text stringlengths 28 2.36M | meta stringlengths 20 188 |
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TITLE: Wavefunction Amplitude Intuition
QUESTION [6 upvotes]: Reading the responses to this question: Contradiction in my understanding of wavefunction in finite potential well
it seems people are pretty confident that, e.g., the wavefunction of a particle in a slanted potential well:
makes physical sense, since the s... | {"set_name": "stack_exchange", "score": 6, "question_id": 711930} |
TITLE: Plugging a circle into real polynomial
QUESTION [5 upvotes]: I have written some code (attached below) that generates a random real polynomial $P$ degree and coefficient within some range.
I then plotted and looked at $im(P(S^1)) $ with $S$ being the unit circle in the complex plane.
To my surprise I got pictur... | {"set_name": "stack_exchange", "score": 5, "question_id": 2432033} |
\begin{document}
\maketitle
\begin{abstract}
A new implementation of the canonical polyadic decomposition (CPD) is presented. It features lower computational complexity and memory usage than the available state of art implementations available. The CPD of tensors is a challenging problem which has been approached in... | {"config": "arxiv", "file": "1912.02366/ex_article.tex"} |
\section{Robustness under Cascading Failure}
\label{sec:cascading-failure}
\input{./cascading-failure/cascading-failure-network-flow}
\input{./cascading-failure/cascading-failure-contagion}
\paragraph{Future Research Directions}
A very few formal stability and robustness analyses exist for cascading failure dynamic... | {"config": "arxiv", "file": "1909.06506/cascading-failure/cascading-failure-main-v2.tex"} |
TITLE: How do I get the value of the structure sheaf on an arbitrary open subset of an affine scheme?
QUESTION [0 upvotes]: In Vakil's notes on the foundations of algebraic geometry, he defines the structure sheaf $
\mathscr{O}_{\text{Spec}A}(D(f))$ for distinguished open sets $D(f)$ by $\mathscr{O}_{\text{Spec}A}(D(f)... | {"set_name": "stack_exchange", "score": 0, "question_id": 4291018} |
TITLE: Finding a critical point
QUESTION [2 upvotes]: Let $X=\{u\in C^2([0,1],\mathbb R),u(0)=u'(0)=0 \}$. Let $F : X \rightarrow \mathbb R$ such as $F(u)=\frac{1}{2}\int_0^1(u')^2dt-\int_0^1\cos(u)dt$.
For which norm $X$ is a Banach space ?
Show that $F$ is differentiable in every point and calculate its differential... | {"set_name": "stack_exchange", "score": 2, "question_id": 2097179} |
\begin{document}
\setcounter{page}{1}
\title{Coded Elastic Computing on Machines with Heterogeneous Storage and Computation Speed}
\author{Nicholas Woolsey,~\IEEEmembership{Student Member,~IEEE}, Rong-Rong Chen,~\IEEEmembership{Member,~IEEE},\\ and Mingyue Ji,~\IEEEmembership{Member,~IEEE}
\thanks{This manuscript was ... | {"config": "arxiv", "file": "2008.05141/Heterogeneous CEC_ArXiv_v2/CEC_HetStorage_v5.tex"} |
TITLE: Can the cardinality of a set $\mathbb{S}^n$ be written in terms of $\lvert S\rvert$?
QUESTION [0 upvotes]: The question is pretty simple:
Can the cardinality of a set $\mathbb{S}^n$ be written in terms of $\lvert \mathbb{S}\rvert$? This includes transfinite cardinals.
Here, $\mathbb{S}^n=\{(x_1,\cdots,x_n):x_i\i... | {"set_name": "stack_exchange", "score": 0, "question_id": 3265911} |
\begin{document}
\maketitle
\begin{abstract}
For the Hermitian inexact Rayleigh quotient iteration (RQI), we
present a new general theory, independent of iterative solvers for
shifted inner linear systems. The theory shows that the method
converges at least quadratically under a new condition, called the
uniform posi... | {"config": "arxiv", "file": "0906.2238/rqiminresv3.tex"} |
\documentclass{article}
\usepackage{graphics,graphicx,psfrag,color,float, hyperref}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm}
\oddsidemargin 0pt
\evensidemargin 0pt
\marginparwidth 40pt
\marginparsep 10pt
\topmargin 0pt
\headsep 10pt
\textheight 8.4in
\textwidth 6.6in
\usepackage{amss... | {"config": "arxiv", "file": "1512.03878/Arxiv-Submission.tex"} |
TITLE: Meaning of the notation $\sigma_{ji,j}$
QUESTION [0 upvotes]: In page 28 of the book Introduction to Linear Elasticity, 4ed by Phillip L. Gould · Yuan Feng, it says
$$
\int_V{\left( f_i+\sigma _{ji,j} \right) \text{d}V=0}
$$
What does it mean by writing $\sigma _{ji,j}$?
Also in equation $(2.32)$,
$$
\int_V{G_{i... | {"set_name": "stack_exchange", "score": 0, "question_id": 599549} |
TITLE: Maximum number of elemets in the range of function $ f{2012} $
QUESTION [0 upvotes]: Consider a family of functions $ f_m : N \cup \{0\} \to N \cup \{0\} $ that follows the relations :
$ f_m(a+b) = f_m(f_m(a)) + f_m(b) $,
$ f_m(km) = 0 $
HERE m is any positive integer apart from 1. How can I find the maxi... | {"set_name": "stack_exchange", "score": 0, "question_id": 563601} |
TITLE: Distribution of a continuous random variable
QUESTION [0 upvotes]: I was reading through my"Random Variability in Business Situations" notes and wanted to enquire about some difficulty I've encountered.
