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\begin{document}
\title[On the Digraph of a Unitary Matrix]{On the Digraph of a Unitary Matrix
}
\author{Simone Severini}
\address{Computer Science, Univ. Bristol, Bristol, U.K.}
\email{severini@cs.bris.ac.uk}
\date{May 2002: Published in SIAM Journal on Matrix Analysis and
Applications, Volume 25, Number 1, pp. 295-30... | {"config": "arxiv", "file": "math0205187.tex"} |
\begin{document}
\title[Parametrized topological complexity]
{Parametrized topological complexity of collision-free motion planning in the plane}
\author[D. Cohen]{Daniel C. Cohen}\thanks{D. Cohen was partially supported by an LSU Faculty Travel Grant}
\address{Department of Mathematics, Louisiana State University, B... | {"config": "arxiv", "file": "2010.09809.tex"} |
TITLE: Definite integrals
QUESTION [1 upvotes]: $$\int_0^{1.5}[x^2]dx$$where [.] denotes the greatest integer function, is equal to :
(1) $\sqrt{2}-2$
(2) $2 –\sqrt{2}$
(3) $2 + \sqrt{2}$
(4) None of these
What I did, I broke the function into two parts..one with limits from 0 to 1.the problem is how should I deal wit... | {"set_name": "stack_exchange", "score": 1, "question_id": 789980} |
TITLE: Two from cubic subgraph hardness
QUESTION [2 upvotes]: The Problem
For a given graph $G$, the cubic subgraph problem asks if there is a subgraph where every vertex has degree 3.
The cubic subgraph problem is NP-hard even in bipartite planar graphs with maximum degree at most 4.
Suppose we have an oracle that dec... | {"set_name": "stack_exchange", "score": 2, "question_id": 374391} |
\onecolumn
\begin{center} { \Large Supplementary Material for:\vspace{2mm} \\
Adaptive Kernel Learning in Heterogeneous Networks}\vspace{2mm} \\
by Hrusikesha Pradhan, Amrit Singh Bedi, Alec Koppel, and Ketan Rajawat
\end{center}
\section{Proof of Corollary \ref{thm:representer} }\label{proof_representthm}
The pr... | {"config": "arxiv", "file": "1908.00510/Supplementary.tex"} |
TITLE: How do I prove inconsistency in FOL?
QUESTION [1 upvotes]: So I have to prove that this set S is inconsistent:
S = {{P(x),P(f(a)), ¬Q(z)}, {P(g(x’,x)),Q(x)},{¬P(y)}}
I just have no idea where to start. The only time I learned about inconsistency is when I learned about Skolem form, but I got no idea how to prov... | {"set_name": "stack_exchange", "score": 1, "question_id": 2210460} |
TITLE: What are the algebras for the ultrafilter monad on topological spaces?
QUESTION [16 upvotes]: Motivation: Let $(X,\tau)$ be a topological space. Then the set $\beta X$ of ultrafilters on $X$ admits a natural topology (cf. Example 5.14 in Adámek and Sousa - D-ultrafilters and their monads), giving rise to a funct... | {"set_name": "stack_exchange", "score": 16, "question_id": 348574} |
TITLE: Let $X$ be a connected metric space and $A, B \subset X$ Show that $d(x,A) = d(x,B)$ doesn't hold without the connectedness assumption.
QUESTION [3 upvotes]: Let $X$ be a connected metric space and $A, B \subset X$ non-empty sets. (i) Show that there exists $x \in X$ such that $d(x,A) = d(x,B).$ (ii) Give an exa... | {"set_name": "stack_exchange", "score": 3, "question_id": 4128472} |
TITLE: Combinatorial question about sets of rational numbers
QUESTION [12 upvotes]: The following question came up in my research. Since lots of clever people post here, I thought I'd ask it.
Recall that the group ring of a group $G$ is the abelian group $\mathbb{Z}[G]$ consisting of linear combinations of formal symb... | {"set_name": "stack_exchange", "score": 12, "question_id": 125306} |
TITLE: Function is equal to its own derivative
QUESTION [7 upvotes]: We all know that derivative of $e^x$ is $e^x$. Is exponential function only function that has such property? If yes how to prove that there are no other functions. If no, what are other functions? Help me please
REPLY [12 votes]: You seek to solve th... | {"set_name": "stack_exchange", "score": 7, "question_id": 644879} |
\begin{definition}[Definition:Completely Additive Function]
Let $\left({R, +, \times}\right)$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $f: R \to R$ be a [[Definition:Mapping|mapping]] on $R$.
Then $f$ is described as '''completely additive''' {{iff}}:
:$\forall m, n \in R: f \left({m \times n}\right) = f \... | {"config": "wiki", "file": "def_21031.txt"} |
TITLE: Prove that $\lim_{x\to a}f(x) = L$ if and only if $\lim_{x\to a^-}f(x) = L$ and $\lim_{x\to a^+}f(x) = L$
QUESTION [0 upvotes]: First, I mentioned that $\lim_{x\to a^-}f(x) = L$ if there exists a $\delta > 0$ such that $a-x < \delta$ where $|f(a)-f(x)| < \epsilon$.
And that $\lim_{x\to a^+}f(x) = L$ if there ex... | {"set_name": "stack_exchange", "score": 0, "question_id": 1557246} |
TITLE: Is it OK to see time dilation and (relativistic) mass increase as phenomena that avoid $c$ being reached? And how about length contraction?
