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\section{Invariance of the mutual information under spatial coupling}\label{sec:subadditivitystyle}
In this section we prove that the mutual information remains unchanged under spatial coupling in a suitable asymptotic limit
(Theorem \ref{LemmaGuerraSubadditivityStyle}).
We will compare the mutual informations of th... | {"config": "arxiv", "file": "1812.02537/v5 arxiv/sections/5_interpolation_subadditivity_jeanRevision.tex"} |
TITLE: Diverging improper integral
QUESTION [6 upvotes]: When asked to evaluate $\int_{a}^{\infty}f(x)dx$, you split the interval based on the improper points.
If there is another improper point other than $\infty$, at $b$, we will write: $\int_{a}^{\infty}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx+\int_{c}^{\infty}f... | {"set_name": "stack_exchange", "score": 6, "question_id": 13802} |
TITLE: If $n$ vectors are linearly independent, is there only one way to write a vector as a linear combination of those vectors?
QUESTION [2 upvotes]: I know the converse is true, because if you can write 0 in two ways you can keep adding 0 to get an infinite number of linear combinations that sum to the same thing.
(... | {"set_name": "stack_exchange", "score": 2, "question_id": 87270} |
TITLE: Permutations and Combinations.
QUESTION [0 upvotes]: There are 4 red and 6 blue marbles in a bag, 3 are picked out at random, what is the probability that atleast one of them is red? My answer is 1/2
First I calculated the total number of ways that I can pick out three marbles at random: 10C3 = 120.
Then I multi... | {"set_name": "stack_exchange", "score": 0, "question_id": 593868} |
TITLE: Difficulty with a tricky matrix exponential step
QUESTION [2 upvotes]: I am having great difficulty checking a step involving operations with 2x2 matrix exponentials.
The expression I would like to simplify is
$$\lim _{x\to \infty} e^{-iHt-Vt}$$
where, for some $\epsilon, x, y \in \mathbb{R}$, we define
$$H=\beg... | {"set_name": "stack_exchange", "score": 2, "question_id": 3617895} |
TITLE: Evaluating the limit $\lim_{n\to+\infty}(\sqrt[n]{n}-1)^n$
QUESTION [6 upvotes]: Evaluate the limit
$$\lim_{n\to+\infty}(\sqrt[n]{n}-1)^n$$
I know the limit is 0 by looking at the graph of the function, but how can I algebraically show that that is the limit?
REPLY [1 votes]: Since, $\frac{\log(x)}x\le\frac1e$,... | {"set_name": "stack_exchange", "score": 6, "question_id": 576009} |
TITLE: $\sum_{k=0}^n {n \choose k} ^{2} = {2n \choose n}$ - Generating function $\sum_{k=0}^\infty \binom nk x^k = (1+x)^n$.
QUESTION [2 upvotes]: As part of a preparatory course in the contest PUTNAM, I have to show $\sum_{k=0}^n {n \choose k} ^{2} = {2n \choose n}$. I know that I can use the identity $\sum_{k=0}^n {n... | {"set_name": "stack_exchange", "score": 2, "question_id": 1473827} |
\begin{document}
\articletype{}
\title{Structure from Appearance: Topology with Shapes, without Points}
\author{
\name{Alexandros Haridis\textsuperscript{a}\thanks{CONTACT A. Haridis. Email: charidis@mit.edu}
}
\affil{\textsuperscript{a}Department of Architecture, Massachusetts Institute of Technology, 77 Massachus... | {"config": "arxiv", "file": "1902.03974/interacttfssample.tex"} |
TITLE: Quillen groupoid of a groupoid.
QUESTION [1 upvotes]: For any category $\mathcal{C}$ we can define its Quillen's groupoid, denoted $\mathcal{Q}(\mathcal{C})$, as the category which have the same objects than $\mathcal{C}$ and the arrows between two objects are $\operatorname{Hom}_{\mathcal{Q}(\mathcal{C})}(c, c^... | {"set_name": "stack_exchange", "score": 1, "question_id": 1101005} |
TITLE: How to prove that the intersection of $L^1(\mathbb{R})$ and $L^2(\mathbb{R})$ is dense in $L^2(\mathbb{R})$
QUESTION [5 upvotes]: How to prove that the intersection of $L^1(\mathbb{R})$ space and $L^2(\mathbb{R})$ space is dense in $L^2(\mathbb{R})$ space?
REPLY [7 votes]: If $f \in L^2(\mathbb R)$, then defin... | {"set_name": "stack_exchange", "score": 5, "question_id": 541576} |
\begin{document}
\title{\small{\textbf{STRONG-VISCOSITY SOLUTIONS: SEMILINEAR PARABOLIC PDEs\\ AND PATH-DEPENDENT PDEs}}}
\author{
\textsc{Andrea Cosso}\thanks{Laboratoire de Probabilit\'es et Mod\`eles Al\'eatoires, CNRS, UMR 7599, Universit\'e Paris Diderot, France, \sf cosso at math.univ-paris-diderot.fr}
\qquad\q... | {"config": "arxiv", "file": "1505.02927/ComparisonViscosityII_April2015Submitted.tex"} |
\begin{document}
\title{ Erd\H{o}s-R\'enyi laws for exponentially and polynomially mixing dynamical systems.}
\author{Nicolai Haydn and Matthew Nicol
\thanks{Department of Mathematics,
University of Southern California; Department of Mathematics, University of Houston.