On the fourth and final column "probability calculated from model X~N (53, 2^2)" how are the values determined... | {"set_name": "stack_exchange", "score": 0, "question_id": 116645} |
\begin{document}
\title[On Leinster groups of order $pqrs$]{On Leinster groups of order $pqrs$}
\author[S. J. Baishya ]{Sekhar Jyoti Baishya}
\address{S. J. Baishya, Department of Mathematics, Pandit Deendayal Upadhyaya Adarsha Mahavidyalaya, Behali, Biswanath-784184, Assam, India.}
\email{sekharnehu@yahoo.com}
... | {"config": "arxiv", "file": "1911.04829.tex"} |
TITLE: Why is dx or dy width during rotation of a function around an axis?
QUESTION [0 upvotes]: For example
Why is dx the width? Isn't dx infinitely small?
Thank you
REPLY [1 votes]: In problems like this, $dx$ is the infinitesimal width. Recall the definition of an integral:
$$\int_a^bf(x)dx= \lim_{n\rightarrow\inf... | {"set_name": "stack_exchange", "score": 0, "question_id": 3757012} |
\begin{document}
\title{Distributed Sequential Detection for Gaussian Shift-in-Mean Hypothesis Testing}
\author{Anit~Kumar~Sahu,~\IEEEmembership{Student Member,~IEEE} and Soummya~Kar,~\IEEEmembership{Member,~IEEE}
\thanks{The authors are with the Department of Electrical and Computer Engineering, Carnegie Mellon Unive... | {"config": "arxiv", "file": "1411.7716/p21_paper_1column.tex"} |
\begin{document}
\title{About tests of the ``simplifying'' assumption for conditional copulas}
\author{Alexis Derumigny\thanks{ENSAE, 3 avenue Pierre-Larousse, 92245 Malakoff cedex, France. alexis.derumigny@ensae.fr}, Jean-David Fermanian\thanks{ENSAE, J120, 3 avenue Pierre-Larousse, 92245 Malakoff cedex, France. je... | {"config": "arxiv", "file": "1612.07349/On_the_simplified_vine_assumption_2017-05-04.tex"} |
TITLE: Can AND, OR and NOT be used to represent any truth table?
QUESTION [0 upvotes]: there are $2^{2^n}$ truth tables for n inputs. We know that NAND and NOR can be used to represent every truth table in two inputs. What about three inputs. Is it possible or not? If not, why?
REPLY [3 votes]: Yes, any truth-function... | {"set_name": "stack_exchange", "score": 0, "question_id": 2646024} |
TITLE: Are there any functions where differentiating the Taylor series yields an accurate approximation for the differentiated function?
QUESTION [0 upvotes]: Considering the Taylor series for $e^x$, you can differentiate it and get the same series. ie $$\frac{d}{dx} (1 + x + \frac{x^2}{2!} + ...) = 1 + x + ...$$
This ... | {"set_name": "stack_exchange", "score": 0, "question_id": 3619217} |
TITLE: Measurable sets of $\mathbb R^n$ forming unique absolutely continuous convex combinations?
QUESTION [1 upvotes]: If we consider a finite set $A\subset\mathbb R^n$, uniqueness of the convex decomposition of points in $A$ is equivalent to the absence of $\mu\neq0$ signed measure supported on $A$ such that $\mu(\ma... | {"set_name": "stack_exchange", "score": 1, "question_id": 422781} |
TITLE: $H:=\{ g^2 : g \in G \}$ is a subgroup of $G$ $\implies $ $H$ is normal in $G$
QUESTION [1 upvotes]: Let $G$ be a group . If $H :=\{ g^2 : g \in G\}$ is a subgroup of $G$ , the how to prove that $H$ is normal in $G$ ?
REPLY [6 votes]: Consider if $h\in G$ then $h^2 \in H$, so for any $g\in G$ we have:
$$gh^2g^{... | {"set_name": "stack_exchange", "score": 1, "question_id": 894102} |
TITLE: West German Mathematics Olympiad 1981
QUESTION [4 upvotes]: If $n=2^k$ then from a set of $2n-1$ elements we can select $n$ numbers such that their sum is divisible by $n$.
I divided it in two case, first if any $n$ of them have the same remainder mod $n$ the we are done, in the other case i am having difficult... | {"set_name": "stack_exchange", "score": 4, "question_id": 2049522} |
TITLE: Conditional distribution of the sum of k, independent ordered draws
QUESTION [1 upvotes]: Suppose I make k independent draws from the same distribution (say uniform).
The distribution of the sum of these order statistics should equal the distribution of the sum of k independent random variables, am I right?
But ... | {"set_name": "stack_exchange", "score": 1, "question_id": 1277145} |
TITLE: Generating infinite index subgroups of a free group
QUESTION [4 upvotes]: Let $F$ be nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of infinite index and let $x,y \in F \setminus H$. Must there be some $a \in F$ such that $[F : \langle H,a,xay\rangle] = \infty$ ?
REPLY... | {"set_name": "stack_exchange", "score": 4, "question_id": 191855} |
\begin{document}
\begin{abstract}
Gröbner bases are a fundamental tool when studying ideals in multivariate
polynomial rings. More recently there has been a growing interest in
transferring techniques from the field case to other
coefficient rings, most notably Euclidean domains and principal ideal rings.
