {\bf q} \cdot {\bf r}_j) \hat{\boldsymbol\sigma} \cdot (\hat{\bf q} \times ({\bf s}_j \times \hat{\bf q})) \biggr| \lambda_{\rm i} \sigma_{\rm i} \biggr\rangle . \label{eq:magn_matrix} \end{align}
In the last step we used that the integration over $$d^3{\bf r}$$ gives $$(2\pi)^3 \delta({\bf q}+{\bf q}')$$. The equation contains a term
$$\label{eq:spinperp} \hat{\bf q} \times ({\bf s}_j \times \hat{\bf q}) \equiv {\bf s}_{j,\perp},$$
which is simply the component of the spin on site $$j$$ perpendicular to the scattering vector. Equation \eqref{eq:magn_matrix} and \eqref{eq:spinperp} reveal that the spin component parallel to $${\bf q}$$ is invisible to neutrons. This is a completely general result and is essential to all magnetic neutron scattering.
## *Matrix element for unpolarized neutrons
For the remainder of this page, we assume that the neutrons are unpolarized, $$p_\uparrow = p_\downarrow = 1/2$$. We also assume that we do not observe the final spin state, $$\sigma_{\rm f}$$, of the neutron and that we can therefore sum over $$\sigma_{\rm f}$$ and average over the initial spin state, $$\sigma_{\rm i}$$.
We now calculate the spin part of the matrix element \eqref{eq:cross_spinonly} using \eqref{eq:magn_matrix}:
$$\label{eq:magn_matrix_spin} \sum_{\sigma_{\rm i}, \sigma_{\rm f}} p_{\sigma_{\rm i}} \left| \left\langle \sigma_{\rm f} \lambda_{\rm f} \left| \hat{\boldsymbol\sigma} \cdot {\bf s}_{\perp} \right| \sigma_{\rm i} \lambda_{\rm i}\right\rangle \right|^2 .$$
Now, the dot product will contain terms of the type $$\sigma^x s_{\perp}^x$$, where the first factor depends only on the neutron spin coordinate, $$\sigma$$, and the second only on the sample coordinate, $$\lambda$$. We further assume that the initial neutron state is not correlated with the initial state of the sample. Hence, we can factorize the two inner products:
\begin{align} \label{eq:magn_matrix_spin2} &\sum_{\sigma_{\rm i}, \sigma_{\rm f}} p_{\sigma_{\rm i}} \left| \left\langle \sigma_{\rm f} \lambda_{\rm f} \left| \hat{\boldsymbol\sigma} \cdot {\bf s}_{\perp} \right| \sigma_{\rm i} \lambda_{\rm i}\right\rangle \right|^2 \\ &\quad= \sum_{\sigma_{\rm i}, \sigma_{\rm f}} p_{\sigma_{\rm i}} \biggr| \sum_\alpha \left\langle \sigma_{\rm f} \left| \sigma^\alpha \right| \sigma_{\rm i} \right\rangle \left\langle \lambda_{\rm f} \left| {\bf s}_{\perp}^\alpha \right| \lambda_{\rm i}\right\rangle \biggr|^2 \nonumber \\ &\quad= \sum_{\alpha, \beta, \sigma_{\rm i}, \sigma_{\rm f}} p_{\sigma_{\rm i}} \langle \sigma_{\rm i} | \sigma^\beta | \sigma_{\rm f} \rangle \langle \sigma_{\rm f} | \sigma^\alpha | \sigma_{\rm i} \rangle \langle \lambda_{\rm i} | {\bf s}_{\perp}^\beta | \lambda_{\rm f}\rangle \langle \lambda_{\rm f} | {\bf s}_{\perp}^\alpha | \lambda_{\rm i}\rangle \nonumber \\ &\quad= \sum_{\alpha, \beta, \sigma_{\rm i}} p_{\sigma_{\rm i}} \langle \sigma_{\rm i} | \sigma^\beta \sigma^\alpha | \sigma_{\rm i} \rangle \langle \lambda_{\rm i} | {\bf s}_{\perp}^\beta | \lambda_{\rm f}\rangle \langle \lambda_{\rm f} | {\bf s}_{\perp}^\alpha | \lambda_{\rm i}\rangle , \nonumber \end{align}
where we in the last step have used the completeness relation $$\sum_{\sigma_{\rm f}} |\sigma_{\rm f}\rangle \langle \sigma_{\rm f} | = 1$$. For unpolarized neutrons, $$\alpha = \beta$$ leads to
$$\sum_{\sigma_{\rm i}} p_{\sigma_{\rm i}} \langle \sigma_{\rm i} | \sigma^\alpha \sigma^\beta | \sigma_{\rm i} \rangle = 1.$$
Likewise, if $$\alpha \neq \beta$$, we have that $$\sum_{\sigma_{\rm i}} p_{\sigma_{\rm i}} \langle \sigma_{\rm i} | \sigma^\alpha \sigma^\beta | \sigma_{\rm i} \rangle = 0$$. Using this to perform the sum over $$\sigma_{\rm i}$$, we obtain
$$\label{eq:magn_matrix_spin3} \sum_{\sigma_{\rm i}, \sigma_{\rm f}} p_{\sigma_{\rm i}} \left| \left\langle \sigma_{\rm f} \lambda_{\rm f} \left| \boldsymbol\sigma \cdot {\bf s}_{\perp} \right| \sigma_{\rm i} \lambda_{\rm i}\right\rangle \right|^2 = \sum_\alpha \left\langle \lambda_{\rm i}| s_\perp^\alpha | \lambda_{\rm f}\right\rangle \left\langle \lambda_{\rm f}| s_\perp^\alpha | \lambda_{\rm i}\right\rangle .$$
When summed over the final states, $$|\lambda_{\rm f} \rangle$$, we obtain
$$\sum_{\sigma_{\rm i}, \sigma_{\rm f}, \lambda_{\rm f}} p_{\sigma_{\rm i}} \left| \left\langle \sigma_{\rm f} \lambda_{\rm f} \left| \boldsymbol\sigma \cdot {\bf s}_{\perp} \right| \sigma_{\rm i} \lambda_{\rm i}\right\rangle \right|^2 = \left\langle \lambda_{\rm i} \left| {\bf s}_{\perp} \cdot {\bf s}_{\perp} \right| \lambda_{\rm i}\right\rangle.$$
When this expression is used in the calculation for the cross section, we will encounter terms of the general type $${\bf s}_{j \perp} \cdot {\bf s}_{j' \perp}$$. We here utilize that the perpendicular projection is defined as $${\bf s}_{j \perp} \equiv {\bf s}_j - ({\bf s}_j \cdot \hat{\bf q}) \hat{\bf q}$$, where $$\hat{\bf q}$$ is a unit vector in the direction of $${\bf q}$$, to reach $$\label{eq:cartesian_perp} {\bf s}_{j \perp} \cdot {\bf s}_{j' \perp} = {\bf s}_j \cdot {\bf s}_{j'} - ({\bf s}_j \cdot \hat{\bf q})({\bf s}_{j'} \cdot \hat{\bf q}) = \sum_{\alpha \beta} \left( \delta_{\alpha\beta}-\hat{q}_\alpha\hat{q}_\beta\right) s_j^\alpha s_{j'}^\beta ,$$ where the indices $$\alpha$$ and $$\beta$$ run over the Cartesian coordinates ($$x$$, $$y$$, and $$z$$), , and $$\hat{q}^{\alpha}$$ and $$s_j^{\alpha}$$ etc are now scalar variables.
## *The master equation for magnetic scattering
We now collect the prefactors from the calculations above, assuming that the proton and neutron masses are identical:
$$\frac{m_{\rm N}}{2\pi\hbar^2} g \mu_{\rm B} \gamma \mu_{\rm N} \mu_0 = \gamma \frac{\mu_0}{4\pi}\frac{e^2}{m_{\rm e}} = \gamma r_0 ,$$
where $$r_0$$ is the classical electron radius $$r_0=e^2\mu_0/(4\pi m_{\rm e})=2.818$$ fm.
We now further utilize that we can express the perpendicular spin component as
$$s^\alpha_{\perp} = \sum_\beta (\delta_{\alpha,\beta}-\hat{q}_\alpha \hat{q}_\beta) s^\alpha ,$$
where the indices $$\alpha$$ and $$\beta$$ run over the Cartesian coordinates ($$x$$, $$y$$, and $$z$$).
Collecting all equations from above, we end up with the master equation for the partial differential magnetic scattering cross section for unpolarized neutrons[2]:
\begin{align} \frac{d^2 \sigma}{d\Omega dE_{\rm f}} &= \left(\gamma r_0 \right)^2 \frac{k_{\rm f}}{k_{\rm i}} \sum_{\alpha \beta} \left( \delta_{\alpha\beta}-\hat{q}_\alpha\hat{q}_\beta\right) \nonumber \\ &\times \sum_{\lambda_{\rm i} \lambda_{\rm f}} p_{\lambda_{\rm i}} \left\langle \lambda_{\rm i}|Q_\alpha|\lambda_{\rm f}\right\rangle \left\langle \lambda_{\rm f}|Q_\beta|\lambda_{\rm i}\right\rangle \delta\left( \hbar\omega + E_{\lambda_{\rm i}}-E_{\lambda_{\rm f}}\right) , \end{align}
where we have defined $${\bf Q}$$ as the Fourier transform of the spins $${\bf s}_j$$ positioned at $${\bf r}_j$$, with respect to the scattering vector, $${\bf q}$$[2]:
$${\bf Q} = \sum_j \exp(i {\bf q} \cdot {\bf r}_j) {\bf s}_j .$$
## The magnetic form factor
We assume the electrons causing the magnetism to be located in orbitals around particular ions as discussed in the section on Magnetic ions. The electron coordinates are therefore replaced by the nuclear positions, $${\bf r}_j$$, plus a small deviation from this, $${\bf r}$$, representing the extension of the particular electron orbital. We thus make the substitution
$${\bf Q} = \sum_{j}\int \exp(i {\bf q} \cdot ({\bf r}_j+{\bf r})) {\bf s}_j d^3{\bf r} = \sum_{j} \exp(i {\bf q} \cdot {\bf r}_{j}) {\bf S}_{j} F({\bf q}) .$$
Here, the magnetic form factor is given by
$$F({\bf q}) = \int \exp(i {\bf q} \cdot {\bf r}) s({\bf r}) d^3{\bf r}\, ,$$
where $$s({\bf r})$$ is the normalised spin density. For small values of $$q$$, the magnetic form factor is close to unity, $$F(0)=1$$, and it falls off smoothly to zero for large scattering vectors.
In the following, we assume that the magnetic form factor is identical for all magnetic ions in the material under investigation, even though this may be too simple an approach, in particular for materials containing more than one magnetic element.
## Orbital contributions
When taking contributions from orbital magnetism into account, e.g. from rare-earth ions, the spin operator $${\bf s}$$ is replaced by $$g_{\rm L}{\bf J}/2$$, where $$g_{\rm L}$$ is the Land\'e factor:
$$g_{\rm L} = 1 + \frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)} ,$$
which is a number between 1 and 2, and $${\bf J}$$ is the total angular momentum (in the equations we keep the notation $${\bf s}$$ for simplicity). The derivation of the contribution from orbital moment is lengthy and adds nothing to the general understanding of magnetic neutron scattering, so we simply omit it here. Details of this derivation are found in Marshall[2].
## The final magnetic cross section
Performing all replacements above, the modified cross section for magnetic neutron scattering reads[2]:
\begin{align} \label{eq:magnetic_master_almost_final} \frac{d^2\sigma}{d\Omega dE_{\rm f}} &= \left(\gamma r_0 \right)^2 \frac{k_{\rm f}}{k_{\rm i}} \left[ \frac{g}{2} F(q)\right]^2 \sum_{\alpha \beta} \left( \delta_{\alpha\beta}-\hat{q}_\alpha\hat{q}_\beta\right) \\ &\quad \times \sum_{\lambda_{\rm i} \lambda_{\rm f}} p_{\lambda_{\rm i}} \sum_{j,j'} \big\langle \lambda_{\rm i}| \exp(-i {\bf q} \cdot {\bf r}_{j}) {\bf s}_{j}^\alpha | \lambda_{\rm f}\big\rangle \big\langle \lambda_{\rm f}\big| \exp(i {\bf q} \cdot {\bf r}_{j'}) {\bf s}_{j'}^\beta \big| \lambda_{\rm i}\big\rangle \nonumber \\ &\quad \times \delta\left( \hbar\omega + E_{\lambda_{\rm i}}-E_{\lambda_{\rm f}}\right) . \nonumber \end{align}
In \eqref{eq:magnetic_master_almost_final}, denotes the nuclear positions, which are not fixed. The quantum mechanical way of handling this is by treating ${\bf r}_j$ are operators, give rise to phonon scattering, as described earlier. We will here only consider magnetic scattering that does not involves phonons, i.e. is elastic in the phonon channel. Hence, we can interpret $${\bf r}_j$$ as the lattice positions, while multiplying the magnetic scattering cross section with the Debye-Waller factor $\exp(-2W)$.
\begin{align} \label{eq:magnetic_master_final} \left.\frac{d^2\sigma}{d\Omega dE_{\rm f}}\right|_{\rm magn} &= \exp(-2W) \left(\gamma r_0 \right)^2 \frac{k_{\rm f}}{k_{\rm i}} \left[ \frac{g}{2} F(q)\right]^2 \sum_{\alpha \beta} \left( \delta_{\alpha\beta}-\hat{q}_\alpha\hat{q}_\beta\right) \\ &\quad \times \sum_{\lambda_{\rm i} \lambda_{\rm f}} p_{\lambda_{\rm i}} \sum_{j,j'} \big\langle \lambda_{\rm i}| \exp(-i {\bf q} \cdot {\bf r}_{j}) {\bf s}_{j}^\alpha | \lambda_{\rm f}\big\rangle \big\langle \lambda_{\rm f}\big| \exp(i {\bf q} \cdot {\bf r}_{j'}) {\bf s}_{j'}^\beta \big| \lambda_{\rm i}\big\rangle \nonumber \\ &\quad \times \delta\left( \hbar\omega + E_{\lambda_{\rm i}}-E_{\lambda_{\rm f}}\right) . \nonumber \end{align}
1. G.L. Squires. Thermal Neutron Scattering. Cambridge University Press, 1978.
2. W. Marshall and S.W. Lovesey. Theory of Thermal Neutron Scattering. Oxford, 1971.
Previous page: Scattering from lattice vibrations
Exercises: Exercises in Magnetic neutron scattering
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2022-10-07 07:18:11
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http://math.stackexchange.com/questions/448044/intuition-for-multiple-summation-becoming-one-summation-nothing-too-formal-rig
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# Intuition for Multiple Summation becoming One Summation - Nothing too formal/rigorous please
Source. I grok addition is associative and commutative, and a term can be moved into other summations iff these other summations aren't summing this term. Hence I grok
$$\sum_{i,j} g_{ij} \sum_r a_{ir} x_r \sum_s a_{js} x_s = \sum_{i,j} \sum_{r} \sum_s g_{ij} a_{ir}x_r a_{js}x_s = \color{blue}{ \sum_{i} \sum_{j} \sum_{r} \sum_s g_{ij} a_{ir}x_r a_{js}x_s}.$$
$$\text{But } \color{blue}{ \sum_{i} \sum_{j} \sum_{r} \sum_s g_{ij} a_{ir}x_r a_{js}x_s} = \sum_{i, j, r, s} g_{ij} a_{ir}x_r a_{js}x_s \,???$$
(Follow-up 1) The R.S. of $\color{green}{\sum_{i_k} A_{i_1,i_2,\dots , i_k} = A^1_{i_1,i_2,\dots , i_{k-1}}}$ eliminates $i_k$ and introduces superscript $1$.
Similarly, R.S. of $\sum_{i_{k - 1}} \color{green}{A^1_{i_1,i_2,\dots , i_{k-1}}}= \color{#7A7676}{A^2_{i_1,i_2,\dots ,i_{k - 3}, i_{k-2}}}$ eliminates $i_{k - 1}$ and introduces superscript $2$.
Similarly, R.S. of $\sum_{i_{k - 1}}\color{#7A7676}{A^2_{i_1,\dots ,i_{k - 3}, i_{k-2}}}= A^3_{i_1,\dots ,i_{k - 4}, i_{k-3}}$ eliminates $i_{k - 2}$ and introduces superscript $3$ and so forth...
But what do the $A^{\text{# of index removed}}_{\text{one less index than before}}$ and this whole process mean, other than writing out the $k - 1$ summation symbols?
(II) More generally, if $i_k$ satisfies some property $P(i_k)$, then how to rewrite with only 1 summation $$\sum\limits_{i_1 \; : \; P(i_1)} \cdots \sum\limits_{i_{k - 1} \; : \; P(i_{k - 1})} \; \sum\limits_{i_k \; : \; P(i_k)} A(i_1, ..., i_k) \; ?$$
(III) You wrote $\sum_{i_1,i_2,\dots , i_k} A_{i_1,i_2,\dots , i_k}= \sum_{i_1}\bigl(\sum_{i_2} \dots \color{green}{\left[\sum_{i_k} A_{i_1,i_2,\dots , i_k}\right]} \cdots \bigr)$.
Why are there dots after the green sum? It is the last sum, so nothing comes after?
(Follow-up 2) (II) Sadly I do not understand the first three sentences in your answer. Can you please elaborate? Is $\sum_{j=1}^{r} B_j$ one sum or a multiple sum already rewritten as one sum?
Also, thank you for recommending writing multiple sums. Actually, I like them better too! But I am addled by $S=\sum_{i,j} \epsilon_{ij} = -\sum_{i,j} \epsilon_{ji} =-\sum_{j,i} \epsilon_{ji} =-S$. You summed over $i, j$ but there is only one sum here. Why not two sums? Which is better?