E-mail: $<$nhaydn@usc.edu$>$,$<$nicol@math.uh... | {"config": "arxiv", "file": "2103.00645.tex"} |
TITLE: Why does ebonite rod gets negatively charged when rubbed with fur
QUESTION [0 upvotes]: What I think is : The protons are present at the centre of the atom with rotating electrons around it so when it is rubbed by fur the electrons get passed from the ebonite rod to the fur leaving the rod negatively charged. B... | {"set_name": "stack_exchange", "score": 0, "question_id": 96390} |
TITLE: Comparing the password strength of random characters to random words.
QUESTION [0 upvotes]: Passwords with any ASCII printable character and passwords containing only words in the English dictionary are attacked equally using a guessing program that cycles between random words and random characters with a set mi... | {"set_name": "stack_exchange", "score": 0, "question_id": 1370054} |
TITLE: Relaxing continuous differentiability in the inverse function theorem
QUESTION [1 upvotes]: Let $f$ be a continuous injective function on a ball centered at the origin of $\bf R^n$ into $\bf R^n$.
Suppose that $f$ is differentiable at the origin with non-zero Jacobian determinant.
Denote the inverse of $f$ by $g... | {"set_name": "stack_exchange", "score": 1, "question_id": 3842873} |
TITLE: why when $X=qq^T \in S^n_+$,then $tr(XY) \lt 0$,it means that $Y \notin \mathbf K^*$?
QUESTION [1 upvotes]: $\mathbf K = S^n_+$ (nxn real symmetric positive semidefinite matrix). Show that the
dual of $\mathbf K$
$\mathbf K^*=\{\mathbf Y | Tr(\mathbf X \mathbf Y) >0, \forall \mathbf X \ge \mathbf 0\}$
is self-d... | {"set_name": "stack_exchange", "score": 1, "question_id": 3019562} |
\begin{document}
\title{Optimal Control of the Laplace-Beltrami operator on compact surfaces -- concept and numerical treatment}
\author{Michael Hinze and Morten Vierling}
\Nr{2011-1}
\date{January 2011}
\maketitle
\newpage
\begin{center}
{\bf \LARGE }
\end{center}
\pagenumbering{arabic}
{\small {\bf A... | {"config": "arxiv", "file": "1101.1385/linear.tex"} |
TITLE: Big O notation for two series
QUESTION [0 upvotes]: I'm having trouble understanding how to analyze these two series for their Big O representations.
I can get the correct answer for this series
$1+2+3+4+...+N$ is $O(N^2)$, which I found by finding the Big O for the equation $N(N+1)/2$.
Apparently the correct an... | {"set_name": "stack_exchange", "score": 0, "question_id": 1981966} |
\section{Application}\label{extra}
We apply Proposition \ref{g} to the following examples in \cite{AGK17}.
By \cite[Ex. 1.4]{AGK17}, the blow-up $\bl_p S$ of the following $S=\PP(a,b,c)$ at the identity point $p$ is not a MDS:
\begin{align}
(a,b,c)=((m+2)^2,(m+2)^3+1,(m+2)^3 (m^2+2m-1)+m^2+3m+1),
\label{higherM}
\en... | {"config": "arxiv", "file": "1803.11536/application.tex"} |
TITLE: How to quickly tell if a set is linearly independent or not?
QUESTION [2 upvotes]: Before proving if a set is a basis for $R^n$, I have to determine if the set is linearly independent or not.
We aren't allowed to use matrices, and I want to save time during a quiz.
What are some things I should watch out for?
... | {"set_name": "stack_exchange", "score": 2, "question_id": 3081500} |
TITLE: Proof for the linear dependence of vectors
QUESTION [0 upvotes]: Let there be a vector space V and a subset A={a1... ar} ⊂ V. Leit it also be the case that 1≤ i ≤ r.
Show that the elements of A are linearly dependent, when there does exist an index i∈{1,...,r} and real numbers λj j∈{1,...,r}, j≠i so that the ... | {"set_name": "stack_exchange", "score": 0, "question_id": 3452934} |
TITLE: angle of an inscribed triangle
QUESTION [1 upvotes]: I have a scalene triangle inscribed in a circle, one of its sides $a$ is $2\sqrt3$ and the length $r$ from that side to the center is $1$. I need to find the angle $x$ opposite to the side given. Here's how it looks:
How to find $x$?
REPLY [2 votes]: Draw in... | {"set_name": "stack_exchange", "score": 1, "question_id": 139932} |
\begin{document}
\theoremstyle{plain}
\newtheorem{thm}{Theorem}[subsection]
\newtheorem{theorem}[thm]{Theorem}
\newtheorem*{theorem*}{Theorem}
\newtheorem*{definition*}{Definition}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{sublemma}[thm]{Sublemma}
\newtheorem{corollary}[thm]{Corollary}
\newtheorem*{corollary*}{Coroll... | {"config": "arxiv", "file": "2010.10825.tex"} |
TITLE: Are normal subgroups of finite index in an absolute Galois groups open?
QUESTION [3 upvotes]: Let $G$ be a profinite group. It is known that open normal subgroups of $G$ have finite index, but in general not every normal subgroup of finite index is open (see https://ncatlab.org/nlab/show/profinite+group#examples... | {"set_name": "stack_exchange", "score": 3, "question_id": 3734250} |
\begin{document}
\begin{abstract}
Let $Q$ be a tame quiver of type $\widetilde{\mathbb{A}}_n$ and
$\Rep(Q)$ the category of finite dimensional representations over
an algebraically closed field. A representation is simply called a
module. It will be shown that a regular string module has, up to isomorphism,
at most... | {"config": "arxiv", "file": "0909.5594.tex"} |
\begin{document}
\begin{abstract}
We investigate the group $G\pv H$ obtained by gluing together two groups $G$ and $H$ at the neutral element. This construction curiously shares some properties with the free product but others with the direct product.