In... | {"config": "arxiv", "file": "1906.08555/pseudo_grobner.tex"} |
\begin{document}
\title[Multi-valued graphs]
{The space of embedded minimal surfaces of fixed genus in a $3$-manifold II;
Multi-valued graphs in disks}
\author{Tobias H. Colding}
\address{Courant Institute of Mathematical Sciences and MIT\\
251 Mercer Street\\
New York, NY 10012 and 77 Mass. Av., Cambridge, MA 02139}... | {"config": "arxiv", "file": "math0210086/blowup106.tex"} |
\section{Complex Integration by Substitution}
Tags: Contour Integration
\begin{theorem}
Let $\left[{a \,.\,.\, b}\right]$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\phi: \left[{a \,.\,.\, b}\right] \to \R$ be a [[Definition:Real Function|real function]] which has a [[Definition:Derivative|der... | {"config": "wiki", "file": "thm_6245.txt"} |
TITLE: Prove roots of quadratic are rational
QUESTION [1 upvotes]: Given the equation:
$$(3m-5)x^2-3m^2x+5m^2=0$$
Prove the roots are rational given $m$ is rational.
I have tried finding the discriminate of the quadratic however this has not been fruitful and ends up being very ugly. I have tried setting the disc... | {"set_name": "stack_exchange", "score": 1, "question_id": 2438708} |
TITLE: Extract a variable from the following summation
QUESTION [1 upvotes]: Is it possible to extract the variable a from the following summation and hence write the summation as a times 'something', or similar?
$$n/a = \sum_{i=1}^n 1/(a+x_i)$$
REPLY [1 votes]: Let $g(a,x_1,\dots,x_n)=\sum_{i=1}^n \frac{1}{(a+x_i)}$... | {"set_name": "stack_exchange", "score": 1, "question_id": 2233617} |
TITLE: Maximum degree of a polynomial $\sum\limits_{i=0}^n a_i x^{n-i}$ such that $a_i = \pm 1$ for every $i$, with only real zeroes
QUESTION [1 upvotes]: What is the maximum degree of a polynomial of the form $\sum\limits_{i=0}^n a_i x^{n-i}$ with $a_i = \pm 1$ for $0 \leq i \leq n$, such that all the zeros are real?
... | {"set_name": "stack_exchange", "score": 1, "question_id": 1967570} |
TITLE: prove : if $G$ is a finite group of order $n$ and $p$ is the smallest prime dividing $|G|$ then any subgroup of index $p$ is normal
QUESTION [10 upvotes]: Prove:
If $G$ is a finite group of order $n$ and $p$ is the smallest prime dividing $ |G| $ then any subgroup of index $p$ is normal
where $ |G| $ is the ord... | {"set_name": "stack_exchange", "score": 10, "question_id": 289305} |
TITLE: Taking a theorem as a definition and proving the original definition as a theorem
QUESTION [46 upvotes]: Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage:
Perform the following thought experiment. Suppose that you are given two formal ... | {"set_name": "stack_exchange", "score": 46, "question_id": 404882} |
\begin{document}
\title[]{Fokker-Planck Equation and Path Integral Representation of Fractional Ornstein-Uhlenbeck Process with Two Indices}
\author{Chai Hok Eab}
\address{
Department of Chemistry
Faculty of Science, Chulalongkorn University
Bangkok 10330, Thailand
}
\ead{Chaihok.E@chula.ac.th}
\author{S.C. Lim}
\ad... | {"config": "arxiv", "file": "1405.0653/main.tex"} |
TITLE: show: $\overline{\overline X} = \overline X$
QUESTION [0 upvotes]: is my proof correct?
Definition:
Let $X\subset\mathbb R$ and let $x'\in\mathbb R$, we say that $x'$ is an adherent point of $X$ iff $\forall\epsilon>0\exists x\in X \text{ s.t. }d(x′,x)≤ε$. the closure of X is denoted as $\overline X$ and is defi... | {"set_name": "stack_exchange", "score": 0, "question_id": 1016682} |
TITLE: Does the spiral Theta = L/R have a name?
QUESTION [0 upvotes]: Note: I intentionally left the equation in the title in plain text instead of MathJax, so it is searchable.
Here is a spiral's equation in polar coordinates:
$$\theta=L/r,$$
and in Cartesian coordinates:
$$(x,y) = \left(r\cdot\cos\frac Lr,\, r\cdot\s... | {"set_name": "stack_exchange", "score": 0, "question_id": 3766344} |
TITLE: Calculus interval problem please explain how to solve it
QUESTION [1 upvotes]: Suppose $f(x)>0$ for all $x$ in $[0,10]$. Express the area under the curve $f(x) $for $0≤x≤10$ as the limit of a sum, using the value of $f(x)$ at right endpoints of your intervals.
Could you explain how to solve it?
REPLY [3 votes]... | {"set_name": "stack_exchange", "score": 1, "question_id": 592187} |
TITLE: Why is this a valid proof for the harmonic series?
