# Questions tagged [lower-bounds]

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### Lower bound on the number of balanced graphs

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 ...
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### Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues

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 $i∈{1,…,n}$ ...
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### Probability of winning a $k$-rounds coin toss game

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 ...
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### An interesting sequence of numbers arising from the Riemann hypothesis

A very good coincidence occurred today with me. While just plotting random functions in Mathematica, I entered this command: ... 234 views

### better lower (and upper) bound for $i$'s moment of function of binomial random variable with $i = \frac{1}{j}, j \in \mathbb{N}$

I want to derive a lower bound for $E\left[\left(\frac{X}{k-X}\right)^{i}\right]$ with $X \sim Bin_{(k-1),p}$ and $k \in \mathbb{N}$. So far I could prove that \begin{equation} E\left[\frac{X}{k-X}\...
Let $\Sigma \in S_{++}^n$ be a symmetric positive definite matrix with all diagonal entries equal to one. Let $U \in \mathbb{R}^{n \times k_1}$, $W \in \mathbb{R}^{n \times k_2}$, $\Lambda \in \mathbb{... 0 votes 1 answer 187 views ### Lower-bound on smallest singular-value of rectangular random matrix Let$X$be a random$N \times n$matrix with iid entries from$\mathcal N(0, 1)$and with$n/N =: \lambda(N,n) \le \lambda_0$, for some$\lambda_0 \in (0, 1)$. That is,$X$is genuinely rectangular (... 3 votes 1 answer 84 views ### If$X \sim N(0,I_m)$, what is a necessary and sufficient condition on$u_m > 0$such that$\lim\sup_{m\to \infty} P(\|X\|^2 \ge u_m|X_1|) = 1$Let$m$be a large positive integer and$X=(X_1,\ldots,X_m) \sim N(0,I_m)$. I wish to show that the squared norm of$X$is much much bigger than the absolute value of any of the$X_j$'s. For example, ... 1 vote 2 answers 198 views ### What is the approximation of$\log(|\zeta'(\frac{1}{2}+it)|)$in Dirichlet polynomial if it is exists? I have done some search many times on web to find any approximation of$\log|(\zeta'(s))|$in Dirichlet polynomial but I didn't got it, Probably that$\log(|\zeta'(s)|$dosn't have a Dirichlet ... 0 votes 0 answers 72 views ### Upper bounding the sum with hypergeometric and binomial probabilities Could you please help me upper bound this tricky expression: $$P(A)=\sum_{i=0}^n{\left( 1 - \dfrac{\binom kq \binom {n-k}{i-q}}{\binom {n}{i}} \right)}^I \binom ni p^i {(1-p)}^{n-i}$$. So far I only ... 1 vote 1 answer 67 views ### How to compare the minimums of two discrete convex functions? I have a question that troubled me for a long time. If I have two convex discrete function$f(·)$and$g(·)$such that$f(·) \ge g(·)$. (may be not necessary?) Let$x_1 = \text{argmin } f(·)$, ... 0 votes 0 answers 171 views ### Bounding the absolute value of a complex integral I'm working on some problems involving Fourier transforms and convolution problems and there is one problem I cannot solve. In my situation we have a complex valued function$f(ix)$, with$x\in\mathbb{...
I have found this lower bound for the size of minimal vertex cover (and proved it). If a simple connected graph G on n vertices has largest and smallest eigenvalues $\lambda_1,\lambda_n$, ...