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I have the following dynamical system:
$\dot{x_1}= -x_2 + (x_1(1-(x_1^2+x_2^2)^2))$ , $ \dot{x_2}= x_1 + (x_2(1-(x_1^2+x_2^2)^2))$, $\dot{x_3}= \epsilon x_3$ . I am required to work out the flow for this system. I have switched it to cylindrical coordinates obtaining $\dot{r}=r(1-r^4)$ , $\dot{\theta}=1$, $\dot{z}=\eps... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
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Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
De Bruijn-Newman constant
For each real number [math]t[/math], define the entire function [math]H_t: {\mathbf C} \to {\mathbf C}[/math] by the formula
[math]\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du[/math]
where [math]\Phi[/math] is the super-exponentially decaying function
[math]\displaystyle... |
Edit: I'm leaving the old post below, but before I want to write the proof as suggested by Bruce from his book, which uses the ideas in a more efficient way.
Assume that $\|p-q\|<1$, with $p,q\in A$, a unital C$^*$-algebra. Let $x=pq+(1-p)(1-q)$. Then, as $2p-1$ is a unitary, $$\|1-x\|=\|(2p-1)(p-q)\|=\|p-q\|<1.$$So $x... |
Here is a problem from Guillemin-Pollack:
The graph of a map $f: X\rightarrow Y$ is the subset of $X\times Y$ defined by $$graph(f)=\{(x,f(x)):x\in X\}.$$ Define $F: X\rightarrow graph(f)$ by $F(x)=(x,f(x))$. Show that if $f$ is smooth, $F$ is a diffeomorphism; thus $graph(f)$ is a manifold if $X$ is.
(Note that the no... |
As asked in the title, is Hamiltonian containing enough information to judge the existence of spontaneously symmetry breaking?
Any examples?
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up.Sign up to join this community
Th... |
When we calculate the dynamic resistance \$r=(\frac{dv}{dI})\$, for any n-p junction, how is it different from the normal resistance \$R=\frac VI\$? Does the equation for the voltage drop (The fermi potential drop, and not the absolute Galvani potential) work if we use the dynamic resistance with the instantaneous curr... |
I have multiple regression with, say 3 independent variables: $Y=B_0+B_1x_1+B_2x_2+B_3x_3$ I would like to test if $B_2+3B_3$ is significantly different from zero, i.e. $$H_0: B_2+3B_3=0$$ $$H_1: B_2+3B_3\neq 0$$ Can you please help to find appropriate way to test for significance of linear functions of two coefficient... |
Can I be a pedant and say that if the question states that $\langle \alpha \vert A \vert \alpha \rangle = 0$ for every vector $\lvert \alpha \rangle$, that means that $A$ is everywhere defined, so there are no domain issues?
Gravitational optics is very different from quantum optics, if by the latter you mean the quant... |
Previously, we used visualizations to see the closest customers to our salesperson, Bob. Now using a simple graph works fine when Molly only has one salesperson. However, as the business grows to have more salespeople, or when it becomes more difficult to just
see who is closest, we need a more formalized approach. So ... |
Quadratic function is also a second degree polynomial function. The graph of a quadratic function is a parabola. The parabola open upwards if a graph is made for the quadratic formula.
The point at which the function attains maximum or minimum value is the vertex of the quadratic function. When we say second degree, th... |
What does $|H(j\omega)|^2$ in $20 \log_{10}|H(j\omega)|$ mean?
Is it some ratio of energy or power? And why? How to derive it?
I'm sorry for curtness of my questin.
Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only take... |
$$\frac{\tan^2(3x)}{1+\tan^2(3x)}+\frac{\tan^2(2x)}{1+\tan^2(2x)}+\frac{\tan^2(x)}{1+\tan^2(x)}=2$$ if we simplify ,we have $$\sin^2(3x)+\sin^2(2x)+\sin^2(x)=2\\\frac{1-\cos(6x)}{2}+\frac{1-\cos(4x)}{2}+\frac{1-\cos(2x)}{2}=2\\\cos(6x)+\cos(4x)+\cos(2x)=-1\\\cos(6x)+\cos(2x)=-(1+\cos(4x))\\2\cos(4x)\cos(2x)=-2\cos^2(2x... |
We touched on quadratic residues when discussing Pythagorean triples. We relied upon the quadratic residues and non-residues in modulo \(4\) for our proof that \(x\) and \(y\) took opposite parities.
