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When was $\sum$ introduced as the notation for a sum and who was the first person to solve a infinite sum other than 0+0+0+...?
According to Florian Cajori's
History of Mathematical Notations Vol II paragraph 438, the symbol $\displaystyle \sum$ for summation is attributed to Leonard Euler. In Euler's own words, "summam indicabimus signo $\sum$". Cajori cites Euler's Institutiones calculi differentialis (1755) in his sourcing.
As for the first known infinite sum, I do see some online sources attributing the first known infinite sum to Archimedes. In particular, on the site A history of the calculus, the authors John J. O'Connor and Edmund Robertson cite Archimedes' showing around 225 BC that "the area of a segment of a parabola is $\frac{4}{3}$ the area of a triangle with the same base and vertex and $\frac{2}{3}$ of the area of the circumscribed parallelogram," as the first known example of an infinite sum. Archimedes lived from 287 to 212 BC.
Personal opinion has it that while this may be the first known infinite summation, there may well have been one before that not known to us today.
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The condition is a bit misleading and incomplete, the gradient of the function will indeed be a linear combination of the gradients of all constraints (equalities as well as inequalities), however there are some additional conditions on this linear combination which might make make things more clear. Assume that $f$ is the function we minimise and $c_i(x)\geq 0$ are the constraints. Then a necessary condition for the minimiser $x_0$ is that $\nabla f(x_0) = \sum_i \lambda_i \nabla c_i(x_0)$, for some $\lambda_i$ with $\lambda_i \geq 0$ and $\lambda_i = 0$ if $c_i(x_0)\neq 0$.
The last condition perhaps already solves your confusion, in a way the linear combination only consists of those $\nabla c_i$ for which the constraints are actually equalities.
This condition maybe gets clearer when looking at the idea of the proof. You will probably remember from your calculus classes, that a (sufficiently smooth) function $f$ can only have a (local) minimiser at $x_0$ if $\nabla f(x_0) = 0$. This is usually shown using contradiction: Assume that $\nabla f(x_0) \neq 0$. Then we can move a tiny bit in direction $a := -\nabla f(x_0)$ and since $f$ locally looks like$$f(x) = f(x_0) + \nabla f(x_0) \cdot (x-x_0) + \text{higher order terms} ,$$you will find that $f(x_0+h a) < f(x_0)$ for small enough $h >0$, which of course is a contradiction.
Now we add constraints. Let us look at a single constraint $c(x) \geq 0$, the extension to multiple constraints is not hard from there. Here also$$c(x) = c(x_0) + \nabla c(x_0) \cdot (x-x_0) + \text{higher order terms}$$assuming that $c$ is sufficiently smooth. Since you seem to be interested in linear constraints, let us ignore the higher order terms, since they do not occur and are a bit technical to deal with. So we assume$$c(x) = c(x_0) + \nabla c(x_0) \cdot (x-x_0).$$
Now what happens, if we try the trick from before? If we move a tiny bit in a direction $a$ again, $c$ changes as well and we have $$c(x_0 + ha) = c(x_0) + h \nabla c(x_0)\cdot a.$$If $c(x_0) > 0$, this is no problem. We just pick $h$ small enough and then we will still have $c(x) \geq 0$. So we can choose $a = -\nabla f(x_0)$ as before and conclude that $\nabla f(x_0) = 0$.
Now the other case is $c(x_0) = 0$. This is essentially the classical case of a Lagrange multiplier, with a small difference. We can only pick a direction $a$ for which $c(x_0+ha) \geq 0$. In other words$$0 \leq c(x_0+ha) = c(x_0) + h \nabla c(x_0) \cdot a.$$If we note that $c(x_0) = 0$ and divide by $h$, we then get the restriction$$0 \leq \nabla c(x_0) \cdot a.$$Thinking geometrically, we are only allowed to move in the half-space around $x_0$ for which the constraint is not violated.
So the question is, when can we still find a direction satisfying $a$ such that $f$ decreases. By the same reasoning as before we want$$0< f(x_0)-f(x_0 + ha) = -h \nabla f(x_0) \cdot a + \text{higher order},$$so $0 > \nabla f(x_0) \cdot a$. Now the rest of the proof is simple linear algebra. If we can pick a vector $a$ perpendicular to $\nabla c(x_0)$ with $\nabla f(x_0) \cdot a < 0$, we get a contradiction. So $\nabla c(x_0) \cdot a = 0$ implies $\nabla f(x_0) \cdot a = 0$. As a result $\nabla f(x_0) = \lambda \nabla c(x_0)$ for some $\lambda$. Now assume that this is the case, but for $\lambda < 0$. Then we could try to pick $a = -\nabla f(x_0)$ again, which will decrease $f$, but for which$$\nabla c(x_0) \cdot a = -\lambda \nabla c(x_0) \cdot \nabla c(x_0) > 0$$ which does not violate our constraint. So $\lambda \geq 0$ is the only possibility which does not lead to a contradiction.
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Chapter 03: General Theorem, Intermediate Forms
Rolle's theorem Geometrical interpretation of Rolle's theorem The mean value theorems Another form of mean value theorem Increasing and decreasing functions Cauchy's mean value theorem Extended mean value theorem Indeterminate form The form $\frac{0}{0}$ The form $\frac{\infty}{\infty}$ The form $0\times \infty$ (or $\infty \times 0$) The form $\infty-\infty$ The forms $0^\infty, 1^\infty, \infty^0$ Use of expansions Solutions
We are very thankful to Muhammad Idrees for providing notes of Exercise 3.3
bsc/notes_of_calculus_with_analytic_geometry/ch03_general_theorem_intermediate_forms Last modified: 3 weeks ago by Dr. Atiq ur Rehman
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The internal definition of the
sagesilent environment involves the use of verbatim text, which never can be written in a macro argument. The same kind of problem arises if you try something like
\AtBeginDocument{\begin{verbatim} Some verbatim text\end{verbatim}}
As a workaround, you can move to an auxiliary
tex file all the code needed to config SageTeX. Then you can input that file in the argument of
\AtBeginDocument. For example, let the contents of
examplepackage.sty be
\NeedsTeXFormat{LaTeX2e}[1994/06/01]\ProvidesPackage{examplepackage}[2018/12/24 examplepackage]\RequirePackage{sagetex}\AtBeginDocument{\input{exampleconfig}}\endinput
The file
exampleconfig.tex contains the code to set up SageTeX:
\begin{sagesilent} # Set up some variables a = 3 b = 5 # Define a function var("x") g(x) = a*sin(x)+b*cos(x)\end{sagesilent}\endinput
The main
tex file could be
\documentclass[12pt]{article}\usepackage{examplepackage}\begin{document}% Do something with a and bThe value of $a$ is $\sage{a}$, while that of $b$is $\sage{b}$. Their sum is $\sage{a+b}$.% Do something with the function\begin{sagesilent} gp(x) = diff(g(x),x) g_int(x) = integrate(g(x), x)\end{sagesilent}Let $g(x)=\sage{g(x)}$. It is easily seen that\[ g'(x)=\sage{gp(x)},\]whereas\[ \int g(x)\,dx=\sage{g_int(x)}.\]Observe also that $g'(2)\approx\sage{gp(2).n()}$.\end{document}
Hope that helps.
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Graphing Quadratic Functions:
Quadratic equations in standard form are represented as \(ax^2~+~bx~+~c~=~0\)
To plot any function \(y~+~f(x)\)
The graph of a quadratic function is a parabola which can be represented in two forms:
Standard Form: \(ax^2~+~bx~+~c\) Vertex Form or \((a~-~h~-~k)\) form: \(f(x)~=~a(x~-~h)^2~+~k\) , where \((h,k)\) is vertex of parabola.
The graph opens upwards if \(a > 0\)
Vertex of a quadratic equation is its minimum or lowest point if the parabola is opening upwards or its highest or maximum point if it opens downwards.
On rearranging the standard form, vertex form is obtained by completing the square method.
On comparing both forms,
\(h\)
\(k\)
When a quadratic function is represented in vertex form, following points are to be noted:
If h > 0, graph shifts right by h units. If h < 0, graph shifts left by h units. If k > 0, graph shifts upwards by k units. If k < 0, graph shifts downwards by k units. (h, k) denotes the vertex of function.
For graphing a quadratic function, above steps are followed and further transformations are used.
Using Transformations to Graph Quadratic Functions: i) Horizontal shifting by m units:
Consider the standard form of quadratic equation \(ax^2~+~bx~+~c~=~0\)
Suppose instead of \(\alpha\)
Similarly, if roots are \((\alpha~-~m)\)
ii) Vertical shifting by k units:
Considering the vertex form i.e.,\(y\)
If k > 0 , graph shifts upwards by k units. If k < 0 , graph shifts downwards by k units.
Let us first consider the graph of \(y\)
Let us go through an example for a better understanding.
Example: Find range of \(x\)
Solution: Let us try to graph the given equations to find out range of \(x\)
Let \(x^2~-~6x~+~5~\leq~0\)
From equation (1),\((x~-~1)(x~-~5)~\leq~0\)
\(\Rightarrow~1~\leq~x~\leq~5\)
From equation (2), \((x~-~0)(x~-~2)~\gt~0\)
\(\Rightarrow~x~\lt~0\)
Thus, range of \(x\)
Thus graphing quadratic functions becomes easy using transformations.’
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Let V be the universe (the class of all sets), let W(0)=V, W(1) be the class of all singletons whose unique member element is a member set of W(0), and for n>0 let W(n+1) be the class of all singletons whose unique member element is a member set of W(n). Let, for every n>=0 S(n)=W(n+1)/W(n) be the class that is the difference of the class W(n+1) and of the class W(n). We are interested with the proposition (S): "For every set x, there exists an unique natural number n such that x is a member set of S(n)" . Question 1: Let ZFC be our set theory; does ZFC prove (S) ? Question 2: Suppose the answer to question 1 is YES, and let now our set theory be ZF-(I mean ZF with the omission of the axiom of Regularity (or Foundation)); does ZF- prove (S) ? Question 3: Suppose the answer to question 2 is NO; does ZF- prove the equivalence of (S) with the axiom of regularity ? Gérard Lang
I assume that you meant to write $S(n)=W(n)-W(n+1)$, rather than what you have written, since inductively one can show $W(n+1)\subseteq W(n)$. With this understanding, all the $S(n)$ are disjoint, and the question is whether every set eventually falls out, or whether there can be a set in every $W(n)$.
In ZFC, there can be no set $x$ in every $W(n)$, since the transitive closure of such an $x$ would consist of the set containing as members the unique element of $x$, the unique element of that set, and so on, and thus have no $\in$-minimal element, contrary to the foundation axiom. So the answer to question 1 is Yes.
Meanwhile, it is relatively consistent with ZF- that there is a set $x$ which has $x=\{x\}$; for example, such sets exist under the Anti-Foundation axiom. Such a set is in every $W(n)$, and so is not in any $S(n)$, and so the answer to question 2 is No.
Question 3 is quite interesting, but I don't know the answer. You may want to add the axiom of Dependent Choices DC, a mild version of AC, for with this axiom a violation of the foundation axiom will give rise to an $\in$-descending $\omega$-sequence $x_0,x_1,\ldots$ with $x_{n+1}\in x_n$, and from such a sequence one might hope to build a set that is in every $W(n)$. But I don't see this how to complete this idea just yet...
Since Joel has already answered Questions (1), and (2), I will only offer an answer for Question 3.
This is a revised version of my answer; thanks to Joel Hamkins for pointing out that my previous construction was not quite right.
Start with a simple graph with 3 elements {$a,b,c$}, where each of the three nodes has an edge to the other two. So in this 3-element model of "set theory", $a$ = {$b$, $c$}, $b$ = {$c$, $a$}, and $c$ = {$a$, $b$}.
Given an extensional digraph $G=(X,E)$, with $X$ as the vertex set and $E$ as the edge set, define the
deficiency set $D(G)$ of $G$ to be the collection of subsets $S$ of $X$ that are not "coded" in $G$, i.e., there is no element $a$ in $X$ such that $S$ = {$x \in X : xEa$}.
We now can define by recursion a digraph $G_\alpha = (X_\alpha, E_\alpha)$ for each ordinal $\alpha$ as follows:
$G_0 = G$;
$G_{\alpha+1} = (X_{\alpha+1}, E_{\alpha+1})$, where $X_{\alpha+1} = X_{\alpha} \cup D(G_{\alpha})$, and $E_{\alpha+1} = E_{\alpha}$ together with edges of the form $(x,X)$, where $x\in X_{\alpha}$, $X \in D(G_{\alpha})$, and $x\in X$.
For limit $\alpha$, $G_\alpha$ is the union of $G_\beta$ for $\beta<\alpha$.
The model/digraph we are interested in is the union of all the $G_\alpha$, as $\alpha$ ranges over the ordinals, and $G$ is the 3-element digraph on {$a,b,c$} mentioned earlier. Let's call this model $V(a,b,c)$. It satisfies all the axioms of $ZF$ with the exception of Foundation.
$V(a,b,c)$ also satisfies $S$ since any infinite descending epsilon chain must eventually hit $a$, $b$, or $c$. So this shows that Question 3 has a negative answer.
Since {$a,b,c$} are indiscernibles in $V(a,b,c)$, I suspect that $DC$ fails in $V(a,b,c)$, but a variation on this theme might produce a model with enough asymmetry for $DC$ to hold as well.
PS. Models of $ZF$ in which the Foundation fails are often constructed using the so-called Bernays-Rieger permutation method (not to be confused with the Fraenkel-Mostowski permutation method of constructing models of $ZF$ in which the axiom of choice fails). The model constructed above is based on a different idea, explored in detail for models of finite set theory in the following paper:
A. Enayat, J. Schmerl, and A. Visser,
Omega Models of Finite Set Theory , to appear in Set theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies (edited by J. Kennedy and R. Kossak), Cambridge University Press, to appear October 2011.
A preprint can be found here.
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I think the missing link for you is not realizing that $Q$ is actually equal to $q_1 + q_2$: in a duopoly, quantity demanded can only be derived for the two firms in question. With that in mind, we can rewrite the profit functions for both firms in terms of $q_1$ and $q_2$ and then optimize with respect to the $q$'s accordingly (since Cournot competition is competition over quantities). We are then going to be left with two
best response functions, one that tells you the optimal level of $q_1$ as a function of $q_2$ and another that tells you the optimal level of $q_2$ as a function of $q_1$. Then it just becomes an algebra problem to find the Nash equilibrium and the relationship of quantities and profits between the firms will become clear.
Firm 1 maximizes profits\begin{align*}\Pi_1(q_1) = (\alpha - (q_1 + q_2))q_1 - c_1 q_1.\end{align*}
The first-order condition is\begin{gather*}\frac{d \Pi_1(q_1)}{dq_1} = 0 \\\implies \alpha - 2q_1 + q_2 = c_1 \\\implies q_1(q_2) = \frac{\alpha - c_1}{2} + \frac{q_2}{2}.\end{gather*}
Similarly, Firm 2 maximizes profits\begin{align*}\Pi_2(q_2) = (\alpha - (q_1 + q_2))q_2 - c_2 q_2.\end{align*}
The first-order condition is\begin{gather*}\frac{d \Pi_2(q_2)}{dq_2} = 0 \\\implies \alpha - 2q_2 + q_1 = c_2 \\\implies q_2(q_1) = \frac{\alpha - c_2}{2} + \frac{q_1}{2}.\end{gather*}
Solving the system of equations for $q_1$ and $q_2$ yields\begin{align*}q_1^* &= \frac{\alpha - 2c_1 + c_2}{3} \\q_2^* &= \frac{\alpha - 2c_2 + c_1}{3}, \end{align*}
which implies that $\mathbf{q_1^* > q_2^*}$ since $c_1 < c_2$.
Note that both firms face the same price $P^* = \alpha - (q^*_1 + q^*_2)$. Therefore, we have that\begin{align*}(P^* - c_1)q_1^* &> (P^* - c_2)q_2^* \\\implies \mathbf{\Pi^*_1} &> \mathbf{\Pi^*_2}.\end{align*}
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Description
Recently, I have been learning a couse called "
Numerical Analysis". The
fixed point iteration theory was introducted to solve the
the approximate root of the equations in this course.
Fixed point iteration theory: $x=Ψ(x) \Rightarrow x_{k+1}=Ψ(x_k)$ Example
Solving the approximate root of $f(x)=x^3+4x^2-10 = 0$
So I give three styles of $x=Ψ(x)$, shown as below:
$Ψ_1(x)=\frac{1}{2}\sqrt{10-x^3}$
$Ψ_2(x)=\sqrt{\frac{10}{x+4}}$
$Ψ_3(x)=x-\frac{f(x)}{f'(x)}=x-\frac{x^3+4x^2-10}{3x^2+8x}$
Using Mathematica to implement it and verify the results of textbook My trial: I set the precision to 10
result1 = FixedPointList[N[1/2 Sqrt[10 - #^3] &, 10], 1.5`10]
{1.500000000, 1.286953768, 1.402540804, 1.345458374, 1.375170253, 1.360094193, 1.367846968, 1.363887004, 1.365916733, 1.364878217, 1.365410061, 1.365137821, 1.365277209, 1.365205850, 1.365242384, 1.365223680, 1.365233256, 1.365228353, 1.365230863, 1.365229578, 1.365230236, 1.365229899, 1.365230072, 1.365229984, 1.365230029, 1.365230006, 1.365230017, 1.365230011, 1.365230014, 1.365230013, 1.365230014}
result2 = FixedPointList[N[Sqrt[10/(# + 4)] &, 10], 1.5`10]
{1.500000000, 1.348399725, 1.367376372, 1.364957015, 1.365264748, 1.365225594, 1.365230576, 1.365229942, 1.365230023, 1.365230012, 1.365230014, 1.365230013}
result3 = FixedPointList[N[# - (#^3 + 4 #^2 - 10)/(3 #^2 + 8 #) &, 10], 1.5`10]
{1.500000000, 1.373333333, 1.365262015, 1.36523001, 1.36523001}
However, the last two results of
the result3 is
1.36523001, 1.36523001, whose precision is
9 rather than
10
TableForm[ Flatten[{result1, result2, result3}, {{2}, {1}}], TableHeadings -> {None, Style[#, 15, Red] & /@ {"\[Phi]1 steps", "\[Phi]2 steps", "\[Phi]3 steps"}}]
As the picture shown, the
result2 ia same as
the result of textbook, however, the
result1 and
result3 are a little different from
the result of textbook. In addition, the precison of
result3 is
9, not
10.
Question Can someone give me a explalation about this trial? For me, I cannot understand this result.
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There are two ways to answer your question. One is direct and has less depth to it, the other is more indirect and has a lot of depth to it. I will begin with the indirect one because it has wide ranging applications beyond finance or economics and because it should serve as a warning to journal editors and so forth. Also, it covers an area of statistics that everyone but statisticians have forgotten about.
Although the idea of a statistic is rather old, the field of statistics is rather new. It is probably the newest or nearly newest of all fields. Aeronautics is older. Genetics is older. It opened up a ton of practical questions that it took time to solve. It has to do with how the field defined as statistic. A statistic is any function of the data. This means that almost every statistic is useless as there are uncountably many functions.
This led to a process to decide which statistics to keep and which to discard. This created unexpected results. If you are finding poor estimators as a result of your theory, then there is a good chance, you are doing it wrong. In the defense of finance, it has been struggling with this since Mandelbrot published the first empirical refutation of mean-variance finance. It tried to solve it in the 1960's but a couple of things got in the way. The first was the use of punch card technology. Even if work by Eugene Fama or Mandelbrot were correct, it would have resulted in problems that would take decades to solve. The second was that there was no reason for them to be correct. There was no theory behind the observations.
The unexpected result, in searching for a statistic, was that all Bayesian statistics were admissible. This was surprising because it was proved with Frequentist axioms. It also found that all other statistics were valid to the extent they either mapped to a Bayesian measure in a particular case or at the limit. It provides a test, however. If you can stochastically dominate a measure, then you drop that measure. If you hunting for accurate measurements that work, then something deeper is going on and you are missing it.
The more direct answer is that the distribution of returns, for the Markowtiz model to be correct, have to have certain properties. The first is that there needs to be a mean in order to have an expectation in the first place. Most standard distributions have a mean, but not all do. The Cauchy distribution and, in general, the Paretian distributions of Mandelbrot's article do not. The Cauchy distribution is $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(x-\mu)^2}.$$
The second is that if a mean exists, a covariance matrix needs to exist. Not all distributions with a variance have a covariance in its multivariate form. The hyperbolic secant distribution is an example of that. It is $$sech\left(\frac{x-\mu}{\sigma}\right).$$
There have been attempts to use both in empirical finance. If either of those distributions are present in the likelihood function, then mean-variance finance is indefensible. The former is problematic because you cannot form an expectation on your returns in the first place. They are excluded by the laws of general summation. The second is a bit more subtle because if the second is present none of the assets can be independent, but none of them can covary either. They can comove, but not covary. It creates a very ugly issue.
There is a paper that derives the distribution of returns at https://ssrn.com/abstract=2828744. It shows that there are many distributions that can be present. The logic of the paper is that returns are not data, rather, prices are data. Returns are transformations of data. In particular, they are the ratio of jointly distributed variables, a present value and a future value. The distribution depends upon the rules in use to create the prices. As a result, stocks have different returns than antiques because the auction process is different.
As it happens, all distributions for equity securities include some mixture of a transformation of the Cauchy distribution. Because the distributions involved lack a sufficient statistic, any point estimator has to lose information, so no non-Bayesian solution exists for projective problems (such as choosing an allocation), and should be avoided for inferential questions if possible. You cannot avoid them in your true hypothesis is a sharp null hypothesis as there is no good Bayesian solution for sharp null hypotheses.
A population test of the paper can be found at https://ssrn.com/abstract=2653151
There are also papers to replace the method of pricing options and the rules of econometrics. Papers to create optimal portfolios and to extend stochastic calculus are in process. The distributions paper will be presented at the Southerwestern Finance Association Conference in March.
Some things will have to change. You cannot make an assumption of i.i.d. variables, for example. The entire discussion of the Solow convergence will have to change in economics and so the core of the whole discussion of capital, physical, financial and human.
A lot of focus will end up on the scale parameter. In the Cauchy distribution, there is no covariance matrix. If you had a one asset portfolio, denoted $a$, then it may have a scale parameter $\gamma_a$. If you switch to a two asset portfolio you do not get two scale parameters, let alone a covariance style matrix. Instead you get a new scale parameter $\gamma_{ab}$. If you got fancy and used a vector process, all the vectors would jointly share a scale parameter $\gamma_v$. Taking the logarithm brings you to the hyperbolic secant distribution and so no gain is had. It also has no covariance matrix, but OLS does. OLS would be measuring something that does not exist.
The headaches are just starting.
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Higher order Melnikov function for a quartic hamiltonian with cuspidal loop
1.
Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China
2.
Department of Mathematics, Zhongshan (Sun Yat-sen) University, 510275, Guangzhou, China
$X_\epsilon=(H_y+\epsilon f(x,y,\epsilon))\frac{\partial}{\partial x}+ (-H_x+\epsilon g(x,y,\epsilon))\frac{\partial}{\partial y},$
where the Hamiltonian $H(x,y)=\frac{1}{2}y^2+U(x)$ has one center and one cuspidal loop, $deg U(x)=4$. In present paper we find an upper bound for the number of zeros of the $k$th order Melnikov function $M_k(h)$ for arbitrary polynomials $f(x,y,\epsilon)$ and $g(x,y,\epsilon)$.
Mathematics Subject Classification:34C07, 34C08, 37G1. Citation:Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995
[1]
Qinian Jin, YanYan Li.
Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space.
[2]
Kazuyuki Yagasaki.
Higher-order Melnikov method and chaos
for two-degree-of-freedom Hamiltonian systems
with saddle-centers.
[3] [4] [5] [6]
Ravi P. Agarwal, Abdullah Özbekler.
Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities.
[7]
Giovanni Colombo, Thuy T. T. Le.
Higher order discrete controllability and the approximation of the minimum time function.
[8]
Guillaume Duval, Andrzej J. Maciejewski.
Integrability of potentials of degree $k \neq \pm 2$. Second
order variational equations between Kolchin solvability and
Abelianity.
[9]
Peng Mei, Zhan Zhou, Genghong Lin.
Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations.
[10] [11] [12] [13] [14] [15]
Olga A. Brezhneva, Alexey A. Tret’yakov, Jerrold E. Marsden.
Higher--order implicit function theorems and degenerate nonlinear boundary-value problems.
[16]
Anurag Jayswala, Tadeusz Antczakb, Shalini Jha.
Second order modified objective function method for twice differentiable vector optimization problems over cone constraints.
[17]
Robert Baier, Thuy T. T. Le.
Construction of the minimum time function for linear systems via higher-order set-valued methods.
[18] [19] [20]
2018 Impact Factor: 1.143
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by authors
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Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation
1.
Université Paris-Est Marne-la-Vallée, Laboratoire d'Analyse et de Mathématiques Appliquées (UMR 8050), 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France
2.
Université Paris 13 Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99, avenue Jean-Baptiste Clément, F-93 430 Villetaneuse, France
$ u_t-D_x^α u_{x} + u_{xyy} = uu_x,\,\,\,\,\,\, (t,x,y)∈\mathbb{R}^3, 1≤ α≤ 2,$
$E^s$
$s > \frac{2}{\alpha } - \frac{3}{4}$
$\|f{{\|}_{{{E}^{s}}}}=\|{{\left\langle {{\left| \xi \right|}^{\alpha }}+{{\mu }^{2}} \right\rangle }^{s}}\hat{f}{{\|}_{{{L}^{2}}({{\mathbb{R}}^{2}})}}.$
$E^{1/2}$
$α>\frac 85$ Mathematics Subject Classification:Primary:35A01, 35Q53;Secondary:35Q6. Citation:Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019
References:
[1] [2] [3] [4]
A. Cunha and A. Pastor, The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in low regularity Sobolev spaces, J. Differential Equations, 261 (2016), 2041–2067, arXiv: 1601.02803. doi: 10.1016/j.jde.2016.04.022. Google Scholar
[5]
A. Esfahani and A. Pastor,
Ill-posseness results for the (generalized) Benjamin-Ono-ZakharovKuznetsov equation,
[6] [7] [8] [9]
S. Herr, A. D. Ionescu, C. E. Kenig and H. Koch,
A para-differential renormalization technique for nonlinear dispersive equations,
[10]
A. D. Ionescu, C. E. Kenig and D. Tataru,
Global well-posedness of the KP-I initial-value problem in the energy space,
[11] [12]
M. C. Jorge, G. Cruz-Pacheco, L. Mier-y-Teran-Romero and N. F. Smyth, Evolution of twodimensional lump nanosolitons for the Zakharov-Kuznetsov and electromigration equations,
[13] [14]
C.E. Kenig, G. Ponce and L. Vega,
Well-posedness of the initial value problem for the Korteweg-de Vries equation,
[15] [16]
D. Lannes, F. Linares and J.-C. Saut,
The Cauchy Problem for the Euler-Poisson System and Derivation of the Zakharov-Kuznetsov Equation,
[17]
J. C. Latorre, A. A. Minzoni, N. F. Smyth and C. A. Vargas, Evolution of Benjamin-Ono solitons in the presence of weak Zakharov-Kutznetsov lateral dispersion,
[18]
L. Molinet and D. Pilod,
Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications,
[19]
L. Molinet, J. C. Saut and N. Tzvetkov,
Ill-posedness issues for the Benjamin-Ono equation and related equations,
[20]
L. Molinet and S. Vento,
Improvement of the energy method for strongly non resonant dispersive equations and applications,
[21] [22] [23]
show all references
References:
[1] [2] [3] [4]
A. Cunha and A. Pastor, The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in low regularity Sobolev spaces, J. Differential Equations, 261 (2016), 2041–2067, arXiv: 1601.02803. doi: 10.1016/j.jde.2016.04.022. Google Scholar
[5]
A. Esfahani and A. Pastor,
Ill-posseness results for the (generalized) Benjamin-Ono-ZakharovKuznetsov equation,
[6] [7] [8] [9]
S. Herr, A. D. Ionescu, C. E. Kenig and H. Koch,
A para-differential renormalization technique for nonlinear dispersive equations,
[10]
A. D. Ionescu, C. E. Kenig and D. Tataru,
Global well-posedness of the KP-I initial-value problem in the energy space,
[11] [12]
M. C. Jorge, G. Cruz-Pacheco, L. Mier-y-Teran-Romero and N. F. Smyth, Evolution of twodimensional lump nanosolitons for the Zakharov-Kuznetsov and electromigration equations,
[13] [14]
C.E. Kenig, G. Ponce and L. Vega,
Well-posedness of the initial value problem for the Korteweg-de Vries equation,
[15] [16]
D. Lannes, F. Linares and J.-C. Saut,
The Cauchy Problem for the Euler-Poisson System and Derivation of the Zakharov-Kuznetsov Equation,
[17]
J. C. Latorre, A. A. Minzoni, N. F. Smyth and C. A. Vargas, Evolution of Benjamin-Ono solitons in the presence of weak Zakharov-Kutznetsov lateral dispersion,
[18]
L. Molinet and D. Pilod,
Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications,
[19]
L. Molinet, J. C. Saut and N. Tzvetkov,
Ill-posedness issues for the Benjamin-Ono equation and related equations,
[20]
L. Molinet and S. Vento,
Improvement of the energy method for strongly non resonant dispersive equations and applications,
[21] [22] [23]
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[3]
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[4] [5]
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[10] [11]
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in four and more spatial dimensions.
[12]
Sergey Zelik, Jon Pennant.
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Luiz Gustavo Farah.
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[17]
Qifan Li.
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[20]
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Equivalence between uniform $L^{2^\star}(\Omega)$ a-priori bounds and uniform $L^{\infty}(\Omega)$ a-priori bounds for subcritical elliptic equations
DOI: http://dx.doi.org/10.12775/TMNA.2018.036
Abstract Keywords References
F.V. Atkinson and L.A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations 70 (1987), no. 3, 349–365.
A. Bahri and J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41(1988), no. 3, 253–294.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential equations, Universitext, Springer, New York, 2011.
H. Brezis and R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), no. 6, 601–614.
A. Castro and A. Kurepa, Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball, Proc. Amer. Math. Soc. 101 (1987), no. 1, 57–64.
A. Castro and R. Pardo, A priori bounds for positive solutions of subcritical elliptic equations, Rev. Mat. Complut. 28 (2015), 715–731.
A. Castro and R. Pardo, Branches of positive solutions of subcritical elliptic equations in convex domains, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, (2015), 230–238.
A. Castro and R. Pardo, Branches of positive solutions for subcritical elliptic equations, Contributions to Nonlinear Elliptic Equations and Systems, Progress in Nonlinear Differential Equations and Their Applications 86 (2015), 87–98.
A. Castro and R. Pardo, A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 3, 783–790.
D.G. de Figueiredo, P.-L. Lions and R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), no. 1, 41–63.
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901.
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer–Verlag, Berlin, second ed., 1983.
Z.-C .Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 2, 159–174.
D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 9 (1972/1973), 241–269.
P.L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441–467.
N. Mavinga and R. Pardo, A priori bounds and existence of positive solutions for subcritical semilinear elliptic systems, J. Math. Anal. Appl. 449 (2017), no. 2, 1172–1188.
R. Nussbaum, Positive solutions of nonlinear elliptic boundary value problems, J. Math. Anal. Appl. 51 (1975), (1975), no. 2, 461–482.
S.I. Pohozaev, On the eigenfunctions of the equation ∆u + λf (u) = 0, Dokl. Akad. Nauk SSSR 165 (1965), 36–39.
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Ok, here's my first attempt. Close scrutiny and comments appreciated!
The Two-Sample Hypotheses If we can frame two-sample one-sided Kolmogorov-Smirnov hypothesis tests, with null and alternate hypotheses along these lines:
H$_{0}\text{: }F_{Y}\left(t\right) \geq F_{X}\left(t\right)$, and
H$_{\text{A}}\text{: }F_{Y}\left(t\right) < F_{X}\left(t\right)$, for at least one $t$, where:
the test statistic $D^{-}=\left|\min_{t}\left(F_{Y}\left(t\right) - F_{X}\left(t\right)\right)\right|$ corresponds to H$_0\text{: }F_{Y}\left(t\right) \geq F_{X}\left(t\right)$;
the test statistic $D^{+}=\left|\max_{t}\left(F_{Y}\left(t\right) - F_{X}\left(t\right)\right)\right|$ corresponds to H$_0\text{: }F_{Y}\left(t\right) \leq F_{X}\left(t\right)$; and
$F_{Y}\left(t\right)$ & $F_{X}\left(t\right)$ are the empirical CDFs of samples $Y$ and $X$,
then it should be reasonable to create a general interval hypothesis for an equivalence test along these lines (assuming that the equivalence interval is symmetric for the moment):
H$^{-}_0\text{: }\left|F_{Y}\left(t\right) - F_{X}\left(t\right)\right| \geq \Delta$, and
H$^{-}_{\text{A}}\text{: }\left|F_{Y}\left(t\right) - F_{X}\left(t\right)\right| < \Delta$, for at least one $t$.
This would translate to the specific two one-sided "negativist" null hypotheses to test for equivalence (these two hypotheses take the same form, since
both $D^{+}$ and $D^{-}$ are strictly non-negative):
H$^{-}_{01}\text{: }D^{+} \geq \Delta$, or
H$^{-}_{02}\text{: }D^{-} \geq \Delta$.
Rejecting
both H$^{-}_{01}$ and H$^{-}_{02}$ would lead one to conclude that $-\Delta < F_{Y}\left(t\right) - F_{X}\left(t\right) < \Delta$. Of course, the equivalence interval need not be symmetric, and $-\Delta$ and $\Delta$ could be replaced with $\Delta_{2}$ (lower) and $\Delta_{1}$ (upper) for the respective one-sided null hypotheses.
The Test Statistics (Updated: Delta is outside the absolute value sign) The test statistics $D^{+}_{1}$ and $D^{-}_{2}$ (leaving the $n_{Y}$ and $n_{X}$ implicit) correspond to H$^{-}_{01}$ and H$^{-}_{02}$, respectively, and are:
$D^{+}_{1} = \Delta - D^{+} = \Delta - \left|\max_{t}\left[\left(F_{Y}\left(t\right) - F_{X}\left(t\right)\right)\right]\right|$, and
$D^{-}_{2} = \Delta - D^{-} = \Delta - \left|\min_{t}\left[\left(F_{Y}\left(t\right) - F_{X}\left(t\right)\right)\right]\right|$
The Equivalence/Relevance Threshold The interval $[-\Delta, \Delta]$—or $[\Delta_{2}, \Delta_{1}]$, if using an asymmetric equivalence interval—is expressed in units of $D^{+}$ and $D^{-}$, or the magnitude of differenced probabilities. As $n_{Y}$ and $n_{X}$ approach infinity, the CDF of $D^{+}$ or $D^{-}$ for $n_{Y},n_{X}$ approaches $0$ for $t<0$, and for $t \ge 0$:
$$\lim_{n_{Y},n_{X}\to \infty}p^{+} = \text{P}\left(\sqrt{\frac{n_{Y}n_{X}}{n_{Y}+n_{X}}}D^{+} \le t\right) = 1 - e^{-2t^{2}}$$
So it seems to me that the PDF for sample size-scaled $D^{+}$ (or sample size-scaled $D^{-}$) must be $0$ for $t<0$, and for $t \ge 0$:
$$f(t) = {1 - e^{-2t^{2}}}\frac{d}{dt} = 4te^{-2t^{2}}$$
Glen_b points out that this is a Rayleigh distribution with $\sigma=\frac{1}{2}$. So the large sample quantile function for sample size-scaled $D^{+}$ and $D^{-}$ is:
$$\text{CDF}^{-1} = Q\left(p\right) = \sqrt{\frac{-\ln{\left(1 - p\right)}}{2}}$$
and a liberal choice of $\Delta$ might be the critical value $Q_{\alpha}+\sigma/2 = Q_{\alpha}+\frac{1}{4}$, and a more strict choice the critical value $Q_{\alpha}+\sigma/4=Q_{\alpha}+\frac{1}{8}$.
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It's hard to say just from the sheet music; not having an actual keyboard here. The first line seems difficult, I would guess that second and third are playable. But you would have to ask somebody more experienced.
Having a few experienced users here, do you think that limsup could be an useful tag? I think there are a few questions concerned with the properties of limsup and liminf. Usually they're tagged limit.
@Srivatsan it is unclear what is being asked... Is inner or outer measure of $E$ meant by $m\ast(E)$ (then the question whether it works for non-measurable $E$ has an obvious negative answer since $E$ is measurable if and only if $m^\ast(E) = m_\ast(E)$ assuming completeness, or the question doesn't make sense). If ordinary measure is meant by $m\ast(E)$ then the question doesn't make sense. Either way: the question is incomplete and not answerable in its current form.
A few questions where this tag would (in my opinion) make sense: http://math.stackexchange.com/questions/6168/definitions-for-limsup-and-liminf http://math.stackexchange.com/questions/8489/liminf-of-difference-of-two-sequences http://math.stackexchange.com/questions/60873/limit-supremum-limit-of-a-product http://math.stackexchange.com/questions/60229/limit-supremum-finite-limit-meaning http://math.stackexchange.com/questions/73508/an-exercise-on-liminf-and-limsup http://math.stackexchange.com/questions/85498/limit-of-sequence-of-sets-some-paradoxical-facts
I'm looking for the book "Symmetry Methods for Differential Equations: A Beginner's Guide" by Haydon. Is there some ebooks-site to which I hope my university has a subscription that has this book? ebooks.cambridge.org doesn't seem to have it.
Not sure about uniform continuity questions, but I think they should go under a different tag. I would expect most of "continuity" question be in general-topology and "uniform continuity" in real-analysis.
Here's a challenge for your Google skills... can you locate an online copy of: Walter Rudin, Lebesgue’s first theorem (in L. Nachbin (Ed.), Mathematical Analysis and Applications, Part B, in Advances in Mathematics Supplementary Studies, Vol. 7B, Academic Press, New York, 1981, pp. 741–747)?
No, it was an honest challenge which I myself failed to meet (hence my "what I'm really curious to see..." post). I agree. If it is scanned somewhere it definitely isn't OCR'ed or so new that Google hasn't stumbled over it, yet.
@MartinSleziak I don't think so :) I'm not very good at coming up with new tags. I just think there is little sense to prefer one of liminf/limsup over the other and every term encompassing both would most likely lead to us having to do the tagging ourselves since beginners won't be familiar with it.
Anyway, my opinion is this: I did what I considered the best way: I've created [tag:limsup] and mentioned liminf in tag-wiki. Feel free to create new tag and retag the two questions if you have better name. I do not plan on adding other questions to that tag until tommorrow.
@QED You do not have to accept anything. I am not saying it is a good question; but that doesn't mean it's not acceptable either. The site's policy/vision is to be open towards "math of all levels". It seems hypocritical to me to declare this if we downvote a question simply because it is elementary.
@Matt Basically, the a priori probability (the true probability) is different from the a posteriori probability after part (or whole) of the sample point is revealed. I think that is a legitimate answer.
@QED Well, the tag can be removed (if someone decides to do so). Main purpose of the edit was that you can retract you downvote. It's not a good reason for editing, but I think we've seen worse edits...
@QED Ah. Once, when it was snowing at Princeton, I was heading toward the main door to the math department, about 30 feet away, and I saw the secretary coming out of the door. Next thing I knew, I saw the secretary looking down at me asking if I was all right.
OK, so chat is now available... but; it has been suggested that for Mathematics we should have TeX support.The current TeX processing has some non-trivial client impact. Before I even attempt trying to hack this in, is this something that the community would want / use?(this would only apply ...
So in between doing phone surveys for CNN yesterday I had an interesting thought. For $p$ an odd prime, define the truncation map $$t_{p^r}:\mathbb{Z}_p\to\mathbb{Z}/p^r\mathbb{Z}:\sum_{l=0}^\infty a_lp^l\mapsto\sum_{l=0}^{r-1}a_lp^l.$$ Then primitive roots lift to $$W_p=\{w\in\mathbb{Z}_p:\langle t_{p^r}(w)\rangle=(\mathbb{Z}/p^r\mathbb{Z})^\times\}.$$ Does $\langle W_p\rangle\subset\mathbb{Z}_p$ have a name or any formal study?
> I agree with @Matt E, as almost always. But I think it is true that a standard (pun not originally intended) freshman calculus does not provide any mathematically useful information or insight about infinitesimals, so thinking about freshman calculus in terms of infinitesimals is likely to be unrewarding. – Pete L. Clark 4 mins ago
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two elements in the subset are incomparable. (Some authors use the term "antichain" to mean strong antichain, a subset such that there is no element of the poset smaller than 2 distinct elements of the antichain.)Let S be a partially ordered set. We say two elements a and b of a partially ordered set are comparable if a ≤ b or b ≤ a. If two elements are not comparable, we say they are incomparable; that is, x and y are incomparable if neither x ≤ y nor y ≤ x.A chain in S is a...
@MartinSleziak Yes, I almost expected the subnets-debate. I was always happy with the order-preserving+cofinal definition and never felt the need for the other one. I haven't thought about Alexei's question really.
When I look at the comments in Norbert's question it seems that the comments together give a sufficient answer to his first question already - and they came very quickly. Nobody said anything about his second question. Wouldn't it be better to divide it into two separate questions? What do you think t.b.?
@tb About Alexei's questions, I spent some time on it. My guess was that it doesn't hold but I wasn't able to find a counterexample. I hope to get back to that question. (But there is already too many questions which I would like get back to...)
@MartinSleziak I deleted part of my comment since I figured out that I never actually proved that in detail but I'm sure it should work. I needed a bit of summability in topological vector spaces but it's really no problem at all. It's just a special case of nets written differently (as series are a special case of sequences).
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What does the classical proof of the proposition "there exists irrational numbers a, b such that $a^b$ is rational" want to reveal? I know it has something to do with the difference between classical and constructive mathematics but am not quite clear about it. Materials I found online does not give quite clear explanations either. Could someone give a better explanation?
Presumably, the proof you have in mind is to use $a=b=\sqrt2$ if $\sqrt2^{\sqrt2}$ is rational, and otherwise use $a=\sqrt2^{\sqrt2}$ and $b=\sqrt 2$. The non-constructivity here is that, unless you know some deeper number theory than just irrationality of $\sqrt 2$, you won't know which of the two cases in the proof actually occurs, so you won't be able to give $a$ explicitly, say by writing a decimal approximation.
The proposition can also be proved without explicitly knowing any particular irrational number:
Fix a positive rational number $c$. Then for any positive irrational real number number $a$, the equation $a^b = c$ has a solution, and the solutions for different $a$ are different. Now assume that there would be no irrational numbers $a$ and $b$ such that $a^b$ is rational. Then all these solutions would be rational, thus we would have an injective mapping from the positive irrational real numbers to the rationals; however we know that the rationals are countable and the positive irrational reals are not, so there is no such injection, and we have a contradiction.
From a more abstract point of view, suppose you have a set $X$, a subset $\mathcal{I} \subset X$, an element $a \subset \mathcal{I}$ and a map $b : X \to X$ satisfying $b^n(a) \not\in \mathcal{I}$ for some $n$.
Then necessarily there exists $m$ such that $b^m(a) \in \mathcal{I}$ and $b^{m+1}(a) \not\in \mathcal{I}$.
The proof you refer to is not really mysterious; it's only presented to look mysterious, specifically by the choice of $\sqrt{2}$. There are plenty of choices for irrational $a$ and $b$, such that $a^{b^n}$ is rational for some $n$, and exactly the same conclusion can be drawn.
The classical proof You have mentioned shows how powerful logic may be. The crucial point of the proof is the use of the law of excluded middle (A or not(A)). In that case we are lucky to have knowledge of which of the two possibilities:
i) $\sqrt{2}^{\sqrt{2}}$ is rational or ii) $\sqrt{2}^{\sqrt{2}}$ is not rational
takes place to be. But You can imagine a statement A about natural numbers such that neither A nor not(A) is provable (in, say, ZFC). Now we may use analogously the particular statement of the law of excluded middle "A or not(A)" in some proof of some statement B. In that case the statement B is proved both by supposing A and by supposing not(A) and at the same time we know in advance that we shall not resolve which of the cases has the place to be. In such a proof the purely logical priciple of excluded middle becomes even more important (or at least as important) than the axioms of actual mathematical structure (in this case that of natural numbers). It is this power that made some mathematicians (e. g. intuitionists under the head of Brouwer) to think that logical principles such as that of exluded middle have caused the paradoxes discovered in the beginning of the XX century. For a classical mathematician the proof of B would be a legitimate proof since he/she believes that either A or not(A) is valid for natural numbers even if we shall never prove neither of them. (This position is often called platonism because it presupposes the "independent existence" of natural numbers.) Intuitionist, however, says that it is a hypothesis and to turn this reasoning into a valid proof one should prove that indeed one of those possiblities has the place to be.
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Entanglement is synonymous to ignorance!