QUESTION [3 upvotes]: The other day, I encountered the Harmonic series which I though is an interesting concept. There were many ways to prove that it's divergent and the one I really liked was rather simple but did the job nevertheless. But a proof used integra... | {"set_name": "stack_exchange", "score": 3, "question_id": 3207598} |
TITLE: Expectation value with plane waves
QUESTION [1 upvotes]: Hey guys Im a little confused with the concept of plane waves and how to perform an expectation value. Let me show you by an example. Suppose you have a wave function of the form
$\psi_{\boldsymbol{p}_{0}}(x)=f(p_{0})e^{\frac{i}{\hbar}\boldsymbol{p}_{0}\cd... | {"set_name": "stack_exchange", "score": 1, "question_id": 162835} |
\begin{document}
\section{Examples of combinatorial projection maps}\label{sec:combinatorial-examples}
In this section we list combinatorial projection maps for several common examples of groups $G\le S_n$. Notice that in each of the four examples of concrete groups below, implementation via a suitable sorting func... | {"config": "arxiv", "file": "2202.02164/sections_arxiv/app-combinatorial-examples.tex"} |
\begin{document}
\title[Hofmann-Lawson duality for locally small spaces]{Hofmann-Lawson duality for locally small spaces}
\author[A. Pi\k{e}kosz]{Artur Pi\k{e}kosz}
\address{{\rm Department of Applied Mathematics, Cracow University of Technology,\\
ul. Warszawska 24, 31-155 Krak\'ow, Poland\\
email: \textit{pupiek... | {"config": "arxiv", "file": "2009.02966.tex"} |
TITLE: How is this function closed?
QUESTION [0 upvotes]: I am reading appendix of convex optimization book. In example A.1 it is written that $$f(x)=\array{x\log(x)\quad \text{if}\quad x>0 \quad \text{and}\quad 0 \quad \text{if} \quad x=0}$$ Its domain is $[0,\infty)$. Now function is continuous. The boundary of the... | {"set_name": "stack_exchange", "score": 0, "question_id": 2618451} |
\begin{document}
\sloppy
\maketitle
\begin{abstract}
Consider a self-similar space $X$. A typical situation is that $X$ looks
like several copies of itself glued to several copies of another space $Y$,
and that $Y$ looks like several copies of itself glued to several copies of
$X$---or the same kind of thing with m... | {"config": "arxiv", "file": "math0411343/gsso.tex"} |
TITLE: Root equation - What am I missing?
QUESTION [3 upvotes]: There's a problem of which I know the solution but not the solving process:
$(\sqrt{x} + 7)(\sqrt{x} - 1) = \frac{105}{4}$
I'm convinced that up to:
$x + 6\sqrt{x} - 7 = \frac{105}{4}$
everything is correct.
But afterwards, I never seem to be able to g... | {"set_name": "stack_exchange", "score": 3, "question_id": 145498} |
\begin{document}
\title[]{Convexity of strata in diagonal pants graphs of surfaces}
\author{J. Aramayona, C. Lecuire, H. Parlier \& K. J. Shackleton}
\address{School of Mathematics, Statistics and Applied Mathematics,
NUI Galway, Ireland}
\email{javier.aramayona@nuigalway.ie}
\urladdr{http://www.maths.nuigalway.i... | {"config": "arxiv", "file": "1111.1187/convex041111.tex"} |
\begin{document}
\selectlanguage{english}
\maketitle
We establish the existence of a global solution to a family of equations, which are obtained in certain regimes in~\cite{DS-16} as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electr... | {"config": "arxiv", "file": "1607.00268/MF-glob-ex_4.tex"} |
\section{One-dimensional PCP Lagrangian schemes}\label{section:1D-PCP}
This section considers the Lagrangian finite volume schemes for the special relativistic hydrodynamics (RHD) equations \eqref{eq1} or \eqref{eq4} with $d=1$, here
the notations $\vec{ F}_1$, $u_1$ and $x_1$ are replaced with $\vec{ F}$, $u$ and $x$... | {"config": "arxiv", "file": "1901.10625/agorithm_1D.tex"} |
TITLE: Fundamental Representation of $SU(3)$ is a complex representation
QUESTION [16 upvotes]: Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule:
$\qquad \qquad \qquad [T^a_R, T^b_R]=if^{abc}T^c_R$.
Now if $-(T^a_R)^* = T^a_R $,... | {"set_name": "stack_exchange", "score": 16, "question_id": 69443} |
TITLE: Smallest integer which leaves remainder 3, 2, 1 when divided by 17, 15, 13
QUESTION [1 upvotes]: Find the smallest positive integer so that the remainder when it is divided by $17,15,13$ is $3,2,1$ respectively.
This question can be solved using Chinese remainder theorem, but the theorem gives any integer, not t... | {"set_name": "stack_exchange", "score": 1, "question_id": 3280460} |
TITLE: Definition of null space
QUESTION [1 upvotes]: I have two definitions of null space. One by Serge Lang
Suppose that for every element $u$ of $V$ we have $\langle u,u\rangle=O$. The
scalar product is then said to be null, and $V$ is called a null space.
and another by David C. Lay
The null space of an $m \ti... | {"set_name": "stack_exchange", "score": 1, "question_id": 863648} |
TITLE: Why does the average z-score for a standardized distribution always equal to zero?
QUESTION [0 upvotes]: My introductory statistics book mentioned this:
"When an entire distribution of scores is standardized, the average (i.e., mean) z score for the standardized distribution will always be 0, and the standard de... | {"set_name": "stack_exchange", "score": 0, "question_id": 3217144} |
TITLE: Prove that it is NOT true that for every integer $n$, 60 divides $n$ if and only if 6 divides $n$ and 10 divides $n$.