These residues have important results in encryption, integer factorisation and sound diffusion.
Definition of quadratic ... |
Skip to 0 minutes and 11 secondsWelcome again to "It's Your Turn" step. You're dealing today with trigonometric inequalities, which is quite a difficult topic. In exercise one, we've got a product of terms, and we want it to be strictly positive. So we will study separately the sign of the two terms. Let us begin with ... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
To understand this you will also need to understand the first FTC, and I will show you this. (As you said there may be more than one proof, I will base mine off one similar given in the stewart calculus books).
You may notice during the proof of the first already the connectedness to the derivative which I assume you a... |
Let $C(r)$ be the origin-centered circle of radius $r$,and let $\beta(r)$ be the
exterior buffer around $C(r)$:the distance from $C(r)$ to the closest lattice point exterior to $C(r)$: For example, $\beta(2) = \sqrt{5}-2 \approx 0.24$ because there are no lattice points strictly between $C(2)$ and $C(\sqrt{5})$, and th... |
Let $$A$$ and $$B$$ be two sets. The set difference of $$A$$ and $$B$$, denoted as $$A - B$$, is the set of all the elements of $$A$$ that are not members of $$B$$.
Let $$A$$ and $$B$$ be two sets. The set difference $$A - B$$ is:
$$$A-B=\{x\in A \ and \ x\notin B\}$$$
Elements belonging to the set difference $$A - B$$... |
Suppose I have the term $t=\sqrt{-a^2-b^2}$, where $a,b\in\mathbb R$. Of course, we know that it holds$$t = \mathbb i \sqrt{a^2+b^2}$$which is (in my opinion) more convenient. How can I get
Mathematica to factor this out? Especially for things like$$\cosh(\sqrt{-a^2-b^2}) = \cos(\sqrt{a^2+b^2})$$it would be very helpfu... |
In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is endless from the top. Even in a straight forward way one can find an equiconsistent theory (respect to $ZFC$) which its w... |
Given a finite subset $S$ not containing the identity element in a residually finite group $G$, does there always exist a normal subgroup of $G$ which has finite index (in $G$) and which avoids $S$? (If $S$ is a singleton, this is of course the definition of a residually finite group.)
Yes, take the intersection of the... |
De Bruijn-Newman constant
For each real number [math]t[/math], define the entire function [math]H_t: {\mathbf C} \to {\mathbf C}[/math] by the formula
[math]\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du[/math]
where [math]\Phi[/math] is the super-exponentially decaying function
[math]\displaystyle... |
LaTeX/Algorithms
LaTeX has several packages for typesetting algorithms in form of "pseudocode". They provide stylistic enhancements over a uniform style (i.e., all in typewriter font) so that constructs such as loops or conditionals are visually separated from other text. The pseudocode is usually put in an
algorithm e... |
But if you don't want to have a Google account: Chrome is really good. Much faster than FF (I can't run FF on either of the laptops here) and more reliable (it restores your previous session if it crashes with 100% certainty).
And Chrome has a Personal Blocklist extension which does what you want.
: )
Of course you alr... |
My question is about the realization of the symmetric group $S_n$ as a galois group of a real and normal field extension $K/\mathbb Q$. As I read, such a field $K$ can be obtained as the splitting field of a polynomial
$$f(x)=\alpha_0 + \alpha_1 x + ... + \alpha_{n-1}x^{n-1}+ x^n \in \mathbb Q[x] $$
in which all the co... |
I'm using Expectation Maximization algorithm to determine the parameters of Gaussian distributions in a mixture. To get a better understanding of the algorithm, I
executed it manually step by step on a small example, following this procedure:
The problem is that in the second iteration log-likelihood decreased.
These a... |
"I am not given wealth $w$ although I suppose I could assume any firm who
is purchasing has some budget."
This is No. exactly where the fundamental microeconomic theory of the firm differs from the microeconomic Consumer Theory: the firm is not constrained by a budget. The reason is that this fundamental theory deals m... |
Let $a$, $b$ and $c$ be real numbers such that $abc=1$. Prove that: $$\frac{7-6a}{2+a^2}+\frac{7-6b}{2+b^2}+\frac{7-6c}{2+c^2}\geq1$$
The equality occurs also for $a=b=2$ and $c=\frac{1}{4}$.