Our results address among others Property~(T), \cat0 cubical compl... | {"config": "arxiv", "file": "2201.03625/produit16.tex"} |
TITLE: Can someone explain how the square root of two negative numbers multiplied together doesn't equal a positive/negative number.
QUESTION [2 upvotes]: On a test I got a question asking to simplify $\sqrt{-25}\times \sqrt{-3}$
I answered $5\sqrt 3$ but apparently it's only $-5\sqrt 3$ because of some rule of imagina... | {"set_name": "stack_exchange", "score": 2, "question_id": 4023471} |
TITLE: Finite fields, trace map and the induce map is being a permutation of $ \mathbb F_{2^n}$
QUESTION [1 upvotes]: $n\ge 2$ is an integer
$$tr : \mathbb F_{2^n} \to \mathbb F_{2} \\ tr(x)=x+x^2+x^{2^2}+\cdots x^{2^{n-1}}$$ is the absolute trace map
For fixed constants $\alpha, \theta \in \mathbb F_{2^n} $ define th... | {"set_name": "stack_exchange", "score": 1, "question_id": 4449215} |
TITLE: Calculate the Rate of Change of the Volume of a Frustum of a Right Circular Cone
QUESTION [2 upvotes]: I have a question for calculus which I'm having trouble with. I've calculated the volume of the frustum with the formula $V=\frac\pi3(R^2+Rr+r^2).$
However, I'm having difficulty deriving this equation in order... | {"set_name": "stack_exchange", "score": 2, "question_id": 2182640} |
TITLE: Quadratic form defined by a permutation of projection matrix
QUESTION [0 upvotes]: Let $M$ be a projection matrix. We know that
$$
v'Mv=(Mv)'(Mv)=\sum^n_{i=1}\sum^n_{j=1}v_iM_{ij}v_j\leq \sum^n_{i=1}v_i^2
$$
where $v$ is a column vector with dimension compatible to $M$.
Can we conclcude the same thing with follo... | {"set_name": "stack_exchange", "score": 0, "question_id": 4513940} |
\begin{document}
\title{Model-Based Iterative Reconstruction \\ for Radial Fast Spin-Echo MRI}
\author{Kai~Tobias~Block,~Martin~Uecker,~and~Jens~Frahm}
\date{}
\maketitle
\begin{abstract}
In radial fast spin-echo MRI, a set of overlapping spokes with an
inconsistent T2 weighting is acquired, which results ... | {"config": "arxiv", "file": "1603.00040/radTSE2.tex"} |
:: Affine Independence in Vector Spaces
:: by Karol P\c{a}k
environ
vocabularies ALGSTR_0, ARYTM_1, ARYTM_3, XBOOLE_0, CARD_1, CONVEX1, CONVEX2,
CONVEX3, FINSEQ_1, FINSEQ_2, FINSEQ_4, FINSET_1, FUNCT_1, FUNCT_2,
MONOID_0, ORDERS_1, RELAT_1, RLSUB_1, RLVECT_1, RLVECT_2, RLVECT_3,
RUSUB_4, SEMI_AF1,... | {"subset_name": "curated", "file": "formal/mizar/rlaffin1.miz"} |
TITLE: Bayes Rule answer verification
QUESTION [1 upvotes]: A screening test has a 90% chance of registering breast cancer if it
exists, as well as a 20% chance of falsely registering cancer when it
does not exist. About one in one hundred women requesting the
screening test end up diagnosed with breast cancer. Ms. X h... | {"set_name": "stack_exchange", "score": 1, "question_id": 2497525} |
TITLE: System of differential equations $x'=f(y-x),\, y'=f(x-y)$
QUESTION [3 upvotes]: I have a problem regarding differential equations, $f$ is increasing. $x,y$ are dependent on $t$
$$ f(0)=0 $$
$$x'=f(y-x)$$
$$y'=f(x-y)$$
$$x(0)=1$$
$$y(0)=0$$
prove that when $x,y$ are the solutions of this equation then $$ \lim_{t... | {"set_name": "stack_exchange", "score": 3, "question_id": 2815692} |
\section{Fractional Derivatives and their Approximations} \label{meth:frac_derivatives}
\subsection{Caputo fractional derivative}
The concept of fractional calculus started with questions about about the generalization of integral and differential operators by L'Hospital and Leibniz~\cite{ross1977development} from th... | {"config": "arxiv", "file": "1910.02141/Content/2-methods-caputo.tex"} |
TITLE: orbits of automorphism group for indefinite lattices
QUESTION [13 upvotes]: I have a question about indefinite lattices.
QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form,
not necessarily unimodular, and $G:=O(\Lambda)$ the g... | {"set_name": "stack_exchange", "score": 13, "question_id": 145672} |
TITLE: Complex integration with Cauchy formula II
QUESTION [0 upvotes]: I want to compute the following integral:
$$\int_{C^{+}} \frac{\text{Log}(z)}{(z+1)(z+2)^2}dz$$
where $C^{+}$ is the circle with center $z=-2$ and radius $3/2$.
I suppose that I have to use Cauchy integral formula but I am lost.
REPLY [0 votes]: H... | {"set_name": "stack_exchange", "score": 0, "question_id": 3654823} |
TITLE: the pseudometric induced by a measure
QUESTION [5 upvotes]: Let $(X, \Sigma, \mu)$ be a measure space.