QUESTION [0 upvotes]: I think I have been exposed since years ago to this line of reasoning:
if $ v\to c $, then $ \Delta t \to \infty $. As $\displaystyle v=\frac{\Delta s}... | {"set_name": "stack_exchange", "score": 0, "question_id": 146038} |
TITLE: Why are monotone functions Riemann integrable on a closed interval?
QUESTION [0 upvotes]: Monotone functions are continuous except countably many points. If function is Riemann integrable it has only a finite number of discontinuity points. So how monotone functions are Riemann integrable on closed interval alwa... | {"set_name": "stack_exchange", "score": 0, "question_id": 920833} |
TITLE: Prove that $1\cdot 1! + 2\cdot 2! +\dots+n\cdot n! = (n + 1)! - 1$
QUESTION [0 upvotes]: (whenever $n$ is a non-negative integer)
I did the basic step $P(1)$ and found the statment $P(n+1)$
I now have $(n+1)! - 1 + (n+1)\cdot(n+1)!$
This should equal $(n+2)! - 1$, but how do I show that?
REPLY [2 votes]: $$(n+... | {"set_name": "stack_exchange", "score": 0, "question_id": 1977040} |
\begin{document}
\title{Sub-Nyquist Sampling for Power Spectrum Sensing in Cognitive Radios: A Unified Approach}
\author{Deborah Cohen, \emph{Student IEEE}
and Yonina C. Eldar, \emph{Fellow IEEE}}
\maketitle
\begin{abstract}
In light of the ever-increasing demand for new spectral bands and the underutilizatio... | {"config": "arxiv", "file": "1308.5149/power_spectrum.tex"} |
\begin{document}
\maketitle
\thispagestyle{empty}
\begin{abstract}
This paper proposes a ``quasi-synchronous'' design approach for signal processing circuits, in which timing violations are permitted, but without the need for a hardware compensation mechanism.
The case of a low-density parity-check (LDPC) decoder is s... | {"config": "arxiv", "file": "1503.03880/LDPC_energy_optim_JNL-rev.tex"} |
TITLE: Explaining The Unbelievable Pendulum Catch
QUESTION [20 upvotes]: What would be a theoretical explanation of an "ideal" 14:1 mass ratio in this experiment, also demonstrated in this video?
The experiment ties one nut to one end of the string and 14 nuts to the other, then holds the string like this and lets go:
... | {"set_name": "stack_exchange", "score": 20, "question_id": 537834} |
TITLE: Sufficient condition for $k$-colorability
QUESTION [4 upvotes]: We know that a graph is $ 2 $-colorable iff it has no odd cycles. I am asked to generalize this statement to the following: a graph is $ k $-colorable if each vertex is in less than $ \binom{k}{2} $ distinct odd cycles.
I am having trouble with this... | {"set_name": "stack_exchange", "score": 4, "question_id": 2520571} |
\begin{document}
\maketitle
\begin{abstract}
We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tail... | {"config": "arxiv", "file": "1908.01181/weighted-sum.tex"} |
TITLE: Show $h_\mu (f^n)=n*h_\mu(f)$
QUESTION [0 upvotes]: In some article about ergodic theory, it said (without proof) that if $\mu$ is a $f$-invariant measure, then $h_\mu(f^k)=k*h_\mu(f)$. I'm looking to prove this.
$h_\mu(f^k,A)=\lim_n (1/n)*H_\mu(\bigvee_{i=0}^{n-1} f^{-ik}A)$, so in some manner we should have $H... | {"set_name": "stack_exchange", "score": 0, "question_id": 1948489} |
TITLE: How to find an upper bound on the number of solutions of $y^3=x^2+4^k$
QUESTION [1 upvotes]: I have solved the first two parts of this question but I am struggling with the remaining section. I can't see any meaningful way to reuse what I did before and/or find a way forward.
Just to be clear it is part c) that ... | {"set_name": "stack_exchange", "score": 1, "question_id": 2170088} |
TITLE: Why is the two sheeted cone not a regular surface?
QUESTION [0 upvotes]: The two sheeted cone is $\{(x,y,z) \in \mathbb R^3 : x^2+y^2-z^2=0\}$.
I would like to use this proposition:
to show that the two sheeted cone is not a regular surface. I know that the point of failure has to be the $(0,0,0)$, but I am n... | {"set_name": "stack_exchange", "score": 0, "question_id": 3088982} |
\section{Transferring vanishing lines}\label{sec:Transferring}
In this section we explain under which circumstances vanishing lines for $E_k$-homology imply vanishing lines for $E_{k-1}$-homology (``transferring down'') or $E_{k+1}$-homology (``transferring up''). Transferring up using the bar constructions of the pr... | {"config": "arxiv", "file": "1805.07184/chap14.tex"} |
TITLE: How to express the set of intersections between two ordered sets by selecting exactly one element per index?
QUESTION [0 upvotes]: Given a set $\mathcal{S} = \{S_1, S_2, S_3\}$, two ordered sets can be produced:
$\mathcal{S}^+$, where $S^+_i \in (S_1,S_2,S_3)$, and
$\mathcal{S}^-$, where $S^-_i \in (U\setminus S... | {"set_name": "stack_exchange", "score": 0, "question_id": 2837314} |
TITLE: Pythagorean triplets
QUESTION [6 upvotes]: Respected Mathematicians,
For Pythagorean triplets $(a,b,c)$, if $c$ is odd then any one of $a$ and $b$ is odd. Here $(a, b, c)$ is a Pythagorean triplet with $c^2 = a^2 + b^2$.