This inequality is a similar to the very many contest's inequalities, but nothing helps.
At least, I don't see how we can prove ... |
Decay of solutions to a water wave model with a nonlocal viscous dispersive term
1.
Department of Mathematics, Purdue University, West Lafayette, IN 47907
2.
LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex
3.
LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Sa... |
What is Galois Theory Anyway? The Basic Idea
Perhaps you've heard of Évariste Galois? (Pronounced "GAL-wah.") You know, the French mathematician who died tragically in 1832 in a duel at the tender age of 20? (Supposedly over a girl! C
'est romantique, n'est-ce pas?) Well, today we're taking a bird's-eye view of his mos... |
Short Answer:
Your work is perfectly fine if your lower and upper integral limits satisfy $0 < a \leq b$. In that case your answer $2u - \ln(u)$ even has a nice closed form purely in terms of $x$:
$$
\int_a^b \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}} = \Big[\big(1 + \sqrt{4x + 1}\big) - \ln\Big(\frac{1}{2}\big(1 + \... |
The Borel-Cantelli Lemma
Today we're chatting about the
Borel-Cantelli Lemma: Let $(X,\Sigma,\mu)$ be a measure space with $\mu(X)< \infty$ and suppose $\{E_n\}_{n=1}^\infty \subset\Sigma$ is a collection of measurable sets such that $\displaystyle{\sum_{n=1}^\infty \mu(E_n)< \infty}$. Then$$\mu\left(\bigcap_{n=1}^\inf... |
The Yoneda Embedding
Last week we began a discussion about the Yoneda lemma. Though rather than stating the lemma (sans motivation)
, we took a leisurely stroll through an implication of its corollaries - the Yoneda perspective, as we called it : An object is completely determined by its relationships to other objects,... |
In this section we will explain how to solve the following type of problems:
Given several restrictions (inequations), we have to determine the area on the plane that satisfies all of them by giving its vertexes.
We usually find more than one simultaneous restriction for the variables in inequations exercises. For exam... |
A system of inequations with one variable is a set of inequations of a variable that act simultaneously, that is, the solution points must satisfy all the inequations of the system.
An example of a system of inequations is:
$$$ \left\{ \begin{array}{l} x-1 > 0 \\ x+2(1-x) < 4+x \\ 2x < 8 \end{array}\right. $$$
In this ... |
expr_t_onesample
expr_t_parametric
expr_anova_parametric
expr_contingency_tab
expr_corr_test
This vignette provides a go-to summary for which test is carried out for each function included in the package and what effect size it returns. Additionally, there are also recommendations on how to interpret those effect sizes... |
In this reference the author states what he calls "
the theory of local solutions" for separable ordinary differential equations of the form $\frac{dy}{dx} = \frac{f(x)}{g(y)}$. He asserts that it suffices for $f$ and $g$ to be continuous and not to vanish simultaneously in a rectangular area $R$ of the plane in order ... |
Is there a "simple" mathematical proof that is fully understandable by a 1st year university student that impressed you because it is beautiful?
closed as primarily opinion-based by Daniel W. Farlow, Najib Idrissi, user91500, LutzL, Jonas Meyer Apr 7 '15 at 3:40
Many good questions generate some degree of opinion based... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Let $f: [a,b] \longrightarrow \mathbb{R}$ be an increasing function. If $x_1, \cdots, x_n \in [a,b]$ are distinct, show that $\sum_{i=1}^{i=n} o(f,x_i) < f(b) - f(a).$
I would like to know if my proof is correct and, if that's the case, an opinion if my proof writting is fine and an advice about how to write this bette... |
I'm currently trying to solve a linear system $Ax = B$, where the matrix $A$ is ill conditioned (i.e. nearly singular), with a condition number of $~10^7$. The aforementioned linear system arises from a finite difference discretization.