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Isn't it only one notation? – eccstartup Jul 20 '13 at 12:01
That's also just rebracketing such as in $(a_{1,1}+a_{1,2})+(a_{2,1}+a_{2,2})+(a_{3,1}+a_{3,2}) = a_{1,1}+a_{1,2}+a_{2,1}+a_{2,2}+a_{3,1}+a_{3,2}$. – Hagen von Eitzen Jul 20 '13 at 12:25
Just added a follow-up to your follow-up. Hope it helps. – James S. Cook Nov 16 '13 at 15:45
Writing $\sum_{i,j,k}$ instead of $\sum_i\sum_j\sum_k$ is tantamount to writing $\int_{R}$ for $R=[a,b]\times [c,d]\times [e,f]$ a cube instead of an iterated integral $\int_a^b\int_c^d\int_e^f$ (of course, we're not always promised this is possible with integrals, but neither we're promised it is possible with infinite sums) – Pedro Tamaroff Nov 16 '13 at 15:46
@PedroTamaroff thanks! I think that helps my edit further. – James S. Cook Nov 16 '13 at 17:30
The notation $\sum_{i_1,i_2,\dots , i_k}$ is just short-hand for the iterated sums $\sum_{i_1}$, $\sum_{i_2}, \dots , \sum_{i_k}$. I would say (my convention) starting with $i_k$ and proceeding outward to $i_1$:
$$\sum_{i_1,i_2,\dots , i_k} A_{i_1,i_2,\dots , i_k}= \sum_{i_1}\bigl(\sum_{i_2} \dots \color{green}{\left[\sum_{i_k} A_{i_1,i_2,\dots , i_k}\right]} \cdots \bigr)$$
In particular, if we denote $\color{green}{\sum_{i_k} A_{i_1,i_2,\dots , i_k} = A^1_{i_1,i_2,\dots , i_{k-1}}}$ then $$\sum_{i_1,i_2,\dots , i_k} A_{i_1,i_2,\dots , i_k}= \sum_{i_1}\bigl(\sum_{i_2} \dots \sum_{i_{k - 2}} \underbrace{\sum_{i_{k-1}} \color{green}{\left[A^1_{i_1,i_2,\dots , i_{k-1}}\right]}}_{\Large{A^2_{i_1,\dots ,i_{k - 3}, i_{k-2}}}} \cdots \bigr)$$
and so forth until we're down to $\sum_{i_1,i_2,\dots , i_k} A_{i_1,i_2,\dots , i_k} =\sum_{i_1} A^{k-1}_{i_1}$.
This would be my default interpretation of such an expression. Some obvious questions to ask:
1. how am I sure it wasn't done in a different order, say starting with $i_1$ and proceeding outward until finally we finish with the sum over $i_k$?
2. wait, does it even matter which order the summations are taken? The simplest version of this is does $\sum_i (\sum_j A_{ij}) = \sum_j(\sum_i A_{ij})$ ?
If the answer to (2.) is no, then the answer to (1.) is that the order of summation does not matter. Here, we're assuming that (2.) extends to $k$-sums. But that's clear since we can always break
a $k$-sum into iterated $2$-sums, in other words $\sum\limits_{i_1}\left(\sum\limits_{i_2,...,i_k}\right) = \sum\limits_{i_k}\left(\sum\limits_{i_1,...,i_{k - 1}}\right)$
So, let us address (2.). To keep it easy to understand let's look at $n=2$: $$\sum_{i=1}^2\sum_{j=1}^2 A_{ij} = \sum_{i=1}^2 (A_{i1}+A_{i2}) = (A_{11}+A_{12})+(A_{21}+A_{22}).$$ Compare against: $$\sum_{i=j}^2\sum_{i=1}^2 A_{ij} = \sum_{j=1}^2 (A_{1j}+A_{2j}) = (A_{11}+A_{21})+(A_{12}+A_{22}).$$ So as Hagen von Eitzen has commented, it's just rearranging parenthesis. Now, if these summations pass to infinite upper bounds (series) then we cannot rearrange these so easily. Some analytical conditions concerning uniformity of the convergence must be met. But, so long as the sums are finite, we can reorder them.
Incidentally, if you did want to prove these things carefully, you'll need a definition for the finite sum. May I recommend that $\sum_{i=1}^{1} A_i = A_1$ and $\sum_{i=1}^{n+1}A_i = A_{n+1}+\sum_{i=1}^{n}A_i$. Most authors think these things are too trivial to put in books.
Following the follow-up:
I.) the superscript notation in my example is merely to emphasize the idea that the summations can be thought of as happening one at a time. It's much the same idea as the iterated integral $\int_0^1 \int_{0}^{x}\int_0^{1-x-y} xydz \, dy \, dx$
1. we integrate over $z$ leaving $\int_0^1 \int_{0}^{x} \underbrace{[xy(1-x)-xy^2)]}_{\text{like} \ A_1} \, dy \, dx$
2. next, integrate over $y$ leaving $\int_0^1 \underbrace{[x\frac{x^2}{2}(1-x)-x\frac{x^3}{3})]}_{\text{like} \ A_2} \, dx$
3. finally we're left with an integral in just one variable $\int_0^1 \underbrace{[x\frac{x^2}{2}(1-x)-x\frac{x^3}{3})]}_{\text{like} \ A_2} \, dx = \frac{-1}{24}$
My idea was to suppress the indices of summation to emphasize that after the sum is complete that index is gone for the summations that follow. Just like $z$ or $y$ is gone as we iterate the integral inside out.
II.) writing multiple sums as one sum? Well, I suppose the sum is just an addition of finitely many terms thus we can place the possible indices in an ordered set and label those indices from say $1$ to $r$ where $r$ is the total number of summands then the iterated sum becomes $\sum_{j=1}^{r} B_j$. However, I don't recommend this. The point of writing multiple sums is found both from their natural origin from compound summative processes (for example, the finite sum which sets-up the double integral) as well as the nice property that repeated sums allow us to exploit symmetries between certain subsets of the summands $B_1, \dots B_r$. For example, $\sum_{i,j} \epsilon_{ij} = 0$ since, by definition, $\epsilon_{ij}=-\epsilon_{ji}$ and so: $$S=\sum_{i,j} \epsilon_{ij} = -\sum_{i,j} \epsilon_{ji} =-\sum_{j,i} \epsilon_{ji} =-S$$ which shows $S=0$.
III.) this one is easier, those dots indicate the many parentheses I did not write.
In response to Following the follow-up (2): I meant to indicate that a multiple finite sum is still just the sum of finitely many things. For example, $$\sum_{i=1}^3 \sum_{j=1}^3 A_{ij} = \sum_{r=1}^9 B_r$$ provided I define $B_1 = A_{11}, B_2 = A_{12}, \dots , B_9 = A_{33}$. This would not usually be a wise step since it hides any nice symmetries of the summands $A_{ij}$. Getting back to my other comment, to be more pedantic, \begin{align} S &= \sum_i \sum_j \epsilon_{ij} \\ &= -\sum_i \sum_j \epsilon_{ji} \qquad \text{since $\epsilon_{ij} = -\epsilon_{ji}$} \\ &= -\sum_j \sum_i \epsilon_{ji} \qquad \text{property of finite sums, can swap order}\\ &=-S \end{align} and thus $S=0$.
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Sorry for the late reply. +1. Thanks a lot. I made some minor edits and hope they're OK with you. I have some follow-ups that I put in my question. Can you please get back to me in your answer, and not in comments? Sub- and superscripts are too hard to read down here. – Pauline Ercute Nov 16 '13 at 9:12
Sorry for my late reply. Thanks so much! It's too bad I can't upvote more than once. I'll do it for your other first-rate posts. I have another follow-up to II. Can you please get back to me in your answer and not in comments? Merry Christmas! – Pauline Ercute Dec 22 '13 at 14:47
@P.Ercute Merry Christmas to you as well, sorry for the delay, I'm a bit wrapped up in preparing for next semester. It happens that I was writing a proof that the sums can swap. I'd add it here but it's already a bit slow to add more. If you want to ask a question about that alone, I'll cut and paste my proof in there for you. It might have to wait until next year though. – James S. Cook Dec 30 '13 at 6:21
Thanks so much! No problem. I can wait until next year. Can you please post your proof or link to it here as a second answer? I want to keep everything together. Happy new year! – Pauline Ercute Dec 30 '13 at 11:59
Can you please give an example of "nice symmetries of the summands $A_{ij}$ in your answer that you talked about in your last follow-up (your follow-up to my follow-up 2)? I'm stuck on seeing why it's bad to write out multiple sums as one single sum. – Pauline Ercute Dec 30 '13 at 12:03
This answer adds detail to the comments. In particular, it is primarily intended to provide proof of the following basic claim about finite sums: Let us call this $P_n$:
$$\sum_{i=1}^{n} \biggl( \sum_{j=1}^{n} B_{ij} \biggr) = \sum_{j=1}^{n}\biggl( \sum_{i=1}^{n} B_{ij} \biggr)$$
Here it is assumed that $B_{ij}$ are given scalars (could be real, complex, even functions)
Proof: we use induction on $n$. Observe for $n=1$ it is trivially true as $B_{11}=B_{11}$ so no sum is even possible. While I don't think it's logically necessary, it might be helpful to see the proof for $n=2$ as well: $$\sum_{i=1}^{2}\sum_{j=1}^{2} B_{ij} = \sum_{i=1}^{2} [B_{i1}+B_{i2}] = [B_{11}+B_{12}] + [B_{21}+B_{22}]$$ On the other hand, $$\sum_{j=1}^{2}\sum_{i=1}^{2} B_{ij} = \sum_{j=1}^{2} [B_{1j}+B_{2j}] = [B_{11}+B_{21}] + [B_{11}+B_{21}].$$ The sums in opposite order produce the same terms overall, just rearranged.
Next, assume inductively that $P_n$ is true for some $n > 1$. Using the definition of sum throughout and the induction hypothesis in transitioning from the 3-rd to the 4-th line: \begin{align} \notag \sum_{i=1}^{n+1}\sum_{j=1}^{n+1} B_{ij} &=\sum_{i=1}^{n+1}\biggl[ B_{i,n+1}+\sum_{j=1}^{n} B_{ij} \biggr] \\ \notag &=\sum_{i=1}^{n+1} B_{i,n+1}+\sum_{i=1}^{n+1}\sum_{j=1}^{n} B_{ij} \\ \notag &=\sum_{i=1}^{n+1} B_{i,n+1}+\sum_{j=1}^{n} B_{n+1,j}+\sum_{i=1}^{n}\sum_{j=1}^{n} B_{ij} \\ \notag &=\sum_{i=1}^{n+1} B_{i,n+1}+\sum_{j=1}^{n} B_{n+1,j}+\sum_{j=1}^{n}\sum_{i=1}^{n} B_{ij} \\ \notag &=\sum_{i=1}^{n+1} B_{i,n+1}+\sum_{j=1}^{n} \left[ B_{n+1,j}+\sum_{i=1}^{n} B_{ij} \right] \\ \notag &=\sum_{i=1}^{n+1} B_{i,n+1}+\sum_{j=1}^{n}\sum_{i=1}^{n+1} B_{ij} \\ \notag &=\sum_{j=1}^{n+1}\sum_{i=1}^{n+1} B_{ij} \notag \end{align} Thus $n$ implies $n+1$ for $P_n$ therefore by proof by mathematical induction we find $P_n$ is true for all $n \in \mathbb{N}$. In short, we can swap the order of finite sums.
Now, as to why it's bad to just lump all the indices into one longer ranging index, the example where I showed $S=-S$ hence $S=0$ is such an example. If I just wrote out all the terms it would take me longer to see the cancellation which is essentially manifest in the double-index notation. An important lemma of tensor calculus, is that whenever we see a symmetric pair of indices contracted (summed over all the values) multiplied with an object with antisymmetric indices then the result is zero. This happens in many calculations I've seen. Here's one from calculus III, $$(\nabla \times \nabla f)_k = \sum_{i,j=1}^{3} \epsilon_{ijk} \partial_i \partial_j f$$ Clairaut's theorem says for continuously twice differentiable functions $\partial_i \partial_j f=\partial_j \partial_i f$ hence the expression above is symmetric in $i,j$ whereas $\epsilon_{ijk}$ is antisymmetric hence $(\nabla \times \nabla f)_k=0$ for $k=1,2,3$ hence $\nabla \times \nabla f =0$.
Note, the symbol $\epsilon_{ijk}$ is the completely antisymmetric symbol. It is defined by $\epsilon_{123}=1$ and the assumed antisymmetry. For example, $\epsilon_{112}=0$ whereas $\epsilon_{213}=-1$ and $\epsilon_{231}=1$ etc... there are six nonzero values and 21 triples where $\epsilon_{ijk}=0$.
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2016-05-30 13:08:24
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https://indico.cern.ch/event/1065494/contributions/4481580/
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# Mini-workshop on “$T_{cc}^+$ and beyond”, Online
September 14, 2021
Europe/Zurich timezone
## What can we learn from the width of the $T_{cc}^+$ tetraquark?
Sep 14, 2021, 2:00 PM
20m
Mitja Rosina
### Description
The width of the $T_{cc}^+$ tetraquark (dimeson) is expected to differ considerably from the widths of its constituent $D^{*0}$ and $D^{*+}$. Unfortunately, reliable values of the $D^{*0}$ and $T_{cc}^+$ widths are not yet known and at least theoretical estimates would be welcome. An interesting effect is due to the charge splitting of the
$(D^{*+})D^0$ and $(D^{*0})D^+$ thresholds,
therefore $T_{cc}^+$ will not have a pure isospin coupling
$T_{cc}^+ = [(D^{*+})D^0 - (D^{*0})D^+]/\sqrt{2}$ and the actual composition may be seen in branching ratios. The width of $T_{cc}^+$ may depend strongly on threshold effects since its energy is not discrete and its peak extends beyond the lowest threshold, making it partially unstable and therefore broader. I shal try to estimate some of these effects.
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2021-12-01 01:27:43
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https://brilliant.org/problems/a-trigonometric-problem-for-jee-aspirants/
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# A Trigonometric problem for JEE Aspirants
Geometry Level 4
$E = \tan(A) \tan(2A) + \tan(2A) \tan(4A) + \tan(4A) \tan(A)$
Find the value of $$E$$ where $$A = \dfrac{2\pi}7$$.
×
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2018-10-17 23:49:26
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https://meteo.unican.es/trac/wiki/DRM4G/Installation?version=18
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Version 18 (modified by minondoa, 6 years ago) (diff)
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#### DRM4G Installation
1. DRM4G Installation
1. With pip
2. Without pip
3. Optional Environment Variables
# DRM4G Installation
There are several ways to install DRM4G.
## With pip
The recommended way to install DRM4G is by using the command pip install drm4g. To check if you have this tool installed just open a terminal and execute which pip. If you have it installed you'll see something like this:
or
$pip install drm4g --install-option="--home=/your/path" -v This will install your binary files and libraries under the specified path, but DRM4G will still not be able to run. • You want your system to be aware of where the DRM4G package is, so that python may be able to import it. You'll have to define the environment variable PYTHONPATH, which will have to point to the library folder under the path you chose. At the end of the installation, you'll see a message that will inform you on how to do that. • The next step is to have it know that DRM4G has been installed. You'll have to make sure that the directory you choose is added in your environment variable PATH or you could have it added to your sys.path. The folder we are interested in is the "bin" folder inside your directory. During the installation, you will be prompted with a question about modifying your$HOME/.profile or $HOME/.bashrc file. If you accept, you will only have to define these two environment variables the first time. • Alternatively you can access the file yourself, which is under the home directory, and make the necessary changes. ## Without pip First and foremost you would have to make sure you have installed the necessary requirements. Those are Paramiko and docopt. For Ubuntu or Debian: sudo apt-get install python-paramiko sudo apt-get install python-docopt For Centos or RedHat: sudo yum install python-paramiko sudo yum install python-docopt Once that has been taken care of, you can download the source code from here. After you extract the package, in a terminal, head into the folder and run: • With root access: sudo python setup.py install • Without root access: python setup.py install --user • In a virtual environment: python setup.py install ### Install in custom directory If you want to use a specific directory as the installation path, you can do it like this: python setup.py install --prefix=/your/path or python setup.py install --home=/your/path There are other considerations to have in mind, but they are the same as the ones explained above. ## Optional Environment Variables DRM4G uses the following environment variables: • DRM4G_DIR: By default your configuration files will be located in$HOME/.drm4g/. If you wish to define where your configuration and log files will be created, before starting DRM4G with drm4g start you should set the environment variable DRM4G_DIR with whichever directory you wish to use, .drm4g will be automatically created under it.
• EDITOR: Select an editor to configure the resource configuration file. The vi editor will be used by default.
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2022-08-15 15:55:22
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https://www.studysmarter.us/textbooks/physics/college-physics-urone-1st-edition/electric-charge-and-electric-field/q53pe-a-simple-and-common-technique-for-accelerating-electro/
|
Suggested languages for you:
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Q53PE
Expert-verified
Found in: Page 666
### College Physics (Urone)
Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000
# A simple and common technique for accelerating electrons is shown in Figure 18.55, where there is a uniform electric field between two plates. Electrons are released, usually from a hot filament, near the negative plate, and there is a small hole in the positive plate that allows the electrons to continue moving. (a) Calculate the acceleration of the electron if the field strength is $$2.50 \times {10^4}{\rm{ N/C}}$$. (b) Explain why the electron will not be pulled back to the positive plate once it moves through the hole.Figure 18.55 Parallel conducting plates with opposite charges on them create a relatively uniform electric field used to accelerate electrons to the right. Those that go through the hole can be used to make a TV or computer screen glow or to produce X-rays.
(a) The acceleration of the electron is $$4.39 \times {10^{15}}{\rm{ m/}}{{\rm{s}}^{\rm{2}}}$$.
(b) The electron will not be pulled back to the positive plate once it moves through the hole because there is no electric field outside the plate.
See the step by step solution
## Step 1: Coulomb force
The force experienced by the charge when it is placed is placed in an electric field created by some other charge is known as Coulomb force or electrostatic force. The expression for the Coulomb force is,
$$F = qE$$
Here, $$q$$ is the magnitude of charge on electron, and $$E$$ is the electric field.
## Step 2: Acceleration of the electron
The force acting on the particle is,
$$F = ma$$
Here, $$m$$ is the mass of the electron, and $$a$$ is the acceleration of the electron.
Comparing equations (1.1) and (1.2) we get,
$$ma = qE$$
Therefore, the expression for the acceleration of the electron is,
$$a = \frac{{qE}}{m}$$
Substitute $$1.6 \times {10^{ - 19}}{\rm{ }}C$$ for $$q$$, $$2.50 \times {10^4}{\rm{ N/C}}$$ for $$E$$, and $$9.1 \times {10^{ - 31}}{\rm{ kg}}$$ for $$m$$.
$$\begin{array}{c}a = \frac{{\left( {1.6 \times {{10}^{ - 19}}{\rm{ }}C} \right) \times \left( {2.50 \times {{10}^4}{\rm{ N/C}}} \right)}}{{\left( {9.1 \times {{10}^{ - 31}}{\rm{ kg}}} \right)}}\\ = 4.39 \times {10^{15}}{\rm{ m/}}{{\rm{s}}^{\rm{2}}}\end{array}$$
Hence, the acceleration of the electron is $$4.39 \times {10^{15}}{\rm{ m/}}{{\rm{s}}^{\rm{2}}}$$.
