The lower-bounds tag has no usage guidance.

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102 views

### Possible lower bound in quantum many body system with non-local terms

I am asking a question related to Lieb-Robinson bound and nonlocality.
As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. ...

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341 views

### Bounding the probability that a random variable is maximal

Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$. I am looking for ...

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599 views

### Lower bounds for linear forms of logarithms (a la Baker)?

Let $\lambda_1$, $\lambda_2$, and $a$ be three fixed complex algebraic
numbers.
For a given integer $n$, write
$\Theta(n) = \arg(a \lambda_1^n + \lambda_2^n)$.
Assuming $\Theta(n)$ is not zero, I am ...

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126 views

### A generalization of coupon collector problem - $\geq1$ pick per experiment

Mix $T\geq1$ coupons numbered $1$ to $T$ with a set of $S\geq0$ number of dummy coupons with no numbers. Select $N\geq1$ coupons at each trial at random and put them back.
$N=1$ is standard coupon ...

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56 views

### Minimal condition $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $, $\mid \rho_{ij} \mid \leq 1$, $s_i \in \mathbb R$ and $\Psi_{ij} \in \{0,1\}$

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below
$$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$
$$G = ...

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106 views

### Minimum number of perfect matchings in a regular bipartite graph

Is there a lower bound on the number of perfect matchings in a $k$-regular bipartite graph?
One can use Hall's marriage theorem and induction on $k$ to derive the lower bound of $k$. I can't come up ...

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92 views

### Lower asymptotic bounds for the derivative of Laguerre polynomials

Let $ L_{d}^{(1)}(x)$ denote the generalized Laguerre polynomial of degree $d$ and order $\alpha=1$. Clearly, since all the roots $r_1,\dots,r_d$ of $L_{d}^{(1)}$ are simple, there exists a strictly ...

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118 views

### Bounding the Schur's complement of similiar matrices

Assume the following:
• $L\leq K$
.
• $\Gamma\in M_{K,L}$ is a $L$ rank ${ 0,1} $ matrix, without identical rows or the zeros row.
• $N\in M_{K,K}$ is a diagonal matrix, whose diagonal is a ...

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94 views

### Bounding a q-expansion on a bounded open subset of the complex upper-half plane

Let $f:\mathbf{H}\to \mathbf{C}$ be a holomorphic function on the complex upper-half plane and let $q:\tau\mapsto \exp(\pi i \tau)$ be the nome on $\mathbf{H}$. Suppose that there are integers $a_j$ ...

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222 views

### Geometric/Analytic techniques for constructive and asymptotic bounds in the Lee metric

Slight extension of cross posting from
http://cstheory.stackexchange.com/questions/7408/lee-metric-gilbert-varshamov-and-hamming-bounds-for-larger-relative-distance-rang (closed there)
The following ...

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21 views

### bounds on $\sum_{j=1}^m\frac{\pi^j}{\Gamma(j)(x^2+(j+1/4)^2)}$

During our search of real rooted entire function approximations to Riemann $\Xi$ function, we need to calculate the upper and lower bounds of
$$f_m(x):=\sum_{j=1}^m ...

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62 views

### Eigenvalue bounds from eigenvalues of Schur complement

Is it possible to lower bound the minimal eigenvalue of a symmetric PSD matrix
$M= \begin{pmatrix}A & C\\ C^*&B\end{pmatrix}$
from the knowledge of the eigenvalues of $M$'s Schur complement ...

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45 views

### Deriving inequalities from a polynomially-bounded derivative

In this paper (p. 2, definition/remark) the following notion of ‘polynomial growth’ is defined for a non-negative real function $g(x)$ and a real constant $b\in(0;1)$:
There exist positive ...

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107 views

### Estimating when does a certain binomial sum exceed an upper bound

Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers
$$f_{n,m} = \sum_{i=\lfloor m/2 \rfloor}^m {n-2i \choose n - m -i}{i+1 \choose m - i +1}.$$
For example $f_{n,0} = ...

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104 views

### partition of an integer n into atmost k =O(log n) parts

Suppose you have a partition p of n into atmost k parts, say $$\{i_1, i_2, ..., i_j, ..., i_{k-1}\}$$
For example $\{1, 4\}$ is a partition of 10 into 3 parts (in this notation i am specifying the ...

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74 views

### Looking for tighter bounds

I have to solve an equation which is $$\sum_{i=1}^N x_i = \sum_{i=1}^N y_i,$$ where
$$x_i = \frac{z_i}{1 + (K_i - 1) w}$$ and $$y_i = \frac{K_i z_i}{1 + (K_i - 1) w}.$$
The $z_i$ are all positive ...