text stringlengths 28 2.36M | meta stringlengths 20 188 |
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TITLE: How to recreate intuitively a geometric series when you know the sum it converges against?
QUESTION [0 upvotes]: Given $x=ax+b$, $0<a<1$ solving for $x$ you get the sum $$\frac{b}{1-a}$$ but how can one intuitively come up with the geometric series $$b+ba+ba^2+...+ba^n $$
REPLY [3 votes]: Is this what you are l... | {"set_name": "stack_exchange", "score": 0, "question_id": 4463503} |
TITLE: Probability household cars problem
QUESTION [1 upvotes]: A survey consists of recording, the number of cars presently owned by a household among six major manufacturers. Order does not matter.
i. Suppose that no household has more than four cars. In how many different ways can the survey sheet be filled out?
I ... | {"set_name": "stack_exchange", "score": 1, "question_id": 1189979} |
\section{Introduction}\label{sec:Introduction}
In \cite{math.QA/9909027}, Jones introduced the notion of a {\it planar algebra} as an axiomatization of the standard invariant of a finite index subfactor.
A planar algebra (in vector spaces) is a sequence of vector spaces $\cP[0]$, $\cP[1]$, $\cP[2], \ldots$ called the... | {"config": "arxiv", "file": "1607.06041/Chapters/Sec_1_IntroductionPAinBTC.tex"} |
TITLE: On the equivalence of representations of Fourier series
QUESTION [5 upvotes]: Let $f : \Bbb{R} \to \Bbb{C}$ be a $2\pi$-periodic function such that
$$
\int_0^{2\pi} |f(t)| \,dt < \infty
$$
Define
$$
\hat{f}(k) := \frac{1}{2\pi} \int_0^{2\pi} f(t) e^{-i k t} \,dt
$$
The Fourier series of $f$ is then
$$
\sum_{k=-\... | {"set_name": "stack_exchange", "score": 5, "question_id": 2182785} |
\begin{document}
\title{Quantum Markov Processes \\(Correspondences and Dilations)}
\author{Paul S. Muhly\thanks{Supported by grants from the U.S. National Science
Foundation and from the U.S.-Israel Binational Science Foundation.}\\Department of Mathematics\\University of Iowa\\Iowa City, IA 52242\\\texttt{muhly@math... | {"config": "arxiv", "file": "math0203193.tex"} |
TITLE: Can an implication be a tautology?
QUESTION [6 upvotes]: I have a problem understanding this proof:
$ F →G \equiv \top \implies F \models G $.
My textbook proceeds as follows:
$ \implies $ : we assume $ F \to G \equiv \top $ . This means that the implication $ F \to G $ holds for every possible truth assignmen... | {"set_name": "stack_exchange", "score": 6, "question_id": 3280640} |
TITLE: Can someone explain me the sentence about ideals?
QUESTION [1 upvotes]: Can someone explain me the sentence:
"If $R=K[x]$ the prime ideals are $\langle f(x)\rangle $ where $f(x)$ is an irreducible polynomial in $K[x]$ and $\langle 0\rangle $, and again $\langle f(x)\rangle $ $\ f(x)$ is also maximal."
?
REPLY... | {"set_name": "stack_exchange", "score": 1, "question_id": 1016648} |
TITLE: Proving set of linear functionals is a basis
QUESTION [0 upvotes]: Let $\mathbb{R}[X]_{\leq 2}$ be the vector space of polynomials (with real coefficients) of degree at most $2$. Consider the functionals $l_1, l_2, l_3$ on this space: $$ l_1: f \mapsto \int_{0}^1 f(t) dt, \quad l_2: f \mapsto f'(1), \quad l_3: f... | {"set_name": "stack_exchange", "score": 0, "question_id": 1790729} |
TITLE: Coloring Complete Graph
QUESTION [3 upvotes]: Let $n$ be a positive odd integer. There are $n$ computers and exactly one cable joining each pair of computers. You are to colour the computers and cables such that no two computers have the same colour, no two cables joined to a common computer have the same colour... | {"set_name": "stack_exchange", "score": 3, "question_id": 3229028} |
TITLE: Movies about mathematics/mathematicians
QUESTION [11 upvotes]: I would like to watch a movie about mathematics/mathematicians (english/french language is OK, italian would be the best! Both real and invented stories are OK, maybe I would prefer something based on a real story). Well, I know maybe just the most f... | {"set_name": "stack_exchange", "score": 11, "question_id": 77279} |
\begin{document}
\title[Small covers and realization of cycles]{Small covers of graph-associahedra and realization of cycles}
\author{Alexander A.~Gaifullin}
\thanks{The work is supported by the Russian Science Foundation under grant 14-11-00414.}
\address{Steklov Mathematical Institute of the Russian Academy of Sc... | {"config": "arxiv", "file": "1611.01816.tex"} |
\begin{document}
\title{Machine Learning for CSI Recreation Based on Prior Knowledge}
\author{
\IEEEauthorblockN{\textit{Brenda Vilas Boas\IEEEauthorrefmark{1}$^,$\IEEEauthorrefmark{2}, Wolfgang Zirwas\IEEEauthorrefmark{1}}, \textit{Martin Haardt\IEEEauthorrefmark{2}}}
\IEEEauthorblockA{\IEEEauthorrefmark{1}Nokia, ... | {"config": "arxiv", "file": "2111.07854/main.tex"} |
\begin{document}
\title{CaSCADE: Compressed Carrier and DOA Estimation}
\author{\thanks{
This project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 646804-ERC-COG-BNYQ, and from the Israel Science Foundation under Grant no. 335/14. Deborah Cohen ... | {"config": "arxiv", "file": "1604.02723/DOA_last_column.tex"} |
TITLE: Open balls appearance
QUESTION [0 upvotes]: We know that with the euclidean metric the open balls in $\mathbb{R}^2$ are circles without the frontier, of course.