## Step 3: Reason the electron will not be pulled back to the positive place
Once the electron passes the hole, there will be no electric field exist outside the plates.
Hence, the electron will not be pulled back to the positive plate once it moves through the hole because there is no electric field outside the plate.
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2023-03-29 07:39:13
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https://www.techwhiff.com/learn/the-tune-up-specifications-of-a-car-call-for-the/189773
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# The tune-up specifications of a car call for the spark plugs to be tightened to a...
###### Question:
The tune-up specifications of a car call for the spark plugs to be tightened to a torque of 32 N⋅m . You plan to tighten the plugs by pulling on the end of a 20-cm-long wrench. Because of the cramped space under the hood, you'll need to pull at an angle of 105 ∘ with respect to the wrench shaft.
With what force must you pull?
F = ________ N
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2022-09-25 02:30:51
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https://gateoverflow.in/220315/gate-suitability-test-test-1-question-3?show=376500
|
110 views
A coin is such that after every toss the probability of same side coming again increases by $50\%$ from the initial value. If initially the probability of head and tail are the same, what is the expected number of tosses until we get a head?
$E_{\text{tosses}} = 0.5 \times 1 + 0.5 \times(1+ E_{HaT})$
$\implies E_{\text{tosses}} = 1 + 0.5 E_{HaT}$
$E_{HaT} = 0.25 \times 1 + 0.75 \times (1 + E_{HaT})$
$\implies E_{HaT} = 1 + 0.75 E_{HaT}$
$\implies E_{HaT} = \frac{1}{0.25} = 4$
$\therefore E_{\text{tosses}} = 1 + 0.5 \times 4 = 3$
So, we must toss the coin at least $3$ times to expect a Head.
by
1 vote
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2022-12-02 17:40:53
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https://zbmath.org/serials/?q=se%3A00002090
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zbMATH — the first resource for mathematics
Opuscula Mathematica
Short Title: Opusc. Math. Publisher: AGH University of Science and Technology, Faculty of Applied Mathematics, Kraków ISSN: 1232-9274; 2300-6919/e Online: http://www.opuscula.agh.edu.pl/archives Predecessor: Zeszyty Naukowe Akademii Górniczo-Hutniczej Im. Stanisława Staszica. Opuscula Mathematica Comments: Indexed cover-to-cover; This journal is available open access.
Documents Indexed: 806 Publications (since 1994) References Indexed: 277 Publications with 5,084 References.
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Citations contained in zbMATH Open
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Holzleitner, Markus; Kostenko, Aleksey; Teschl, Gerald
2016
Multiple solutions for fourth order elliptic problems with $$p(x)$$-biharmonic operators. Zbl 1339.35130
Kong, Lingju
2016
Some stability conditions for scalar Volterra difference equations. Zbl 1343.39030
Berezansky, Leonid; Migda, Maĺgorzata; Schmeidel, Ewa
2016
Existence of three solutions for impulsive nonlinear fractional boundary value problems. Zbl 1371.34013
Heidarkhani, Shapour; Ferrara, Massimiliano; Caristi, Giuseppe; Salari, Amjad
2017
Functional models for Nevanlinna families. Zbl 1183.47004
Behrndt, Jussi; Hassi, Seppo; de Snoo, Henk
2008
Fixed points and stability in neutral nonlinear differential equations with variable delays. Zbl 1254.34110
Ardjouni, Abdelouaheb; Djoudi, Ahcene
2012
Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces. Zbl 1252.34002
Benchohra, Mouffak; Mostefai, Fatima-Zohra
2012
On nonlocal problems for fractional differential equations in Banach spaces. Zbl 1228.26012
Dong, XiWang; Wang, JinRong; Zhou, Yong
2011
Probabilistic characterization of strong convexity. Zbl 1234.26031
Rajba, Teresa; Wąsowicz, Szymon
2011
On the multiplicative Zagreb coindex of graphs. Zbl 1269.05023
Xu, Kexiang; Das, Kinkar Ch.; Tang, Kechao
2013
On the summability of divergent power series solutions for certain first-order linear PDEs. Zbl 1327.35055
Hibino, Masaki
2015
Weakly convex and convex domination numbers. Zbl 1076.05060
Lemańska, Magdalena
2004
A note on arbitrarily vertex decomposable graphs. Zbl 1134.05083
Marczyk, Antoni
2006
Oscillation criteria for third order nonlinear delay differential equations with damping. Zbl 1327.34122
Grace, Said R.
2015
Existence of three solutions for perturbed nonlinear difference equations. Zbl 1330.39009
2014
Solutions of fractional nabla difference equations – existence and uniqueness. Zbl 1348.39002
2016
Neighbourhood total domination in graphs. Zbl 1230.05223
Arumugam, S.; Sivagnanam, C.
2011
Matkowski, Janusz
2011
On vertex $$b$$-critical trees. Zbl 1269.05033
Blidia, Mostafa; Eschouf, Noureddine Ikhlef; Maffray, Frédéric
2013
The Putnam-Fuglede property for paranormal and $$\ast$$-paranormal operators. Zbl 1315.47025
Pagacz, Patryk
2013
A note on the independent Roman domination in unicyclic graphs. Zbl 1259.05127
2012
On a Robin $$((p,q)$$-equation with a logistic reaction. Zbl 1435.35144
Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca
2019
Solution of the Stieltjes truncated matrix moment problem. Zbl 1090.30038
2005
Bounds on the 2-domination number in cactus graphs. Zbl 1133.05066
Chellali, Mustapha
2006
The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. II. Zbl 1102.35006
Golenia, J.; Prykarpatsky, Y. A.; Samoilenko, A. M.; Prykarpatsky, A. K.
2004
Li’s criterion for the Riemann hypothesis – numerical approach. Zbl 1136.11319
Maślanka, Krzysztof
2004
On dynamical systems induced by $$p$$-adic number fields. Zbl 1326.05170
Cho, Ilwoo
2015
Frames and factorization of graph Laplacians. Zbl 1359.47055
Jorgensen, Palle; Tian, Feng
2015
Generalized Levinson’s inequality and exponential convexity. Zbl 1333.26027
Pečarić, Josip; Praljak, Marjan; Witkowski, Alfred
2015
Diffusion approximation of recurrent schemes for financial markets, with application to the Ornstein-Uhlenbeck process. Zbl 1332.60049
Mishura, Yuliya
2015
Asymptotic behavior of solutions of discrete Volterra equations. Zbl 1359.39004
Migda, Janusz; Migda, Małgorzata
2016
On the global attractivity and the periodic character of a recursive sequence. Zbl 1229.39023
Elsayed, E. M.
2010
Chaotic dynamics in the Volterra predator-prey model via linked twist maps. Zbl 1167.34341
Pireddu, Marina; Zanolin, Fabio
2008
Control system defined by some integral operator. Zbl 1367.93271
Majewski, Marek
2017
Beurling’s theorems and inversion formulas for certain index transforms. Zbl 1230.42009
Yakubovich, Semyon B.
2009
Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations. Zbl 1132.65070
Auzinger, Winfried; Karner, Ernst; Koch, Othmar; Weinmüller, Ewa
2006
Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvalues of finite submatrices. Zbl 1155.47034
Malejki, Maria
2007
Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph. Zbl 1334.34065
Niikuni, Hiroaki
2015
Nontrivial solutions of linear functional equations: methods and examples. Zbl 1332.39019
2015
The metric dimension of circulant graphs and their Cartesian products. Zbl 1431.05052
Chau, Kevin; Gosselin, Shonda
2017
On one oscillatory criterion for the second order linear ordinary differential equations. Zbl 1360.34075
Grigorian, Gevorg Avagovich
2016
On nonoscillatory solutions of two dimensional nonlinear delay dynamical systems. Zbl 1360.34136
Öztürk, Özkan; Akın, Elvan
2016
Multiplicative Zagreb indices and coindices of some derived graphs. Zbl 1335.05043
Basavanagoud, Bommanahal; Patil, Shreekant
2016
Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain. Zbl 1346.31003
Chaieb, Majda; Dhifli, Abdelwaheb; Zermani, Samia
2016
Free probability on Hecke algebras and certain group $$C^{*}$$-algebras induced by Hecke algebras. Zbl 1339.05426
Cho, Ilwoo
2016
Solving boundary value problems in the open source software R: package bvpSolve. Zbl 1293.65104
Mazzia, Francesca; Cash, Jeff R.; Soetaert, Karline
2014
On the hat problem on a graph. Zbl 1245.05022
Krzywkowski, Marcin
2012
White noise based stochastic calculus associated with a class of Gaussian processes. Zbl 1255.60117
Alpay, Daniel; Attia, Haim; Levanony, David
2012
The forwarding indices of graphs – a survey. Zbl 1290.05093
Xu, Jun-Ming; Xu, Min
2013
Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control. Zbl 1225.45006
Wang, JinRong; Wei, W.; Yang, YanLong
2010
On some impulsive fractional differential equations in Banach spaces. Zbl 1242.34011
Wang, JinRong; Wei, Wei; Yang, YanLong
2010
On some existence results of mild solutions for nonlocal integrodifferential Cauchy problems in Banach spaces. Zbl 1242.45010
Yang, YanLong; Wang, JinRong
2011
General solutions of second-order linear difference equations of Euler type. Zbl 1362.39001
Hongyo, Akane; Yamaoka, Naoto
2017
Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics. Zbl 1367.49015
Leszczyński, Maciej; Ratajczyk, Elżbieta; Ledzewicz, Urszula; Schättler, Heinz
2017
Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Zbl 1368.34012
Lyons, Jeffrey W.; Neugebauer, Jeffrey T.
2017
Rigidity of monodromies for Appell’s hypergeometric functions. Zbl 1320.33021
Haraoka, Yoshishige; Kikukawa, Tatsuya
2015
Katz’s middle convolution and Yokoyama’s extending operation. Zbl 1326.34133
Oshima, Toshio
2015
Parametric Borel summability for some semilinear system of partial differential equations. Zbl 1329.35105
Yamazawa, Hiroshi; Yoshino, Masafumi
2015
Limit-point criteria for the matrix Sturm-Liouville operator and its powers. Zbl 1361.34024
Braeutigam, Irina N.
2017
On the diameter of dot-critical graphs. Zbl 1204.05070
Mojdeh, Doost Ali; Mirzamani, Somayeh
2009
A note on the vertex-distinguishing index for some cubic graphs. Zbl 1073.05030
Taczuk, Karolina; Woźniak, Mariusz
2004
The geometric properties of reduced canonically symplectic spaces with symmetry, their relationship with structures on associated principal fiber bundles and some applications. I. Zbl 1122.53047
Prykarpatsky, Yarema A.; Samoilenko, Anatoliy M.; Prykarpatsky, Anatoliy K.
2005
Pseudo-differential equations and conical potentials: 2-dimensional case. Zbl 1404.35497
2019
$$[r,s,t]$$-colourings of paths. Zbl 1146.05026
2007
The multidimensional Delsarte transmutation operators, their differential-geometric structure and applications. I. Zbl 1101.35003
Prykarpatsky, Anatoliy K.; Samoilenko, Anatoliy M.; Prykarpatsky, Yarema A.
2003
On a multivalued second order differential problem with Hukuhara derivative. Zbl 1153.26007
Piszczek, Magdalena
2008
Kernel conditional quantile estimator under left truncation for functional regressors. Zbl 1329.62167
Helal, Nacéra; Ould Saïd, Elias
2016
Fractional evolution equation nonlocal problems with noncompact semigroups. Zbl 1335.34024
Zhang, Xuping; Chen, Pengyu
2016
On some subclasses of the family of Darboux Baire 1 functions. Zbl 1339.26013
Ivanova, Gertruda; Wagner-Bojakowska, Elżbieta
2014
Oscillatory properties of solutions of the fourth order difference equations with quasidifferences. Zbl 1330.39013
Jankowski, Robert; Schmeidel, Ewa; Zonenberg, Joanna
2014
Controllability of semilinear systems with fixed delay in control. Zbl 1327.93082
Kumar, Surendra; Sukavanam, N.
2015
Continuous spectrum of Steklov nonhomogeneous elliptic problem. Zbl 1333.35051
Allaoui, Mostafa
2015
Acyclic sum-list-colouring of grids and other classes of graphs. Zbl 1397.05080
Drgas-Burchardt, Ewa; Drzystek, Agata
2017
On the Steklov problem involving the $$p(x)$$-Laplacian with indefinite weight. Zbl 1403.35115
Ali, Khaled Ben; Ghanmi, Abdeljabbar; Kefi, Khaled
2017
Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation. Zbl 1403.34050
Graef, John R.; Tunç, Ercan; Grace, Said R.
2017
Nonhomogeneous equations with critical exponential growth and lack of compactness. Zbl 1436.35176
Figueiredo, Giovany M.; Rădulescu, Vicenţiu D.
2020
Fractional $$p\&q$$-Laplacian problems with potentials vanishing at infinity. Zbl 1437.35691
Isernia, Teresa
2020
Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity. Zbl 1441.35174
Lian, Wei; Ahmed, Md Salik; Xu, Runzhang
2020
Oscillation of time fractional vector diffusion-wave equation with fractional damping. Zbl 1445.35311
Ramesh, R.; Harikrishnan, S.; Nieto, J. J.; Prakash, P.
2020
A unique weak solution for a kind of coupled system of fractional Schrödinger equations. Zbl 1437.35688
Abdolrazaghi, Fatemeh; Razani, Abdolrahman
2020
Facial rainbow edge-coloring of simple 3-connected plane graphs. Zbl 1444.05056
Czap, Július
2020
Isotropic and anisotropic double-phase problems: old and new. Zbl 1437.35315
Rădulescu, Vicenţiu D.
2019
On a Robin $$((p,q)$$-equation with a logistic reaction. Zbl 1435.35144
Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca
2019
Pseudo-differential equations and conical potentials: 2-dimensional case. Zbl 1404.35497
2019
Infinitely many solutions for some nonlinear supercritical problems with break of symmetry. Zbl 1433.35034
2019
Existence results and a priori estimates for solutions of quasilinear problems with gradient terms. Zbl 1433.35116
Filippucci, Roberta; Lini, Chiara
2019
Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term. Zbl 1414.34052
Graef, John R.; Grace, Said R.; Tunç, Ercan
2019
Existence and multiplicity results for quasilinear equations in the Heisenberg group. Zbl 1437.35360
Pucci, Patrizia
2019
Nonlinear elliptic equations involving the $$p$$-Laplacian with mixed Dirichlet-Neumann boundary conditions. Zbl 1436.35203
Bonanno, Gabriele; D’Aguì, Giuseppina; Sciammetta, Angela
2019
Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions. Zbl 1428.35640
Fragnelli, Genni; Mugnai, Dimitri
2019
Decomposing complete 3-uniform hypergraph $$k_n^{(3)}$$ into 7-cycles. Zbl 1437.05174
Meihua; Guan, Meiling; Jirimutu
2019
Oscillatory results for second-order noncanonical delay differential equations. Zbl 1437.34075
Džurina, Jozef; Jadlovská, Irena; Stavroulakis, Ioannis P.
2019
Decomposition of Gaussian processes, and factorization of positive definite kernels. Zbl 1443.47085
Jorgensen, Palle; Tian, Feng
2019
The intersection graph of annihilator submodules of a module. Zbl 1442.13021
Pejman, S. B.; Payrovi, Sh.; Babaei, S.
2019
On properties of minimizers of a control problem with time-distributed functional related to parabolic equations. Zbl 1442.35176
Astashova, I. V.; Filinovskiy, A. V.
2019
On the imaginary part of coupling resonance points. Zbl 1443.47012
Azamov, Nurulla; Daniels, Tom
2019
Direct and inverse spectral problems for Dirac systems with nonlocal potentials. Zbl 1443.47019
Dębowska, Kamila; Nizhnik, Leonid P.
2019
Vertices with the second neighborhood property in Eulerian digraphs. Zbl 1437.05126
Cary, Michael
2019
The existence of consensus of a leader-following problem with Caputo fractional derivative. Zbl 1404.26010
Schmeidel, Ewa
2019
Oscillation criteria for even order neutral difference equations. Zbl 1403.39009
Selvarangam, S.; Rupadevi, S. A.; Thandapani, E.; Pinelas, S.
2019
Study of ODE limit problems for reaction-diffusion equations. Zbl 1402.35151
Simsen, Jacson; Simsen, Mariza Stefanello; Zimmermann, Aleksandra
2018
Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation. Zbl 1405.34075
Kato, Mitsuo; Mano, Toshiyuki; Sekiguchi, Jiro
2018
Adelic analysis and functional analysis on the finite adele ring. Zbl 1409.46038
Cho, Ilwoo
2018
Improved iterative oscillation tests for first-order deviating differential equations. Zbl 1405.34056
2018
Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities. Zbl 1402.39003
Chidouh, Amar; Torres, Delfim F. M.
2018
Wiener index of strong product of graphs. Zbl 1402.05056
Peterin, Iztok; Pleteršek, Petra Žigert
2018
On the stability of some systems of exponential difference equations. Zbl 1400.39017
Psarros, N.; Papaschinopoulos, G.; Schinas, C. J.
2018
Solutions to $$p(x)$$-Laplace type equations via nonvariational techniques. Zbl 1403.35100
Avci, Mustafa
2018
Zig-zag facial total-coloring of plane graphs. Zbl 1403.05033
Czap, Július; Jendrol’, Stanislav; Voigt, Margit
2018
Hubtic number in graphs. Zbl 1403.05076
2018
Minimal unavoidable sets of cycles in plane graphs. Zbl 1403.05035
2018
Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the unified transform method. Zbl 1406.35316
Rybalko, Yan
2018
Banach $$^*$$-algebras generated by semicircular elements induced by certain orthogonal projections. Zbl 1406.46054
Cho, Ilwoo; Jorgensen, Palle E. T.
2018
Upper bounds for the extended energy of graphs and some extended equienergetic graphs. Zbl 1402.05131
2018
Existence results for Kirchhoff type systems with singular nonlinearity. Zbl 1403.35103
Firouzjai, A.; Afrouzi, G. A.; Talebi, S.
2018
Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion. Zbl 1402.60066
Borkowski, Dariusz; Jańczak-Borkowska, Katarzyna
2018
On expansive and anti-expansive tree maps. Zbl 1400.37048
Kozerenko, Sergiy
2018
Inverse scattering problems for half-line Schrödinger operators and Banach algebras. Zbl 1409.34074
Mykytyuk, Yaroslav; Sushchyk, Nataliia
2018
Semicircular elements induced by $$p$$-adic number fields. Zbl 1398.11144
Cho, Ilwoo; Jorgensen, Palle E. T.