We can define a pseudometric $d$ on $\Sigma$ in the following way:
$$d(A, B) = \mu(A\bigtriangleup{}B)$$
where $A\bigtriangleup{}B = (A\cup{}B)\setminus{}(A\cap{}B)$ is the symmetric difference.
Does this metri... | {"set_name": "stack_exchange", "score": 5, "question_id": 44171} |
TITLE: Integral $\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$
QUESTION [20 upvotes]: Last year I wondered about this integral:$$\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$$
That is because it looks very similar to this integral
and this one. Surprisingly the result is quite nice and an approach can be fo... | {"set_name": "stack_exchange", "score": 20, "question_id": 3071027} |
TITLE: Question about the chain rule in this problem.
QUESTION [3 upvotes]: This is a problem taken from Caluclus Early Transcendentals by James Stewart 7th edition.
My question is about how the chain rule is used to get from $m(dv/dt)$ to $mv(dv/dx)$?
REPLY [3 votes]: Suppose $x$ is a function of time $t$ so that $x=... | {"set_name": "stack_exchange", "score": 3, "question_id": 1238417} |
\begin{document}
\title{Message Passing Algorithms for Phase Noise Tracking Using
Tikhonov Mixtures}
\author{Shachar~Shayovitz,~\IEEEmembership{Student~Member,~IEEE,}
and~Dan~Raphaeli,~\IEEEmembership{Member,~IEEE}
\thanks{S. Shayovitz and D. Raphaeli are with the Department
of EE-Systems, Tel Aviv University... | {"config": "arxiv", "file": "1306.3693/final.tex"} |
\begin{document}
\maketitle
\begin{abstract}
\noindent
In this article we show that extendability from one side of a simple analytic curve
is a rare phenomenon in the topological sense in various spaces of functions. Our
result can be proven using Fourier methods combined with other facts or by complex
analytic method... | {"config": "arxiv", "file": "1511.08584.tex"} |
\begin{document}
\title{The ${[46,9,20]_2}$ code is unique}
\author{Sascha Kurz}
\address{Sascha Kurz, University of Bayreuth, 95440 Bayreuth, Germany}
\email{sascha.kurz@uni-bayreuth.de}
\abstract{The minimum distance of all binary linear codes with dimension at most eight is known. The smallest open case fo... | {"config": "arxiv", "file": "1906.02621.tex"} |
TITLE: Prove that **PTIME** has no complete problems with respect to linear-time reductions.
QUESTION [1 upvotes]: Prove that PTIME has no complete problems with respect to
linear-time reductions.
I know that PTIME means all these problems that we can solve them in polynomial time. I suppose that I should use here t... | {"set_name": "stack_exchange", "score": 1, "question_id": 2196898} |
TITLE: free product with amalgamation is correspondingly a pushout
QUESTION [5 upvotes]: I'm trying to proof that the following diagram in the category of groups with $i_1$ and $i_2$ being inclusions is a pushout iff $G$ is the free product with amalgamation (up to isomorphism). It should be sufficient to proof $"\Left... | {"set_name": "stack_exchange", "score": 5, "question_id": 1261767} |
TITLE: Number of totally ramified extensions of $\mathbb{Q}_p$ of degree $n$
QUESTION [2 upvotes]: I just read the proof of this theorem that $\mathbb{Q}_p$ has finitely many totally ramified extensions of any degree $n$. The proof uses Krasner's lemma and the compactness of a space which corresponds to Eisenstein poly... | {"set_name": "stack_exchange", "score": 2, "question_id": 78211} |
TITLE: Discontinuous linear operator e and its core
QUESTION [1 upvotes]: Let $T:E\to \mathbb{R}$ nonzero linear operator with $E$ vector space. So, show the following equivalence: $T$ is discontinuous if and only if $Ker(T)$ is dense in $E$.
REPLY [2 votes]: I was trying to come up with something elegant, but all I c... | {"set_name": "stack_exchange", "score": 1, "question_id": 513753} |
TITLE: Mass equals Moment of inertia when constant density?
QUESTION [0 upvotes]: I have found equation for moment of inertia $(J)$. I'm calculating $J$ for hemisphere, with rotational axis $Z$.
$$ J = \iiint\limits_V r^2 \cdot \rho \cdot dV $$
But if $\rho$ is constant (homogenous), I can do:
$$ J = \rho \cdot \iiint\... | {"set_name": "stack_exchange", "score": 0, "question_id": 61170} |
TITLE: For $a,b,c$. Prove that $\frac{a^2}{a+\sqrt[3]{bc}}+\frac{b^2}{b+\sqrt[3]{ca}}+\frac{c^2}{c+\sqrt[3]{ab}}\ge\frac{3}{2}$
QUESTION [0 upvotes]: For $a,b,c$ are positive real number satisfy $a+b+c=3$. Prove that $$\dfrac{a^2}{a+\sqrt[3]{bc}}+\dfrac{b^2}{b+\sqrt[3]{ca}}+\dfrac{c^2}{c+\sqrt[3]{ab}}\ge\dfrac{3}{2}$$
... | {"set_name": "stack_exchange", "score": 0, "question_id": 2177039} |
TITLE: Why does brushless motors heat up?
QUESTION [0 upvotes]: I would like to know what makes a brushless motor heat up. I'm so aware of the problems generating the high temperetature such as wires contact problems or over loaded motors. I want to know what makes a normal motor in normal conditions get warm, is it re... | {"set_name": "stack_exchange", "score": 0, "question_id": 384563} |
TITLE: Norm estimate for a product of two orthogonal projectors
QUESTION [2 upvotes]: Let $H$ denote a Hilbert space.