My question is if there exist a known metric in $\mathbb{R}^2$ such that the open balls have the appearance as triangles or hexagons?
Thank you for the ... | {"set_name": "stack_exchange", "score": 0, "question_id": 2956919} |
TITLE: Does this matrix identity hold?
QUESTION [4 upvotes]: For invertible matrices A and B does the identity:
$$
(A^{-1} + B^{-1})^{-1} = A - A(A+B)^{-1}A
$$
hold? My supervisor suggested that they are equal but I haven't been able to prove this and in the matrix cookbook (http://www.math.uwaterloo.ca/~hwolkowi/matr... | {"set_name": "stack_exchange", "score": 4, "question_id": 1617991} |
\begin{document}
\begin{titlepage}
\title{\vspace{-3cm}{\huge On Factorizable S-matrices, Generalized TTbar, \break and the Hagedorn Transition}}
\author{Giancarlo Camilo$^{1,2\musDoubleFlat}$, Thiago Fleury$^{1\musFlat}$, M\'{a}t\'{e} Lencs\'{e}s$^{3\musNatural}$,\\[0.1cm] Stefano Negro$^{4\musSharp}$ and Alexander ... | {"config": "arxiv", "file": "2106.11999/main.tex"} |
TITLE: How to change $ Cx^2 + Dy^2 + Ex + Fy + G = 0$ to$ (x-h)^2/a^2 ± (y-k)^2/b^2=1 $ using only the variables C, D, E, F, and G
QUESTION [0 upvotes]: Or, state the terms a,b,h,and k in terms of C, D, E, F, and/or G
$Cx^2 + Dy^2 + Ex + Fy + G = 0$
$(x-h)^2/a^2 ± (y-k)^2/b^2=1$
REPLY [0 votes]: $$\begin{array}{rcl}
\... | {"set_name": "stack_exchange", "score": 0, "question_id": 1623932} |
TITLE: Evaluate the partial derivatives of $g(x)$
QUESTION [0 upvotes]: $g\begin{pmatrix}x\\y \end{pmatrix} = \begin{pmatrix} \sqrt{x^2 + y^2} \\ arctan\frac{y}{x} \end{pmatrix}$
I got $g'\begin{pmatrix}x\\y \end{pmatrix} = \begin{pmatrix} 2x\sqrt{x^2 + y^2} & 2y\sqrt{x^2 + y^2} \\ \frac{1}{1 + y/x^2} & \frac{1}{1 + y^... | {"set_name": "stack_exchange", "score": 0, "question_id": 3422375} |
TITLE: Integral closure in an infinite algebraic extension
QUESTION [1 upvotes]: If $A$ is a principal ideal domain and $L/Q(A)$ a finite field extension, then it follows from Krull-Akizuki theorem that the integral closure of $A$ in $L$ is a Dedekind domain. Now if $L/Q(A)$ is an infinite algebraic extension, can we s... | {"set_name": "stack_exchange", "score": 1, "question_id": 4460052} |
TITLE: What does this proof of Fermat's little theorem mean for Euler's theorem?
QUESTION [0 upvotes]: The following proof of Fermat's little theorem is semi-standard: We prove that $a^p-a \equiv 0 \mod p$ by induction on $a.$ For $a = 2,$ we write
$2^p = (1+1)^p = 2 + \sum_{i=1}^{p-1} \binom{p}{i},$ and since each of ... | {"set_name": "stack_exchange", "score": 0, "question_id": 235929} |
In this chapter, we study multi-objective power allocation for energy-efficient SWIPT with separated receivers. In a MISO system, information signals and energy beams are transmitted simultaneously to jointly support information delivery to a information receiver and energy supply to a energy harvester. Under a maximum... | {"config": "arxiv", "file": "1504.02360/2_WIPT_sepuser.tex"} |
TITLE: >Let $X_1,\ldots,X_n$ indepent variable RV $X_n\sim Bern(1/n)$ Does the $X_n\underset{a.s}{\to}0$
QUESTION [0 upvotes]: Let $X_1,\ldots,X_n$ indepent variable RV $X_n\sim Bern(1/n)$
Does the $X_n\underset{a.s}{\to}0$
I tried to use Borel-Cantelli but I get that $\sum_{i=1}^{\infty}\mathbb{P}(A_n^{\epsilon})=\in... | {"set_name": "stack_exchange", "score": 0, "question_id": 3968862} |
TITLE: A magician places $n$ coins on a table and walks down off the stage.
QUESTION [5 upvotes]: A magician places n coins on a table and walks down off the stage. A volunteer comes, turns over whichever coins he wishes, selects one coin and whispers its number to the apprentice. Then the apprentice turns over one coi... | {"set_name": "stack_exchange", "score": 5, "question_id": 3292668} |
TITLE: Probability of randomly choosing all elements fulfilling a certain condition
QUESTION [0 upvotes]: Assume you have a bag containing $m$ marbles, of $c$ different colors, where the number of marbles of each color is equal to $\frac mc$.
If $n$ marbles are drawn from the bag, without replacement, what is the prob... | {"set_name": "stack_exchange", "score": 0, "question_id": 3261641} |
TITLE: Proving big-O notation?