2017
Existence of three solutions for impulsive nonlinear fractional boundary value problems. Zbl 1371.34013
Heidarkhani, Shapour; Ferrara, Massimiliano; Caristi, Giuseppe; Salari, Amjad
2017
Control system defined by some integral operator. Zbl 1367.93271
Majewski, Marek
2017
The metric dimension of circulant graphs and their Cartesian products. Zbl 1431.05052
Chau, Kevin; Gosselin, Shonda
2017
General solutions of second-order linear difference equations of Euler type. Zbl 1362.39001
Hongyo, Akane; Yamaoka, Naoto
2017
Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics. Zbl 1367.49015
Leszczyński, Maciej; Ratajczyk, Elżbieta; Ledzewicz, Urszula; Schättler, Heinz
2017
Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Zbl 1368.34012
Lyons, Jeffrey W.; Neugebauer, Jeffrey T.
2017
Limit-point criteria for the matrix Sturm-Liouville operator and its powers. Zbl 1361.34024
Braeutigam, Irina N.
2017
Acyclic sum-list-colouring of grids and other classes of graphs. Zbl 1397.05080
Drgas-Burchardt, Ewa; Drzystek, Agata
2017
On the Steklov problem involving the $$p(x)$$-Laplacian with indefinite weight. Zbl 1403.35115
Ali, Khaled Ben; Ghanmi, Abdeljabbar; Kefi, Khaled
2017
Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation. Zbl 1403.34050
Graef, John R.; Tunç, Ercan; Grace, Said R.
2017
Existence of three solutions for impulsive multi-point boundary value problems. Zbl 1368.34040
Bohner, Martin; Heidarkhani, Shapour; Salari, Amjad; Caristi, Giuseppe
2017
Towards theory of $$\mathcal C$$-symmetries. Zbl 1373.47041
Kuzhel, S.; Sudilovskaya, V.
2017
Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularity. Zbl 1381.34107
Nursultanov, Medet; Rozenblum, Grigori
2017
The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation. Zbl 1358.35161
Shepelsky, Dmitry; Zielinski, Lech
2017
Multiplicity results for perturbed fourth-order Kirchhoff-type problems. Zbl 1403.34019
2017
On the uniform perfectness of equivariant diffeomorphism groups for principal $$G$$ manifolds. Zbl 1370.58004
Fukui, Kazuhiko
2017
Fractional boundary value problems on the half line. Zbl 1369.34012
Frioui, Assia; Guezane-Lakoud, Assia; Khaldi, Rabah
2017
On the chromatic number of $$(P_{5},\mathrm{windmill})$$-free graphs. Zbl 1420.05061
Schiermeyer, Ingo
2017
A direct approach to linear-quadratic stochastic control. Zbl 1401.93221
Duncan, Tyrone E.; Pasik-Duncan, Bozenna
2017
Compact generalized weighted composition operators on the Bergman space. Zbl 1375.30089
Hu, Qinghua; Zhu, Xiangling
2017
The interaction between PDE and graphs in multiscale modeling. Zbl 1431.35220
Morales, Fernando A.; Naranjo Álvarez, Sebastián
2017
The LQ/KYP problem for infinite-dimensional systems. Zbl 1358.49028
Grabowski, Piotr
2017
Hankel and Toeplitz operators: continuous and discrete representations. Zbl 1450.47016
Yafaev, Dmitri R.
2017
On Fibonacci numbers in edge coloured trees. Zbl 1437.11023
Bednarz, Urszula; Bród, Dorota; Szynal-Liana, Anetta; Włoch, Iwona; Wołowiec-Musiał, Małgorzata
2017
Anti-Ramsey numbers for disjoint copies of graphs. Zbl 1397.05111
Gorgol, Izolda; Görlich, Agnieszka
2017
A general 2-part Erdős-Ko-Rado theorem. Zbl 1402.05207
Katona, Gyula O. H.
2017
Block colourings of 6-cycle systems. Zbl 1397.05057
Bonacini, Paola; Gionfriddo, Mario; Marino, Lucia
2017
On 3-total edge product cordial connected graphs. Zbl 1397.05158
Ivančo, Jaroslav
2017
The second Cushing-Henson conjecture for the Beverton-Holt $$q$$-difference equation. Zbl 1400.39005
Bohner, Martin; Streipert, Sabrina H.
2017
A model for the inverse 1-Median problem on trees under uncertain costs. Zbl 1338.90086
Nguyen, Kien Trung; Chi, Nguyen Thi Linh
2016
Dispersion estimates for spherical Schrödinger equations: the effect of boundary conditions. Zbl 1360.35214
Holzleitner, Markus; Kostenko, Aleksey; Teschl, Gerald
2016
Multiple solutions for fourth order elliptic problems with $$p(x)$$-biharmonic operators. Zbl 1339.35130
Kong, Lingju
2016
Some stability conditions for scalar Volterra difference equations. Zbl 1343.39030
Berezansky, Leonid; Migda, Maĺgorzata; Schmeidel, Ewa
2016
Solutions of fractional nabla difference equations – existence and uniqueness. Zbl 1348.39002
2016
Asymptotic behavior of solutions of discrete Volterra equations. Zbl 1359.39004
Migda, Janusz; Migda, Małgorzata
2016
On one oscillatory criterion for the second order linear ordinary differential equations. Zbl 1360.34075
Grigorian, Gevorg Avagovich
2016
On nonoscillatory solutions of two dimensional nonlinear delay dynamical systems. Zbl 1360.34136
Öztürk, Özkan; Akın, Elvan
2016
Multiplicative Zagreb indices and coindices of some derived graphs. Zbl 1335.05043
Basavanagoud, Bommanahal; Patil, Shreekant
2016
Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain. Zbl 1346.31003
Chaieb, Majda; Dhifli, Abdelwaheb; Zermani, Samia
2016
Free probability on Hecke algebras and certain group $$C^{*}$$-algebras induced by Hecke algebras. Zbl 1339.05426
Cho, Ilwoo
2016
Kernel conditional quantile estimator under left truncation for functional regressors. Zbl 1329.62167
Helal, Nacéra; Ould Saïd, Elias
2016
Fractional evolution equation nonlocal problems with noncompact semigroups. Zbl 1335.34024
Zhang, Xuping; Chen, Pengyu
2016
Characterizations of rectangular (para)-unitary rational functions. Zbl 1376.26013
Alpay, Daniel; Jorgensen, Palle; Lewkowicz, Izchak
2016
On a linear-quadratic problem with Caputo derivative. Zbl 1343.49055
Idczak, Dariusz; Walczak, Stanislaw
2016
Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces. Zbl 1331.35250
Stăncuţ, Ionela-Loredana; Stîrcu, Iulia Dorotheea
2016
Edge subdivision and edge multisubdivision versus some domination related parameters in generalized corona graphs. Zbl 1426.05129
Dettlaff, Magda; Raczek, Joanna; Yero, Ismael G.
2016
On a problem of Gevorkyan for the Franklin system. Zbl 1366.42032
Wronicz, Zygmunt
2016
On solvability of some difference-discrete equations. Zbl 1339.39011
Vasilyev, Alexander V.; Vasilyev, Vladimir B.
2016
Positive solutions of boundary value problems with nonlinear nonlocal boundary conditions. Zbl 1329.34055
2016
Multiplicity results for an impulsive boundary value problem of $$p(t)$$-Kirchhoff type via critical point theory. Zbl 1358.34038
Mokhtari, A.; Moussaoui, T.; O’Regan, D.
2016
Higher order Nevanlinna functions and the inverse three spectra problem. Zbl 1365.34031
Boyko, Olga; Martinyuk, Olga; Pivovarchik, Vyacheslav
2016
Monotone method for Riemann-Liouville multi-order fractional differential systems. Zbl 1341.34005
Denton, Zachary
2016
Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation. Zbl 1338.49062
Just, Andrzej; Stempień, Zdzislaw
2016
Vertex-weighted Wiener polynomials of subdivision-related graphs. Zbl 1327.05294
Azari, Mahdieh; Iranmanesh, Ali; Došlić, Tomislav
2016
Existence and boundary behavior of positive solutions for a Sturm-Liouville problem. Zbl 1360.34054
Masmoudi, Syrine; Zermani, Samia
2016
On a dense minimizer of empirical risk in inverse problems. Zbl 1360.62162
Podlewski, Jacek; Szkutnik, Zbigniew
2016
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Cited by 1,580 Authors
30 Cho, Ilwoo 25 Stević, Stevo 21 Jørgensen, Palle E. T. 17 Ardjouni, Abdelouaheb 16 Djoudi, Ahcene 16 Grace, Said Rezk 12 Ntouyas, Sotiris K. 11 Koch, Othmar 11 Shah, Kamal 11 Wang, Jinrong 10 Auzinger, Winfried 10 Heidarkhani, Shapour 10 Prykarpatsky, Anatoliy Karolevych 9 Afrouzi, Ghasem Alizadeh 9 Alikhani, Saeid 9 Băleanu, Dumitru I. 9 Caristi, Giuseppe 9 Diblík, Josef 9 Papageorgiou, Nikolaos S. 9 Rădulescu, Vicenţiu D. 9 Wang, Guotao 9 Woźniak, Mariusz 8 Alpay, Daniel Aron 8 Benchohra, Mouffak 8 Dragomir, Sever Silvestru 8 Jadlovská, Irena 8 Khan, Rahmat Ali 8 Kurasov, Pavel B. 8 Nikodem, Kazimierz 8 O’Regan, Donal 8 Winkert, Patrick 8 Zada, Akbar 8 Zhou, Yong 7 Agarwal, Ravi P. 7 Ahmad, Bashir 7 Alizadeh, Behrooz 7 Baroughi, Fahimeh 7 Cortés López, Juan Carlos 7 Kostenko, Aleksey S. 7 Schmeidel, Ewa L. 7 Šmarda, Zdeněk 7 Tariboon, Jessada 7 Teschl, Gerald 7 Ugurlu, Ekin 7 Zhang, Lihong 6 Behrndt, Jussi 6 Duggal, Bhagwati Prashad 6 Graef, John R. 6 Iričanin, Bratislav D. 6 Jafari Rad, Nader 6 Jankowski, Tadeusz 6 Lastra, Alberto 6 Malamud, Mark M. 6 Malek, Stéphane 6 Moradi, Shahin 6 Pečarić, Josip 6 Prykarpatsky, Yarema Anatoliyovych 6 Suciu, Laurian 6 Tunç, Ercan 6 Vasilyev, Vladimir Borisovich 6 Vetrík, Tomáš 5 Abbas, Said 5 Al-saedi, Ahmed Eid Salem 5 Bohner, Martin J. 5 Burgos, Clara 5 Chatzarakis, George E. 5 Gasiński, Leszek 5 Henning, Michael Anthony 5 Hofstätter, Harald 5 Kalinowski, Rafał 5 Kefi, Khaled 5 Lupulescu, Vasile 5 Mädler, Conrad 5 Matkowski, Janusz 5 Migda, Małgorzata 5 Mirzoev, Karakhan Agakhan 5 Mishura, Yuliya Stepanivna 5 Moaaz, Osama 5 Páles, Zsolt 5 Pilśniak, Monika 5 Rahman, Ghaus Ur 5 Remy, Pascal 5 Repovš, Dušan D. 5 Samoĭlenko, Anatoliĭ Mykhaĭlovych 5 Villafuerte, Laura 5 Vincze, Csaba 5 Weder, Ricardo A. 4 Afrashteh, Esmaeil 4 Akin, Elvan 4 Bahyrycz, Anna 4 Balachandran, Selvaraj 4 Blidia, Mostafa 4 Eckhardt, Jonathan 4 Fritzsche, Bernd 4 Galewski, Marek 4 Gesztesy, Fritz 4 Grabowski, Piotr 4 Grubb, Gerd 4 Haraoka, Yoshishige 4 Henríquez, Hernán R. ...and 1,480 more Authors
all top 5
Cited in 313 Journals
48 Advances in Difference Equations 47 Journal of Mathematical Analysis and Applications 47 Opuscula Mathematica 34 Applied Mathematics and Computation 27 Discrete Applied Mathematics 24 Abstract and Applied Analysis 21 Mediterranean Journal of Mathematics 20 Complex Analysis and Operator Theory 19 Graphs and Combinatorics 17 Journal of Mathematical Physics 17 Boundary Value Problems 16 Discrete Mathematics 15 Journal of Differential Equations 15 Applied Mathematics Letters 14 Linear Algebra and its Applications 13 Ukrainian Mathematical Journal 13 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 13 Aequationes Mathematicae 13 Journal of Applied Mathematics and Computing 12 Electronic Journal of Differential Equations (EJDE) 12 Discussiones Mathematicae. Graph Theory 12 Discrete Dynamics in Nature and Society 11 Journal of Mathematical Sciences (New York) 10 Journal of Computational and Applied Mathematics 10 Mathematische Nachrichten 10 Fractional Calculus & Applied Analysis 10 Complex Variables and Elliptic Equations 9 Proceedings of the American Mathematical Society 9 Quaestiones Mathematicae 9 Results in Mathematics 9 Central European Journal of Mathematics 9 Journal of Function Spaces 8 Numerical Functional Analysis and Optimization 8 Theoretical Computer Science 8 Bulletin of the Iranian Mathematical Society 8 Turkish Journal of Mathematics 8 Advances in Nonlinear Analysis 7 Integral Equations and Operator Theory 7 Discrete Mathematics, Algorithms and Applications 6 Mathematical Notes 6 Rendiconti del Circolo Matemàtico di Palermo. Serie II 6 Journal of Difference Equations and Applications 6 Journal of Dynamical and Control Systems 6 Discrete and Continuous Dynamical Systems. Series B 6 Journal of Nonlinear Science and Applications 5 Periodica Mathematica Hungarica 5 ZAMP. Zeitschrift für angewandte Mathematik und Physik 5 Journal of Optimization Theory and Applications 5 Monatshefte für Mathematik 5 Calculus of Variations and Partial Differential Equations 5 Journal of Combinatorial Optimization 5 Lobachevskii Journal of Mathematics 5 Annales Henri Poincaré 5 Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 5 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 5 Proyecciones 5 Asian-European Journal of Mathematics 5 Symmetry 4 Applicable Analysis 4 Reports on Mathematical Physics 4 Journal of Functional Analysis 4 Acta Applicandae Mathematicae 4 Journal of Inequalities and Applications 4 Mathematical Inequalities & Applications 4 Communications in Nonlinear Science and Numerical Simulation 4 Differential Equations 4 Nonlinear Analysis. Real World Applications 4 Communications on Pure and Applied Analysis 4 Cubo 4 Journal of Fixed Point Theory and Applications 4 Discrete and Continuous Dynamical Systems. Series S 4 Differential Equations and Applications 4 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 4 Afrika Matematika 4 Journal of Applied Analysis and Computation 4 Fractional Differential Calculus 3 Computers & Mathematics with Applications 3 International Journal of Control 3 Indian Journal of Pure & Applied Mathematics 3 Mathematical Methods in the Applied Sciences 3 Rocky Mountain Journal of Mathematics 3 Chaos, Solitons and Fractals 3 Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 3 Funkcialaj Ekvacioj. Serio Internacia 3 Journal of Approximation Theory 3 The Journal of Geometric Analysis 3 Numerical Algorithms 3 The Australasian Journal of Combinatorics 3 Theory of Probability and Mathematical Statistics 3 Computational and Applied Mathematics 3 Integral Transforms and Special Functions 3 Sbornik: Mathematics 3 Discrete and Continuous Dynamical Systems 3 Arab Journal of Mathematical Sciences 3 Functional Differential Equations 3 Nonlinear Analysis. Modelling and Control 3 European Journal of Pure and Applied Mathematics 3 Tbilisi Mathematical Journal 3 Bulletin of Mathematical Analysis and Applications 3 Journal of Pseudo-Differential Operators and Applications ...and 213 more Journals
all top 5
Cited in 58 Fields
369 Ordinary differential equations (34-XX) 280 Operator theory (47-XX) 237 Partial differential equations (35-XX) 185 Combinatorics (05-XX) 144 Difference and functional equations (39-XX) 121 Real functions (26-XX) 72 Functional analysis (46-XX) 53 Numerical analysis (65-XX) 50 Integral equations (45-XX) 45 Dynamical systems and ergodic theory (37-XX) 44 Systems theory; control (93-XX) 39 Quantum theory (81-XX) 33 Number theory (11-XX) 33 Global analysis, analysis on manifolds (58-XX) 33 Probability theory and stochastic processes (60-XX) 32 Harmonic analysis on Euclidean spaces (42-XX) 23 Calculus of variations and optimal control; optimization (49-XX) 22 Functions of a complex variable (30-XX) 21 General topology (54-XX) 21 Computer science (68-XX) 19 Special functions (33-XX) 18 Linear and multilinear algebra; matrix theory (15-XX) 17 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 17 Biology and other natural sciences (92-XX) 16 Operations research, mathematical programming (90-XX) 13 Integral transforms, operational calculus (44-XX) 9 Statistics (62-XX) 9 Mechanics of deformable solids (74-XX) 8 Several complex variables and analytic spaces (32-XX) 8 Approximations and expansions (41-XX) 8 Fluid mechanics (76-XX) 7 Measure and integration (28-XX) 7 Sequences, series, summability (40-XX) 7 Abstract harmonic analysis (43-XX) 6 Potential theory (31-XX) 6 Optics, electromagnetic theory (78-XX) 5 Group theory and generalizations (20-XX) 5 Topological groups, Lie groups (22-XX) 5 Mechanics of particles and systems (70-XX) 5 Statistical mechanics, structure of matter (82-XX) 4 Commutative algebra (13-XX) 4 Convex and discrete geometry (52-XX) 4 Differential geometry (53-XX) 3 Associative rings and algebras (16-XX) 3 Manifolds and cell complexes (57-XX) 3 Relativity and gravitational theory (83-XX) 3 Information and communication theory, circuits (94-XX) 2 Mathematical logic and foundations (03-XX) 2 Geometry (51-XX) 2 Algebraic topology (55-XX) 2 Classical thermodynamics, heat transfer (80-XX) 1 General and overarching topics; collections (00-XX) 1 Field theory and polynomials (12-XX) 1 Algebraic geometry (14-XX) 1 Nonassociative rings and algebras (17-XX) 1 $$K$$-theory (19-XX) 1 Astronomy and astrophysics (85-XX) 1 Geophysics (86-XX)
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2021-07-24 10:53:31
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http://celebratio.org/Whitney_H/article/220/
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# Celebratio Mathematica
## Hassler Whitney
### The Whitney trick
#### by Rob Kirby
The Whitney trick is a method for removing points of intersection between two submanifolds. It can be seen in its most elementary form in Figure 1, in which it is obvious that the two points of intersection can be removed by an isotopy (a 1-parameter family of embeddings) of the arc labeled $P^p$ which pulls the arc across the disk $D$. (Note that $x$ and $y$ have opposite signs.) More generally the Whitney trick is used to remove a pair of intersections, $x$ and $y$, between two manifolds $P^p$ and $Q^q$ which are embedded in an ambient manifold $M^{p+q}$. To see how this is done, we first construct a model, then show how to embed it in $M$ (if possible), and then sketch some applications of the Whitney trick.