In general, it arises whenever trying to break down a quantum system into subsystems. In more mathematical terms of quantum mechanics, this means to decompose the Hilbert space $\mathcal{H}$ of the full system into a Hilbert space $\mathcal{H}_{A}$ of a subsystem $A$ and a Hilbert space $\mathcal{H}_{\bar{A}}$ of the complimentary subsystem $\bar{A}$. As a result, given a pure state $|\Psi\rangle\in\mathcal{H}$ of the full system, it can be written in terms of pure states $|\psi^{A}_{i}\rangle\in\mathcal{H}_{A}$ and $|\psi^{\bar{A}}_{\bar{i}}\rangle\in\mathcal{H}_{\bar{A}}$ of the subsystems as a mixed state,
\begin{equation}|\Psi\rangle = \sum_{i,\bar{i}} c_{i\bar{i}} |\psi^{A}_{i}\rangle \otimes |\psi^{\bar{A}}_{\bar{i}}\rangle\end{equation}
Entanglement is present whenever $|c_{i\bar{i}}|\ne1$, $\forall i,\bar{i}$. To put it into perspective, the fact that the entire universe cannot be measured at the same time forces us to consider subsystems that are more controllable. However, this procedure of breaking down the system comes with unavoidable ignorance of the true nature of the full system, i.e. entanglement.
So, to answer your question,
initially un-entangled, i.e. pure, states cannot become entangled because this would invoke an information loss paradox since the initial pure states that are equipped with "deterministic" information suddenly become mixed states whose information content has reduced due to ignorance. To give you more understanding of your own inquiry, you are asking the same question that caused the black hole war between Leonard Susskind and Stephen Hawking. After Hawking discovered his Hawking radiation, the information loss paradox came into play: "Throw a pure state in the black hole and wait long enough to obtain the Hawking radiation which describes now a thermal, mixed state. Where did the information go!?"
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This question already has an answer here:
Apparently my question
is different from Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}δq=δ\frac{dq}{dt}$. I hadn't noticed because the answer given in the comments to this question was satisfactory for my immediate purposes. My question pertains to time-independent variation, whereas the other question pertains to time-dependent variations.
By
time-independent variation I mean an active variation of the generalized position coordinates which does not change with time for a given configuration point. That is, one that may be written without an explicit time argument. Position as a function of time $\left\{q\right\}=\left\{q^{i}\left[t\right]\right\}$ is assumed to be some sufficiently smooth trajectory in configuration space representing the actual physical system being modeled.
At the point in the discussion where the question occurred to me, the only variations considered are active rigid transformations. These are assumed "infinitesimal" in the case of rotations. No further definitions for $\delta{q^i}$ and $\delta{\dot{q}^i}$ were given.
See: The Theoretical Minimum: What You Need to Know to Start Doing Physics, by Susskind and Harbovsky.
Apparently, in the case of time-independent variations, $\delta{\dot{q}^i}$ has to be given independently of $\delta{q^i},$ or be defined using the chain-rule, treating the argument $\left\{q\right\}$ in $\delta q^{i}=f^{i}\left[\left\{q\right\}\right]\varepsilon$ as a function of time.
In the case where $\delta{q^i}$ and $\dot{\delta{q^i}}$ are given independently the commutativity has to be part of the definition.
Following Susskind, a variation of the form
$$\delta q^{i}=f^{i}\left[\left\{q\right\}\right]\varepsilon,$$
such that the consequent variation of the Lagrangian is $\delta{L}=0,$ is said to be a symmetry. Using Einstein summation, the variation of the Lagrangian may be written
$$\delta{L}=\delta L=\frac{\partial L}{\partial q^{i}}\delta{q^{i}}+\frac{\partial L}{\partial\dot{q}^{i}}\delta\dot{q}^{i}.$$
Assuming the trajectory of the configuration point has a stationary action integral, and therefore satisfies the Euler-Lagrange differential equation, we may write
$$\delta L=\dot{p_{i}}\delta q^{i}+p_{i}\delta\dot{q}^{i}.$$
If we can justify its application in this context, the basic calculus rule for the derivative of the product gives
$$\delta L=\frac{d}{dt}\left[p_{i}\delta q^{i}\right].$$ Apparently, this applicability depends upon the validity of
$$\frac{d}{dt}\delta q^{i}=\delta \frac{dq^{i}}{dt}.$$
Within the context of discussing the rectilinear motion of a single particle and variations which are purely translational or purely rotational, the equation
$$\delta \dot{q}^{i}=f^{i}\left[\left\{\dot{q}\right\}\right]\varepsilon$$
was asserted to follow from the corresponding variation of the position coordinates. The claim is fairly obvious for those circumstances. But it is less obvious for more general variations such as those applied to a path when seeking a stationary action integral. I believe the product rule is applicable to all "well-behaved" variations, but I'm having difficulty formulating a satisfactory argument (informal or formal proof) of that fact.
I am assuming that the operators $f^i$ are point functions of the coordinate system and are time-independent. I believe they will always be affine transformations.
So, my question is:
in the context of time-independent variations of the configuration of a system, when and why is it true that
$$\frac{d}{dt}\delta q^{i}=\delta \frac{dq^{i}}{dt}?$$
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Good Morning!
I am new in signal processing and I am trying to do a work in noise control of an electronic steering lock device (ESL). My aim is to calculate the loudness (Zwicker Method- ISO 532 B) of this device. To do so, first I need to obtain the 1/3 octave spectrum of a time signal that I measure with a microphone. The problem is I keep getting negative values in $\textrm{dB}$ for the 1/3 Octave bands after filtering the signal in the time domain to obtain the spectrum. I will explain here the procedure I have used and hope that anyone sees what I am doing wrong. Thanks in advance.
I have done the following procedure by now:
Sampled the noise signal (impulsive noise) by using a microphone and a data logger (to record the data), which has a sample frequency of $50\textrm{ kHz}$. Then, after this step I have a Curve that it is Amplitude ($\textrm{dBA}$) vs. time ($\textrm{s}$), as shown below. Once the ($\textrm{dBA}$) value of a sound level meter is calculated by: $$10\log_{10}\left( \dfrac{p^2}{p_0^2} \right)$$ where $p_0$ is $2\cdot 10^{-6}\textrm{ Pa}$. I am able to evaluate the pressure variation ($\textrm{ Pa}$) vs. time and use it as INPUT of the 1/3 Octave filters.
I get the vector INPUT (with $250000$ points of pressure ($\textrm{ Pa}$)-measurements of $5\textrm{ s}$) and use a function in MATLAB, in order to filter the signal in each each 1/3 octave band.
Then, the program calculates the RMS value of the OUTPUT (after filtering). And this is the value that represents each frequency band.
Finally, I use the same expression used before to calculate the Magnitude in $\textrm{ dB}$ for each 1/3 Octave band. $10\log_{10}\left( \dfrac{p^2}{p_0^2} \right)$, where $p_0$ is $2\cdot 10^{-6}\textrm{ Pa}$.
The thing is the obtained 1/3 Octave is lower then $0\textrm{ dB}$ and this doesn't make sense once I can hear the noise when I run the device, moreover it doesn't make sense to calculate the loudness following the ISO 532 B if we have negative third octave bands. It seems like the pressure that I have in time domain that is higher then the reference pressure somehow is attenuated and gets lower than the reference pressure after filtering.
Does anybody know what I am doing wrong?
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Most engineers have seen the moment-to-moment fluctuations that are common with instantaneous measurements of a supposedly steady spectrum. You can see these fluctuations in magnitude and phase for each frequency bin of your spectrogram. Although major variations are certainly reason for concern, recall that we don’t live in an ideal, noise-free world. After verifying the integrity of your measurement setup by checking connections, sensors, wiring, and the like, you might conclude that the...
This is an article that is the last of my digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). It is along the lines of the last two.
In those articles, I presented exact formulas for calculating the frequency of a pure tone signal as instantaneously as possible in the time domain. Although the formulas work for both real and complex signals (something that does not happen with frequency domain formulas), for real signals they...
This is an article that is a continuation of a digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). It is recommended that my previous article "Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 1)"[1] be read first as many sections of this article are directly dependent upon it.
A second family of formulas for calculating the frequency of a single pure tone in a short interval in the time domain is presented. It...
$$ atan(z) \approx \dfrac{z}{1.0 +...
This is an article that is a another digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). Although it is not as far off as the last blog article.
A new family of formulas for calculating the frequency of a single pure tone in a short interval in the time domain is presented. They are a generalization of Equation (1) from Rick Lyons' recent blog article titled "Sinusoidal Frequency Estimation Based on Time-Domain Samples"[1]. ...
Some weeks ago a question appeared on the dsp.related Forum regarding the notion of translating a signal down in frequency and lowpass filtering in a single operation [1]. It is possible to implement such a process by embedding a discrete cosine sequence's values within the coefficients of a traditional lowpass FIR filter. I first learned about this process from Reference [2]. Here's the story.Traditional Frequency Translation Prior To Filtering
Think about the process shown in...
I'm working on a video, tying the Fibonacci sequence into the general subject of difference equations.
Here's a fun trick: take any two consecutive numbers in the Fibonacci sequence, say 34 and 55. Now negate one and use them as the seed for the Fibonacci sequence, larger magnitude first, i.e.
$-55, 34, \cdots$
Carry it out, and you'll eventually get the Fibonacci sequence, or it's negative:
$-55, 34, -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1 \cdots$
This is NOT a general property of difference...
My basement is covered with power lines and florescent lights which makes collecting ECG and EEG data rather difficult due to the 60 cycle hum. I found the following notch filter to work very well at eliminating the background signal without effecting the highly amplified signals I was looking for.
The notch filter is based on the a transfer function with the form $$H(z)=\frac{1}{2}(1+A(z))$$ where A(z) is an all pass filter. The original paper [1] describes a method to...
FFMPEG is a set of libraries and a command line tool for encoding and decoding audio and video in many different formats. It is a free software project for manipulating/processing multimedia data. Many open source media players are based on FFMPEG libraries.
FFMPEG is developed under Linux but it can be compiled under most operating systems including Mac OS, Microsoft Windows. For more details about FFMPEG please refer
$$ atan(z) \approx \dfrac{z}{1.0 +...
Some days ago I read a post on the comp.dsp newsgroup and, if I understood the poster's words, it seemed that the poster would benefit from knowing how to compute the twiddle factors of a radix-2 fast Fourier transform (FFT).
Then, later it occurred to me that it might be useful for this blog's readers to be aware of algorithms for computing FFT twiddle factors. So,... what follows are two algorithms showing how to compute the individual twiddle factors of an N-point decimation-in-frequency...
In the last posts I reviewed how to use the Python scipy.signal package to design digital infinite impulse response (IIR) filters, specifically, using the iirdesign function (IIR design I and IIR design II ). In this post I am going to conclude the IIR filter design review with an example.
Previous posts:
The problem of "spectral inversion" comes up fairly frequently in the context of signal processing for communication systems. In short, "spectral inversion" is the reversal of the orientation of the signal bandwidth with respect to the carrier frequency. Rick Lyons' article on "Spectral Flipping" at http://www.dsprelated.com/showarticle/37.php discusses methods of handling the inversion (as shown in Figure 1a and 1b) at the signal center frequency. Since most communication systems process...
This blog is not about signal processing. Rather, it discusses an interesting topic in number theory, the magic of the number 9. As such, this blog is for people who are charmed by the behavior and properties of numbers.
For decades I've thought the number 9 had tricky, almost magical, qualities. Many people feel the same way. I have a book on number theory, whose chapter 8 is titled "Digits — and the Magic of 9", that discusses all sorts of interesting mathematical characteristics of the...
This blog describes several DC removal networks that might be of interest to the dsprelated.com readers.
Back in August 2007 there was a thread on the comp.dsp newsgroup concerning the process of removing the DC (zero Hz) component from a time-domain sequence [1]. Discussed in that thread was the notion of removing a signal's DC bias by subtracting the signal's moving average from that signal, as shown in Figure 1(a).
Figure 1.
At first I thought...
Most engineers have seen the moment-to-moment fluctuations that are common with instantaneous measurements of a supposedly steady spectrum. You can see these fluctuations in magnitude and phase for each frequency bin of your spectrogram. Although major variations are certainly reason for concern, recall that we don’t live in an ideal, noise-free world. After verifying the integrity of your measurement setup by checking connections, sensors, wiring, and the like, you might conclude that the...
This article relates to the Matlab / Octave code snippet: Delay estimation with subsample resolution It explains the algorithm and the design decisions behind it.Introduction
There are many DSP-related problems, where an unknown timing between two signals needs to be determined and corrected, for example, radar, sonar,...
In many applications the detection or processing of signals in the frequency domain offers an advantage over performing the same task in the time-domain. Sometimes the advantage is just a simpler or more conceptually straightforward algorithm, and often the largest barrier to working in the frequency domain is the complexity or latency involved in the Fast Fourier Transform computation. If the frequency-domain data must be updated frequently in a...
It is possible to compute N-point discrete Fourier transforms (DFTs) using radix-2 fast Fourier transforms (FFTs) whose sizes are less than N. For example, let's say the largest size FFT software routine you have available is a 1024-point FFT. With the following trick you can combine the results of multiple 1024-point FFTs to compute DFTs whose sizes are greater than 1024.
The simplest form of this idea is computing an N-point DFT using two N/2-point FFT operations. Here's how the trick...
The problem of "spectral inversion" comes up fairly frequently in the context of signal processing for communication systems. In short, "spectral inversion" is the reversal of the orientation of the signal bandwidth with respect to the carrier frequency. Rick Lyons' article on "Spectral Flipping" at http://www.dsprelated.com/showarticle/37.php discusses methods of handling the inversion (as shown in Figure 1a and 1b) at the signal center frequency. Since most communication systems process...
In the last posts I reviewed how to use the Python scipy.signal package to design digital infinite impulse response (IIR) filters, specifically, using the iirdesign function (IIR design I and IIR design II ). In this post I am going to conclude the IIR filter design review with an example.
Previous posts:
Some days ago I read a post on the comp.dsp newsgroup and, if I understood the poster's words, it seemed that the poster would benefit from knowing how to compute the twiddle factors of a radix-2 fast Fourier transform (FFT).
Then, later it occurred to me that it might be useful for this blog's readers to be aware of algorithms for computing FFT twiddle factors. So,... what follows are two algorithms showing how to compute the individual twiddle factors of an N-point decimation-in-frequency...
This article relates to the Matlab / Octave code snippet: Delay estimation with subsample resolution It explains the algorithm and the design decisions behind it.Introduction
There are many DSP-related problems, where an unknown timing between two signals needs to be determined and corrected, for example, radar, sonar,...
In many applications the detection or processing of signals in the frequency domain offers an advantage over performing the same task in the time-domain. Sometimes the advantage is just a simpler or more conceptually straightforward algorithm, and often the largest barrier to working in the frequency domain is the complexity or latency involved in the Fast Fourier Transform computation. If the frequency-domain data must be updated frequently in a...
$$ atan(z) \approx \dfrac{z}{1.0 +...
This blog describes several DC removal networks that might be of interest to the dsprelated.com readers.
Back in August 2007 there was a thread on the comp.dsp newsgroup concerning the process of removing the DC (zero Hz) component from a time-domain sequence [1]. Discussed in that thread was the notion of removing a signal's DC bias by subtracting the signal's moving average from that signal, as shown in Figure 1(a).
Figure 1.
At first I thought...
Some time ago I was studying various digital differentiating networks, i.e., networks that approximate the process of taking the derivative of a discrete time-domain sequence. By "studying" I mean that I was experimenting with various differentiating filter coefficients, and I discovered a computationally-efficient digital differentiator. A differentiator that, for low fequency signals, has the power of George Foreman's right hand! Before I describe this differentiator, let's review a few...
There are two code snippets associated with this blog post:
and
This blog discusses an accurate method of estimating time-domain sinewave peak amplitudes based on fast Fourier transform (FFT) data. Such an operation sounds simple, but the scalloping loss characteristic of FFTs complicates the process. We eliminate that complication by...
I just learned a new method (new to me at least) for computing the group delay of digital filters. In the event this process turns out to be interesting to my readers, this blog describes the method. Let's start with a bit of algebra so that you'll know I'm not making all of this up.
Assume we have the N-sample h(n) impulse response of a digital filter, with n being our time-domain index, and that we represent the filter's discrete-time Fourier transform (DTFT), H(ω), in polar form...
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math.DG
Change to browse by: References & Citations Bookmark(what is this?) Mathematics > Differential Geometry Title: Convergence of vector bundles with metrics of Sasaki-type
(Submitted on 2 Nov 2010 (v1), last revised 18 Mar 2011 (this version, v2))
Abstract: If a sequence of Riemannian manifolds, $X_i$, converges in the pointed Gromov-Hausdorff sense to a limit space, $X_\infty$, and if $E_i$ are vector bundles over $X_i$ endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the $E_i$ converges in the pointed Gromov-Hausdorff sense to a metric space, $E_\infty$. The projection maps $\pi_i$ converge to a limit submetry $\pi_\infty$ and the fibers converge to its fibers; the latter may no longer be vector spaces but are homeomorphic to $\R^k/G$, where $G$ is a closed subgroup of $O(k)$ ---called the {\em wane group}--- that depends on the basepoint and that is defined using the holonomy groups on the vector bundles. The norms $\mu_i=\|\cdot\|_i$ converges to a map $\mu_{\infty}$ compatible with the re-scaling in $\R^k/G$ and the $\R$-action on $E_i$ converges to an $\R-$action on $E_{\infty}$ compatible with the limiting norm. In the special case when the sequence of vector bundles has a uniform lower bound on holonomy radius (as in a sequence of collapsing flat tori to a circle), the limit fibers are vector spaces. Under the opposite extreme, e.g. when a single compact $n$-dimensional manifold is re-scaled to a point, the limit fiber is $\R^n/H$ where $H$ is the closure of the holonomy group of the compact manifold considered. An appropriate notion of parallelism is given to the limiting spaces by considering curves whose length is unchanged under the projection. The class of such curves is invariant under the $\R$-action and each such curve preserves norms. The existence of parallel translation along rectifiable curves with arbitrary initial conditions is also exhibited. Uniqueness is not true in general, but a necessary condition is given in terms of the aforementioned wane groups $G$. Submission historyFrom: Pedro Solórzano [view email] [v1]Tue, 2 Nov 2010 05:22:26 GMT (28kb) [v2]Fri, 18 Mar 2011 06:45:58 GMT (120kb)
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A differential equation can be easily converted into an integral equation just by integrating it once or twice or as many times, if needed. Let’s start with an example. Let $$\frac{dy}{dx} + 5y+1=0 \ldots (1)$$ be a simple first order differential equation. We can integrate it one time with respect to $x$ , to obtain $$\int \frac{dy}{dx} dx + 5 \int y dx +\int 1 \cdot dx=c$$
Or, $$y + 5 \int y dx +x =c \ldots (2)$$
If we arrange equation (2) in standard integral equation forms, as studied in very first part of this series, we get
$$y=(c-x)- 5 \int y dx$$
or, $$y(x)=(c-x)-5\int y(t) dt \ldots (3)$$
We can remove the arbitrary constant $c$ from the above integral equation by applying a boundary condition. For example, if we have $$y(0)=1$$ , then it can be easily seen that
$$y(0)=(c-0)-5\int y(0) dt$$
or, $$c=y(0)+5\int y(0) dt$$ $$ \Rightarrow c=1+5 \int 1 \cdot dt$$
$$\Rightarrow c=1+5 \int dt \ldots (4)$$
At this instance, we see that if the limits of the integration could have known, the value of $c$ should have been easier to interpret. Still we can convert the given differential equation into integral equation by substituting the value of $c$ in equation (3) above:
$$y(x)=(1-x+5 \int dt)-5\int y(t) dt $$
$$y(x)=(1-x)+5 \int (1-y(t)) dt \ldots (5)$$
Equation(5) is the resulting integral equation converted from equation (1). $\Box$
We see that there is one boundary condition required to obtain the single constant $c$ in First Order differential equation. In the same way, there are two boundary conditions needed in a second order differential equation.
Problems in second order differential equation with boundary conditions, are of two types.
Initial value problem
For some finite value of variable $x$, the value of function $y$ and its derivative $dy/dx$ is given in an initial value differential equation problem.
For example $$\frac{d^2y}{dx^2} +ky=tx $$ with $$y(0)=2$$ and $$y'(0)=5$$ is an initial value problem. Just try to see how, point $x=0$ is used for both $y$ and $y’$, which is called the initial value of the differential equation. This initial value changes into the lower limit when we try to derive the integral equation. And, also, the integral equation derived from an initial value problem is of Volterra type, i.e., having upper limit as variable $x$. Boundary value problem
For different values of variable $x$, the value of function given in a boundary value condition.
For example $$ \frac{d^2y}{dx^2}+ly=mx$$ with $$y(a)=A$$ and $$y(b)=B$$ is a boundary value problem. Generally, we chose the lower limit of the integration as zero and integrate the differential equation within limit $(0,x)$. After the boundary values are substituted, we obtain a Fredholm integral equation, i.e., having upper limit as a constant $b$ (say).
All doubts may cleared by working out the following two examples
Converting initial value problem into a Volterra integral equation
Example 1 : Convert the following differential equation into integral equation:
$$y”+y=0$$ when $$y(0)=y'(0)=0$$
Solution: Given
$$y”(x)+y(x)=0 \ldots (6)$$ with $$y(0)=0 \ldots (7)$$ and $$y'(0)=0 \ldots(8)$$ From (1), $$y”(x)=-y(x) \ldots (9)$$ Integrating (9) with respect to $x$ from $0$ to $x$. $$\int_{0}^{x} y”(x) dx =-\int_{0}^{x} y(x) dx$$
$(y'(x))_0^x= -\int{0}^{x} y(x) dx$
$\Rightarrow y'(x)-y'(0) = -\int_{0}^{x} y(x) dx$
Since, $$y'(x)=0$$ ,
$$\Rightarrow y'(x)-0 =-\int_{0}^{x} y(x) dx$$
$$\Rightarrow y'(x) =-\int_{0}^{x} y(x) dx \ldots (10)$$
Integrating both sides of (10) with respect to $x$ from $0$ to $x$ –
$$\int_{0}^{x} y'(x) dx =-\int_{0}^{x} \left(\int_{0}^{x} y(x) dx\right) dx $$
$$\int_{0}^{x} y'(x) dx =-\int_{0}^{x} y(x) dx^2 $$
$$\Rightarrow (y(x))_0^x=-\int_{0}^{x} y(t) dt^2 $$
$$y(x)-y(0)=-\int_{0}^{x} (x-t) y(t) dt$$
$$\Rightarrow y(x)-0 =-\int_{0}^{x} (x-t) y(t) dt$$
$$\Rightarrow y(x)=-\int_{0}^{x} (x-t) y(t) dt \ldots (11)$$
This equation (11) is the resulting integral equation derived from the given second order differential equation. $\Box$
Converting boundary value problem into a Fredholm integral equation
Example 2: Reduce the following boundary value problem into an integral equation $$\frac{d^2y}{dx^2} +\lambda y =0$$ with $$y(0)=0$$ and $$y(l)=0$$
Solution: Given differential equation is $$y”(x)+\lambda y(x)=0 \ldots (12)$$ with $$y(0)=0 \ldots (13)$$ and $$y(l)=0 \ldots (14)$$ Since, $$(12) \Rightarrow y”(x) = -\lambda y(x) \ldots (15)$$ Integrating both sides of (15) w.r.t. $x$ from $0$ to $x$ $$\int_{0}^{x}y”(x) dx = -\lambda \int_{0}^{x}y(x) dx \ldots (16)$$
$$ {(y'(x))}_0^x=-\lambda \int_{0}^{x}y(x) dx$$
$$ y'(x) -y'(0)=-\lambda \int_{0}^{x}y(x) dx \ldots (17)$$
Let $y'(0)=\mathbf{ constant}=c$, then
$$ y'(x) -c=-\lambda \int_{0}^{x}y(x) dx $$
$$ y'(x)=c-\lambda \int_{0}^{x}y(x) dx \ldots (18)$$
Integrating (18) again with respect to $x$ from 0 to $x$
$$\int_{0}^{x} y'(x)dx=c\int_{0}^{x} dx-\lambda \int_{0}^{x} \left({ \int_{0}^{x}y(x) dx}\right) dx$$
or, $$ (y(x))_0^x=cx-\lambda \int_{0}^{x} y(x) dx^2$$
$$y(x)-y(0)=cx-\lambda \int_{0}^{x}y(t) dt^2$$
Putting $y(0)=0$
$$y(x)=cx-\lambda \int_{0}^{x} (x-t) y(t) dt \ldots (19)$$
Now, putting $x=l$ in (19):
$$y(l)=cl-\lambda \int_{0}^{l} (l-t) y(t) dt$$
$$0=cl-\lambda \int_{0}^{l} (l-t) y(t) dt$$
$$c=\frac{\lambda}{l} \int_{0}^{l} (l-t) y(t) dt \ldots (20)$$
Putting this value of $c$ in (19), (19) reduces to:
$$y(x) =\frac{\lambda x}{l} \int_{0}^{l} (l-t) y(t) dt-\lambda \int_0^x (x-t) y(t) dt \ldots (21) $$
On simplifying (21) we get
$$y(x) =\lambda (\int_{0}^{x} \frac{(l-x)t}{l} y(t) dt + \int_{x}^{l} \frac{x(l-t)}{l} y(t) dt) \ldots (22)$$
Which is the required integral equation derived from the given differential equation.
The solution can also be written as $$y(x) =\lambda \int_{0}^{l} K(x,t) y(t) dt$$
where $$K(x,t)=\frac{t(l-x)}{l} \qquad \mathbf{0<t<x}$$ and $$K(x,t)=\frac{x(l-t)}{l} \qquad \mathbf{x<t<l}$$
$\Box$
We can now define a strategy for changing the ordinary differential equations of second order into an integral equation.
Step 1: Write the differential equation and its boundary conditions.
Step 2: Now re-write the differential equation in its normal form, i.e., highest derivatives being on one side and other, all values on the other side. For example, $ y”=-\frac{\alpha}{2} xy’ +ny$ is the normal form of $ 2y”+\alpha xy’ -2ny=0$ .
Step 3: Integrate the normal form of the differential equation, from 0 to $x$. Use applicable rules and formulas to simplify it.
Step 4: If substitutable, substitute the values of the boundary conditions. In boundary value problems, take $y'(0)=c$ a constant.
Step 5: Again integrate, the, so obtained differential-integral equation, within the limits $(0,x)$ with respect to $x$.
Step 6: Substitute the values of given boundary conditions.
Step 7: Simplify using essential integration rules, change the variable inside the integration sign to $t$ . Use the ‘multiple integral‘ rules to change multiple integral into linear integral, as we discussed in Part(1).
Why should I convert a differential equation into an integral equation?
DoTheOrdersStilStand asks:
Hi Gaurav,
This is all good, but it would help if you added some context on why you’d want to convert differential equations into integral equations.
Finding analytical or numerical solutions in the former case is often easier, also qualitative analysis of the asymptotic and singularity behavior in the phase space.
Possible Answer:
Returning to basics of differential equation, we know that the values of $y(x)$ which satisfy above differential equations are their solutions. Performing a conversion from differential equation in $y$ to integral equation in $y$ is nothing but solving the differential equation for $y$. After converting an initial value or boundary value problem into an integral equation, we can solve them by shorter methods of integration. This conversion may also be treated as another representation formula for the solution of an ordinary differential equation.
Feel free to ask questions, send feedback and even point out mistakes. Great conversations start with just a single word.
How to write better comments?
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Terms sourced from: http://iupac.org/publications/pac/51/12/2451/
"Quantities and Units in Clinical Chemistry", IUPAC and IFCC,
Pure and Applied Chemistry 1979, 51(12), 2451
base kind of quantity catalysed rate of reaction \(\xi_{\text{cat}}\)derived coherent unit derived kind of quantity derived non-coherent unit mean catalytic activity rate \(\frac{\Delta z}{\Delta t}\)mean mass rate \(\frac{\Delta m}{\Delta t}\)mean substance rate \(\frac{\Delta n}{\Delta t}\)mean volume rate \(\frac{\Delta V}{\Delta t}\)relative volumic mass specific gravity specific weight volume content \(V_{\text{c}}/m_{\text{s}}\)
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Fraction & Decimals
Category : 5th Class
FRACTION AND DECIMALS
FUNDAMENTALS
Types of Fraction
Example: \[\frac{1}{3},\frac{2}{3},\frac{4}{5}\]
Example:\[\frac{5}{3},\frac{6}{5},\frac{7}{4},\frac{7}{7}\] etc.
Example:\[1+\frac{2}{3}\] is written as\[1\frac{2}{3}\],\[2+\frac{1}{5}\] is written as\[2\frac{1}{5}\]
Example: \[\frac{1}{8},\frac{2}{8},\frac{5}{8}\] etc.
In all the above fraction denominators are equal, so they are like fraction.
Example: \[\frac{1}{3},\frac{1}{5},\frac{5}{8},\frac{3}{7}\] etc.
Example: \[\frac{1}{5},\frac{2}{10},\frac{3}{15}\] etc.
In above fractions value of each fraction is equal so they are equipment fractions.
Example:\[\frac{3}{10},\frac{1}{100},\frac{1}{1000}\]
Example: \[\frac{1}{2}\]of\[\frac{3}{8}\],\[\frac{1}{3}\] of \[\frac{4}{7}\] etc.
Example: \[4+\frac{1}{1+\frac{1}{1+\frac{2}{3}}},\,\,2+\frac{1}{1-\frac{1}{1-\frac{1}{3}}}\]etc.
Additional of fractions
Sum of like fractions\[=\frac{\text{sum}\,\,\text{of}\,\,\text{numerators}}{\text{sum}\,\,\text{of}\,\,\text{denominators}}\]
Example:\[\frac{3}{7}+\frac{4}{7}=\frac{3+4}{7}=\frac{7}{7}=1,\]
\[\frac{3}{8}+\frac{7}{8}=\frac{10}{8}=\frac{5}{4}=1\frac{1}{4}\]etc.
Sum of\[\frac{2}{5}\]and \[\frac{1}{3}\]
LCM of 5 and 3= 15
Now,\[\frac{2}{5}\times \frac{5}{3}=\frac{6}{15}\]and \[\frac{1\times 5}{3\times 5}=\frac{5}{15}\]
Then,\[\frac{6}{15}+\frac{5}{15}=\frac{11}{15}\]
Subtraction of lie fractions
\[=\frac{\text{Difference between the numerators}}{\text{common denominator}}\]
Examples: \[\frac{6}{5}-\frac{2}{5}=\frac{6-2}{5}=\frac{6-2}{5}=\frac{4}{5}\]
\[\frac{8}{3}-\frac{1}{3}=\frac{7}{3}=2\frac{1}{3}\] etc.
Difference of \[\frac{3}{5}\]and\[\frac{1}{2}\]
LCM of 5 and 2=10
\[\frac{3}{5}=\frac{3\times 2}{5\times 2}=\frac{6}{10}\]
\[\frac{1}{2}=\frac{1\times 5}{2\times 5}=\frac{5}{10}\]
Now, \[\frac{6}{10}-\frac{5}{10}=\frac{6-5}{10}=\frac{1}{10}\]
Multiplication of a fraction by a whole number.
\[=\frac{\text{ Numerator of fractionwhole number }\!\!~\!\!\text{ }}{\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ Denominator of the fraction}}\]
Multiply \[\frac{2}{5}\] by 4
We have, \[\frac{2}{5}\times 4=\frac{2\times 4}{5}=\frac{8}{5}=1\frac{3}{5}\]
Multiplication of a fraction by a fraction
\[\frac{\text{Product of their numerators}}{\text{Product of their denominators}}\]
Multiply \[\frac{3}{10}\] by \[\frac{7}{8}\]
We have, \[\frac{3}{10}\times \frac{7}{8}=\frac{21}{80}\]
Division of a fraction by a fraction
Example: Divide \[\frac{3}{5}\div \frac{7}{8}=\frac{3}{5}=\frac{8}{7}=\frac{24}{35}\]
Division of a fraction by a fraction
Example: Divide\[\frac{7}{8}\div 14\] \[\therefore \]\[\frac{7}{8}\times \frac{1}{14}=\frac{1}{16}\]
Decimals
Example: 0.7, 1.68, 9.357
Example: In 468.23, whole part is 468 and decimal part is 0. 23. It is read as Four Sixty eight point two three.
Example: 1. 23 have two decimal places and 1. 417 have three decimal places
Example: 5.321, 6.932, 5.834 are like decimals because of each having 3 places of decimals.
Example: 5.41, 6.232, 9.2314 are unlike decimals because of each having different number of decimals.
Equivalent Decimals
Let there be two decimal numbers having different numbers of decimal after decimal point to the number having less number of decimal places, we add appropriate number of zeros at the extreme right so that the two numbers have same number of decimal places. Then two numbers called equivalent decimals.
Example: Let 9.6 and 8.324 ne two number.
Now, 9.6 can be written as 9.600 so that 9.600 and 8.324 have both 3 decimal places.
Hence 9.600 and 8.324 are equivalent decimals
Similarly, 10.32 = 10.320
7.3 = 7.300
9.142 = 9.142
All the above decimals are equivalent decimals.
Additional of Decimal Numbers
Add 7.35 and 5.26
We have,
Subtraction of Decimal Numbers
Step I: If the given decimal numbers are unlike decimals, write them into like decimals.
Step II: Write the smaller decimal number under the larger decimal number.
Step III: Subtract as usual ignoring the decimal points.
Step IV: Finally, put the decimal point in the difference under the decimal points of the given number.
Example: Subtract 13.74 from 80.4
On converting the given numbers into like decimals we get, 13.74 and 80.40. Writing the decimals in column and on subtracting, we get,
\[\therefore \]\[80.40-13.74=66.66\]
Multiplication of Decimal Numbers by a whole number
Step I: Multiply like whole numbers, ignoring the decimals.
Step II: Count the number of decimal places in the decimal numbers
Step III: Show the same number of decimal places in the product.
Example: Multiply 6.238 by 6
First we multiply 6.238 by 6
Since given decimal number has 3 decimal places. So, the product will have 3 decimals places.
So,\[6.238\times 6=37.428\]
Multiplication of two Decimal Numbers
Example: Multiply 12.8 by 1.2
Sum of decimal places in the given decimals\[=1+1=2\]
So, place the decimal point in the product so as to have 2 decimal points
\[\therefore \]\[12.8\times 1.2=15.36\]
(a) On multiplying a decimal number by 10, the decimal point moves one place to the right \[10\times 0.212=2012,\,\,10\times 3.163=31.63.\]
(b) On multiplying a decimal number by 100, the decimal point moves two place to the right\[100\times 0.4321=43.21,\,\,100\times 7.832=783.2\].
(c) On multiplying a decimal number by\[1000\], the decimal point moves three places to the right.
\[1000\times 0.2312=231.2\]
\[1000\times 0.12=120\]
\[1000\times 7.3=7300\]
Decimal of Decimal Numbers
Consider the dividend as a whole number and perform the division, when the division of whole number part of the decimal is complete. Place the decimal point in the question and continue with the division as in the case of whole numbers.
Example: Divide 337.5 by 15
\[\therefore \,\,337.5\div 15=22.5\]
Example: \[\frac{13}{10}=1.3,\,\,\frac{151}{100}=1.51,\,\,\frac{1321}{1000}=1.321\]
Division of a decimal number by another decimal number
Example: Divided 21.97 by 1.3
\[21.97\div 13=\frac{21.97\times 10}{1.3\times 10}=\frac{219.7}{13}=16.9\]
Division of a whole numbers by a decimal number
Example: Divided 68 by 0.17
We have
\[\frac{68}{0.17}=\frac{68\times 100}{0.17\times 100}=\frac{6800}{17}=400\]
Converting a Decimal into a vulgar fraction
Example: \[0.13=\frac{13}{100},\,\,0.17=\frac{17}{100}\]etc.
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I have done question on frequency response of RLC it is easy to find whether a given circuit is high pass filter or low pass filter. But I am wondering how to determine for band pass or band reject filters. Please help me and I would be obliged if someone explain it by considering an example of a passive filter (containing all R,L,C).
A "parallel" band pass filter constructed from R,L and C has a centre frequency determined largely by the formula below: -
Fc = \$\dfrac{1}{2\pi\sqrt{LC}}\$
Given the following circuit: -
The impedance reaches a maximum at resonance and current I will only flow thru the resistor at resonance. Clearly, if R is big less current flows and if the frequency is moved away from Fc then the impedance drops rapidly. This type of circuit is used to let one frequency through (Fc) and rapidly attenuate frequencies that are not at resonance.
More typically the parallel RLC circuit looks like this (because it takes into account the biggest losses that tend to occur in the inductor): -
Now the frequency of resonance is slightly shifted from the previous formula to take into account R: -
Fc = \$\dfrac{1}{2\pi}\sqrt{\dfrac{1}{LC} - \dfrac{R^2}{L^2}}\$
The resistance in series with the coil (L) also reduces the Q of the circuit which makes the filter less peaky.
There is a lot more to these types of circuits and I'd take a look here - wiki page for RLC circuits (includes series RLC).
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This is a famous problem of intermediate analysis, also known as ‘Archimedes’ Cattle Problem Puzzle’, sent by Archimedes to Eratosthenes as a challenge to Alexandrian scholars. In it one is required to find the number of bulls and cows of each of four colors, the eight unknown quantities being connected by nine conditions. These conditions ultimately form a Pell equation which solution is necessary in case of finding the answer of the puzzle. The Greek puzzle is stated below with a little deviation. I have just tried to make the language simpler than the original, hope you’ll be able to grasp the puzzle easily.
O Stranger! If you are intelligent and wise, find the number of cattle of the Sun, who once upon a time grazed on the fields of an Island, divided into four groups (herds) of different colors, one white, another a black, a third yellow and the last dappled color.In each herd were bulls, mighty in number according to these proportions:
White bulls were equal to a half and a third of the black together with the whole of the yellow. The black bulls were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. The dappled bulls, were equal to a sixth part of the white and a seventh, together with all of the yellow.
So, these were the proportions of bulls, now the proportions of the cows were as following:
White cows were equal to the third part and a fourth of the whole herd of the black. Black cows were equal to the fourth part once more of the dappled and with it a fifth part, when all cattle, including the bulls, went to pasture together. Now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Yellow cows were in number equal to a sixth part and a seventh of the white herd.
Keeping above conditions in focus, find the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each color. But come, this solution is not complete unless you understand all these conditions regarding the cattle of the Sun:
When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth. Number of bulls in a row were equal to the number of columns. When the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colors in their midst nor none of them lacking.
Find the number of cows and bulls of each color separately.
$ W$ = number of white bulls $ B$ = number of black bulls $ Y$ = number of yellow bulls $ D$ = number of dappled bulls $ w$ = number of white cows $ b$ = number of black cows $ y$ = number of yellow cows $ d$ = number of dappled cows
The relations come as:
$ W = (\frac{1}{2} + \frac{1}{3})B + Y$ The white bulls were equal to a half and a third of the black bulls together with the whole of the yellow bulls. $ B = (\frac{1}{4} + \frac{1}{5})D + Y$ The black [bulls] were equal to the fourth part of the dappled bulls and a fifth, together with, once more, the whole of the yellow bulls $ D = (\frac{1}{6} + \frac{1}{7})W + Y$ The remaining bulls, the dappled, were equal to a sixth part of the white bulls and a seventh, together with all of the yellow bulls $ w = (\frac{1}{3} + \frac{1}{4})(B + b)$ The white cows were equal to the third part and a fourth of the whole herd of the black. $ b = (\frac{1}{4} + \frac{1}{5})(D + d)$ The black cows were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. $ d = (\frac{1}{5} + \frac{1}{6})(Y + y)$ the dappled cows in four parts [in totality] were equal in number to a fifth part and a sixth of the yellow herd. $ y = (\frac{1}{6} + \frac{1}{7})(W + w)$ the yellow cows were in number equal to a sixth part and a seventh of the white herd.
The arrangement on solving gives following relations in W,B,D,Y,w,b,d and y. which is a system of seven equations with eight unknowns. It is indeterminate, and has infinitely many solutions and form the following matrix:
6 -5 -6 0 0 0 0 0 0 20 -20 -9 0 0 0 0 -13 0 -42 42 0 0 0 0 0 -7 0 0 12 -7 0 0 0 0 0 -9 0 20 0 -9 0 0 -11 0 0 0 -11 30 -13 0 0 0 -13 0 42 0
Which yields the following solutions
W = 10,366,482k B = 7,460,514k Y = 4,149,387k D = 7,358,060k w = 7,206,360k b = 4,893,246k y = 5,439,213k d = 3,515,820k
where $ k$ is an arbitrary constant, which can be equal to either 1 or 2 or 3 … etc. Again, from the second part of the problem:
White bulls + black bulls = a square number, $ W+B=10366482k +7460514k$= a square number. or $ W+B=17,826,996k$ =a square number$ 2 \cdot 2\cdot 3 \cdot 11 \cdot 29 \cdot 4657 k = \textrm{a square number}$ . Thus $ k$ atleast be $ 3 \cdot 11 \cdot 29 \cdot 4657 $ or in general be $ 3\cdot 11\cdot 29 \cdot 4657 \cdot r^2=4456749r^2$ where $ r$ is any integer. Again, Dappled bulls + yellow bulls = a triangular number.or, $ Y + D = \textrm{a triangular number}$ where triangular numbers are numbers of the form $ 1 + 2 + 3 + 4 + 5 + \ldots + m =\frac{m(m+1)}{2}$ . where $ m$ is some positive integer. Thus $ 4,149,387k + 7,358,060k =\frac{m(m+1)}{2}$ or $ 11,507,447k =\frac{m(m+1)}{2}$ . Putting $ k=4456749 r^2$ we have $ 11,507,447 \times 4,456,749 r^2 = \frac{m(m+1)}{2}$ or $ 102,571,605,819,606 r^2 = m(m + 1)$ . The problem is now to find the values of $ r$ and $ m$ that we can find the value of $ k$ and thus the solution of the problem. The computer generated answers for smallest solutions are at my Pastebin Account. Recently, Ilan Vardi of Occidental College (Los Angeles, California, USA) developed simple explicit formulas to generate solutions to the cattle problem.Click here to read his paper on the cattle problem.
References and Further Readings: Weisstein, Eric W. “Archimedes’ Cattle Problem.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/ArchimedesCattleProblem.html Archimedes’s Cattle Problem http://en.wikipedia.org/wiki/Archimedes%27_cattle_problem Archemedes’s Cattle Problem http://math.nyu.edu/~crorres/Archimedes/Cattle/Statement.html The Archemedes’s Cattle Problem http://www.maa.org/devlin/devlin_02_04.html
Feel free to ask questions, send feedback and even point out mistakes. Great conversations start with just a single word.
How to write better comments?
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Lengthy post ahead! Look toward the end for a quick recap.
The relationship between energy and temperature is a little more complicated than what you sketched here.
Boltzmann Distribution
The atoms in a gas don't all have the same amount of energy. In fact, their energies greatly vary. It is true that temperature is the measure of the average energy of the atoms. But temperature can be more precisely defined in terms of how this energy is actually distributed.
There is a distribution of energy known as the Boltzmann-distribution. It says that there is a reverse exponential relationship between the energy of an atom (or whatever thing you are considering) and the chances of finding an atom with that energy. So the chance of finding an atom with an energy in an interval $dE$ arround $E$ equals:
$$P(E)dE=A\cdot e^{-\beta E}dE$$
Where $\beta$ is some positive factor of proportionality. $A$ is chosen such that the total chance of finding an atom at any energy is $1$.
We already know that $\beta$ will be a function of the temperature $T$. We know that a higher temperature means more energetic atoms. So the higher the temperature, the better the chances of finding an atom with high energy. This means that a high $T$, must mean a low $\beta$. A possible expression for $\beta$ would therefore be:
$$\beta=\frac{1}{k_BT}$$
Where $k_B$ is some positive factor of proportionality, known as Boltzmann's constant.
In fact, the above equation is the
definition of temperature.
Now then, we have an equation relating energy and temperature, but its factor isn't $R$, it's $k_B$. What's going on here? Well, the above equations talk about the energy of a single atom, which isn't the kind of scale we usually talk about. Therefore, we have also defined a factor $R$, which talks about the energy of a mole of gas:
$$R=N_Ak_B$$
Where $N_A$ is the number of particles in a mole, known as Avogrado's constant.
But that still doesn't help us with the factor of $\frac{3}{2}$!
Equipartition of Energy
To explain this, let's do a little though experiment:
Imagine a cubic gas chamber. On one side of this gas chamber we have a very thin, but long, tube. The tube is so thin, that the gas atoms in this tube can only move forward and backward through the tube, not up and down or left and right. On the other hand, the gas atoms in the cubic part of the chamber can move in all three directions. The gas in the tube and in the chamber are in thermal contact. That means they have the same temperature. Now let's think about how both groups of atoms are behaving.
Let's imagine for a moment that the atoms in the cubic part of the chamber are doing their normal thing, all moving at different speeds, but with a well defined average temperature, in accordance with the Boltzmann-distribution. Now lets imagine that the atoms in the tube are sitting totally still. At some point, an atom from the cubic chamber will hit one of the atoms in the tube. Let's pretend the moving atom gives off exactly half its energy tot the idle atom. The idle atom probably wants to move in some diagonal direction, but it can only move forward and backward. While bouncing off of the tube's walls the atom will start to lose its velocity in the sideways and up- and downwards directions, and instead start to move forward and backward more quickly. Over time, the same happens to all of the other atoms in the tube.