Consider two orthogonal projectors $\,P,Q\in\mathscr L(H)\,$ such that $H=\operatorname{Im}P\oplus\operatorname{Im}Q\,,$ that is
both $\,\operatorname{Im}Q\,$ and $\,\operatorname{Im}P\,$ are closed su... | {"set_name": "stack_exchange", "score": 2, "question_id": 3092323} |
TITLE: What is the name of this digraph created from other digraphs?
QUESTION [5 upvotes]: Let $D_1, D_2, ..., D_n$ be digraphs of various sizes and let $C$ be a diagraph with $n$ vertices $\{1,...,n\}$. Construct a new digraph, $D$, whose vertex set is $V(D) = V(D_1) \cup \cdots \cup V(D_n)$ and whose edge set is defi... | {"set_name": "stack_exchange", "score": 5, "question_id": 4360185} |
TITLE: Torsion Modules Annihilators
QUESTION [2 upvotes]: Let $R$ be a PID, let $B$ be a torsion $R$-module, and let $p$ be a prime in $R$. Show that if $pb = 0$ for some nonzero $b \in B$, then $\operatorname{Ann}(B)$ is a subset of $(p)$.
Well we can let $\operatorname{Ann}(B) = (t)$ for some $t \in R$ since it is an... | {"set_name": "stack_exchange", "score": 2, "question_id": 2179743} |
TITLE: Are coherent states destroyed by measurements?
QUESTION [2 upvotes]: For a quantum harmonic oscillator in a coherent superposition, what happens if the energy is measured? Would it collapse to an energy eigenstate (a single excitation) corresponding to the result of the measurement, thus destroying the coherence... | {"set_name": "stack_exchange", "score": 2, "question_id": 567697} |
TITLE: Does the shuffling of a sequence of measurements produce an i.i.d. sequence?
QUESTION [0 upvotes]: Given a stationary sequence of measurements, say, of response times (i.e., sojourn times), of a queue (e.g., M/M/1), if we randomly reshuffle such samples, will we get a sequence of i.i.d. samples? If so, how to ... | {"set_name": "stack_exchange", "score": 0, "question_id": 2965518} |
TITLE: If sin B = 4/5 with B in Q1, find the following. sin (B/ 2)
QUESTION [0 upvotes]: If
$\sin B = 4/5$
with $B$ in Q1, find the following:
$\sin (B/2)$.
Well, I know the formula is $$\sin{\frac{b}{2}} = \sqrt{\frac{1+ \cos A}{2}}$$ and I know its a $3$-$4$-$5$ triangle. I keep getting $\frac{2\sqrt{5}}{5}$.
REP... | {"set_name": "stack_exchange", "score": 0, "question_id": 2495734} |
\chapter{Complexes with \texorpdfstring{$\psi$}{psi}-operator}\label{chapter Complex with psi-operator}
In this chapter, we construct a complex with $\psi$ operator (similar as in \cite[\S 3]{Her98}) that computes the continuous Galois cohomology.
\medskip
As $F_{\tau}$ is perfect, we cannot have such an operator on... | {"config": "arxiv", "file": "2201.11235/Chapter3.tex"} |
\begin{document}
\title{Stabilization of Second Order Nonlinear Equations with Variable Delay}
\author{Leonid Berezansky$^{\rm a}$,
Elena Braverman$^{\rm b}$$^{\ast}$\thanks{$^\ast$Corresponding author. Email: maelena@ucalgary.ca}
and Lev Idels$^{\rm c}$ \\ \vspace{6pt}
$^{a}${\em{Dept. of Math, Ben-Gurion Univers... | {"config": "arxiv", "file": "1606.03045/control_formatted_08_01_15.tex"} |
\begin{document}
\title{\Large Properties of the $\epsilon$-Expansion, Lagrange Inversion and Associahedra and the $O(1)$ Model. }
\author{Thomas A. Ryttov$^{\varheartsuit}$}\email{ryttov@cp3.sdu.dk}
\affiliation{
$^{\varheartsuit}${ \rm CP}$^{\bf 3}${ \rm-Origins}, University of Southern Denmark, Campusvej 55, 523... | {"config": "arxiv", "file": "1910.12631/EpsilonExpansion.tex"} |
TITLE: Verifying inverse Laplace transformation of an expression
QUESTION [0 upvotes]: I tried solving the next inverse Laplace transformation myself:
$$f(t)=L^{-1}\Bigl({{s}\over {s-C}}X(s)\Bigl)={dx(t)\over{dt}}*e^{Ct}$$
but I am not sure if it is correct. I can not find a similar example in my text books. Can anyone... | {"set_name": "stack_exchange", "score": 0, "question_id": 4203862} |
\begin{document}
\maketitle
\begin{abstract}
Numerical schemes {\em provably} preserving
the positivity of density and pressure
are highly desirable
for ideal
magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) analysis remains
challenging.
The difficulties mainly arise fr... | {"config": "arxiv", "file": "1802.02278/PPDGMHD12.tex"} |
TITLE: Given a person's age in years, what is the best way to estimate their age correctly in the future?
QUESTION [1 upvotes]: I have a database of people who have only given their age in years (and the date that they specified it). Because I want to keep this number accurate over time, I want to convert ages to birth... | {"set_name": "stack_exchange", "score": 1, "question_id": 863265} |
\section{Experiment Details}
\label{app:experiments}
We tested the proposed class of optimizers on different tasks that have potentially different loss surfaces in large parameter spaces.