QUESTION [2 upvotes]: This is Velleman's exercise 3.4.26 (b):
Prove that it is NOT true that for every integer $n$, 60 divides $n$ iff 6 divides $n$ and 10 divides $n$.
I do understand that a ... | {"set_name": "stack_exchange", "score": 2, "question_id": 1403337} |
TITLE: Finding $\lim_{n \to \infty} \prod_{r=1}^n (1+ \frac{1}{a_r})$
QUESTION [0 upvotes]: Evaluate the following infinite product:
$$\lim_{n\to \infty} \prod_{r=1}^n (1+ \frac{1}{a_r}),\\
a_1=1; \quad a_r=r(1+a_{r-1}).$$
I thought of two ways in which this might be evaluated: (1) By taking log and converting into ... | {"set_name": "stack_exchange", "score": 0, "question_id": 2392104} |
TITLE: Find matrix rank according to parameter value
QUESTION [0 upvotes]: I've been trying to solve this "simple" problem using the gaussian elimination method, but I don't get the right reduction steps to simplify the matrix and left a simple parameter term in the last row.
So I can't get the matrix rank according to... | {"set_name": "stack_exchange", "score": 0, "question_id": 1513555} |
TITLE: Let $f\in\mathbb{Q}[X]$ such that $f(1)=-2, f(2)=1$ and $f(-1)=0$. Find the remainder of the division of $f$ by $X^3-2X^2-X+2$.
QUESTION [1 upvotes]: Let $f\in\mathbb{Q}[X]$ such that $f(1)=-2, f(2)=1$ and $f(-1)=0$. Find the remainder of the division of $f$ by $X^3-2X^2-X+2$.
So, I figured: $f=(X+1)q$. Assummin... | {"set_name": "stack_exchange", "score": 1, "question_id": 4189376} |
\appendix
\section{Symbols}
\label{app:symbols}
List of symbols used in the paper with their brief description.
\begin{table}[h]
\centering
\small
\begin{tabular}{c|l}
\toprule
\multicolumn{2}{c}{Two-stage stochastic program}\\
\midrule
$x$ & First-stage decision vector \\
... | {"config": "arxiv", "file": "2205.12006/sections/08_Appendix.tex"} |
TITLE: Corners are cut off from an equilateral triangle to produce a regular hexagon. Are the sides trisected?
QUESTION [8 upvotes]: The corners are cut off from an equilateral triangle to produce a regular hexagon. Are the sides of the triangle trisected?
In the actual question, the first line was exactly the same. I... | {"set_name": "stack_exchange", "score": 8, "question_id": 3726183} |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import data.finsupp.order
/-!
# Equivalence between `multiset` and `ℕ`-valued finitely supported functions
This defines `finsupp.to_multiset` the equivalence betwee... | {"subset_name": "curated", "file": "formal/lean/mathlib/data/finsupp/multiset.lean"} |
TITLE: What is dark matter composed of?
QUESTION [1 upvotes]: They say that about 80% of the matter in the universe is dark matter -- some unknown substance that has mass. But what is this matter expected or speculated to be composed of? For example might it exist as neutrinos or protons or atoms of hydrogen or somethi... | {"set_name": "stack_exchange", "score": 1, "question_id": 166254} |
\begin{document}
\title[]
{A lower bound of the $L^2$ norm error estimate for the Adini element of the
biharmonic equation}
\author[J. Hu]{Jun Hu}
\address{LMAM and School of Mathematical Sciences,
Peking University, Beijing 100871, P. R. China}
\email{hujun@math.pku.edu.cn}
\author[Z. C. Shi]{Zhongci Shi}
\addres... | {"config": "arxiv", "file": "1211.4677.tex"} |
TITLE: Disproving Cantor's diagonal argument
QUESTION [2 upvotes]: I am familiar with Cantor's diagonal argument and how it can be used to prove the uncountability of the set of real numbers. However I have an extremely simple objection to make. Given the following:
Theorem: Every number with a finite number of digits ... | {"set_name": "stack_exchange", "score": 2, "question_id": 2694917} |
TITLE: Fourier transform
QUESTION [0 upvotes]: I find some problems about pde, I guess, concerning Distribution solution.
Define $$<T,\phi> = \frac{1}{\pi}\lim_{\epsilon \rightarrow 0} \int_{|x|\geq \epsilon} \frac{\phi(x)}{x} dx.$$ Then $$ \mathcal{F}({T})(t) = -i \ \mbox{sgn} (t),$$ where $\mathcal{F}(T)$ is the Four... | {"set_name": "stack_exchange", "score": 0, "question_id": 1739556} |
\begin{document}
\title[On $q$-Euler numbers related to the modified $q$-Bernstein polynomials]
{On $q$-Euler numbers related to the modified $q$-Bernstein polynomials}
\author{Min-Soo Kim, Daeyeoul Kim and Taekyun Kim}
\begin{abstract}
We consider $q$-Euler numbers and polynomials and $q$-Stirling numbers of first ... | {"config": "arxiv", "file": "1007.3317.tex"} |
TITLE: Is the image of an injective immersion automatically both closed and open, depending on what topology you use?