The atoms in the tube now all have the same energy as the atoms in the cubic chamber. But all this energy is 'channeled' into just one direction (let's say the x-direction), whereas the energy of the other atoms is distributed equally over all three directions. In other words, the x-velocity of the atoms in the tube, is much greater than the x-velocity of the atoms in the cubic chamber.
Now when an atom from the cubic chamber hits an atom from the tube, it will get a push in the x-direction. Because of this, the cube-atom gains some energy, and the tube-atom loses some. This will keep happening, until the x-velocity of the atoms is the same in both parts of the gas chamber. The entire system must eventually get to this state of equilibrium.
But now the energy isn't evenly distributed between the atoms! Instead the energy is evenly distributed among the different directions atoms can move along. We call these directions
degrees of freedom. The atoms in the cubic chamber have three degrees of freedom, the atoms in the tube just one. The fact that energy is evenly distributed over different degrees of freedom is called the Equipartition Theorem. It is entirely possible to have more than three degrees of freedom. For example, complex molecules do not only move through the chamber, they also rotate and vibrate in all sorts of ways. These are all degrees of freedom, and energy will be distributed among them equally. But in the case of a mono-atomic gas, there are exactly three degrees of freedom, each getting their fair share of energy.
This explains the $3$ in $\frac{3}{2}$, but how about the $\frac{1}{2}$?
Average Energy
This half comes falling out when you apply all of this knowledge and calculate the integral to actually find the average energy. At this point, I'm not really willing to get into the algebra, but perhaps this is a more intuitive explanation:
To find the average energy $\langle E\rangle $, we add the energies in each of the degrees of freedom. So we split up the velocity $\mathscr{v}$ into its components $v_x$, $v_y$ and $v_z$.
$$\mathscr{v}=\sqrt{v_x^2+v_y^2+v_z^2}$$
$$\langle E\rangle =\langle \frac{1}{2}m\mathscr{v}^2\rangle =\langle \frac{1}{2}m(v_x^2+v_y^2+v_z^2)\rangle =\frac{1}{2}m(\langle v_x^2\rangle +\langle v_y^2\rangle +\langle v_z^2\rangle )$$
Now, because of the equipartition theorem we know that $\langle v_x^2\rangle =\langle v_y^2\rangle =\langle v_z^2\rangle $, let's call it $\langle v^2 \rangle$. So:
$$\langle E\rangle=\frac{3}{2}m\langle v^2 \rangle$$
Calculating $\langle v^2 \rangle$ is not so simple. But I think it now looks a lot more likely that this equation will come out to be $\frac{3}{2}k_BT$. If you then multiply by $N$ to get the total energy in the gas, you get:
$$U=N\langle E \rangle = N\cdot \frac{3}{2}k_b T = \frac{3}{2} nRT$$
Summary
To recap:
Temperature is defined not in terms of the average energy, but the distribution of energy. Energy distributes equally between all degrees of freedom. A simple mono-atomic gas has three degrees of freedom. This gives rise to the factor $3$. The factor $\frac{1}{2}$ has to do with the translation of the energy distribution to the average energy. An intuitive way to think of this is to think of the factor $\frac{1}{2}$ in $\frac{1}{2}mv^2$.
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To round off our series on round objects (see the first and second posts), let’s compute the sphere’s surface area. We can compute this in the same way we related the area and circumference of a circle two weeks ago. Approximate the surface of the sphere with lots of small triangles, and connect these to the center of the sphere to create lots of triangular pyramids. Each pyramid has volume \(\frac{1}{3}(\text{area of base})(\text{height})\), where the heights are all nearly \(r\) and the base areas add to approximately the surface area. By using more and smaller triangles these approximations get better and better, so the volume of the sphere is $$\frac{4}{3}\pi r^3 = \frac{1}{3}(\text{surface area})\cdot r,$$ meaning the surface area is \(4\pi r^2\). (This and previous arguments can be made precise with the modern language of integral calculus.)
Here’s an elegant way to rephrase this result: The surface area of a sphere is equal to the area of the curved portion of a cylinder that exactly encloses the sphere. In fact, something very surprising happens here!:
Archimedes’ Hat-Box Theorem: If we draw any two horizontal planes as shown below, then the portions of the sphere and the cylinder between the two planes have the same surface area.
We can prove this with (all!) the methods in the last few posts; here’s a quick sketch. To compute the area of the “spherical band” (usually called a spherical zone), first consider the solid spherical sector formed by joining the spherical zone to the center:
By dividing this into lots of triangular pyramids as we did with the sphere above, we can compute the area of the spherical zone by instead computing the sector’s volume. This volume can be computed by breaking it into three parts: two cones and the spherical segment between the two planes (on the left of the next figure). Compute the volume of the spherical segment by comparing (via Cavalieri’s Principle) to the corresponding part of the vase (from the previous post), which can be expressed with just cylinders and cones.
Every geometry textbook has formulas for the circumference (\(C = 2 \pi r\)) and area (\(A = \pi r^2\)) of a circle. But where do these come from? How can we prove them?
Well, the first is more a definition than a theorem: the number \(\pi\) is usually defined as the ratio of a circle’s circumference to its diameter: \(\pi = C/(2r)\). Armed with this, we can compute the area of a circle. Archimedes’ idea (in 260 BCE) was to approximate this area by looking at regular \(n\)-sided polygons drawn inside and outside the circle, as in the diagram below. Increasing \(n\) gives better and better approximations to the area.
Look first at the inner polygon. Its perimeter is slightly less than the circle’s circumference, \(C = 2 \pi r\), and the height of each triangle is slightly less than \(r\). So when reassembled as shown, the triangles form a rectangle whose area is just under \(C/2\cdot r = \pi r^2\). Likewise, the outer polygon has area just larger than \(\pi r^2\). As \(n\) gets larger, these two bounds get closer and closer to \(\pi r^2\), which is therefore the circle’s area.
Archimedes used this same idea to approximate the number \(\pi\). Not only was he working by hand, but the notion of “square root” was not yet understood well enough to compute with. Nevertheless, he was amazingly able to use 96-sided polygons to approximate the circle! His computation included impressive dexterity with fractions: for example, instead of being able to use \(\sqrt{3}\) directly, he had to use the (very close!) approximation \(\sqrt{3} > 265/153\). In the end, he obtained the bounds \( 3\frac{10}{71} < \pi < 3\frac{1}{7} \), which are accurate to within 0.0013, or about .04%. (In fact, he proved the slightly stronger but uglier bounds \(3\frac{1137}{8069} < \pi < 3\frac{1335}{9347}\). See this translation and exposition for more information on Archimedes’ methods.)
These ideas can be pushed further. Focus on a circle with radius 1. The area of the regular \(n\)-sided polygon inscribed in this circle can be used as an approximation for the circle’s area, namely \(\pi\). This polygon has area \(A_n = n/2 \cdot \sin(360/n)\) (prove this!). What happens when we double the number of sides? The approximation changes by a factor of $$\frac{A_{2n}}{A_n} = \frac{2\sin(180/n)}{\sin(360/n)} = \frac{1}{\cos(180/n)}.$$ Starting from \(A_4 = 2\), we can use the above formula to compute \(A_8,A_{16},A_{32},\ldots\), and in the limit we find that $$\pi = \frac{2}{\cos(180/4)\cdot\cos(180/8)\cdot\cos(180/16)\cdots}.$$ Finally, recalling that \(\cos(180/4) = \cos(45) = \sqrt{\frac{1}{2}}\) and \(\cos(\theta/2) = \sqrt{\frac{1}{2}(1+\cos\theta)}\) (whenever \(\cos(\theta/2) \ge 0\)), we can rearrange this into the fun infinite product $$\frac{2}{\pi} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}} \cdots$$ (which I found at Mathworld). (It’s ironic that this formula for a circle uses so many square roots!)
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[This is the 6th post in the current series about Wythoff’s game: see posts #1, #2, #3, #4, and #5.
Caveat lector: this post is a bit more difficult than usual. Let me know what you think in the comments!]
Our only remaining task from last week was to prove the mysterious Covering Theorem: we must show that there is exactly one dot in each row and column of the grid (we already covered the diagonal case). Since the rows and columns are symmetric, let’s focus on columns.
The columns really only care about the
x-coordinates of the points, so let’s draw just these x-coordinates on the number-line. We’ve drawn \(\phi,2\phi,3\phi,\ldots\) with small dots and \(\phi^2,2\phi^2,3\phi^2,\ldots\) with large dots. We need to show that there’s exactly one dot between 1 and 2, precisely one dot between 2 and 3, just one between 3 and 4, and so on down the line. For terminology’s sake, break the number line into length-1 intervals [1,2], [2,3], [3,4], etc., so we must show that each interval has one and only one dot:
Why is this true? One explanation hinges on a nice geometric observation: Take any small dot
s and large dot t on our number-line above, and cut segment st into two parts in the ratio \(1:\phi\) (with s on the shorter side). Then the point where we cut is always an integer! For example, the upper-left segment in the diagram below has endpoints at \(s=2\cdot\phi\) and \(t=1\cdot\phi^2\), and its cutting point is the integer 3:
In general, if
s is the jth small dot—i.e., \(s=j\cdot\phi\)—and \(t=k\cdot\phi^2\) is the kth large dot, then the cutting point between s and t is \(\frac{1}{\phi}\cdot s+\frac{1}{\phi^2}\cdot t = j+k\) (Why?! [1]). But more importantly, this observation shows that no interval has two or more dots: a small dot and a large dot can’t be in the same interval because they always have an integer between them! [2]
So all we have to do now is prove that no interval is
empty: for each integer n, some dot lies in the interval [ n, n+1]. We will prove this by contradiction. What happens if no dot hits this interval? Then the sequence \(\phi,2\phi,3\phi,\ldots\) jumps over the interval, i.e., for some j, the jth dot in the sequence is less than n but the ( j+1)st is greater than n+1. Likewise, the sequence \(\phi^2,2\phi^2,3\phi^2,\ldots\) jumps over the interval: its kth dot is less than n while its ( k+1)st dot is greater than n+1:
By our observation above on segment \(s=j\phi\) and \(t=k\phi^2\), we find that the integer
j+ k is less than n, so \(j+k\le n-1\). Similarly, \(j+k+2 > n+1\), so \(j+k+2 \ge n+2\). But together these inequalities say that \(n\le j+k\le n-1\), which is clearly absurd! This is the contradiction we were hoping for, so the interval [ n, n+1] is in fact not empty. This completes our proof of the Covering Theorem and the Wythoff formula!
It was a long journey, but we’ve finally seen exactly why the Wythoff losing positions are arranged as they are. Thank you for following me through this!
A Few Words on the Column Covering Theorem
Using the
floor function \(\lfloor x\rfloor\) that rounds x down to the nearest integer, we can restate the Column Covering Theorem in perhaps a more natural context. The sequence of integers $$\lfloor\phi\rfloor = 1, \lfloor 2\phi\rfloor = 3, \lfloor 3\phi\rfloor = 4, \lfloor 4\phi\rfloor = 6, \ldots$$ is called the Beatty sequence for the number \(\phi\), and similarly, $$\lfloor\phi^2\rfloor = 2, \lfloor 2\phi^2\rfloor = 5, \lfloor 3\phi^2\rfloor = 7, \lfloor 4\phi^2\rfloor = 8,\ldots$$ is the Beatty sequence for \(\phi^2\). Today we proved that these two sequence are complementary, i.e., together they contain each positive integer exactly once. We seemed to use very specific properties of the numbers \(\phi\) and \(\phi^2\), but in fact, a much more general theorem is true: Beatty’s Theorem: If \(\alpha\) and \(\beta\) are any positive irrational numbers with \(\frac{1}{\alpha}+\frac{1}{\beta}=1\), then their Beatty sequences \(\lfloor\alpha\rfloor, \lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\ldots\) and \(\lfloor\beta\rfloor, \lfloor 2\beta\rfloor, \lfloor 3\beta\rfloor,\ldots\) are complementary sequences.
Furthermore, our same argument—using \(\alpha\) and \(\beta\) instead of \(\phi\) and \(\phi^2\)—can be used to prove the more general Beatty’s Theorem!
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Answer
$f=64.3249482 \dfrac{\sqrt T}{Ld}$
Work Step by Step
General equation for inverse variation is given by $y=\dfrac{k}{x}$ Here, we have $f=k \dfrac{\sqrt T}{Ld}$ ...(1) Plug the data, we have $262=k \dfrac{\sqrt {670}}{(62)(0.1025)}$ and $K \approx 64.3249482$ Thus, the equation (1) becomes: $f=64.3249482 \dfrac{\sqrt T}{Ld}$
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To round off our series on round objects (see the first and second posts), let’s compute the sphere’s surface area. We can compute this in the same way we related the area and circumference of a circle two weeks ago. Approximate the surface of the sphere with lots of small triangles, and connect these to the center of the sphere to create lots of triangular pyramids. Each pyramid has volume \(\frac{1}{3}(\text{area of base})(\text{height})\), where the heights are all nearly \(r\) and the base areas add to approximately the surface area. By using more and smaller triangles these approximations get better and better, so the volume of the sphere is $$\frac{4}{3}\pi r^3 = \frac{1}{3}(\text{surface area})\cdot r,$$ meaning the surface area is \(4\pi r^2\). (This and previous arguments can be made precise with the modern language of integral calculus.)
Here’s an elegant way to rephrase this result: The surface area of a sphere is equal to the area of the curved portion of a cylinder that exactly encloses the sphere. In fact, something very surprising happens here!:
Archimedes’ Hat-Box Theorem: If we draw any two horizontal planes as shown below, then the portions of the sphere and the cylinder between the two planes have the same surface area.
We can prove this with (all!) the methods in the last few posts; here’s a quick sketch. To compute the area of the “spherical band” (usually called a spherical zone), first consider the solid spherical sector formed by joining the spherical zone to the center:
By dividing this into lots of triangular pyramids as we did with the sphere above, we can compute the area of the spherical zone by instead computing the sector’s volume. This volume can be computed by breaking it into three parts: two cones and the spherical segment between the two planes (on the left of the next figure). Compute the volume of the spherical segment by comparing (via Cavalieri’s Principle) to the corresponding part of the vase (from the previous post), which can be expressed with just cylinders and cones.
Last week we saw how to compute the area of a circle from first principles. What about spheres?
To compute the volume of a sphere, let’s show that a hemisphere (with radius \(r\)) has the same volume as the vase shown in the figure below, formed by carving a cone from the circular cylinder with radius and height \(r\). Why this shape? Here’s why: if we cut these two solids at any height \(h\) (between 0 and \(r\)), the areas of the two slices match. Indeed, the slice—usually called cross section—of the sphere is a circle of radius \(\sqrt{r^2-h^2}\), which has area \(\pi(r^2-h^2)\). Similarly, the vase’s cross section is a radius \(r\) circle with a radius \(h\) circle cut out, so its area is \(\pi r^2-\pi h^2\), as claimed.
If we imagine the hemisphere and vase as being made from lots of tiny grains of sand, then we just showed, intuitively, that the two solids have the same number of grains of sand in every layer. So there should be the same number of grains in total, i.e., the volumes should match. This intuition is exactly right:
Cavalieri’s Principle: any two shapes that have matching horizontal cross sectional areas also have the same volume.
So the volumes are indeed equal, and all that’s left is to compute the volume of the vase. But we can do this! Recall that the cone has volume \(\frac{1}{3} (\text{area of base}) (\text{height}) = \frac{1}{3}\pi r^3\) (better yet, prove this too! Hint: use Cavalieri’s Principle again to compare to a triangular pyramid). Likewise, the cylinder has volume \((\text{area of base}) (\text{height}) = \pi r^3\), so the vase (and hemisphere) have volume \(\pi r^3 – \frac{1}{3} \pi r^3 = \frac{2}{3}\pi r^3\). The volume of the whole sphere is thus \(\frac{4}{3}\pi r^3\). Success!
The following visualization illustrates what we have shown, namely $$\text{hemisphere} + \text{cone} = \text{cylinder}.$$ The “grains of sand” in the hemisphere are being displaced horizontally by the stabbing cone, and at the end we have exactly filled the cylinder.
If you put a quarter flat on a table, you can surround it with six other quarters that all touch the original. Is this the best you can do, if the coins can’t overlap? Well, yes: the coins are packed as tightly as possible, so there is no room for a seventh coin to fit around the middle one. We call this the Kissing Number Problem, and it gets much more interesting if you jump from two dimensions to three: given a sphere of radius 1, how many nonoverlapping spheres (of radius 1) can you place around it such that they all “kiss” the original sphere? Said differently, what is the kissing number in 3 dimensions?
Here’s a pretty good attempt: If you put one sphere at each vertex of a cuboctahedron like this,
you’ll end up with 12 that seem pretty tightly packed. In another try, you could put a sphere at each vertex of an icosahedron:
You’ll again get 12, but this time there is noticable wiggle room to move the spheres without losing contact with the central sphere—so much wiggle room, in fact, that any two spheres can switch places! Is there enough extra space for a 13th sphere to fit? This was a topic of heated debate between Isaac Newton and David Gregory (who believed 12 and 13, respectively), but it was not resolved until much later, in 1953, in favor of Newton’s 12. The proof is highly nontrivial.
What about still higher dimensions? What is the kissing number in four dimensions? (What does a 4-dimensional sphere even look like? Perhaps a topic of a future post…) This too is known, and the answer is 24—the proof is again quite difficult. See if you can figure out the arrangement! What about dimension 5, 6, 7, … ? As might be expected, the problem gets even harder in higher dimensions, and no other exact kissing numbers are known—except for two. Due to some veritable mathematical miracles (namely, the highly symmetric structures called the E8 lattice and the Leech lattice), the kissing numbers in dimensions 8 and 24 have been shown to be 240 and 196560, respectively.
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I'm a student in electrical engineering in Belgium and I use a multiple feedback bandpass filter in one of my projects.
I tried to find out the transfer function of this filter and this is what I found:
$$ \frac{V_o}{V_i}=-\left( j\omega C+ \frac{2}{R_2}+\frac{R_1+R_3}{R_1R_2R_3}\cdot\frac{1}{j\omega C} \right) ^{-1} $$
With this expression I made a Bode diagram:
Is that normal that the frequencies other than 1kHz are still amplified (not as much but still) ? Is my transfer function right ?
Thanks !
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Lasted edited by Andrew Munsey, updated on June 15, 2016 at 2:04 am.
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The Lorentz force can act on a current carrying conductor, in this case called
There was an error working with the wiki: Code[17], by the interaction of the conduction electrons with the atoms of the conductor material. This force is used in many devices including :
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The particle will experience a force due to electric field of q'E, and due to the magnetic field q'v × B. Combined they give the
There was an error working with the wiki: Code[4] force equation (or law):
: {F} = q ({E} + {v} \times {B}),
where
:F is the Force (in
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:B is the
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:q is the Electric charge of the particle (in
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:v is the instantaneous Velocity of the particle (in meters per
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:and × is the
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Thus a positively charged particle will be accelerated in the same linear orientation as the E field, but will curve perpendicularly to the B field according to the
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Alternative form
Equivalently, we can express the Lorentz force law in terms of the electric
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:{F} = \int_V ( \rho {E} + {J} \times {B}) dV
Lorentz force in special relativity
When particle speeds approach the speed of light, the Lorentz force equation must be modified according to
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: {d \left ( \gamma m {v} \right ) \over dt } = {F} = q ({E} + {v} \times {B}),
where
:\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}
is called the
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This expression differs from the expression obtained from the Lorentz force by a factor of \gamma .
The change of energy due to the fields is
: {d \left ( \gamma m c^2 \right ) \over dt } = q {E} \cdot {v} .
Covariant form of the Lorentz force
The Lorentz force equation can be written in
There was an error working with the wiki: Code[7] in terms of the field strength tensor (cgs units).
: m c { d u^{\alpha} \over { d \tau } } = { {} \over {} }F^{\alpha \beta} q u_{\beta}
where m is the particle
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: u_{\beta} = \eta_{\beta \alpha } u^{\alpha } = \eta_{\beta \alpha } { d x^{\alpha } \over {d \tau} }
is the
There was an error working with the wiki: Code[9] of the particle. Here, \tau is c times the
There was an error working with the wiki: Code[29] of the particle and \eta is the
There was an error working with the wiki: Code[30] tensor. The fields can be transformed to a frame moving with constant relative velocity by:
: \acute{F}^{\mu \nu} = {\Lambda^{\mu}}_{\alpha} {\Lambda^{\nu}}_{\beta} F^{\alpha \beta}
,
where {\Lambda^{\mu}}_{\alpha} is a
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General internet
There was an error working with the wiki: Code[1], Wikipedia: The Free Encyclopedia. Wikimedia Foundation.
news://sci.physics Physical laws, properties, etc. Activity: High
news://sci.physics.electromag Electromagnetic theory and applications. Activity: Medium
news://sci.physics.relativity The theory of relativity. Activity: High
Books
Serway and Jewett, "Physics for Scientists and Engineers with Modern Physics" Thomson Brooks/Cole, 2004 ISBN 0-534-40846-X
Andre Koch and Torres Assis, "Weber's Electrodynamics" (Lorentz force laws), Springer, 1994. ISBN 0792331370
H.C. Sharma, Rajesh Kharb, Rajesh Sharma, "Comprehensive Physics for Engineers" (Lorentz force and magnetic induction). Laxmi Publications, 2005.
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One shouldn't imagine the T-duality between the two heterotic strings to be a $Z_2$ group, like in the case of type II string theories' T-duality. In type II string theory, there is only one relevant scalar field, the radius of the circle producing T-duality, and it gets reverted $R\to 1/R$ under T-duality.
In the heterotic case, it's more complicated because more scalar fields participate in the T-duality. Instead of a $Z_2$ map acting on one scalar field, one must correctly adjust the moduli, especially the Wilson lines generically breaking the 10D gauge group to $U(1)^{16}$, and find an identification between the points of the moduli space of the two heterotic string theories: there is one theory at the end.
A fundamental reason why the T-duality holds is that one may define the heterotic string theories in the bosonic representation, using 16-dimensional lattice $\Gamma^{16}$ which is the weight lattice of $Spin(32)/Z_2$, and $\Gamma^{8}\oplus \Gamma^8$ which is the root lattice of $E_8\times E_8$.
One may also describe the compactification on a circle in terms of lattices. It corresponds to adding ($\oplus$) the lattice $\Gamma^{1,1}$ of the indefinite signature to the original lattice. The extra 1+1 dimensions correspond to the compactified left-moving and right-moving boson (of the circle), respectively.
Now, the key mathematical fact is that the 17+1-dimensional even self-dual lattice exists and is unique which really means$$\Gamma^{16}\oplus \Gamma^{1,1} = \Gamma^{8}\oplus \Gamma^8\oplus \Gamma^{1,1}$$Even self-dual lattices in $p+q$ dimensions (signature) exist whenever $p-q$ is a multiple of eight and if both $p$ and $q$ are nonzero, the lattice is unique.
It's unique up to an isometry – a Lorentz transformation of a sort – and that's how the identity above should be understood, too. So there is a way to linearly redefine the 17+1 bosons on the heterotic string world sheet so that a basis that is natural for the $E_8\times E_8$ heterotic string gets transformed to the $Spin(32)/Z_2$ string or vice versa. The compactified boson has to be nontrivially included in the transformation – the 17+1-dimensional Lorentz transformation that makes the T-duality manifest mixes the 16 chiral bosons with the 1+1 boson from the compactified circle.
A different derivation of the equivalence may be found e.g. in Polchinski's book. One may start with one of the heterotic strings and carefully adjust the Wilson lines to see that at a special point, the symmetry broken to $U(1)^{17+1}$ is enhanced once again to the other gauge group.This post imported from StackExchange Physics at 2014-03-07 13:40 (UCT), posted by SE-user Luboš Motl
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I have come across two similar definitions of
primary fields in conformal field theory. Depending on what I am doing each definition has its own usefulness. I expect both definitions to be compatible but I can't seem to be able to show it. By compatible I mean definition 1 $\iff$ definition 2. I will write both definitions in the two-dimensional case and restricting to holomorphic transformations.
Def #1 from D'Francesco et al's CFT: A field $f(z)$ if it transforms as $f(z) \rightarrow g(\omega)=\left( \frac{d\omega}{dz}\right)^{-h}f(z), h\in\mathbb{R}$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$.
Def #2 from Blumenhagen et al's Intro to CFT: A field $f(z)$ is primary if it transforms as $f(z) \rightarrow g(z)=\left( \frac{d\omega}{dz}\right)^{h}f(\omega), h\in\mathbb{R}$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$.
This post imported from StackExchange MathOverflow at 2014-08-31 09:09 (UCT), posted by SE-user Daniel
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I have been reviewing general equilibrium models and was trying to find an efficient method for computing the core of a cooperative game. I was taught this topic in a very poor way so I believe I still have some conceptual errors.
Here is a thought I had:
Suppose we are in an economy with three consumers, $A$, $B$, and $C$, with utility $u_{i}(x)$ defined over bundles $x \in \mathbb{R}^{2}$ and endowments $\omega_{i}$ for $i = A, B ,C$. I want to compute the core for this economy.
I know the core must satisfy: \begin{align} u_{A}(x_{A}) &\geq u_{A}(\omega_{A})\\ u_{B}(x_{B}) &\geq u_{B}(\omega_{B})\\ u_{C}(x_{C}) &\geq u_{C}(\omega_{C})\\ \end{align} i.e. the core must be individually rational. So let $$D =\{ x \in \mathbb{R}^{2} : x \text{ is individually rational for $A$, $B$ and $C$} \}$$ I also know that the core is a subset of the pareto efficient outcomes, so let $$E =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient} \}$$ Now here is the part I am not sure about: I know that the core is also not blocked by any two-person coalition. I think this means that any core allocation is pareto efficient for any two-person game. Thus I define: \begin{align} F_{1} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $A$ and $B$ } \}\\ F_{2} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $A$ and $C$ } \}\\ F_{3} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $B$ and $C$ } \} \end{align}
Here are my questions:
Is the above analysis correct? Can I write the set of core allocations $\mathcal{C}$ as $$\mathcal{C} = D \cap E \cap F_{1} \cap F_{2} \cap F_{3}\text{?}$$ Can this method of solving be generalized to a game with $n$ players and $m$ goods?
Let me know if anything is not clear!
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Now showing items 1-10 of 27
Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider
(American Physical Society, 2016-02)
The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has been studied using the ALICE detector at the LHC. The spectra exhibit a significant hardening with ...
Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV
(Elsevier, 2016-02)
Two-particle angular correlations between trigger particles in the forward pseudorapidity range ($2.5 < |\eta| < 4.0$) and associated particles in the central range ($|\eta| < 1.0$) are measured with the ALICE detector in ...
Measurement of D-meson production versus multiplicity in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
(Springer, 2016-08)
The measurement of prompt D-meson production as a function of multiplicity in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV with the ALICE detector at the LHC is reported. D$^0$, D$^+$ and D$^{*+}$ mesons are reconstructed ...
Measurement of electrons from heavy-flavour hadron decays in p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV
(Elsevier, 2016-03)
The production of electrons from heavy-flavour hadron decays was measured as a function of transverse momentum ($p_{\rm T}$) in minimum-bias p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV with ALICE at the LHC for $0.5 ...
Direct photon production in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV
(Elsevier, 2016-03)
Direct photon production at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV was studied in the transverse momentum range $0.9 < p_{\rm T} < 14$ GeV/$c$. Photons were detected via conversions in the ALICE ...
Multi-strange baryon production in p-Pb collisions at $\sqrt{s_\mathbf{NN}}=5.02$ TeV
(Elsevier, 2016-07)
The multi-strange baryon yields in Pb--Pb collisions have been shown to exhibit an enhancement relative to pp reactions. In this work, $\Xi$ and $\Omega$ production rates have been measured with the ALICE experiment as a ...
$^{3}_{\Lambda}\mathrm H$ and $^{3}_{\bar{\Lambda}} \overline{\mathrm H}$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Elsevier, 2016-03)
The production of the hypertriton nuclei $^{3}_{\Lambda}\mathrm H$ and $^{3}_{\bar{\Lambda}} \overline{\mathrm H}$ has been measured for the first time in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV with the ALICE ...
Multiplicity dependence of charged pion, kaon, and (anti)proton production at large transverse momentum in p-Pb collisions at $\sqrt{s_{\rm NN}}$= 5.02 TeV
(Elsevier, 2016-09)
The production of charged pions, kaons and (anti)protons has been measured at mid-rapidity ($-0.5 < y < 0$) in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV using the ALICE detector at the LHC. Exploiting particle ...
Jet-like correlations with neutral pion triggers in pp and central Pb–Pb collisions at 2.76 TeV
(Elsevier, 2016-12)
We present measurements of two-particle correlations with neutral pion trigger particles of transverse momenta $8 < p_{\mathrm{T}}^{\rm trig} < 16 \mathrm{GeV}/c$ and associated charged particles of $0.5 < p_{\mathrm{T}}^{\rm ...
Centrality dependence of charged jet production in p-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 5.02 TeV
(Springer, 2016-05)
Measurements of charged jet production as a function of centrality are presented for p-Pb collisions recorded at $\sqrt{s_{\rm NN}} = 5.02$ TeV with the ALICE detector. Centrality classes are determined via the energy ...
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I am attempting to investigate transformations between two distinct sets of vertices on n-dimensional manifolds with a minimal change in the fundamental shape of the vertices. I will give some background so that when my question is asked it will be fully understood, the introductory explanation describes how the metric for configuration space is generated, and then my first question is in regards to computing explicit examples and minimizing my procedure. My second question is in regards to finding constraint conditions for my functional $L$, with the Euler-Lagrange equation, or something else. My main question though, is how to investigate the same type of geodesic equation in configuration space when the manifold is not endowed with a simple Pythagorean metric. Given an n-dimensional manifold with any metric, I want to find a way to compute the geodesic for a set of $I$-vertices traveling from one position to another. If we then consider these vertices as charges of masses, we have just found a way of minimizing the change in the electric field potential or gravitational field potential given any motion whatsoever.
NOTE: Do a quick check of the questions to see if you happen to know the answer right away in order to avoid reading something irrelevant if you are familiar with this type of stuff.
Part 1
The non-abelian similarity group $G_{\phi}$ with $\phi$ generations is given by, $$d + \frac{d(d-1)}{2} + 1$$where $d$ is the dimension as given by the translation $T^{d}$ and rotation $SO(d)$ subgroups with $+1$ for the constant scaling matrix $k$. This gives $\nu = dn - \phi$ shape degrees of freedom where $n$ is the number of vertices. My first approach to understanding the transformations between two distinct sets of vertices on an n-dimensional manifold is when there is no curvature and the system is in 2-dimensions. This process must be split into two explicit stages, the first defines the vertices $q_{i}(x_{i},y_{i})$ undergoing all possible gauge transformations to end up with an arbitrary system $f_{i}(g_{i},h_{i})$. The second minimizes the distance between $f_{i}(g_{i},h_{i})$ and $p_{i}(w_{i},z_{i})$, resulting in $\bar{q}(k_{i},j_{i})$. Representing the operations of rotation and translation explicitly with matrices we get that for the translation vector $\begin{pmatrix} c & d \end{pmatrix}$, $$\begin{pmatrix} g_{i} & h_{i} \end{pmatrix} = \begin{pmatrix} x_{i}k\cos(\theta) + y_{i}k\sin(\theta) + c & -x_{i}k\sin(\theta) + y_{i}k\cos(\theta) + d \end{pmatrix}$$The result is the following function which can, given explicit $(x,y)$ coordinates, be simplified to give a solution by Mathematica, $$f(\theta,k,c,d) = \sum_{i=1}^{n} (w_{i} - g_{i})^2 + (z_{i} - h_{i})^2$$ So, we have that our metric $\omega$ is given by, $$\omega = \sum\limits_{i=1}^{n} \sqrt{(k_{i} - w_{i})^2 + (j_{i} - z_{i})^2}$$
Question 1:How would I compute this procedure explicitly with say a $47$-dimensional system of particles? Could someone reference me a paper or give an explanation of how to find a general $n$-dimensional rotation matrix? Everything else seems possible in principle, as Mathematica can likely compute $47$ partial derivatives concurrently.
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Generalizing, the gauge transformations done to a system of vertices $q_{I}^{i}$ in configuration space, where $I$ is the number of vertices, and $i$ is the number of dimensions is:$$q_{I}^{i} \rightarrow \sum_{j=1}^{d} s R_{j}^{i}(q_{I}^{j} + a^{j})$$Where $s$ is the scaling matrix, $R_{j}^{i}$ is the rotation matrix, $a^{j}$ is the translation row vector, and $j$ is an index defining the dimension. So, we then have that for the group action $G(\phi)$, the explicit process in 2-dimensions described above can be represented in n-dimensions as $$\min|G(\phi(\lambda + \delta \lambda))(q_{I}^{j}(\lambda + \delta \lambda)) - p_{I}^{i}|$$To illustrate, this process in 3-dimensions is shown by the following figure:
With some effort, the following functional that gives the length of all possible paths between two distinct sets of $I$-vertices in $n$-dimensional space is as follows,$$L = \int\limits_{\lambda_{1}}^{\lambda_{2}} \sqrt{\sum_{I=1}^{n} \sum_{i=1}^{d}\left(\frac{d}{d\lambda}\left(\sum_{j=1}^{d} s(\lambda)R_{j}^{i}(\lambda)(q_{I}^{j}(\lambda) + a^{j}(\lambda))\right)\right)^2} d\lambda$$
So, we have a system of vertices $q_{I}^{j}$ parametrized by $\lambda$ and we want to find when this functional is stationary. Graphically, we can think of this as the functional which when minimized by the Euler-Lagrange equation gives (with a standard pythagorean metric) the straight line distance between two systems of vertices (points in configuration space) as,
Question 2:How can I, in practice, use the Euler-Lagrange equation to find the constraint conditions for when the functional is stationary? I have tried many different approaches and I just get dauntingly long computations where nothing simplifies. Are there any other methods for finding the constraint conditions for a functional to be stationary? Even if I somehow find constraint conditions, what will they look like and how can I compute an explicit example in say 3-dimensions?
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Part 2
My ideas for what I want to do if I figure out the difficulties I am having in my "Question 2" are as follows, I apologize for any blatant misconceptions about manifolds, singularities or curvature, as I have not fully formulated my ideas yet, I am looking for a direction and papers to read on closely related work. Essentially, I want to then find out how I can treat these "vertices" as above as actual masses or charges, and when the metric is not the pythagorean one. How will the same sort of geodesic equation in configuration space (in similar style to $L$) respond to these changes in the metric? Succinctly, my question can be posed as follows,
For a system of $I$-vertices (with the possibility of being interpreted as charges or masses given the appropriate Lagrangian) in $n$-dimensions, how can a geodesic equation in configuration space be developed to give a comprehensive account of the dynamics of the vertices as they traverse through space defined by the metric of any smooth manifold?
That is, I want to know how to generalize some of the procedures explained here for manifolds with metrics other than the standard pythagorean one. My motivation for wanting to know this, is I want to know how black holes interact with the gravitational field potential and electric field potential of charges and masses nearby. Maybe, sometime in the future, experiments can be done on the dynamics of neutron stars interacting closely with black holes.
Any responses are really appreciated for those of you who braved the wall of text! Comprehensive explanations of Questions 1 and 2 are required for an accepted answer and any suggestions or references to papers that might relate to part 3 are always welcome.This post imported from StackExchange MathOverflow at 2014-10-31 07:40 (UTC), posted by SE-user Samuel Reid
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FSc Part 1 (KPK Boards)
Notes of FSc Part 1 of “A Textbook of Mathematics For Class XI” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.
Author: Engr. Majid Amin Type: Solutions only Sender: Muhammad Kareem Format: PDF Scanned (Handwritten) Chapter 01: Complex Numbers Objectives
After reading this unit the students will be able to:
know complex numbers, its conjugate and absolute value. understand algebraic properties of complex numbers. recongnize real and imaginary parts of different types of complex numbers. know the solution of simultaneous linear equations with complex co-efficients. write the polynomial $P(z)$ as product of linear factors. solve quadratic equations in complex variable with real co-efficients. Download Chapter 02: Matrices and Determinants Objectives
After reading this unit the students will be able to:
know a matrix and its notations, order of a matrix and equality of two matrices. understand types of matrices, algebra of matrices and some properties of matrix addition and scalar multiplication. describe determinant of a square matrix and its evaluation using cofactors. know adjoint of a square matrix and use of adjoint method to calculate inverse of a square matrix. state and prove properties of determinants. know elementary row and column operations on matrices. recognize echelon and reduced echelon form of a matrix and rank of a matrix. solve a system of linear equations of both homogeneous and non-homogeneous equations. Download Chapter 03: Vectors Objectives
After reading this unit the students will be able to:
differentiate between scalar and vector quantities. give geometrical representation of a vector in a space. know the fundamental defintion of vector using geometrical as well as analytical representation. use vector to prove simple theorems of descriptive geometry. recognize rectangular coordinate system in space. define unit vectors i, j and k. repeat all fundamental definitions of vector in plane for space. know properties of vector addition and cross or vector product. define scalar triple product of vectors. express scalar triple product of vectors in terms of components (determinantal form). Download Chapter 4: Sequences Objectives
After reading this unit the students will be able to:
define a sequence and its terms. recognize triangle, factorial and Pascal sequence. know the definition of an arithmetic sequence. define arithmetic mean and arithmetic series. solve real life problems involving arithmetic series. define a geometric sequence. solve problems involving geometric sequences. define geometric mean and geometric series. solve real life problems involving geometric series. recognize a harmonic sequence and find its nth term. define harmonic mean and insert n harmonic means between two numbers. Download Chapter 5: Miscellaneous Series Objectives
After reading this unit the students will be able to:
know sigma $(\sum)$ sign and evaluation of $\sum n$, $\sum n^2$ and $\sum n^3$. understand arithmetical-geometric series and its sum of $n$ terms. know method of differences and its uses. use the partial fraction to find the sum to $n$ terms and to infinity of the series of the type $$\frac{a}{a(a+d)}+\frac{a}{(a+d)(a+2d)}+\ldots$$ Download Chapter 6: Permutation, Combination and Probability Objectives
After reading this unit the students will be able to:
know Kramp's factorial notation to express the product of first n natural numbers by n!. recognize the fundamental principle of counting and its illustration by using tree diagram. understand the concept of permutation and know the notation $^nP_r$. Prove the formula ^nP_r=n(n-1)(n-2)…(n-r+1), its deductions and application to solve relevant problems define combination and know the notation $^nC_r=\left(\begin{smallmatrix}n\\ r\end{smallmatrix} \right)=\frac{n!}{r!(n-r)!}$, its deduction and application to solve relevant problems. define kind of events. recognize the formula $P(E)=\frac{n(E)}{n(S)}$, $0\leq P(E)\leq1$ for probability of occurrence of an event $E$ and to know its application. recognize the addition theorem (or law) of probability and its deduction. recognize the multiplication theorem (or law) of probability and its deduction. Use theorem of addition and multiplication of probability to solve related problems. Download Chapter 7: Mathematical Induction and Binomial Theorem Objectives
After reading this unit the students will be able to:
know the principle of mathematical induction. apply the principle to prove the statements, identities or formulae. state and prove binomial theorem for positive integral index. expand $(x+y)^n$ using binomial theorem and find its general form. understand pascal's triangle and its use to obtain the coefficients of the binomial expansion $(x+y)^n$ when $n$ is a small number. know binomial series and its use to find the sum of the given series. Download Chapter 8: Functions and Graphs Objectives
After reading this unit the students will be able to
know linear, quadratic and square root functions. define inverse functions and find their domain and range. sketch the graph of the function $y=x^n$ for different values of $x$. sketch the graph of quadratic function. predict function from their graph. find the intersecting point of intersecting graphs of a linear functions and coordinate axes, two linear functions and a linear and quadratic function. solve graphically appropriate problems from daily life. Download Chapter 9: Linear Programming Objectives
After reading this unit the students will be able to
define linear programming (LP) as planning of allocation of limited resources to obtain optimal result. understand linear inequalities in one and two variables and its importance in real life problems. know the feasible region and identification of feasible region of simple LP problems. define optimal solution of an LP problem. find optimal solution graphically of LP problems. solve real life simple LP problems. Download Chapter 10: Trigonometric Identities of Sum and Difference of Angles Objectives
After reading this unit the students will be able to
know the fundamental law of trigonometry and deduction of trigonometric identities from it. understand trigonometric ratios and allied angles. use fundamental law and its deduction to derive trigonometric ratios of allied angles. derive double, half and triple angle identities from fundamental law and its deduction. express the product of sines and cosines as sum or differences of sines and cosines. express the sums or differences of sines and cosines as product. Download Chapter 11: Application of Trigonometry Objectives
After reading this unit the students will be able to
find the solution of right angles triangle. understand oblique triangles and find solution of such triangles, using the law of sines, cosines and tangents. derive the formula for finding the areas of triangles. know circum-circle, in-circle and escribed circle. derive the formula for finding circum-radius, in-radius, escribed radii and deduction of different identities. Download Chapter 12: Graph of Trigonometric and Inverse Trigonometric Functions and Solutions of Trigonometric Equations Objectives
After reading this unit the students will be able to
know trigonometric functions and their domain and range. define periodic, even/odd and translation properties of the graph of $\sin \theta$, $\cos \theta$ and $\tan \theta$. solve trigonometric equations of the type $\sin\theta=k$, $\cos\theta=k$ and $\tan\theta=k$. solve graphically the trigonometric equations of the type $\sin\theta=\frac{\theta}{2}$, $\cos\theta=\theta$ and $\tan\theta=2\theta$ when $-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$. define inverse trigonometric functions and their domain and range. prove the addition and subtraction formulae of inverse trigonometric functions and know their applications. solve general trigonometric equations. Download fsc/kpk_fsc_part_1 Last modified: 12 months ago by Administrator
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Blatantly ripped off from chem.SE, this post is meant to help people understand how to use MathJax formatting of mathematical expressions here on
Robotics. Getting started with MathJax
On
Robotics Stack Exchange, we use MathJax to format mathematical expressions. MathJax is a tool that lets us display LaTeX expressions on a browser.
To use MathJax, enclose your mathematical expressions within single(
$...$) or double(
$$...$$) dollar signs. Single dollar signs make the expression
inline, for example,
Let $x$ be a variable gives:
Let $x$ be a variable.
On the other hand, double dollar signs make the expression a block element. It gets its own line, and is slightly larger. For example,
The equation of motion is as follows: $$v=u+at$$ It is a SUVAT equation gives:
The equation of motion is as follows: $$v=u+at$$ It is a SUVAT equation
Note that the extra spaces in LaTeX do not render, use
\: or
~ for a space.
Basic MathJax Superscripts and subscripts
You can denote superscripts via the
^ character, and subscripts via
_. For example,
x^2 renders as $x^2$,
x_1 renders as $x_1$, and
x_1^3 renders as $x_1^3$.
If you want to include more than one character in the super/sub script, enclose it in curly braces (
{...}).
For example,
x^10 renders as $x^10$, but
x^{10} renders as $x^{10}$
Fractions and square roots
Fractions can be easily displayed using
\frac{..}{..}. For example,
\frac{a+b^c}{de+f} renders as $\frac{a+b^c}{de+f}$
Protip: You can exclude the braces for single-character numerators/denominators (if the first character is a letter, you need to use a space after
\frac, though). For example
\frac12 renders as $\frac12$, and
\frac ab renders as $\frac ab$
Square roots can be added in a similar manner, via
\sqrt{....}. For example,
\sqrt{x+y} renders as $\sqrt{x+y}$.
Greek letters
Greek letters can be added usung a backslash ('\'), followed by the name of the letter. Captialise the first letter of the name for greek capital letters.
Eg
\alpha \beta \gamma \Omega renders as $\alpha \beta \gamma \Omega$.
Make sure that you put spaces after these if you are typing normal alphabet characters. Eg
e^{\pii} gives an error, you need to use
e^{\pi i} for $e^{\pi i}$.
Note that there are special commands
\varepsilon , \varsigma , \varrho , and \varpi to distinguish between the lunate Greek letters ($\varepsilon \varsigma \varrho \varpi$ rather than $\epsilon \sigma \rho \pi$).
Further reading Wikipedia TeX help page (extremely useful as a reference, useless as a tutorial) Harvard intro to TeX LaTeX wikibook, Math section LaTeX wikibook, Advanced Math section
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Typesetting and Converting Mathematics¶
There are two main uses for MathJax:
Typesetting all the mathematics within a web page, and Converting a string containing mathematics into another form.
In version 2, MathJax could perform the first function very well, but it was much harder to do the second. MathJax version 3 makes both easy to do. Both these tasks are described below.
Typesetting Math in a Web Page¶
MathJax makes it easy to typeset all the math in a web page, and in fact it will do this automatically when it is first loaded unless you configure it not to. So this is one of the easiest actions to perform in MathJax; if your page is static, there is nothing to do but load MathJax.
