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Shmuel Friedland

Publications and source records attributed to Shmuel Friedland.

At least 91 records · Page 5Linked to original sources
arXiv2008

Asymptotic Positivity of Hurwitz Product Traces: Two Proofs

Consider the polynomial $tr (A + tB)^m$ in $t$ for positive hermitian matrices $A$ and $B$ with $m \in \N$. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative coefficients only. We prove that they are at least asymptotically positive, for the nontrivial case of $AB \neq 0$. More precisely, we show - once complex-analytically, once combinatorially - that the $k$-th coefficient is positive for all integer $m \geq m_0$, where $m_0$ depends on $A$, $B$ and $k$.

math-ph↗
arXiv2008

On the First Eigenvalue of Bipartite Graphs

In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which is an analog of the Brualdi- Hoffman conjecture for general graphs, and prove the conjecture in some special cases.

math.CO↗
arXiv2008

Maximizing Sum Rates in Gaussian Interference-limited Channels

We study the problem of maximizing sum rates in a Gaussian interference-limited channel that models multiuser communication in a CDMA wireless network or DSL cable binder. Using tools from nonnegative irreducible matrix theory, in particular the Perron-Frobenius Theorem and the Friedland-Karlin inequalities, we provide insights into the structural property of optimal power allocation strategies that maximize sum rates. Our approach is similar to the treatment of linear models in mathematical economies, where interference is viewed in the context of competition. We show that this maximum problem can be restated as a maximization problem of a convex function on a closed convex set. We suggest three algorithms to find the exact and approximate values of the optimal sum rates. In particular, our algorithms exploit the eigenspace of specially crafted nonnegative {\it interference matrices}, which, with the use of standard optimization tools, can provide useful upper bounds and feasible solutions to the nonconvex problem.

math.OC↗
arXiv2008

Remarks on BMV conjecture

We show that for fixed $A,B$, hermitian nonnegative definite matrices, and fixed $k$ the coefficients of the $t^k$ in the polynomial $\tr (A+tB)^m$ is positive if $\tr AB >0$ and $m>N(A,B,k)$.

math-ph↗
arXiv2008

Validations of the Asymptotic Matching Conjectures

In this paper we review the asymptotic matching conjectures for $r$-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in $d$-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of $r$-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density $p$, for any $p\in [0,1]$, for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for certain infinite $r$-regular bipartite graphs.

math.CO↗
arXiv2008

An upper bound for the number of perfect matchings in graphs

We give an upper bound on the number of perfect matchings in an undirected simple graph $G$ with an even number of vertices, in terms of the degrees of all the vertices in $G$. This bound is sharp if $G$ is a union of complete bipartite graphs. This bound is a generalization of the upper bound on the number of perfect matchings in bipartite graphs on $n+n$ vertices given by the Bregman-Minc inequality for the permanents of $(0,1)$ matrices.

math.CO↗
arXiv2008

On the graph isomorphism problem

We relate the graph isomorphism problem to the solvability of certain systems of linear equations with nonnegative variables. This version replaces the two previous versions of this paper.

cs.CC↗
arXiv2006

Generalized Friedland-Tverberg inequality: applications and extensions

We derive here the Friedland-Tverberg inequality for positive hyperbolic polynomials. This inequality is applied to give lower bounds for the number of matchings in $r$-regular bipartite graphs. It is shown that some of these bounds are asymptotically sharp. We improve the known lower bound for the three dimensional monomer-dimer entropy. We present Ryser-like formulas for computations of matchings in bipartite and general graphs. Additional algorithmic applications are given.

math.CO↗
arXiv2006

A polynomial-time approximation algorithm for the number of k-matchings in bipartite graphs

We show that the number of $k$-matching in a given undirected graph $G$ is equal to the number of perfect matching of the corresponding graph $G_k$ on an even number of vertices divided by a suitable factor. If $G$ is bipartite then one can construct a bipartite $G_k$. For bipartite graphs this result implies that the number of $k$-matching has a polynomial-time approximation algorithm. The above results are extended to permanents and hafnians of corresponding matrices.

cs.CC↗
arXiv2006

Entropy of holomorphic and rational maps: a survey

We give a brief survey on the entropy of holomorphic self maps $f$ of compact Kähler manifolds, and rational dominating self maps $f$ of smooth projective varieties. We emphasize the connection between the entropy and the spectral radii of the induced action of $f$ on the homology of the compact manifold. The main conjecture for the rational maps states that modulo birational isomorphism all various notions of entropy and the spectral radii are equal.

math.DS↗
arXiv2006

Generalized rank-constrained matrix approximations

In this paper we give an explicit solution to the rank constrained matrix approximation in Frobenius norm, which is a generalization of the classical approximation of an m by n matrix A by a matrix of rank k at most.

math.OC↗
arXiv2006

Exact conditions for countable inclusion-exclusion identity and extensions

We give simple necessary and sufficient conditions for the inclusion-exclusion identity to hold for an infinite countable number of sets. In terms of a random variable, whose range are nonnegative integers, this condition is equivalent to the convergence to zero of binomial moments. Some standard extensions of the countable inclusion-exclusion identity are also given.

math.PR↗
arXiv2005

Fast Monte-Carlo Low Rank Approximations for Matrices

In many applications, it is of interest to approximate data, given by mxn matrix A, by a matrix B of at most rank k, which is much smaller than m and n. The best approximation is given by singular value decomposition, which is too time consuming for very large m and n. We present here a Monte Carlo algorithm for iteratively computing a k-rank approximation to the data consisting of mxn matrix A. Each iteration involves the reading of O(k) of columns or rows of A. The complexity of our algorithm is O(kmn). Our algorithm, distinguished from other known algorithms, guarantees that each iteration is a better k-rank approximation than the previous iteration. We believe that this algorithm will have many applications in data mining, data storage and data analysis.

math.NA↗
arXiv2005

An Algorithm for Missing Value Estimation for DNA Microarray Data

Gene expression data matrices often contain missing expression values. In this paper, we describe a new algorithm, named improved fixed rank approximation algorithm (IFRAA), for missing values estimations of the large gene expression data matrices. We compare the present algorithm with the two existing and widely used methods for reconstructing missing entries for DNA microarray gene expression data: the Bayesian principal component analysis (BPCA) and the local least squares imputation method (LLS). The three algorithms were applied to four microarray data sets and two synthetic low-rank data matrices. Certain percentages of the elements of these data sets were randomly deleted, and the three algorithms were used to recover them. In conclusion IFRAA appears to be the most reliable and accurate approach for recovering missing DNA microarray gene expression data, or any other noisy data matrices that are effectively low rank.

q-bio.GN↗
arXiv2005

2-adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman--Friedland--Loewy on the parity of the degrees of projective varieties of $n\times n$ complex symmetric matrices of rank at most $k$. We also characterize the parity of the degrees of projective varieties of $n\times n$ complex skew symmetric matrices of rank at most $2p$. We give recursive relations which determine the parity of the degrees of projective varieties of $m\times n$ complex matrices of rank at most $k$. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of $n\times n$ skew symmetric real matrices and of $m\times n$ real matrices containing a nonzero matrix of rank at most $k$. The parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box, on the one hand, and the parity of the number of symplectic tableaux of rectangular shape, on the other hand.

math.NT↗