QUESTION [0 upvotes]: $2n^2 \in O(n^2-19n)$
This was proven in my lecture notes but it didn't make sense to me.
I tried solving for c like this:
$n_0 = 1$
$2n^2 ≤ c * n^2 - 19n$
$2 ≤ c * (1-19)$
$2 ≤ c * -18$
$-36 \leq c$, but $c$ has to be a positive constant.
REPLY [1 votes]: Often it ... | {"set_name": "stack_exchange", "score": 0, "question_id": 3219273} |
\section{Introduction \label{sec:introduction}}
Recently, there has been an immense surge of interest in the use of unmanned aerial systems (UASs) for civil applications \cite{Tice91, Debusk10, Amazon16, AUVSI16, BBC16}, which will involve unmanned aerial vehicles (UAVs) flying in urban environments, potentially in clo... | {"config": "arxiv", "file": "1711.02540/introduction.tex"} |
TITLE: Find the density function from a joint density function
QUESTION [0 upvotes]: I try to solve the following task and I don't know what the correct way to do is.
Let $p\in(0,1)$ and $(X,Y)$ be a pair of random variables with distribution density function $$f(x,y)=\frac{1}{2\pi\sqrt{1-p^2}}
\exp\left(-\frac{1}{2(1... | {"set_name": "stack_exchange", "score": 0, "question_id": 1762534} |
TITLE: Homology spheres and fundamental group
QUESTION [16 upvotes]: I have a curiosity about homology spheres: I was wondering if they were uniquely characterized by their fundamental group. I.e. given two $n-$dimensional (integral) homology spheres with isomorphic fundamental groups, are they homeomorphic? If not, ho... | {"set_name": "stack_exchange", "score": 16, "question_id": 211160} |
TITLE: Algorithm behind Sin Function
QUESTION [1 upvotes]: The relationship between the hypothenuse and the opposite cathetus of an angle can be described by $$\sin\theta=\frac{a}{h}$$ , where $a$ is the opposite cathetus, and $h$ is the hypothenuse.
But I am curious about the sine function, it is not a variable and... | {"set_name": "stack_exchange", "score": 1, "question_id": 4378919} |
TITLE: How can we apply this simple eigenvector expression 'repeatedly'?
QUESTION [3 upvotes]: Let $A,B$ be linear operators on a complex vector space $V$ and suppose
$$ABu = (\alpha + 2)Bu$$
where $u \in V$ is an eigenvector of $A$ with eigenvalue $\alpha$ and $\alpha \in \mathbb{C}$. We can interpret this as either $... | {"set_name": "stack_exchange", "score": 3, "question_id": 4263040} |
\begin{document}
\maketitle
\begin{abstract}
We consider the {\em Deligne-Simpson problem (DSP) (resp. the weak DSP):
Give necessary and sufficient conditions upon the choice of the $p+1$
conjugacy
classes $c_j\subset gl(n,{\bf C})$ or $C_j\subset GL(n,{\bf C})$ so that
there exist irreducible $(p+1)$-tuples (res... | {"config": "arxiv", "file": "math0310441.tex"} |
TITLE: Group as direct sum of cyclic groups
QUESTION [2 upvotes]: What are necessary conditions for a cyclic group $G$ to be a direct sum of cyclic groups?
I saw somewhere that $G$ must be a non $p$-group. But I couldn't prove it.
Thank you for your hints/help
REPLY [2 votes]: I think the only necessary condition is t... | {"set_name": "stack_exchange", "score": 2, "question_id": 3187934} |
\section{Spectral Gap Undecidability of a Continuous~Family~of~Hamiltonians}
In this section, we combine the 2D Marker Hamiltonian with the QPE History State construction.
Despite the two-dimensional marker Hamiltonian, the setup is very reminiscent of the 1D construction; the crucial difference being the more finely-g... | {"config": "arxiv", "file": "1910.01631/ch-together.tex"} |
TITLE: Solve $ 11 \cdot 16^{1/(n-1)} = 16^{n/(n-1)} - 10 $
QUESTION [2 upvotes]: This is probably an easy task for the users here, but I could not solve it.
$$
11 \cdot 16^{1/(n-1)} = 16^{n/(n-1)} - 10
$$
Wolfram Alpha gives the result $ n= 5 $.
What are the steps to solve this?
REPLY [6 votes]: $11\times 16^{\frac{1}... | {"set_name": "stack_exchange", "score": 2, "question_id": 257044} |
TITLE: $\int fd\mu=\sup\{\int_{E}fd\mu:E\in S,\mu(E)<+\infty\}$
QUESTION [1 upvotes]: I'm trying to prove the next proposition:
Let $(X,S)$ be a measurable space. Let $f$ be a $S-$measurable function non negative such that $\int fd\mu<+\infty.$ Then
$$\int fd\mu=\sup\{\int_{E}fd\mu:E\in S,\mu(E)<+\infty\}.$$
Because of... | {"set_name": "stack_exchange", "score": 1, "question_id": 2601771} |
\begin{document}
\maketitle
\begin{abstract}
A common problem in applied mathematics is to find a function
in a Hilbert space with prescribed best approximations from a finite
number of closed vector subspaces. In the present paper we study the
question of the existence of solutions to such problems. A finite
famil... | {"config": "arxiv", "file": "0905.3520.tex"} |
TITLE: Night Louder Than The Day
QUESTION [0 upvotes]: Have you guys ever felt that it is quiet at night than in comparison to day?I'm known with the fact that reduction in people's activity makes night quiet...but is there something else which in a way amplifies the sound wave in night?