The mathematical model for my problem is a PDE with derivatives of $x$ and $t$. The... |
So I need to cave in and learn some software package ... Until now I have been a paper and pen-mathematician and nearly managed to avoid mathematical software beyond free Wolfram Alpha. To my surprise, I don't find pieces of code snips or instructions that are close enough to my application, so I want to make sure I ch... |
Define
$$\hat{X}(Y) = [X,Y] $$
I have known matrices $S_i$ and $V$. I am trying to use Mathematica to define a function which calculates
$$ \sum_{\substack{n_1, \ldots, n_k>1\\ n_1+\ldots n_k = m}} \hat{S}_{n_1}(\hat{S}_{n_2}(\ldots (\hat{S}_{n_k} (V))\ldots )) $$
for arbitrary integers $k$ and $m$ (with $k,m < 10$ or ... |
Suppose I had the following problem:
$U_{tt}=U_{xx}+U_{yy}$ in $\Omega=[0,1]\times[0,1]$
$U(x,y,0)=f(x,y)$ $U_{t}(x,y,0)=g(x,y)$ $U=0$ on $\partial \Omega$
I know that there is an explicit finite difference scheme to solve this problem of the form:
$\frac{U(i,j,k+1) - 2U(i,j,k)+U(i,j,k-1)}{\Delta t^2} = \frac{U(i+1,j,k... |
What is a Natural Transformation? Definition and Examples, Part 2
Continuing our list of examples of natural transformations, here is...
Example #2: double dual space
This is really the archetypical example of a natural transformation. You'll recall (or let's observe) that every finite dimensional vector space $V$ over... |
I think the usual proof for the asymptotic number of zeros of the Riemann zeta function$$N(T) = \#\left\{\rho : \ \zeta(\rho)=0, \begin{array}{l}\scriptstyle Im(\rho)\ \in\ [0,T]\\ \scriptstyle Re(\rho) \ \in\ (-1,2)\end{array}\right\} = \frac{ T\ln T}{2\pi}-\frac{1+\log 2\pi}{2\pi}T+\mathcal{O}(\ln T)$$ works for any ... |
You can achieve the same as in egreg's answer, without
amsmath nor
AtBeginDocument trick:
\documentclass{article}
\newcommand\plim{\mathop{p\mkern2mu\mathrm{\mathchar"702D lim}}}
\begin{document}
$\plim_{a\to\infty}$
$\displaystyle\plim_{a\to\infty}$
\end{document}
Sometimes the math alphabet command
\mathrm does not u... |
Finitely Generated Modules Over a PID The Basic Idea
We know what it means to have a module $M$ over a (commutative, say) ring $R$. We also know that if our ring $R$ is actually a field, our module becomes a vector space. But what happens if $R$ is "merely" a PID? Answer:
A lot.
Recall from group theory that all finite... |
In the theory of finite undirected graph, a basic notion that is broadly studied is the notion of matching cut. A
matching cut in a graph $G$ is defined to be a matching subgraph of $G$ that is also the edge set of a cut. A $\it cut$ of a graph $G = (V, E)$ is a partition of the vertex set $V$ into two sets $(A, B)$ su... |
Let an asset follow a Brownian motion $$dS = \mu dt + \sigma dW$$ with $\mu$ and $\sigma$ constant. The constant interest rate is $r$. What process does $S$ follow in the risk-neutral measure? Develop a formula for the price of a call option and for the price of a digital call option.
In chapter 6, Mark Joshi states th... |
I finally solved problem G of GP of Dolgoprudny (also problem 83 in New Year Contest 2018). Though there are discussions on Codeforces, they all look very abstract to me. Hence, I decided to write down a full editorial.
First, we need some definitions. For given string $$$s$$$, we write $$$\mathrm{occ}_s(p) = {i : s[i ... |
This notebook is the first in a series of tutorials highlighting various aspects of seismic inversion based on Devito operators. In this first example we aim to highlight the core ideas behind seismic modelling, where we create a numerical model that captures the processes involved in a seismic survey. This forward mod... |
In the documentclass I am writing, the user will input an integer such as
\lessonnumber as an integer with possibly a number of leading zeroes, e.g. 4 as
04. At some places in the document I will need the full form (
04) and at others I only want to common form (
4). How can I accomplish this?
In the documentclass I am... |
Answer
In July, the average monthly temperature is $64^\circ F$.
Work Step by Step
$$f(x)=14\sin[\frac{\pi}{6}(x-4)]+50$$ The question asks when the average temperature is $64^\circ F$ In other words, find $x$ so that $f(x)=64$. Therefore, we can replace $f(x)=64$ to the given equation and find $x$: $$14\sin[\frac{\pi}... |
Dear OR experts,
I am confronted with the following condition in my problem, but I have no idea how to linearize it.