QUESTION [1 upvotes]: I know I must be missing something somewhere; at the risk of being laughed at (a lot) I am asking if someone could point out to me exactly where. The question has to do with subman... | {"set_name": "stack_exchange", "score": 1, "question_id": 4515791} |
TITLE: How to solve $x^{x^x}=(x^x)^x$?
QUESTION [2 upvotes]: How can we solve the equation :
$x^{x^x}=(x^x)^x$
with $x \in {\mathbb{R}+}^*$
Thanks for heping me :)
REPLY [1 votes]: Applying $\log$ to both sides
$$
x^x\log x = x^2\log x\Rightarrow (x^x-x^2)\log x=0
$$
so we have $\log x= 0\Rightarrow x = 1$ and $x^x=x... | {"set_name": "stack_exchange", "score": 2, "question_id": 3184200} |
\begin{document}
\title[Nodal deficiency]{Nodal deficiency of random spherical harmonics in presence of boundary}
\author{Valentina Cammarota\textsuperscript{1}}
\email{valentina.cammarota@uniroma1.it}
\address{\textsuperscript{1}Department of Statistics, Sapienza University of Rome}
\author{Domenico Marinucci\texts... | {"config": "arxiv", "file": "2011.01571.tex"} |
TITLE: Derivative of angular velocity in a rotating frame
QUESTION [0 upvotes]: Taylor Relies on these relations
$v = \omega \times r$
$\frac{d}{dt}Q = \omega \times Q$
To show that
$a = a' + 2 \omega \times v' + \omega \times \omega \times r' + \alpha \times r' $
So we take the product rule of (1) and get:
$... | {"set_name": "stack_exchange", "score": 0, "question_id": 535207} |
\begin{document}
\title{Maximal dissipation in Hunter-Saxton equation for bounded energy initial data.}
\author[1]{Tomasz Cie\'slak}
\author[1,2]{Grzegorz Jamr\'oz}
\affil[1]{\small Institute of Mathematics, Polish Academy of Sciences, \'Sniadeckich 8, 00-656 Warszawa, Poland \newline{\small e-mail: T.Cieslak@impan... | {"config": "arxiv", "file": "1405.7558/TCGJ9.tex"} |
TITLE: Definition of the Limes Superior
QUESTION [0 upvotes]: I understand what the limes superior is insofar as I know that for example for an alternating series as $a_n = (-1)^n$ which does not diverge to $\infty$, we can define the bigger accumulation point 1 as the limes superior.
What I don't understand yet though... | {"set_name": "stack_exchange", "score": 0, "question_id": 344770} |
TITLE: The dual of the dual of a group $G$ is isomorphic to $G$
QUESTION [4 upvotes]: This is exercise 3.3 of Serre's book on Representation theory of finite groups: Let $G$ be a finite abelian group and $G'$ the set of irreducible characters of $G$. It is easy to prove that $G'$ is also a finite abelian group of the s... | {"set_name": "stack_exchange", "score": 4, "question_id": 1934915} |
TITLE: Zeros of the derivative of a function are real
QUESTION [0 upvotes]: I'm reviewing Complex Analysis from Ahlfors' book and stuck at this question.
"Show that if $f(z)$ is of genus $0$ or $1$ with real zeros, and if $f(z)$ is real
for real $z$, then all zeros of $f'(z)$ are real. Hint: Consider $Im (f'(z)/f(z))$.... | {"set_name": "stack_exchange", "score": 0, "question_id": 3590217} |
TITLE: Why is $\sum_{i=1}^n\frac n{n-i+1}=n\sum_{i=1}^n\frac1i$?
QUESTION [2 upvotes]: While reading through my (algorithms and) probability script, I have seen this equality for calculating the first moment of the coupon-collector problem. However, I don't quite see how the sum of the fraction on the left can be split... | {"set_name": "stack_exchange", "score": 2, "question_id": 3610538} |
TITLE: What problems are easier to solve in a higher dimension, i.e. 3D vs 2D?
QUESTION [12 upvotes]: I'd be interested in knowing if there are any problems that are easier to solve in a higher dimension, i.e. using solutions in a higher dimension that don't have an equally optimal counterpart in a lower dimension, par... | {"set_name": "stack_exchange", "score": 12, "question_id": 693485} |
TITLE: Notation to edges set incidents in a vertice $v$
QUESTION [1 upvotes]: How I can denote the set of edges incident a vertice $v$ in a graph? Any sugestion/reference that has this?
I knowed that the set of vertices incidents in other vertices is N(v) (set of neighbor of vertice), but the edges set i not knowed!
R... | {"set_name": "stack_exchange", "score": 1, "question_id": 4385376} |
TITLE: Number of permutations of r items from T total items of which n are distinct.
QUESTION [2 upvotes]: Lets say I have T total items and out of them n are distinct. So n <= T.
In other words we can have m1 items of item 1 and m2 items of item 2....and so on, and that $m_1 + m_2 +\cdots +m_n = T$.
What are the numbe... | {"set_name": "stack_exchange", "score": 2, "question_id": 3258675} |
TITLE: Given $f(x)=-x^2+1$ and $g(x)=\sqrt{x+1}$, find $k(x)=(g\circ f)(x)$?
QUESTION [2 upvotes]: Given $f(x)=-x^2+1$ and $g(x)=\sqrt{x+1}$, find $k(x)=(g \circ f)(x)$?