Now, I will consider $c = b + \Omega$. The reason for considering $c = b + \Omega$ is, $c$... | {"set_name": "stack_exchange", "score": 6, "question_id": 101550} |
TITLE: tangent space at origin of a variety
QUESTION [2 upvotes]: Could any one explain me how to show that the tangent space at origin of the variety $V=\mathbb{V}(y^2-x^3)$ is equal to full affine plane? They have defined $l$ is a tangent line at $p$ if the multiplicity of $l\cap V$ at $p$ exceeds one. The tangent ... | {"set_name": "stack_exchange", "score": 2, "question_id": 143948} |
TITLE: ODE compartmental model: waiting time
QUESTION [1 upvotes]: Here is a ODE compartmental model made of 3 equations :
$\frac{dX}{dt}=-\alpha X$
$\frac{dY}{dt}=\alpha X-\beta Y$
$\frac{dZ}{dt}=\beta Y$
$X$, $Y$, $Z$ represents, in my case, the total number of people being in the state/compartment/case $X$, $Y$ or $... | {"set_name": "stack_exchange", "score": 1, "question_id": 2540988} |
TITLE: What formula do I use for factoring these?
QUESTION [0 upvotes]: An elementary question, but I am having a lot of discrepancies identifying the correct formula to use, I can do more complex ones but not the simple ones if that makes sense.
a) $8x^3 + 1$
b) $m^2 - 100n^2$
Thank you, regards.
REPLY [1 votes]: $\!... | {"set_name": "stack_exchange", "score": 0, "question_id": 731959} |
TITLE: Asymptotic boundary on Fourier coefficients of absolutely continuous function
QUESTION [1 upvotes]: Let $f$ be absolutely continuous. Prove that $\hat{f}(n)=o\left(\frac{1}{n}\right)$.
Any hint will be appreciated, thanks.
REPLY [1 votes]: Hint:
Absolute continuous means that $f'\in L^1$. What can you say abo... | {"set_name": "stack_exchange", "score": 1, "question_id": 90197} |
TITLE: Given that $\cos\left(\dfrac{2\pi m}{n}\right) \in \mathbb{Q}$ prove $\cos\left(\dfrac{2\pi}{n}\right) \in \mathbb{Q}$
QUESTION [4 upvotes]: Given that $\cos\left(\dfrac{2\pi m}{n}\right) \in \mathbb{Q}$, $\gcd(m,n) = 1$, $m \in \mathbb{Z}, \, n \in \mathbb{N}$ prove that $\cos\left(\dfrac{2\pi}{n}\right) \in \m... | {"set_name": "stack_exchange", "score": 4, "question_id": 4023263} |
TITLE: PM is supersolvable group
QUESTION [1 upvotes]: $G$ is a finite group, $G = PM$, where $P$ is a Sylow $p$-subgroup of $G$, $p$ is the largest prime dividing the order of $G$, $P$ is normal in $G$, $M$ is maximal subgroup of $G$, $M$ is supersolvable and $|G/M| = p$.
Is $G$ supersolvable?
(The group $G$ is said ... | {"set_name": "stack_exchange", "score": 1, "question_id": 501114} |
TITLE: $X$ is the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$, and $M=\{f\in X:f(0)=0\}$, show that $M$ is not closed.
QUESTION [1 upvotes]: Here is my question:
Let $X$ be the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$. Let
$$M=\{f\in X:f(0)=0\}$$
Show that $M$ is not closed. Sho... | {"set_name": "stack_exchange", "score": 1, "question_id": 999293} |
TITLE: How can I prove, that the 2nd Bergman space is a Hilbert space?
QUESTION [0 upvotes]: We consider the analytic functions on an open set $U\subset\mathbb{C}$, which are also in $L^2(U)$. I've found several posts in this topic, but all of them references to classical complex analysis results.
It's obvious, that th... | {"set_name": "stack_exchange", "score": 0, "question_id": 1922611} |
TITLE: Is "False" in logic analogous to "Null set" in set theory?
QUESTION [4 upvotes]: I have been doing proofs in elementary set theory, and so far, just using definitions (like below) and applying propositional logic has sufficed.
A ⋃ B = e ∈ A ∨ e ∈ B
A ⊂ B = e ∈ A ⟹ e ∈ B
A' = e ∉ A = ¬(e ∈ A)
So the proofs are ... | {"set_name": "stack_exchange", "score": 4, "question_id": 3664146} |
TITLE: Calculation of heat flux on a surface
QUESTION [0 upvotes]: I have a basic question about calculating heat flux applied to a surface.
Suppose you have a solid cylinder that has a height of 1 $\text{cm}$ and a radius of 1 $\text{cm}$, giving it a lateral surface area of 2π $\text{cm}^2$.
You take a heat tape devi... | {"set_name": "stack_exchange", "score": 0, "question_id": 652102} |
TITLE: Approximation Property: Characterization
QUESTION [3 upvotes]: As reference the german wiki: Approximationseigenschaft
Problem
Given a Banach space.