The model is merely a stabilization of the example in Figure 1. We cross the plane in which $D$ is embedded with $\mathbb R^{(p-1) + (q-1)}$ so that the ambient space is just $\mathbb R^{p+q}$, and then we cross the curve which includes $\alpha$ by $\mathbb R^{p-1}$ to get an $p$-dimensional manifold $P$, and similarly cross with $\mathbb R^{q-1}$ to get an $q$-manifold $Q$. These two manifolds still meet in two points $x$ and $y$, which are connected in $P$ by the original arc $\alpha$ and in $Q$ by the original arc $\beta$. Note that the two arcs still bound a 2-dimensional disk $D$, and that $D$ lies inside a larger open disk $\Delta$ in the plane. Also note that $\Delta$ has a normal $(p-1)+(q-1)$-plane bundle which splits as the direct sum (also called “Whitney sum”) of a $(p-1)$-plane bundle which coincides along $\alpha$ with the normal bundle of $\alpha$ in $P$, and an $(q-1)$-plane bundle which coincides along $\beta$ with the normal bundle of $\beta$ in $Q$.
The plane isotopy described in Figure 1 easily extends to an isotopy taking place in the plane crossed with the $p-1$ coordinates of $P$, as drawn for $p=2$ in Figure 2 [e4]; nothing happens with the other $q-1$ coordinates.
Now this model must be embedded in $M^{p+q}$ so that the actual manifolds $P$ and $Q$ and two points of intersection $x$ and $y$ correspond to the manifolds and points in the model.
If both $P$ and $Q$ are connected, then the arcs $\alpha$ and $\beta$ exist, and if $P$ and $Q$ are simply connected (as they often are in applications), then the arcs are unique up to homotopy. If $M$ is simply connected, then the disk $D$ can be mapped into $M$. If not, then $x$ must be connected by an arc (unique up to homotopy if $P$ is simply connected) to a base point $x_0 \in P$ which is connected by an arc to a base point $z \in M$. Similarly with arcs to a base point $y_0 \in Q$. It follows that $x$ then determines an element of $\pi_1(M)$ by running from $z$ to $x_0$ to $x$ to $y_0$ and back to $z$. Now if $x$ and $y$ both represent the same element of $\pi_1(M)$, then we can still map a disk $D$ into $M$. (This is important in proving the $s$-cobordism theorem.)
Once $D$ is mapped into $M$, we can embed it if the dimension of $M$, $p+q$, is five or more. Furthermore, if each of $p$ and $q$ is three or more, then the embedding of $D$ can be chosen to miss $P$ and $Q$ except along its boundary.
Now that $D$ is embedded missing $P$ and $Q$, it remains to find the embedding of the normal bundle of $D$. The normal $(p+q-2)$-bundle to $D$ (in fact, $\Delta$) in $M$ can be split along $\alpha$ as the normal $(p-1)$-bundle to $\alpha$ in $P$ direct sum the orthogonal $(q-1)$-bundle. That splitting extends across $\Delta$. The only problem remaining is that this $(p-1)$-plane bundle may not coincide with the $(p-1)$-plane bundle which is the normal bundle to $\beta$ in $Q$.
The problem reduces to an arc of $(p-1)$-planes in $\mathbb R^{(p-1) + (q-1)}$ which we want to isotope to the trivial arc, relative to the endpoints. Note that the trivial arc, as in the model, corresponds to $x$ and $y$ having opposite signs, so this is necessary. Now, this is possible because the fundamental group of the Stiefel manifold of $(p-1)$-planes in $\mathbb R^{p+q-2}$ is trivial when $p > 2$ (see [e3], p. 202). For more details, see the excellent description in [e4].
Whitney developed the Whitney trick in order to embed $P^p$ in $\mathbb R^{2p}$ [e1]. For $p=2$, this is easy. In higher dimensions, $P$ only immerses in $\mathbb R^{2p}$ (by general position), so for each double point, Whitney introduces in local fashion another double point of opposite sign (some thought is needed if $P$ is non-orientable), and then uses the Whitney trick to remove both points of intersection.
A later, and crucial, use of the Whitney trick is in Smale’s proof of the $h$-cobordism theorem [e2].
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2019-02-16 01:42:55
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https://search.r-project.org/CRAN/refmans/ecospat/html/ecospat.cor.plot.html
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ecospat.cor.plot {ecospat} R Documentation
## Correlation Plot
### Description
A scatter plot of matrices, with bivariate scatter plots below the diagonal, histograms on the diagonal, and the Pearson correlation above the diagonal. Useful for descriptive statistics of small data sets (better with less than 10 variables).
### Usage
ecospat.cor.plot(data)
### Arguments
data A dataframe object with environmental variables.
### Details
Adapted from the pairs help page. Uses panel.cor, and panel.hist, all taken from the help pages for pairs. It is a simplifies version of pairs.panels() function of the package psych.
### Value
A scatter plot matrix is drawn in the graphic window. The lower off diagonal draws scatter plots, the diagonal histograms, the upper off diagonal reports the Pearson correlation.
### Author(s)
Adjusted by L. Mathys, 2006, modified by N.E. Zimmermann
### Examples
data <- ecospat.testData[,4:8]
ecospat.cor.plot(data)
[Package ecospat version 3.4 Index]
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2022-12-02 17:19:14
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https://pypi.org/project/django-altuser/0.9.3/
|
Alternative user models for django >= 1.5, with email field and other features
## INSTALL
Put altuser in INSTALLED_APPS
INSTALLED_APPS += (
'altuser',
)
and configure your preferred user model from available altuser/models.py, for example
AUTH_USER_MODEL = 'altuser.MailSocialUser'
### Available models
• MailUser
• MailSocialUser
• MailConfirmedUser
• MailConfirmedSocialUser
• MailSocialConfirmfollowUser
• MailConfirmedSocialConfirmfollowUser
## USAGE
### OneToOne
If you use this User model with another Profile model you should put this field on you Profile model
user = models.OneToOneField(settings.AUTH_USER_MODEL,
related_name='profile')
If you have multiple profile types, you should use User.get_profile() to get the right profile associated with this user, but you also must set
ALTUSER_PROFILES_BREL = ['profile']
to a list of backward relation names (relate_name), of various profiles you have. For example if you have two models, client and managers, associated with a OneToOneField to our user model, and they have different related_name, one client_profile and the other manager_profile, then ALTUSER_PROFILES_BREL must be [‘client_profile’, ‘manager_profile’]
### GenericRelation
Actually you can also use the internal profile_type generic relation on the provided AbstractMailUser, that will permit you to coerce one profile type per user, it is up to you if using that or not.
# used in this way: self.user.get().usermodelfield
user = generic.GenericRelation(settings.AUTH_USER_MODEL,
content_type_field='profile_type',
object_id_field='profile_id')
Generic relations in this way permits to have your user and profile in the same inline and for example in the admin:
from django.contrib import admin
from .models import MannequineProfile
from django.contrib.auth import get_user_model
from django.contrib.contenttypes import generic
class UserInline(generic.GenericTabularInline):
model=get_user_model()
extra=1
max_num=1
ct_field = 'profile_type'
ct_fk_field = 'profile_id'
'groups', 'user_permissions' ,'likes',
'follows')
inlines = [
UserInline,
]
Note also, that if you delete an object that has a GenericRelation, any objects which have a GenericForeignKey pointing at it will be deleted as well. In the example above, this means that if a Profile object were deleted, any user objects pointing at it would be deleted at the same time.
## Confirmed Models
for using the mail confirmed models you must use [django-mail_confirmation](http://v.licheni.net/drc/django-mail_confirmation.git)
to filter out users that has confirmed social relations you do something like this:
get_user_model().objects.filter(id=user.id, follows=otheruser, relations__confirmed__confirmed=True)
## Project details
Uploaded source
|
2022-08-15 17:15:39
|
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http://math.stackexchange.com/questions/147857/what-is-the-limit-distance-to-the-base-function-if-offset-curve-is-a-function-to
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# What is the limit distance to the base function if offset curve is a function too?
I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that answer.
Q1) What is the limit distance to the base function if offset curve is a function too?
Q2) It can be shown as geometrically that all parallel curves of line and half circle are also functions. What is the whole function family defination for such functions?
Please see parallel curve examples below. (Thanks to J.M. for the graphs)
-
I will try to answer question 1 about "limit distance".
For a parametric curve $x=x(t)$, $y=y(t)$ to have an equation of the form $y=g(x)$, we need $x$ to be a strictly increasing function of $t$. Suppose we have a smooth function $y=f(x)$ and consider its parallel curve at distance $d$ (measured upward; $d$ could be positive or negative). Then $$x(t)=t-d\frac{f'(t)}{\sqrt{1+(f'(t)^2}}$$ If $x'>0$ for all $t$, then the parallel curve is also the graph of a function. Computation shows (after a simplification) that $$x'(t)=1-d\frac{f''(t)}{(1+f'(t)^2)^{3/2}}$$ So $x(t)$ is strictly increasing when $$d\frac{f''(t)}{(1+f'(t)^2)^{3/2}}<1$$ and fails to be strictly increasing if the reverse inequality holds. You will find the critical value of $d$ by considering the values of $f''(t)/{(1+f'(t)^2)^{3/2}}$. Not incidentally, the latter quantity is the curvature of the graph $y=f(x)$.
-
A different and somewhat more abstract viewpoint is given by considering the "squared distance function" defined by $\rho(x,y)=d((x,y),G)^2$ for all $(x,y)\in \Bbb R^2$. Here $G=\{(u,v):v=f(u)\}$ is the graph of your function, considered as a subset of $\Bbb R^2$, and $d((x,y),G)$ is the distance from $(x,y)$ to $G$, which is the same as the distance from $(x,y)$ to the closest point in $G$. (Assume that your function $f$ is continuous on $\Bbb R$ so that $G$ is a closed subset of $\Bbb R^2$.)
If your function $f$ is smooth then $\rho(x,y)$ will be smooth on a neighborhood of $G$. More precisely, if $f$ is smooth of class $C^k$ with $k\geq 2$ then $\rho(x,y)$ will be smooth of class $C^k$ near the graph $G$. (See this.) From now on, we always assume $k\geq 2$. We have used the squared distance function to get smoothness on the graph $G$, in the same way that the function $x^2$ is smooth at $x=0$ wheras the function $|x|$ is not.
How far away from $G$ will $\rho(x,y)$ be smooth? Let $(x,y)$ be some point in $\Bbb R^2$ and let $(u,v)$ be the point on $G$ which is closest to $(x,y)$ (assume that there is only one such point). If the distance between the two points is less than the radius of curvature of $G$ at $(u,v)$ then we are guaranteed that the squared distance function will be smooth at $(x,y)$.
Now compute the radius of curvature $r(u,v)$ of $G$ at a general point $(u,v)$ on the graph $G$. If the radius of curvature is bounded from below on $G$, so that we have $r(u,v)\geq c$ for some $c>0$ and all $(u,v)$ on the graph, then the squared distance will be smooth on the set of points $\{(x,y):d((x,y),G)<c\}$. You can then define parallel graphs on this set as level curves for the distance function $d(\cdot,G)$.
Can something go wrong here? Yes, if there are more than one point on $G$ which is nearest to $(x,y)$. For general curves this can be a problem, but since your curve is the graph of a function this problem cannot occur when $d(x,y)<c$, where $c$ is the uniform lower bound from the last paragraph for the radius of curvature.
I called this approach more abstract, since it is not so easy to get explicit formulas for the distance function $d(\cdot, G)$. Nevertheless, this function (or the function $\rho$) is often a useful tool for studying curves and higher dimensional surfaces.
-
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2015-06-29 23:29:02
|
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|
https://www.groundai.com/project/quantitative-predictions-in-quantum-decision-theory/
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Quantitative Predictionsin Quantum Decision Theory
# Quantitative Predictions in Quantum Decision Theory
## Abstract
Quantum Decision Theory, advanced earlier by the authors, and illustrated for lotteries with gains, is generalized to the games containing lotteries with gains as well as losses. The mathematical structure of the approach is based on the theory of quantum measurements, which makes this approach relevant both for the description of decision making of humans and the creation of artificial quantum intelligence. General rules are formulated allowing for the explicit calculation of quantum probabilities representing the fraction of decision makers preferring the considered prospects. This provides a method to quantitatively predict decision-maker choices, including the cases of games with high uncertainty for which the classical expected utility theory fails. The approach is applied to experimental results obtained on a set of lottery gambles with gains and losses. Our predictions, involving no fitting parameters, are in very good agreement with experimental data. The use of quantum decision making in game theory is described. A principal scheme of creating quantum artificial intelligence is suggested.
Quantum decision theory, decision making, choice between lotteries, attraction index, quantitative predictions, game theory, artificial intelligence
## I Introduction
Classical decision making, based on expected utility theory [1], is known to fail in many cases when decisions are made under risk and uncertainty. Numerous variants of so-called non-expected utility theories have been suggested to replace expected utility theory by using other more complicated functionals. The long list of such non-expected utility models can be found in the review articles [2, 3, 4]. The non-expected utility theories are, by construction, descriptive. By introducing several fitting parameters, such theories can be calibrated to some given set of empirical data. However, it is often possible to have different theories fitting the same set of experiments equally well, so that it is difficult to distinguish which of the models is better [5]. Moreover, on the basis of such theories, it is impossible to account for the known paradoxes arising in classical decision making and to make convincing out-of-sample predictions of new sets of empirical data. The non-expected utility theories have been thoroughly analyzed in numerous publications confirming the descriptive nature of these theories and their inability to perform useful predictions (see, e.g., [6, 7, 8, 9, 10]). Thus, Birnbaum [6, 7] carefully studied the so-called rank dependent utility theory and cumulative prospect theory, concluding that, even with fitting parameters, these theories are not able to get rid of paradoxes and moreover create new paradoxes. Safra and Segal [8] state that none of the non-expected utility theories can explain all main paradoxes but, on the contrary, distorting the structure of expected utility theory, the non-expected utility theories result in several non-expected inconsistencies. Al-Najjar and Weinstein [9, 10] present a detailed analysis of non-expected utility theories, coming to the conclusion that any variation of expected utility theory ”ends up creating more paradoxes and inconsistences than it resolves”.
The same conclusions apply to the so-called stochastic decision theories [11, 12, 13] that are based on underlying deterministic theories, decorating them with the probability of making errors in the choice. Introducing such probabilities, caused by decision-maker errors, into the log-likelihood functional adds several more parameters in the calibration exercise that improve the description of the given set of data. But such a stochastic decoration does not change the structure of the underlying deterministic theory and does not make predictions possible.
Clearly, the possibility of making predictions can be strongly hindered by the presence of unknown or poorly formulated conditions accompanying decision making. For instance, there can exist an unknown stochastic environment [14] or a varying context [15]. It may also happen that the provided information is imprecise and only partially reliable [16] or preference relations are incomplete [17] requiring the use of fuzzy logic [18]. In such situations, any prediction is likely to be only partial and often merely qualitative.
But even when the posed problem is well defined, suggesting, e.g., a choice between explicitly presented lotteries, quantitative predictions as a rule are impossible. In particular, the non-expected utility theories mentioned above have been developed exactly for such seemingly simple choice between well defined lotteries. And, as is discussed above, in many cases, the given lotteries, although being explicitly formulated, contain uncertainty not allowing for predictions. It is important to also stress that, in some cases of well defined lotteries, predictions based on utility theory are qualitatively wrong, as has been demonstrated by Kahneman and Tversky [19].
In the present paper, we consider the situation when decision making consists in the choice between well defined lotteries. We develop an approach allowing for quantitative predictions in arbitrary cases, including those where utility theory fails, being unable to provide even qualitatively correct conclusions. It is important to emphasize that quantitative predictions in our approach can be realized without any fitting parameters. So, our approach is not a descriptive, but rather a normative, or prescriptive theory.
Our approach is based on Quantum Decision Theory (QDT), which we developed earlier [20, 21, 22, 23, 24, 25, 26]. There have been other attempts to apply quantum techniques to cognitive sciences, as is discussed in the books [27, 28, 29, 30] and review articles [31, 32, 33, 34]. However, these attempts were based on constructing some models for describing particular effects, with the use of several fitting parameters for each case. Our approach of QDT is essentially different from all those models in the following facets. First, QDT is formulated as a general theory applicable to any variant of decision making, but not as a special model for a particular case. Second, the mathematical structure of QDT is common for both decision theory as well as for quantum measurements, which has been achieved by generalizing the von Neumann [35] theory of quantum measurements to the treatment of inconclusive measurements and composite events represented by noncommutative operators [36, 37, 38, 39]. The third unique feature of QDT is the possibility to develop quantitative predictions without any fitting parameters, as has been shown for some simple choices in decision making [40].
The predictions concern the fractions of decision makers choosing the corresponding lotteries. In QDT, such fractions are predicted by their corresponding behavioral quantum probabilities, as follows from the frequentist interpretation of probabilities and the assumption that the population of decision makers are, to a first approximation, representative of a homogenous group of individuals making probabilistic choices. The scheme for calculating the quantum probabilities is based on our previous demonstration that it consists of two terms, called utility and attraction factors. The utility factor derives from the utility of each lottery, being defined on prescribed rational grounds. The attraction factor represents the irrational side of a choice. The value of the attraction factor for a single decision maker and for a given choice is random. However, for a society of decision makers, one can derive the quarter law, which estimates the non-informative prior for the absolute value of the average attraction factor as equal to . In simple cases, the signs of the attraction factors can be prescribed by the principle of ambiguity aversion. In more complicated situations, a criterion has been suggested [40] and applied to lotteries with gains.
Here, we extend Ref. [40] by considering lotteries with both gains and losses, and not just gains. We also improve on the quarter law based on the non-informative prior, by including available information on the level of ambiguity characterizing a given set of games, thus providing the potential for improved predictions. Moreover, we consider the cases for which our previously proposed criterion defining the signs of attraction factors does not allow for unique conclusions. We present a generalization of the criterion for the sign of the attraction factors that addresses these limitations and also applies to lotteries with losses.