Following the step's my teach told me to do this type of equation I did this... I feel like I'am not showing enough work and I'am going to college nex... | {"set_name": "stack_exchange", "score": 2, "question_id": 425547} |
\begin{document}
\begin{abstract}
We introduce three new nonlinear continuous data assimilation algorithms. These models are compared with the linear continuous data assimilation algorithm introduced by Azouani, Olson, and Titi (AOT). As a proof-of-concept for these models, we computationally investigate these alg... | {"config": "arxiv", "file": "1703.03546/KSE1D_DA_NL_arXiv.tex"} |
TITLE: Can you add an extra $e^x$ when integrating?
QUESTION [0 upvotes]: So I've been given this problem to solve (pretend it's a fraction or click the link to see the question please)
$$\int \frac{-26e^x-144}{e^{2x} + 13e^x + 36}dx$$
and I got this far:
$$-2\int\frac{13e^x + 72}{e^{2x} + 13e^x + 36}dx$$
The next step... | {"set_name": "stack_exchange", "score": 0, "question_id": 3135813} |
TITLE: Decidability and Cluster algebras
QUESTION [6 upvotes]: Recall the definition of a cluster algebra,
which can be seen as a (possibly infinite) graph, where each vertex is a tuple of a quiver and Laurent expressions at some of the vertices of the quiver. The edges of this graph is given by mutations, and every ve... | {"set_name": "stack_exchange", "score": 6, "question_id": 271192} |
TITLE: Would an atom OR its nucleus alone affect its surrounding empty space, with respect to the particles that come and go of existence within that space?
QUESTION [0 upvotes]: I understand that the particles come and go of existence in the empty space. So, what effect could a stable particle (one that do not go out ... | {"set_name": "stack_exchange", "score": 0, "question_id": 325880} |
TITLE: Separating the topics of general and special relativity
QUESTION [2 upvotes]: So, I have managed to confuse myself beyond the point of repair. I am not a physics student, so my physical knowledge is limited.
Anyway, there are a few topics of relativity, which I can not seem to be able to seperate into either the... | {"set_name": "stack_exchange", "score": 2, "question_id": 480401} |
\section{NP-hardness of Quasi MLE}
\label{section_appendix_nphard}
\begin{theorem}\label{mle_alignmentproblem}
Consider the problem $\mathsf{ALIGN}(y_1, \ldots, y_N)$: for vectors $y_1,\dots,y_N \in \mathbb{R}^L$, find the shifts $\boldsymbol{\ell} = (l_1,\dots,l_N)$ which maximize
\[
\mathcal{A}(l_1, \ldots, l_N) = ... | {"config": "arxiv", "file": "1308.5256/appendix_nphard.tex"} |
\begin{document}
\title{Gromov-Witten theory of locally conformally symplectic
manifolds}
\author{Yasha Savelyev}
\thanks {Partially supported by PRODEP grant of SEP, Mexico}
\email{yasha.savelyev@gmail.com}
\address{University of Colima, CUICBAS}
\keywords{locally conformally symplectic manifolds, conformal symplec... | {"config": "arxiv", "file": "1609.08991/conformalsymplectic21.tex"} |
TITLE: Solving limits by Taylor series
QUESTION [0 upvotes]: Perhaps I didn't fully understand the concept of Taylor series. I would like to compute
$$\lim_{x \to 1} \frac{\ln x}{x^{2}-1}$$
using Taylor expansion around the right point (point is not given).
So far I only solved 'series problems' when the point was giv... | {"set_name": "stack_exchange", "score": 0, "question_id": 3265842} |
TITLE: About generalized planar graphs and generalized outerplanar graphs
QUESTION [16 upvotes]: Any planar, respectively, outerplanar graph $G=(V,E)$ satisfies $|E'|\le 3|V'|-6$,
respectively, $|E'|\le 2|V'|-3$, for every subgraph $G'=(V',E')$ of $G$.
Also, (outer-)planar graphs can be recognized in polynomial time.
W... | {"set_name": "stack_exchange", "score": 16, "question_id": 18517} |
TITLE: If $A$, $B$ are both pairwise independent of $C$, do we have that $A \cap B$ is also independent of $C$?
QUESTION [0 upvotes]: If $A$, $B$ are both pairwise independent of $C$, do we have that $A \cap B$ is also independent of $C$?
REPLY [0 votes]: Toss two fair coins. Let $A$ = "first toss heads", $B$= "secon... | {"set_name": "stack_exchange", "score": 0, "question_id": 3679281} |
TITLE: If $\pi(x,y)=x$ and $F$ is closed, $\pi(F)$ is closed.
QUESTION [1 upvotes]: I have a problem with this because I didn't use one of the hypotheses (compactness of $K$) during the demonstration. Thank you in advance for your help
Question: Let $K\subset \mathbb{R}^n$ a compact. Show that the projection $\pi: \ma... | {"set_name": "stack_exchange", "score": 1, "question_id": 3179138} |
As pointed out in~\cite{bottou}, one is usually not interested in the
minimization of an \emph{empirical cost} on a finite training set, but instead
in minimizing an \emph{expected cost}. Thus, we assume from now on that $f$ has the form of an expectation:
\begin{equation}
\min_{\theta \in \Theta} \left[ f(\theta) \... | {"config": "arxiv", "file": "1306.4650/stochastic.tex"} |
\subsection{The idea of proof}
We return to the proof of the estimates obtained in \ref{vns_solving}.