To cast a wide net and ensure that we capture a plethora of neural architectures, we vary the network designs by using fully-conne... | {"config": "arxiv", "file": "2106.10708/Appendix-Sections/ExperimentDetails.tex"} |
TITLE: Creating A Function For The Following:
QUESTION [0 upvotes]: In trying to manipulate a function arrived at in a prior post, I've arrived at this (as a goal):
$$\begin{array}{c|c}
x & y \\ \hline
0.3^{-1} & -1 \\
0.3^{-1}+0.3^{0} & 0 \\
0.3^{-1}+0.3^{0}+0.3^{1}& 1 \\
0.3^{-1}+0.3^{0}+0.3^{1}+0.3^{2} & 2 \\
0.3^{-... | {"set_name": "stack_exchange", "score": 0, "question_id": 3378363} |
\begin{document}
\author{Prasanta Kumar Barik$^1$, {Ankik Kumar Giri$^1$}\footnote{Corresponding author. Tel +91-1332-284818 (O); Fax: +91-1332-273560 \newline{\it{${}$ \hspace{.3cm} Email address: }}ankikgiri.fma@iitr.ac.in/ankik.math@gmail.com} and Philippe Lauren\c{c}ot$^2$\\
\footnotesize ${}^1$Department of Ma... | {"config": "arxiv", "file": "1804.00853/PKB_AKG_PL_20180328.tex"} |
TITLE: Proof of chain rule on wikipedia: what does this sentence mean?
QUESTION [1 upvotes]: On wikipedia it says the following on the first chain rule proof:
$\lim_{x \to a} \frac{f(g(x)) - f(g(a))}{g(x) - g(a)} \cdot \frac{g(x) - g(a)}{x - a}$
When $g$ oscillates near $a$, then it might happen that no matter how clo... | {"set_name": "stack_exchange", "score": 1, "question_id": 2079790} |
TITLE: Compactness and Limit points
QUESTION [3 upvotes]: If $A$ is compact how can it be shown that any sequence $\{x_n\}$ in $A$ has a limit point in $A$?
I know this is proven in a lot of textbooks but I'm finding this hard to conceptualise.
REPLY [2 votes]: I assume that $\{x_n\}_{n \in \mathbb{N}}$ is infinite, ... | {"set_name": "stack_exchange", "score": 3, "question_id": 424473} |
TITLE: Summation of series in powers of x with certain combinations as coefficients
QUESTION [6 upvotes]: How can I find the sum: $$\sum_{k=0}^{n} \binom{n-k}{k}x^{k}$$
Edit: The answer to this question is: $$\frac{{(1+\sqrt{1+4x})}^{n+1}-{(1-\sqrt{1+4x})}^{n+1}}{2^{n+1}\sqrt{1+4x}}$$ I don't know how to arrive at this... | {"set_name": "stack_exchange", "score": 6, "question_id": 1324616} |
\begin{document}
\title[On integrality properties of hypergeometric series]{On integrality properties of \\ hypergeometric series}
\author{Alan Adolphson}
\address{Department of Mathematics\\
Oklahoma State University\\
Stillwater, Oklahoma 74078}
\email{adolphs@math.okstate.edu}
\author{Steven Sperber}
\address{Schoo... | {"config": "arxiv", "file": "1905.03235.tex"} |
\begin{document}
\title{Optimal Focusing for Monochromatic Scalar and
Electromagnetic Waves}
\author{\sc \small Jeffrey Rauch\thanks{University of Michigan, Ann Arbor 48109 MI, USA. email: rauch@umich.edu. Research partially supported by NSF under grant NSF DMS-0807600}
}
\date{}
\maketitle
\begin{abstract}Fo... | {"config": "arxiv", "file": "1008.3571/optimalfocus2.tex"} |
\begin{document}
\begin{frontmatter}
\title{Zastavnyi Operators and Positive Definite Radial Functions}
\author[aff63c5b8f064b9863807f549819f278608]{Tarik Faouzi}
\ead{tfaouzi@ubiobio.cl}
\author[aff980d40830c7f8bb81f02e241db520037]{Emilio Porcu \corref{contrib-49136fc397578fcb577fd032b6140a3b}}
\ead{emilio.porcu@... | {"config": "arxiv", "file": "1811.09266/Arxiv-Zastavnyi-operator-09-07-2019.tex"} |
\section{Infimum of Power Set}
Tags: Power Set, Empty Set, Order Theory
\begin{theorem}
Let $S$ be a [[Definition:Set|set]].
Let $\mathcal P \left({S}\right)$ be the [[Definition:Power Set|power set]] of $S$.
Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the [[Definition:Relational Structure|relationa... | {"config": "wiki", "file": "thm_2233.txt"} |
TITLE: Minimal polynomial with $f(\alpha)=0$
QUESTION [4 upvotes]: I'm learning this stuff for the first time so please bear with me.
I'm given $\alpha=\sqrt{3}+\sqrt{2}i$ and asked to find the minimal polynomial in $\mathbb{Q}[x]$ which has $\alpha$ as a root.
I'm ok with this part, quite sure I can find such a polyno... | {"set_name": "stack_exchange", "score": 4, "question_id": 3480242} |
\section{Numerical Results}\label{sect:Num}
In this section we provide numerical results that support the assumptions made and the results deduced from these assumptions. Moreover we describe the method used to obtain these results.