REPLY [1 votes]: The differenc... | {"set_name": "stack_exchange", "score": 0, "question_id": 410940} |
TITLE: Check whether the following is a subgroup of S4
QUESTION [1 upvotes]: Let $G=S_4$ and $U=\{\sigma \in S_4 | σ^2=(1)\}$
Prove or disprove that $U$ is a subgroup of $G$.
I tried this:
Let $a,b \in U$. Because $(ab)^2 =(1):$
$$ab=(ab)^{-1}=abab=ab(ab)^{-1} =(1)$$
Is my proof totally wrong , can it be saved somehow?... | {"set_name": "stack_exchange", "score": 1, "question_id": 2026371} |
\begin{document}
\title{Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients}
\author[Liu X. e t.~al.]{Xinliang Liu\affil{1}, Lei Zhang\affil{1}\comma\corrauth
and
Shengxin Zhu\affil{2}\comma\affil{3}$^*$
}
\address{\affilnum{1}\ Institute of Natural Sciences, S... | {"config": "arxiv", "file": "2103.01788/rps_main_cicp.tex"} |
TITLE: Chevalley basis for $G_2$
QUESTION [0 upvotes]: I want to find the Chevalley basis for the exceptional group $G_2$.
Could you point to literature where the computation is done in detail or show me how to do it?
REPLY [1 votes]: One can find such a basis in Humphreys' Introduction to Lie Algebras and Representa... | {"set_name": "stack_exchange", "score": 0, "question_id": 1682183} |
TITLE: Three variable, second-degree symmetric Diophantine equation
QUESTION [10 upvotes]: Find integers $f,g,h$ such that $3(f^2+g^2+h^2)=14(fg+gh+hf)$.
You can do it using a computer or by hand.
I tried this problem for ages, got nowhere. Unfortunately I don't know how to program, but I thought it would help a lot he... | {"set_name": "stack_exchange", "score": 10, "question_id": 1133888} |
\begin{definition}[Definition:Well-Ordered Integral Domain/Definition 1]
$\struct {D, +, \times \le}$ is a '''well-ordered integral domain''' {{iff}} the [[Definition:Total Ordering Induced by Strict Positivity Property|ordering $\le$]] is a [[Definition:Well-Ordering|well-ordering]] on the [[Definition:Set|set]] $P$ o... | {"config": "wiki", "file": "def_30031.txt"} |
TITLE: Is a function with everywhere discontinuities of the second kind always measurable?
QUESTION [1 upvotes]: Let $f : [0,1] \to \left\{ 0, 1 \right\}$ be a function that has at each point a discontinuity of the second kind. Is $f$ measurable if we equip the domain with the Borel or even Lebesgue $\sigma$-algebra an... | {"set_name": "stack_exchange", "score": 1, "question_id": 987339} |
TITLE: Inequality Proof by Induction involving Euler Totient function and Summation of Euler's Phi function
QUESTION [4 upvotes]: I want to Prove the following result using induction:
Show that P$_n$:
\begin{equation}
2\sum_{k=1}^n \left(\prod_{p \vert k}\left(1-\frac{1}{p}\right)\right)+2 \geq \frac{12(n-1)^2}{\pi^2}\... | {"set_name": "stack_exchange", "score": 4, "question_id": 4570609} |
TITLE: Element of a ring acting as a permutation on an ideal
QUESTION [4 upvotes]: I am investigating cases when $r \cdot I = I$ for some element $r$ and an ideal $I$ of a commutative ring or rng R.
Clearly, $r \cdot \langle 0 \rangle = \langle 0 \rangle$ for any element $r$ of $R$, and $u \cdot R = R$ for any unit $u$... | {"set_name": "stack_exchange", "score": 4, "question_id": 3538503} |
TITLE: Finding the domain of a natural log composite function
QUESTION [0 upvotes]: If $f(x) = ln(x)$, what is the largest possible domain of $f(f(x))$?
I only know that the input for $ln(x)$ must always be greater than 0, so the domain must be $x$ > $0$ for $ln(x)$ alone. However, once I plug in $f(x)$ into $f(x)$ to... | {"set_name": "stack_exchange", "score": 0, "question_id": 3336661} |
\begin{document}
\title{ Weak-star point of continuity property and Schauder bases }
\author{Gin{\'e}s L{\'o}pez-P{\'e}rez and Jos{\'e} A. Soler Arias}
\address{Universidad de Granada, Facultad de Ciencias.
Departamento de An\'{a}lisis Matem\'{a}tico, 18071-Granada
(Spain)} \email{glopezp@ugr.es, jasoler@ugr.es}
\t... | {"config": "arxiv", "file": "1309.3862.tex"} |
TITLE: Question on Power Spectral Density and Wiener-Khinchin theorem
QUESTION [3 upvotes]: I tried asking this question on stackexchange and I also extensively researched it online without results so I will ask here.
In my textbook the Wiener-Khinchin theorem is used to connect the auto-correlation definition of PSD ... | {"set_name": "stack_exchange", "score": 3, "question_id": 297866} |
TITLE: Is Von Neumann's function application operation defined for all functions and all arguments?
QUESTION [1 upvotes]: Von Neumann's original axioms of what became von Neumann–Bernays–Gödel set theory took as primitive notions function and argument.
Accompanying these was a two-variable operation, denoted $[x,y]$, c... | {"set_name": "stack_exchange", "score": 1, "question_id": 2739972} |
TITLE: Given $f'(x)=\csc{x}(\cot{x}–\sin{2x})$, find $f(x)$
QUESTION [1 upvotes]: I have a question on my integration assignment that I am not quite sure how to approach. I've looked at it and can't seem to think of a suitable place to use u-substitution.