If your page is dynamic, and you may be adding math after the page isloaded, then you will need to tell MathJax to typeset the mathematicsonce it has been inserted into the page. There are two methods fordoing that:
MathJax.typeset() and
MathJax.typesetPromise().
The first of these,
MathJax.typeset(), typesets the page, anddoes so immediately and synchronously, so when the call finishes, thepage will have been typeset. Note, however, that if the math includesactions that require additional files to be loaded (e.g., TeX inputthat uses require, or that includes autoloaded extensions), thenan error will be thrown. You can use the
try/catch command totrap this condition.
The second,
Mathjax.typesetPromise(), performs the typesettingasynchronously, and returns a promise that is resolved when thetypesetting is complete. This properly handles loading of externalfiles, so if you are expecting to process TeX input that can includerequire or autoloaded extensions, you should use this form oftypesetting. It can be used with
await as part of a larger
async function.
Both functions take an optional argument, which is an array of elements whose content should be processed. An element can be either an actual DOM element, or a CSS selector string for an element or collection of elements. Supplying an array of elements will restrict the typesetting to the contents of those elements only.
Handling Asynchronous Typesetting¶
It is generally a bad idea to try to perform multiple asynchronoustypesetting calls simultaneously, so if you are using
MathJax.typesetPromise() to make several typeset calls, youshould chain them using the promises they return. For example:
MathJax.typesetPromise().then(() => { // modify the DOM here MathJax.typesetPromise();}).catch((err) => console.log(err.message));
This approach can get complicated fast, however, so you may want to maintain a promise that can be used to chain the later typesetting calls. For example,
let promise = Promise.resolve(); // Used to hold chain of typesetting callsfunction typeset(code) { promise = promise.then(() => {code(); return MathJax.typesetPromise()}) .catch((err) => console.log('Typeset failed: ' + err.message)); return promise;}
Then you can use
typeset() to run code that changes the DOMand typesets the result. The
code() that you pass it does theDOM modifications and returns the array of elements to typeset, or
null to typeset the whole page. E.g.,
typeset(() => { const math = document.querySelector('#math'); math.innerHTML = '$$\\frac{a}{1-a^2}$$'; return math;});
would replace the contents of the element with
id="math" with thespecified fraction and have MathJax typeset it (asynchronously).Because the
then() call returns the result of
MathJax.typesetPromise(), which is itself a promise, the
then() will not resolve until that promise is resolved; i.e.,not until the typesetting is complete. Finally, since the
typeset() function returns the
promise, you can use
await in an
async function to wait for the typesetting tocomplete:
await typeset(...);
Note that this doesn’t take the initial typesetting that MathJaxperforms into account, so you might want to use
MathJax.startup.promise in place of
promise above.I.e., simply use
function typeset(code) { MathJax.startup.promise = MathJax.startup.promise .then(() => {code(); return MathJax.typesetPromise()}) .catch((err) => console.log('Typeset failed: ' + err.message)); return MathJax.startup.promise;}
This avoids the need for the global
promise variable, andmakes sure that your typesetting doesn’t occur until the initialtypesetting is complete.
Resetting Automatic Equation Numbering¶
The TeX input jax allows you to automatically number equations. When modifying a page, this can lead to problems as numbered equations may be removed and added; most commonly, duplicate labels lead to issues.
You can reset equation numbering using the command
MathJax.texReset([start])
where
start is the number at which to start equation numbering.
If you have inserted new content, that may require the entire page to be reprocessed in order to get the automatic numbering, labels, and references to be correct. In that case, you can do
MathJax.startup.document.state(0);MathJax.texReset();MathJax.typeset();
to force MathJax to reset the page to the state it was before MathJax processed it, reset the TeX automatic line numbering and labels, and then re-typeset the contents of the page from scratch.
Loading MathJax Only on Pages with Math¶
The MathJax combined configuration files are large, and so you maywish to include MathJax in your page only if it is necessary. If youare using a content-management system that puts headers and footersinto your pages automatically, you may not want to include MathJaxdirectly, unless most of your pages include math, as that would loadMathJax on
all your pages. Once MathJax has been loaded, it shouldbe in the browser’s cache and load quickly on subsequent pages, butthe first page a reader looks at will load more slowly. In order toavoid that, you can use a script like the following one that checks tosee if the content of the page seems to include math, and only loadsMathJax if it does. Note that this is not a very sophisticated test,and it may think there is math in some cases when there really isn’tbut it should reduce the number of pages on which MathJax will have tobe loaded.
Create a file called
check-for-tex.js containing the following:
(function () { var body = document.body.textContent; if (body.match(/(?:\$|\\\(|\\\[|\\begin\{.*?})/)) { if (!window.MathJax) { window.MathJax = { tex: { inlineMath: {'[+]': [['$', '$']]} } }; } var script = document.createElement('script'); script.src = 'https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js'; document.head.appendChild(script); }})();
and then use
<script src="check-for-tex.js" defer></script>
in order to load the script when the page content is ready. Notethat you will want to include the path to the location where youstored
check-mathjax.js, that you should change
tex-chtml.js to whatever component file you want to use, and thatthe
window.MathJax value should be set to whatever configurationyou want to use. In this case, it just adds dollar signs to thein-line math delimiters. Finally, adjust the
body.match() regularexpression to match whatever you are using for math delimiters.
This simply checks if there is something that looks like a TeX in-line or displayed math delimiter, and loads MathJax if there is. If you are using different delimiters, you will need to change the pattern to include those (and exclude any that you don’t use). If you are using AsciiMath instead of TeX, then change the pattern to look for the AsciiMath delimiters.
If you are using MathML, you may want to use
if (document.body.querySelector('math')) {...}
for the test instead (provided you aren’t using namespace prefixes, like <m:math>).
Converting a Math String to Other Formats¶
An important use case for MathJax is to convert a string containing mathematics (in one of the three forms that MathJax understands) and convert it into another form (either MathML, or one of the output formats that MathJax supports). This was difficult to do in MathJax version 2, but easy to do in version 3.
When MathJax startup up, it creates methods for converting from theinput format(s) to the output format(s) that you have loaded, and toMathML format. For example, if you have loaded the MathML input jaxand the SVG output jax (say by using the
mml-svg component), thenMathJax will create the following conversion methods for you:
MathJax.mathml2svg(math[,options])
MathJax.mathml2svgPromise(math[,options])
MathJax.mathml2mml(math[,options])
MathJax.mathml2mmlPromise(math[,options])
If you had loaded the TeX input jax as well, you would also get fourmore methods, with
tex in place of
mathml.
As the names imply, the
Promise functions perform the conversionasynchronously, and return promises, while the others operatesynchronously and return the converted form immediately. The firsttwo functions (and any others like them) produce DOM elements as theresults of the conversion, with the promise versions passing that totheir
then() functions as their argument (see the section onAsynchronous Conversion below), and the non-promise versions returningthem directly. You can insert these DOM elements into the documentdirectly, or you can use their
outerHTML property to obtaintheir serialized string form.
The functions that convert to MathML produce serialized MathML stringsautomatically, rather than DOM elements. (You can use the browser’s
DOMParser object to convert the string into a MathML DOM treeif you need one.)
Conversion Options¶
All four of these functions require an argument that is the mathstring to be converted (e.g., the serialized MathML string, or in thecase of
tex2chtml(), the TeX or LaTeX string). You can alsopass a second argument that is an object containing options thatcontrol the conversion process. The options that can be included are:
display, a boolean specifying whether the math is in display-mode or not (for TeX input). Default is
true.
em, a number giving the number of pixels in an
emfor the surrounding font. Default is
16.
ex, a number giving the number of pixels in an
exfor the surrounding font. Default is
8.
containerWidth, a number giving the width of the container, in pixels. Default is 80 times the
exvalue.
lineWidth', a number giving the line-breaking width in
emunits. Default is a very large number (100000), so effectively no line breaking.
scale, a number giving a scaling factor to apply to the resulting conversion. Default is 1.
For example,
let html = MathJax.tex2chtml('\\sqrt{x^2+1}', {em: 12, ex: 6, display: false});
would convert the TeX expression
\sqrt{x^2+1} to HTML as anin0line expression, with
em size being 12 pixels and
ex sizebeing 6 pixels. The result will be a DOM element containing the HTMLfor the expression. Similarly,
let html = MathJax.tex2chtml('\\sqrt{x^2+1}', {em: 12, ex: 6, display: false});let text = html.outerHTML;
sets
text to be the serialized HTML string for the expression.
Obtaining the Output Metrics¶
Since the
em,
ex, and
containerWidth alldepend on the location where the math will be placed in the document(they are values based on the surrounding text font and the containerelements width), MathJax provides a method for obtaining these valuesfrom a given DOM element. The method
MathJax.getMetricsFor(node, display)
takes a DOM element (
node) and a boolean (
display), indicatingif the math is in display mode or not, and returns an objectcontaining all six of the options listed above. You can pass thisobject directly to the conversion methods discussed above. So you cando something like
let node = document.querySelector('#math');let options = MathJax.getMetricsFor(node, true);let html = MathJax.tex2svg('\\sqrt{x^2+1}', options);node.appendChild(html);
in order to get get the correct metrics for the (eventual) location ofthe math that is being converted. Of course, it would be easier tosimply insert the TeX code into the page and use
MathJax.typeset() to typeset it, but this is just an exampleto show you how to obtain the metrics from a particular location inthe page.
Note that obtaining the metrics causes a page refresh, so it is expensive to do this. If you need to get the metrics from many different locations, there are more efficient ways, but these are advanced topics to be dealt with elsewhere.
Obtaining the Output Stylesheet¶
The output from the SVG and CommonHTML output jax both depend on CSS stylesheets in order to properly format their results. You can obtain the SVG stylesheet element by calling
MathJax.svgStylesheet();
and the HTML stylesheet from
MathJax.chtmlStylesheet();
The CommonHTML output jax CSS can be quite large, so the output jaxtries to minimize the stylesheet by including only the styles that areactually needed for the mathematics that has been processed by theoutput jax. That means you should request the stylesheet only
afteryou have typeset the mathematics itself.
Moreover, if you typeset several expressions, the stylesheet will include everything needed for all the expressions you have typeset. If you want to reset the stylesheet, then use
MathJax.startup.output.clearCache();
if the output jax is the CommonHTML output jax. So if you want toproduce the style sheet for a single expression, issue the
clearCache() command just before the
tex2chtml() call.
Asynchronous Conversion¶
If you are converting TeX or LaTeX that might use require to load extensions, or where extensions might be autoloaded, you will either need to use one of the “full” components that include all the extensions, or preload all the extensions you need if you plan to use the synchronous calls listed above. Otherwise, you can use the promise-based calls, which handle the loading of extensions transparently.
For example,
let node = document.querySelector('#math');let options = MathJax.getMetricsFor(node, true);MathJax.tex2chtmlPromise('\\require{bbox}\\bbox[red]{\\sqrt{x^2+1}}', options) .then((html) => { node.appendChild(html); let sheet = document.querySelector('#MJX-CHTML-styles'); if (sheet) sheet.parentNode.removeChild(sheet); document.head.appendChild(MathJax.chtmlStylesheet()); });
would get the metrics for the element with
id="math", convertthe TeX expression using those metrics (properly handling theasynchronous load needed for the
\require command); then when theexpression is typeset, it is added to the document and the CHTMLstylesheet is updated.
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Mathematically, the intuition is simple. (This is not a formal proof, though it can easily be turned into one. It's just another dimension of intuition -- symbolic intuition.)
The price elasticity of demand at $(P_0, Q_0)$ is the infinitesimal ratio of percentage change in quantity demanded ($dQ/Q_0$) to percentage change in price ($dP/P_0$).
$$\epsilon_d=\left|\frac{dQ/Q}{dP/P}\right|_{(P_0, Q_0)}=\color{red}{\left|\frac{dQ}{dP}\right|_{(P_0,Q_0)}}\cdot\color{blue}{\frac{P_0}{Q_0}}$$
When the demand curve is linear, the red expression is constant: it's just the slope of the demand curve. The blue expression, however, depends on the point at which the elasticity is calculated; even for a linear demand curve, it is
not constant.
In fact, if you draw a typical demand diagram, you plot the
inverse demand function, with $P$ on the vertical axis and $Q$ on the horizontal axis. In this case, $P_0/Q_0>1$ when the point $(P_0, Q_0)$ lies above the line $P=Q$ $P_0/Q_0=1$ when the point $(P_0, Q_0)$ lies on the line $P=Q$ $P_0/Q_0<1$ when the point $(P_0, Q_0)$ lies below the line $P=Q$
Thus, it is easy to see why $\epsilon_d$ changes along even a linear demand curve. The inverse demand curve will cut through Quadrant I of the $QP$-plane. The closer the point $(P_0, Q_0)$ gets to the vertical axis, the larger the elasticity.
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Equivalence of Definitions of Polynomial Function on Subset of Ring Contents Theorem
Let $R$ be a commutative ring with unity.
Let $S \subset R$ be a subset.
A
polynomial function on $S$ is a mapping $f : S \to R$ for which there exist: a natural number $n \in \N$ $a_0, \ldots, a_n \in R$
such that for all $x\in S$:
$\map f x = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$
where $\sum$ denotes indexed summation.
Let $R \sqbrk X$ be the polynomial ring in one variable over $R$.
Let $R^S$ be the ring of mappings from $S$ to $R$.
Let $\iota \in R^S$ denote the inclusion $S \hookrightarrow R$.
Outline of proof
This follows by interpreting both definitions as $f$ being a polynomial in the element $\iota$.
Proof 1 implies 2
Let $\map P X \in R \sqbrk X$ be the polynomial:
$P = \displaystyle \sum_{k \mathop = 0}^n a_k \cdot X^k$
where $\sum$ denotes indexed summation.
We show that $\map P \iota = f$.
2 implies 1
Let $P \in R \sqbrk X$ be a polynomial and $f = \map P \iota \in R^S$.
By Polynomial is Linear Combination of Monomials, there exist:
$n \in \N$ $a_0, \ldots, a_n \in R$
such that $P = \displaystyle \sum_{k \mathop = 0}^n a_k \cdot X^k$ where $\sum$ denotes indexed summation.
Let $\operatorname{ev}_{\iota}$ denote the evaluation homomorphism at $\iota$.
Then $\map {\operatorname{ev}_{\iota} } P = f$.
We have:
\(\displaystyle f\) \(=\) \(\displaystyle \map {\operatorname{ev}_{\iota} } {\displaystyle \sum_{k \mathop = 0}^n a_k \cdot X^k}\) \(\displaystyle \) \(=\) \(\displaystyle \displaystyle \sum_{k \mathop = 0}^n \map {\operatorname{ev}_{\iota} } {a_k \cdot X^k}\) Ring Homomorphism Preserves Indexed Summations \(\displaystyle \) \(=\) \(\displaystyle \displaystyle \sum_{k \mathop = 0}^n a_k \cdot \map {\operatorname{ev}_{\iota} } {X^k}\) Definition of Polynomial Evaluation Homomorphism \(\displaystyle \) \(=\) \(\displaystyle \displaystyle \sum_{k \mathop = 0}^n a_k \cdot \map {\operatorname{ev}_{\iota} } {X^k }\) Definition of Polynomial Evaluation Homomorphism
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Since I spend a lot of time on solving sparse linear equation systems then I am also a user of sparse matrix reordering methods. My claim to fame is that I have implemented approximate minimum degree myself and it is used in MOSEK.
Below I summarize some interesting link to graph partitioning software:
It is very common to use a BLAS library to perform linear algebra operations such as dense matrix times dense matrix multiplication which can be performed using the dgemm function. The advantage of BLAS is
it is well a defined standard.
and hardware vendors such as Intel supplies a tuned version.
Now at MOSEK my employeer we use the Intel MKL library that includes a BLAS implementation. It really helps us deliver good floating point performance. Indeed we use a sequential version of Intel MKL but call it from potentially many threads using Clik plus. This works well due to the well designed BLAS interface. However, there is one rotten apple in the basket and that is error handling.
Here I will summarize why the error handling in the BLAS standard is awful from my perspective.
First of all why can errors occur when you do the dgemm operation if we assume the dimensions of the matrices are correct and ignoring issues with NANs and the like. Well, in order to obtain good performance the dgemm function may allocate additional memory to store smallish matrices that fit into the cache. I.e. the library use a blocked version to improve the performance.
Oh wait that means it can run of memory and then what? The BLAS standard error handling is to print a message to stderr or something along that line.
Recall that dgemm is embedded deep inside MOSEK which might be embedded deep inside a third party program. This implies an error message printed to stderr does not make sense to the user. Also the user would NOT like us to terminate the application with a fatal error. Rather we want to know that an out of space situation happened and terminate gracefully. Or doing something to lower the space requirement. E.g. use a fewer threads.
What is the solution to this problem? The only solution offered is to replace a function named xerbla that gets called when an error happens. The idea is that the function can set a global flag indicating an error happened. This might be a reasonable solution if the program is single threaded. Now instead assume you use a single threaded dgemm (from say Intel MKL) but call it from many threads. Then first of all you have to introduce a lock (a mutex) around the global error flag leading to performance issues. Next it is hard to figure out which of all the dgemm calls that failed. Hence, you have to fail them all. What pain.
Why is the error handling so primitive in BLAS libraries. I think the reasons are:
BLAS is an old Fortran based standard.
For many years BLAS routine would not allocate storage. Hence, dgemm would never fail unless the dimensions where wrong.
BLAS is proposed by academics which does not care so much about error handling. I mean if you run out of memory you just buy a bigger supercomputer and rerun your computations.
If the BLAS had been invented today it would have been designed in C most likely and then all functions would have returned an error code. I know dealing with error codes is a pain too but that would have made error reporting much easier for those who wanted to do it properly.
I found the talk: Plain Threads are the GOTO of todays computing by Hartmut Kaiser very interesting because I have been working on improving the multithreaded code in MOSEK recently. And is also thinking how MOSEK should deal with all the cores in the CPUs in the future.I agree with Hartmut something else than plain threads is needed.
First a clarification conic quadratic optimization and second order cone optimization is the same thing. I prefer the name conic quadratic optimization though.
Frequently it is asked on the internet what is the computational complexity of solving conic quadratic problems. Or the related questions what is the complexity of the algorithms implemented in MOSEK, SeDuMi or SDPT3.
To the best of my knowledge almost all open source and commercial software employ a primal-dual interior-point algorithm using for instance the so-called Nesterov-Todd scaling.
A conic quadratic problem can be stated on the form
\[\begin{array}{lccl}\mbox{min} & \sum_{j=1}^d (c^j)^T x^j & \\\mbox{st} & \sum_{j=1}^d A^j x^j & = & b \\& x^j \in K^j & \\\end{array}\]where \(K_j\) is a \(n^j\) dimensional quadratic cone.Moreover, I will use \(A = [A^1,\ldots, A^d ]\) and \(n=\sum_j n^j\). Note that \(d \leq n\).First observe the problem cannot be solved exactly on a computer using floating numbers since the solution might be irrational. This is in contrast to linear problems that always have rational solution if the data is rational.
Using for instance the primal-dual interior point algorithm the problem can be solved to \(\varepsilon\) accuracy in \(O(\sqrt{d} \ln(\varepsilon^{-1}))\) interior-point iterations, where \(\varepsilon\) is the accepted duality gap. The most famous variant having that iteration complexity is based on Nesterov and Todds beautiful work on symmetric cones.
Each iteration requires solution of a linear system with the coefficient matrix\[ \label{neweq}\left [ \begin{array}{cc}H & A^T \\A & 0 \\\end{array}\right ] \mbox{ (*)}\]This is the most expensive operation and that can be done in \(O(n^3)\) complexity using Gaussian elimination so we end at the complexity \(O(n^{3.5}\ln(\varepsilon^{-1}))\).
That is the theoretical result. In practice the algorithms usually works much better because they normally finish in something like 10 to 100 iterations and rarely employs more than 200 iterations. In fact if the algorithm requires more than 200 iterations then typically numerical issues prevent the software from solving the problem.
Finally, typically conic quadratic problem is sparse and that implies the linear system mentioned above can be solved must faster when the sparsity is exploited. Figuring our to solve the linear equation system (*) in the lowest complexity when exploiting sparsity is NP hard and therefore optimization only employs various heuristics such minimum degree order that helps cutting the iteration complexity. If you want to know more then read my Mathematical Programming publication mentioned below. One important fact is that it is impossible to predict the iteration complexity without knowing the problem structure and then doing a complicated analysis of that. I.e. the iteration complexity is not a simple function of the number constraints and variables unless A is completely dense.
To summarize primal-dual interior-point algorithms solve a conic quadratic problem in less 200 times the cost of solving the linear equation system (*) in practice.
So can the best proven polynomial complexity bound be proven for software like MOSEK. In general the answer is no because the software employ an bunch of tricks that speed up the practical performance but unfortunately they destroy the theoretical complexity proof. In fact, it is commonly accepted that if the algorithm is implemented strictly as theory suggest then it will be hopelessly slow.
I have spend of lot time on implementing interior-point methods as documented by the Mathematical Programming publication and my view on the practical implementations are that are they very close to theory.
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If you take a look at the documentation, Mathematica's symbolic Fourier transform function,
FourierTransform, computes
$$\hat f(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{ikx}\mathrm{d}x$$
You can discretize some piece of this integral by limiting $x$ and $k$ to values $x_1 + (r-1)\Delta x$ and $(s-1)\Delta k$ respectively, where $\Delta x\Delta k = 2\pi/N$, giving
$$\begin{align}\hat f_s &= \frac{1}{\sqrt{2\pi}}\sum_{r=1}^N f(x_1 + (r-1)\Delta x)e^{i(s-1)\Delta k[x_1 + (r-1)\Delta x]}\Delta x\\
&= \frac{1}{\sqrt{2\pi}}e^{i(s-1)\Delta k x_1/N}\sum_{r=1}^N f_r\ e^{2\pi i(r-1)(s-1)/N}\Delta x\end{align}$$
Now compare this with the
Fourier function, which calculates
$$\frac{1}{\sqrt{N}}\sum_{r=1}^{N}u_r\ e^{2\pi i(r-1)(s-1)/N}$$
Evidently
$$u_r = \sqrt{\frac{N}{2\pi}}\ f_r\exp\biggl(\frac{2\pi i(s-1)x_1}{N\Delta x}\biggr)$$
The $2\pi/\Delta x$ in that exponent is a translation generator, and $(s-1)x_1/N$ is the corresponding parameter.
So here's what you need to do to compute a numerical Fourier transform: first, choose a grid on which to sample your function, consisting of equally spaced points $x_1,\ldots,x_N$.
grid = Range[x1, x1+(n-1)*deltax, deltax]
Here
x1 and
x1+n*deltax should be the endpoints of the region over which you are going to compute the Fourier transform, and
deltax should be some interval of $x$ small enough to capture the smallest details in your function.
Then sample your function on this grid,
samples = f/@grid
Notice that the factor $\exp\bigl(\frac{2\pi i(s-1)x_1}{N\Delta x}\bigr)$ depends only on $s$, which is a frequency space index. But it's independent of $r$, which is a position space index. So you can compute that factor separately and merge it in to the result of the FFT.
factor = Sqrt[n/(2 Pi)] Exp[2 Pi I x1/(n*deltax) Range[0, n-1]]
transform = Chop[Fourier[samples] * factor]
In order to actually turn this into the Fourier transformation of your function, you need to know that the frequencies which your transformed values correspond to start at zero, increase up to some maximum value which occurs in the middle of the array, then jump down to a negative value and increase up to zero again. (Actually, the FFT assumes momentum space is periodic and calculates from $k=0$ to $k=(N-1)\Delta k$, and you then need to map the second half of this to $k=-\frac{N}{2}\Delta k$ to $k = -\Delta k$.) You can create the frequency array using
freq = #~Join~Most@Reverse[-#]&@Range[0, Pi/deltax, 2 Pi/(n*deltax)]
and then you should be able to, say, plot your Fourier transform using
ListPlot[Transpose[freq, transform]]
(actually my expression for
freq seems a little off in tests, but I'll see if I can fix it up).
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Roll Your Own Differentiation Filters Introduction
There are many times in digital signal processing that it is necessary to obtain estimates of the derivative of some signal or process from discretely sampled values. Such numerical derivatives are useful in applications such as edge detection, rate of change estimation and optimization, and can even be used as part of quick and efficient dc-blocking filters.
Suppose then, that we want to approximate the derivative of a function at a set of points $\{x_i:i=1,...,N\}$ from a set of sampled values $\{f(x_i):i=1,...,N\}$. A common approach is to use Taylor series to build a few approximations around a point, then combine them. For example, if the $x_i$ are evenly spaced, with say $h = x_{i+1}-x_i$, then we may write $$f(x_{i+1}) \approx f(x_i) + h f'(x_i) + \frac{h^2}{2}f''(x_i)$$ and $$f(x_{i-1}) \approx f(x_i) - h f'(x_i) + \frac{h^2}{2} f''(x_i).$$ If we drop the $h^2$ term, we can solve for $f'(x_i)$ approximately as $$f'(x_i) \approx \frac{f(x_{i+1}) - f(x_i)}{h}.$$ If we subtract $f(x_{i-1})$ from $f(x_{i-1})$ then the even terms cancel, leaving $$f'(x_i) \approx \frac{f(x_{i+1}) - f(x_{i-1})}{2 h}.$$
Lagrange Polynomials
Building differentiation filters using Taylor series is all well and good, but what if we want higher order approximations, or maybe even formulas that are not central? We could continue to use Taylor series, but the math gets pretty messy. A more concise way is to make use of the Lagrange polynomials$$l_j(x) = \prod_{i \neq j}\frac{(x - x_i)}{(x_j - x_i)}$$ and write $$f(x) \approx \sum_{i=1}^N f(x_i)l_i(x).$$ Substituting three points $\{x_{i-1},x_i,x_{i+1}\}$ into the above formula gives us the following approximation $$f(x) \approx \frac{(x-x_i)(x-x_{i+1})}{(x_{i-1}-x_i)(x_{i-1}-x_{i+1})}f(x_{i-1}) +\frac{(x-x_{i-1})(x-x_{i+1})}{(x_i-x_{i-1})(x_i-x_{i+1})}f(x_i)$$ $$+\frac{(x-x_{i-1})(x-x_i)}{(x_{i+1}-x_{i-1})(x_{i+1}-x_i)}f(x_{i+1}).$$ Differentiating, yields the first derivative approximation $$f'(x) = \frac{ 2x - x_i - x_{i+1}}{(x_{i-1} - x_i)(x_{i-1}-x_{i+1})}f(x_{i-1}) + \frac{2x-x_{i-1}-x_{i+1}}{(x_i-x_{i-1})(x_i-x_{i+1})}f(x_i)$$ $$+ \frac{2x-x_{i-1}-x_i}{(x_{i+1}-x_{i-1})(x_{i+1}-x_i)}f(x_{i+1}).$$ If we evaluate at $x = x_i$ and choose a uniform grid spacing $h = (x_{i+1}-x_i)$ then our formula simplifies into the familiar form $$f'(x_i) \approx \frac{f(x_{i+1}) - f(x_{i-1})}{2 h}.$$ There's no particular reason why we need to evaluate at $x_i$ though, we could have instead chosen $x_{i+1}$ and obtained the formula $$f'(x_{i+1}) \approx \frac{- 3f(x_{i-1}) + 4f(x_i) - f(x_{i+1})}{2 h}$$
which allows us to approximate the derivative at the current sample using only information from the previous sample.
A More General Formula for Interpolating Polynomials
A nice property of polynomials is that they are entirely defined by their roots. This means that no matter what form you write your interpolating polynomial in, if there's another form of the same degree that has the same roots, the two forms define the same polynomial. Because of this, we can assume that our polynomial has the simpler form $$p(x) = \sum_{i=0}^n a_n x^n.$$ If p(x) interpolates some function f(x) at $n+1$ nodes so that $p(x_i) = f(x_i)$, then we can write the matrix equation $$ \begin{bmatrix} 1 & x_0 & x_0^2 & \cdots & x_0^n \\ 1 & x_1 & x_1^2 & \cdots & x_1^n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^n \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_n \end{bmatrix} = \begin{bmatrix} f(x_0) \\ f(x_1) \\ \vdots \\ f(x_n) \end{bmatrix} $$ The matrix on the left is the generalized Vandermonde matrix. Let's just write this as $$X \mathbf{a} = \mathbf{f}$$ so then $$\mathbf{a} = X^{-1} \mathbf{f}.$$ If we differentiate the interpolating polynomial analytically we can write the matrix equation $$ \begin{bmatrix} 0 & 1 & 2x_0 & \cdots & nx_0^{n-1} \\ 0 & 1 & 2x_1 & \cdots & nx_1^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 1 & 2x_n & \cdots & nx_n^{n-1} \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_n \end{bmatrix} = \begin{bmatrix} f'(x_0) \\ f'(x_1) \\ \vdots \\ f'(x_n) \end{bmatrix} $$ or, more compactly, as $$\mathbf{f}' = X' \mathbf{a}.$$ We can then write the matrix equation that maps a vector of function values to a vector of approximate derivative values $$\mathbf{f}' = X' X^{-1} \mathbf{f}$$ The matrix $D = X' X^{-1}$ is commonly called the pseudospectral differentiation matrix in the literature. Here's a little snippet of Matlab code to compute $D$:
function [D,V,Vx] = getDiffMat(x) x = x(:).'; N = numel(x)-1; V = zeros(N+1); Vx = zeros(N+1); for j=0:N V(j+1,:) = x(j+1).^(0:N); Vx(j+1,:) = (0:N).*(x(j+1).^(-1:N-1)); end D = Vx/V;
Now, say we want to define a few differentiation filters for 3 evenly-space points. We could then call the above function like so:
D = getDiffMat(1:3) D = -1.5000 2.0000 -0.5000 -0.5000 0 0.5000 0.5000 -2.0000 1.5000The rows of the resulting matrix are our differentiation filters (which should look familiar if you've been following along). We can make $D$ as large as we want now. For example, we could use 5 points and get something like: D = getDiffMat(1:5) D = -2.0833 4.0000 -3.0000 1.3333 -0.2500 -0.2500 -0.8333 1.5000 -0.5000 0.0833 0.0833 -0.6667 0 0.6667 -0.0833 -0.0833 0.5000 -1.5000 0.8333 0.2500 0.2500 -1.3333 3.0000 -4.0000 2.0833
Note that the matrix $D$ is entirely defined by the choice of the $x_i$ used to define it, and essentially you can choose whatever $x_i$ you like and define a pseudospectral differentiation matrix, but there are some choices which are better than others. Evenly-spaced values are the most convenient when defining differentiation filters, but for large N, accuracy becomes an increasing concern as the Vandermonde matrix becomes ill-conditioned. Ways around this problem are to choose the $x_i$ to be the singular values of some Sturm-Liouville polynomial and common choices are the Legendre, or Chebyshev Gauss-Lobatto points; unfortunately, these points are not equidistant and so usually require another interpolation step.
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Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type
DOI: http://dx.doi.org/10.12775/TMNA.2006.003
Abstract
New criteria are established for the existence of multiple positive
solutions of a Hammerstein integral equation of the form $$ u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))ds \equiv Au(t) $$ where $k$ can have discontinuities in its second variable and $g \in L^{1}$. These criteria are determined by the relationship between the behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the principal (positive) eigenvalue of the linear Hammerstein integral operator $$ Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)ds. $$ We obtain new results on the existence of multiple positive solutions of a second order differential equation of the form $$ u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1], $$ subject to general separated boundary conditions and also to nonlocal $m$-point boundary conditions. Our results are optimal in some cases. This work contains several new ideas, and gives a {\it unified} approach applicable to many BVPs.
solutions of a Hammerstein integral equation of the form
$$
u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))ds \equiv Au(t)
$$
where $k$ can have discontinuities in its second variable and $g \in
L^{1}$.
These criteria are determined by the relationship between the
behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the
principal (positive) eigenvalue of the linear Hammerstein integral
operator
$$
Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)ds.
$$
We obtain new results
on the existence of multiple positive solutions of a second order
differential equation of the form
$$
u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1],
$$
subject to general separated boundary conditions and also to nonlocal
$m$-point boundary conditions. Our results are optimal in some cases.
This work contains several new ideas, and gives a {\it unified}
approach applicable to many BVPs.
Keywords
Fixed point index; positive solution; eigenvalue criteria
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I know that if we move a rectangular wire from no magnetic field to through a magnetic field, there would be an induced voltage because there is change in flux (b∆x). However, if we moved a wire/rod in the same situation, it will also induce a voltage but is it due to the change in flux (b∆x) or charge separation?
The induced voltage depends on the change in flux according to faraday's law $$volt=-N\frac{\mathrm{d}\phi}{\mathrm{d}t}=-N\mathop{\iint}_{S^{'}}\frac{\mathrm{d}\vec{B}}{\mathrm{d}t}\cdot\mathrm{d}\vec{S}=-N\mathop{\oint}_{C^{'}}\frac{\mathrm{d}}{\mathrm{d}t}\left\{\nabla\times\vec{B}\right\}\cdot\mathrm{d}\vec{l}$$
So as you can see that the flux or the math associated with it needs a defined closed surface for inspection, since the potential induced in your case is NOT electrostatic, but electrodynamic. Electrodynamic potentials are NOT absolute and need a defined closed circuit for their realization. Hence, in this case, the question doesn't make sense since you don't have a closed circuit and are asking for an analysis of electrodynamic potential.
What would make sense is to ask that if you have a multimeter attached to the rod and then continuously monitor the potential, would the potential change? The answer depends on the state of motion of the multimeter. If the multimeter is static and the rod moves and while the rod moves the wires from the multimeter to the rod ends unfurl, then yes, you would see a voltage.
If, on the other hand the multimeter is soldered to the rod with two other rods so that they form a static hoop, then you will not see any voltage since, in that case, you'd have $$\mathop{\oint}_{C^{'}}\nabla\times\vec{B}\cdot\mathrm{d}\vec{l}=const$$ and the rate of change of that quantity wrt time is zero.
Now, when you talk of a Lorentz force, then $\vec{F}=q\left(\vec{v}\times\vec{B}\right)$ and you see the perpendicularity of the velocity and the magnetic field hence creating a force on the electrons which sift towards a given side. This creates a charge seggregation and hence a potential difference on the ends of the rod.
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The simplest approach (in terms of programming effort) might be to try using an existing graph layout tool. Those solve a related problem: given a graph with distances on the edges, try to find the best layout to draw the graph on the plane. You can treat your problem as an instance of the graph layout problem: we have one vertex per point, and for each pair of points $v,w$ with distance bounds $[\ell,u]$, we create an edge $v \to w$ with length $(\ell+u)/2$. However, this does have some limitations: typical graph layout algorithms try to get the distances between vertices correct, but also try to avoid edges that cross each other; whereas in your case you don't care about crossings. So, your problem might be easier.
Another possibility is to apply the ideas used for graph layout to your problem. There are several algorithmic techniques for graph layout. For instance, you could use a spring-based model, where you have a spring between each pair of vertices that have a distance bound, and the spring tries to keep those pair of vertices a suitable distance apart.
A third approach is to use black-box mathematical optimization. Introduce an objective function $\Phi$ which, given a set of locations for the points, calculates a penalty value (how "badly" the arrangement violates your constraints), and then try to find an arrangement that minimizes $\Phi$.
For instance, suppose for each pair $v,w$ of points, you have a lower bound $\ell_{v,w}$ and an upper bound $u_{v,w}$. You could define
$$\Phi(x_1,\dots,x_n) = \sum_{i,j} \frac{[||x_i - x_j||_2 - (u_{i,j} + \ell_{i,j})/2]^2}{(u_{i,j} - \ell_{i,j})^2},$$
and then use some optimization technique to find an arrangement $x_1,\dots,x_n$ that minimizes $\Phi(x_1,\dots,x_n)$. For instance, you could try using hillclimbing, gradient descent, or other convex optimization methods. This approach might be sensitive to the initial values for $x_1,\dots,x_n$, so you might want to repeat it multiple times with different random choices for the initial value, and take the best result.
Finally, you could try using simulated annealing.
The latter two approaches can be easily adjusted to incorporate angle constraints, simply by modifying the objective function appropriately to add a term that penalizes angles that differ from the desired value.
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The model you referenced in your question is called the "one-way model." It assumes that random row effects are the only systematic source of variance. In the case of inter-rater reliability, rows correspond to objects of measurement (e.g., subjects).
One-way model: $$ x_{ij} = \mu + r_i + w_{ij} $$ where $\mu$ is the mean for all objects, $r_i$ is the row effect, and $w_{ij}$ is the residual effect.
However, there are also "two-way models." These assume that there is variance associated with random row effects as well as random or fixed column effects. In the case of inter-rater reliability, columns correspond to sources of measurement (e.g., raters).
Two-way models: $$ x_{ij} = \mu + r_i + c_j + rc_{ij} + e_{ij} $$$$ x_{ij} = \mu + r_i + c_j + e_{ij} $$ where $\mu$ is the mean for all objects, $r_i$ is the row effect, $c_j$ is the column effect, $rc_{ij}$ is the interaction effect, and $e_{ij}$ is the residual effect. The difference between these two models is the inclusion or exclusion of the interaction effect.
Given a two-way model, you can calculate one of four ICC coefficients: the single score consistency ICC(C,1), the average score consistency ICC(C,k), the single score agreement ICC(A,1), or the average score agreement ICC(A,k). Single score ICCs apply to single measurements $x_{ij}$ (e.g., individual raters), whereas average score ICCs apply to average measurements $\bar{x}_i$ (e.g., the mean of all raters). Consistency ICCs exclude the column variance from the denominator variance (e.g., allowing raters to vary around their own means), whereas agreement ICCs include the column variance in the denominator variance (e.g., requiring raters to vary around the same mean).
Here are the definitions if you assume a random column effect:
Two-way Random-Effects ICC Definitions (with or without interaction effect):
$$
ICC(C,1) = \frac{\sigma_r^2}{\sigma_r^2 + (\sigma_{rc}^2 +
\sigma_e^2)}\text{ or }\frac{\sigma_r^2}{\sigma_r^2 + \sigma_e^2}
$$
$$
ICC(C,k) = \frac{\sigma_r^2}{\sigma_r^2 + (\sigma_{rc}^2 + \sigma_e^2)/k}\text{ or }\frac{\sigma_r^2}{\sigma_r^2 + \sigma_e^2/k}
$$
$$
ICC(A,1) = \frac{\sigma_r^2}{\sigma_r^2 + (\sigma_c^2 + \sigma_{rc}^2 + \sigma_e^2)}\text{ or }\frac{\sigma_r^2}{\sigma_r^2 + (\sigma_c^2 + \sigma_e^2)}
$$
$$
ICC(A,k) = \frac{\sigma_r^2}{\sigma_r^2 + (\sigma_c^2 + \sigma_{rc}^2 + \sigma_e^2)/k}\text{ or }\frac{\sigma_r^2}{\sigma_r^2 + (\sigma_c^2 + \sigma_e^2)/k}
$$
You can also estimate these values using mean squares from ANOVA:
Two-way ICC Estimations: $$ ICC(C,1) = \frac{MS_R - MS_E}{MS_R + (k-1)MS_E} $$ $$ ICC(C,k) =
\frac{MS_R-MS_E}{MS_R} $$ $$ ICC(A,1) = \frac{MS_R-MS_E}{MS_R +
(k-1)MS_E + k/n(MS_C-MS_E)} $$ $$ ICC(A,k) = \frac{MS_R-MS_E}{MS_R +
(MS_C-MS_E)/n} $$
You can calculate these coefficients in R using the irr package:
icc(ratings, model = c("oneway", "twoway"),
type = c("consistency", "agreement"),
unit = c("single", "average"), r0 = 0, conf.level = 0.95)
References
McGraw, K. O., & Wong, S. P. (1996). Forming inferences about some intraclass correlation coefficients.
Psychological Methods, 1(1), 30–46.
Shrout, P. E., & Fleiss, J. L. (1979). Intraclass correlations: Uses in assessing rater reliability.
Psychological Bulletin, 86(2), 420–428.
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Convergence of measures
$\newcommand{\abs}[1]{\left|#1\right|}$ A concept in measure theory, determined by a certain topology in a space of measures that are defined on a certain $\sigma$-algebra $\mathcal{B}$ of subsets of a space $X$ or, more generally, in a space $\mathcal{M} (X, \mathcal{B})$ of charges, i.e. countably-additive real (resp. complex) functions $\mu: \mathcal{B}\to \mathbb R$ (resp. $\mathbb C$), often also called $\mathbb R$ (resp. $\mathbb C$) valued or signed measures. The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as: \[ \abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}: \text{'"`UNIQ-MathJax13-QINU`"' is a countable partition of '"`UNIQ-MathJax14-QINU`"'}\right\}. \] In the real-valued case the above definition simplifies as \[ \abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (X\setminus B)}\right). \] The total variation of $\mu$ is then defined as $\left\|\mu\right\|_v := \abs{\mu}(X)$. The space $\mathcal{M}^b (X, \mathcal{B})$ of $\mathbb R$ (resp. $\mathbb C$) valued measure with finite total variation is a Banach space and the following are the most commonly used topologies.
1) The norm or strong topology: $\mu_n\to \mu$ if and only if $\left\|\mu_n-\mu\right\|_v\to 0$.
2) The weak topology: a sequence of measures $\mu_n \rightharpoonup \mu$ if and only if $F (\mu_n)\to F(\mu)$ for every bounded linear functional $F$ on $\mathcal{M}^b$.
3) When $X$ is a topological space and $\mathcal{B}$ the corresponding $\sigma$-algebra of Borel sets, we can introduce on $X$ the narrow topology. In this case $\mu_n$ converges to $\mu$ if and only if \begin{equation}\label{e:narrow} \int f\, \mathrm{d}\mu_n \to \int f\, \mathrm{d}\mu \end{equation} for every bounded continuous function $f:X\to \mathbb R$ (resp. $\mathbb C$). This topology is also sometimes called the weak topology, however such notation is inconsistent with the Banach space theory, see below. The following is an important consequence of the narrow convergence: if $\mu_n$ converges narrowly to $\mu$, then $\mu_n (A)\to \mu (A)$ for any Borel set such that $\abs{\mu}(\partial A) = 0$.
4) When $X$ is a locally compact topological space and $\mathcal{B}$ the $\sigma$-algebra of Borel sets yet another topology can be introduced, the so-called wide topology, or sometimes referred to as weak$^\star$ topology. A sequence $\mu_n\rightharpoonup^\star \mu$ if and only if \eqref{e:narrow} holds for continuous functions which are compactly supported.
This topology is in general weaker than the narrow topology. If $X$ is compact and Hausdorff the Riesz representation theorem shows that $\mathcal{M}^b$ is the dual of the space $C(X)$ of continuous functions. Under this assumption the narrow and weak$^\star$ topology coincides with the usual weak$^\star$ topology of the Banach space theory. Since in general $C(X)$ is not a reflexive space, it turns out that the narrow topology is in general weaker than the weak topology.
A topology analogous to the weak$^\star$ topology is defined in the more general space $\mathcal{M}^b_{loc}$ of locally bounded measures, i.e. those measures $\mu$ such that for any point $x\in X$ there is a neighborhood $U$ with $\abs{\mu}(U)<\infty$.
References
[AmFuPa] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001 [Bo] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001 [DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523 [Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201 [Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005 How to Cite This Entry:
Convergence of measures.
Encyclopedia of Mathematics.URL: http://www.encyclopediaofmath.org/index.php?title=Convergence_of_measures&oldid=27239
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Two different documents about how to edit are being combined here. Each starts with the XXXX title. At present the simple outline is first; it should keep users productive for their first few days.
XXXXXXXXXEdit The basicsEdit
The
bare essential points of editing are: Type in the edit box, much the same as in any wordprocessing program. For a new paragraph, you need a complete blank line ORa line starting with either a colon (which produces an indent) or an asterisk (which produces a bullet). Doubling those characters gives extra indent. Fractionally more adventurous thingsEdit
Other effects may be achieved very quickly if you can see a toolbar (row of little boxes) just above the edit box. But here are the slow ways of doing the most important:
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add a few words of explanation in the "Summary:" line hit the "Show preview" button edit further if desired, then if really OK hit the "Save page" button. More detailEdit
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* Lists are easy to do: ** start every line with a star *** more stars means deeper levels *A newline *in a list marks the end of the list. *Of course *you can *start again.
marks the end of the list.
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Sue is reading the [[video policy]].
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Subscript: x
Subscript: x<sub>2</sub><br/> Superscript: x<sup>2</sup> or x² or in projects with the templates sub and sup: Subscript: x{{sub|2}}<br/> Superscript: x{{sup|2}} ε<sub>0</sub> = 8.85 × 10<sup>−12</sup> C² / J m. 1 [[hectare]] = [[1 E4 m²]] α β γ δ ε ζ<br/> η θ ι κ λ μ ν<br/> ξ ο π ρ σ ς<br/> τ υ φ χ ψ ω<br/> Γ Δ Θ Λ Ξ Π<br/> Σ Φ Ψ Ω ∫ ∑ ∏ √ − ± ∞<br/> ≈ ∝ ≡ ≠ ≤ ≥ ←<br/> × · ÷ ∂ ′ ″<br/> ∇ ‰ ° ∴ ℵ ø<br/> ∈ ∉ ∩ ∪ ⊂ ⊃ ⊆ ⊇<br/> ¬ ∧ ∨ ∃ ∀ ⇒ ⇔<br/> → ↔ <i>x</i><sup>2</sup> ≥ 0 true. <math>\sum_{n=0}^\infty \frac{x^n}{n!}</math> arrow → ''italics'' [[link]] <nowiki>arrow → ''italics'' [[link]] </nowiki> arrow → ''italics'' [[link]] <nowiki> <pre>arrow → ''italics'' [[link]]
</nowiki></pre>
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Expected Index of Coincidence usually refers to a language's expected index of coincidence (1.73 for English, or 0.067 if you're not normalising).