Suppose it has the approximation property:
$$C\in\mathcal{C}:\quad\|T_N-1\|_C\to0\quad(T_N\in\mathcal{F}(E))$$
Then every compact operator is of almost finite rank:... | {"set_name": "stack_exchange", "score": 3, "question_id": 1122760} |
\begin{document}
\maketitle
\markboth{Florin Diacu and Ernesto P\'erez-Chavela}{Homographic solutions of the curved $3$-body problem}
\author{\begin{center}
Florin Diacu\\
\smallskip
{\footnotesize Pacific Institute for the Mathematical Sciences\\
and\\
Department of Mathematics and Statistics\\
University of Victoria\... | {"config": "arxiv", "file": "1001.1789/Homographic2.tex"} |
TITLE: $\gamma$ and an examination of its composition
QUESTION [0 upvotes]: Ok, so the Euler Mascheroni constant is defined as $$\sum_{k=1}^{x} \frac1k - \ln x$$ as $x\rightarrow\infty$. However, through some fancy l'Hôpital footwork, I've discovered that the harmonic series grows at a faster rate than the natural log ... | {"set_name": "stack_exchange", "score": 0, "question_id": 2156817} |
TITLE: Topological Version of First Isomorphism Theorem
QUESTION [5 upvotes]: Given a set $X$ and an equivalence relation $\sim$ on $X$, we can define the set $X_\sim=\left\lbrace\left[x\right]:x\in X\right\rbrace$ of equivalence classes, and we can define a projection map $\pi:X\rightarrow X_\sim$ defined by $\pi(x)=\... | {"set_name": "stack_exchange", "score": 5, "question_id": 1416709} |
TITLE: How to calculate the average molar mass of the atmosphere?
QUESTION [1 upvotes]: The task: Determine the average molar mass of the atmosphere - each for the moist air and the dry air. One example that I have calculated is this one: I have the gas Nitrogen and the percentage for the moist air(77,0%) and the perce... | {"set_name": "stack_exchange", "score": 1, "question_id": 2480084} |
TITLE: How to solve for angles $4\theta = \theta$?
QUESTION [0 upvotes]: I want to find all the angles in $[0, 2\pi)$ for which $4\theta = \theta$ is true. I can obviously get $\theta = 0$, but the other solutions are $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$. How do I find these particular ones?
REPLY [0 votes]: The writ... | {"set_name": "stack_exchange", "score": 0, "question_id": 1203217} |
TITLE: Prove that the set of open spheres is countable.
QUESTION [1 upvotes]: Can someone help me with the following problem:
I'm trying to prove that the following set of open spheres of $\mathbb R^2$ with $x_1,y_1,r ∈ \mathbb Q$ is countable:
$$S[(x_1,y_1),r]=\{(x,y∈\mathbb R^2: \sqrt {(x-x_1)^2+(y-y_1)^2}<r\}$$
I ... | {"set_name": "stack_exchange", "score": 1, "question_id": 3531009} |
TITLE: Maximum likelihood estimation 3
QUESTION [0 upvotes]: if I have a simple random sample $Y_{1},...,Y_{n}$ of an uniform distribution over interval $(0,2\theta+1)$, how can i compute the maximum likelihood estimation of $\theta$?
Thank you for your time.
REPLY [0 votes]: The m.l.e. for $2\theta+1$ is the highest... | {"set_name": "stack_exchange", "score": 0, "question_id": 2569110} |
TITLE: Congruence for Bernoulli numbers
QUESTION [0 upvotes]: It appears that for every odd prime $p$, the following congruence holds for Bernoulli numbers:
$$
2pB_{p-1}-pB_{2p-2}\equiv p-1\mod p^2\mathbb{Z}_{(p)}.
$$
The weaker statement that $2pB_{p-1}-pB_{2p-2}\equiv -1\mod p\mathbb{Z}_{(p)}$ follows from the von St... | {"set_name": "stack_exchange", "score": 0, "question_id": 1804666} |
TITLE: Problem with showing that operations of defined set are $n$-transitive
QUESTION [0 upvotes]: I have two problems which I have been thinking about since several days. They are connected with transitive operations.
We are considering the group $G$ of all linear transformations of real line in a form $x\rightarrow... | {"set_name": "stack_exchange", "score": 0, "question_id": 3811145} |
TITLE: What is a decision threshold and how does it apply to a statistical power?
QUESTION [0 upvotes]: I'm pretty confused on what is actually going on in this section with hypothesis testing. As another note, the values below are computed using R.
I have a homework problem that says:
From the perspective of a cereal ... | {"set_name": "stack_exchange", "score": 0, "question_id": 1292231} |
TITLE: Need help solving a problem on arranging balls
QUESTION [1 upvotes]: One of my friends gave me an interesting problem yesterday...Please omit the first four lines...start from the second paragraph...
What is the underlying principle that affects the fifth line?How to do it?
Thanks a lot in advance!!
REPLY [1 ... | {"set_name": "stack_exchange", "score": 1, "question_id": 1449173} |
\begin{document}
\begin{abstract}
The starting point of this article is a decades-old yet little-noticed sufficient condition, presented by Sassenfeld in 1951, for the convergence of the classical Gau\ss-Seidel method. The purpose of the present paper is to shed new light on \emph{Sassenfeld's criterion} and to demons... | {"config": "arxiv", "file": "2201.05628/paper9.tex"} |
\begin{document}
\title{Dilution, decorrelation and scaling in radial growth}
\author{Carlos Escudero}
\affiliation{ICMAT (CSIC-UAM-UC3M-UCM), Departamento de
Matem\'{a}ticas, Facultad de Ciencias, Universidad Aut\'{o}noma de
Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain}
\begin{abstract}
The dyna... | {"config": "arxiv", "file": "0909.5304.tex"} |
TITLE: Did Supernova 2007bi really explode due to antimatter creation?