Variable \(x\) and \(z_i\) are binary, \(f(y)\) is a linear function of \(y = \sum\limits_{i}{z_i}\), \(c\) is an integer constant. \(z_i=1\) means facility \(i\) is open. So \(y\) is the number of open ... |
A quadratic inequation is an inequation in which we find numbers, a variable (which we call $$x$$) that can be multiplied by itself, and an inequality symbol.
An example of a quadratic inequation might be: $$$ 2x^2-x < 2x-1$$$
where we can observe that the term $$2x^2$$ is the quadratic term, characteristic of the quad... |
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The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron... |
The equation $$2^x\left(x\ln(2)-1\right)=n$$ does not show solution in terms of elementary function and, in principle, only numerical methods (such as Newton) would need to be used.
However, for your curiosity, there is a solution in terms of Lambert function which is such that $z=W(z)\, e^{W(z)}$. In the case of the c... |
I need to draw this (likely very simple) diagram shown in the image but I've never used Tikz before.
My problem is basiclly how to draw these vertical dotted lines and mainly arrange the math in blocks like this with these arrows pointing from one to the other - of course I have no problem writing the math, the needed ... |
The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that $$ \mathcal W\subset C^0_{(0)}(\mathbb R)=\{\phi\text{ continuous on }\mathbb R, \lim_{\vert \xi\vert\rightarrow+\infty}\phi(\xi)=0\} . $$
I ... |
As described in the Rich Output tutorial, the IPython display system can display rich representations of objects in the following formats:
This Notebook shows how you can add custom display logic to your own classes, so that they can be displayed using these rich representations. There are two ways of accomplishing thi... |
This seems an excellent classroom experiment to reinforce the practical meaning of the confidence level of an interval estimate, and I wish more instructors would do this kind of thing.
First, let's just check to make sure we're on the same page about the experiment.Each of the 16 groups rolls a fair die $n = 25$ times... |
PCTeX Talk Discussions on TeX, LaTeX, fonts, and typesetting
Author Message zedler Joined: 03 Mar 2006 Posts: 15
Posted: Mon Mar 27, 2006 11:53 am Post subject: spacing of mathrm Hello,
\documentclass{book}
\usepackage{times,mtpro2}
\begin{document}
$(\mathrm j$
\end{document}
gives touching glyphs.
Michael Michael Spi... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. ... |
The principles of simple linear regression lay the foundation for more sophisticated regression methods used in a wide range of challenging settings. In this section, we explore multiple regression, which introduces the possibility of more than one predictor, and logistic regression, a technique for predicting categori... |
Let $K$ be a convex body and let $\| \cdot \|_{K}$ be the correspoding Minkowski functional $$\| x \|_{K} = \inf\{\lambda > 0 : x \in \lambda K \}$$
Let us consider the following map $f: K \rightarrow \partial \mathring K$ such that $f(x) = \nabla \| x \|_{K}$ Here $\partial \mathring K$ stands for the polar set, i.e. ... |
There are several options for integrating your R workspace with LaTeX. One of these is the R package
tikzDevice that allows you to export images created in R as tikz code in a .tex file, for immediate use in a LaTeX document via the line
\include{diagrams}.
A simpler way, the one we all start out with, is to export an ... |
A global bifurcation theorem for a positone multiparameter problem and its application
1.
Fundamental General Education Center, National Chin-Yi University of Technology, Taichung 411, Taiwan
2.
Center for General Education, National Formosa University, Yunlin 632, Taiwan
3.
Department of Mathematics, National Tsing Hu... |
Difference between revisions of "Remarkable"
(wPFA)
(→Results: +1)
Line 32: Line 32:
* If $κ$ is virtually [[measurable]], then either $κ$ is remarkable in $L$ or $L_κ \models \text{“there is a proper class of virtually measurables”}$.<cite>NielsenWelch2018:GamesRamseylike</cite>
* If $κ$ is virtually [[measurable]], t... |
Assuming we have a convergent nozzle I've read that the maximum thrust is achieved just in the moment the nozzle exit -minimum area- is chocked, i.e., the nozzle is adapted in the sense the pressure in the exit area equals the ambient pressure. How can I demostrate this with formulas?
Let's look at two cases, choked ex... |
For every $ N \in \mathbb Z$ there exists an integer $n$ such that $ \sqrt N \in \mathbb Q(\zeta_n)$.