The formula in question is usually used to determine the length of the key ($t$) given the (measured) $IC$ of received cipher-text. $IC$ is the probability that two randomly-selected letters from the cipher text are identical.
Let $X=\{ x_1,x_2,x_3,...x_L\}$ be the cipher-text. If we think the the poly-alphabetic cipher has period $t$, then we would expect each of the following:
\begin{eqnarray}& X_1 =\{x_1, x_{t+1},x_{2t+1},...\}\\ & X_2 =\{x_2, x_{t+2},x_{2t+2},...\}\\& \vdots \\ &X_t =\{x_t, x_{2t},x_{3t},...\}\end{eqnarray}
to exhibit the same index as the plaintext ($\kappa_p$). So we can reconstruct $IC(X)$ as follows. Pick two letters at random, we want the probability that they match.
The probability that they are in the same $X_i$ is:
$$\frac{tC(\frac{L}{t},2)}{C(L,2)} = \frac{L*(\frac{L}{t}-1)}{L(L-1)}$$
The probability they are in different $X_i$s is:$$\frac{C(t,2)*\frac{L}{t}*\frac{L}{t}}{C(L,2)} = \frac{t(t-1)*\frac{L}{t}*\frac{L}{t}}{L(L-1)}$$
If they are in the same $X_i$ then they are both enciphered using the same alphabet: so the probability is $\kappa_p$ If they are in different $X_i$s then they are enciphered using different alphabets, so we can assume they are randomly distributed: so the probability is $\kappa_r$
So the probability of two random letters matching is approximately
\begin{eqnarray}IC(X) & \approx \frac{L*(\frac{L}{t}-1)}{L(L-1)}\kappa_p + \frac{t(t-1)*\frac{L}{t}*\frac{L}{t}}{L(L-1)}\kappa_r\\ & = \frac{(L-t)}{t(L-1)}\kappa_p + \frac{(t-1)*L}{t(L-1)}\kappa_r\end{eqnarray}
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I am reading about Riemann-Stieltjes Integrals in Carother's Real Analysis. I am trying to prove the following Theorem:
Theorem: $14.9$
Let $\alpha$ be continuous and increasing. Given $f$ $\in$ $R_{\alpha}([a,b])$ and $\epsilon >0 $.
There exists a step function $h$ on $[a,b]$ with $||h||_{\infty}$ $\leq$ $||f||_{\infty}$ such that $\int_{a}^{b}|f - h| d\alpha$ $<$ $\epsilon$.
Carother's uses the Riemann conditions to obtain a partition $P$ such that $U(f:P) - L(f:P)$ $<$ $\epsilon$. Using this partition, Carother's selects $t_i$ $\in$ $[x_{i-1}, x_i)$ and define the step function $h(x) = f(t_{i})$ for $i = 1,2,3,...n$. It follows clearly that $||h||_{\infty}$ $\leq$ $\|f||_{\infty}$ however, he makes the following statement that confuses me and concludes the proof using the statement:
"Since $\alpha$ is continuous, we have h $\in$ $R_{\alpha}([a,b])$"
I tried showing this myself by showing $U(h:P) - L(h:p) = \sum_{i=1}^{n}(M_{i} - m_{i})\Delta\alpha_{i} < \epsilon$ where $M_{i}$ and $m_{i}$ are the sup and inf of interval $i$ respectively. However, if $h(x)$ is a step function we have $M_{i} = m_{i}$ which implies $\sum_{i=1}^{n}(M_{i} - m_{i})\Delta\alpha_{i} = 0$. Therefore, I would not even consider the continuity of $\alpha$. Evidently, this is incorrect so if anyone could help explain it would be greatly appreciated.
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A tl;dr of the below is basically what was echoed in the comments: it's not even remotely close to Graham's number, nor even close to the "building block" of it that is $3 \uparrow \uparrow \uparrow \uparrow 3$. Truthfully, pretty much any analogy grounded in reality would only serve to show "yeah Graham's number is way bigger."
Still, for its own sake and because I found this a little fun, I felt like demonstrating this properly.
Let's take a result from here as granted: the number of legal positions of a Rubik's cube of arbitrary dimension. I don't know why it's called "combinations" there; in my opinion it's misleading, but it matches up with the number of legal positions and in turn symmetries of a $3\times 3 \times 3$ cube if you plug in $n=3$, well-known to be $43,252,003,274,489,856,000$:
I feel it is fairly self-evident that the dimension of the resulting cube is an even number, let it be $k$; this allows for a little simplifying. Then $k \equiv 0 \pmod 2$ and the resulting formula for the number of symmetries for a $k \times k \times k$ Rubik's cube is:
$$7! \cdot 3^6 \cdot \frac{24! ^{\; \text{floor}[(k^2 - 2k)/4]}}{4!^{ \; 6 \cdot \text{floor}[(k-2)^2/4]}}$$
So what is $k$?
You have $4.6 \times 10^{185}$ cubes, in each of which you have $10^{300}$ (a googol cubed) cubes. Let's make the math simpler (and $k$ larger) by basically just assuming each cube is unique from the rest - don't put them together in one cube. This prevents nonsense about "well the inside faces of the smaller cubes won't affect the overall cube," simplifying it all greatly. Also, this way, each symmetry of one cube counts as one for the overall "universe-cube" and ultimately the "universe-cube" need not even be a cube. We now just have a bunch of Rubik's cubes.
Anyhow we see $k = 4.6 \times 10^{485}$ with these considerations.
Plugging this into WolframAlpha gives us that the number of symmetries of the universe-cube is roughly $10 \uparrow 10 \uparrow 973$ if we round up. Thus, the number of the symmetries of the universe-cube has roughly $10^{972}$ digits ... so far.
This neglects the cube-switching symmetry. Let's assume the worst-case value again, and basically line up each cube in order, and consider the overall configuration as one single symmetry. Thus if cube $1$ and cube $2$ individually show the exact same symmetry (i.e. look identical aside from the numbering), swapping them still counts as a new symmetry for the overall "universe-cube" you propose, even if it looks identical. Again, all in the interest of simplifying the math (even if it hugely overestimates the actual number).
Thus we have $(4.6 \times 10^{185})!$ more symmetries to account for, this being the number of ways to arrange that many objects. Ironically, that doesn't matter in the end: Wolfram gives that factor to be $10 \uparrow 10 \uparrow 188$, approximately. So it doesn't even meaningfully affect the overall number (though it is still calculated here if interested).
Still, a number with $10^{972}$ digits, pretty hefty, right?
Okay, but how many digits does Graham's number have? I'll largely follow the explanation given by him on Numberphile.
Let us recall its construction with the up-arrow notation. $3 \uparrow 3$ is basically $3^3$, and $3 \uparrow \uparrow 3 = 3 \uparrow (3 \uparrow 3)$. Proceeding numbers of arrows "reduce" similarly to the two-arrow case:
$3 \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow (3 \uparrow \uparrow 3)$ $3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow 3)$
To see the escalation:
$3 \uparrow 3 = 3^3 = 27$ $3 \uparrow \uparrow 3 = 3^{27} > 7$ trillion $3 \uparrow \uparrow \uparrow 3 > 10^{\text{3.6 trillion}}$ $3 \uparrow \uparrow \uparrow \uparrow 3$ amounts to amounts to a "power tower" of $3$ to its own power $7,625,597,484,987$ times per Wikipedia.
Starting at this last one, let's try to even approach that with the universe cube. After all, Wolfram can at least approximate fairly high. Again, not even in the realm of Graham's number, just a building block of it, but it's a good place to start.
Sadly, we're already well beyond the number of symmetries of your universe-cube at just calculating $3 \uparrow 3 \uparrow 3 \uparrow 3 \uparrow 3$ (a power tower of five $3$'s), which I chose as our random starting point. Wolfram claims this number by itself to simply have about $6×10^{3,638,334,640,023}$ decimal digits. We're well, well past the universe-cube already.
Its suffices to say, if this isn't even in the realm of $3 \uparrow \uparrow \uparrow \uparrow 3$, and $3 \uparrow \uparrow \uparrow \uparrow 3$ isn't even in the realm of Graham's number, the universe-cube is just not enough. Graham's number pretty much defies analogies based in reality. Its status as one of the biggest (meaningful) numbers is not given lightly.
For fun, just to continue seeing just how bad off we are, I continued approximating the power tower of $n$ threes for higher $n$ than $5$. For simplicity I keep using $x \uparrow y$ as a substitute for $x^y$ for readability - that I even have to do so shows just how nuts this gets.
$n=10$ yielded about $\underbrace{10 \uparrow \cdots \uparrow 10}_{\text{seven 10's}} \uparrow 12$ digits (Wolfram link) $n=15$ yielded about $\underbrace{10 \uparrow \cdots \uparrow 10}_{\text{twelve 10's}} \uparrow 12$ digits (Wolfram link)
Wolfram wouldn't even return anything in a meaningful timeframe for $n=20$ but I imagine the idea is the number of $10$'s in these is roughly $n-3$.
And then bear in mind what Graham's number is. Let $G_1 = 3 \uparrow^4 3$ (where the $4$ means "put four arrows here"). Then Graham's number is defined recursively by
$$G_{n+1} = 3 \uparrow^{G_n} 3$$
Each time a number is giving the number of arrows in the next, and $G_{64}$ is Graham's number. And remember just how going from one to four arrows blew up to the point of $G_1$ being incalculable for all intents and purposes?
To say it is big, large, huge, monstrous, or any single adjective in the English language doesn't do it justice.
It just straight up defies reality and practicality; good luck at even trying to approach it with an analogy grounded in reality and not something else abstract and mathematical in nature like the problem it was designed for. At best all the analogy would indicate is just how massive Graham's number is by the sheer way in which Graham's number dwarfs it.
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I'm trying to learn how to apply WKB. I asked a similar question already, but that question was related to finding the energies. Here, I would like to understand how to find the wave functions using WKB.
An electron, say, in the nuclear potential$$U(r)=\begin{cases} & -U_{0} \;\;\;\;\;\;\text{ if } r < r_{0} \\ & k/r \;\;\;\;\;\;\;\;\text{ if } r > r_{0} \end{cases}$$ What is the wave function inside the barrier region ($r_{0} < r < k/E$)?
Shouldn't the wave function have the following form?
$$\psi(r)=\frac{A}{\sqrt{2m(E-U(r))}}e^{\phi(r)}+\frac{B}{\sqrt{2m(E-U(r))}}e^{-\phi(r)}$$ where $$\phi(r)=\frac{1}{\hbar}\int_{0}^{r} \sqrt{2m(E-U(r))} dr'$$
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Sandbox From AbInitio
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This is a "sandbox" page where you can practice editing. Feel free to scribble any nonsense you want. (The contents of this page may be deleted at any time.)
Please only scribble below this line, however.
Scribble below
MediaWiki allows you to enter LaTeX equations using the <math> tag. For example:
∠
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X F Wang
Articles written in Pramana – Journal of Physics
Volume 71 Issue 1 July 2008 pp 167-173 Research Articles
The EPR 𝑔 factors $g_{i}$ $(i = x, y, z)$ for the interstitial Ti
3+ in rutile are theoretically studied from the perturbation formulas of these parameters for a 3d 1 ion in rhombically compressed octahedra. The ligand octahedron in the impurity center is found to be less compressed than that on the host interstitial site due to the Jahn–Teller effect. The local compression parameter $(\approx 0.026)$ and the rhombic distortion angle $\delta \phi'$ $(\approx 0.7^{\circ})$ around the impurity Ti 3+ are smaller than the host values ($\approx 0.091$ and 3.5°). The theoretical 𝑔 factors based on the above local structural parameters are in good agreement with the experimental data. In addition, the 𝑔 factors for a tetragonal interstitial Ti 3+ center are also reasonably interpreted.
Volume 72 Issue 6 June 2009 pp 989-997 Research Articles
The 𝑔 factors and the ligand superhyperfine parameters $A'$ and $B'$ for Ni
3+ in KMgF 3, CsCaF 3 and RbCaF 3 are theoretically studied from the formulas of these parameters for a 3d 7 ion under octahedral environments in the weak field scheme. The unpaired spin densities for the fluorine 2s, 2p$_{\sigma}$ and 2p$_{\pi}$ orbitals are quantitatively determined from the molecular orbital and configuration interaction coefficients based on the cluster approach. The calculated results show good agreement with the experimental data, based on only one adjustable parameter (i.e., the proportionality factor 𝜌 related to the ligand s- and p-orbitals). The superhyperfine parameters for the axial and planar ligands in RbCaF 3:Ni 3+ are satisfactorily interpreted from the different impurity–ligand distances due to the elongation of the ligand octahedron during cubic-to-tetragonal phase transition.
Current Issue
Volume 93 | Issue 5 November 2019
Click here for Editorial Note on CAP Mode
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I am working on a cryptographic scheme and I need to rely on the following problem, which I have nicknamed the
"Hybrid Decisional Bilinear Diffie Hellman (hDBDH)" problem:
Let $e: \mathbb G_1 \times \mathbb G_1 \rightarrow \mathbb G_T$ be an efficient bilinear pairing. Given the tuple $(g,g^x,e(g,g)^y,Q)$ as input, the problem is to decide whether $Q = e(g,g)^{xy}$
First, has this problem already been defined? Now, I want to prove that this problem is equivalent to the DBDH problem. I
have the feeling that they are equivalent, but so far I only have that $DBDH \leq hDBDH$:
Proof:From a hDBDH solver, we can construct a DBDH solver in the following way:
Input: DBDH tuple = $(g, g^a, g^b, g^c, Q)$
Output: Return $hDBDH_{\mathsf{solver}}(g,g^a,e(g^b,g^c),Q)$
Now I want to prove the other direction. Any ideas of how a DBDH oracle could be used to solve the hDBDH problem? Any ideas of why it is not possible?
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Generally speaking functions of operators are defined as
spectral functions, making use of the spectral theorem machinery.
The approaches based of power series usually have not a rigorous basis: Even identities like $$e^A = \sum_n \frac{1}{n!}A^n$$ are generally false if $A$ is an
unbounded operator.
Nevertheless in most relevant physical cases some of the results obtained by formal (non-rigorous) manipulations can be obtained following alternative rigorous ways.
However the case you are considering is very easy. Suppose that $F$ is a smooth, generally
non-analytic, complex-valued function bounded by some polynomial of $\vec{x}$. In this case both $F(\vec{x})$ and $p_i$ admit the space of Schwartz functions ${\cal S}(\mathbb R^3)$ as invariant space, so that $[p_i, F(\vec{x})]\psi$is well defined for $\psi \in {\cal S}(\mathbb R^3)$. On that space $p_i$coincides to $-i \hbar\frac{\partial}{\partial x_i}$. By direct computation:$$[p_i, F(\vec{x})]\psi = -i \hbar\frac{\partial}{\partial x_i} F(\vec{x}) \psi(x) + i \hbar F(\vec{x})\frac{\partial \psi}{\partial x_i}= -i\hbar \frac{\partial F}{\partial x_i} \psi\:.$$We have obtained that$$\left([p_i, F(\vec{x})] + i\hbar \frac{\partial F}{\partial x_i} \right) \psi =0 \quad \forall \psi \in {\cal S}(\mathbb R^3)\:.$$In other words, at least on the domain ${\cal S}(\mathbb R^3)$: $$[p_i, F(\vec{x})] = -i\hbar \frac{\partial F}{\partial x_i} $$In general, this identity holds on larger domains and, in fact, it can be extended exploiting some further know property of $F$ and some other properties like the fact that $\cal S(\mathbb R^3)$ is a core for $p_i$...
One could assume weaker hypotheses. The self-adjointness domain of $p_j$ is so made.$$D(p_j) := \{\psi \in L^2(\mathbb R^3)\:|\: \exists\: w\mbox{-}\partial_{x_j}\psi \in L^2(\mathbb R^3)\}$$where $w\mbox{-}\partial_{x_j}\psi$ denotes the weak partial $j$-derivative of $\psi$. On that domain $p_j = w\mbox{-}\partial_{x_j}$ as expected.
If $F$ is, for instance, just $C^1$ and compactly supported, all above reasoning can be re-implemented with $\psi \in D(p_j)$, obtaining exactly the same result, using the fact that the weak derivative of a compactly supported $C^1$ function $F$ coincides with the standard one.
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Valter Moretti's answer is very nice, and I learned some things from reading it. This answer is meant to be a lower-level explanation of the basics of this topic, giving a treatment in the same style that is found when most GR texts introduce this topic.
Definition of a Killing vector
A Killing vector $\xi$ is vector field describing a symmetry of a spacetime. If we move every point in the spacetime by an infinitesimal amount, the direction and amount being determined by the Killing vector, then the metric gives the same results. A Killing vector can be defined as a solution to Killing's equation,
$$ \nabla_a \xi_b + \nabla_b \xi_a = 0,$$
i.e., the covariant derivative is asymmetric on the two indices.
Now suppose that $p$ is a tangent vector along a geodesic. By this we mean that it satisfies the geodesic equation $p^a \nabla_a p^b = 0$. This equation states not just that $p$ stays tangent to a geodesic, i.e., is transported parallelly to itself, but also that it's parallel-transported along this geodesic in an affine manner, so that it doesn't "change length" as we go along. This is an affine notion of "not changing length," not a metrical one, so it applies equally well if $p$ is null rather than timelike. For a massive or massless particle moving inertially, the momentum is a tangent vector to the geodesic in this sense.
The conserved quantity associated with a Killing vector
Theorem: Under the assumptions given above, $p_b \xi^b$ is a conserved quantity along the geodesic. That is, it is constant for a test particle.
Proof: We prove this by showing that
$$p^a\nabla_a (p_b \xi^b)=0.$$
Application of the product rule gives
$$p^a \xi^b \nabla_a p_b+p^a p_b \nabla_a \xi^b.$$
The first term vanishes by the geodesic equation, and the second term by the antisymmetry expressed by the Killing equation, combined with the symmetry of $p^a p_b$.
None of the above is changed if we scale $p$ by some factor. In the case of a massive particle, it may be more convenient to let $p$ be the momentum per unit mass, so that all expressions depend only on the geodesic.
Nowhere in this argument was it necessary to make any assumptions about whether $\xi$ was timelike, null, or spacelike.
Some special cases
Some special cases are of interest. Suppose that the metric is independent of one coordinate $x^\mu$. Then $\partial_\mu$ is a killing vector , and $p_\mu$ is conserved.
A series of further restrictions to more and more special cases: ---
If the metric is independent of $t$, where $t$ is a timelike coordinate, it's $p_t$ (a component of the covariant momentum vector) that's conserved, but it's usually $p^t$ that we call "the" energy.
When the metric is also diagonal, it's $g_{tt} p^t$ that is conserved. For the Schwarzschild metric, this is $(1-2M/r)E$, where E is the energy measured by a static observer, i.e., an observer whose velocity vector is parallel to the Killing vector. (Note that this sequence of interpretations doesn't work when the Killing vector isn't timelike.)
Null or spacelike Killing vectors: an example
To see what happens when $\xi$ isn't timelike, it's helpful to look at the special case of a photon infalling from $r=+\infty$ into a Schwarzschild black hole. Then in Schwarzschild coordinates, $ds^2=0$ gives $dr/dt=\pm A$, where $A=1-2M/r=g_{tt}$.
Outside the horizon, $dr/dt=-A$, $p^t$ is the energy measured by a static observer hovering at $r$, and the conserved quantity $p_t=p^tA$ is the redshifted energy seen by an observer at infinity.
As the same particle passes into the interior of the horizon, its trajectory now has $dr/dt=+A$, which is still negative. There are no static observers here, so $p^t$, which is negative, is not the energy seen by any observer.
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No laws are broken, but Planck's constant is higher than in our world.
I would say that no physical laws are broken, it's just that the double jump is performed in a universe where Planck's constant has a (much) higher value.
Looking at the uncertainty principle for energy:$$\Delta E \Delta t \geq \frac{\hbar}{2}$$Just like a virtual particle can break energy conservation for a short time, so can your jumper. When he jumps from the wall, he converts $\Delta E $ muscle energy into $\Delta E/2$ kinetic energy. And a short time later, mid-air, he converts the other $\Delta E/2$ into kinetic energy. In the time period between the two jumps, energy conservation was broken, but because Planck's constant is so large, it is ok.
Another way to look at it, using the uncertainty principle for position-momentum:$$\Delta q \Delta p \geq \frac{\hbar}{2}$$
You say that at the second jump, he is mid-air. Are you sure? If you are very sure that he is not close to the wall, you can not be certain about his momentum, so how do you know that his momentum changed upwards?And if you are certain that he jumped upwards, according to the uncertainty principle you don't know much about his position, so he might still be at the wall the second time...
By the way: for this to be true, Planck's constant needs to be at least $10^{34}$ times higher than it currently is. The universe would look very different.
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2018-08-25 06:58
Recent developments of the CERN RD50 collaboration / Menichelli, David (U. Florence (main) ; INFN, Florence)/CERN RD50 The objective of the RD50 collaboration is to develop radiation hard semiconductor detectors for very high luminosity colliders, particularly to face the requirements of the possible upgrade of the large hadron collider (LHC) at CERN. Some of the RD50 most recent results about silicon detectors are reported in this paper, with special reference to: (i) the progresses in the characterization of lattice defects responsible for carrier trapping; (ii) charge collection efficiency of n-in-p microstrip detectors, irradiated with neutrons, as measured with different readout electronics; (iii) charge collection efficiency of single-type column 3D detectors, after proton and neutron irradiations, including position-sensitive measurement; (iv) simulations of irradiated double-sided and full-3D detectors, as well as the state of their production process.. 2008 - 5 p. - Published in : Nucl. Instrum. Methods Phys. Res., A 596 (2008) 48-52 In : 8th International Conference on Large Scale Applications and Radiation Hardness of Semiconductor Detectors, Florence, Italy, 27 - 29 Jun 2007, pp.48-52 详细记录 - 相似记录 2018-08-25 06:58 详细记录 - 相似记录 2018-08-25 06:58
Performance of irradiated bulk SiC detectors / Cunningham, W (Glasgow U.) ; Melone, J (Glasgow U.) ; Horn, M (Glasgow U.) ; Kazukauskas, V (Vilnius U.) ; Roy, P (Glasgow U.) ; Doherty, F (Glasgow U.) ; Glaser, M (CERN) ; Vaitkus, J (Vilnius U.) ; Rahman, M (Glasgow U.)/CERN RD50 Silicon carbide (SiC) is a wide bandgap material with many excellent properties for future use as a detector medium. We present here the performance of irradiated planar detector diodes made from 100-$\mu \rm{m}$-thick semi-insulating SiC from Cree. [...] 2003 - 5 p. - Published in : Nucl. Instrum. Methods Phys. Res., A 509 (2003) 127-131 In : 4th International Workshop on Radiation Imaging Detectors, Amsterdam, The Netherlands, 8 - 12 Sep 2002, pp.127-131 详细记录 - 相似记录 2018-08-24 06:19
Measurements and simulations of charge collection efficiency of p$^+$/n junction SiC detectors / Moscatelli, F (IMM, Bologna ; U. Perugia (main) ; INFN, Perugia) ; Scorzoni, A (U. Perugia (main) ; INFN, Perugia ; IMM, Bologna) ; Poggi, A (Perugia U.) ; Bruzzi, M (Florence U.) ; Lagomarsino, S (Florence U.) ; Mersi, S (Florence U.) ; Sciortino, Silvio (Florence U.) ; Nipoti, R (IMM, Bologna) Due to its excellent electrical and physical properties, silicon carbide can represent a good alternative to Si in applications like the inner tracking detectors of particle physics experiments (RD50, LHCC 2002–2003, 15 February 2002, CERN, Ginevra). In this work p$^+$/n SiC diodes realised on a medium-doped ($1 \times 10^{15} \rm{cm}^{−3}$), 40 $\mu \rm{m}$ thick epitaxial layer are exploited as detectors and measurements of their charge collection properties under $\beta$ particle radiation from a $^{90}$Sr source are presented. [...] 2005 - 4 p. - Published in : Nucl. Instrum. Methods Phys. Res., A 546 (2005) 218-221 In : 6th International Workshop on Radiation Imaging Detectors, Glasgow, UK, 25-29 Jul 2004, pp.218-221 详细记录 - 相似记录 2018-08-24 06:19
Measurement of trapping time constants in proton-irradiated silicon pad detectors / Krasel, O (Dortmund U.) ; Gossling, C (Dortmund U.) ; Klingenberg, R (Dortmund U.) ; Rajek, S (Dortmund U.) ; Wunstorf, R (Dortmund U.) Silicon pad-detectors fabricated from oxygenated silicon were irradiated with 24-GeV/c protons with fluences between $2 \cdot 10^{13} \ n_{\rm{eq}}/\rm{cm}^2$ and $9 \cdot 10^{14} \ n_{\rm{eq}}/\rm{cm}^2$. The transient current technique was used to measure the trapping probability for holes and electrons. [...] 2004 - 8 p. - Published in : IEEE Trans. Nucl. Sci. 51 (2004) 3055-3062 In : 50th IEEE 2003 Nuclear Science Symposium, Medical Imaging Conference, 13th International Workshop on Room Temperature Semiconductor Detectors and Symposium on Nuclear Power Systems, Portland, OR, USA, 19 - 25 Oct 2003, pp.3055-3062 详细记录 - 相似记录 2018-08-24 06:19
Lithium ion irradiation effects on epitaxial silicon detectors / Candelori, A (INFN, Padua ; Padua U.) ; Bisello, D (INFN, Padua ; Padua U.) ; Rando, R (INFN, Padua ; Padua U.) ; Schramm, A (Hamburg U., Inst. Exp. Phys. II) ; Contarato, D (Hamburg U., Inst. Exp. Phys. II) ; Fretwurst, E (Hamburg U., Inst. Exp. Phys. II) ; Lindstrom, G (Hamburg U., Inst. Exp. Phys. II) ; Wyss, J (Cassino U. ; INFN, Pisa) Diodes manufactured on a thin and highly doped epitaxial silicon layer grown on a Czochralski silicon substrate have been irradiated by high energy lithium ions in order to investigate the effects of high bulk damage levels. This information is useful for possible developments of pixel detectors in future very high luminosity colliders because these new devices present superior radiation hardness than nowadays silicon detectors. [...] 2004 - 7 p. - Published in : IEEE Trans. Nucl. Sci. 51 (2004) 1766-1772 In : 13th IEEE-NPSS Real Time Conference 2003, Montreal, Canada, 18 - 23 May 2003, pp.1766-1772 详细记录 - 相似记录 2018-08-24 06:19
Radiation hardness of different silicon materials after high-energy electron irradiation / Dittongo, S (Trieste U. ; INFN, Trieste) ; Bosisio, L (Trieste U. ; INFN, Trieste) ; Ciacchi, M (Trieste U.) ; Contarato, D (Hamburg U., Inst. Exp. Phys. II) ; D'Auria, G (Sincrotrone Trieste) ; Fretwurst, E (Hamburg U., Inst. Exp. Phys. II) ; Lindstrom, G (Hamburg U., Inst. Exp. Phys. II) The radiation hardness of diodes fabricated on standard and diffusion-oxygenated float-zone, Czochralski and epitaxial silicon substrates has been compared after irradiation with 900 MeV electrons up to a fluence of $2.1 \times 10^{15} \ \rm{e} / cm^2$. The variation of the effective dopant concentration, the current related damage constant $\alpha$ and their annealing behavior, as well as the charge collection efficiency of the irradiated devices have been investigated.. 2004 - 7 p. - Published in : Nucl. Instrum. Methods Phys. Res., A 530 (2004) 110-116 In : 6th International Conference on Large Scale Applications and Radiation Hardness of Semiconductor Detectors, Florence, Italy, 29 Sep - 1 Oct 2003, pp.110-116 详细记录 - 相似记录 2018-08-24 06:19
Recovery of charge collection in heavily irradiated silicon diodes with continuous hole injection / Cindro, V (Stefan Inst., Ljubljana) ; Mandić, I (Stefan Inst., Ljubljana) ; Kramberger, G (Stefan Inst., Ljubljana) ; Mikuž, M (Stefan Inst., Ljubljana ; Ljubljana U.) ; Zavrtanik, M (Ljubljana U.) Holes were continuously injected into irradiated diodes by light illumination of the n$^+$-side. The charge of holes trapped in the radiation-induced levels modified the effective space charge. [...] 2004 - 3 p. - Published in : Nucl. Instrum. Methods Phys. Res., A 518 (2004) 343-345 In : 9th Pisa Meeting on Advanced Detectors, La Biodola, Italy, 25 - 31 May 2003, pp.343-345 详细记录 - 相似记录 2018-08-24 06:19
First results on charge collection efficiency of heavily irradiated microstrip sensors fabricated on oxygenated p-type silicon / Casse, G (Liverpool U.) ; Allport, P P (Liverpool U.) ; Martí i Garcia, S (CSIC, Catalunya) ; Lozano, M (Barcelona, Inst. Microelectron.) ; Turner, P R (Liverpool U.) Heavy hadron irradiation leads to type inversion of n-type silicon detectors. After type inversion, the charge collected at low bias voltages by silicon microstrip detectors is higher when read out from the n-side compared to p-side read out. [...] 2004 - 3 p. - Published in : Nucl. Instrum. Methods Phys. Res., A 518 (2004) 340-342 In : 9th Pisa Meeting on Advanced Detectors, La Biodola, Italy, 25 - 31 May 2003, pp.340-342 详细记录 - 相似记录 2018-08-23 11:31
Formation and annealing of boron-oxygen defects in irradiated silicon and silicon-germanium n$^+$–p structures / Makarenko, L F (Belarus State U.) ; Lastovskii, S B (Minsk, Inst. Phys.) ; Korshunov, F P (Minsk, Inst. Phys.) ; Moll, M (CERN) ; Pintilie, I (Bucharest, Nat. Inst. Mat. Sci.) ; Abrosimov, N V (Unlisted, DE) New findings on the formation and annealing of interstitial boron-interstitial oxygen complex ($\rm{B_iO_i}$) in p-type silicon are presented. Different types of n+−p structures irradiated with electrons and alpha-particles have been used for DLTS and MCTS studies. [...] 2015 - 4 p. - Published in : AIP Conf. Proc. 1583 (2015) 123-126 详细记录 - 相似记录
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While vertical asymptotes describe the behavior of a graph as the
output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
There are three distinct outcomes when checking for horizontal asymptotes:
Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at y = 0.
In this case, the end behavior is [latex]f\left(x\right)\approx \frac{4x}{{x}^{2}}=\frac{4}{x}[/latex]. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\left(x\right)=\frac{4}{x}[/latex], and the outputs will approach zero, resulting in a horizontal asymptote at
y = 0. Note that this graph crosses the horizontal asymptote.
Figure 12. Horizontal Asymptote y = 0 when [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}, q\left(x\right)\ne{0}\text{ where degree of }p<\text{degree of }q[/latex]. Case 2: If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.
In this case, the end behavior is [latex]f\left(x\right)\approx \frac{3{x}^{2}}{x}=3x[/latex]. This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\left(x\right)=3x[/latex]. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of [latex]g\left(x\right)=3x[/latex] looks like a diagonal line, and since
f will behave similarly to g, it will approach a line close to [latex]y=3x[/latex]. This line is a slant asymptote.
To find the equation of the slant asymptote, divide [latex]\frac{3{x}^{2}-2x+1}{x - 1}[/latex]. The quotient is [latex]3x+1[/latex], and the remainder is 2. The slant asymptote is the graph of the line [latex]g\left(x\right)=3x+1[/latex].
Figure 13. Slant Asymptote when [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0[/latex] where degree of [latex]p>\text{degree of }q\text{ by }1[/latex]. Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at [latex]y=\frac{{a}_{n}}{{b}_{n}}[/latex], where [latex]{a}_{n}[/latex] and [latex]{b}_{n}[/latex] are the leading coefficients of [latex]p\left(x\right)[/latex] and [latex]q\left(x\right)[/latex] for [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0[/latex].
In this case, the end behavior is [latex]f\left(x\right)\approx \frac{3{x}^{2}}{{x}^{2}}=3[/latex]. This tells us that as the inputs grow large, this function will behave like the function [latex]g\left(x\right)=3[/latex], which is a horizontal line. As [latex]x\to\pm\infty ,f\left(x\right)\to 3[/latex], resulting in a horizontal asymptote at
y = 3. Note that this graph crosses the horizontal asymptote.
Figure 14. Horizontal Asymptote when [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0\text{ where degree of }p=\text{degree of }q[/latex].
Notice that, while the graph of a rational function will never cross a
vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.
It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the
end behavior of the graph will mimic the behavior of the reduced end behavior \fraction. For instance, if we had the function
with end behavior
the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.
A General Note: Horizontal Asymptotes of Rational Functions
The
horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator. Degree of numerator is less thandegree of denominator: horizontal asymptote at y= 0. Degree of numerator is greater than degree of denominator by one: no horizontal asymptote; slant asymptote. Degree of numerator is equal todegree of denominator: horizontal asymptote at ratio of leading coefficients. Example 7: Identifying Horizontal and Slant Asymptotes
For the functions below, identify the horizontal or slant asymptote.
[latex]g\left(x\right)=\frac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[/latex] [latex]h\left(x\right)=\frac{{x}^{2}-4x+1}{x+2}[/latex] [latex]k\left(x\right)=\frac{{x}^{2}+4x}{{x}^{3}-8}[/latex] Solution
For these solutions, we will use [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}, q\left(x\right)\ne 0[/latex].
[latex]g\left(x\right)=\frac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[/latex]: The degree of [latex]p=\text{degree of } q=3[/latex], so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at [latex]y=\frac{6}{2}[/latex] or [latex]y=3[/latex]. [latex]h\left(x\right)=\frac{{x}^{2}-4x+1}{x+2}[/latex]: The degree of [latex]p=2[/latex] and degree of [latex]q=1[/latex]. Since [latex]p>q[/latex] by 1, there is a slant asymptote found at [latex]\frac{{x}^{2}-4x+1}{x+2}[/latex].
The quotient is [latex]x - 6[/latex] and the remainder is 13. There is a slant asymptote at [latex]y=x - 6[/latex].
[latex]k\left(x\right)=\frac{{x}^{2}+4x}{{x}^{3}-8}[/latex]: The degree of [latex]p=2\text{ }<[/latex] degree of [latex]q=3[/latex], so there is a horizontal asymptote y= 0. Example 8: Identifying Horizontal Asymptotes
In the sugar concentration problem earlier, we created the equation [latex]C\left(t\right)=\frac{5+t}{100+10t}[/latex].
Find the horizontal asymptote and interpret it in context of the problem.
Solution
Both the numerator and denominator are linear (degree 1). Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. In the numerator, the leading term is
t, with coefficient 1. In the denominator, the leading term is 10 t, with coefficient 10. The horizontal asymptote will be at the ratio of these values:
This function will have a horizontal asymptote at [latex]y=\frac{1}{10}[/latex].
This tells us that as the values of
t increase, the values of C will approach [latex]\frac{1}{10}[/latex]. In context, this means that, as more time goes by, the concentration of sugar in the tank will approach one-tenth of a pound of sugar per gallon of water or [latex]\frac{1}{10}[/latex] pounds per gallon. Example 9: Identifying Horizontal and Vertical Asymptotes
Find the horizontal and vertical asymptotes of the function
Solution
First, note that this function has no common factors, so there are no potential removable discontinuities.
The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at [latex]x=1,-2,\text{and }5[/latex], indicating vertical asymptotes at these values.
The numerator has degree 2, while the denominator has degree 3. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as [latex]x\to \pm \infty , f\left(x\right)\to 0[/latex]. This function will have a horizontal asymptote at [latex]y=0[/latex].
Try It 6
Find the vertical and horizontal asymptotes of the function:
[latex]f\left(x\right)=\frac{\left(2x - 1\right)\left(2x+1\right)}{\left(x - 2\right)\left(x+3\right)}[/latex]
A General Note: Intercepts of Rational Functions
A
rational function will have a y-intercept when the input is zero, if the function is defined at zero. A rational function will not have a y-intercept if the function is not defined at zero.
Likewise, a rational function will have
x-intercepts at the inputs that cause the output to be zero. Since a \fraction is only equal to zero when the numerator is zero, x-intercepts can only occur when the numerator of the rational function is equal to zero. Example 10: Finding the Intercepts of a Rational Function
Find the intercepts of [latex]f\left(x\right)=\frac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)}[/latex].
Solution
We can find the
y-intercept by evaluating the function at zero
The
x-intercepts will occur when the function is equal to zero:
[latex]0=\frac{(x-2)(x+3)}{(x-1)(x+2)(x-5)}[/latex] This is zero when the numerator is zero.
[latex]0=(x-2)(x+3)[/latex]
[latex]x=2, -3[/latex]
The
y-intercept is [latex]\left(0,-0.6\right)[/latex], the x-intercepts are [latex]\left(2,0\right)[/latex] and [latex]\left(-3,0\right)[/latex]. Try It 7
Given the reciprocal squared function that is shifted \right 3 units and down 4 units, write this as a rational function. Then, find the
x– and y-intercepts and the horizontal and vertical asymptotes.
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Injection iff Left Cancellable Theorem Proof $\forall X: \forall g_1: X \to Y, g_2: X \to Y: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$
Let $f: Y \to Z$ be an injection.
Let $X$ be a set
Let $g_1: X \to Y, g_2: X \to Y$ be mappings such that:
$f \circ g_1 = f \circ g_2$ Then $\forall x \in X$:
\(\displaystyle \map f {g_1 \paren x}\) \(=\) \(\displaystyle \map {f \circ g_1} x\) Definition of Composite Mapping \(\displaystyle \) \(=\) \(\displaystyle \map {f \circ g_2} x\) By Hypothesis \(\displaystyle \) \(=\) \(\displaystyle \map f {\map {g_2} x}\) Definition of Composite Mapping
$\Box$
We use a Proof by Contraposition.
Hence, suppose $f: Y \to Z$ is not injective.
Then:
$\exists y_1 \ne y_2 \in Y: f \left({y_1}\right) = f \left({y_2}\right)$
Let the two mappings $g_1: Y \to Y, g_2: Y \to Y$ be defined as follows:
$\forall y \in Y: g_1 \left({y}\right) = y$ $\forall y \in Y: g_2 \left({y}\right) = \begin{cases} y_2 & : y = y_1 \\ y & : y \ne y_1 \end{cases}$
Thus we have $g_1 \ne g_2$ such that $f \circ g_1 = f \circ g_2$.
That is, $f$ is not left cancellable.
From Rule of Transposition it follows that if $f$ is left cancellable, it is injective.
$\blacksquare$
Also see Sources 1965: Seth Warner: Modern Algebra... (previous) ... (next): Exercise $8.10 \ \text{(a)}$ 1967: George McCarty: Topology: An Introduction with Application to Topological Groups... (previous) ... (next): $\text{I}$: Problem $\text{BB}$ 1968: Ian D. Macdonald: The Theory of Groups... (previous) ... (next): Appendix: Elementary set and number theory 1975: T.S. Blyth: Set Theory and Abstract Algebra... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Theorem $5.4$ 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability... (previous) ... (next): Appendix $\text{A}.5$: Identity, One-one, and Onto Functions: Proposition $\text{A}.5.1: 1 \ \text{(d)}$
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Now we know how to do some math with vectors, and the question arises, “If we can add and subtract vectors, can we also multiply them?” The answer is yes and no. It turns out that there is not one unique way to define a product of two vectors, but instead there are two…
Scalar Product
Soon we will be looking at how we can describe the effect that a force pushing on an object has on its speed as it moves from one position to another. The force is a vector, because it has a magnitude (the amount of the push) and a direction (the way the push is directed). And the movement of the object is also a vector (tail is at the object’s starting point, and head is at its ending point). It will turn out that this effect is describable mathematically as the product of the amount of force and the amount of movement. This is simple to compute if the push is along the direction of movement, but what if it is not? It turns out that only the amount of push that
acts in the direction of the movement will affect the object's speed.
We therefore would like to introduce the notion of the
projection of one vector onto another. The best description of this is, "the amount of a given vector that points along the other vector." This could be imagined as the “shadow” one vector casts upon another vector: Figure 1.2.1 – Projecting One Vector Onto Another
So if we want to multiply the length of a vector by the amount of a second vector that is projected onto it we get:
\[ ( \text{projection of } \overrightarrow A \text{ onto } \overrightarrow B )( \text{magnitude of } \overrightarrow B ) = (A \cos \theta) (B) = AB \cos \theta\]
This is the first of the two types of vector multiplication, and it is called a
scalar product, because the result of the product is a scalar. We usually write the product with a dot (giving its alternative name of dot product):
Example \(\PageIndex{1}\)
The vector \(\overrightarrow A\) has a magnitude of 120 units, and when projected onto \(\overrightarrow B\), the projected portion has a value of 105 units. Suppose that \(\overrightarrow B\) is now projected onto \(\overrightarrow A\), and the projected length is 49 units. Find the magnitude of \(\overrightarrow B\). Solution The factor that determines the length of the projection is \(\cos \theta\). The angle between the two vectors is the same regardless of which vector is projected, so the factor is the same in both directions. The projection of \(\overrightarrow A\) onto \(\overrightarrow B\) is 7/8 of the magnitude of \(\overrightarrow A\), so the magnitude of \(\overrightarrow B\) must be 8/7 of its projection, which is 56 units. Note that when the projection of one vector is multiplied by the magnitude of the other, the same product results regardless of which way the projection occurs. That is, the scalar product is the same in either order (i.e. it is commutative).
Note that a scalar product of a vector with itself is the square of the magnitude of that vector:
\[\overrightarrow A \cdot \overrightarrow A = A^2 \cos 0 = A^2\]
It should be immediately clear what the scalar products of the unit vectors are. They have unit length, so a scalar product of a unit vector with itself is just 1.
\[\widehat i \cdot \widehat i = \widehat j \cdot \widehat j = \widehat k \cdot \widehat k = 1\]
They are also
mutually orthogonal, so the scalar products with each other are zero:
\[\widehat i \cdot \widehat j = \widehat j \cdot \widehat k = \widehat k \cdot \widehat i = 0\]
This gives us an alternative way to look at components, which are projections of a vector onto the coordinate axes. Since the unit vectors point along the \(x\), \(y\), and \(z\) directions, the components of a vector can be expressed as a dot product. For example:
\[ \begin{align} \overrightarrow A \cdot \widehat i &= (A_x \widehat i + A_y \widehat j + A_z \widehat k) \cdot \widehat i \nonumber\\[5pt] &= A_x \cancelto{1}{\widehat i \cdot \widehat i } + A_y \cancelto{0} {\widehat j \cdot \widehat i} +A_z \cancelto{0}{ \widehat k \cdot \widehat i} \end{align}\]
Unit vectors also show us an easy way to take the scalar product of two vectors whose components we know. Start with two vectors written in component form:
\[\overrightarrow A = A_x\widehat i + A_y \widehat j\nonumber\]
\[\overrightarrow B = B_x\widehat i + B_y \widehat j\nonumber\]
then just do "normal algebra," distributing the dot product as you would with normal multiplication:
\[ \begin{align} \overrightarrow A \cdot \overrightarrow B &= (A_x \widehat i + A_y \widehat j) \cdot (B_x \widehat i + B_y \widehat j) \nonumber\\[5pt] &= (A_x B_x) \cancelto{1}{\widehat i \cdot \widehat i} + (A_y B_x) \cancelto{0} {\widehat j \cdot \widehat i} + (A_x B_y ) \cancelto{0}{ \widehat i \cdot \widehat j} + (B_x B_x) \cancelto{1}{\widehat j \cdot \widehat j} \nonumber\\[5pt] &= A_x B_x + A_yB_y \end{align} \]
If we didn’t have this simple result, think about what we would have to do: We would need to calculate the angles each vector makes with (say) the \(x\)-axis. Then from those two angles, we need to figure out the angles between the two vectors. Then we would need to compute the magnitudes of the two vectors. Finally, with the magnitudes of the vectors and the angle between the vectors, we could finally plug into our scalar product equation.
Alert
With two different ways to compute a scalar product, it should be clear that the simplest method to use will depend upon what information is provided. If you are given (or can easily ascertain) the magnitudes of the vectors and the angle between them, then use Equation 1.2.2, but if you are given (or can easily ascertain) the components of the vectors, use Equation 1.2.7.