QUESTION [2 upvotes]: I was watching a video (https://www.youtube.com/watch?v=IZ59_akUUBs) about massive explosions and came across 2007bi.
The video stated that this SN happened due to gamma-ray driven antimatter creation.
Apparently, its core bei... | {"set_name": "stack_exchange", "score": 2, "question_id": 464016} |
TITLE: Are all orbits of the conservative pendulum homoclinic?
QUESTION [1 upvotes]: I don't understand this statement:
"The homoclinic orbit is characterized by $E = mgl$. When $E < mgl$, the pendulum is tracing other orbits."
If energy is conserved, then $E_0 = E$ ($E$ is shorthand for $E(t)$, the energy at time $t... | {"set_name": "stack_exchange", "score": 1, "question_id": 559202} |
TITLE: If $\mathbf{AA}^T=\mathbf{I}$, is $\mathbf A$ necessarily square?
QUESTION [3 upvotes]: If $\mathbf{AA}^T=\mathbf{I}$, is $\mathbf A$ necessarily square?
I am starting to learn about matrices, and had the above question. When I have tried to think about this, I have not been able to progress using matrix multipl... | {"set_name": "stack_exchange", "score": 3, "question_id": 2257809} |
TITLE: Graph The Solution Of First Order Linear ODE
QUESTION [0 upvotes]: Graph all of the solutions of $y'=-\frac{x}{y}$
2.find the value of $x_0$ and $y_0$ such there is one and only solution, defined in the area that includes $x_0$ such that $y(x_0)=y_0$
$$\frac{dy}{dx}=-\frac{x}{y}$$
$$\frac{ydy}{dx}=-x$$
$${ydy}... | {"set_name": "stack_exchange", "score": 0, "question_id": 2008079} |
TITLE: For a number field $K$, does there exist totally splitting prime?
QUESTION [3 upvotes]: Let $K$ be a number field.
Then does there exist a rational prime number $l$ which splits completely in $K$?
I think this follows from Cebotarev density theorem.
But I think there exists more elementary proof.
REPLY [1 votes... | {"set_name": "stack_exchange", "score": 3, "question_id": 3100398} |
TITLE: Are there negative prime numbers?
QUESTION [1 upvotes]: It seems generally admitted that there are no negative prime numbers.
What are the rules that can affirm this?
Thanks in advance and happy new year to all.
Best regards,
REPLY [2 votes]: This is false. $-2$ is prime. One of the two following statements (de... | {"set_name": "stack_exchange", "score": 1, "question_id": 2605120} |
TITLE: Is the path space of a space homeomorphic to the disjoint union of the path spaces of the path components
QUESTION [2 upvotes]: Let $X$ be an arbitrary wild topological space. Equip the space $\mathcal{C}([0,1],X)$ of continuous paths $[0,1]\rightarrow X$ with the compact open topology. As every path lands in ex... | {"set_name": "stack_exchange", "score": 2, "question_id": 1576887} |
\begin{document}
\newcommand{\T}{\mathbb{T}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\tx}[1]{\quad\mbox{#1}\quad}
\title [Escaping Set of Hyperbolic Semigroup]{Escaping Set of Hyperbolic Semigroup}
\author[Bishnu Hari... | {"config": "arxiv", "file": "1803.10381.tex"} |
TITLE: Integration by Trig Substution - completely stuck
QUESTION [4 upvotes]: I'm trying to solve this integral, but after more than an hour I can't figure it out. I've outlined my thinking below.
$$
\int \dfrac{dx}{x^2\sqrt{4x^2+9}}
$$
If we let $\ a=3 $ and $\ b=2 $, the radical in the denominator fits the form $... | {"set_name": "stack_exchange", "score": 4, "question_id": 684459} |
TITLE: Proving $A^n=\left[\begin{smallmatrix}1&(2^n-1)a\\0&2^n\end{smallmatrix}\right]$
QUESTION [0 upvotes]: Given the matrix
\begin{align}
A = \begin{bmatrix}1&a\\0&2\end{bmatrix} ,
\end{align}
is it true that
\begin{align}
A^n=\begin{bmatrix}1&(2^n-1)a\\0&2^n\end{bmatrix}
\end{align}
for all $n \geq 0$?
I f... | {"set_name": "stack_exchange", "score": 0, "question_id": 3159115} |
TITLE: Reweighting probability measures by convex potentials, and contraction in transport distance
QUESTION [2 upvotes]: Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu_y$ by
$$\mu_y (dx) = \exp( - W (x - y)) \,dx,$$
i.e. each $\mu_y$ is a t... | {"set_name": "stack_exchange", "score": 2, "question_id": 393537} |
TITLE: Prove the following Lemma in the polynomial rings.
QUESTION [1 upvotes]: Let $R$ be a ring. Then, the natural inclusion
$R \to R[x]$
which just sends an element $r \in R$ to the constant polynomial $r$, is a ring homomorphism.
Attempts:
Let $r \in R$ and define $g : R \to R[x]$ as $g(r) = f(x)$ where $f(x) = r$ ... | {"set_name": "stack_exchange", "score": 1, "question_id": 3893414} |
TITLE: ordinary differential equations solution needed
QUESTION [0 upvotes]: How to find non-trivial solution $y$ of BVP $$y''+xy=0, x\in [a,b],$$ $$y(a)=y(b)=0.$$
Till time no method which i know works. Please help some one.
REPLY [1 votes]: The general solution of $[ \partial_{x} + k^{2} x ] y = 0$ is
\begin{... | {"set_name": "stack_exchange", "score": 0, "question_id": 792190} |
TITLE: How to derive end-correction value relationship for open-ended air columns?