All I can figure is that I can expand it out to be $f'(x)=\csc{x... | {"set_name": "stack_exchange", "score": 1, "question_id": 298153} |
TITLE: Countable partition in atoms
QUESTION [4 upvotes]: Let $\mu: \Sigma \to [0, \infty)$ a measure over $\Omega$. We say a set $A \in \Sigma$ is an atom if for all $B \in \Sigma$ with $B \subset A$, $\mu(B)=\mu(A)$ or $\mu(B)=0$. We say that $\mu$ is atomic if every set $A \in \Sigma$ with positive measure contains ... | {"set_name": "stack_exchange", "score": 4, "question_id": 2025975} |
TITLE: Understanding vector space
QUESTION [0 upvotes]: I was reading an example from a textbook giving a vector space. It goes with all properties to check if it is a vector space, and finally concludes it is. I am worried about existence of zero.
Now, let V be R(+) and define addition and scalar multiplication as fol... | {"set_name": "stack_exchange", "score": 0, "question_id": 3022767} |
TITLE: Is indefinite integration non-linear?
QUESTION [27 upvotes]: Let us consider this small problem:
$$
\int0\;dx = 0\cdot\int1\;dx = 0\cdot(x+c) = 0
\tag1
$$
$$
\frac{dc}{dx} = 0 \qquad\iff\qquad
\int 0\;dx = c, \qquad\forall c\in\mathbb{R}
\tag2
$$
These are two conflicting results. Based on this other question, S... | {"set_name": "stack_exchange", "score": 27, "question_id": 1069664} |
TITLE: Spivak problem on Schwarz inequality
QUESTION [2 upvotes]: I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality:
$$
x_1y_1 + x_2y_2 \leq \sqrt{x_1^2+x_2^2}\sqrt{y_1^2+y_2^2} \ .
$$
It says to prove that if $x_1=\lambda y_... | {"set_name": "stack_exchange", "score": 2, "question_id": 392070} |
TITLE: Find $G$ and $H$ such that $dF$ has given form
QUESTION [1 upvotes]: Let $F$ be a cumulative distribution function and $$dF(x)=\begin{cases}dx/3&x\in (0,1)\cup(2,3)\\1/6&x \in \{1,2\}\\0&\mathrm{elsewhere}\end{cases}$$
Find a continuous cdf $H$, a discrete cdf $G$ and a real constant $c$ such that $$F(x)=cG(x)+(... | {"set_name": "stack_exchange", "score": 1, "question_id": 3884457} |
TITLE: Integral points on varieties
QUESTION [19 upvotes]: I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties.
For example, consider $X = \mathbb{A}_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine $n$-space over $\mathbb{Z}$ with the origin removed. Naively,... | {"set_name": "stack_exchange", "score": 19, "question_id": 28485} |
TITLE: Every neighbourhood of $1$ is of form $U\setminus \mathbb{N} \cup \{1\}$?
QUESTION [1 upvotes]: From Topology without tears:
Let $X$ be a set $(\mathbb{R}\setminus\mathbb{N})\cup {1}$.
Define a function $f:\mathbb{R}\to X$:
\begin{equation}
f(x) =
\begin{cases}
x & \text{if $x \in \mathbb{R}\setminus\m... | {"set_name": "stack_exchange", "score": 1, "question_id": 4050216} |
TITLE: Get a third point (lat, lng) from two given
QUESTION [0 upvotes]: I have two points as follow (the distance between them is variable):
I need to get a third as shown:
The two first points change all the time, including the distance between them. My problem: I have many points in a road got from a GPS, and a ph... | {"set_name": "stack_exchange", "score": 0, "question_id": 842634} |
\begin{document}
\title{The $L^2$ restriction norm of a Maass form on $SL_{n+1}(\mathbb{Z})$.}
\author{Xiaoqing Li}
\address{Deptartment of Mathematics \\
State University of New York at Buffalo \\
Buffalo, NY, 14260
}
\email{XL29@buffalo.edu}
\author{Sheng-Chi Liu}
\author{Matthew P. Young}
\address{Department of M... | {"config": "arxiv", "file": "1212.4002/GLnRestrictionSubmit3.tex"} |
TITLE: Shortest path in linear time
QUESTION [3 upvotes]: Suppose each edge can receive one of two weights $\{r_1,r_2\}$ where $r_1$ and $r_2$ are real and non-negative. And suppose $r_1 \leq r_2$. How do you find the shortest path from a given vertex s to every other vertex in the graph in linear time? ($O(V+E)$)
REP... | {"set_name": "stack_exchange", "score": 3, "question_id": 97244} |
TITLE: Prove $ \int _0^1\:e^xf\left(1-x\right)dx=\int _0^1\:e^x\left(f′\left(1-x\right)\right)dx$.