Example \(\PageIndex{2}\)
The two vectors given below are perpendicular to each other. Find the unknown \(z\)-component.
\[\overrightarrow A = +5 \widehat i - 4 \widehat j - \widehat k \;\;\;\;\;\; \overrightarrow B = +2 \widehat i + 3 \widehat j + B_z \widehat k \nonumber\]
Solution The scalar product of two vectors is proportional to the cosine of the angle between them. This means that if they are orthogonal, the scalar product is zero. The dot product is easy to compute when given the components, so we do so and solve for \(B_z\): \[0 = \overrightarrow A \cdot \overrightarrow B = \left( +5 \right) \left( +2 \right) + \left( -4 \right) \left( +3 \right) \ + \left( -1 \right) \left( B_z \right) \;\;\; \Rightarrow \;\;\; B_z = -2 \nonumber\]
The scalar product of two vectors in terms of column vectors works exactly how you would expect – simply multiply the similar components and sum all the products.
Vector Product
As mentioned earlier, there are actually two ways to define products of vectors. If the scalar product involves the amount of one vector that is
parallel to the other vector, then it should not be surprising that our other product involves the amount of a vector that is perpendicular to the other vector. Figure 1.2.2 – Portion of One Vector Perpendicular to Another
If we take a product like before, multiplying this perpendicular piece by the magnitude of the other vector, we get an expression similar to what we got for the scalar product, this time with a sine function rather than a cosine. For reasons that will be clear soon, this type of product is referred to as a
vector product. Because this is distinct from the scalar product, we use a different mathematical notation as well – a cross rather than a dot (giving it an alternative name of cross product). This has a simple (though not entirely useful, at least not in physics) geometric interpretation in terms of the parallelogram defined by the two vectors: Figure 1.2.3 – Constructing a Vector Product (Magnitude)
\[ magnitude\; of\; \overrightarrow A \times \overrightarrow B = \left| \overrightarrow A \times \overrightarrow B \right| = \left| \overrightarrow A\right| \left| \overrightarrow B \right| \sin\theta = AB\sin\theta \]
But there is another even bigger difference between the vector and scalar products. While the projection always lands parallel to the second vector, the perpendicular part implies an orientation, since the perpendicular part can point in multiple directions. Any quantity that has an orientation has the potential to be a vector, and in fact we will define a vector that results from this type of product as follows: If we follow the perimeter of the parallelogram above in the direction given by the two vectors, we get a clockwise orientation [
Would we get the same orientation if the product was in the opposite order: \(\overrightarrow B\) \(\times\) \(\overrightarrow A\)?]. We turn this circulation direction into a vector direction (which points in a specific direction in space) using a convention called the right-hand-rule:
Convention: Right-hand-rule
Point the fingers of your right hand in the direction of the first vector in the product, then orient your hand such that those fingers curl naturally into the direction of the second vector in the product. As your fingers curl, your extended thumb points in a direction that is perpendicular to both vectors in the product. This is the direction of the vector that results from the cross product.
If we perform the cross product with the vectors in the opposite order, our fingers curl in the opposite direction, which makes our thumb point in the opposite direction in space. This means that the cross product has an
anticommutative property:
\[ \overrightarrow A \times \overrightarrow B = -\overrightarrow B \times \overrightarrow A\]
A cross product of any vector with itself gives zero, since the part of the first vector that is perpendicular to the second vector is zero:
\[ | \overrightarrow A \times \overrightarrow A | = A^2 \sin 0 = 0\]
As with the scalar product, the vector product can be easily expressed with components and unit vectors. The vector products of the unit vectors with themselves are zero. Each of the unit vectors is at right angles with the other two unit vectors, so the magnitude of the cross product of two unit vectors is also a unit vector (since the sine of the angle between them is 1).
Convention: Right-handED Coordinate Systems
We will always choose a right-handed coordinate system, which means that using the right-hand-rule on the +\(x\) to +\(y\) axis yields the +\(z\) axis.
In terms of the unit vectors, we therefore have:
\[ \widehat i \times \widehat i = \widehat j \times \widehat j = \widehat k \times \widehat k = 0\]
and
\[\widehat i \times \widehat j =-\widehat j \times \widehat i = \widehat k ,\;\;\;\;\;\; \widehat j \times \widehat k =-\widehat k \times \widehat j =\widehat i ,\;\;\;\;\;\; \widehat k \times \widehat i =-\widehat i \times \widehat k =\widehat j \]
This allows us to do cross products purely mathematically (without resorting to the right-hand-rule) when we know the components, as we did for the scalar product. Again start with two vectors in component form:
\[\overrightarrow A = A_x\widehat i + A_y \widehat j\nonumber\]
\[\overrightarrow B = B_x\widehat i + B_y \widehat j\nonumber\]
then, as in the case of the scalar product, just do "normal algebra," distributing the cross product, and applying the unit vector cross products above:
\[ \begin{align} \overrightarrow A \times \overrightarrow B &= (A_x \widehat i + A_y \widehat j) \times (B_x \widehat i + B_y \widehat j) \\[5pt] &= (A_x B_x) \cancelto{0}{\widehat i \times \widehat i} + (A_yB_x) \cancelto{- \widehat k} {\widehat j \times \widehat i} + (A_x B_y ) \cancelto{+\widehat k}{ \widehat i \times \widehat j} + (A_y B_y) \cancelto{0}{\widehat j \times \widehat j} \\[5pt] &= (A_x B_y - A_yB_x ) \widehat k \end{align}\]
It is not obvious right now how we will use the dot and cross product in physics, but it is coming, so it’s a good idea to get a firm grasp on these important tools now.
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Faddeeva Package From AbInitio
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Line 26: Line 26: :<math>\mathrm{erfi}(x) = -i\mathrm{erf}(ix) = -i[e^{x^2} w(x) - 1]</math> (imaginary error function) :<math>\mathrm{erfi}(x) = -i\mathrm{erf}(ix) = -i[e^{x^2} w(x) - 1]</math> (imaginary error function) :<math>F(x) = \frac{i\sqrt{\pi}}{2} \left[ e^{-x^2} - w(x) \right]</math> ([[w:Dawson function|Dawson function]]) :<math>F(x) = \frac{i\sqrt{\pi}}{2} \left[ e^{-x^2} - w(x) \right]</math> ([[w:Dawson function|Dawson function]]) - :<math>\mathrm{Voigt}(x,y) = \mathrm{Re}[w(x+iy)] \!</math> (real [[w:Voigt function|Voigt function]], up to scale factor) + :<math>\mathrm{Voigt}(x,y) = \mathrm{Re}[w(x+iy)] \!</math> (real [[w:Voigt function|Voigt function]], up to a scale factor) Note that in the case of erf and erfc, we provide different equations for positive and negative ''x'', in order to avoid numerical problems arising from multiplying exponentially large and small quantities. Note that in the case of erf and erfc, we provide different equations for positive and negative ''x'', in order to avoid numerical problems arising from multiplying exponentially large and small quantities. Revision as of 22:53, 29 October 2012
Contents Faddeeva / complex error function
Steven G. Johnson has written free/open-source C++ code (with wrappers for other languages) to compute the
scaled complex error function w( z) = e − z2erfc(− iz), also called the Faddeeva function(and also the plasma dispersion function), for arbitrary complex arguments zto a given accuracy. Given the Faddeeva function, one can easily compute Voigt functions, the Dawson function, and similar related functions.
Download the source code from:
http://ab-initio.mit.edu/Faddeeva_w.cc (updated 29 October 2012) Usage
To use the code, add the following declaration to your C++ source (or header file):
#include <complex> extern std::complex<double> Faddeeva_w(std::complex<double> z, double relerr=0);
The function
Faddeeva_w(z, relerr) computes
w( z) to a desired relative error
relerr.
Omitting the
relerr argument, or passing
relerr=0 (or any
relerr less than machine precision ε≈10
−16), corresponds to requesting machine precision, and in practice a relative error < 10 −13 is usually achieved. Specifying a larger value of
relerr may improve performance (at the expense of accuracy).
You should also compile
Faddeeva_w.cc and link it with your program, of course.
In terms of
w( z), some other important functions are: (scaled complementary error function) (complementary error function) (error function) (imaginary error function) (Dawson function) (real Voigt function, up to a scale factor)
Note that in the case of erf and erfc, we provide different equations for positive and negative
x, in order to avoid numerical problems arising from multiplying exponentially large and small quantities. Wrappers: Matlab, GNU Octave, and Python
Wrappers are available for this function in other languages.
Matlab (also available here): A function
Faddeeva_w(z, relerr), where the arguments have the same meaning as above (the
relerrargument is optional) can be downloaded from Faddeeva_w_mex.cc (along with the help file Faddeeva_w.m. Compile it into an octave plugin with:
mex -output Faddeeva_w -O Faddeeva_w_mex.cc Faddeeva_w.cc GNU Octave: A function
Faddeeva_w(z, relerr), where the arguments have the same meaning as above (the
relerrargument is optional) can be downloaded from Faddeeva_w_oct.cc. Compile it into a MEX file with:
mkoctfile -DMPICH_SKIP_MPICXX=1 -DOMPI_SKIP_MPICXX=1 -s -o Faddeeva_w.oct Faddeeva_w_oct.cc Faddeeva_w.cc Python: Our code is used to provide
scipy.special.wofzin SciPy starting in version 0.12.0 (see here).
Algorithm
This implementation uses a combination of different algorithms. For sufficiently large |
z|, we use a continued-fraction expansion for w( z) similar to those described in Walter Gautschi, "Efficient computation of the complex error function," SIAM J. Numer. Anal. 7(1), pp. 187–198 (1970). G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Soft. 16(1), pp. 38–46 (1990); this is TOMS Algorithm 680.
Unlike those papers, however, we switch to a completely different algorithm for smaller |
z|: Mofreh R. Zaghloul and Ahmed N. Ali, "Algorithm 916: Computing the Faddeyeva and Voigt Functions," ACM Trans. Math. Soft. 38(2), 15 (2011). Preprint available at arXiv:1106.0151.
(I initially used this algorithm for all z, but the continued-fraction expansion turned out to be faster for larger |
z|. On the other hand, Algorithm 916 is competitive or faster for smaller | z|, and appears to be significantly more accurate than the Poppe & Wijers code in some regions, e.g. in the vicinity of | z|=1 [although comparison with other compilers suggests that this may be a problem specific to gfortran]. Algorithm 916 also has better relative accuracy in Re[ z] for some regions near the real- z axis. You can switch back to using Algorithm 916 for all z by changing
USE_CONTINUED_FRACTION to
0 in the code.)
Note that this is SGJ's
independent re-implementation of these algorithms, based on the descriptions in the papers only. In particular, we did not refer to the authors' Fortran or Matlab implementations (respectively), which are under restrictive "semifree" ACM copyright terms and are therefore unusable in free/open-source software.
Algorithm 916 requires an external complementary error function erfc(
x) function for real arguments x to be supplied as a subroutine. More precisely, it requires the scaled function erfcx( x) = e erfc( x2 x). Here, we use an erfcx routine written by SGJ that uses a combination of two algorithms: a continued-fraction expansion for large xand a lookup table of Chebyshev polynomials for small x. (I initially used an erfcx function derived from the DERFC routine in SLATEC, modified by SGJ to compute erfcx instead of erfc, by the new erfcx routine is much faster.) Test program
To test the code, a small test program is included at the end of
Faddeeva_w.cc which tests
w( z) against several known results (from Wolfram Alpha) and prints the relative errors obtained. To compile the test program,
#define FADDEEVA_W_TEST in the file (or compile with
-DFADDEEVA_W_TEST on Unix) and compile
Faddeeva_w.cc. The resulting program prints
SUCCESS at the end of its output if the errors were acceptable.
License
The software is distributed under the "MIT License", a simple permissive free/open-source license:
Copyright © 2012 Massachusetts Institute of Technology Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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I would say because of the way you efficiently solve problems as well as pedagogy. Both are used in both cases though.
The Hamiltonian operator approach emphasises the spectrum aspects of quantum mechanics, which the student is introduced to at this point $-$ but here is a Lagrangian
$$\mathcal{L}\left(\psi, \mathbf{\nabla}\psi, \dot{\psi}\right) = \mathrm i\hbar\, \frac{1}{2} (\psi^{*}\dot{\psi}-\dot{\psi^{*}}\psi) - \frac{\hbar^2}{2m} \mathbf{\nabla}\psi^{*} \mathbf{\nabla}\psi - V( \mathbf{r},t)\,\psi^{*}\psi$$
for the Schrödinger equation$$\frac{\partial \mathcal{L}}{\partial \psi^{*}} - \frac{\partial}{\partial t} \frac{\partial \mathcal{L}}{\partial\frac{\partial \psi^{*}}{\partial t}} - \sum_{j=1}^3
\frac{\partial}{\partial x_j} \frac{\partial \mathcal{L}}{\partial\frac{\partial \psi^{*}}{\partial x_j}} = 0.$$
The Lagrangian (density) is especially relevant for the path integral formulation, and in some way closer to bring out symmetries of a field theory. Noether theorem and so on. $-$ but I remember Peskin & Schröders book on quantum field theory starts out with the Hamiltonian approach and introduces path integral methods only 300 pages in.
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ROOT 6.19/01
Reference Guide
These examples show the functionalities of the VecOps utilities.
file vo001_AdoptOrOwnMemory.C In this tutorial we learn how the RVec class can be used to adopt existing memory or allocate some. file vo001_AdoptOrOwnMemory.py In this tutorial we learn how the RVec class can be used to adopt existing memory or allocate some. file vo002_VectorCalculations.C In this tutorial we learn how the RVec class can be used to express easily mathematical operations involving arrays and scalars. file vo003_LogicalOperations.C In this tutorial we learn how the RVec class can be used to express logical operations. file vo004_SortAndSelect.C In this tutorial we learn how elements of an RVec can be easily sorted and selected. file vo004_SortAndSelect.py In this tutorial we learn how elements of an RVec can be easily sorted and selected. file vo005_Combinations.C In this tutorial we learn how combinations of RVecs can be build. file vo005_Combinations.py In this tutorial we learn how combinations of RVecs can be build. file vo006_IndexManipulation.C In this tutorial we demonstrate RVec helpers for index manipulation. file vo007_PhysicsHelpers.C In this tutorial we demonstrate RVec helpers for physics computations such as angle differences \(\Delta \phi\), the distance in the \(\eta\)- \(\phi\) plane \(\Delta R\) and the invariant mass.
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ok, suppose we have the set $U_1=[a,\frac{a+b}{2}) \cup (\frac{a+2}{2},b]$ where $a,b$ are rational. It is easy to see that there exists a countable cover which consists of intervals that converges towards, a,b and $\frac{a+b}{2}$. Therefore $U_1$ is not compact. Now we can construct $U_2$ by taking the midpoint of each half open interval of $U_1$ and we can similarly construct a countable cover that has no finite subcover.
By induction on the naturals, we eventually end up with the set $\Bbb{I} \cap [a,b]$. Thus this set is not compact
I am currently working under the Lebesgue outer measure, though I did not know we cannot define any measure where subsets of rationals have nonzero measure
The above workings is basically trying to compute $\lambda^*(\Bbb{I}\cap[a,b])$ more directly without using the fact $(\Bbb{I}\cap[a,b]) \cup (\Bbb{I}\cap[a,b]) = [a,b]$ where $\lambda^*$ is the Lebesgue outer measure
that is, trying to compute the Lebesgue outer measure of the irrationals using only the notions of covers, topology and the definition of the measure
What I hope from such more direct computation is to get deeper rigorous and intuitve insight on what exactly controls the value of the measure of some given uncountable set, because MSE and Asaf taught me it has nothing to do with connectedness or the topology of the set
Problem: Let $X$ be some measurable space and $f,g : X \to [-\infty, \infty]$ measurable functions. Prove that the set $\{x \mid f(x) < g(x) \}$ is a measurable set. Question: In a solution I am reading, the author just asserts that $g-f$ is measurable and the rest of the proof essentially follows from that. My problem is, how can $g-f$ make sense if either function could possibly take on an infinite value?
@AkivaWeinberger For $\lambda^*$ I can think of simple examples like: If $\frac{a}{2} < \frac{b}{2} < a, b$, then I can always add some $\frac{c}{2}$ to $\frac{a}{2},\frac{b}{2}$ to generate the interval $[\frac{a+c}{2},\frac{b+c}{2}]$ which will fullfill the criteria. But if you are interested in some $X$ that are not intervals, I am not very sure
We then manipulate the $c_n$ for the Fourier series of $h$ to obtain a new $c_n$, but expressed w.r.t. $g$.
Now, I am still not understanding why by doing what we have done we're logically showing that this new $c_n$ is the $d_n$ which we need. Why would this $c_n$ be the $d_n$ associated with the Fourier series of $g$?
$\lambda^*(\Bbb{I}\cap [a,b]) = \lambda^*(C) = \lim_{i\to \aleph_0}\lambda^*(C_i) = \lim_{i\to \aleph_0} (b-q_i) + \sum_{k=1}^i (q_{n(i)}-q_{m(i)}) + (q_{i+1}-a)$. Therefore, computing the Lebesgue outer measure of the irrationals directly amounts to computing the value of this series. Therefore, we first need to check it is convergent, and then compute its value
The above workings is basically trying to compute $\lambda^*(\Bbb{I}\cap[a,b])$ more directly without using the fact $(\Bbb{I}\cap[a,b]) \cup (\Bbb{I}\cap[a,b]) = [a,b]$ where $\lambda^*$ is the Lebesgue outer measure
What I hope from such more direct computation is to get deeper rigorous and intuitve insight on what exactly controls the value of the measure of some given uncountable set, because MSE and Asaf taught me it has nothing to do with connectedness or the topology of the set
Alessandro: and typo for the third $\Bbb{I}$ in the quote, which should be $\Bbb{Q}$
(cont.) We first observed that the above countable sum is an alternating series. Therefore, we can use some machinery in checking the convergence of an alternating series
Next, we observed the terms in the alternating series is monotonically increasing and bounded from above and below by b and a respectively
Each term in brackets are also nonegative by the Lebesgue outer measure of open intervals, and together, let the differences be $c_i = q_{n(i)-q_{m(i)}}$. These form a series that is bounded from above and below
Hence (also typo in the subscript just above): $$\lambda^*(\Bbb{I}\cap [a,b])=\sum_{i=1}^{\aleph_0}c_i$$
Consider the partial sums of the above series. Note every partial sum is telescoping since in finite series, addition associates and thus we are free to cancel out. By the construction of the cover $C$ every rational $q_i$ that is enumerated is ordered such that they form expressions $-q_i+q_i$. Hence for any partial sum by moving through the stages of the constructions of $C$ i.e. $C_0,C_1,C_2,...$, the only surviving term is $b-a$. Therefore, the countable sequence is also telescoping and:
@AkivaWeinberger Never mind. I think I figured it out alone. Basically, the value of the definite integral for $c_n$ is actually the value of the define integral of $d_n$. So they are the same thing but re-expressed differently.
If you have a function $f : X \to Y$ between two topological spaces $X$ and $Y$ you can't conclude anything about the topologies, if however the function is continuous, then you can say stuff about the topologies
@Overflow2341313 Could you send a picture or a screenshot of the problem?
nvm I overlooked something important. Each interval contains a rational, and there are only countably many rationals. This means at the $\omega_1$ limit stage, thre are uncountably many intervals that contains neither rationals nor irrationals, thus they are empty and does not contribute to the sum
So there are only countably many disjoint intervals in the cover $C$
@Perturbative Okay similar problem if you don't mind guiding me in the right direction. If a function f exists, with the same setup (X, t) -> (Y,S), that is 1-1, open, and continous but not onto construct a topological space which is homeomorphic to the space (X, t).
Simply restrict the codomain so that it is onto? Making it bijective and hence invertible.
hmm, I don't understand. While I do start with an uncountable cover and using axiom of choice to well order the irrationals, the fact that the rationals are countable means I eventually end up with a countable cover of the rationals. However the telescoping countable sum clearly does not vanish, so this is weird...
In a schematic, we have the following, I will try to figure this out tomorrow before moving on to computing the Lebesgue outer measure of the cantor set:
@Perturbative Okay, kast question. Think I'm starting to get this stuff now.... I want to find a topology t on R such that f: R, U -> R, t defined by f(x) = x^2 is an open map where U is the "usual" topology defined by U = {x in U | x in U implies that x in (a,b) \subseteq U}.
To do this... the smallest t can be is the trivial topology on R - {\emptyset, R}
But, we required that everything in U be in t under f?
@Overflow2341313 Also for the previous example, I think it may not be as simple (contrary to what I initially thought), because there do exist functions which are continuous, bijective but do not have continuous inverse
I'm not sure if adding the additional condition that $f$ is an open map will make an difference
For those who are not very familiar about this interest of mine, besides the maths, I am also interested in the notion of a "proof space", that is the set or class of all possible proofs of a given proposition and their relationship
Elements in a proof space is a proof, which consists of steps and forming a path in this space For that I have a postulate that given two paths A and B in proof space with the same starting point and a proposition $\phi$. If $A \vdash \phi$ but $B \not\vdash \phi$, then there must exists some condition that make the path $B$ unable to reach $\phi$, or that $B$ is unprovable under the current formal system
Hi. I believe I have numerically discovered that $\sum_{n=0}^{K-c}\binom{K}{n}\binom{K}{n+c}z_K^{n+c/2} \sim \sum_{n=0}^K \binom{K}{n}^2 z_K^n$ as $K\to\infty$, where $c=0,\dots,K$ is fixed and $z_K=K^{-\alpha}$ for some $\alpha\in(0,2)$. Any ideas how to prove that?
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I'd like to tell you about two kinds of logic. In both, we start with a set \( X \) of "states of the world" and build a set of statements about the world, also known as "propositions". In the first, propositions correspond to subsets of \( X \). In the second, propositions correspond to partitions of \( X \). In both approaches we get a
poset of propositions where the partial order is "implication", written \( \implies \).
The first kind of logic is very familiar. We could call it "subset logic", but it's part of what people usually call "classical logic". This is the sort of logic we learn in school, assuming we learn any at all. The second kind of logic is less well known: it's called "partition logic". Interestingly, Fong and Spivak spend more time on the second kind.
I'll start by talking about the first kind.
Most of us learn the relation between propositions and subsets, at least implicitly, when we meet Venn diagrams:
This is a picture of some set \( X \) of "states of the world". But the world here is very tiny: it's just a letter. It can be any letter in the Latin, Greek or Cyrillic alphabet. Each region in the Venn diagram is subset of \( X \): for example, the upper left circle contains all the letters in the Greek alphabet. But each region can also be seen as a proposition: a statement about the world. For example, the upper left circle corresponds to the proposition "The letter belongs to the Greek alphabet".
As a result, everything you can do with subsets of \( X \) turns into something you can do with propositions. Suppose \( P, Q, \) and \( R \) are subsets of \( X \). We can also think of these as propositions, and:
if \( P \subseteq Q \) we say the proposition \( P \)
implies the proposition \( Q \), and we write \( P \implies Q \).
If \( P \cap Q = R \) we say \(P \textbf { and } Q = R \).
If \( P \cup Q = R \) we say the proposition \( P \textbf{ o r} Q = R \).
All the rules obeyed by "subset", "and" and "or" become rules obeyed by "implies", "and" and "or".
I hope you know this already, but if you don't, you're in luck:
this is this most important thing you've heard all year! Please think about it and ask questions until it revolutionizes the way you think about logic.
But really, all this stuff is about one particular way of getting a poset from the set \( X \).
For any set \( X \) the
power set of \( X \) is the collection of all subsets of \( X \). We call it \( P(X) \). It's a poset, where the partial ordering is \( \subseteq \).
For example, here is a picture of the poset \( P(X) \) when \( X = \{x,y,z\} \):
As you can see, it looks like a 3-dimensional cube. Here's a picture of \( P(X) \) when \( X \) has 4 elements:
In this picture we say whether each element is in or out of the subset by writing a 1 or 0. This time we get a 4-dimensional cube.
What's the union of two subsets \( S, T \subseteq X \)? It's the smallest subset of \( X \) that contains both \( S \) and \( T \) as subsets. This is an example of a concept we can define in any poset:
Definition. Given a poset \( (A, \le) \), the join of \( a, b \in A \), if it exists, is the least element \( c \in A \) such that \( a \le c \) and \( b \le c \). We denote the join of \( a \) and \( b \) as \( a \vee b \).
Quite generally we can try to think of
any poset as a poset of propositions. Then \( \vee \) means "or". In the logic we're studying today, this poset is \( P X \) and \( \vee \) is just "union", or \( \cup \).
Similarly, what's the intersection of two subsets \( S, T \subseteq X \)? Well, it's the largest subset of \( X \) that is contained as a subset of both \( S \) and \( T \). Again this is an example of a concept we can define in any poset:
Definition. Given a poset \( (A, \le) \), the meet of \( a, b \in A \), if it exists, is the greatest element \( c \in A \) such that \( c \le a \) and \( c \le b \). We denote the meet of \( a \) and \( b \) as \( a \wedge b \).
When we think of a poset as a poset of propositions, \( \wedge \) means "and". When our poset is \( P(X) \), \( \wedge \) is just "intersection". \( \cap \).
We could go on with this, and if this were a course on classical logic I would. But this is a course on applied category theory! So, we shouldn't just stick with a fixed set \( X \). We should see what happens when we let it vary! We get a poset of propositions for each set \( X \), but all these posets are related to each other.
I'll talk about this more next time, but let me give you a teaser now. Say we have two sets \( X \) and \( Y \) and a function \( f : X \to Y \). Then we get a monotone map from the poset \( P(Y) \) to the poset \( P(X) \), called
$$ f^* : P(Y) \to P(X) $$ For any \( S \in P(Y) \), the set \( f^*(S) \in P(X) \) is defined like this:
$$ f^*(S) = \{ x \in X : \; f(x) \in S \} $$ Next time, I'll show you this monotone map has both a left and a right adjoint! And these turn out to be connected to the logical concepts of "there exists" and "for all". I believe this was first discovered by the great category theorist Bill Lawvere.
So you see, I haven't given up talking about left and right adjoints. I'm really just getting ready to explain how they show up in logic: first in good old classical "subset logic", and then in the weird new "partition logic".
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[This is the 6th post in the current series about Wythoff’s game: see posts #1, #2, #3, #4, and #5.
Caveat lector: this post is a bit more difficult than usual. Let me know what you think in the comments!]
Our only remaining task from last week was to prove the mysterious Covering Theorem: we must show that there is exactly one dot in each row and column of the grid (we already covered the diagonal case). Since the rows and columns are symmetric, let’s focus on columns.
The columns really only care about the
x-coordinates of the points, so let’s draw just these x-coordinates on the number-line. We’ve drawn \(\phi,2\phi,3\phi,\ldots\) with small dots and \(\phi^2,2\phi^2,3\phi^2,\ldots\) with large dots. We need to show that there’s exactly one dot between 1 and 2, precisely one dot between 2 and 3, just one between 3 and 4, and so on down the line. For terminology’s sake, break the number line into length-1 intervals [1,2], [2,3], [3,4], etc., so we must show that each interval has one and only one dot:
Why is this true? One explanation hinges on a nice geometric observation: Take any small dot
s and large dot t on our number-line above, and cut segment st into two parts in the ratio \(1:\phi\) (with s on the shorter side). Then the point where we cut is always an integer! For example, the upper-left segment in the diagram below has endpoints at \(s=2\cdot\phi\) and \(t=1\cdot\phi^2\), and its cutting point is the integer 3:
In general, if
s is the jth small dot—i.e., \(s=j\cdot\phi\)—and \(t=k\cdot\phi^2\) is the kth large dot, then the cutting point between s and t is \(\frac{1}{\phi}\cdot s+\frac{1}{\phi^2}\cdot t = j+k\) (Why?! [1]). But more importantly, this observation shows that no interval has two or more dots: a small dot and a large dot can’t be in the same interval because they always have an integer between them! [2]
So all we have to do now is prove that no interval is
empty: for each integer n, some dot lies in the interval [ n, n+1]. We will prove this by contradiction. What happens if no dot hits this interval? Then the sequence \(\phi,2\phi,3\phi,\ldots\) jumps over the interval, i.e., for some j, the jth dot in the sequence is less than n but the ( j+1)st is greater than n+1. Likewise, the sequence \(\phi^2,2\phi^2,3\phi^2,\ldots\) jumps over the interval: its kth dot is less than n while its ( k+1)st dot is greater than n+1:
By our observation above on segment \(s=j\phi\) and \(t=k\phi^2\), we find that the integer
j+ k is less than n, so \(j+k\le n-1\). Similarly, \(j+k+2 > n+1\), so \(j+k+2 \ge n+2\). But together these inequalities say that \(n\le j+k\le n-1\), which is clearly absurd! This is the contradiction we were hoping for, so the interval [ n, n+1] is in fact not empty. This completes our proof of the Covering Theorem and the Wythoff formula!
It was a long journey, but we’ve finally seen exactly why the Wythoff losing positions are arranged as they are. Thank you for following me through this!
A Few Words on the Column Covering Theorem
Using the
floor function \(\lfloor x\rfloor\) that rounds x down to the nearest integer, we can restate the Column Covering Theorem in perhaps a more natural context. The sequence of integers $$\lfloor\phi\rfloor = 1, \lfloor 2\phi\rfloor = 3, \lfloor 3\phi\rfloor = 4, \lfloor 4\phi\rfloor = 6, \ldots$$ is called the Beatty sequence for the number \(\phi\), and similarly, $$\lfloor\phi^2\rfloor = 2, \lfloor 2\phi^2\rfloor = 5, \lfloor 3\phi^2\rfloor = 7, \lfloor 4\phi^2\rfloor = 8,\ldots$$ is the Beatty sequence for \(\phi^2\). Today we proved that these two sequence are complementary, i.e., together they contain each positive integer exactly once. We seemed to use very specific properties of the numbers \(\phi\) and \(\phi^2\), but in fact, a much more general theorem is true: Beatty’s Theorem: If \(\alpha\) and \(\beta\) are any positive irrational numbers with \(\frac{1}{\alpha}+\frac{1}{\beta}=1\), then their Beatty sequences \(\lfloor\alpha\rfloor, \lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\ldots\) and \(\lfloor\beta\rfloor, \lfloor 2\beta\rfloor, \lfloor 3\beta\rfloor,\ldots\) are complementary sequences.
Furthermore, our same argument—using \(\alpha\) and \(\beta\) instead of \(\phi\) and \(\phi^2\)—can be used to prove the more general Beatty’s Theorem!
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On spherical cycles in the complement to complex hypersurfaces Natalia A. Bushueva Institute of Mathematics, Siberian Federal University, Krasnoyarsk, Russia Abstract: It is known due to S. Yu. Nemirovski, that for $n\geq3$ and generic hypersurface $V\subset\mathbb C^n$ of degree $d\geq3$ there exists a sum of the Whitney spheres homotopic to an embedded sphere, which represents a nontrivial homological class of the homology group $H_n(\mathbb C^n\setminus V)$. We discuss whether a linear combination of the Whitney spheres can be represented as an embedded sphere. Keywords: homology group, embedding, Whitney sphere. Full text: PDF file (158 kB) References: PDF file HTML file UDC: 517.55+512.7 Received: 10.09.2010 Received in revised form: 10.10.2010 Accepted: 20.11.2010 Language: Citation: Natalia A. Bushueva, “On spherical cycles in the complement to complex hypersurfaces”, J. Sib. Fed. Univ. Math. Phys., 4:1 (2011), 11–17 Citation in format AMSBIB
\Bibitem{Bus11}
\by Natalia~A.~Bushueva \paper On spherical cycles in the complement to complex hypersurfaces \jour J. Sib. Fed. Univ. Math. Phys. \yr 2011 \vol 4 \issue 1 \pages 11--17 \mathnet{http://mi.mathnet.ru/jsfu157} \elib{http://elibrary.ru/item.asp?id=15540045} Linking options: http://mi.mathnet.ru/eng/jsfu157 http://mi.mathnet.ru/eng/jsfu/v4/i1/p11 Citing articles on Google Scholar: Russian citations, English citations Related articles on Google Scholar: Russian articles, English articles
Number of views: This page: 180 Full text: 45 References: 16
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I did some algebra...
In Planck unit, if make $\mu_0 = 4\pi$ and $\epsilon_0 = \frac{1}{4\pi}$ you get:
$$\mu_0 = 4\pi \cdot \frac{m_p l_p}{t_p^2 I_p^2} = 1.2566368452237765 \cdot 10^{-6} N \cdot A^{-2} $$
(where $4\pi$ is the supposed value of $\mu_0$, $m_p$ is Planck mass, $l_p$ is Planck length, $t_p$ is Planck time and $I_p$ is Planck current).
Which is very near to the CODATA value in SI and probably is the correct value.
CODATA: $1.25663706212(19) \cdot 10^{-6} N A^{-2}$
Similar for ε of vacuum:
$$\epsilon_0 = \frac{1}{4\pi} \cdot \frac{t_p^4 I_p^2}{m l_p^3} = 8.85419142073371 \cdot 10^{-12} F/m$$
CODATA value: $8.854 187 8128(13) x 10^{-12} F m^{-1}$
It is clear to me that the measurement are approximations of this perfect mathematical values... $4\pi$ and $\frac{1}{4\pi}$, so that $\mu_0\epsilon_0c^2=1$, and $c^2 = \frac{1}{\mu_0\epsilon_0}$ and $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$, in facts:
$$\mu_0\cdot\epsilon_0 = 4\pi \cdot \frac{m_p l_p}{t_p^2 I_p^2} \cdot \frac{1}{4\pi} \cdot \frac{t_p^4 I_p^2}{m l_p^3} = \frac{t_p^2}{l_p^2} = \frac{1}{c^2}$$ in Planck units.
Couloumb constant $k_C$, at this point, is:
$$k_C = \frac{1}{4\pi\epsilon_0} = \frac{c^2\mu_0}{4\pi} = c^2 \cdot 10^{-7} H m^{-1} = 8987548129.98536 N m^2 C^{-2}$$
So, we have correct and exact values for $\epsilon_0, \mu_0, c, k_C$ in Planck units, that is, respectively: $\epsilon_0 = \frac{1}{4\pi}, \mu_0 = 4\pi, c=1, k_C=1$, and by multiplying for their dimensions expressed in Planck Units we obtain the correct, exact, values in SI.
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Looping Directions and Integrals of Eigenfunctions over Submanifolds
Article
First Online:
57 Downloads Abstract
Let \((M,\,g)\) be a compact
n-dimensional Riemannian manifold without boundary and \(e_\lambda \) be an \(L^2\)-normalized eigenfunction of the Laplace–Beltrami operator with respect to the metric g, i.e.,
Let \(\varSigma \) be a
$$\begin{aligned} -\Delta _g e_\lambda = \lambda ^2 e_\lambda \quad \text {and} \quad \left\| e_\lambda \right\| _{L^2(M)} = 1. \end{aligned}$$
d-dimensional submanifold and \(\mathrm{d}\mu \) a smooth, compactly supported measure on \(\varSigma .\) It is well known (e.g., proved by Zelditch, Commun Partial Differ Equ 17(1–2):221–260, 1992 in far greater generality) that
We show this bound improves to \(o\left( \lambda ^\frac{n-d-1}{2}\right) \) provided the set of looping directions,
$$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\mu = O\left( \lambda ^\frac{n-d-1}{2}\right) . \end{aligned}$$
has measure zero as a subset of \(\mathrm{SN}^*\varSigma ,\) where here \(\varPhi _t\) is the geodesic flow on the cosphere bundle \(S^*M\) and \(\mathrm{SN}^*\varSigma \) is the unit conormal bundle over \(\varSigma .\)
$$\begin{aligned} {{\mathcal {L}}}_{\varSigma } = \{ (x,\,\xi ) \in \mathrm{SN}^*\varSigma : \varPhi _t(x,\,\xi ) \in \mathrm{SN}^*\varSigma \text { for some } t > 0 \} \end{aligned}$$
KeywordsSubmanifolds Eigenfunctions Kuznecov formulae Mathematics Subject Classification58J50 35P20 Notes Acknowledgements
The author would like to thank Yakun Xi for pointing out an error in an earlier draft of this paper.
References 1. 2. 3.Hezari, H., Riviere, G.: Equidistribution of toral eigenfunctions along hypersurfaces (01 2018)Google Scholar 4. 5. 6.Sogge, C.D.: Fourier Integrals in Classical Aanalysis. Cambridge Tracts in Mathematics, 2nd edn, vol 210. Cambridge University Press, Cambridge (2017)Google Scholar 7. 8. 9. 10.Wyman, E.: Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature (preprint, 2017)Google Scholar 11.Wyman, E.: Integrals of eigenfunctions over curves in surfaces of nonpositive curvature (preprint, 2017)Google Scholar 12. Copyright information
© Mathematica Josephina, Inc. 2018
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Hom-set adjunction Collection
context ${\bf C},{\bf D}$ … small category context $F$ in ${\bf D}\longrightarrow{\bf C}$ context $G$ in ${\bf C}\longrightarrow{\bf D}$ definiendum $\Phi$ in $\mathrm{it}$ postulate $\Phi$ in $\mathrm{Hom}_{\bf C}(F-,=)\cong\mathrm{Hom}_{\bf D}(-,G=)$ Discussion
Here $\mathrm{Hom}_{\bf C}(F-,=),\mathrm{Hom}_{\bf D}(-,G=)$ in ${\bf Set}^{{\bf D}\times {\bf C}}$.
Observe that if $\mathrm{Hom}_{\bf C}(F-,=)\cong\mathrm{Hom}_{\bf C}(-,G=)$, then $\mathrm{Hom}_{\bf C}(F-,B)$ in ${\bf Set}^{\bf D}$ is represented by $GB$ and $\mathrm{Hom}_{\bf C}(A,G-)$ in in ${\bf Set}^{\bf C}$ is represented by $FA$.
Examples An example in the category of sets
Let both ${\bf C}$ and ${\bf D}$ be the category ${\bf Set}$, which has products and exponential objects. Fix some objects (sets) $A$ and $Y$. Many examples can be thought of as variation of the pretty obvious relation
$\mathrm{Hom}_{\bf Set}(*\times A,Y)\cong\mathrm{Hom}_{\bf Set}(A,Y):= Y^A\cong\mathrm{Hom}_{\bf Set}(*,Y^A)$
where $*$ is a one-element set, but that's an unnecessary restriction:
Consider any set $X$. Indeed, we have
$\mathrm{Hom}_{\bf Set}(X\times A,Y)\cong\mathrm{Hom}_{\bf Set}(X,Y^A)$
and this is a hom-set adjunction
$\mathrm{Hom}_{\bf Set}(FX,Y)\cong\mathrm{Hom}_{\bf Set}(X,GY)$
if we define the Action of $F$ on object via $FX:=X\times A$ (Cartesian product) and let the action of $G$ on object be given by $GY:=Y^A$ (function space from $A$ to $Y$).
== Idea == More generally, view the left adjoint $F$ as A-“thickening” of ist argument ($X$), enabling to attack data, and view $G$ as the A-indexing's of aspects of it's argument $Y$, enabling to consider processes. If ${\bf C}\neq{\bf D}$, then viewing $G$ as indexing may be harder. Currying
Similarly, for propositions
$\left((X\land A)\implies Y\right)\leftrightarrow\left(X\implies(A\implies Y)\right)$
Here the A-“thickening” side says you have more arugments to prove $Y$ to begin with, while the “$A$-indexing's” side means you only demonstrate A-conditional truth of $Y$.
Example from Algebra
For example in the category of groups
$\mathrm{Hom}(X\otimes A,Y)\cong\mathrm{Hom}(X,\mathrm{Hom}(A,Y))$
Galois connection
$\langle A,\le\rangle$, $\langle B,\le'\rangle$ … posets, and $F:A\to B,G:B\to A$ … monotone functions, then Galois connection =
$(F(a)\le b)\leftrightarrow(a\le'G(b))$
Counit-unit adjunction
If we look at the morphisms from the corresponding Counit-unit adjunction,
$\eta_Y:{\mathrm{Hom}}(Y, GFY)$
resp.
$\epsilon_Y:{\mathrm{Hom}}(FGY, Y)$
at least for sets the way in which those must be defined should be clear from how they map
$Y$ to $(A\times Y)^A$
resp.
$(A\times Y^A)$ to $Y$.
The first can only be a direct embedding
$\eta_Y(y):=\lambda a.\, \langle a,y\rangle$
and the second is an evaluation
$\epsilon_Y(\langle a,f\rangle) := f(a)$
Reference
Wikipedia: Adjoint functors (category theory),
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Writing Mathematics for MathJax¶ Putting mathematics in a web page¶
To put mathematics in your web page, you can use TeX and LaTeX notation, MathML notation, AsciiMath notation, or a combination of all three within the same page; the MathJax configuration tells MathJax which you want to use, and how you plan to indicate the mathematics when you are using TeX/LaTeX or AsciiMath notation. These three formats are described in more detail below.
TeX and LaTeX input¶
Mathematics that is written in TeX or LaTeX format is indicated using
math delimiters that surround the mathematics, telling MathJax whatpart of your page represents mathematics and what is normal text.There are two types of equations: ones that occur within a paragraph(in-line mathematics), and larger equations that appear separated fromthe rest of the text on lines by themselves (displayed mathematics).
The default math delimiters are
$$...$$ and
\[...\] fordisplayed mathematics, and
\(...\) for in-line mathematics. Notein particular that the
$...$ in-line delimiters are
not usedby default. That is because dollar signs appear too often innon-mathematical settings, which could cause some text to be treatedas mathematics unexpectedly. For example, with single-dollardelimiters, “… the cost is $2.50 for the first one, and $2.00 foreach additional one …” would cause the phrase “2.50 for the firstone, and” to be treated as mathematics since it falls between dollarsigns. See the section on TeX and LaTeX Math Delimiters for more information on using dollar signs asdelimiters.
Here is a complete sample page containing TeX mathematics (see the MathJax Web Demos Repository for more).
<!DOCTYPE html><html><head><title>MathJax TeX Test Page</title><script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script><script type="text/javascript" id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script></head><body>When \(a \ne 0\), there are two solutions to \(ax^2 + bx + c = 0\) and they are$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$</body></html>
Since the TeX notation is part of the text of the page, there are some caveats that you must keep in mind when you enter your mathematics. In particular, you need to be careful about the use of less-than signs, since those are what the browser uses to indicate the start of a tag in HTML. Putting a space on both sides of the less-than sign should be sufficient, but see TeX and LaTeX support for more details.
If you are using MathJax within a blog, wiki, or other content management system, the markup language used by that system may interfere with the TeX notation used by MathJax. For example, if your blog uses Markdown notation for authoring your pages, the underscores used by TeX to indicate subscripts may be confused with the use of underscores by Markdown to indicate italics, and the two uses may prevent your mathematics from being displayed. See TeX and LaTeX support for some suggestions about how to deal with the problem.
There are a number of extensions for the TeX input processor that areloaded by combined components that include the TeX input format (e.g.,
tex-chtml.js), and others that are loaded automatically whenneeded. See TeX and LaTeX Extensions fordetails on TeX extensions that are available.
MathML input¶
For mathematics written in MathML notation, you mark your mathematicsusing standard
<math> tags, where
<math display="block">represents displayed mathematics and
<math display="inline"> orjust
<math> represents in-line mathematics.
MathML notation will work with MathJax in HTML files, not just XHTMLfiles, even in older browsers and that the web page need not be servedwith any special MIME-type. Note, however, that in HTML (as opposed toXHTML), you should
not include a namespace prefix for your
<math>tags; for example, you should not use
<m:math> except in an XHTML filewhere you have tied the
m namespace to the MathML DTD by adding the
xmlns:m="http://www.w3.org/1998/Math/MathML" attribute to your file’s
<html> tag.
In order to make your MathML work in the widest range of situations,it is recommended that you include the
xmlns="http://www.w3.org/1998/Math/MathML" attribute on all
<math> tags in your document (and this is preferred to the use ofa namespace prefix like
m: above, since those are deprecated inHTML5), although this is not strictly required.
Here is a complete sample page containing MathML mathematics (see the MathJax Web Demos Repository for more).
<!DOCTYPE html><html><head><title>MathJax MathML Test Page</title><script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script><script type="text/javascript" id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/mml-chtml.js"></script></head><body><p>When<math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>a</mi><mo>≠</mo><mn>0</mn></math>,there are two solutions to<math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>a</mi><msup><mi>x</mi><mn>2</mn></msup> <mo>+</mo> <mi>b</mi><mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn></math>and they are<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mi>x</mi> <mo>=</mo> <mrow> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>±</mo> <msqrt> <msup><mi>b</mi><mn>2</mn></msup> <mo>−</mo> <mn>4</mn><mi>a</mi><mi>c</mi> </msqrt> </mrow> <mrow> <mn>2</mn><mi>a</mi> </mrow> </mfrac> </mrow> <mtext>.</mtext></math></p></body></html>
When entering MathML notation in an HTML page (rather than an XHTMLpage), you should
not use self-closing tags, as these are not partof HTML, but should use explicit open and close tags for all your mathelements. For example, you should use
<mspace width="5pt"></mspace>
rather than
<mspace width="5pt" /> in an HTML document. If youuse the self-closing form, some browsers will not build the math treeproperly, and MathJax will receive a damaged math structure, whichwill not be rendered as the original notation would have been.Typically, this will cause parts of your expression to not bedisplayed. Unfortunately, there is nothing MathJax can do about that,since the browser has incorrectly interpreted the tags long beforeMathJax has a chance to work with them.