QUESTION [5 upvotes]: According to Young and Freedman's Physics textbook, in open-ended air columns like some woodwind instruments, the position of the displacement antinode extends a tiny amount beyond the end of the column.
UCONN's we... | {"set_name": "stack_exchange", "score": 5, "question_id": 214330} |
\begin{document}
\begin{abstract}
Internal diffusion-limited aggregation is a growth model based on random
walk in~$\Z^d$. We study how the shape of the aggregate depends on the law
of the underlying walk, focusing on a family of walks in $\Z^2$ for which
the limiting shape is a diamond. Certain of these walks-... | {"config": "arxiv", "file": "0905.1361/ddiamond.tex"} |
\begin{document}
\maketitle
\begin{abstract}
\noi
{\it It is shown that irreducible two-state continuous-time Markov chains interacting on a network in a bilinear fashion have a unique stable steady state. The proof is elementary and uses the relative entropy function.}
\end{abstract}
\bigskip
\section{{\bf Descrip... | {"config": "arxiv", "file": "1412.0700.tex"} |
TITLE: Upper-bound for nuclear norm of $A \circ (v \otimes v)$ in terms of operator norm (or nuclear norm) of matrix $A$ and $L_\infty$-norm of vector $v$.
QUESTION [1 upvotes]: Let $A \in \mathbb R^{n \times }$ be a psd matrix such that $\|A\|_{op} \le r_1$ and $\|A\|_{*} \le r_2$. Let $v \in \mathbb R^n$ such that $\... | {"set_name": "stack_exchange", "score": 1, "question_id": 4319197} |
TITLE: Proof integration identity $\int_{0}^{1}dx\int_{0}^{x}e^{x^2}dy=\int_{0}^{1}dy\int_{y}^{1}e^{x^2}dx$
QUESTION [0 upvotes]: I have to prove this identity:
$$\int_{0}^{1}dx\int_{0}^{x}e^{x^2}dy=\int_{0}^{1}dy\int_{y}^{1}e^{x^2}dx$$
I've shown that:
$$\int_{0}^{1}dx\int_{0}^{x}e^{x^2}dy=\int_{0}^{1}xe^{x^2}dx=\frac... | {"set_name": "stack_exchange", "score": 0, "question_id": 3084349} |
TITLE: Finding the Density Change of a Fluid
QUESTION [1 upvotes]: Consider the motion of a fluid with velocity field defined in Eulerian variables by the following equations
$$u=kx,\,\,v=-ky,\,\,w=0$$
where $k$ is a constant. Also assume that the density is given by
$$\rho = \rho_0 + Aye^{kt}$$
What is the rat... | {"set_name": "stack_exchange", "score": 1, "question_id": 2654317} |
\begin{document}
\begin{center}
\Large{\bf Minimal non-orientable matroids in a projective plane}
\normalsize
{\sc Rigoberto Fl\'orez}
\footnote {The work of the first author was
performed at the State University of New York at Binghamton.}
\footnotesize
{\sc University of South Carolina Sumter\\
Sumter, SC, ... | {"config": "arxiv", "file": "2202.09621/MinimalNon-OrientableMatroidsPP_ARXIV.tex"} |
TITLE: intersection of maximal ideals in a polynomial ring
QUESTION [2 upvotes]: Given $A=K[x_1,\dots,x_n]$ a polynomial ring on a field $K$, let $p(x)\in A$ be an element, and $M_1,\dots,M_s$ some maximal ideals.
Is it true that
$$\cap(M_i,p) = (\cap M_i,p)?$$
I obtained that it's true if $K$ is algebraically closed, ... | {"set_name": "stack_exchange", "score": 2, "question_id": 1358717} |
TITLE: Evaluating a logarithmic integral in terms of trilogarithms
QUESTION [3 upvotes]: For $a,c\in\mathbb{R}\land-1\le a\land-1<c$, define the function $J{\left(a,c\right)}$ to be the value of the dilogarithmic integral
$$J{\left(a,c\right)}:=\int_{0}^{1}\mathrm{d}y\,\frac{\operatorname{Li}_{2}{\left(\frac{c}{1+c}\r... | {"set_name": "stack_exchange", "score": 3, "question_id": 1428987} |
TITLE: Reference request: the theory of currents
QUESTION [14 upvotes]: I am a graduate student and want to study the theory of currents. What is a good reference for a beginner?
I should be familiar with the theory of distributions or generalized functions on $\mathbb R^n$.
REPLY [18 votes]: The theory of currents is... | {"set_name": "stack_exchange", "score": 14, "question_id": 372196} |
TITLE: Show the set of the limits of all the subsequence of a bounded set contains sup and inf.
QUESTION [5 upvotes]: Let $(X_n)$ be a bounded sequence, and let $E$ be the set of subsequential limits of $(X_n)$. Prove that $E$ is bounded and contains $\sup E$ and $\inf E$.
Does this ask us to prove limsup and liminf ex... | {"set_name": "stack_exchange", "score": 5, "question_id": 701846} |
TITLE: Question about Wilson's theorem, when $n = 4$.