All computations are done using the following six new Hecke eigenforms $f_N \in S_2... | {"config": "arxiv", "file": "1508.07206/LTGNum.tex"} |
TITLE: Isomorphism of quotient fields
QUESTION [1 upvotes]: Consider $\mathbb R $ and $\mathbb Q$ with usual meanings.Which of the following rings are isomorphic?
a. $\mathbb Q[x]/\langle x^2+1\rangle $ and $\mathbb Q[x]/\langle x^2+x+1\rangle$
b. $\mathbb R[x]/\langle x^2+1\rangle $ and $\mathbb R[x]/\langle x^2+x+1\... | {"set_name": "stack_exchange", "score": 1, "question_id": 1008410} |
TITLE: Subgroup Lattice of D14
QUESTION [1 upvotes]: I've managed to find all of the subgroups of D14 (6 of order 7, 7 of order 2 and 1 of order 1) but I'm really struggling to put this into a lattice as I've never done it for dihedral groups before, only for symmetric groups. How do I find which of my subgroups are su... | {"set_name": "stack_exchange", "score": 1, "question_id": 1050158} |
theory Derivations
imports CFG ListTools InductRules
begin
(* leftderives and leftderives1 *)
context CFG begin
lemma [simp]: "is_terminal t \<Longrightarrow> is_symbol t"
by (auto simp add: is_terminal_def is_symbol_def)
lemma [simp]: "is_sentence []" by (auto simp add: is_sentence_def)
lemma [simp]: "is_word []... | {"subset_name": "curated", "file": "formal/afp/LocalLexing/Derivations.thy"} |
TITLE: Prove $\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$
QUESTION [4 upvotes]: I want to prove
$$\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$$
if $\sum_{k=1}^n a_k\leq1$ and $a_k\in[0,+\infty)$
I have no idea where to start, any advice would be greatly appreciated!
REPLY [1 votes]: David C. Ullrich's answer shows a ... | {"set_name": "stack_exchange", "score": 4, "question_id": 1772905} |
TITLE: What if the net force provided for a circular motion is larger than the required centripetal force?
QUESTION [5 upvotes]: Will an object be pulled towards the centre linearly if the net force provided for a circular motion is larger than the required centripetal force? And why?
For example, if the object in a ci... | {"set_name": "stack_exchange", "score": 5, "question_id": 534748} |
\begin{document}
\title{Representation type of ${}^{\infty}_{\hspace{2mm}\lambda}\mathcal{H}_{\mu}^1$}
\author{Yuriy Drozd and Volodymyr Mazorchuk}
\date{}
\maketitle
\begin{abstract}
For a semi-simple finite-dimensional complex Lie algebra $\mathfrak{g}$ we classify the
representation type of the associative alge... | {"config": "arxiv", "file": "math0412301.tex"} |
TITLE: Set theory: Lexicographic ordering and chain proof
QUESTION [0 upvotes]: Let $A$ and $B$ be partially ordered sets, and let $C$ and $D$ be chains of $A$ and $B$, respectively. If $A \times B$ is ordered lexicographically, prove that $C \times D$ is a chain of $A \times B$.
My attempt
for all $c_1,c_2 \in C, c_1 ... | {"set_name": "stack_exchange", "score": 0, "question_id": 2241817} |
TITLE: Will these frogs ever meet?
QUESTION [8 upvotes]: Two frogs are on an eternal stairway. Will they ever be able to meet?
Anton is on step 14 and jumps 4 steps.
Billy is on step 16 and jumps 6 steps.
The way I look at this is that as long as they're not on the same step, the one on the lowest jumps next, so:
Thei... | {"set_name": "stack_exchange", "score": 8, "question_id": 1498587} |
TITLE: Checking a fundamental unit of a real quadratic field
QUESTION [4 upvotes]: I just want to check whether I have got the fundamental unit of a certain real quadratic field, but I can't find how.
For instance, if I am working in $\mathbb{Q}(\sqrt{2})$ then $\mathcal{O}_K=\mathbb{Z}[\sqrt{2}]$ and so the fundamenta... | {"set_name": "stack_exchange", "score": 4, "question_id": 1220602} |
\begin{document}
\title[\it Spin-Boson model]{Exact solution of the Schr\"{o}dinger equation with the spin-boson Hamiltonian}
\author{Bart{\l}omiej Gardas}
\address{Institute of Physics, University of Silesia, PL-40-007 Katowice, Poland}
\ead{bartek.gardas@gmail.com}
\begin{abstract}
We address the problem of o... | {"config": "arxiv", "file": "1103.4383/spin_boson.tex"} |
\section{Notation and Problem Formulation}
\label{sec:problem_formulation}
In this section the minimax optimal sequential testing problem is defined, and some common notations are introduced. Notations not covered here are defined when they occur in the text.
\subsection{Notation}
\label{ssec:notation}
Random var... | {"config": "arxiv", "file": "1811.04286/02-problem_formulation.tex"} |
TITLE: Why intuitively do the quaternions satisfy the mixture of geometric and algebraic properties that they do?
QUESTION [9 upvotes]: [I completely rewrote the question to see if I could make it clearer. The comments below won't make any sense. In fact, my original question has been answered by Eric Wolfsey, so I may... | {"set_name": "stack_exchange", "score": 9, "question_id": 3196043} |
TITLE: how to calculate the quotient group of $U/DU$ where $U$ is an arbitrary unitary matrices, and $DU$ is diagonal unitary matrices?