I am struggling where to start this question, please suggest me few hints.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only tak... |
First consider the single torus, represented as the square with opposite edges identified, call the edges $a$ and $b$. $U$ is the torus with a single point removed, this retracts to just the borders of the square, so it is the union of two copies of $S^1$ and it has fundamental group $<a, b>$, the free group in two ele... |
Source transformation lets you change a voltage source into a current source, or the other way around. It is a way to simplify a circuit. The method is based on Thévenin’s theorem and Norton’s theorem.
Two simple circuits have special names,
The
Thévenin form is a voltage source in series with a resistor. The Norton fo... |
Hierarchical Binominal Model: Rat Tumor Example¶
In [1]:
%matplotlib inlineimport pymc3 as pmimport numpy as npimport matplotlib.pyplot as pltimport seaborn as snsimport pandas as pdimport pymc3.distributions.transforms as trimport theano.tensor as ttfrom scipy.special import gammalnplt.style.use('seaborn-darkgrid')pri... |
101 people flip a fair coin. Everyone who tosses heads is on one team and everyone who tosses tails is on another other team. The team with more people on it wins. What are the odds that, given you are one of the 101 players, you will win? (101 players and coins eliminates ties but I am also interested the case where t... |
LaTeX is a great tool for printable professional-looking documents, but can be also used to generate PDF files with excellent navigation tools. This article describes how to create hyperlinks in your document, and how to set up LaTeX documents to be viewed with a PDF-reader.
Contents
Let's start with a minimal working ... |
Consider a column of fluid of length $L$, with initial density $\rho_0$ and initial velocity ($u_0 =0$) everywhere. Now at time $t=0$ gravity is switched on. No-slip boundary conditions are assumed at both end of the fluid column.
We know that after a while column will attain a steady state with fluid everywhere at res... |
What is $O_{Proj S_*}(n) = O_S(n)$? Whatever it is, you can describe it and thus work with it by knowing the following facts:
$O_S(n)$ is isomorphic to the structure sheaf on each open set of the cover $D(s)$, where $s \in S_n$, via some $\phi_s: O(n)|_{D(s)} \to O|_{D(s)}$. In fact the structure sheaf over $D(s)$ is j... |
We begin with Adam Tooze laying out the issues:
Adam Tooze: Italy: How Does the E.U. Think This Is Going to End?: "Over the past 10 years, Italy’s gross domestic product per capita has fallen... unique among large advanced economies...
...More than 32 percent of Italy’s young people are unemployed. The gloom, disappoin... |
We are doing an assignment for our Advanced Econometrics course for which we are trying to illustrate Gordin's Central Limit Theorem by simulation. We used an AR(1) process to show that if the conditions of the theorem are satisfied, its implications hold too. Now we are trying to find counter-examples, and we were won... |
It is common to measure the peak absorbance of the plasmonic nanoparticles in solution and then get the following: Molarity (moles/liter) Number of particles per ml Weight concentration (ug/ml) I wrote a small python function to extract these parameters. The following inputs are needed for the function d_nm : diameter ... |
In this very first post of the Connect the Dots series, I set up the supervised learning problem from a function fitting perspective and discuss the objective of function fitting.
In a supervised learning problem, we have a bunch of input (also referred to as feature, independent variable, predictor) and the correspond... |
$$\lim_{(x,y)\to(0,0)}\frac{(2x+y)\sin(x+y)}{(x+y)\sin(2x+y)}$$
Take $x+y=z$ and $2x+y=w$, then $z$ and $w$ tend to zero as $x$ and $y$ tend to zero and use the above-mentioned limit $\lim_{t\to 0}(\sin t) / t = 1$.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and profe... |
The Pseudo-Hyperbolic Metric and Lindelöf's Inequality (cont.)
Last time, we proved that the pseudo-hyperbolic metric on the unit disc $\Delta\subset \mathbb{C}$ given by $$d(z,a)=|\varphi_a(z)|=\bigg|\frac{z-a}{1-\bar a z}\bigg|$$ is indeed a metric. As an application, our next goal is to verify the following inequali... |
Hokkaido Mathematical Journal Hokkaido Math. J. Volume 46, Number 3 (2017), 487-512. Growth of meromorphic solutions of some linear differential equations Abstract
In this paper, we investigate the order and the hyper-order of meromorphic solutions of the linear differential equation \begin{equation*} f^{(k)}+\sum^{k-1... |
I'm working with some data from a cell analyzer, which takes millions of measurements and returns mean, median and 5% / 95% quantiles for the distribution. Preliminary model comparisons have indicated good fit to the data for a gamma distribution.