QUESTION [0 upvotes]: Recently I was looking up on Wilson's theorem to find out what are the values of a function $f(n) \equiv (n-1) \pmod{n}$ for any $n$. So I know that for prime numbers $\geq $ 2 that would be -1, and it looks like for composite values $(n-1)! \e... | {"set_name": "stack_exchange", "score": 0, "question_id": 2327817} |
\section{Third Sylow Theorem}
Tags: Sylow Theorems, Third Sylow Theorem
\begin{theorem}
All the [[Definition:Sylow p-Subgroup|Sylow $p$-subgroups]] of a [[Definition:Finite Group|finite group]] are [[Definition:Conjugate of Group Subset|conjugate]].
\end{theorem}
\begin{proof}
Suppose $P$ and $Q$ are [[Definition:Syl... | {"config": "wiki", "file": "thm_942.txt"} |
TITLE: How many units squares can fit on $\mathbb R^2$?
QUESTION [0 upvotes]: How many units squares can fit on $\mathbb R^2$?
I have thought about this problem for a while and I've come to a solution, however, I am not sure whether my reasoning is good.
Filling the plane with unit squares can be done following this ... | {"set_name": "stack_exchange", "score": 0, "question_id": 2631080} |
\begin{document}
\title{Infinite-dimensional analyticity in quantum physics}
\author{Paul~E.~Lammert}
\email{lammert@psu.edu}
\affiliation{Department of Physics, 104B Davey Lab \\
Pennsylvania State University \\ University Park, PA 16802-6300}
\begin{abstract}
A study is made, of families of Hamiltonians parame... | {"config": "arxiv", "file": "2108.10094/arxiv-210821.tex"} |
TITLE: Prove the tautology (⋀( → ¬)) → ¬
QUESTION [1 upvotes]: I must prove this tautology using logical equivalences but I can't quite figure it out. I know it has something to do with the fact that not p and p have opposite truth values at all times. Any help would be appreciated.
REPLY [0 votes]: Use the fact that ... | {"set_name": "stack_exchange", "score": 1, "question_id": 3109036} |
\begin{document}
\baselineskip=16pt
\title[Relations in the tautological ring]
{Relations in the tautological ring of the
moduli space of $K3$ surfaces}
\author{Rahul Pandharipande}
\address{Department of Mathematics, ETH Z\"urich}
\email {rahul@math.ethz.ch}
\author{Qizheng Yin}
\address{Department of Mathematics, ... | {"config": "arxiv", "file": "1607.08758.tex"} |
\begin{document}
\title[class numbers along a Galois representation]{
Asymptotic lower bound of class numbers along a Galois representation}
\author{Tatsuya Ohshita}
\address{Graduate School of Science and Engineering, Ehime University 2--5, Bunkyo-cho, Matsuyama-shi, Ehime 790--8577, Japan}
\email{ohshita.tatsuya.n... | {"config": "arxiv", "file": "1805.03850.tex"} |
TITLE: Can special relativity be deduced from $E=mc^2$?
QUESTION [1 upvotes]: So instead of assuming that the velocity $c$ is a maximal velocity, proving that while assuming $E=mc^2$.
REPLY [0 votes]: Here is a quick result that might interest you:
Let's begin from the equation of relativistic energy in terms of the r... | {"set_name": "stack_exchange", "score": 1, "question_id": 1273} |
\begin{document}
\begin{frontmatter}
\title{A heat flow for the mean field equation on a finite graph}
\author{Yong Lin}
\ead{yonglin@mail.tsinghua.edu.cn}
\address{Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China}
\author{Yunyan Yang\footnote{Corresponding author}}
\ead{yunyanyang@ruc.e... | {"config": "arxiv", "file": "2108.01416/Flow-Graph.tex"} |
TITLE: Existence and Uniqueness Theorem
QUESTION [2 upvotes]: I had a question about how to do one of these problems. So here's the question:
Given this equation $y'=\frac{-\cos(t)y(t)}{(t+2)(t-1)}+t$, find if the initial conditions $y(0)=10, y(2)=-1, y(-10)=5$ exist.
So I think the first step is just to take the parti... | {"set_name": "stack_exchange", "score": 2, "question_id": 296796} |
TITLE: Prove: If $A$ and $B$ are closed subsets of $[0,\Omega]$ then at least $A$ or $B$ is bounded
QUESTION [0 upvotes]: As usual, I am self studying topology and my knowledge of ordinals is meagre. Have
done some research on it.
Theorem 5.1 Any countable subset of $[0,\Omega)$ is bounded above.
(This exercise require... | {"set_name": "stack_exchange", "score": 0, "question_id": 4351932} |
TITLE: Finding the inverse of a function using bisection method
QUESTION [0 upvotes]: It is said that we can find $f^{-1}(y)$ by solving the equation $y-f(x)=0$ using bisection method. But all sources that I can find use bisection to find roots, so I can't figure how and why. Could you explain it?
REPLY [3 votes]: For... | {"set_name": "stack_exchange", "score": 0, "question_id": 1440818} |
TITLE: How to expand constant function in fourier sine series?
