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Let $(a_n)_{n\geq 1}$ be a sequence of integers such that for some constant $N$ and all $n>N$ we have $$ a_n = -\max_{i+j=n}(a_i + a_j). $$ Is it true that any such sequence is eventually periodic? ...
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Let $\mathcal{M}_{n,m}(\mathbb{F_q})$ be the set of $n$ by $m$ matrices over the finite field of order $q$, with $n \geq m$. For a (non-trivial) subspace $V \subset \mathcal{M}_{n,m}(\mathbb{F_q})$, ...
Ward Beullens's user avatar
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Let $\pi: Y\to X$ be a modification of a compact complex manifold $X$ under successive blow ups along smooth centers. Let $\omega$ be a Kähler metric on $X$ and $\eta$ be a Kähler metric on $Y$. Put $\...
Skywalker's user avatar
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For any Banach space $X$ of dimension greater than or equal 2 we denote by $\beta(X)$ the number $$\hbox{inf} \ d(\ell_1^2, X_2),$$ where $X_2$ runs through all 2-dimensional subspaces of $X$ and $\...
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The James constant of a normed space $X$ is defined by $$J(X)=\hbox{sup}\{\hbox{inf}\{\|x+y\|, \|x-y\|\}: \|x\|=\|y\|=1\}.$$ It is well-known that $J(X)=0$ or $J(X) \geq \sqrt{2}$. On the other hand, ...
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This is related to another question I just asked 1, after running some experiments. Let $X$ and $Y$ be two independent uniform sets of points in the unit interval, with $|X| = 2|Y|$. By running ...
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Let $X$ and $Y$ denote two sets of $m$ and $n$ points distributed uniformly at random in the unit interval. When $m$ and $n$ are both large, is there a bound for the expected cost of a minimum-weight ...
Tom Solberg's user avatar
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I am having some trouble with this (undergrad) problem. The Twitter account I found this from was deleted so I unfortunately have not found an answer. I have tried decomposing into Riemann sum and ...
Eftew's user avatar
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I am interested in upper and lower bounding probability that one component of a multivariate Gaussian exceeds all others. For instance, say we have a multivariate Gaussian RV $X \sim N(\mu, \Sigma)$ ...
ted's user avatar
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Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows $$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$ where $0<C_i<1$ and $0\leq D_i<2\pi$ are real-...
loizuf's user avatar
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Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. symmetric random variables, with $-1\leq X_i\leq 1$, $\mathbb{E}(X_i) =0$, $\mathbb{E}(X_i^2) = 1$. We have that: $$ P\left(\bigcap_{k = 1}^{n}\frac{|\sum_{i = ...
MathRevenge's user avatar
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Let $W_n = \sum_{i = 1}^{n}X_i$ be a random walk on $\mathbb{R}$, where the increments $X_i$ are i.i.d., symmetric around the origin ($X\sim -X$), such that $-1\leq |X(\omega)| \leq 1$ $\forall\omega\...
MathRevenge's user avatar
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Background The Norwegian mathematician and astronomer Carl Størmer did important work on the equation $$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$ ...
Max Lonysa Muller's user avatar
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Edit: Apologies, as mentioned in the comments I failed to notice the analytic algorithms that take $O(n^{1/2+\epsilon})$ time, so this question doesn’t make much sense. It’s possible there is a ...
Geoffrey Irving's user avatar
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For $n,k\in\mathbb{N}$, let $f(n,k)$ be defined as follows. If $n \geq k$ and $n > 2$, then $$ f(n,k) = \frac{k(n-k)}{n(n-1)}f(n-2,k-1) + \frac{k(k-1)}{n(n-1)}f(n-2,k-2) + \frac{n-k}{n}f(n-1,k) + \...
macat's user avatar
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Given a positive integer $k$, is there a positive real number $c(k)$ such that $\mathbb{E}\left(\prod_{i=1}^k X_i\right)\geq c(k)$ for any $k$-random variables $X_1,X_2,\ldots,X_k$ which all have the ...
fengzju's user avatar
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In a paper I found a strange application of Markovs inequality which I couldn't follow maybe you can help. $X_k$ is the set of $k$-element Subsets of $\mathbb{Z}_d^n$ we fix a $C^{-1} \in X_{k-1}$ and ...
Mathhead123's user avatar
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Motivation In the following question posted here on MO and over at MSE, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the ...
Max Lonysa Muller's user avatar
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Consider a binary classification problem for $(X,Y)$, and let $\hat{f}$ be a proposed classifier. We wish to bound the misclassification rate $P(\hat{f}(X)\ne Y)$. There are many known lower bounds on ...
tim523's user avatar
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The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument. For a binomially distributed variable $X \sim \text{Bin} \...
reservoir's user avatar
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In this post I proposed a formulation of Mertens function $M(n)$ using the inclusion-exclusion principle, as follows: $$M(n)=-\pi\left(n\right)+\left(\sum_{p_{i}<\sqrt{n}}\pi\left(\lfloor\frac{n}{...
Juan Moreno's user avatar
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Suppose $(Z_1, Z_2)$ is the zero-mean bivariate normal distribution with covariance $\left( \begin{matrix} 1 & \rho; \\ \rho & 1\end{matrix} \right)$ with positive $\rho > 0$. What I want ...
香结丁's user avatar
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Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$? For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...
Charles's user avatar
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I came up with the following thought experiment in my research in order to better understand the way Turing machines can transfer information through their tapes (the motivation is detailed below, isn'...
exfret's user avatar
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I have a sum of $n$ i.i.d random variables $X_i$ such that $E[X_i] = 0$,$\mathrm{E}[|X_i|^{1 + \delta}]$ exists for some $0 < \delta < 1$ but $\mathrm{E}[|X_i|^{1 + \delta+ \epsilon}]$ does not ...
Kaiyue Wen's user avatar
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Consider bivariate copulas $C_1$ and $C_2$ with $\max\{C_1(u,v), C_2(u,v)\}< M_2(u,v)$ for all $u,v \in(0,1)$, where $M_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound. Is there a ...
Corram's user avatar
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Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
Tom Solberg's user avatar
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Consider a $k$ regular graph of $n$ vertices, where $3 \leq k \leq (n-1)$. Is there any upper or lower bound, in the worst case, known for either the tree-width or the clique width of each $k$ regular ...
RandomMatrices's user avatar
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I'm hoping that this is an easy question for someone. How many elements can we expect to have multiplicative order at most $n^{1/c}$ in one of the finite fields $\mathbb{F}_p$ with $p$ prime with $n \...
Matt Groff's user avatar
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1 answer
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Let $\alpha>1$ be a constant and define $B_n$ as the number of (labeled) balanced graphs with $n$ vertices and $\left\lceil \alpha n\right\rceil $ edges. The paper Strongly Balanced Graphs and ...
35T41's user avatar
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Let $X_i$ be the $i$-th element of the vector $X=(X_1, ..., X_m)$ of i.i.d. random variables. I am looking for a lower bound for the expression $\mathbb{P}((\sum^n_{i=1}\prod^{m_i}_{j=1}(X_j))^2 \geq ...
Scriddie's user avatar
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1 answer
431 views