QUESTION [0 upvotes]: I'm having trouble with following problem, thinking about integration by parts but just getting circular answer:
Let $f$ by continuous on $[0,1]$ and differentiable on $(0,1)$, and also $f(0)=f(1)=0... | {"set_name": "stack_exchange", "score": 0, "question_id": 2440456} |
\begin{document}
\title[ Controlled rectangular metric type spaces and some applications to polynomial equations]
{Controlled rectangular metric type spaces and some applications to polynomial equations}
\author[Nabil Mlaiki]
{Nabil Mlaiki}
\address{Nabil Mlaiki \newline
Department of Mathematics and General Scien... | {"config": "arxiv", "file": "1910.13704.tex"} |
TITLE: Continuity of the inverse map
QUESTION [1 upvotes]: If we have a function $F(x): \mathbb{R^4} \rightarrow \mathbb{R^3}$. Defined as
\begin{align}
x_1\, x_4&=y_1 \\
x_2\, x_4&=y_2 \\
x_1^2+x_2^2-x_3^2&=y_3
\end{align}
Can a continuous inverse map exist? I'm intuitively guessing that the problem with continuity ca... | {"set_name": "stack_exchange", "score": 1, "question_id": 1386721} |
TITLE: Traffic lights probability
QUESTION [0 upvotes]: Ive been asked the following :
Two consecutive traffic lights have been synchronized to make a run of green lights more likely. In particular, if a driver finds the first light to be red, the second light will be green with probability 0.9, and if the first light... | {"set_name": "stack_exchange", "score": 0, "question_id": 2993630} |
TITLE: Homotopy equivalence in terms of strong deformation retract
QUESTION [2 upvotes]: Visualizing homotopy equivalence maps are not so easy. I thought before that $f:X\to Y$ and $g:Y\to X$ are homotopy equivalence iff one can deform $X$ continuously to $Y$. But this is wrong in general. So I tried the following:
Q1:... | {"set_name": "stack_exchange", "score": 2, "question_id": 3849785} |
\begin{document}
\title{Extinction probabilities of branching processes with countably infinitely many types}
\authorone[The University of Melbourne]{S. Hautphenne}
\authortwo[Universit\'e libre de Bruxelles]{G. Latouche}
\authorthree[The University of Adelaide]{G. T. Nguyen}
\addressone{Department of Mathemati... | {"config": "arxiv", "file": "1211.4129/HLN_multitypeBP.tex"} |
TITLE: How to guarantee if a real function is $C^\infty$ without using $f^{(n)}$
QUESTION [0 upvotes]: I know that function $f(x)=x+\frac{1}{1+e^x}$ is $C^\infty$ but I wanna prove that.
Question
Is there something that I could use to guarantee this without actually calculating $f^{(n)}$ ?
REPLY [2 votes]: It is the s... | {"set_name": "stack_exchange", "score": 0, "question_id": 3456321} |
\begin{document}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\newenvironment{proof}{{\bf Proof}.}{\hspace{3mm}\rule{3mm}{3mm}}
\newenvironment{mainproof}{{\bf Proof of Theorem 7}.}{\hspace{3mm}\rule{3mm}{3mm}}
\newenvironment{cor1proof}{{\bf Proof of Corollary 1}.}{\hspace{... | {"config": "arxiv", "file": "1012.1231/0201124.tex"} |
\section{Numerical results}\label{sec:results}
We now present the results for a series of simulations chosen to demonstrate the properties of our numerical approach. We first consider the INS equations on a sequence of refined unit square meshes to study the accuracy of the scheme; cases with and without the divergen... | {"config": "arxiv", "file": "1902.06773/texFiles/results.tex"} |
TITLE: I get 2 possibilities for one problem. In first one, x=y for an equation but in another one x is not equal to y for same equation. Why?
QUESTION [0 upvotes]: Equation is
$$9x^2+6x-5=9y^2+6y-5$$
I was solving this to prove $x=y$ for one-one function. Although I proved it but here
$$ (x-y)(9x+9y+6)=0$$
Both sides ... | {"set_name": "stack_exchange", "score": 0, "question_id": 4190295} |
\begin{document}
\fontsize{12pt}{14pt} \textwidth=14cm \textheight=21 cm
\numberwithin{equation}{section}
\title{Estimates of sections of determinant line bundles on Moduli spaces of pure sheaves on algebraic surfaces}
\author{Yao YUAN}
\date{\small\textsc SISSA, Via Bonomea 265, 34136, Trieste, ITALY
\\ yuayao@gmail.c... | {"config": "arxiv", "file": "1010.1815.tex"} |
TITLE: Understanding the $\mathcal{L}^0(\mu)$ space
QUESTION [1 upvotes]: After having introduced the $\mathcal{L}^p(\mu)$ spaces for $p\in(0,\infty)$, my book has a little remark that we can expand the definition to $p=0$.
Let $(X,\mathcal{E},\mu)$ be a measure space.
Let $\mathcal{M}(\mathcal{E})=\{f:X\to \mathbb{R}\... | {"set_name": "stack_exchange", "score": 1, "question_id": 3410955} |
TITLE: What is the probability that exactly two of them are trout if we know that at least three of them are not?
QUESTION [1 upvotes]: In a small lake, it is estimated that there are 105 fish, of which 40 are trout and 65 are of another species. A fisherman catches 8 fish. What is the probability that exactly two of t... | {"set_name": "stack_exchange", "score": 1, "question_id": 2746456} |
TITLE: How to show that a multivariable function is not differentiable?
QUESTION [0 upvotes]: How would I show that $$f(x,y) = \frac{2xy}{x^2 + y^2}$$
is not differentiable at the origin?