QUESTION [1 upvotes]: If a function is constant, by orthogonality $$\int_{-L}^L C \cdot \sin(n\pi x/L) \, dx = C \cdot \int_{-L}^{L} \sin(n\pi x/L) \, dx=0\text{ ??}$$
REPLY [5 votes]: Extend the constant function C into odd function:
$$f(x) = \begin{cases... | {"set_name": "stack_exchange", "score": 1, "question_id": 1699638} |
TITLE: Factoring the polynomial $3(2x+3)^2 + 7(2x+3) - 6$
QUESTION [1 upvotes]: Factor $3(2x+3)^2 + 7(2x+3) - 6$
What I did:
With the substitution $X=2x+3$:
\begin{align}
3(2x+3)^2 + 7(2x+3) - 6
&= 3X^2-9X+2X-6 \\
&= 3X(X-3)+2(X-3) \\
&= (X-3)(3X+2) \\
&=((2x+3)-3)(3(2x+3)+2) \\
&=(2x+0)(6x+9+2) \\
&=(2x)(6x+11)
\end{... | {"set_name": "stack_exchange", "score": 1, "question_id": 2994809} |
\begin{document}
\font\ef=eufm10
\def\fraktur#1{{\ef {#1}}}
\def\bb#1{{\mathbb{#1}}}
\def\cal#1{{\mathcal{#1}}}
\def\fk#1{{\hbox{\fraktur#1}}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{axiom}{Axiom}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposi... | {"config": "arxiv", "file": "0812.1575.tex"} |
\section{Introduction}
This is the third of the series concerning a localization of the
index of elliptic operators.
The localization of integral is a mechanism by which
the integral of a differential form on a manifold
becomes equal to the integral of another differential
form on a submanifold, which has been form... | {"config": "arxiv", "file": "1008.5007/ss1.tex"} |
TITLE: Scalar product of Gaussian random vector with projection matrix is chi-squared
QUESTION [1 upvotes]: We define the $n$ chi-square random variable this way : if $Z \sim N(0,I_n)$ is multivariate Gaussian random vector, then $\lVert Z \rVert ^2 = \sum_{i=1}^n Z_i^2$ (sum of $n$ standard gaussian RV squared) is sai... | {"set_name": "stack_exchange", "score": 1, "question_id": 4547610} |
TITLE: Solve for unknown in exponential equation
QUESTION [0 upvotes]: How do I solve for an unknown in the base of an exponential equation. In my example W is unknown: $$PPV=K\cdot\left(\frac{D}{W^{0.5}}\right)^{-1.6}$$
REPLY [1 votes]: I will do the first step.
$$PPV=K\cdot\left(\frac{D}{W^{0.5}}\right)^{-1.6} \im... | {"set_name": "stack_exchange", "score": 0, "question_id": 1908147} |
TITLE: AMC 2011 Coloring Problem
QUESTION [9 upvotes]: A 40 X 40 white square is divided into 1 X 1 squares by lines parallel to its sides. Some of these 1 X 1 squares are coloured red so that each of the 1 X 1 squares, regardless of whether it is coloured red or not, shares a side with at most one red square (not coun... | {"set_name": "stack_exchange", "score": 9, "question_id": 3488926} |
\begin{document}
\maketitle
\newpage
\begin{abstract}
We consider planar rotors (XY spins) in $\mathbb{Z}^d$,
starting from an initial Gibbs measure and evolving with infinite-temperature
stochastic (diffusive) dynamics. At intermediate times, if the system
starts at low temperature,
Gibbsianness can be lost. Due to t... | {"config": "arxiv", "file": "0808.4092/enruyepE.tex"} |
\begin{document}
\author{Camilo Hern\'andez}
\address{
IEOR Departament\\
Columbia University\\
NY, USA.
}
\email{camilo.hernandez@columbia.edu}
\author{Mauricio Junca}
\address{
Mathematics Department\\
Universidad de los Andes\\
Bogot\'a, Colombia.
}
\email{mj.junca20@uniandes.edu.co}
\author{Harold Moreno-Franco... | {"config": "arxiv", "file": "1608.02550/DividendsConstraint.tex"} |
TITLE: Algebraic of degree $n$ over $F$ implies algebraic of degree at most $n$ over $F(a)$?
QUESTION [0 upvotes]: Suppose $a, b$ in the field $K$ are algebraic over the subfield $F \subseteq K$. Suppose $a$ is algebraic of degree $m$ over $F$ and $b$ is algebraic of degree $n$ over $F$. Let $F(a)$ be the field obtaine... | {"set_name": "stack_exchange", "score": 0, "question_id": 4001192} |
TITLE: Derivative of dot product?
QUESTION [2 upvotes]: What's the derivative ${\partial \over \partial x} \langle x, f(x)\rangle$?
According to the product rule it should be $1\cdot f(x) + x \cdot f'(x) $ but in my previous post I was told that this makes no sense.
Here $f: \mathbb R \to \mathbb R^2$ and $1$ is the c... | {"set_name": "stack_exchange", "score": 2, "question_id": 1362611} |
TITLE: Does sound waves pick up the speed of its source?
QUESTION [5 upvotes]: I googled the speed of sound and found that it only depends on the medium (just like the speed of light but with different parameters). I can't see how it doesn't pick up the speed of its source! I mean for the constancy of the speed of ligh... | {"set_name": "stack_exchange", "score": 5, "question_id": 237676} |
TITLE: Which schemes can be presented as limits of smooth varieties?
QUESTION [4 upvotes]: I can prove a certain statement for any scheme that can be presented as the limit of an essentially affine (filtering) projective system of smooth varieties over a perfect field such the connecting morphisms are dominant. In this... | {"set_name": "stack_exchange", "score": 4, "question_id": 123716} |
TITLE: What is the analytical expression which shows the convergence of a 6 sided fair dice's expected value to 3.5 as a function of the number of rolls(N)?
QUESTION [1 upvotes]: What is the analytical expression which shows the convergence of a 6 sided fair dice's expected value to 3.5 as a function of the number of r... | {"set_name": "stack_exchange", "score": 1, "question_id": 1933507} |
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