QUESTION [0 upvotes]: Question is the calculation of $U/DU$ where $U$ is a unitary matrices with non-degenerate eigenvalues, and $DU$ are diagonal unitary matrices with non-degenerate ... | {"set_name": "stack_exchange", "score": 0, "question_id": 2686342} |
\section{Random walk bridges}\label{sec:ale}
\subsection{Bridge of the random walk on \texorpdfstring{${\{0,1 \}}^d$}{}}\label{subsec:bridge01}
\subsubsection{Setting and notation}\label{subsubsec:not_hyper}
In this Subsection we are interested in studying a continuous time random walk on the hypercube ${\{0,\,1\}}^d$... | {"config": "arxiv", "file": "1710.08856/Bridges.tex"} |
\begin{document}
\title[Recombination and adaptation]{An Eco-Evolutionary approach of Adaptation and Recombination in a large population of varying size}
\author{Charline Smadi}
\address{Universit\'e Paris-Est, CERMICS (ENPC), F-77455 Marne La Vall\'ee, France and CMAP UMR 7641, \'Ecole Polytechnique CNRS, Route de... | {"config": "arxiv", "file": "1402.4104/revisedversion.tex"} |
TITLE: Consider $u_t - \Delta u = f(u)$ and $u=0$ on $\partial\Omega \times (0,\infty)$. Show if $u(x,0) \geq 0$, then $u(x,t) \geq 0$
QUESTION [1 upvotes]: The question was asked here ($u$ is a $C^2$ solution of $u_t - \Delta u = f(u)$ and $u=0$ on $\partial\Omega \times (0,\infty)$. Show if $u(x,0) \geq 0$, then $u(x... | {"set_name": "stack_exchange", "score": 1, "question_id": 2380914} |
TITLE: Find the relation between the volumes of a cone and inscribed sphere
QUESTION [0 upvotes]: I have a question that I've been working upon for a long time but in vain. Can you help me.
Determine the relation between the volume of a con circumscribed to a regular tethadron and the volume of a sphere inscribed in t... | {"set_name": "stack_exchange", "score": 0, "question_id": 725001} |
TITLE: Adjunct for solution operator of Poisson problem
QUESTION [0 upvotes]: Consider the Poisson problem with Neumann boundary conditions
$\Delta u = f$ on $\Omega \subset \mathbb{R}^d$
$\partial_{\nu}u = 0$ on $\partial \Omega$
where $\Omega$ is Lipschitz and $f \in L^2(\Omega)$. Now we know that the problem has uni... | {"set_name": "stack_exchange", "score": 0, "question_id": 3936830} |
TITLE: How to Integrate $ \int^{\pi/2}_{0} x \ln(\cos x) \sqrt{\tan x}\,dx$
QUESTION [11 upvotes]: Evaluate
$$\displaystyle \int^{\frac{\pi}{2}}_{0} x \ln(\cos x) \sqrt{\tan x}dx$$
Unfortunately, I have no idea on how to integrate this and thus cannot provide any inputs on my own. The only thing I noticed was that s... | {"set_name": "stack_exchange", "score": 11, "question_id": 1319849} |
\begin{document}
\maketitle
\begin{abstract}
A proper vertex coloring of a graph is said to be \emph{locally
identifying} if the sets of colors in the closed neighborhood of
any two adjacent non-twin vertices are distinct. The lid-chromatic
number of a graph is the minimum number of colors used by a locally... | {"config": "arxiv", "file": "1212.5468.tex"} |
TITLE: How to change the order of integration of $\int_{-1}^1dx \int_{1-x^2}^{2-x^2}f(x,y)dy$
QUESTION [0 upvotes]: How to change the order of integration:
$$\int_{-1}^1dx \int_{1-x^2}^{2-x^2}f(x,y)dy$$
I tried to sketch the area and got:
Where red line is $x^2< 1$, green lines are $1-y$ and $2-y$. However I can't see... | {"set_name": "stack_exchange", "score": 0, "question_id": 4379693} |
TITLE: Don't understand an induction proof for odometer principle
QUESTION [1 upvotes]: Prove by induction that the Odometer Principle with base b does indeed give the representation $$\text{$x_{n-1}...x_1x_0$ for the natural number $N = x_{n-1}b^{n-1}+...+x_1b+x_0$} $$.
So my question is, in the bracketed section of t... | {"set_name": "stack_exchange", "score": 1, "question_id": 3270190} |
\begin{document}
\title{DOA Estimation for Transmit Beamspace MIMO Radar via Tensor Decomposition with Vandermonde Factor Matrix}
\author{Feng Xu, \IEEEmembership{Student Member, IEEE}, Matthew W. Morency, \IEEEmembership{Student Member, IEEE}, and Sergiy A. Vorobyov, \IEEEmembership{Fellow, IEEE}
\thanks{This work w... | {"config": "arxiv", "file": "2101.12666/AngleEstimationforBeamspaceMIMOwithVandermonde.tex"} |
TITLE: Evaluating $\int_0^2 \frac{x^2}{x^3 + 1} \,\mathrm{d}x$
QUESTION [0 upvotes]: I don't understand how to get from the first to the second step here and get $1/3$ in front.
In the second step $g(x)$ substitutes $x^3 + 1$.
\begin{align*}
\int_0^2 \frac{x^2}{x^3 + 1} \,\mathrm{d}x
&= \frac{1}{3} \int_{0}^{2} ... | {"set_name": "stack_exchange", "score": 0, "question_id": 2909022} |
import data.real.nnreal
import topology.algebra.infinite_sum
import topology.instances.ennreal
import topology.algebra.monoid
open_locale nnreal big_operators
open finset
/-!
# A technical lemma on the way to `lem98`
The purpose of this file is to prove the following lemma:
```
lemma exists_partition (N : ℕ) (hN : ... | {"subset_name": "curated", "file": "formal/lean/liquid/combinatorial_lemma/partition.lean"} |
\begin{document}
\title[Multicomponent compressible fluids]{Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models}
\author[D. Bothe]{Dieter Bothe}
\address{Mathematische Modellierung und Analysis, Technische Universit\"at Darm... | {"config": "arxiv", "file": "2001.08970.tex"} |
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