The experimental design is a straightforward ANOVA-type hierarchical mod... |
In atmospheric physics we use often a variety of vertical coordinates, $z$ the height above some reference surface, or $r$ as distance from the planetary center.However sometimes it simplifies the ...
What is the difference between the Hydraulic diffusion equation and the Richards equation in groundwater dynamics? I wa... |
After my series of post on classification algorithms, it’s time to get back to R codes, this time for quantile regression. Yes, I still want to get a better understanding of optimization routines, in R. Before looking at the quantile regression, let us compute the median, or the quantile, from a sample.
Median
Consider... |
Suppose that we have a sequence $\{a_n\}$ with $a_n\subseteq\mathbb{R}$ for all $n$. What I wonder is that is it possible to define a 'limit' for this kind of sequences?
Of course this 'limit' should make some sense. For example intuitively if $a_n=(\frac{-1}{n},\frac{1}{n})$, then $\lim a_n$ should be $\{0\}$, or if $... |
If I have two functions in a convolution like
$$X*Y=1$$
$$X*Z=1$$
then it means (trivially) $Y=Z$.
Is this correct or are there subtleties in the convolution theorem where $Y=Z$ isn't always true?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related... |
\begin{align}Z_1 & = X+Y \\[4pt]Z_2 & = \frac X {X+Y} \\[12pt]X & = Z_1 Z_2 \\[4pt]Y & = Z_1(1-Z_2)\end{align}
Suppose you've figured out that the support of $Z_1$ is $[0,\infty)$ and the support of $Z_2$ is $[0,1]$. Then certainly the support of $(Z_1,Z_2)$ is a
subset of $[0,\infty)\times[0,1]$. The question is wheth... |
The Annals of Probability Ann. Probab. Volume 24, Number 1 (1996), 237-267. Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions Abstract
We consider the supercritical finite measure-valued diffusion, $X(t)$, whose log-Laplace equation is associated wi... |
I try to find a good proof for invertibility of strictly diagonally dominant matrices (defined by $|m_{ii}|>\sum_{j\ne i}|m_{ij}|$). There is a proof of this in this paper but I'm wondering whether there are are better proof such as using determinant, etc to show that the matrix is non singular.
The proof in the PDF (T... |
I had a system of three PDEs $$\frac{\partial \theta_h}{\partial x}+\beta_h (\theta_h-\theta_w) = 0$$
$$\frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c-\theta_w) = 0$$
$$ \lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V\frac{\partial^2 \theta_w}{\partial y^2}-\frac{\partial \theta_h}{\partial x... |
I'll be teaching an introductory course in algebraic number theory this fall (stopping before class field theory). I'm looking for a good list of "special topics" I can include to illustrate the general theory. In other words, attractive theorems (preferably off the beaten path) that can be proved using the basic resul... |
Good day,
I have an old exam without solutions and there is the following exercise:
Let $(Y_n)_{n \in \mathbb{N}}$ be a sequence of independent identically distributed (i.i.d.) random variables which are uniformely distributed on $[0,1]$. Further define $X_n :=\min\{Y_1,...,Y_n\}$. Show:
(i) $F_n(t)=[1-(1-t)^n] 1_{[0,1... |
I have been fiddling around trying to understand coordinate transformations in dynamical systems. I know the standard way to convert the system from cartesian coordinates to polar, but am having trouble reversing it.
I know the standard transformation equations: $$ r\cos(\theta)=x,\quad r\sin(\theta)=y,\quad r^2 = x^2+... |
Why is $e^{i2\pi Nf_snT}=1$ in snippet from book?
Because $Tf{_s}$=1 and $N$ and $n$ are integers. So the exponent becomes 2$\pi$ times an integer, say $K$. But $e{^{i\theta}} = \cos(\theta)+i\sin(\theta)$. Therefore $e{^{i2K\pi}} = 1$.
A try without maths: $x(t)$ gives you your location in the complex plane at time $t... |
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