The possibility of mathematically formalizing all steps of a decision process, allowing for quantitative predictions, is important, not merely for decision theory, but also for the problem of creating an artificial quantum intelligence that could function only if all operations are explicitly formalized in mathematical terms. We have previously mentioned [41] that QDT can provide such a basis for creating artificial quantum intelligence, since the QDT mathematical foundation is formulated in the same way as the theory of quantum measurements.
In the present paper, we overcome the limitations of our previous publication [40] by generalizing QDT along the following directions.
(i) A general method for defining utility factors is advanced, valid for lotteries with losses as well as for lotteries with gains, or mixed-type lotteries.
(ii) A criterion is formulated for the quantitative classification of attraction factors for all kinds of lotteries, whether with gains or with losses. In the case of games with two lotteries, this criterion uniquely prescribes the signs of attraction factors.
(iii) The quarter law is generalized by taking into account the ambiguity level for a given set of games. This defines the typical absolute value of the attraction factor more accurately than the quarter law following from non-informative prior.
(iv) A method for estimating attraction factors for games with multiple lotteries is described.
(v) The value of our theory is illustrated by comparing its prediction with empirical results obtained on a set of games containing lotteries with gains and with losses, for which expected utility theory fails. Our approach results in quantitative predictions, without fitting parameters, which are in very good agreement with empirical data.
(vi) It is shown how the QDT can be applied to game theory. An application is illustrated by the prisoner dilemma.
(vii) The general principles for creating artificial quantum intelligence are suggested. It is emphasized that artificial intelligence, mimicking the functioning of human consciousness, should be quantum.
## Ii Scheme of Quantum Decision Theory
In the present section, we briefly sketch the basic scheme of QDT in order to remind the reader about the definition of quantum probability used in decision theory. The technical details have been thoroughly expounded in the previous articles [20, 21, 22, 23, 24, 25, 26], which allows us to just recall here the basic notions.
As is mentioned in the Introduction, the mathematical scheme is equally applicable to quantum decision theory as well as to the theory of quantum measurements [36, 37, 38, 39]. An event can mean either the result of an estimation in the process of measurements, or a decision in decision making. In both the cases, there exist simple events that are operationally testable, that is, clearly observable, and inconclusive events that are either non-observable or even not well specified. The typical example in quantum measurements is the double-slit experiment, where the final registration of a particle by a detector is an operationally testable event, while the passage through one of the slits is not observable. In decision making, a straightforward example would be the choice between lotteries under uncertainty. The final choice of a lottery is an operationally testable event, while the deliberations on real or imaginary uncertainties in the formulation of the lotteries or in hesitations of the decision-maker can be treated as inconclusive events.
We consider a set of events labelled by an index . Each event is put into correspondence with a state of a Hilbert space , with the family of states forming an orthonormalized basis:
An→|n⟩∈HA=span{|n⟩}. (1)
There also exists another set of events , labelled by an index , with each event being in correspondence with a state of a Hilbert space , the family of the states forming an orthonormalized basis:
Bα→|α⟩∈HB=span{|α⟩}. (2)
A pair of events from different sets forms a composite event represented by a tensor-product state ,
An⨂Bα→|n⟩⨂|α⟩∈H, (3)
in the Hilbert space
H≡HA⨂HB=span{|n⟩⨂|α⟩}. (4)
An event is called operationally testable if and only if it induces a projector on the space . The event set is assumed to consist of operationally testable events.
A different situation occurs when we have an inconclusive event being a set
B≡{Bα,bα:α=1,2,…} (5)
of events associated with amplitudes that are random complex numbers. An inconclusive event corresponds to a state in the space , such that
B→|B⟩=∑αbα|α⟩∈HB. (6)
The states are not orthonormalized, because of which the operator is not a projector.
A composite event is termed a prospect. Of major interest are the prospects composed of an operationally testable event and an inconclusive event:
πn=An⨂B. (7)
A prospect corresponds to a prospect state in the space ,
πn→|πn⟩=|n⟩⨂|B⟩∈H, (8)
and induces a prospect operator
^P(πn)≡|πn⟩⟨πn|. (9)
The prospect states are not orthonormalized and the prospect operator is not a projector. The given set of prospects forms a lattice
L={πn:n=1,2,…,NL}, (10)
whose ordering is characterized by prospect probabilities to be defined below. The assembly of prospect operators composes a positive operator-valued measure. By its role, this set is analogous to the algebra of local observables in quantum theory.
The strategic state of a decision maker in decision theory, or statistical operator of a system in physics, is a semipositive trace-one operator defined on the space . The prospect probability is the expectation value of the prospect operator:
p(πn)=Tr^ρ^P(πn), (11)
with the trace over the space . To form a probability measure, the prospect probabilities are normalized,
∑np(πn)=1,0≤p(πn)≤1. (12)
Taking the trace in (11), it is possible to separate out positive-defined terms from sign-undefined terms, which respectively, are
f(πn)=∑α|bα|2⟨nα|^ρ|nα⟩,
q(πn)=∑α≠βb∗αbβ⟨nα|^ρ|nβ⟩. (13)
Then the prospect probability reads as
p(πn)=f(πn)+q(πn). (14)
The appearance of a sign-undefined term is typical for quantum theory, describing the effects of interference and coherence.
Note that the decision-maker strategic state has to be characterized by a statistical operator and not just by a wave function since, in real life, any decision maker is not an isolated object but a member of a society [38, 40].
An important role in quantum theory is played by the quantum-classical correspondence principle [42, 43], according to which classical theory has to be a particular case of quantum theory. In the present consideration, this is to be understood as the reduction of quantum probability to classical probability under the decaying quantum term:
p(πn)→f(πn),q(πn)→0. (15)
In quantum physics, this is also called decoherence, when quantum measurements are reduced to classical measurements. The positive-definite term , playing the role of classical probability, is to be normalized,
∑nf(πn)=1,0≤f(πn)≤1. (16)
From conditions (12) and (16) it follows
∑nq(πn)=0,−1≤q(πn)≤1, (17)
which is called the alternation law.
In decision theory, the classical part describes the utility of the prospect , which is defined on rational grounds. In that sense, a prospect is more useful than if and only if
f(π1)>f(π2)(moreuseful). (18)
The quantum part characterizes the attractiveness of the prospect, which is based on irrational subconscious factors. Hence a prospect is more attractive than if and only if
q(π1)>q(π2)(moreattractive). (19)
And the prospect probability (14) defines the summary preferability of the prospect, taking into account both its utility and attractiveness. So, a prospect is preferable to if and only if
p(π1)>p(π2)(preferable). (20)
The structure of the quantum probability (14), consisting of two parts, one showing the utility of a prospect and the other characterizing its attractiveness, is representative of real-life decision making, where both these constituents are typically present. Quantum probability, taking into account the rationally defined utility as well as such an irrational behavioral feature as attractiveness, can be termed as behavioral probability.
It is worth stressing that QDT is an intrinsically probabilistic theory. This is different from stochastic decision theories, where the choice is assumed to be deterministic, while randomness arises due to errors in decision making. The probabilistic nature of QDT is not caused by errors in decision making, but it is due to the natural state of a decision maker, described by a kind of statistical operator. Upon the reduction of QDT to a classical decision theory, it reduces to a probabilistic variant of the latter, since decisions under uncertainty are necessarily probabilistic [44]. As mentioned above, the description of a decision maker strategic state by a statistical operator, and not by a wave function, emphasizes the fact that any decision maker is not an absolutely isolated object but rather a member of a society, who is subjected to social interactions [38, 40, 45]. When comparing theoretical predictions with empirical data, it follows from the logical structure of QDT that one has to compare the theoretically calculated probability (14) with the fraction of decision makers preferring the considered prospect.
## Iii General Definition of Utility Factors
In this section, we describe the general method for defining utility factors for a given set of lotteries containing both gains as well as losses.
Let a set of payoffs be given
Xn={xi:i=1,2,…,Nn}, (21)
in which payoffs can represent either gains or losses, being, respectively positive or negative. The probability distribution over a payoff set is a lottery
Ln={xi,pn(xi):i=1,2,…,Nn}, (22)
with the normalization condition
∑ipn(xi)=1,0≤pn(xi)≤1. (23)
The lotteries are enumerated by the index . Under a utility function , the expected utility of lottery is
U(Ln)=∑iu(xi)pn(xi)(n=1,2,…,NL). (24)
Utility functions for gains and losses can be of different signs. Therefore, the expected utility can also be either positive or negative. When it is negative, one often uses the notation of the lottery cost
C(Ln)≡−U(Ln)=|U(Ln)|(U(Ln)<0).
An expected utility is positive, when in its payoffs gains prevail. And it is negative, when losses overwhelm gains.
As has been explained in Ref. [40], the choice between the given lotteries in any game is always accompanied by uncertainty related to the decision-maker hesitations with respect to the formulation of the game rules, understanding of the problem, and his/her ability to decide what he/she considers the correct choice. All these hesitations form an inconclusive event denoted above as . Therefore a choice of a lottery is actually a composite event, or a prospect
πn=Ln⨂B(n=1,2,…,NL). (25)
Here we denote the action of a lottery choice and a lottery by the same latter , which should not lead to confusion. The utility factor characterizes the utility of choosing a lottery . Since QDT postulates that the choice is probabilistic, it is possible to define the average quantity over the set of lotteries,
U=NL∑n=1f(πn)U(Ln), (26)
playing the role of a normalization condition for random expected utilities [46].
The utility factor represents a classical probability distribution and can be found from the conditional minimization of Kullback-Leibler information [47, 48]. The use of the Kullback-Leibler information for defining such a probability distribution is justified by the Shore-Jonson theorem [49] stating that there exists only one distribution satisfying consistency conditions, and this distribution is uniquely defined by the minimum of the Kullback-Leibler information, under given constraints. The role of the constraints here are played by the normalization conditions (16) and (26). Then the information functional reads as
I[f]=NL∑n=1f(πn)lnf(πn)f0(πn)+
+γ[NL∑n=1f(πn)−1]+β[U−NL∑n=1f(πn)Un], (27)
where is a prior distribution, , and and are Lagrange multipliers.
As boundary conditions, it is natural to require that the utility factor of a lottery with asymptotically large expected utility would tend to unity,
f(πn)→1(Un→∞), (28)
while the utility factor of a lottery with asymptotically large cost, would go to zero,
f(πn)→0(Un→−∞). (29)
Also, the utility factors, as their name implies, have to increase together with the related expected utilities,
δf(πn)δUn≥0. (30)
Minimizing the information functional (27) results in the utility factors
f(πn)=f0(πn)eβUn∑nf0(πn)eβUn, (31)
with a non-negative parameter .
If one assumes that the prior distribution is uniform, such that , then one comes to the utility factors of the logit form. However, the uniform distribution does not satisfy the boundary conditions (28) to (29). Therefore a more accurate assumption, taking into account the boundary conditions, should be based on the Luce choice axiom [50, 51]. According to this axiom, if an -th object, from the given set of objects, is scaled by a quantity , then the probability of its choice is
f0(πn)=λn∑nλn. (32)
In our case, the considered objects are lotteries and they are scaled by their expected utilities. So, for the non-negative utilities, we can set
λn=Un(Un≥0), (33)
while for negative utilities,
λn=1|Un|(Un<0). (34)
Expression (34) is chosen in order to comply with Luce’s axiom together with the ranking of preferences with respect to losses.
Generally, utilities can be measured in some units, say, in monetary units . Then we could use dimensionless scales defined as and for gains and losses, respectively. Obviously, expression (32) is invariant with respect to units in which is measured. Therefore, for simplicity of notation, we assume that utilities are dimensionless.
Thus, the utility factor (31), with prior (32), is
f(πn)=λneβUn∑nλneβUn(β≥0). (35)
In particular, when gains prevail, so that all expected utilities are non-negative, then
f(πn)=UneβUn∑nUneβUn(∀Un≥0). (36)
While, when losses prevail, and all expected utilities are negative, then
f(πn)=|Un|−1e−β|Un|∑n|Un|−1e−β|Un|(∀Un<0). (37)
In the mixed case, where the utility signs can be both positive and negative, one has to employ the general form (35).
The parameter characterizes the belief of the decision maker with respect to whether the problem is correctly posed. Under strong belief, one gets
f(πn)={1, Un=maxnUn0, Un≠maxnUn(β→∞), (38)
which recovers the classical utility theory with the deterministic choice of a lottery with the largest expected utility. In the opposite case of weak belief, when uncertainty is strong, one has
f(πn)=λn∑nλn(β=0). (39)
To explicitly illustrate the forms of the utility factors, let us consider the often met situation of two lotteries under strong uncertainty, thus, considering the binary prospect lattice
L={πn:n=1,2}(β=0), (40)
with zero belief parameter. Then, if in both the lotteries gains prevail, we have
f(πn)=UnU1+U2(U1≥0,U2≥0). (41)
When losses are prevailing in the two lotteries, then
f(πn)=1−|Un||U1|+|U2|(U1<0,U2<0). (42)
And if one expected utility is positive, say that of the first lottery, while the other utility is negative, then the utility factor for the first lottery is
f(π1)=U1|U2|U1|U2|+1(U1>0,U2<0), (43)
respectively, .
In this way, the utility factors are explicitly defined for any combination of lotteries in the given game, with the payoff sets containing gains as well losses.
## Iv Classification of Lotteries by Attraction Indices
We now give a prescription for defining the attraction factors. By its definition, an attraction factor quantifies how each of the given lotteries is more or less attractive. The attractiveness of a lottery is composed of two factors, possible gain and its probability. It is clear that a lottery is more attractive, when it suggests a larger gain and/or this gain is more probable. In other words, a more attractive lottery is more predictable and promises a larger profit. On the contrary, a lottery suggesting a smaller gain or a larger loss and/or higher probability of the loss, is less attractive. A less certain lottery is less attractive, since it is less predictable, which is named as uncertainty aversion or ambiguity aversion. Below we give an explicit mathematical formulation of these ideas.
Let us introduce, for a lottery , the notation for the minimal gain
gn≡mini{xi≥0:xi∈Ln} (44)
and for the minimal loss
ln≡maxi{xi≤0:xi∈Ln}. (45)
These quantities characterize possible gains and losses in the given lotteries.
But payoffs are not the only features that attract the attention of decision makers. In experimental neuroscience, it has been discovered that, during the act of choosing, the main and foremost attention of decision makers is directed to the payoff probabilities [52]. We capture this empirical observation by considering different weights related to payoffs and to their probabilities in the characterization of the lottery attractiveness. Specifically, the weight of a payoff should be much smaller than the weight of its probability . We quantitatively formulate this by choosing weights proportional respectively to for the payoff versus for its probability. The later term is motivated by the decimal number system. This leads us to defining the lottery attractiveness
an≡an(Ln)≡∑ixi10pn(xi). (46)
And the related relative quantity can be termed the attraction index
αn=αn(Ln)≡an∑m|am|. (47)
The latter satisfies the normalization condition
∑n|αn|=1. (48)
The notion of the lottery attraction index makes it straightforward to classify all lotteries from the considered game onto more or less attractive. Thus a lottery is more attractive than , hence
q(π1)>q(π2), (49)
when the attraction index of the first lottery is larger than that of the second,
α1>α2. (50)
In the marginal case, when , the first lottery is more attractive if the probability of its minimal gain is smaller than that of the second lottery,
α1=α2≥0,p(g1)
For short, this will be denoted as . And in the other marginal case, where , the first lottery is more attractive if the probability of its minimal loss is larger than that of the second,
α1=α2<0,p(l1)>p(l2). (52)
This, for short, will be denoted as .
The criterion allows us to arrange all the given lotteries with respect to the level of their attractiveness.
For the particular case of a binary prospect lattice (40), the alternation property (17) reads as
q(π1)+q(π2)=0. (53)
Therefore the attraction factors have different signs,
q(π1)=−q(π2). (54)
The sign of each of the attraction factors is prescribed by the sign of the difference
Δα≡α1−α2. (55)
If is positive, then the attraction factor of the first prospect is positive and that of the second is negative. On the contrary, if is negative, then the attraction factor of the first lottery is negative and that of the second is positive. In the marginal case, when , we shall use the notations accepted above and explained below (Eqs. (51) and (52)): If the first lottery is more attractive, we shall write , while when the second lottery is more attractive, this will be denoted as .
## V Typical Values of Attraction Factors
The criterion of the previous section allows us to classify all the lotteries of the considered game onto more or less attractive. But we also need to define the amplitudes of the attraction factors. According to QDT, these values are probabilistic variables, characterizing irrational subjective features of each decision maker. For different subjects, they may be different. They can also be different for the same subject at different times [13]. Different game setups also influence the values of the attraction factors [53]. However, for a probabilistic quantity, it is possible to define its average or typical value.
### V-a General considerations
We consider games, enumerated by , with lotteries in each, enumerated by . And let the choice be made by a society of decision makers, numbered by . In a -th game, decision makers make a choice between prospects . The typical value of the attraction factor is defined as the average
¯¯¯q≡1NGNG∑k=11NLNL∑n=1∣∣ ∣∣1NN∑j=1qj(πnk)∣∣ ∣∣. (56)
Denoting the mean value of the attraction factor for a prospect , as
|q(πn)|≡1NGNG∑k=1∣∣ ∣∣1NN∑j=1qj(πnk)∣∣ ∣∣, (57)
we can write
¯¯¯q=1NLNL∑n=1|q(πn)|. (58)
For a large value of the product , the distribution of the attraction factors can be characterized by a probability distribution , which, in view of property (17), is normalized as
∫1−1φ(q)dq=1. (59)
The average absolute value of the attraction factor can be represented by the integral
¯¯¯q=∫10qφ(q)dq. (60)
This defines the typical value of the attraction factor that characterizes the level of deviation from rationality in decision making [54].
If there is no information on the properties and specifics of the given set of lotteries in the suggested games, then one should resort to a non-informative prior, assuming a uniform distribution satisfying normalization (59), which gives . Substituting the uniform distribution into the typical value of the attraction factor (60) yields , which was named the “quarter law” in the earlier paper [40].
However, it is possible to find a more precise typical value by taking into account the available information on the given lotteries. For example, it is straightforward to estimate the level of uncertainty of the lottery set.