We will consider the matrices of the following kind
$$
A_* =
\left(
\begin{array}{ccccccccc}
a & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 \\
0 & a & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 \\
... | {"config": "arxiv", "file": "1804.05385/vns_proof_concept.tex"} |
TITLE: Modification of the Ramsey number
QUESTION [0 upvotes]: Let us denote by $n=r(k_1,k_2,\ldots,k_s)$ the minimal number of vertices such that for every coloring of the edges of the complete graph $K_n$ by $s$ different colors, there is some color $1\le i\le s$ such that the $i$-th graph contains a $k_i$ clique. By... | {"set_name": "stack_exchange", "score": 0, "question_id": 1055435} |
TITLE: Morphisms between affine schemes
QUESTION [1 upvotes]: Suppose we have two affine schemes $X=\operatorname{Spec} A$ and $Y=\operatorname{Spec} B$ for commutative rings $A,B$. I encountered this statement in my homework that $\operatorname{Mor}(X,Y)=\operatorname{Hom}(B,A)$. I understand that a ring homomorphism ... | {"set_name": "stack_exchange", "score": 1, "question_id": 4090290} |
TITLE: Find and prove a formula for the order of an element $k\in\mathbb Z_n$
QUESTION [3 upvotes]: I've looked at the orders of all of the elements in the group $\mathbb Z_{12}$ and from that guessed a formula that might be right: $$\mid k\mid =\frac{n}{hcf(k,n)}$$ where $\mid k \mid$ denotes the order of the element ... | {"set_name": "stack_exchange", "score": 3, "question_id": 1021526} |
\begin{document}
\maketitle
\begin{abstract}
In a recent paper, Chapuy conjectured that, for any positive integer $k$, the law for the fractions of total area covered by the $k$ Vorono\"\i\ cells
defined by $k$ points picked uniformly at random in the Brownian map of any fixed genus is the same law as that of a unifo... | {"config": "arxiv", "file": "1703.02781/voronoi.tex"} |
TITLE: Why is $(\forall x \in \mathbb{Z})(\exists y \in \mathbb{Z}) (2x + y = 3 \to x + 2y = 3)$ true?
QUESTION [2 upvotes]: Is the statement true or false? $$(\forall x \in \mathbb{Z})(\exists y \in \mathbb{Z}) (2x + y = 3 \to x + 2y = 3)$$ The answer is that the entire statement is true, but I do not see why.
The f... | {"set_name": "stack_exchange", "score": 2, "question_id": 2906826} |
TITLE: Find functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $ f(m+n)=f(m)+f(n)+2mn$
QUESTION [5 upvotes]: Find functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $ f(m+n)=f(m)+f(n)+2mn$
Setting $m=n$ we see that:
$$ f(2n)=2f(n)+2n^2$$
Setting $m=1$:
$$ f(n+1)=f(1)+f(n)+2n$$
Then, for example when $n=1$:... | {"set_name": "stack_exchange", "score": 5, "question_id": 2183340} |
TITLE: $f \in L^1(\mathbb R), f>0$ then $|\hat f(y)| < \hat f(0), y \ne 0$
QUESTION [3 upvotes]: Suppose $f$ is a strictly positive function in $L^1(\mathbb R)$. Show
$$
|\hat f(y)| < \hat f(0) \text{, for all } y \ne 0.
$$
Using monotonicity of the integral, I can show $|\hat f(y)| \le \hat f(0)$.
I don't see how to m... | {"set_name": "stack_exchange", "score": 3, "question_id": 218671} |
TITLE: Understanding units mod $n$ are relatively prime to $n$
QUESTION [1 upvotes]: I am trying to prove that the only invertible elements in $\mathbb{Z}_n$ are those that are relatively prime to $n$.
The first half is straightforward. If $i$ is relatively prime to $n$, so $\gcd(i,n) = 1$, we have $\alpha i + \beta n... | {"set_name": "stack_exchange", "score": 1, "question_id": 3708637} |
\begin{document}
\title{On the generalized Helmholtz conditions for Lagrangian systems with dissipative forces}
\author{M.\ Crampin, T.\ Mestdag and W.\ Sarlet\\[2mm]
{\small Department of Mathematics, Ghent University}\\
{\small Krijgslaan 281, B-9000 Ghent, Belgium}}
\date{}
\maketitle
\begin{quote}
{\small {\b... | {"config": "arxiv", "file": "1003.1840.tex"} |
TITLE: partition to min the max number of intersections
QUESTION [2 upvotes]: Given $n$ items and $m$ customers, each of whom is interested in some subset of the items, partition the set of items among $k$ different stores so that the maximum number of customers visiting any store is minimized.
Does anyone recognize th... | {"set_name": "stack_exchange", "score": 2, "question_id": 17628} |
TITLE: What are the computationally useful ways of thinking about Killing fields?
QUESTION [3 upvotes]: One definition of the Killing field is as those vector fields along which the Lie Derivative of the metric vanishes. But for very many calculation purposes the useful way to think of them when dealing with the Riema... | {"set_name": "stack_exchange", "score": 3, "question_id": 31384} |
\section{Introduction\label{sec:intro_three_users}}
In this paper, we study the erasure source-broadcast problem with feedback for the case of three users in what is the sequel to~\cite{TMKS_TIT20}, which studied the analogous problem for the case of \emph{two} users. In~\cite{TMKS_TIT20}, it was shown that the sour... | {"config": "arxiv", "file": "2105.00353/intro/intro_three_users.tex"} |
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