See the MathML page for more on MathJax’s MathML support.
AsciiMath input¶
MathJax v2.0 introduced a new input format, AsciiMath notation, by incorporating ASCIIMathML. This input processor has not been fully ported to MathJax version 3 yet, but there is a version of it that uses the legacy version 2 code to patch it into MathJax version 3. None of the combined components currently include it, so you would need to specify it explicitly in your MathJax configuration in order to use it. See the AsciiMath page for more details.
By default, you mark mathematical expressions written in AsciiMath bysurrounding them in “back-ticks”, i.e.,
`...`.
Here is a complete sample page containing AsciiMath notation:
<!DOCTYPE html><html><head><title>MathJax AsciiMath Test Page</title><script>MathJax = { loader: {load: ['input/asciimath', 'output/chtml']}}</script><script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script><script type="text/javascript" id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/startup.js"></script><body><p>When `a != 0`, there are two solutions to `ax^2 + bx + c = 0` andthey are</p><p style="text-align:center"> `x = (-b +- sqrt(b^2-4ac))/(2a) .`</p></body></html>
See the AsciiMath support page for more on MathJax’s AsciiMath support and how to configure it.
Putting Math in Javascript Strings¶
If your are using javascript to process mathematics, and need to put aTeX or LaTeX expression in a string literal, you need to be aware thatjavascript uses the backslash (
\) as a special character instrings. Since TeX uses the backslash to indicate a macro name, youoften need backslashes in your javascript strings. In order toachieve this, you must double all the backslashes that you want tohave as part of your javascript string. For example,
var math = '\\frac{1}{\\sqrt{x^2 + 1}}';
This can be particularly confusing when you are using the LaTeX macro \, which must both be doubled, as \. So you would do
var array = '\\begin{array}{cc} a & b \\\\ c & d \\end{array}';
to produce an array with two rows.
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Mahalanobis distance is unperturbed by inclusion of highly correlated additional variable, unlike Euclidean distance (Dasgupta 2008). With an application of law of iterated logarithm, an almost sure result is proved in this direction for convergence of sample
$$D^2$$
statistic to
$$\Delta ^2,$$
the population Mahalanobis distance squared under mild assumptions, improving earlier results. A rate of convergence
$$O_e( n^{-1}(\log \log n)^2(1 -\rho ^2)^{-1})$$
a.s., depending on sample size
n and magnitude of correlation
$$|\rho |$$
, is proved. The results are of relevance in analysis of longitudinal growth data where some of the variables may be highly correlated. Growth experiments on coconut trees are continued in Sunderban to examine its adaptability in saline soil. Moderate salinity of soil is known to be conducive for coconut tree growth. To examine adaptability of coconut trees in saline soil of Sunderban, longitudinal growth of 42 plants at two time points is observed on six growth characteristics per plant and relevant
$$D^2$$
statistic is computed to assess tree growth with gradual proximity of plantations towards a saline water river Bidyadhari. Principal components of
$$p=6$$
variables at two time points on the year 2015 and 2017 of
$$n=42$$
trees are compared to examine the growth status. The angle
$$\theta $$
between two growth vectors x and y defined by the relation
$$\cos \theta = \frac{(x.y)}{||x||.||y||}\in [-1,1]$$
is examined to check direction of growth variability. The values of
$$\cos \theta $$
quantify directional variation present in principal components. These are used to study the growth patterns of coconut trees on a number of aspects, e.g. growth in size, shape. As the variables are scaled in computation of angle, relative variability of the characteristics is examined. When stability of growth is attained, variation over time of each coordinate variable would be negligible after scaling, and the angle between the two growth vectors corresponding to two time points would be small. We compute
$$n=42$$
angles corresponding to 42 trees, between two growth vectors of
$$p=6$$
principal components at two time points, and examine whether growth has reached stability. Angle based on principal components may detect minute growth variations, as principal components maximise variance maintaining orthogonality of components to each other. We examine growth in different orthogonal directions, as the first principal component usually represents the size; the remaining components represent shape and other characteristics. The estimates of angles are independent for 42 plants, and large values of the angles indicate variability of growth direction over time. The angles computed from the growth vectors of original variables are also considered. The angles made by growth vectors of original variables are usually of less variation than those based on principal components. With increase in number of growth variables, variation in growth is seen to be more prominent. In Dasgupta (2017), moderate salinity of soil is found to be conducive for growth of coconut trees. In the present study, in conformity with previous findings, we observe that for some plants the angle between growth vectors increases, when proximity of plants to the river of saline water increases to a moderate level. This is so reflected in variation of angles with respect to the increase in plant’s serial numbers from 1 to 30, in 42 plants. Higher serial numbers represent gradual proximity of plants towards saline water river in the group of 42 plants. Clustering of points with large values of angle is observed in the scatter diagrams near serial number 30, where salinity is moderate. The presence of ‘whiskers’ in a specific region is indicated. The outliers seem to be present near the plant no. 30 with moderate salinity of soil. The plant with largest identification serial number 42 in the riverbank is closest to the river with salinity of water at about 33 g/l, before onset of monsoon.
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In my current research, I'm confronted with the justification of some facts, and I don't know how to proceed in proving them, so I need to know if there exist some theorems (precisely three theorems) which allow me to do so.
The problem I am investigating is the following: I have an explicit real valued function $f$, DEFINED and CONTINUOUS on each point of $D=]0,1[^4 $. As it is customary to do, let $f(u,v,w,t)$ be the value that this function takes at point of $(u,v,w,t)\in D$: $f$ cannot be defined at the boundary of $D$ and I can't extend its domain of definition $D$ in order to define it on the whole $\overline{D}=[0,1]^4 $, the closure of $D$. I know that $$ f(u,v,w,t)=\sum_{n=0}^{+\infty} f_n(u,v,w,t)\quad\forall (u,v,w,t)\in D $$ where $\{f_n\}_{n\in\Bbb N}$ is a sequence of functions defined and continuous over $D$ which can be extended as continuous function on $\overline{D}$. This makes me think that, for all integers $n$, $$ \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 | f_n(u,v,w,t)| \mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t\quad \text{ exists.} $$ And now the questions.
What theorem (be it a necessary and sufficient or only a sufficient condition) would allow me to prove the following formula? $$ \begin{split} \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 & f(u,v,w,t)\mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t \\ =&\sum_{n=0}^{+\infty} \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f_n(u,v,w,t)\mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t \end{split} $$
What theorem (be it a necessary and sufficient or only a sufficient condition) would allow me to perform any change of the order of integration respect to any of the variables involved, in order to have for example that $$ \begin{split} \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 &f(u,v,w,t)\mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t\\ = & \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f(u,v,w,t) \mathrm{d}w \mathrm{d}t \mathrm{d}u \mathrm{d}v \\ = & \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f(u,v,w,t) \mathrm{d}v \mathrm{d}u \mathrm{d}t \mathrm{d}w \quad ? \end{split} $$
Finally, suppose that one further hypothesis is made over $f$: $f$ depend on a parameter $ a \geq 0$, call it $f_a$ and suppose that $\forall (u,v,w,t) \in ]0,1[^4$ the mapping $a \mapsto f_a(u,v,w,t)$ is $C^{\infty}$ over $ \mathbb{R}_+$: what theorem (again be it a necessary and sufficient or only a sufficient condition) allow me to say that $$ g:a \mapsto \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f_a(u,v,w,t) \mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t \in C^{2}(\Bbb R^+) $$ and $$ g''(a)=\displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 \displaystyle \frac{\mathrm{d}^2}{\mathrm{d}a^2} f_a(u,v,w,t) \mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t $$ i.e. would allow me to differentiate twice under the integral symbol?
I know what theorem allowing me to have 1) 2) and 3) in case that $f$ is defined over an interval of $\Bbb{R}$ and so in case a simple integral
Could anybody help me please? Does there exist a freely accessible reference over the internet where I can find such theorems?
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It might help to have an example of each of the three types of statement. So to be concrete, let $F(\vec{v})$ be the probability that a given particle has speed $\vec{v}\in[\vec{v},\vec{v}+d\vec{v}]$ and thus the differential particle flux is given by your expression:
$$ d\vec{\phi}(\vec{v}) = n\vec{v}F(\vec{v})|d\vec{v}| $$
This is some function on position space (as is $F$). Integrating over all positions gives us the total particle flux:
$$\vec{\phi} = \int d\vec{\phi} = \int_{R^3}n\vec{v}F$$
Here the particle number density is $n$ and its transport is entirely captured by $\phi$. We might want to consider other quantities such as $g$ the momentum density. Then it has its own flux $\phi_g$. Actually, momentum is a vector $\vec{g}$ and its flux must be a rank 2 tensor $\hat{\Pi}$ but we can consider a 1D case for simplicity.
For an ideal gas we have that $\vec{g}=nm\vec{v}$ and (in 1D) that $d\phi_g = mvd\phi$. This is because momentum is only transported ballistically by particles carrying their own momentum from place to place. Thus we get expressions like:
$$\phi_g = \int (mv)d\phi $$
Hopefully this clarifies the second of your three relations by giving a specific example. I think we have assumed their is only ballistic transport (on the kinetic level I'm not sure what else there could be. Perhaps transport due to long ranged fields?)
The first equation you give is much more general, and is an expression of continuity for the quantity $A$. It is not an equation for $\phi_A$ but a seperate statement that $A$ is conserved locally. We can use it in conjunction with the third statement to get a diffusion equation for $A$. The third statement is I think just the evaluation of $\phi_A$ within some specific model of the gas - it's worth noting that the one you have presented is the result of a very standard textbook calculation that ignores many subtleties and as such the exact factor of $1/3$ is highly dubious.
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In QFT, a representation of the Lorentz group is specified as follows:$$U^\dagger(\Lambda)\phi(x) U(\Lambda)= R(\Lambda)~\phi(\Lambda^{-1}x)$$Where $\Lambda$ is an element of the Lorentz group, $\phi(x)$ is a quantum field with possibly many components, $U$ is unitary, and $R$ an element in a representation of the Lorentz group.
We know that a representation is a map from a Lie group on to the group of linear operators on some vector space.
My question is, for the representation specified as above, what is the vector space that the representation acts on?
Naively it may look like this representation act on the set of field operators, for $R$ maps some operator $\phi(x)$ to some other field operator $\phi(\Lambda^{-1}x)$, and if we loosely define field operators as things you get from canonically quantizing classical fields, we can possibly convince ourselves that this is indeed a vector space.
But then we recall that the dimension of a representation is simply the dimension of the space that it acts on. This means if we take $R$ to be in the $(1,1)$ singlet representation, this is a rep of dimension 1, hence its target space is of dimension 1. Then if we take the target space to the space of fields, this means $\phi(x)$ and $\phi(\Lambda x)$ are related by linear factors, which I am certainly not convinced of.
EDIT: This can work if we view the set of all $\phi$ as a field over which we define the vector space, see the added section below.
I guess another way to state the question is the following: we all know that scalar fields and vector fields in QFT get their names from the fact that under Lorentz transformations, scalars transform as scalars, and vectors transform like vectors.
I would like to state the statement "a scalar field transforms like a scalar" by precisely describing the target vector space of a scalar representation of the Lorentz group, how can this be done?
ADDED SECTION:
Let me give an explicit example of what I'm trying to get at:Let's take the left handed spinor representation, $(2,1)$. This is a 2 dimensional representation. We know that acts on things like $(\phi_1,\phi_2)$.
Let's call the space consisting of things of the form $(\phi_1,\phi_2)$ $V$. Is $V$ 2 dimensional?
Viewed as a classical field theory, yes, because each $\phi_i$ is just a scalar. As a quantum field theory, each $\phi_i$ is an operator.
We see that in order for $V$ space to be 2 dimensional after quantization, we need to able to view the scalar quantum fields as scalar multipliers of vectors in $V$. i.e. we need to view $V$ as a vector space defined over a (mathematical) field of (quantum) fields.
We therefore have to check whether the set of (quantum) fields satisfy (mathematical) field axioms. Can someone check this? Commutativity seem to hold if we, as in quantum mechanics, take fields and their complex conjugates to live in adjoint vector spaces, rather than the same one. Checking for closure under multiplication would require some axiomatic definition of what a quantum field is. This post imported from StackExchange Physics at 2014-08-07 15:37 (UCT), posted by SE-user bechira
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In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (p.135):
$\begin{cases}Max\text{ }p*y\\s.t.\text{ }F(y)\leq0\end{cases}$
Where y is production vector and
F(.) is a transformation function.
The only information I have found about this function states that $F:R^L\to R$ and that production set of a firm could be represented as $Y=\{y\in R^L:F(y)\leq0\}$. Also if $F(y)=0$ it is named
transformation frontier (p.562)
So I am interested in what concrete are these transformation function and transformation frontier. Intuitively it must be related to some feasibility restrictions but I whant to have some full and strickt explanation.
Will be very greatfull for help!
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As I've already mentioned in my previous answer, the method of using the QZ algorithm for matrix pencils on the Frobenius companion linearization of the polynomial eigenproblem is not always the most efficient approach. To illustrate this, I'll outline a general method for solving a
hyperbolic quadratic eigenvalue problem, which is known to have all its eigenvalues real. (Encapsulating the strategy as a working Mathematica routine is left as an exercise.)
The method also makes use of a linearization; this linearization, as described in this paper by Higham, Mackey, Mackey, and Tisseur, makes use of block symmetric matrices whose blocks are block Hankel and block antitriangular. A
Mathematica routine for constructing the matrix $X_m(P(\lambda))$ (in the notation of section 3.3 of that paper) follows:
BlockSymmetricBasis[k_Integer?NonNegative, matCof : {__?MatrixQ}] :=
Module[{p = Length[matCof] - 1, lm, um},
Switch[k,
0, -ArrayFlatten[Partition[PadRight[Take[matCof, -p], 2 p - 1], p, 1]],
p, ArrayFlatten[Partition[PadLeft[Take[matCof, p], 2 p - 1], p, 1]],
_,
lm = ArrayFlatten[Partition[PadLeft[Take[matCof, k], 2 k - 1], k, 1]];
um = ArrayFlatten[Partition[PadRight[Take[matCof, k - p], 2 (p - k) - 1], p - k, 1]];
ArrayFlatten[{{lm, 0}, {0, -um}}]]] /;
SameQ @@ (Dimensions /@ matCof) && k < Length[matCof]
From here, one now has a family of pencils $X_{m-1}(P(\lambda))-\lambda X_m(P(\lambda))$ at disposal; the key is to choose among these pencils such that $X_m$ is positive definite (i.e., we want to pick out which of the $X_{m-1}(P(\lambda))-\lambda X_m(P(\lambda))$ is a
symmetric-definite pencil).
To continue further with the discussion, here is a concrete example of a hyperbolic quadratic eigenvalue problem:
polycof = N@{{{3, 2, 1}, {2, 3, 2}, {1, 2, 3}},
{{-2, -1, -1}, {-1, -3, 2}, {-1, 2, -1}},
{{-5, 1, -2}, {1, -4, -3}, {-2, -3, -5}}};
We check which of the $X_m$ are positive definite:
Table[PositiveDefiniteMatrixQ[BlockSymmetricBasis[k, polycof]], {k, 0, 2}]
{False, True, False}
We thus continue further with the pencil $X_0-\lambda X_1$. Now, we check the condition number $\kappa$ of $X_1$:
X0 = BlockSymmetricBasis[0, polycof]; X1 = BlockSymmetricBasis[1, polycof];
LinearAlgebra`MatrixConditionNumber[X1]
22.2727
The value of the condition number obtained is rather modest in size, so we can continue with one of the usual approaches for symmetric-definite pencils, which uses Cholesky decomposition. First, build the intermediate matrix:
ℳ = CholeskyDecomposition[X1]; lft = LinearSolve[Transpose[ℳ]];
ℋ = lft[Transpose[lft[X0]]];
Here are the eigenvalues:
λ = Eigenvalues[ℋ]
{6.61035, -1.8856, 1.37725, 1.21165, -1.06445, -0.124207}
Check the eigenvalues:
Table[Det[Fold[#1 λ + #2 &, 0, polycof]] // Chop, {λ, %}]
{3.47379*10^-10, 0, 0, 0, 0, 0}
Here are the eigenvectors:
lf = LinearSolve[ℳ];
\[ScriptV] = Take[lf[#], 3] & /@ Eigenvectors[ℋ]
{{-0.46788, 0.903438, -0.600705}, {0.420022, -0.335861, -0.176273},
{-0.425712, -0.113381, 0.331892}, {-0.0699161, -0.208497, -0.204744},
{0.196805, -0.0360833, 0.26503}, {0.069413, 0.115626, -0.103772}}
Check eigenvalues and eigenvectors:
MapThread[Function[{λ, \[ScriptV]}, Chop[Fold[#1 λ + #2 &, 0, polycof].\[ScriptV]]],
{λ, \[ScriptV]}]
{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}}
Had the result of
LinearAlgebra`MatrixConditionNumber[X1] been rather large (say, $\approx 10^7$), an alternative route would have been the eigendecomposition of
X1, which would have gone like this:
ℳ = #2.#1 & @@ MapThread[#1@#2 &, {{Composition[DiagonalMatrix, Sqrt], Transpose},
Eigensystem[X1]}];
lf = LinearSolve[ℳ];
ℋ = lf[Transpose[lf[X0]]]
λ = Eigenvalues[ℋ] (* eigenvalues *)
lft = LinearSolve[Transpose[ℳ]];
\[ScriptV] = Take[lft[#], 3] & /@ Eigenvectors[ℋ] (* eigenvectors *)
See this reference for more information on definite matrix polynomials, which is the general class of matrix polynomials with real eigenvalues.
To summarize: the Frobenius linearization + QZ route works generally, but one should be on the lookout for methods that exploit structure if you will be solving a lot of structured problems, since they will usually be more efficient with space, computational effort, or both.
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On well-posedness of vector-valued fractional differential-difference equations
1.
Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain
2.
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile
3.
Escuela Técnica Superior de Ingeniería y Sistemas de Teleomunicación, Universidad Politécnica de Madrid, C/Nikola Tesla, s/n 28031 Madrid, Spain
$ \begin{equation*} (*) \left\{\begin{array}{rll} \Delta^{\alpha} u(n) & = & Au(n+2) + f(n,u(n)), \quad n \in \mathbb{N}_0, \,\, 1< \alpha \leq 2; \\ u(0) & = & u_0;\\ u(1) & = & u_1, \end{array}\right. \end{equation*} $
$ A $
$ X $ *) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on sequences of strongly continuous families of bounded operators generated by
$ A, $
$ f $ Keywords:Difference equations, fractional and nonlinear PDE, Poisson distribution, weighted Lebesgue space, well posedness. Mathematics Subject Classification:Primary: 35R11; Secondary: 39A14, 47D06. Citation:Luciano Abadías, Carlos Lizama, Pedro J. Miana, M. Pilar Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2679-2708. doi: 10.3934/dcds.2019112
References:
[1] [2]
L. Abadias and C. Lizama,
Almost automorphic mild solutions to fractional partial difference-differential equations,
[3] [4]
L. Abadias and P. J. Miana, A subordination principle on Wright functions and regularized resolvent families,
[5] [6]
G. Akrivis, B. Li and C. Lubich,
Combining maximal regularity and energy estimates for the discretizations of quasilinear parabolic equations,
[7]
W. Arendt, C. Batty, M. Hieber and F. Neubrander,
[8] [9] [10] [11] [12]
Y. Bai, D. Baleanu and G. C. Wu,
Existence and discrete approximation for optimization problems governed by fractional differential equations,
[13] [14]
E. Bazhlekova,
[15]
J. Cermák, T. Kisela and L. Nechvátal,
Stability and asymptotic properties of a linear fractional difference equation,
[16]
J. Cermák, T. Kisela and L. Nechvátal,
Stability regions for linear fractional differential systems and their discretizations,
[17]
E. Cuesta, C. Lubich and C. Palencia,
Convolution quadrature time discretization of fractional diffusion-wave equations,
[18]
E. Cuesta and C. Palencia,
A numerical method for an integro-differential equation with memory in Banach spaces: Qualitative properties,
[19]
E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations,
[20]
I. K. Dassios, D. I. Baleanu and G. I. Kalogeropoulos,
On non-homogeneous singular systems of fractional nabla difference equations,
[21]
K. J. Engel and R. Nagel,
[22] [23]
V. Keyantuo, C. Lizama and M. Warma,
Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations,
[24] [25] [26] [27]
C. Lizama and M. Murillo-Arcila,
$\ell_p$-maximal regularity for a class of fractional difference equations on $UMD$ spaces: The case $1 < \alpha \leq 2,$,
[28]
C. Lizama and M. Murillo-Arcila,
Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations,
[29]
K. S. Miller and B. Ross, Fractional difference calculus, In:
[30]
A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev,
[31]
A. M. Sinclair,
Continuous Semigroups in Banach Algebras, London Mathematical Society, Lecture Notes Series 63, Cambridge University Press, New York, 1982.
Google Scholar
[32] [33]
L. W. Weis,
A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to Transport Theory,
[34]
G. C. Wu, D. Baleanu and L. L. Huang,
Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse,
[35] [36] [37] [38]
A. Zygmund,
Trigonometric Series, 2nd ed. Vols. Ⅰ, Ⅱ, Cambridge University Press, New York, 1959.
Google Scholar
show all references
References:
[1] [2]
L. Abadias and C. Lizama,
Almost automorphic mild solutions to fractional partial difference-differential equations,
[3] [4]
L. Abadias and P. J. Miana, A subordination principle on Wright functions and regularized resolvent families,
[5] [6]
G. Akrivis, B. Li and C. Lubich,
Combining maximal regularity and energy estimates for the discretizations of quasilinear parabolic equations,
[7]
W. Arendt, C. Batty, M. Hieber and F. Neubrander,
[8] [9] [10] [11] [12]
Y. Bai, D. Baleanu and G. C. Wu,
Existence and discrete approximation for optimization problems governed by fractional differential equations,
[13] [14]
E. Bazhlekova,
[15]
J. Cermák, T. Kisela and L. Nechvátal,
Stability and asymptotic properties of a linear fractional difference equation,
[16]
J. Cermák, T. Kisela and L. Nechvátal,
Stability regions for linear fractional differential systems and their discretizations,
[17]
E. Cuesta, C. Lubich and C. Palencia,
Convolution quadrature time discretization of fractional diffusion-wave equations,
[18]
E. Cuesta and C. Palencia,
A numerical method for an integro-differential equation with memory in Banach spaces: Qualitative properties,
[19]
E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations,
[20]
I. K. Dassios, D. I. Baleanu and G. I. Kalogeropoulos,
On non-homogeneous singular systems of fractional nabla difference equations,
[21]
K. J. Engel and R. Nagel,
[22] [23]
V. Keyantuo, C. Lizama and M. Warma,
Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations,
[24] [25] [26] [27]
C. Lizama and M. Murillo-Arcila,
$\ell_p$-maximal regularity for a class of fractional difference equations on $UMD$ spaces: The case $1 < \alpha \leq 2,$,
[28]
C. Lizama and M. Murillo-Arcila,
Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations,
[29]
K. S. Miller and B. Ross, Fractional difference calculus, In:
[30]
A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev,
[31]
A. M. Sinclair,
Continuous Semigroups in Banach Algebras, London Mathematical Society, Lecture Notes Series 63, Cambridge University Press, New York, 1982.
Google Scholar
[32] [33]
L. W. Weis,
A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to Transport Theory,
[34]
G. C. Wu, D. Baleanu and L. L. Huang,
Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse,
[35] [36] [37] [38]
A. Zygmund,
Trigonometric Series, 2nd ed. Vols. Ⅰ, Ⅱ, Cambridge University Press, New York, 1959.
Google Scholar
[1] [2]
Junxiong Jia, Jigen Peng, Kexue Li.
Well-posedness of abstract distributed-order fractional diffusion equations.
[3]
Can Li, Weihua Deng, Lijing Zhao.
Well-posedness and numerical algorithm for the tempered fractional differential equations.
[4]
George Avalos, Roberto Triggiani.
Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction.
[5]
P. Blue, J. Colliander.
Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS.
[6]
Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee.
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations.
[7] [8] [9]
Hartmut Pecher.
Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions".
[10]
Wei Qu, Siu-Long Lei, Seak-Weng Vong.
A note on the stability of a second order finite difference scheme for space fractional diffusion equations.
[11]
Gabriela Marinoschi.
Well posedness of a time-difference scheme for a degenerate fast
diffusion problem.
[12]
Xing Huang, Michael Röckner, Feng-Yu Wang.
Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs.
[13]
Hiroyuki Hirayama, Mamoru Okamoto.
Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity.
[14]
Changxing Miao, Bo Zhang.
Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations.
[15]
Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli.
Well-posedness and long-time behavior for a class of doubly nonlinear equations.
[16] [17]
Hiroyuki Hirayama.
Well-posedness and scattering for a system of quadratic derivative
nonlinear Schrödinger equations with low regularity initial data.
[18]
Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa.
Global well-posedness of critical nonlinear Schrödinger equations below $L^2$.
[19]
Takeshi Wada.
A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach.
[20]
Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu.
On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities.
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Let $X$, $Y$ be i.i.d. random variables with distribuition $\mathcal{N}(0,1/2)$ and $Z = X^2 + Y^2$. I'd like to prove based on $X$ and $Y$ pdf's that $Z$ has exponential distribuition.
First use the joint pdf of $X$ and $Y$ and switch to polar coordinates, then $$ \mathbb{P}(Z\leq z)=\mathbb{P}(X^2+Y^2\leq z)=\frac{1}{\pi}\int_{x^2+y^2\leq z}e^{-x^2+y^2}\;dxdy=\frac{1}{\pi}\int_{0}^{2\pi}\int_0^{\sqrt{z}}e^{-r^2}r\;drd\theta$$ $$=2\int_0^{\sqrt{z}}re^{-r^2}\;dr $$ Now if we set $u=r^2$ then we get $$ \mathbb{P}(Z\leq z)=\int_0^ze^{-u}\;du$$
so $Z$ is exponentially distributed with rate parameter $\lambda = 1$.
Z has a chi square distribution with the number of degrees of freedom to make it the special case of the exponential. X and Y are required to be independent.
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Quick recap’ of April on the Micro blog.
2019-04-01: Arno,
Human Incognito. 2019-04-01: > You are being watched. Private and state-sponsored organizations are monitoring> and recording your online activities. privacytools.io provides knowledge and> tools to protect your privacy against global mass surveillance. 2019-04-01: Another screenshot in the dark night… 2019-04-01: Beta-blockers are so much fun. I just stayed out of it for almost four hoursafter I got home from work. And it’s been going on for weeks. Time for anotherbeer I guess. 2019-04-01: Built on top of Bootstrap, the Argon Design System comes with hundred of fullfeatured components and is compatible with mainstream browsers. And thedocumentation is gorgeous. 2019-04-01: I don’t really like editing code in a browser. This is why I rarely use Jupyternotebooks. Likewise, I found that the Emacs EIN package was clunky. This wasyears ago, and I don’t think the project has evolved so much in the recentyears. TIL there’s a new interface to communicate with Jupyter kernels:emacs-jupyter.
#emacs
2019-04-01: The borage has finally bloomed again! 2019-04-01: This is the 1000th post! 💪👈
textBuilding sites …| EN+------------------|------+Pages | 1000Paginator pages | 160Non-page files | 0Static files | 617Processed images | 0Aliases | 34Sitemaps | 1Cleaned | 0Total in 2825 ms
2019-04-01: Visual Statistics. Use R! (PDF, 429 pp.)
#rstats
2019-04-03: Cigarettes After Sex, Cigarettes After Sex. 2019-04-03: Another command-line utility to deal with flat files: xsv (available onHomebrew). 2019-04-03: Convert images to LaTeX using Mathpix snipping tool. Maybe one day I will learnthat I was wrong about the way I wrote some equations. A little testdemonstrated that it works right out of the box with simple expressions: Mine was, in this particular case:
text$$ \hat f(x) = \frac{1}{nb}\sum_{j=1}^n K\left(\frac{x-x_j}{b}\right) $$
2019-04-03: I somehow forgot about GitUp. A simple
git log and more Magit when required, andI’m generally done with tracking what I’ve done. Of course, this is because Imostly work alone. Anyway, I like it when you have a such a clean and minimal UIwhich helps to visualize at a glance (or even act on) a full Git repo.
2019-04-03: Oldie but goodie: normal random number generator in R. (via @BrodieGaslam)
#rstats
2019-04-03: Un excellent tutorielEmacs en français !
#emacs
#fr
2019-04-03: Statistics for Hackers. Be sure to also check the GH repo of the author, e.g.code_py.
#python
2019-04-04: Alain Bashung, Bleu pétrole. 2019-04-05: 93% of Paint Splatters are Valid Perl Programs. So funny that you probablyhave to read the paper as well. 2019-04-06: 🎥 Matrix Reloaded. 2019-04-06: Arno, Human Incognito. 2019-04-06: I saw this pandoc LaTeX template popping out on Twitter yesterday. I remembertrying it out at some point when I was looking for some good hand craftedtemplates for R/statistical reports. At that time, many people were also usingthe Metropolis theme for building slides. I wrote custom templates for bothsituations, finally. 2019-04-06: TIL about Bulma, which is a free, open source CSS framework based on Flexbox.Looks like a solid and lightweight alternative to Bootstrap. 2019-04-06: IB Foundations of Data Science (PDF, 106 pp.). Yet another textbook for learningDS “quickly”. Beware, it relies on Python and is quite mathy. 2019-04-06: Python for Epidemiologists: Tutorial in Python targeted at Epidemiologists. Overthe last few years, I came across several attempts at bringing the Pythonstatistical ecosystem to the Epi domain. Now trying this one.
#python
2019-04-06: Writing a sqlite clone from scratch in C. I already learned a lot of cool stuffby reading part of this tutorial. I’m eager to learn more.
#database
#clang
2019-04-07: 📖 Marina Tsvetaeva, Vivre dans le feu (Robert Laffont, 2005) 2019-04-07: Friday burger party for my son: 2019-04-07: Recently, on BSAG’s website: Rethinking my dotfiles setup. Every time it seems Ilearn a new thing about the open source world when reading her blog posts. 2019-04-08: Diana Krall, The Girl in the Other Room. 2019-04-08: Nothing fancy on Twitter today. The web UI is just poor shit. Try to go back inyour browser history, and a different series of posts appears on the very samepage you landed on just a few seconds before. Oh, and did you ever manage tofind an old posts of yours? 2019-04-08: Eight Obscure Bash Options You Might Want to Know About. (via O’ReillyProgramming Newletter) 2019-04-12: Folllowing the recent scandal around DataCamp, I think DataQuest and Exercismremain good online training centers for anyone eager to learn. 2019-04-12: Great. Just when I was trying to recompile my Hugo website I noticed that the0.54->0.55 upgrade just broke everything! And we can see comments like this. Idon’t care about performance issue or variable name changes or even Go itself. Ijust want a tool to build my static pages, using an old theme that I customizedto my liking. Anyway, since I no time to fix the Go code for the modifiedtemplate I use, let’s go back to the previous release (0.54 has SHA
6c0c7919de42ee5d629d3a9786fb111f4498dab3) and pin it for good!
shellbrew uninstall hugobrew install https://bit.ly/2UxujU1brew pin hugo
2019-04-12: Composing Programs and From Python to Numpy are two of the most invaluableressources I know if you want to learn Python for real.
#python
2019-04-13: 🎥 Matrix Revolutions. 2019-04-14: 🎥 Highlander. 2019-04-14: Week-end miscellanies… 2019-04-15: Fred Hersch, Solo. 2019-04-15: I’m slowly, very slowly, updating the stata-sk project. on the one hand I am not in the best possible state of health, on the other hand my degree of motivation bought dramatically in recent weeks. I’m afraid it may have to end up in the garbage one day or the other. 2019-04-15: At this point (order doesn’t matter, or I guess so), I wonder how this whole DC thing is going to end. 2019-04-17: I really like Yann Holtz’s teaching material, especially his Data analytics and visualization track.
#dataviz
2019-04-17: The Carpentries Handbook is live. Go check it if you’ve even been interested in teaching ressources. 2019-04-17: A whirlwind intro to Python: A very nice intro to Python, written using Org.
#python
2019-04-18: 🎥 Wolverine. 2019-04-18: Very nice showcase by the Stitch team. 2019-04-18: With all DC teachers who have been out of business, we now have a plethora of nice tutorial comping up here and there. Here is one by Julia Silge on “tidy” text mining, and here is Ines Montani’s Advanced NLP course with spaCy.
#rstats
#python
2019-04-19: Looks like I just spend my full working day using VS code for Python coding again. It’s perfectly fine for building website (small codebase) or playing interactive script (< 200 LOC). Now, are there any better ways to integrate Jupyter notebooks than this?
#python
2019-04-20: 🎥 The Bourne Identity. 2019-04-20: 📖 Iegor Gran, L’Écologie en bas de chez moi (P.O.L., 2011) 2019-04-20: Always nice to read Zachary Tellman’s code on impure functional data structures, even if Java is not my business. 2019-04-20: If you are looking to convert your Python notebook to a slide deck (and you like JS-based slideshow, of course), RISE looks like a good option. There’s also a PDF backend.
#python
2019-04-21: > Domain-specific languages are the ultimate abstractions. — Paul Hudak (1998) 2019-04-21: I guess when you have some time to spare on a Sunday evening, you can just push a bunch of commits using the wrong
user <email> values (probably updated by SourceTree that I just relaunched for the first time two days ago), right? In this case, this was very helpful. For fancier version, see How to change the commit author for one specific commit?
2019-04-21: Visualising intersecting sets of twitter followers.
#dataviz
2019-04-21: hub is “an extension to command-line git that helps you do everyday GitHub tasks without ever leaving the terminal.” I have a working copy, of course, but I don’t use it often – StackOverflow to the rescue. Maybe it’s time to take a closer look… 2019-04-22: Owen Pallett, In Conflict. 2019-04-22: The joy of deploying on Heroku, when everything works on the first try… 2019-04-22: Embracing Swift for Deep Learning.
#swift
2019-04-22: Why Are Big Data Matrices Approximately Low Rank?. (via @@GCLinderman) 2019-04-23: 🎥 The Bourne Supremacy. 2019-04-23: Guess who tried to customize lintr for more than 30 minutes with syntastic while the plugin was globally disabled and I am using ale instead? 2019-04-23: Just curious so I installed the datatable Python package. I’ve been very happy with R
data.table package so I hope it will not be too difficult to switch over Python. BTW, the h2o.ai team is doing great things for data science.
#python
2019-04-23: Long time Emacs user here: I’ve been using VS Code and NeoVim for five days. I’m perfectly fine. 2019-04-23: news-please – an integrated web crawler and information extractor for news that just works.
#python
2019-04-24: 🎥 The Bourne Ultimatum. 2019-04-24: Foundations of Database. Didn’t know we had this from the INRIA team! (via @CompSciFact) 2019-04-25: Keith Jarrett Trio, The Out-of-Towners. 2019-04-25: Since I’ve been using VS Code for a few days, this probably is a good read for tonight:- Python in Visual Studio Code- Clang/LLVM Support in Visual Studio 2019-04-25: Sample code for the Model-Based Machine Learning book. 2019-04-26: 🎥 Seven Sisters. Because I loved her role role as Lisbeth Salander (and because this one of the few films I’ve seen alone in the cinema). 2019-04-26: Belle and Sebastian, If You’re Feeling Sinister. 2019-04-26: I may have already posted this: Relearning Matrices as Linear Functions. If not, my bad, and here you are! 2019-04-26: While I loved using nteract, and latter hydrogen on Atom, I’m back to basic Jupyter notebook when I have to. I noticed that the interact team recently released papermill, which can help parameterizing, executing, and analyzing Jupyter Notebooks.
#python
2019-04-26: Principles and Techniques of Data Science.
#python
2019-04-26: The Tidynomicon, by Greg Wilson. Must-have ressource for those interested in switching to R, with a background in Python (or basically any scientific PL).
#rstats
2019-04-27: Tidyverse pipes in Pandas.
#python
2019-04-29: 🎥 The Bourne Legacy. 2019-04-30: 🎥 Jason Bourne. 2019-04-30: Generative and Analytical Models for Data Analysis. Nice read!> For both the generative model and the analytical model of data analysis, the missing ingredient was a clear definition of what made a data analysis successful.Also, be sure to check P values are just the tip of the iceberg (PDF). 2019-04-30: Spatial Population Genetics: It’s About Time. (via @strnr) 2019-04-30: Vim anti-patterns, via @_wilfredh.
#vim
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What's different in MultiMarkdown 3.0? What's different in MultiMarkdown 3.0? Why create another version of MultiMarkdown?
Maintaining a growing collection of nested regular expressions was going to become increasingly difficult. I don't plan on adding much (if any) in the way of new syntax features, but it was a mess.
Performance on longer documents was poor. The nested perl regular expressions was slow, even on a relatively fast computer. Performance on something like an iPhone would probably have been miserable.
The reliance on Perl made installation fairly complex on Windows. That didn't bother me too much, but it is a factor.
Perl can't be run on an iPhone/iPad, and I would like to be able to have MultiMarkdown on an iOS device, and not just regular Markdown (which exists in C versions).
I was interested in learning about PEG's and revisiting C programming.
The syntax has been fairly stable, and it would be nice to be able to formalize it a bit --- which happens by definition when using a PEG.
I wanted to revisit the syntax and features and clean things up a bit.
Did I mention how much faster this is? And that it could (eventually) run on an iPhone?
"Complete" documents vs. "snippets"
A "snippet" is a section of HTML (or LaTeX) that is not a complete, fully-formed document. It doesn't contain the header information to make it a valid XML document. It can't be compiled with LaTeX into a PDF without further commands.
For example:
# This is a header #And a paragraph.
becomes the following HTML snippet:
<h1 id="thisisaheader">This is a header</h1><p>And a paragraph.</p>
and the following LaTeX snippet:
\part{This is a header}\label{thisisaheader}And a paragraph.
It was not possible to create a LaTeX snippet with the original MultiMarkdown, because it relied on having a complete XHTML document that was then converted to LaTeX via an XSLT document (requiring a whole separate program). This was powerful, but complicated.
Now, I have come full-circle. peg-multimarkdown will now output LaTeX directly, without requiring XSLT. This allows the creation of LaTeX snippets, or complete documents, as necessary.
To create a complete document, simply include metadata. You can include a title, author, date, or whatever you like. If you don't want to include any real metadata, including "format: complete" will still trigger a complete document, just like it used to.
NOTE: If the only metadata present is
Base Header Level then acomplete document will not be triggered. This can be useful when combiningvarious documents together.
The old approach (even though it was hidden from most users) was a bit of a kludge, and this should be more elegant, and more flexible.
Metadata Differences
When metadata was repeated in MultiMarkdown 2.0, the second instance"overwrote" the first instance. In MultiMarkdown 3.0, each instance of arepeated metadata key is used. This is necessary for something like
LaTeX Input which can require multiple instances.
This means you can't "erase" an unnecessary metadata value by including a second, empty, copy. This was a trick used, for example, to erase undesired style information inserted into the document by older versions of Scrivener.
Creating LaTeX Documents
LaTeX documents are created a bit differently than under the old system. You no longer have to use an XSLT file to convert from XHTML to LaTeX. You can go straight from MultiMarkdown to LaTeX, which is faster and more flexible.
To create a complete LaTeX document, you can process your file as a snippet,and then place it in a LaTeX template that you already have. Alternatively,you can use metadata to trigger the creation of a complete document. You canuse the
LaTeX Input metadata to insert a
\input{file} command. You canthen store various template files in your texmf directory and call them withmetadata, or with embedded raw LaTeX commands in your document. For example:
LaTeX Input: mmd-memoir-header Title: Sample MultiMarkdown Document Author: Fletcher T. Penney LaTeX Mode: memoir LaTeX Input: mmd-memoir-begin-doc LaTeX Footer: mmd-memoir-footer
This would include several template files in the order that you see. The
LaTeX Footer metadata inserts a template at the end of your document. Notethat the order and placement of the
LaTeX Include statements is important.
The
LaTeX Mode metadata allows you to specify that MultiMarkdown should usethe
memoir or
beamer output format. This places subtle differences in theoutput document for compatibility with those respective classes.
This system isn't quite as powerful as the XSLT approach, since it doesn'talter the actual MultiMarkdown to LaTeX conversion process. But it is probablymuch more familiar to LaTeX users who are accustomed to using
\input{}commands and doesn't require knowledge of XSLT programming.
I recommend checking out the default LaTeX Support Files that are available on github. They are designed to serve as a starting point for your own needs.
Note: You can still use this version of MultiMarkdown to convert text intoXHTML, and then process the XHTML using XSLT to create a LaTeX document, justlike you used to in MMD 2.0. Images
In HTML, images have three pieces of "metadata" relevant to MMD ---
alt,
title, and
id. In Markdown and MultiMarkdown 2.0, these were structured inthe following manner:
This is an image 
In MultiMarkdown 2.0, the "alt text" was processed to "alttext" and used as an
id attribute.
The problem was that LaTeX, and later on ODF, didn't really need the
alt or
title metadata --- those formats really needed a
caption instead. Using acaption in some formats, but not others, leads to strange inconsistencies inthe documents created from the same MultiMarkdown source. And while Iunderstand the differences between the
alt and
title attributes, it reallydoesn't make a lot of sense.
So instead, in MultiMarkdown 3.0, the following is used instead:
This is an image orThis is an image ![Another *alt*][fig]![This is a *caption*][fig2]The following image has no `alt` or `caption`:![][fig2][fig]: file2.png "This is another title"[fig2]: file3.png "This is another title"
The
id attribute comes into play when an image is specified as a reference.This allows you to link to the image from elsewhere in your document.
When an image is the only thing constituting a paragraph, it is becomeswrapped in a
<figure> tag, and instead of an
alt attribute, it has acaption. This is demonstrated by
fig2 above. This caption is used regardlessof output format, providing consistency between HTML, LaTeX, and ODF. Whenthis happens, the alt tag is a stripped down version of the caption, since itcan't have any markup applied.
So, in MultiMarkdown 3.0, there are now 4 pieces of metadata ---
alt,
title,
id, and an optional
caption. There are only three places todescribe this metadata, so one piece has to be duplicated. Currently, the
alt is a duplicate of the
caption, sans any markup. If an image iscontained within a paragraph, an
alt attribute is created, but no
caption.
An alternative plan I am considering is to use what currently generates the
title attribute to instead generate the
alt attribute in all instances,and the caption can be ignored if an image is not considered a figure.
Footnotes
Footnotes work slightly differently than before. This is partially on purpose, and partly out of necessity. Specifically:
Footnotes are anchored based on number, rather than the label used in the MMD source. This won't show a visible difference to the reader, but the XHTML source will be different.
Footnotes can be used more than once. Each reference will link to the same numbered note, but the "return" link will only link to the first instance.
Footnote "return" links are a separate paragraph after the footnote. This is due to the way peg-markdown works, and it's not worth the effort to me to change it. You can always use CSS to change the appearance however you like.
Footnote numbers are surrounded by "[]" in the text.
Raw HTML
Because the original MultiMarkdown processed the text document into XHTMLfirst, and then processed the entire XHTML document into LaTeX, it couldn'ttell the difference between raw HTML and HTML that was created from plaintext.This version, however, uses the original plain text to create the LaTeXdocument. This means that any raw HTML inside your MultiMarkdown document is
not converted into LaTeX.
The benefit of this is that you can embed one piece of the document in two formats --- one for XHTML, and one for LaTeX:
<blockquote><p>Release early, release often!</p><blockquote><p>Linus Torvalds</p></blockquote></blockquote><!-- \epigraph{Release early, release often!}{Linus Torvalds} -->
In this section, when the document is converted into XHTML, the
blockquotesections will be used as expected, and the
epigraph will be ignored since itis inside a comment. Conversely, when processed into LaTeX, the raw HTML willbe ignored, and the comment will be processed as raw LaTeX.
You shouldn't need to use this feature, but if you want to specify exactly how a certain part of your document is processed into LaTeX, it's a neat trick.
Math Support
MultiMarkdown 2.0 supported ASCIIMathML embedded with MultiMarkdown documents. This syntax was then converted to MathML for XHTML output, and then further processed into LaTeX when creating LaTeX output. The benefit of this was that the ASCIIMathML syntax was pretty straightforward. The downside was that only a handful of browsers actually support MathML, so most of the time it was only useful for LaTeX. Many MMD users who are interested in LaTeX output already knew LaTeX, so they sometimes preferred native math syntax, which led to several hacks.
MultiMarkdown 3.0 does not have built in support for ASCIIMathML. In fact, I would probably have to write a parser from scratch to do anything useful with it, which I have little desire to do. So I came up with a compromise.
ASCIIMathML is no longer supported by MultiMarkdown. Instead, you
can useLaTeX to code for math within your document. When creating a LaTeX document,the source is simply passed through, and LaTeX handles it as usual. If youdesire, you can add a line to your header when creating XHTML documents thatwill allow MathJax to appropriately display your math.