Is it enough to show that as the function tends to the origin along the paths $y = x$ and $y=2x$ that we get different limits and h... | {"set_name": "stack_exchange", "score": 0, "question_id": 1488049} |
\begin{document}
\title[A surjection theorem]{A surjection theorem for maps with singular perturbation and loss of derivatives}
\author[I. Ekeland]{Ivar Ekeland}
\address{Universit\'e Paris-Dauphine, PSL University, CNRS, CEREMADE, Place de Lattre de Tassigny, 75016 Paris, France}
\email{ekeland@ceremade.dauphine.fr... | {"config": "arxiv", "file": "1811.07568.tex"} |
TITLE: How to do modular arithmetic with a negative n
QUESTION [0 upvotes]: Playing with Python and the mod operation I encountered that (5 % -3) = -1. This is confirmed by WolframAlpha, and I have not been able to find any simple explanation for this online, mostly because all I can find about modular arithmetic uses ... | {"set_name": "stack_exchange", "score": 0, "question_id": 3438107} |
TITLE: Eigenfunctions for the symmetric kernel of an integral equation
QUESTION [1 upvotes]: The solution of the symmetric integral equation below:
$$g(s) = f(s) + \lambda \int_{-1}^{1} (st +s^2t^2)g(t)dt \tag{$*$}$$
with separable kernels method is
$$g(s) = f(s) + \lambda \int_{-1}^{1} (\frac{st}{(1-\frac{2}{3}\lambda... | {"set_name": "stack_exchange", "score": 1, "question_id": 1600936} |
TITLE: Do I have these recursive and closed forms correct?
QUESTION [0 upvotes]: For the sequence: $0,1,5,12,22,35,51,70,92,117,145,176$
I have the closed form (dashes indicate subtext): $$a_n=\frac{n(3n-1)}{2}$$
For recursive: $$a_{n+1}=\frac{a_n+3n^2+5n+2}{2}$$
If they are wrong, please explain how to solve.
REPLY [... | {"set_name": "stack_exchange", "score": 0, "question_id": 1709179} |
TITLE: Topology on the space of foliations
QUESTION [7 upvotes]: Let $(M^3,g)$ be a closed Riemannian manifold. Is there a “natural” topology on the space $\operatorname{Fol}(M)$ of smooth codimension $1$ foliations on $M$? Is there any other relevant structure on this set?
REPLY [4 votes]: A smooth codimension 1 foli... | {"set_name": "stack_exchange", "score": 7, "question_id": 3728363} |
\begin{document}
\title[Strichartz estimates w/o loss outside many convex obstacles]{Strichartz estimates without loss outside many strictly convex obstacles}
\author{David Lafontaine {*}}
\thanks{{*} d.lafontaine@bath.ac.uk, University of Bath, Department of Mathematical
Sciences}
\begin{abstract}
We prove Strichartz ... | {"config": "arxiv", "file": "1811.12357/many_pre_fin.tex"} |
TITLE: Pugh's exercise on Dedekind cuts addition
QUESTION [0 upvotes]: I am trying to solve the following exercise:
Let $x=A|B$ and $x'=A'|B'$ be cuts in $\mathbb{Q}$. Show that although $B+B'$ is disjoint from $A+A'$, it may happen in degenerate cases that $\mathbb{Q}$ is not the union of $A+A'$ and $B+B'$.
The first ... | {"set_name": "stack_exchange", "score": 0, "question_id": 1869911} |
TITLE: Let $E \subset M$ and $x \in M$ then $x$ is a limit point of $E$ iff every $B_r(x)$ contains at least one point of $E$
QUESTION [1 upvotes]: Let $E \subset \left<M,\rho\right>$ and $x \in M$ then $x$ is a limit point of $E$ iff
every $B_r(x)$ contains at least one point of $E$
I am reading Methods of Real Ana... | {"set_name": "stack_exchange", "score": 1, "question_id": 3367645} |
TITLE: Paradigm shifts from $2\to 3$
QUESTION [3 upvotes]: Recently, I've been thinking about a common theme that I've seen all over mathematics. One often finds, that when the number of dimensions/degrees of freedom in a given scenario/problem changes from $2$ to $3$, that some fundamental shifts in the solution or re... | {"set_name": "stack_exchange", "score": 3, "question_id": 4131334} |
\begin{document}
\title[The maximum of the zeta function on the 1-line]{A note on the maximum of the Riemann zeta function on the 1-line}
\author{Winston Heap}
\address{Department of Mathematics, University College London, 25 Gordon Street, London WC1H.}
\email{winstonheap@gmail.com}
\thanks{Research supported by Europ... | {"config": "arxiv", "file": "1812.01415.tex"} |
TITLE: Creating gradient functions based on model parameters?
QUESTION [1 upvotes]: I am using a software library (Math.Net) to try to fit two Lorentzians to a curve. I have found some example software which shows the fitting out a few various types of curves (Line, Parabola, Power Function, and a Sum of Trigonometric ... | {"set_name": "stack_exchange", "score": 1, "question_id": 1769993} |
TITLE: Does there exist a way to find the sum of the digits of the result from a large cubic root?
QUESTION [1 upvotes]: The problem is as follows:
Find the sum of the digits of $B$:
$B=\sqrt[3]{1224 \times 1225\times 1226+35\sqrt[3]{34\times35\times36+35}}$
The possible answers given in my book are as follows:
$\begin... | {"set_name": "stack_exchange", "score": 1, "question_id": 3588675} |
TITLE: Reference request: an elementary result on characters of finite abelian groups
QUESTION [6 upvotes]: The referee of a paper I submitted to a journal asked me to include a reference of the following elementary result on characters of finite abelian groups:
Let $A$ be a finite abelian group of order $N$ and let
$\... | {"set_name": "stack_exchange", "score": 6, "question_id": 312576} |
TITLE: logical quantifiers on sets question
QUESTION [0 upvotes]: I have to prove whether or not these statements are true/false but I'm having trouble understanding it.