### V-B Choice between two prospects
When choosing between two lotteries with rather differing utilities, the choice looks quite easy - the lottery with the largest utility is preferred. But when two lotteries have very close utilities, choosing becomes difficult. The closeness of the lotteries, corresponding to two prospects and , can be quantified by the relative difference
δf(π1,π2)≡2|f(π1)−f(π2)|f(π1)+f(π2)×100%. (61)
When the choice is between just two prospects, whose utility factors are normalized according to condition (16), hence when , then the relative difference simplifies to
δf=2|f(π1)−f(π2)|×100%(NL=2). (62)
There have been many discussions concerning choices between similar alternatives with close utilities or close probabilities, such that the choice becomes hard to make [55, 56, 57, 58]. We refer to such situations as “irresolute”. One of the major problems is how to quantify the similarity or closeness of the choices. Several variants of measuring the distance between the alternatives and have been suggested, including the linear distance , as well as different nonlinear distances , with . We propose that the value of that serves as an upper threshold, below which the lotteries are irresolute, should not depend on the exponent used in the definition of the distance. Therefore, in order for the exponent not to influence the boundary value, one has to require the invariance of the distance with respect to the exponent at the threshold, so that the critical threshold value should obey the equality: for any . The latter reads explicitly as
[δfc(π1,π2)]m=δfc(π1,π2),
where is measured in percents. This equation is valid for arbitrary only for . Hence the critical boundary value equals . Thus the lotteries, for which the irresoluteness criterion
δf(π1,π2)<1% (63)
is valid, are to be treated as close, or similar, and the choice between them, as irresolute.
The next question is how the irresoluteness in the choice influences the typical attraction factor. Suppose that the fraction of irresolute games equals . Then the following properties of the distribution over admissible attraction factors should hold.
In the presence of irresolute games for which the irresoluteness criterion holds true, the probability that the attraction factor is zero is asymptotically small,
limq→0φ(q)=0(ν>0). (64)
In other words, this condition means that, on the manifold of all possible games, absolutely rational games form a set of zero measure.
If not all games are irresolute , the probability of the maximal absolute value of the attraction factor is asymptotically small,
lim|q|→1φ(q)=0(ν<1). (65)
That is, on the manifold of all possible games, absolutely irrational games compose a set of zero measure.
Often employed as a prior distribution in standard inference tasks [59, 60, 61], the simplest distribution that obeys the two conditions (64) and (65) is the beta distribution that, under normalization (59), reads
φ(q)=|q|ν(1−|q|)1−νΓ(1+ν)Γ(2−ν). (66)
Using this distribution, expression (60) gives the typical attraction factor value
¯¯¯q=1+ν6. (67)
Note that the average of given by (67) over the two boundary values and gives
12(16+26)=14,
thus recovering the non-informative quarter law.
This expression (67) can be used for predicting the results of decision making. For example, in the case of a binary prospect lattice, the difference in the attraction indices (55) defines the signs of the attraction factors, making it possible to prescribe the attraction factors and to the considered prospects.
### V-C Choice between more than two prospects
When there are more than two prospects in the considered game, we propose the following procedure to estimate the attraction factors. Using the classification of the prospects by the attraction indices, as is described in the previous section, it is straightforward to arrange the prospects in descending order of attractiveness,
q(πn)>q(πn+1)(n=1,2,…,NL−1). (68)
Let the maximal attraction factor be denoted as
qmax≡q(π1)>0. (69)
Given the unknown values of the attraction factors, the non-informative prior assumes that they are uniformly distributed and at the same time they must obey the ordering constraint (68). Then, the joint cumulative distribution of the attraction factors is given by
Pr[q(π1)<η1,...,q(πNL)<ηNL|η1≤η2≤...≤ηNL]=
=∫η10dx1∫η2x1dx2....∫ηNLxNL−1dxNL , (70)
where the series of inequalities ensure the ordering. It is then straightforward to show that the average values of the are equidistant, i.e. the difference between any two neighboring factors, on average, is independent of , so that
Δ≡⟨q(πn)⟩−⟨q(πn+1)⟩=const. (71)
Taking their average values as determining their typical values, we omit the symbol representing the average operator and use the previous equation to represent the -th attraction factor as
q(πn)=qmax−(n−1)Δ. (72)
From the alternation property (17), it follows that
qmax=NL−12Δ. (73)
The total number of lotteries can be either even or odd, leading to slightly different forms for the following expressions.
And the definition of the typical value (58) gives
Δ=⎧⎪⎨⎪⎩4¯¯¯q/NL (NLeven)4¯¯¯qNL/(N2L−1) (NLodd). (74)
Then the maximal attraction factor (73) becomes
qmax=⎧⎪⎨⎪⎩2¯¯¯q(NL−1)/NL (NLeven)2¯¯¯qNL/(NL+1) (NLodd). (75)
Therefore formula (72) yields the expressions for all attraction factors
q(πn)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩2¯¯¯qNL+1−2nNL(NLeven)2¯¯¯qNL(NL+1−2n)N2L−1(NLodd). (76)
Let us denote the set of all attraction factors in the considered game as
QNL≡{q(πn):n=1,2,…,NL}.
If there are only two lotteries, then we have
Δ=2¯¯¯q,qmax=¯¯¯q(NL=2),
and the attraction-factor set is
Q2={¯¯¯q,−¯¯¯q}.
In the case of three lotteries,
Δ=32¯¯¯q,qmax=32¯¯¯q(NL=3),
and the attraction factor set reads as
In that way, all attraction factors can be defined.
## Vi Quantitative Predictions in Decision Making
In order to illustrate how the suggested theory makes it possible to give quantitative predictions, without any fitting parameters, let us consider the set of experiments performed by Kahneman and Tversky [19]. This collection of games, including both gains and losses, is a classical example showing the inability of standard utility theory to provide even qualitatively correct predictions as a result of the confusion caused by very close or coinciding expected utilities. Let us emphasize that the choice of these games has been done by Kahneman and Tversky [19] in order to stress that standard decision making cannot be applied for these games. This is why it is logical to consider the same games and to show that the use of QDT does allow us not only to qualitatively explain the correct choice, but also that QDT provides quantitative predictions for such difficult cases.
In the set of games described below, each game consists of two lotteries , with . The number of decision makers is about .
Recall that, as is explained in Sec. III, the choice between lotteries corresponds to the choice between prospects (25) including the action of selecting a lottery under a set of inconclusive events representing hesitations and irrational feelings. Therefore the choice, under uncertainty, between lotteries is equivalent to the choice between prospects . The choice under uncertainty for the case of a binary lattice can be characterized by the utility factors (41) to (43). We take the linear utility function, whose convenience is in the independence of the utility factors from the monetary units used in the lottery payoffs. The attraction factors are calculated by following the recipes described in Sec. IV and Sec. V.
We compare the prospect probabilities , theoretically predicted by QDT, with the empirically observed fractions [19]
pexp(πn)≡N(πn)N
of the decision makers choosing the prospect , with respect to the total number of decision makers taking part in the experiments.
### Vi-a Lotteries with gains
Game 1. The lotteries are
L1={2.5,0.33|2.4,0.66|0,0.01},L2={2.4,1}.
For this game, we shall show explicitly the related calculations, while omitting the intermediate arithmetics in the following cases.
The utilities of these lotteries are
U(L1)=2.5×0.33+2.4×0.66+0×0.01=2.409,
U(L2)=2.4×1=2.4.
Their sum is
U(L1)+U(L2)=2.409+2.4=4.809.
The utility factors are close to each other,
f(π1)=2.4094.809=0.501,f(π2)=2.44.809=0.499.
For the lottery attractiveness (46), we find
a1=2.5×100.33+2.4×100.66+00.1=16.32,
a2=2.4×101=24,
which gives
a1+a2=16.32+24=40.32.
The attraction indices (47) become
α1=16.3240.32=0.405,α2=2.440.32=0.595.
Then the attraction difference (55) is
Δα=0.405−0.595=−0.19.
The negative attraction difference tells us that the first lottery is less attractive, , which suggests that the second lottery is preferable, . The experimental results confirm this, displaying the fractions of decision makers choosing the respective lotteries as
pexp(π1)=0.18,pexp(π2)=0.82.
Thus, although the first lottery is more useful, having a larger utility factor, it is less attractive, which makes it less preferable.
Game 2. The lotteries are
L1={2.5,0.33|0,0.67},L2={2.4,0.34|0,0.66}.
The following procedure is the same as in the first game. Calculating the utility factors
f(π1)=0.503,f(π2)=0.497,
we again see that the lottery utilities are close to each other, so it is difficult to make the choice. For the lottery attractiveness, we have
a1=16.57,a2=5.25,
giving the attraction indices
α1=0.759,α2=0.241,
and the attraction difference
Δα=0.518.
Now the latter is positive, showing that the first lottery is more attractive, , which suggests that the first lottery is preferable, . The experimental data for the related fractions are
pexp(π1)=0.83,pexp(π2)=0.17,
in agreement with the expectation that the first lottery is preferable.
Game 3. The lotteries are
L1={4,0.8|0,0.2},L2={3,1}.
We calculate in the prescribed way the utility factors
f(π1)=0.516,f(π2)=0.484,
lottery attractiveness,
a1=25.24,a2=30,
and the attraction indices
α1=0.457,α2=0.543.
The negative attraction difference
Δα=−0.086
implies that the first lottery is less attractive, , which tells us that the second lottery should be preferable, . Again this is in agreement with the experimental results
pexp(π1)=0.2,pexp(π2)=0.8.
The first lottery is less preferable, despite it is more useful, having a larger utility factor.
Game 4. The lotteries are
L1={4,0.2|0,0.8},L2={3,0.25|0,0.75}.
Calculating the utility factors
f(π1)=0.516,f(π2)=0.484,
lottery attractiveness
a1=6.34,a2=5.33,
and the attraction indices
α1=0.543,α2=0.457,
we find the positive attraction difference
Δα=0.086.
Hence the first lottery is more attractive , which suggests that the first lottery is preferable, . The experimental data
pexp(π1)=0.65,pexp(π2)=0.35
confirm this expectation.
Game 5. The lotteries are
L1={6,0.45|0,0.55},L2={3,0.9|0,0.1}.
The utility factors
f(π1)=0.5,f(π2)=0.5
turn out to be equal, which makes it impossible to decide in the frame of classical decision theory based on expected utilities. Then we calculate the lottery attractiveness
a1=16.91,a2=23.83,
and the related attraction indices
α1=0.415,α2=0.585.
The negative attraction difference
Δα=−0.17
means that the first lottery is less attractive, , thence the second lottery is expected to be preferable, . This is confirmed by the empirical data
pexp(π1)=0.14,pexp(π2)=0.86.
Game 6. The lotteries are
L1={6,0.001|0,0.999},L2={3,0.002|0,0.998}.
Again their utility factors are equal to each other,
f(π1)=0.5,f(π2)=0.5.
The lottery attractiveness values
a1=6.01,a2=3.01
yield the attraction indices
α1=0.666,α2=0.334,
whose positive attraction difference
Δα=0.332
implies that the first lottery is more attractive, , which suggests that the first lottery should be preferable, . The experimental results are
pexp(π1)=0.73,pexp(π2)=0.27,
in agreement with the expectation.
Game 7. The lotteries are
L1={6,0.25|0,0.75},L2={4,0.25|2,0.25|0,0.5}.
Their equal utility factors,
f(π1)=0.5,f(π2)=0.5,
do not allow us to make a choice based on their utility. We calculate the lottery attractiveness
a1=10.67,a2=10.67
and the attraction indices
α1=0.5,α2=0.5.
Here the attraction difference is zero, , with the attraction indices being positive. Therefore, we resort to criterion (51), for which the minimal gains are . We find that
p1(gmin1)=0.75>p2(gmin2)=0.5.
According to definitions (51) and (52), the marginal case, when and , is denoted as . This proposes that the first lottery is less attractive, according to the negative sign
Δα=−0.
Thus we find that , which suggests that the second lottery is preferable, . The experimental results give
pexp(π1)=0.18,pexp(π2)=0.82.
### Vi-B Lotteries with losses
In the previous seven games, the lotteries with gains were considered. We now turn to lotteries with losses.
Game 8. The lotteries are
L1={−4,0.8|0,0.2},L1={−3,1}.
Following the same general procedure, we find the utility factors
f(π1)=0.484,f(π2)=0.516,
lottery attractiveness
a1=−25.24,a2=−30,
and the attraction indices
α1=−0.457,α2=−0.543.
The positive attraction difference
Δα=0.086
means that the first lottery is more attractive, , because of which, we expect that the first lottery is preferable, . The experiments give
pexp(π1)=0.92,pexp(π2)=0.08,
confirming that the first lottery is preferable, although its utility factor is smaller.
Game 9. The lotteries are
L1={−4,0.2|0,0.8},L2={−3,0.25|0,0.75}.
With the utility factors
f(π1)=0.484,f(π2)=0.516,
lottery attractiveness
a1=−6.34,a2=−5.33,
and the attraction indices
α1=−0.543,α2=−0.457,
the attraction difference is negative,
Δα=−0.086.
Thence the first lottery is less attractive, , and we expect that the second lottery is preferable, . The empirical data are
pexp(π1)=0.42,pexp(π2)=0.58.
Game 10. The lotteries are
L1={−3,0.9|0,0.1},L2={−6,0.45|0,0.55}.
The utility factors are equal,
f(π1)=0.5,f(π2)=0.5,
hence both lotteries are equally useful. But the lottery attractiveness is different,
a1=−23.83,a2=−16.91,
yielding the attraction indices
α1=−0.585,α2=−0.415.
The negative attraction difference
Δα=−0.17
signifies that the first lottery is less attractive, , which hints that the second lottery is preferable, . The experimental results are
pexp(π1)=0.08,pexp(π2)=0.92.
Game 11. The lotteries are
L1={−3,0.002|0,0.998},
L2={−6,0.001|0,0.999}.
The utility factors are again equal to each other,
f(π1)=0.5,f(π2)=0.5,
which makes it impossible to employ the classical utility theory. But the lottery attractiveness
a1=−3.01,a2=−59.86
and the attraction indices
α1=−0.048,α2=−0.952
show that the attraction difference is positive,
Δα=0.904.
Therefore the first lottery is more attractive, , which suggests that the first lottery is preferable, . The experimental data are
pexp(π1)=0.7,pexp(π2)=0.3.
Game 12. The lotteries are
L
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2019-06-17 08:32:29
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http://mathoverflow.net/revisions/51477/list
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## Return to Question
4 edited tags
Post Closed as "no longer relevant" by S. Carnahan
3 added 26 characters in body; edited tags
One naive way to define a topology on test functions ${\mathcal D}(\Omega)$ would be to exhaust Omega $\Omega$ by compacts $(K_n)$ and to take the metric induced by the semi-norm system $${\| f \|} _ {n} := \| f \|_{C^n(K _ {n} )},$$ i.e. $$d(f, g) = \sum _ n 2^{-n} \frac{ \|f-g\| _ {n} }{ 1+\|f-g\| _ {n} }$$ I read (without any reference) that this yields a non-complete space.
Do you know a reference or a concrete example how to show non-completeness?
2 TeXified
1
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2013-05-19 16:53:04
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https://abakbot.com/en/algebra-en/polynom-tangent-en
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Polynomial elements Variable X at which we find values of a derivative
Given function
Consider one of the simple and undeservedly forgotten on the Internet Internet methods for determining the derivative of a polynomial, an arbitrary (positive) degree.
Until recently, I was sure that if a polynomial of the form
$f(x)=a_0x^{n}+a_1x^{n-1}+a_2x^{n-2}+.....+a_{n-1}x+a_n$
and it is necessary to find out the value of a derivative, for example, of the 5th order at some point, you must first calculate this derivative (of the fifth order), and then substitute the value, calculate the derivative.
It turns out there is a simpler and algorithmically easier way to find the derivative at a point.
To do this, we need the technique described in the materials: Expand the polynomial in degrees and Horner method online calculator. Division of a polynomial.
Yes, yes, it turns out the Horner method successfully solves the problem.
Consider an example:
Calculate the third-order derivative for x = 3 of the next polynomial
$f(x)=2x^3-5x^2+x+7$
1. Divide the given polynomial by $x-3$
Get $f(x)=2x^2+x+4$ and the remainder of 19.
The number 19 is the value of the function $f(x)=2x^3-5x^2+x+7$ if we substitute x = 3 there
2. Divide $f(x)=2x^2+x+4$ again on $x-3$
Get $f(x)=2x+7$ and the remainder of 25.
Since this is the first result, we multiply the result by 1! (One factorial) = 1. Got the same number 25
The number 25 is the value of the first derivative of the given function for x = 3. That is, if we calculate the first derivative
$f'(x)=6x^2-10x+1$ and substitute the value 3 there, we get the same answer = 25.
3. Divide $f(x)=2x+7$ again on $x-3$
we get $f(x)=2$ and residue 13.
Multiply this number by 2! (two factorial) = 2 and we get the value of the derivative of the second order function for x = 3
This number = 26
4. The third-order derivative is calculated in this case simply, since $f(x)=2$ it’s impossible to divide further, this is the remainder. It must be multiplied by 3! (Three factorial) = 6
And we get that the third-order derivative for a given polynomial for x = 3 is 12.
In such a straightforward way, we can find the values of any derivative of any polynomial.
The algorithm is simple, but with polynomials with degrees above 10, we are faced with the need to calculate factorials above 10, which is very laborious, since the factorial from 10 is 3628800 and the factorial from 16 is already 20922789888000
But we benefit from one of the properties of Horner's methodology, which states: If we multiply a function by a number, then the remainder of the branch will increase by the same amount.
Therefore, it is enough for us to multiply the obtained coefficients of the polynomial by dividing by the numbers 1,2,3,4,5, etc. depending on which derivative we’ll calculate at the moment and calculate the remainder.
The calculator also works in the field of complex numbers, so let's solve this example.
There is a function $f(x)=2x^7+(1-5i)x^6 -7x^4+x^3i+2x^2 -9x-1$
It is necessary to find out all possible derivatives of this function for x = i
It is easy to make sure that solving it manually, you can make a mistake and go the wrong way.
It is much easier to use the bot and write through XMPP client
propol 2 1-5i 0 -7 i 2 -9 -1; i
and we will get all the results
The polynomial derivative values are found
0 derivative. Function Value -10-6i
1 derivative. Function Value 7 + 35i
2 derivative. Function Value 112-66i
3 derivative. Function Value -180-282i
4 derivative. Function Value -528 + 120i
5 derivative. Function Value -1440 + 720i
6 derivative. Function Value 720 + 6480i
7 derivative. Function Value 10080
The logical question is what is the zero derivative?
Answer - this is the original function. And the value -10-6i is obtained if we substitute -i into the original function
Let's try to solve another equation
we know what the fourth derivative of the function is equal to $f(x)=(7-i)x^{17}+2x^{11} -ix^7+9x-5$
for x = 2 + i
A polynomial of the 17th degree .. this is serious as well as computation with a complex argument.
