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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. $H=\... 0answers 376 views ### Bounding the probability that a random variable is maximal Suppose we have$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$. I am looking for ... 0answers 607 views ### Lower bounds for linear forms of logarithms (a la Baker)? Let$\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)$is not zero, I am ... 0answers 132 views ### A generalization of coupon collector problem -$\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 ... 0answers 58 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 $$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+},$$ $$G = ... 0answers 138 views ### Minimum number of perfect matchings in a regular bipartite graph Is there a lower bound on the number of perfect matchings in a k-regular bipartite graph? One can use Hall's marriage theorem and induction on k to derive the lower bound of k. I can't come up ... 0answers 106 views ### Lower asymptotic bounds for the derivative of Laguerre polynomials Let L_{d}^{(1)}(x) denote the generalized Laguerre polynomial of degree d and order \alpha=1. Clearly, since all the roots r_1,\dots,r_d of L_{d}^{(1)} are simple, there exists a strictly ... 0answers 60 views ### Finding the infimum of the range of a certain non-negative function associated to a C^{*} -algebra Let A be a non-trivial C^{*} -algebra and n \in \mathbb{N} . Setting \mathcal{D} \stackrel{\text{df}}{=} A^{n} \setminus \{ (0_{A},\ldots,0_{A}) \} , we can define a function f: \... 0answers 45 views ### Bounds on \sum_{j=1}^m\frac{\pi^j}{\Gamma(j)(x^2+(j+1/4)^2)} During our search of real rooted entire function approximations to Riemann \Xi function, we need to calculate the upper and lower bounds of$$f_m(x):=\sum_{j=1}^m b_j(x):=\sum_{j=1}^m\frac{\pi^j}{\... 0answers 122 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}$is a diagonal matrix, whose diagonal is a ... 0answers 94 views ### Bounding a q-expansion on a bounded open subset of the complex upper-half plane Let$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$... 0answers 229 views ### Geometric/Analytic techniques for constructive and asymptotic bounds in the Lee metric Slight extension of cross posting from http://cstheory.stackexchange.com/questions/7408/lee-metric-gilbert-varshamov-and-hamming-bounds-for-larger-relative-distance-rang (closed there) The following ... 0answers 40 views ### When does the equality hold in Dias da Silva - Hamidoune Theorem? Let$p$be prime number and let$A$be a$k$-elements subset of$\mathbb{Z}/p\mathbb{Z}$. Dias da Silva - Hamidoune Theorem states that$|h^{\hat{}}A| \geq \min(p, hk -h^2 + 1)$, where$h$is an ... 0answers 122 views ### Eigenvalue bounds from eigenvalues of Schur complement Is it possible to lower bound the minimal eigenvalue of a symmetric PSD matrix$M= \begin{pmatrix}A & C\\ C^*&B\end{pmatrix}$from the knowledge of the eigenvalues of$M$'s Schur complement ... 0answers 111 views ### Estimating when does a certain binomial sum exceed an upper bound Given a fixed integer$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} = 1,f_{...
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 ...