Normally, MathJax
and LaTeX supported using
\[ math \] or
\( math \) toindicate that math was included. MMD stumbled on this due to some issues withescaping, so instead we use
\\[ math \\] and
\\( math \\). See anexample:
latex input: mmd-article-header Title: MultiMarkdown Math Example latex input: mmd-article-begin-doc latex footer: mmd-memoir-footer xhtml header: <script type="text/javascript" src="http://localhost/~fletcher/math/mathjax/MathJax.js"> </script> An example of math within a paragraph --- \\({e}^{i\pi }+1=0\\)--- easy enough.And an equation on it's own:\\[ {x}_{1,2}=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a} \\]That's it.
You would, of course, need to change the
xhtml header metadata to point toyour own installation of MathJax.
Note: MultiMarkdown doesn't actually do anything with the code insidethe brackets. It simply strips away the extra backslash and passes the LaTeXsource unchanged, where it is handled by MathJax if it's properly installed,or by LaTeX. If you're having trouble, you can certainly email theMultiMarkdown Discussion List, but I do not provide support for LaTeX code.
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Correlation vs Regression
I am still confused as to how correlation differs to regression,
technically. I understand that one is a measure of association and one a measure of causation
This is incorrect, the difference between correlation and regression is not causal. Both measures are
associational measures. I will elaborate on the difference between associational and causal quantities below, but let's quickly answer this part of your question.
Mathematically, the correlation of $X$ an $Y$ is a symmetric quantity (that is, $cor(Y, X) = cor(X, Y)$) and it's given by:
$$ cor(Y, X) = \frac{cov(Y, X)}{sd(Y)sd(X)}$$
Let $R_{yx}$ denote the regression coefficient of regressing $Y$ on $X$. This is usually not symmetric and it's given by:
$$ R_{yx} = \frac{cov(Y,X)}{var(X)} $$
Notice that if $var(X) = var(Y) = 1$ then the correlation coefficient and the regression coefficient will be the same.
These are pure associational quantities and you can see more about them here. Now let's move on to the causal inference part.
Association vs Causation
Let's start by stating what's the difference between association and causation. Consider two random variables, $X$ and $Y$ and now consider these two different questions:
What's the expected value of $Y$ if I see $X = x$? Let's denote this by $\mathbb{E}[Y|X=x]$. What's the expected value of $Y$ if I set $X = x$? Let's denote this by $\mathbb{E}[Y|do(X = x)]$
The first question is what regression can always give you. It's an associational question. The second question is an interventional question --- what would happen if you could set the value of $X$ to whatever you please? Usually, this is not the same as regression.
Let's see an example.
Consider the following
structural equations:
$$U = \epsilon_{u}\\X = \delta U + \epsilon_{x}\\Y = \beta X + \gamma U + \epsilon_y$$
Where all terms denoted by $\epsilon$ are mean zero and mutually independent. To simplify computation, also assume $U$, $X$ and $Y$ have been standardized (mean zero and unit variance). Suppose $U$ is unobserved.
What is the expected value of $X$ if we observe $X = x$? This is a traditional statistics questions and it's simply:
$$\mathbb{E}[Y|X=x] = \left(\beta + \gamma\delta\right)x$$
And you can estimate $b =\left(\beta + \gamma\delta\right)$ with a linear regression of $Y$ on $X$.
But what is the expected value of $Y$ if we
set $X$ to $x$? Setting $X$ to $x$ means erasing the structural equation for $X$ and substituting for $X = x$. Hence:
$$\mathbb{E}[Y|do(X=x)] = \beta x$$
And $\beta$ will be different from the regression coefficient $b$ unless either $\gamma$ or $\delta$ are equal to zero.
When can you estimate causal quantities with regression?
Now let's answer your question:
how can you actually measure causation mathematically, without
actually conducting a real life experiment.
In our example, we can't estimate the causal effect of $X$ on $Y$ because there is an open confounding path ($X \leftarrow U \rightarrow Y$), shown in red in the causal diagram below. We usually call this a
backdoor path.
$\hskip2.5in$
However,notice that if we could observe $U$ we could recover the causal parameter $\beta$ with a regression conditioning on $X$ and $U$.
More generally, the problem of recovering causal effects with adjustment from regression has been
mathematically solved. You can recover the structural coefficient from observational data if you can find a set of variables that: (i) blocks all backdoor paths from $X$ to $Y$; and, (ii) do not open other confounding paths (if you want total effects, you also do not want to control for mediators).
But how can you know which variables satisfy (i) and (ii)? You can only know that with
causal assumptions. For example, in the model below you don't want to adjust for $Z$!
$\hskip1in$
Take a look here for another discussion about confounders.
That is, we need to know the causal graph (or equivalently, a set of structural equations) in order to tell which set of variables you can use to identify the effect
via regression (if that set exists). You need causal assumptions to draw causal conclusions from observational data.
So to sum up Correlation coefficients and regression coefficients are different associational quantities, their difference has nothing to do with causality; Also, regression is usually not equal to causal quantities. Regression asks: what if I observe X? Causal inference asks: what if I manipulate X? But, under some circumstances, you can use regression to identify causal quantities with observational data. In order to do that you need some causal assumptions, for example, to identify when a group of variables satisfy the back-door (or single-door) criterion.
It's also worth noticing that adjustment via regression is not the only way to identify causal effects. For example, two other widely known methods are the front-door criterion and instrumental variables.
If you want to learn more about this, you should check the references here.
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One of the basic concepts of calculus is the correspondence between sums and integrals, which is easily evaluated with the help of Faulhaber’s formula. The Euler–Maclaurin formula is considered as a powerful connection between integrals. It is mostly used to approximate integrals by finite sums, or conversely to calculate or evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber’s formula for the sum of powers is an immediate consequence.
\[\large \sum_{k=p}^{m-1}\phi (k)=\int_{p}^{m}\phi (t)dt+\sum_{v=1}^{n-1}\frac{B_{v}}{v!}(\phi^{v-1}m-\phi^{v-1}p)+R_{n}\]
Where,
$B_{v}$=Bernoulli numbers $R_{n}$=remainder
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№ 9
All Issues Volume 64, № 10, 2012
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1299-1313
We propose a version of the Gauss - Ostrogradskii formula for a Banach manifold with uniform atlas.
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1314-1329
We study the asymptotic behavior of the eigenvalues and eigenfunctions of a singularly perturbed boundary-value problem for a second-order elliptic operator. The problem describes the eigenmodes of an elastic system with finite number of stiff light-weight inclusions of arbitrary shape. The leading terms of the asymptotic representation of eigenelements are constructed with regard for their multiplicity. The justification of the asymptotic formulas is based on the uniform resolvent convergence of a certain family of unbounded self-adjoint operators.
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1330-1329
The object of the present paper is to study a transformation called the $D$-homothetic deformation of normal almost contact metric manifolds. In particular, it is shown that, in a $(2n + 1)$-dimensional normal almost contact metric manifold, the Ricci operator $Q$ commutes with the structure tensor $\phi$ under certain conditions, and the operator $Q\phi - \phi Q$ is invariant under a $D$-homothetic deformation. We also discuss the invariance of $\eta$-Einstein manifolds, $\phi$-sectional curvature, and the local $\phi$-Ricci symmetry under a $D$-homothetic deformation. Finally, we prove the existence of such manifolds by a concrete example.
Asymptotics of Solutions of Nonautonomous Second-Order Ordinary Differential Equations Asymptotically Close to Linear Equations
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1346-1364
Asymptotic representations are obtained for a broad class of monotone solutions of nonautonomous binary differential equations of the second order that are close in a certain sense to linear equations.
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1365-1372
A new algebraic transformation method is constructed for finding traveling-wave solutions of complicated nonlinear wave equations on the basis of simpler ones. The generalized Dullin - Gottwald - Holm (DGH) equation and mKdV equations are chosen to illustrate our method. The solutions of the DGH equation can be obtained directly from solutions of the mKdV equation. Conditions under which different solutions appear are also given. Abundant traveling-wave solutions of the generalized DGH equation are obtained, including periodic solutions, smooth solutions with decay, solitary solutions, and kink solutions.
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1373-1380
We obtain new sufficient conditions for Fourier multipliers in the Hardy spaces $H_p(\mathbb{R}^n),\; 0 < p < 2$. These conditions are given in terms of the simultaneous behavior of a function and its derivatives. The results of this paper generalize the corresponding theorems of A. Miachi.
Solution of a Linear Second-Order Differential Equation with Coefficients Analytic in the Vicinity of a Fuchsian Zero Point
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1381-1393
We obtain a solution of a second-order differential equation with coefficients analytic near a Fuchsian zero point. This solution is expressed via the hypergeometric functions and the fractional-order hypergeometric functions introduced in this paper.
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1394-1415
We propose a functional-discrete method for solving the Goursat problem for the nonlinear Klein-Gordon equation. Sufficient conditions for the superexponential convergence of this method are obtained. The obtained theoretical results are illustrated by a numerical example.
Kolmogorov widths of the classes $B^{\Omega}_{p, \theta}$ of periodic functions of many variables in the space $L_q$
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1416-1425
We obtain exact-order estimates for the Kolmogorov widths of the classes $B^{\Omega}_{p, \theta}$ of periodic functions of many variables in the space $L_q$ for $1 ≤ p, q ≤ ∞$.
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1426-1431
We determine the exact values of the uniformly distributed ridge approximation of some classes of harmonic functions of two variables.
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1432-1437
Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas.
International conference "Theory of approximation of functions and its applications" dedicated to the 70 th birthday of the corresponding member of NASU Professor O. I. Stepanets (1942 - 2007)
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1438-1440
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Mott effect and \(J/\psi\) dissociation at the quark-hadron phase transition 25 Downloads Citations Abstract.
We investigate the in-medium modification of pseudoscalar and vector mesons in a QCD-motivated chiral quark model by solving the Dyson-Schwinger equations for quarks and mesons at finite temperature for a wide mass range of meson masses, from light (\(\pi\), \(\rho\)) to open-charm (
D, D *) states. At the chiral/deconfinement phase transition, the quark-antiquark bound states enter the continuum of unbound states and become broad resonances (hadronic Mott effect). We calculate the in-medium cross-sections for charmonium dissociation due to collisions with light hadrons in a chiral Lagrangian approach, and show that the D- and D *-meson spectral broadening lowers the threshold for charmonium dissociation by \(\pi\)- and \(\rho\)-mesons. This leads to a step-like enhancement in the reaction rate. We suggest that this mechanism for enhanced charmonium dissociation may be the physical mechanism underlying the anomalous \(J/\psi\) suppression observed by NA50. KeywordsPhase Transition Vector Meson Quark Model Finite Temperature Mott Effect Preview
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References 1. 2.T. Matsui, H. Satz, Phys. Lett. B, 178, 416 (1986).Google Scholar 3. 4. 5. 6.X.M. Xu, C.Y. Wong, T. Barnes, Phys. Res. C, 67, 014907 (2003), arXiv:nucl-th/0207018.Google Scholar 7. 8. 9. 10. 11.D. Blaschke, G. Burau, M.K. Volkov, V.L. Yudichev, Eur. Phys. J. A 11, 319 (2001).Google Scholar 12. 13.V. Ivanov, Yu. Kalinovsky, D. Blaschke, G. Burau, Chiral Lagrangian approach to the \(J/\psi\) breakup cross sectionGoogle Scholar 14.D. Blaschke, G. Burau, T. Barnes, Yu. Kalinovsky, E. Swanson, in Proceedings of the Budapest ‘02 Workshop on Quark & Hadron Dynamics (2002) 261, arXiv:hep-ph/0210265.Google Scholar 15. 16.G. Burau, Charmonium Dissoziation am QCD Phasenübergang PhD Thesis, Rostock (2002), unpublished.Google Scholar 17.
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Some degenerate parabolic problems: Existence and decay properties
1.
Dipartimento di Matematica, Sapienza Universitá di Roma, Piazzale A. Moro 5, 00185 Roma, Italy, Italy
Keywords:unbounded solutions, 1}(\Omega))$-solutions, T;W_0^{1, $L^1(0, Nonlinear parabolic problems, decay estimates, degenerate parabolic equations, existence results.. Mathematics Subject Classification:Primary: 35K65, 35K55; Secondary: 35K10, 35K15, 35K2. Citation:Lucio Boccardo, Maria Michaela Porzio. Some degenerate parabolic problems: Existence and decay properties. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 617-629. doi: 10.3934/dcdss.2014.7.617
References:
[1]
L. Boccardo and H. Brezis, Some remarks on a class of elliptic equations with degenerate coercivity,,
6 (2003), 521.
Google Scholar
[2] [3]
L. Boccardo, A. Dall'Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity,,
46 (1998), 51.
Google Scholar
[4] [5]
L. Boccardo, T. Gallouet and F. Murat, Unicité de la solution pour des equations elliptiques non linéaires,,
315 (1992), 1159.
Google Scholar
[6]
Bonforte and G. Grillo, Super and ultracontractive bounds for doubly nonlinear evolution equations,,
22 (2006), 111.
Google Scholar
[7]
H. Brezis and T. Cazenave, Notes,,
[8] [9]
A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations,,
170 (1996), 207.
doi: 10.1007/BF01758989.
Google Scholar
[10] [11] [12]
G. Grillo, On the equivalence between p-Poincaré inequalities and $L^r-L^q$ regularization and decay estimates of certain nonlinear evolution,,
249 (2010), 2561.
doi: 10.1016/j.jde.2010.05.022.
Google Scholar
[13]
G. Grillo, M. Muratori and M. M. Porzio, Porous media equations with two weights: Existence, uniqueness, smoothing and decay properties of energy solutions via Poincaré inequalities,,
33 (2013), 3599.
doi: 10.3934/dcds.2013.33.3599.
Google Scholar
[14]
O. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva,
[15]
J. L. Lions,
[16]
A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form,,
[17] [18] [19]
M. M. Porzio, On uniform estimates,,
[20]
M. M. Porzio and M. A. Pozio, Parabolic equations with non-linear, degenerate and space-time dependent operators,,
8 (2008), 31.
doi: 10.1007/s00028-007-0317-8.
Google Scholar
[21]
M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations,,
210 (2013), 713.
doi: 10.1007/s00205-013-0666-0.
Google Scholar
[22]
M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations,,
[23]
M. M. Porzio and F. Smarrazzo, Radon Measure-Valued Solutions for some quasilinear degenerate elliptic equations,,
[24] [25]
show all references
References:
[1]
L. Boccardo and H. Brezis, Some remarks on a class of elliptic equations with degenerate coercivity,,
6 (2003), 521.
Google Scholar
[2] [3]
L. Boccardo, A. Dall'Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity,,
46 (1998), 51.
Google Scholar
[4] [5]
L. Boccardo, T. Gallouet and F. Murat, Unicité de la solution pour des equations elliptiques non linéaires,,
315 (1992), 1159.
Google Scholar
[6]
Bonforte and G. Grillo, Super and ultracontractive bounds for doubly nonlinear evolution equations,,
22 (2006), 111.
Google Scholar
[7]
H. Brezis and T. Cazenave, Notes,,
[8] [9]
A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations,,
170 (1996), 207.
doi: 10.1007/BF01758989.
Google Scholar
[10] [11] [12]
G. Grillo, On the equivalence between p-Poincaré inequalities and $L^r-L^q$ regularization and decay estimates of certain nonlinear evolution,,
249 (2010), 2561.
doi: 10.1016/j.jde.2010.05.022.
Google Scholar
[13]
G. Grillo, M. Muratori and M. M. Porzio, Porous media equations with two weights: Existence, uniqueness, smoothing and decay properties of energy solutions via Poincaré inequalities,,
33 (2013), 3599.
doi: 10.3934/dcds.2013.33.3599.
Google Scholar
[14]
O. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva,
[15]
J. L. Lions,
[16]
A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form,,
[17] [18] [19]
M. M. Porzio, On uniform estimates,,
[20]
M. M. Porzio and M. A. Pozio, Parabolic equations with non-linear, degenerate and space-time dependent operators,,
8 (2008), 31.
doi: 10.1007/s00028-007-0317-8.
Google Scholar
[21]
M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations,,
210 (2013), 713.
doi: 10.1007/s00205-013-0666-0.
Google Scholar
[22]
M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations,,
[23]
M. M. Porzio and F. Smarrazzo, Radon Measure-Valued Solutions for some quasilinear degenerate elliptic equations,,
[24] [25]
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Alexandre N. Carvalho, Jan W. Cholewa.
Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$.
[2]
Xin Li, Chunyou Sun, Na Zhang.
Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$.
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Denis R. Akhmetov, Renato Spigler.
$L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation.
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On certain nonlinear parabolic equations with singular diffusion and data in $L^1$.
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Canonical dual approach to solving 0-1 quadratic programming problems.
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completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$.
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Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations.
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We often use Ricker wavelets to model seismic, for example when making a synthetic seismogram with which to help tie a well. One simple way to guesstimate the peak or central frequency of the wavelet that will model a particlar seismic section is to count the peaks per unit time in the seismic. But this tends to overestimate the actual frequency because the maximum frequency of a Ricker wavelet is more than the peak frequency. The question is, how much more?
To investigate, let's make a Ricker wavelet and see what it looks like in the time and frequency domains.
>>> T, dt, f = 0.256, 0.001, 25>>> import bruges>>> w, t = bruges.filters.ricker(T, dt, f, return_t=True)>>> import scipy.signal>>> f_W, W = scipy.signal.welch(w, fs=1/dt, nperseg=256)
When we count the peaks in a section, the assumption is that this apparent frequency — that is, the reciprocal of apparent period or distance between the extrema — tells us the dominant or peak frequency.
To help see why this assumption is wrong, let's compare the Ricker with a signal whose apparent frequency does match its peak frequency: a pure cosine:
>>> c = np.cos(2 * 25 * np.pi * t)>>> f_C, C = scipy.signal.welch(c, fs=1/dt, nperseg=256)
Notice that the signal is much narrower in bandwidth. If we allowed more oscillations, it would be even narrower. If it lasted forever, it would be a spike in the frequency domain.
Let's overlay the signals to get a picture of the difference in the relative periods:
The practical consequence of this is that if we estimate the peak frequency to be \(f\ \mathrm{Hz}\), then we need to reduce \(f\) by some factor if we want to design a wavelet to match the data. To get this factor, we need to know the apparent period of the Ricker function, as given by the time difference between the two minima.
Let's look at a couple of different ways to find those minima: numerically and analytically.
Find minima numerically
We'll use
scipy.optimize.minimize to find a numerical solution. In order to use it, we'll need a slightly different expression for the Ricker function — casting it in terms of a time basis
t. We'll also keep
f as a variable, rather than hard-coding it in the expression, to give us the flexibility of computing the minima for different values of
f.
Here's the equation we're implementing:
$$ w(t, f) = (1 - 2\pi^2 f^2 t^2)\ e^{-\pi^2 f^2 t^2} $$
In Python:
>>> def ricker(t, f):>>> return (1 - 2*(np.pi*f*t)**2) * np.exp(-(np.pi*f*t)**2)
Check that the wavelet looks like it did before, by comparing the output of this function when
f is 25 with the wavelet
w we were using before:
>>> f = 25>>> np.allclose(w, ricker(t, f=25))True
Now we call SciPy's
minimize function on our
ricker function. It itertively searches for a minimum solution, then gives us the
x (which is really
t in our case) at that minimum:
>>> import scipy.optimize>>> f = 25>>> scipy.optimize.minimize(ricker, x0=0, args=(f))fun: -0.4462603202963996 hess_inv: array([[1]]) jac: array([-2.19792128e-07]) message: 'Optimization terminated successfully.' nfev: 30 nit: 1 njev: 10 status: 0 success: True x: array([0.01559393])
So the minimum amplitude, given by
fun, is -0.44626 and it occurs at an
x (time) of \(\pm 0.01559\ \mathrm{s}\).
In comparison, the minima of the cosine function occur at a time of \(\pm 0.02\ \mathrm{s}\). In other words, the period appears to be \(0.02 - 0.01559 = 0.00441\ \mathrm{s}\) shorter than the pure waveform, which is...
>>> (0.02 - 0.01559) / 0.020.22050000000000003
...about 22% shorter. This means that if we naively estimate frequency by counting peaks or zero crossings, we'll tend to overestimate the peak frequency of the wavelet by about 22% — assuming it is approximately Ricker-like; if it isn't we can use the same method to estimate the error for other functions.
This is good to know, but it would be interesting to know if this parameter depends on frequency, and also to have a more precise way to describe it than a decimal. To get at these questions, we need an analytic solution.
Find minima analytically
Python's SymPy package is a bit like Maple — it understands math symbolically. We'll use
sympy.solve to find an analytic solution. It turns out that it needs the Ricker function writing in yet another way, using SymPy symbols and expressions for \(\mathrm{e}\) and \(\pi\).
import sympy as spt, f = sp.Symbol('t'), sp.Symbol('f')r = (1 - 2*(sp.pi*f*t)**2) * sp.exp(-(sp.pi*f*t)**2)
Now we can easily find the solutions to the Ricker equation, that is, the times at which the function is equal to zero:
>>> sp.solvers.solve(r, t)[-sqrt(2)/(2*pi*f), sqrt(2)/(2*pi*f)]
But this is not quite what we want. We need the minima, not the zero-crossings.
Maybe there's a better way to do this, but here's one way. Note that the gradient (slope or derivative) of the Ricker function is zero at the minima, so let's just solve the first time derivative of the Ricker function. That will give us the three times at which the function has a gradient of zero.
>>> dwdt = sp.diff(r, t)>>> sp.solvers.solve(dwdt, t)[0, -sqrt(6)/(2*pi*f), sqrt(6)/(2*pi*f)]
In other words, the non-zero minima of the Ricker function are at:
$$ \pm \frac{\sqrt{6}}{2\pi f} $$
Let's just check that this evaluates to the same answer we got from
scipy.optimize, which was 0.01559.
>>> np.sqrt(6) / (2 * np.pi * 25)0.015593936024673521
The solutions agree.
While we're looking at this, we can also compute the analytic solution to the amplitude of the minima, which SciPy calculated as -0.446. We just plug one of the expressions for the minimum time into the expression for
r:
>>> r.subs({t: sp.sqrt(6)/(2*sp.pi*f)})-2*exp(-3/2)
Apparent frequency
So what's the result of all this? What's the correction we need to make?
The minima of the Ricker wavelet are \(\sqrt{6}\ /\ \pi f_\mathrm{actual}\ \mathrm{s}\) apart — this is the apparent period. If we're assuming a pure tone, this period corresponds to an apparent frequency of \(\pi f_\mathrm{actual}\ /\ \sqrt{6}\ \mathrm{Hz}\). For \(f = 25\ \mathrm{Hz}\), this apparent frequency is:
>>> (np.pi * 25) / np.sqrt(6)32.06374575404661
If we were to try to model the data with a Ricker of 32 Hz, the frequency will be too high. We need to multiply the frequency by a factor of \(\sqrt{6} / \pi\), like so:
>>> 32.064 * np.sqrt(6) / (np.pi)25.00019823475659
This gives the correct frequency of 25 Hz.
To sum up, rearranging the expression above:
$$ f_\mathrm{actual} = f_\mathrm{apparent} \frac{\sqrt{6}}{\pi} $$
Expressed as a decimal, the factor we were seeking is therefore \(\sqrt{6}\ /\ \pi\):
>>> np.sqrt(6) / np.pi0.779696801233676
That is, the reduction factor is 22%.
Curious coincidence: in the recent Pi Day post, I mentioned the Riemann zeta function of 2 as a way to compute \(\pi\). It evaluates to \((\pi / \sqrt{6})^2\). Is there a million-dollar connection between the humble Ricker wavelet and the Riemann hypothesis?
I doubt it.
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Is it correct to represent Higgs VEV as the coherent state?
I mean, suppose translational invariant coherent state
$$
|\alpha\rangle = Ne^{\alpha \hat{a}^{\dagger}_{\mathbf p =0}}|\alpha =0\rangle, \quad N: \quad \langle \alpha|\alpha\rangle = 1 $$
of massless particles with zero dispersion relation. In principle, it is possible to state that the non-shifted Higgs doublet operator has VEV on $|\alpha \rangle$ state. But is this correct?
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The functional derivative $\frac{\delta}{\delta \phi}$ acts on
functionals, things that map functions to real numbers. That is, they act on actions $S$, not lagrangians $L$. I don't know where you got your original question, but there indeed should be a minus sign! Altogether, I think what you're asking is why:$$\frac{\delta}{\delta \phi} \int d^4x \left(\frac{1}{2} \partial^\mu \phi \partial_\mu \phi\right) = - \partial^\mu \partial_\mu \phi \ .$$
There are quick formulas you can look up, but for understanding I always find it easiest to work through the variation directly. First, take your term $\frac{1}{2} \partial^\mu \phi \partial_\mu \phi$ and do the transformation $\phi \to \phi + \delta \phi$:
$$\begin{align*}\frac{1}{2} \partial^\mu \phi \partial_\mu \phi &\to \frac{1}{2} \partial^\mu (\phi+\delta \phi) \partial_\mu (\phi + \delta \phi) \\&= \frac{1}{2} \partial^\mu \phi \partial_\mu \phi + \frac{1}{2} \partial^\mu (\delta \phi) \partial_\mu \phi + \frac{1}{2} \partial^\mu \phi \partial_\mu (\delta \phi) + \mathcal{O}(\delta\phi^2)\end{align*}$$
Now you can probably see where this is going, the $1/2$ will be accounted for by the two $\delta \phi$ terms in the expansion. This is really just product rule!
To extract the $\delta \phi$ you can integrate each term by parts, dropping the total derivative because this is physics and everything is 0 on the boundary :)
$$\begin{align*}\delta \int d^4x \left(\frac{1}{2} \partial^\mu \phi \partial_\mu \phi\right) &= \int d^4 x\left(-\frac{1}{2}\delta \phi \ \partial^\mu \partial_\mu \phi - \frac{1}{2}\partial_\mu\partial^\mu \phi \ \delta \phi + \partial_\mu(\dots) \right )\\&= \int d^4 x \ \delta \phi \left( -\partial^\mu \partial_\mu \phi \right)\end{align*}$$The answer is just the integrand, without the $\delta \phi$, so finally we write:$$\frac{\delta}{\delta \phi} \int d^4x \left(\frac{1}{2} \partial^\mu \phi \partial_\mu \phi\right) = - \partial^\mu \partial_\mu \phi \ .$$
Section 9.2 of Peskin & Schroeder runs through the axioms of functional integration if you'd like to see a more formal take on it.
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Unit 02: Differentiation
Notes (Solutions) of Unit 02: Differentiation, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.
Here are few online resource, which are very helpful to find derivative.
Contents & summary Introduction Average Rate of Change Derivative of a Function Finding $f'(x)$ from Definition of Derivative Derivative of $x^n$ where $n \in \mathbb{Z}$ Exercise 2.1 Differentiation of Expressions of the types Exercise 2.2 Theorems on Differentiation Exercise 2.3 The Chain Rule Derivatives of Inverse Function Derivative of a Function given in form of parametric Equations Differentiation of Implicit Relation Exercise 2.4 Derivatives of Trigonometric Function Derivatives of Inverse Trigonometric Functions Exercise 2..5 Derivative of Exponential Functions Derivative of the Logarithmic Function Logarithmic Differentiation Derivative of Hyperbolic Function Derivative of the Inverse Hyperbolic Function Exercise 2.6 Successive Differentiation ( or Derivatives) Exercise 2.7 Series Expansions of Function Tailor Series Expansions of Function Exercise 2.8 Geometrical Interpretation of a Derivative Increasing and Decreasing Function Relative Extrema Critical Values and Critical Points Exercise 2.9 Exercise 2.10 Which method is better
In this chapter many questions can be solved in much easier way. Actually in every exercise some formula/method is introduced to solve the question. In examination it is not necessary to do the same method as given in exercise. Here is one example:
We have to find the derivative of $\frac{x+1}{x-1}$ with respect to $x$.
Method 1
$$ \begin{aligned} \frac{d}{dx}\left(\frac{x+1}{x-1}\right) &= \frac{(x-1)\frac{d}{dx}(x+1)-(x+1)\frac{d}{dx}(x-1)}{(x-1)^2}\\ &= \frac{(x-1)(1)-(x+1)(1)}{(x-1)^2}\\ &= \frac{x-1-x-1}{(x-1)^2}\\ &= \frac{-2}{(x-1)^2} \end{aligned} $$
Method 2
By converting improper to proper fraction $$ \frac{x+1}{x-1}= 1+\frac{2}{x-1}=1+2(x-1)^{-1} $$ Now $$ \begin{aligned} \frac{d}{dx}\left(\frac{x+1}{x-1}\right) &=\frac{d}{dx}\left(1+2(x-1)^{-1}\right)\\ &= 0-2(x-1)^{-2}(1)\\ &= \frac{-2}{(x-1)^2} \end{aligned} $$
This was a simple example but try it to find the derivative of $\frac{x^2+1}{x^2-1}$.
Solutions Here are previous and next chapters fsc/fsc_part_2_solutions/ch02 Last modified: 4 weeks ago by Dr. Atiq ur Rehman
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Is it possible to partition $\mathbb R^3$ into unit circles?
The construction is based on a well ordering of $R^3$ into the least ordinal of cardinality continuum. Let $\phi$ be that ordinal and let $R^3=\{p_\alpha:\alpha<\phi\}$ be an enumeration of the points of space. We define a unit circle $C_\alpha$ containing $p_\alpha$ by transfinite recursion on $\alpha$, for some $\alpha$ we do nothing. Here is the recursion step. Assume we have reached step $\alpha$ and some circles $\{C_\beta:\beta<\alpha\}$ have been determined. If some of them contains (=covers) $p_\alpha$, we do nothing. Otherwise, we choose a unit circle containing $p_\alpha$ that misses all the earlier circles. For that, we first choose a plane throu $p_\alpha$ that is distinct from the planes of the earlier circles. This is possible, as there are continuum many planes throu $p_\alpha$ and less than continuum many planes which are the planes of those earlier circles. Let $K$ be the plane chosen. The earlier circles intersect $K$ in less than continuum many points, so it suffices to find, in $K$, a unit circle going throu $p_\alpha$ which misses certain less than continuum many points. That is easy: there are continuum many unit circles in $K$ taht pass throu $p_\alpha$ and each of the bad points disqualifies only 2 of them.
Even though this question is old, I'd like to give what I regard as a very beautiful solution. It is different from the others in that the circles used are not round (but they are unlinked). First observe that the circles $x^2 + y^2 = r^2$, $z = c$, for $r \geq 1$ and $c$ any real number, decompose all of $\Bbb{R}^3$ except an open cylinder into circles. At first glance, this seems to have accomplished nothing, since the open cylinder is homeomorphic to $\Bbb{R}^3$, so we have reduced the original problem to an equivalent problem. However, look at the left-hand figure of the included image, which shows an open cylinder embedded as a U shape, with the ends going to infinity in the same direction. Since this is just a deformation of the original embedding, we can decompose the complement into circles. To handle the interior, embed an open cylinder into it, as shown in the right-hand figure. We can decompose the complement of the smaller U-shaped cylinder into circles. We continue in this way, making sure that the embedded cylinders go off to infinity, so that every point of $\Bbb{R}^3$ is included at some finite stage.
It seems like we have never really solved the problem, but instead have just pushed it away so much that it vanishes into thin air!
Péter's proof is very clever and, while there is no real need to resurrect this thread, the following is quite straightforward in case one is not inclined to hunt for it in the literature on this subject:
Observe that you can cover a two-punctured sphere with circles. Now consider a family of circles lying in the $xy$ plane, radii 1, centred at the points $(4k+1,0,0)$ for $k \in \mathbb{Z}$. Each sphere about the origin intersects this family in exactly two places.
In this article, the authors prove that not only can you partition $R^3$ into congruent circles, but you can do so into
unlinked congruent circles. They also prove a variety of other similar results: $R^3$ can be partitioned into isometric copies of any family of continuum many real analytic curves. And they consider the question in higher dimensions, and also the role of AC in the proofs: for example, in $R^3$ no AC is needed for circles, if different sizes are allowed.
A very nice, explicit, elementary partition (without the Axiom of Choice) of $\mathbb{R}^3$ into geometric circles of variable radii is given in: [MR0719756] Szulkin, Andrzej. $\mathbb{R}^3$
is the union of disjoint circles. Amer. Math. Monthly 90 (1983), no. 9, 640–641.
Evelyn Sander says here, "Geometric circles of unit radius are called hoops. Using the Axiom of Choice, J.H. Conway and H.T. Croft showed that it is nevertheless possible to discontinuously fill three-space using disjoint hoops." The "nevertheless" was to contrast with filling continuously. This was a report on a talk by Daniel Asimov in 1994, who showed that it is not possible to fill continuously with hoops.
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2 Methods for Simulating Radiated Fields in COMSOL Multiphysics®
In Part 2 of our blog series on multiscale modeling in high-frequency electromagnetics, we discuss a practical implementation of multiscale techniques in the COMSOL Multiphysics® software. We will simulate radiated fields using two different techniques and verify our results with theory. While these methods can be generally applied, we will always revolve around the practical issue of antenna-to-antenna communication. For a review of the theory and terms, you can refer to the first post in the series.
Simulating a Radiating Antenna
Let’s begin by discussing a traditional antenna simulation using COMSOL Multiphysics and the RF Module. When we simulate a radiating antenna, we have a local source and are interested in the subsequent electromagnetic fields, both nearby and outgoing from the antenna. This is fundamentally what an antenna does. It converts local information (e.g., voltage or current) into propagating information (e.g., outgoing radiation). A receiving antenna inverts this operation and changes incident radiation into local information. Many devices, such as a cellphone, act as both receiving and emitting antennas, which is what enables you to make a phone call or browse the web.
Antennas of the Atacama Large Millimeter Array (ALMA) in Chile. ALMA detects signals from space to help scientists study the formation of stars, planets, and galaxies. Needless to say, the distance these signals travel is much greater than the size of an antenna. Image licensed under CC BY 4.0, via ESO/C. Malin.
In order to keep the required computational resources reasonable, we model only a small region of space around the antenna. We then truncate this small simulation domain with an absorbing boundary, such as a perfectly matched layer (PML), which absorbs the outgoing radiation. Since this will solve for the complex electric field everywhere in our simulation domain, we will refer to this as a
Full-Wave simulation.
We then extract information about the antenna’s emission pattern using a
Far-Field Domain node, which performs a near-to-far-field transformation. This approach gives us information about the electromagnetic field in two regions: the fields in the immediate vicinity of the antenna, which are computed directly, and the fields far away, which are calculated using the Far-Field Domain node. This is demonstrated in a number of RF models in the Application Gallery, such as the Dipole Antenna tutorial model, so we will not comment further on the practical implementation here. Using the Far-Field Domain Node
One question that occasionally comes up in technical support is: “How do I use the
Far-Field Domain node to calculate the radiated field at a specific location?” This is an excellent question. As stated in the RF Module User’s Guide, the Far-Field Domain node calculates the scattering amplitude, and so determining the complex field at a specific location requires a modification for distance and phase. The expression for the x-component of the electric field in the far field is:
and similar expressions apply to the
y– and z-component, where r is the radial distance in spherical coordinates, k is the wave vector for the medium, and emw.Efarx is the scattering amplitude. It is worth pointing out that emw.Efarx is the scattering amplitude in a particular direction, and so it depends on angular position (\theta, \phi), but not radial position. The decrease in field strength is solely governed by the 1/ r term. There are also variables emw.Efarphi and emw.Efartheta, which are for the scattering amplitude in spherical coordinates.
To verify this result, we simulate a perfect electric dipole and compare the simulation results with the analytical solution, which we covered in the previous blog post. As we stated in that post, we split the full results into two terms, which we call the near- and far-field terms. We briefly restate those results here.
\overrightarrow{E} & = \overrightarrow{E}_{FF} + \overrightarrow{E}_{NF}\\
\overrightarrow{E}_{FF} & = \frac{1}{4\pi\epsilon_0}k^2(\hat{r}\times\vec{p})\times\hat{r}\frac{e^{-jkr}}{r}\\
\overrightarrow{E}_{NF} & = \frac{1}{4\pi\epsilon_0}[3\hat{r}(\hat{r}\cdot\vec{p})-\vec{p}](\frac{1}{r^3}+\frac{jk}{r^2})e^{-jkr}
\end{align}
where \vec{p} is the dipole moment of the radiation source and \hat{r} is the unit vector in spherical coordinates.
Below, we can see the electric fields vs. distance calculated using the
Far-Field Domain node for a dipole at the origin with \vec{p}=\left(0,0,1\right)A\cdot m. For comparison, we have included the Far-Field Domain node, the full theory, as well as the near- and far-field terms individually. The fields are evaluated along an arbitrary cut line. As you can see, there is overlap between the Far-Field Domain node and the far-field theory plots, and they agree with the full theory as the distance from the antenna increases. This is because the Far-Field Domain node will only account for radiation that goes like 1/ r, and so the agreement improves with increasing distance as the contribution of the 1/ r 2 and 1/ r 3 terms go to zero. In other words, the Far-Field Domain node is correct in the far field, which you probably would have guessed from the name. A comparison of the Far-Field Domain node vs. theory for a point dipole source. Using the Electromagnetic Waves, Beam Envelopes Interface
For most simulations, the near-field and far-field information is sufficient and no further work is necessary. In some cases, however, we also want to know the fields in the intermediate region, also known as the induction or transition zone. One option is to simply increase the simulation size until you explicitly calculate this information as part of the simulation. The drawback of this technique is that the increased simulation size requires more computational resources. We recommend a maximum mesh element size of \lambda/5 for 3D electromagnetic simulations. As the simulation size increases, the number of mesh elements increases, and so do the computational requirements.
Another option is to use the
Electromagnetic Waves, Beam Envelopes interface, which here we will simply refer to as Beam-Envelopes. As discussed in a previous blog post, Beam-Envelopes is an excellent choice when the simulation solution will have either one or two directions of propagation, and will allow us to use a much coarser mesh. Since the phase of the emission from an antenna will look like an outgoing spherical wave, this is a perfect solution for determining these fields. We perform a Full-Wave simulation of the fields near the source, as before, and then use Beam-Envelopes to simulate the fields out to an arbitrary distance, as required. The simulation domain assignments. If the outer region is assigned to PML, then a Full-Wave simulation is performed everywhere. It is also possible to solve the inner region using a Full-Wave simulation and the outer region using Beam-Envelopes , as we will discuss below. Note that this image is not to scale, and we have only modeled 1/8 of the spherical domain due to symmetry.
How do we couple the
Beam-Envelopes simulation to our Full-Wave simulation of the dipole? This can be done in two steps involving the boundary conditions at the interface between the Full-Wave and Beam-Envelopes domains. First, we set the exterior boundary of the Full-Wave simulation to PMC, which is the natural boundary condition for that simulation. The second step is to set that same boundary to an Electric Field boundary condition for Beam-Envelopes. We then specify the field values in the Beam-Envelopes Electric Field boundary condition according to the fields computed from the Full-Wave simulation, as shown here. The Electric Field boundary condition in Beam-Envelopes . Note that the image in the top right is not to scale.
A Matched Boundary Condition is applied to the exterior boundary of the
Beam-Envelopes domain to absorb the outgoing spherical wave. The remaining boundaries are set to PEC and PMC according to symmetry. We must also set the solver to Fully Coupled, which is described in more detail in two blog posts on solving multiphysics models and improving convergence from a previous blog series on solvers.
If we again examine the comparison between simulation and theory, we see excellent agreement over the entire simulation range. This shows that the PMC and Electric Field boundary conditions have enforced continuity between the two interfaces and they have fully reproduced the analytical solution. You can download the model file in the Application Gallery.
A comparison of the electric field of the Full-Wave and Beam-Envelopes simulations vs. the full theory. Concluding Thoughts on Simulating a Radiating Source in COMSOL Multiphysics®
In today’s blog post, we examined two ways of computing the electric field at points far away from the source antenna and verified the results using the analytical solution for an electric point dipole. These two techniques are using the
Far-Field Domain node from a Full-Wave simulation and linking a Full-Wave simulation to a Beam-Envelopes simulation. In both cases, the fields near the source and in the far field are correctly computed. The coupled approach using Beam-Envelopes has the additional advantage in that it also computes fields in the intermediate region. In the next post in the series, we will combine the calculated far-field radiation with a simulation of a receiving antenna and determine the received power. Stay tuned! Read the Full Series Browse the other posts in the Multiscale Modeling for High-Frequency Electromagnetics blog series Comments (6) CATEGORIES Chemical COMSOL Now Electrical Fluid General Interfacing Mechanical Today in Science TAGS CATEGORIES Chemical COMSOL Now Electrical Fluid General Interfacing Mechanical Today in Science
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This is a comment on a post by Andreas Blass here.
I have not read Kripke's statement, apart from Andreas's sketch above, but it sounds familiar from elsewhere, so let me comment a bit on the family of results and the many discussions about a family of unprovable statements of the form "for every $n$ there is a finite set that approximates a model of your theory T to the degree $n$" (where "to the degree $n$" is specified separately each time).
The first such statement found in print belongs to Paris and Harrington (see the original article in the Handbook for Mathematical Logic), and is sometimes referred to as "half-baked Paris-Harrington Principle". It says "for every $n$ there exists a finite sequence of points that acts as diagonal indiscernibles for the first $n$ $\Delta_0$ formulas", where "diagonal indiscernibility" or "Paris indiscernibility" is the condition that for any $c_{i_0} < c_{i_1}<\dots< c_{i_k} < c_{j_1} <\dots< c_{j_k}$ in the sequence, we have: for every parameter $a < c_{i_0}$, the following holds:
$\forall x_1 < c_{i_1} \exists x_2 < c_{i_2}\dots \phi(a, x_1, x_2... x_k)$ < -- > $\forall x_1 < c_{j_1} \exists x_2 < c_{j_2}\dots \phi(a, x_1, x_2, ...x_k)$.
Since that time statements of this form became routine intermediate steps in unprovability proofs. For example Shelah tried to modify this statement and came up with something that should be of strength $\Pi_1^1\text{-CA}_0$ (see "On logical sentences in PA").
The same idea is the core of most of Harvey Friedman's proofs in the last 25 years, but at higher levels of sophistication. For example for Proposition C, Friedman has 9 intermediate statements of this shape ("the Transmutations"), where the notion of indiscernibility changes at each step. And for Proposition B you need perhaps only 6 transmutations.
When I was entering the subject in 2002 -- 2003, I wrote a naive article draft on half-baked PH and beyond, tinkering and trying to generalize. But then I realized that this topic is widely developed and discussed in the unprovability community, so since this piece already entered the unprovability community's knowledge pot long before me, perhaps none of it is publishable.
One more thought: there are two or three ways of dressing unprovability proofs. Paris's original dressing was via cuts in models of arithmetic, but after Harrington's simplification of n-densities, they wrote the proof for the Handbook in the finitistic way, via half-baked Paris-Harrington + compactness as I sketched above. After that Paris and his co-workers returned to thinking in terms of cuts in models of arithmetic. Both ways of dressing the proof are equally good, but each person usually chooses one.
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The qualifier "natural" is meant to exclude examples like "PA + P=NP" or "PA + True $\Pi_1$".
For concreteness, let's say that "natural" = sound, computably enumerable, with a feasible proof-checker. Context of the question.
A naive way to approach the P vs NP problem, from the logical point of view, could go like this:
Show that if P=NP, then P=NP is provable is some fixed system $S$, such as ZFC or ZFC+large cardinals; Improve the previous result for weaker and weaker $S$ $-$ for example, go from ZFC down to PA2, PA, $I\Sigma_1$, etc.; Once $S$ is as elementary as possible, use an ad-hoc argument to show $\not \vdash_S$ P=NP.
Of course, as all known approaches to the problem, this one quickly falls upon itself.
Proposition. ( folklore?)For every function $f$ which is computed by a Turing machine $M$, and for every natural formal system $S$, which proves that $M$ computes $f$, there exists a Turing machine $M'$ which computes $f$, such that $S$ does not prove that $M'$ computes $f$. The runtime of $M'$ is $O(n+Time(M))$.
Given input $x$, the machine $M'$ searches for a contradiction in $S$ for $|x|$ many steps. If no contradiction is found, it runs $M$ on $x$, returning the result; otherwise, it launches all nuclear missiles at once.
Of course, $M'$ computes $f$. But $S$ can never know this, because then it would know that there is no contradiction from $S$, contradicting Gödel's theorem.
The point of the above observation is that no formal system can make inferences about correctness of algorithms from runtime constraints alone. If we assume that S knows that some machine M decides SAT in polynomial time, there will always be another M for which S will not know this.
Motivation.
This seems troubling, since it can in principle be conceived that, while P=NP, the logical complexity of proving any polynomial-time satisfiability algorithm $M$ to be correct can be larger than the consistency strength of any formal theory $T$ that may be considered in say, the next 100 years:
$(\forall x\ M(x){\in}\{0,1\}\ \&\ (M(x)=1 \iff x \in SAT)) \Rightarrow \mathsf{Con}(T)$
Can such a situation be ruled out, for some natural extension $T$ of ZFC? This would mean exactly that $T$ answers the question posed in the title.
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