$\forall x \in \mathbb{Z}^5, \forall y \in \mathbb{Z}^5. \exists z \in \mathbb{Z}^5, \forall j \in \{1,2,3,4,5\},x_{j} \leq z_{j} \leq y_{j}$.
I thin... | {"set_name": "stack_exchange", "score": 0, "question_id": 2161397} |
\begin{document}
\title[]{Index and nullity of proper biharmonic maps in spheres}
\author{S.~Montaldo}
\address{Universit\`a degli Studi di Cagliari\\
Dipartimento di Matematica e Informatica\\
Via Ospedale 72\\
09124 Cagliari, Italia}
\email{montaldo@unica.it}
\author{C.~Oniciuc}
\address{Faculty of Mathematics\\ `... | {"config": "arxiv", "file": "1902.01621.tex"} |
\begin{document}
\maketitle
\begin{abstract}
We characterize fundamental domains of affine reflection groups as those polyhedral convex bodies which support a continuous billiard dynamics. We interpret this characterization in the broader context of Alexandrov geometry and prove an analogous characterization for isos... | {"config": "arxiv", "file": "2202.11624/Billiards.tex"} |
TITLE: Complete Linear system on Del Pezzo surfaces
QUESTION [2 upvotes]: Is there always a reducible curve (EDIT: with exactly two irreducible components intersecting in at least 2 points) in a complete linear system (EDIT: of dimension at least 2 with curves of genus at least 1) on a Del Pezzo surface?
REPLY [7 vote... | {"set_name": "stack_exchange", "score": 2, "question_id": 144638} |
TITLE: Normal subgroup of $A_4$
QUESTION [1 upvotes]: Prove that $\{id, (12)(34), (13)(24), (14)(23)\}$ is a normal subgroup of $A_4$.
I see that it's possible to prove this with the following theorem: $$N \trianglelefteq G \text{ if } N\leq G \text{ and } \forall n\in N, g\in G: g^{-1}ng \in N.$$ But is there an easi... | {"set_name": "stack_exchange", "score": 1, "question_id": 3949412} |
TITLE: Simple inequality with a,b,c
QUESTION [3 upvotes]: I'm looking for proof of
$$\sqrt{a(b+c)}+\sqrt{b(c+a)}+\sqrt{c(a+b)} \leq \sqrt{2}(a+b+c)$$
I tried using $m_g \leq m_a$, generating the permutations of $\sqrt{a(b+c)} \leq \frac{a+(b+c)}{2}$ and adding them together, but I get $\frac{3}{2}(a+b+c)$, and obviousl... | {"set_name": "stack_exchange", "score": 3, "question_id": 1489533} |
TITLE: Compact operator on $l^2$
QUESTION [4 upvotes]: Let $A$ be a bounded linear operator on $l^2$ defined by $A(a_n)= \left(\frac{1}{n} a_n\right)$. Would you help me to prove that $A$ is compact operator. I guess the answer using an approximation by a sequences of finite range operator.
REPLY [6 votes]: Let $$T_k(... | {"set_name": "stack_exchange", "score": 4, "question_id": 229602} |
TITLE: Can the gauge boson respective of a spontaneously broken generator remain massless in the context of the Higgs Mechanism?
QUESTION [3 upvotes]: I'm studying a 3-3-1 model which is an extension of the standard model. The breaking $$SU(3)\times U(1) \to SU(2)\times U(1)'$$ occurs in a single step and through a sin... | {"set_name": "stack_exchange", "score": 3, "question_id": 653048} |
\section{The menagerie: notations, background, and constructions}
In this section, we will lay out the fundamental definitions and constructions that will be used in the proof of the main result. Along the way, we will also prove basic relations between these definitions, to alleviate the density of later arguments.
\... | {"config": "arxiv", "file": "1905.06671/menagerie.tex"} |
TITLE: Discrete Proof with Irrational Numbers
QUESTION [0 upvotes]: I've been stuck on this problem below and have a few more identical to it on my assignment. And yes I've Googled around and found a few links on here regarding to the density of $\mathbb{Q} \subseteq \mathbb{R}$, but I'm just not getting it.
Can somebo... | {"set_name": "stack_exchange", "score": 0, "question_id": 2459728} |
\begin{document}
\title{Landis-type conjecture for the half-Laplacian}
\author{Pu-Zhao Kow}
\address{Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyv\"{a}skyl\"{a}, Finland.}
\email{pu-zhao.pz.kow@jyu.fi}
\author{Jenn-Nan Wang}
\address{Institute of Applied Mathematical Sciences, ... | {"config": "arxiv", "file": "2106.06120/arXiv-preprint_ver4_half-Lap-with-drift.tex"} |
TITLE: If $abc\neq 0$, then $ \frac{(a+b)^2}{c^2}+\frac{(a+c)^2}{b^2}+\frac{(b+c)^2}{a^2}\geq2 $
QUESTION [2 upvotes]: Let $a$, $b$ and $c$ be real numbers such that $abc\neq0$. Prove that:
$$ \frac{(a+b)^2}{c^2}+\frac{(a+c)^2}{b^2}+\frac{(b+c)^2}{a^2}\geq2 $$
I checked for some values and it seems to be true. But no p... | {"set_name": "stack_exchange", "score": 2, "question_id": 836986} |
TITLE: question in discrete mathematics
QUESTION [1 upvotes]: I have questions. Can anyone help me to get the idea or figure out this problem.
Find a recurrence relation.
If an denote the number of words from the alphabet W={A,B,C} of length n with no two adjacent letters being A'S.
thank you very much
REPLY [1 votes... | {"set_name": "stack_exchange", "score": 1, "question_id": 320182} |
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