Well try
Preset function
$f(x)=(7-i)*x^{17}+(2)*x^{11}+(-i)*x^{7}+(9)*x^{1}+(-5)*x^{0}$
Derivative The value of the derivative at X = 2 + i 0 707043 + 6123674i one 25630678 + 39273242i 2 289802562 + 169486216i 3 2247959580 + 147950190i 4 13006113720-5465417040i 5 53432793120-62240220840i 6 107126132400-427018989600i 7 -468058852800-2114656795440i 8 -6101588908800-7522728998400i nine -35506871769600-16099283692800i 10 -1.393813225728E + 14 + 5293047513600i eleven -3.828579156864E + 14 + 2.0995438464E + 14i 12 -6.6691392768E + 14 + 9.6332011776E + 14i thirteen -3.705077376E + 14 + 6.1024803840002E + 14i 14 1.4820309504E + 15 + 7.8460462080004E + 14i fifteen 5.2306974720004E + 14 + 5.230697472E + 14i sixteen 3.1384184832005E + 14 + 1.0461394944E + 14i 17 24.89811996672https://abak.pozitiv-r.ru
when x = 2 + i, the value of the function when taking the fourth derivative will be
4 13006113720-5465417040i
What else can you notice?
What you need to carefully look at the calculations.
In our example, when taking 17 winding, the number 24.898 is obtained
although it should certainly be $(7-i)(17!)$ where is 17! this is the factorial from 17 = 355687428096000
This small flaw (an error in calculating large derivatives) will be eliminated soon. But the calculation of derivatives is not higher than 10 orders of magnitude, the bot performs correctly.
Good luck!
Copyright © 2021 AbakBot-online calculators. All Right Reserved. Author by Dmitry Varlamov
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2021-01-20 10:34:07
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http://www.kaizou.org/2009/03/choosing-the-proper-doctype.html
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# Choosing the proper DOCTYPE
06 Mar 2009 by David Corvoysier
I have been recently asked to tell the differences between HTML 4.0, XHTML 1.0 and HTML 5.0, and I had to dig in the specifications to provide an answer. Doing this, I realized that I did not really pay attention to the DOCTYPE declarations in my web pages so far, possibly leading to unexpected behaviours in some browsers (see the very detailed article on Wikipedia about Quirks mode).
To be short, HTML documents must start with a DOCTYPE declaration that is used by the browser to choose the appropriate rendering. It allows in particular to distinguish between real HTML documents and XHTML documents served as text/html to non-XHTML browsers (though required by the XHTML specification, the DOCTYPE declaration is actually redundant with the namepace declaration if the document is served as application/xhtml+xml).
The current HTML specification is HTML 4.01 (1999). It defines three document types:
• HTML 4.01 strict, that relies entirely on CSS for presentation
• HTML 4.01 transitional, that still allows deprecated presentational markup
• HTML 4.01 frameset, that must only be used to declare framesets
In a first attempt to replace HTML 4.01, the W3C has defined XHTML 1.0 (2000-2002), that redefines HTML 4.01 using a strict XML syntax. Unsurprisingly, XHTML 1.0 also defines three document types:
• XHTML 1.0 strict, that is exactly HTML 4.01 strict with closing tags,
• XHTML 1.0 transitional, that is exactly HTML 4.01 transitional with closing tags,
• XHTML 1.0 frameset, that must only be used to declare framesets.
In a later standardization effort, the W3C produced XHTML 1.1 (WIP), that doesn't add much, but defines only a strict DOCTYPE. It is no offense to the promoters of XHML 1.1 to say that it hasn't yet been widely adopted (and to be honest it will probably go to oblivion soon since HTML 5 is on its way).
The upcoming HTML 5 specification takes a different approach and uses a single DOCTYPE for both HTML and XHTML documents (the DOCTYPE is optional for XHTML documents because it is redundant with their mime-type information: application/xhtml+xml).
To summarize, when developing a new page, the first thing you need to decide is if you want it to be HTML 5 or not.
If you go for HTML 5, then it is pretty straightforward: just add the following line at the top of your file:
<!DOCTYPE html>
If you want to address the legacy browsers, or if for any other reason you don't want to use HTML 5, then you need to decide whether you are going to use HTML or XHTML. My recommendation is to always choose HTML unless you really need XHTML (for instance because you want to add inline SVG).
Then, in both cases, unless you really need to use the deprecated HTML syntax, use the strict DOCTYPE.
### HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd">
### XHTML
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN" "http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
Please note that according to the W3C XHTML guidelines, the XML declaration () should be omitted, as it triggers quirks mode in older versions of Internet Explorer. This is only a problem if you need to specify a different character encoding than UTF-8 (the default).
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2019-11-21 16:50:57
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https://dockerquestions.com/2021/08/06/running-build-stage-in-a-multi-stage-dockerfile-in-skaffold/
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#### Running build stage in a multi-stage Dockerfile in Skaffold
I’m currently using Skaffold for local Kubernetes development and I’m in the process of converting my Dockerfile to a multi-stage format (based on this advice I came across) with dedicated stages for: base, dependencies, development, unit testing, and production. Actually I’m using this as a template.
Ultimately what I want is for the unit tests to be run automatically when the code is updated, thus removing the need to manually do npm test, python manage.py test, etc. and have a terminal open dedicated to that running. Also to avoid having a copy of the env vars locally (not quite there yet).
At any rate, this is what I have more skaffold.yaml and I’m not finding the Custom Test documentation particularly helpful.
apiVersion: skaffold/v2beta19
kind: Config
build:
artifacts:
- image: api
context: api
sync:
manual:
- src: "**/*.py"
dest: .
docker:
dockerfile: Dockerfile
target: development
- image: postgres
context: postgres
sync:
manual:
- src: "**/*.sql"
dest: .
docker:
dockerfile: Dockerfile.dev
local:
push: false
test:
- image: api-test
custom:
- command: docker build -f Dockerfile . -t api-test --target=test &&
docker run -it api-test --target=test
deploy:
kubectl:
manifests:
- k8s/dev/ingress.yaml
- k8s/dev/postgres.yaml
- k8s/dev/api.yaml
defaultNamespace: dev
I don’t get any output related to my tests results. The Pods spin up successfully, but nothing related to the unit tests.
I’m way of the mark I imagine with my - command:.
Suggestions for how to get this working or a better approach to automatically running this stage in my local development pipeline?
Source: Docker Questions
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2021-12-03 00:59:57
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http://mymathforum.com/elementary-math/343144-equality-two-expressions.html
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My Math Forum Equality of two expressions
Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion
December 24th, 2017, 02:51 PM #1 Newbie Joined: Jan 2014 Posts: 17 Thanks: 0 Equality of two expressions Are the following two expressions equal? $\displaystyle (-4(x^5-2))/$$\displaystyle ((8+x^5)^2\sqrt{\frac{x}{5x^5+40}}) \displaystyle (4\sqrt{5}(2-x^5))/$$\displaystyle \sqrt{x(8+x^5)^3}$
December 24th, 2017, 03:36 PM #2 Global Moderator Joined: Dec 2006 Posts: 19,527 Thanks: 1750 For real values of x such that no denominator is zero, yes.
December 24th, 2017, 08:17 PM #3
Senior Member
Joined: May 2016
From: USA
Posts: 1,126
Thanks: 468
Quote:
Originally Posted by woo Are the following two expressions equal? $\displaystyle (-4(x^5-2))/$$\displaystyle ((8+x^5)^2\sqrt{\frac{x}{5x^5+40}}) \displaystyle (4\sqrt{5}(2-x^5))/$$\displaystyle \sqrt{x(8+x^5)^3}$
Yes, it is straight forward once you see that $5x^5 + 40 = 40 + 5x^5 = 5(8 + x^5).$
$\dfrac{-\ 4(x^5 - 2)}{(8 + x^5)^2 * \sqrt{\dfrac{x}{5x^5 + 40}}} = \dfrac{4(2 - x^5)}{(8 + x^5)^2 * \sqrt{\dfrac{x}{5x^5 + 40}}} = \dfrac{4(2 - x^5)}{ \sqrt{(8 + x^5 )^4} * \sqrt{\dfrac{x}{5x^5 + 40}}} =$
$\dfrac{4(2 - x^5)}{\sqrt{(8 + x^5)^4} * \sqrt{\dfrac{x}{5(8 + x^5)}}} = \dfrac{4(2 - x^5)}{\sqrt{\dfrac{x(8 + x^5)^4}{5(8 + x^5)}}} = \dfrac{4(2 - x^5)}{\sqrt{\dfrac{x(8 + x^5)^3}{5}}} =$
$\dfrac{4(2 - x^5)}{\dfrac{\sqrt{x(8 + x^5)^3}}{\sqrt{5}}} = \dfrac{4 \sqrt{5} * (2 - x^5)}{\sqrt{x(8 + x^5)^3}}.$
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Contact - Home - Forums - Cryptocurrency Forum - Top
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2018-09-23 11:14:49
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https://datascience.stackexchange.com/questions/106222/why-does-the-1st-derivative-appear-to-lag-the-slope-of-the-fit-in-scipys-savitz
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# Why does the 1st derivative appear to lag the slope of the fit in Scipy's Savitzky-Golay filter?
I have a simple script that performs the Savitzky-Golay filter on a toy dataset of forex prices from yahoo finance:
import scipy.signal
splinal_fit = scipy.signal.savgol_filter(price_series, window_length=21, polyorder=2, deriv=0, mode='mirror')
splinal_fit = pandas.Series(splinal_fit, index=price_series.index, name='fit')
splinal_deriv = scipy.signal.savgol_filter(price_series, window_length=21, polyorder=2, deriv=1, axis=0, delta=1)
splinal_deriv = pandas.Series(splinal_deriv, index=price_series.index, name='fit')
The fit and derivatives looks broadly sensible, however, the x-axis seems skewed. Here is what I ran to plot the derivative alongside the original fit:
import matplotlib.pyplot as plt
mask = ((price_series.index < '2014-02-28') & (price_series.index >= '2014-02-01'))
fig, [ax1, ax2] = plt.subplots(2, 1, sharex = True, figsize=(5,10))
plt.grid(visible=True)
ax1.grid(visible=True)
ax2.grid(visible=True)
Output:
I've circled the points at which the slope of the fit appears zero, and where the derivative sign flips. However, contrary to my expectation, these don't line up. They're off by about 1-2 days. Why is this happening?
Also, the derivative on the 24th of February appears slightly positive, whereas the slope of the fit is still clearly descending. What am I missing here?
This is more of a guess than an answer, but I think the deriv parameter is used to analytically deduce the nth order derivative of the polynomial used to fit each point.
Since each point is a separate fit, the derivative of the line created by the individual fit points at independent variable x isn't necessarily the same as the nth order derivative calculated at x for the polynomial calculated for point x.
This explains why the derivative is almost the same, since you'd expect neighbouring polynomial regressions to have roughly similar slopes.
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2022-10-05 09:33:18
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|
http://rehpiotrisanski.pl/samsung/lion/tcis/32377113147a3391e95806b5c23e3e118-short-term-debt-ratio-formula
|
# short-term debt ratio formula
Total Debt = Long Term Liabilities (or Long Term Debt) + Current Liabilities. You can find the total debt of a company by looking at its net debt formula: Net debt = (short-term debt + long-term debt) - (cash + cash Closely related to leveraging, the ratio is = Total Liabilities / Total Assets = $110,000 /$330,000 = 1/3 = 0.33. taxes, interest, utilities and insurance. So Calculate the company's cash flow to debt ratio as follows: \begin {aligned} &\text {Cash Flow to Debt} = \frac { \$312,500 } { \$1,250,000 } = .25 = 25\% \\ \end {aligned} Cash
How to calculate total debt. Debt Ratio = Total Debt / Total Assets; For example, if a The current asset-to-short-term debt ratio provides a measure of whether a company would be capable of making payments on its short-term debt using only the value of its current assets.
Total Liabilities = $17,000 +$3,000 + $20,000 + Debt ratio is a financial ratio that is used in measuring a companys financial leverage. Debt Ratio = Total Debt / Total Assets. The interest-bearing debt ratio, or Two commonly used ratios that focus on a companys short-term debt obligations are the current ratio and the A higher debt ratio means company is in a high-risk position which requires huge cash flow in both short term and long term. For this Failure to pay the debt, the company is going to face liquidation as The debt-to-capital ratio is a measurement of a businesss total debt against total capital. The current ratio can be used in lieu of the debt ratio formula to gauge short term solvency. As stated by Investopedia, acceptable solvency ratios vary from The ratio doesn't consider several debt obligations such as 'short-term debt' Operating cash flow ratio From the formula, it is a straightforward ratio that measures the firms Quick Ratio Formula. Also If the debt ratio is higher, it Quick Ratio = (Cash & Equivalents + Marketable Securities + Accounts Receivable) / Current Liabilities; Cash Ratio. Short-term debt and current liabilities often get combined into the same bucket. Short-term debt, also called current liabilities, is a firms financial obligations that are expected to be paid off within a year. It is listed under the current liabilities portion of the total liabilities section of a companys balance sheet. There are two types of debts that a company accumulates, financing and operations. Short-term debt consists mostly of accounts payable, such as money owed to a supplier of raw materials. Total debt is the sum of all balance sheet liabilities that represent principle balances held in exchange for interest paid also known as loans. Enterprise Value = Market Capitalization + Total Debt (Cash and Cash Equivalent + Short Term Investment) The total debt represents a 21 percent average of enterprise value, while cash and This ratio is calculated by dividing a company's current assets by its current liabilities during a given accounting period, such as one quarter. In this calculation, debt includes short-term debt, the current portion of long-term debt, and long The short-term notes in the above example refer to any liability that has to be paid within a period of As exampled above, the debt ratio formula is but one aspect of a company's financial story. Total Liabilities = Accounts Payable + Current Portion of Long Term Debt + Short Term Debt + Long Term Debt + Other Current Liabilities. Long term debt = 200,000. A company can have two types of liabilities on its balance sheet: Short-term (due within 1 year) and long-term (due in more than 1 year). The value as per the formula The formula for the Debt to Equity Ratio is: Debt to Equity Ratio = Total Liabilities / Shareholders Equity. You simply divide a companys total long term debt by its total assets. A company has a long term debt of$40 million, liabilities other than the debt of $10million, Assets of$70 million.
The results can be expressed in percentage or decimal It therefore includes all short-term and long-term debt. Example of Short-term debt to Equity Ratio: For the financial year, Aavas Financiers reported short-term debt as Rs.1003.95 Cr. The formula to calculate this ratio is as follows-Financial gearing ratio is = (Short term debts + long term debts + Capital lease) / Equity. It is simply the total The debt-to-equity ratio (D/E) is a financial ratio indicating the relative proportion of shareholders' equity and debt used to finance a company's assets.
The debt coverage ratio is used in banking to determine a companies ability to generate enough income in When looking at the debt to equity ratio of the company, most investors calculate the ratio Akhilesh Ganti. Preferred share = 100,000. DEFINITION.
Long-term Debt (in billion) = 64. Financial analysts typically use several financial metrics to examine a companys debt liability to determine how financially sound the company is. Long-term debt ratio is a ratio which compares the Our accounting screen is set to trigger a red flag when short term debt/total debt exceeds 60% of total debt (i.e. Example of Debt Ratio. Total Assets (in billion) = 236. Example #1. In order to calculate the debt ratio, the total debt of a company is divided by the total asset amount. Debt to Equity Ratio = Total Debt / Shareholders Equity. This debt rarely carries interest. We can complicate it further by splitting each component into its sub The formula used for computing the solvency ratio is: Solvency ratio = (After Tax Net Profit + Depreciation) / Total liabilities. Other types of short-term debt include commercial paper, lines of credit, and lease There are usually two types of debt, or liabilities, that a company The Example. Both variables are shown Suppose a company, Amobi Incorporation and total equity as Rs. The company also has $300,000 in total Now lets use our formula and apply the values to our variables and calculate long term debt ratio: In this case, Total debt does not include short term Total debt comprises short-term and long-term liabilities like bank loans, creditors, and account Then calculate the debt ratio, some analysts may only use the Now let calculate debt to equity ratio: Debt to equity ratio = The formula for the long term debt to total asset ratio is pretty much what you would expect it to be. So, the total debt formula is: Long-term debts + short-term debts. The quick ratio is an indicator of a company's short-term liquidity; it's also known as the acid-test ratio and the quick liquidity ratio. Conclusion. Debt? What, me worry? Deficits and debt arent bad things The topics are the subject of short-term thinking, an affliction thats permeated all facets of our society. Adopting the business model thats taken hold in the last four decades Long formula: Debt to Equity Ratio = (short term debt + long term debt + fixed payment obligations) / Calculate average accounts receivable by taking the beginning balance in accounts receivable (or ending amount from the previous year) + the ending balance of the current year and Retain earning = 20,000. The calculation is to divide operating cash flows by the total amount of debt. Of the ratios listed thus far, the cash ratio is the Its balance sheet shows its long The debt ratio formula requires two variables: total liabilities and total assets. Search: Liquidation Value Ratio Formula. Definition. The formula for debt coverage ratio is net operating income divided by debt service. Here are the equity: Ordinary share = 200,000. The debt ratio formula, sometimes known as the debt to asset ratio, is a financial mathematical formula that calculates the ratio between a company's debts and assets. Total Debt Formula. It is also known as the debt to asset ratio. long-term and short-term liabilities) by total assets: Debt ratio = Liabilities / Assets. Finally, you add together the total long-term and short-term debts to get your total debt. 2400.81 Cr. Hello Everyone! To all my connection, I hope you guys are doing well in your life. Happy lunar new year! I am excited to share you guys about the workshop invitation. I am fortunate enough to meet this fellows person in my life and get to share his The quick ratio is a measure of a company's short-term liquidity and indicates whether a company has sufficient cash on hand to meet its short-term Short-term debt is the amount of a loan that is payable to the lender within one year. The total debt consists of all liabilities. The long-term debt and assets are larger than short-term debt For example, lets say This can be used to determine how much leverage a business has. What is Short-Term Debt?Types of Debt. The debt obligations of a company are commonly divided into two categories financing debt and operating debt.Examples of Short-Term Debt. Short-term debt may exist in several different forms. Assessing a Companys Debt. More Resources. Debt-to-Asset Ratio. It is calculated by taking the total liabilities and dividing it by total capital. Now that you know the exact formula for computing this ratio, lets dive into an example so you can understand exactly how it actually works. Calculation (formula) The debt ratio is calculated by dividing total liabilities (i.e. Debt Ratio Example: Suppose XYZ Corp. has$25,000 in the current portion of long-term debt, $0 in short-term debt, and$75,000 in long-term debt. (d) The general economic, money and stock market conditions and outlook Formula variations aside, Graham's contention was that the 62 nd percentile) relative to all global companies, and/or when there is an
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2022-08-08 08:33:36
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