Let $$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$ $$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$ $$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...
macat's user avatar
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5 votes
4 answers
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Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$ $$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$ $$C_k = \frac{\sum_{i=1}^k(...
macat's user avatar
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4 votes
2 answers
366 views

I came across the following inequality, which should hold for any integer $k\geq 1$: $$\sum_{j=0}^{k-1}\frac{(-1)^{j}2^{k-1-j}\binom{k}{j}(k-j)}{2k+1-j}\leq \frac{1}{3}.$$ I have been struggling with ...
macat's user avatar
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0 answers
282 views

In the context of statistical estimation or distribution testing, Le Cam's method is a way to prove lower bounds on the minimax sample complexity ([1,2,3,4], further details below). My question is: ...
π314's user avatar
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I am currently reading On Uniqueness Properties of Solutions of Schrödinger Equations and a having trouble understanding a claim made on page 1819. Context from the paper: let $g\in C^\infty_0(\...
Dispersion's user avatar
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In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
ayr's user avatar
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Let $p,q \in [0,1]$ with $p>q$. I denote by $B_k(p), B_k(q)$ two independent random variables following the binomial distribution, with parameters $(k,p)$ and $(k,q)$ respectively. Informal ...
Argemione's user avatar
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1 answer
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We consider a sequence $u = (u_k)_{k\geq 1}$ such that $u_k \geq 0$ for any $k \geq 1$. We assume that there exists a critical $p_c \in \mathbb{R}$ such that, for any $q<p_c <p$, $$\sum_{k=1}^\...
Goulifet's user avatar
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Let $X_1,\dots,X_N$ denote a collection of independent samples of a uniform multinomial random variable in $\mathbb{Z}^k$, with the number of trials equal to $n\ll k$. (By "uniform", I mean ...
Tom Solberg's user avatar
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The chromatic polynomial $P(G, \lambda)$ gives the number of proper vertex colorings of the graph $G$ with $\lambda$ colors. I'm interested in how many possible colorings you loose when you add an ...
M. Stern's user avatar
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I would like to know if there are known results for bounding eigenvalues of positive-definite matrices, in particular gram matrices $AA^\top$ based on easily computable functions of $A$. Gershgorin ...
trenta3's user avatar
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This is a question I originally asked on math.stackexchange, but didn't receive a satisfying answer. Let $\epsilon>0$ and consider constructing a set $S_\epsilon\subseteq S^{d-1}$ of points on the ...
R B's user avatar
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Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
dohmatob's user avatar
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Let $C$ be an $n \times p$ matrix and $b$ be a column vector of length $n$, and $c>0$. Let $E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position. Question 1. ...
dohmatob's user avatar
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11 votes
3 answers
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Let $a(n) = f(n,n)$ where $f(m,n) = 1$ if $m < 2 $ or $ n < 2$ and $f(m,n) = f(m-1,n-1) + f(m-1,n-2) + 2 f(m-2,n-1)$ otherwise. What is the limit of $a(n + 1) / a (n)$? $(2.71...)$
José María Grau Ribas's user avatar
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This is a question I originally posted to math.stackexchange.com but it didn't attract any answers, and I was wondering if someone here can help. Consider a unit vector $x\in\mathbb R^d$ ($\|x\|_2=1$)...
M A's user avatar
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8 votes
2 answers
384 views

I am trying to get a lower bound (or even the exact value) of $$ \min_{X \in \mathbb{R}^{n\times n}} \|X - I_n\|_{\infty} \enspace \text{s.t.} \enspace \mbox{Rank}(X) = m $$ where $m \leq n$, and the ...
PAb's user avatar
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Let $n$ be a large positive integer. Let $A$ be a positive-definite matrix such with eigenvalues $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n$ such that $\lambda_n = o(1) \to 0$ and $\lambda_i=\...
dohmatob's user avatar
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I learned about the following result from one of Peter Winkler's books: It is impossible to infect the entire $n\times n$ chessboard (usual chessboard) starting from fewer than $n$ infected cells. The ...
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