Shmuel Friedland

Dynamical properties of plane polynomial automorphisms

This note studies the dynamical behavior of polynomial mappings with polynomial inverse from the real or complex plane to itself.

Perron--Frobenius theorem for nonnegative multilinear forms and extensions

We prove an analog of Perron-Frobenius theorem for multilinear forms with nonnegative coefficients, and more generally, for polynomial maps with nonnegative coefficients. We determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector.

Thr formulation and analysis of numerical methods for inverse Eigenvalue problems

We consider the formulation and local analysis of various quadratically convergent methods for solving the symmetric matrix inverse eigenvalue problem. One of these methods is new. We study the case where multiple eigenvalues are given: we show how to state the problem so that it is not overdetermined, and describe how to modify the numerical methods to retain quadratic convergence on the modified problem. We give a general convergence analysis, which covers both the distinct and the multiple eigenvalue cases. We also present numerical experiments which illustrate our results.

Inverse eigenvalue problems

Abstract The classical inverse additive and multiplicative inverse eigenvalue problems for matrices are studied. Using general results on the solvability of polynomial systems it is shown that in the complex case these problems are always solvable by a finite number of solutions. In case of real symmetric matrices the inverse problems are reformulated to have a real solution. An algorithm is given to obtain this solution.

On an inverse problem for nonnegative and eventually nonnegative matrices

Let σ= (λ1,···λn)⊂C. We discuss conditions for which σ is the spectrum of a nonnegative or eventually nonnegative matrix. This brings us to study rational functions with nonnegative Maclaurin coefficients. A conjecture for special sets σ is stated and some evidence in support of this conjecture is given.

Convex spectral functions

In this paper we characterize all convex functionals defined on certain convex sets of hermitian matrices and which depend only on the eigenvalues of matrices. We extend these results to certain classes of non-negative matrices. This is done by formulating some new characterizations for the spectral radius of non-negative matrices, which are of independent interest.

Spectrum Management in Multiuser Cognitive Wireless Networks: Optimality and Algorithm

Spectrum management is used to improve performance in multiuser communication system, e.g., cognitive radio or femtocell networks, where multiuser interference can lead to data rate degradation. We study the nonconvex NP-hard problem of maximizing a weighted sum rate in a multiuser Gaussian interference channel by power control subject to affine power constraints. By exploiting the fact that this problem can be restated as an optimization problem with constraints that are spectral radii of specially crafted nonnegative matrices, we derive necessary and sufficient optimality conditions and propose a global optimization algorithm based on the outer approximation method. Central to our techniques is the use of nonnegative matrix theory, e.g., nonnegative matrix inequalities and the Perron-Frobenius theorem. We also study an inner approximation method and a relaxation method that give insights to special cases. Our techniques and algorithm can be extended to a multiple carrier system model, e.g., OFDM system or receivers with interference suppression capability.

Cusps, θ trajectories, and the complex virial theorem

The conditions are discussed under which cusps in ϑ trajectories of complex resonance energies correspond to eigenvalues which satisfy the complex virial theorem. (AIP)

Regular Subgraphs of Almost Regular Graphs

Suppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k ≥ 2q − 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r ((2q − 2)(2q − 1))(Δ + 1), where q is a prime power, contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r ≡ q (mod 2)). These results follow from Chevalleys and Olsons theorems on congruences.

Explicit construction of families of LDPC codes with no 4-cycles

Low-density parity-check (LDPC) codes are serious contenders to turbo codes in terms of decoding performance. One of the main problems is to give an explicit construction of such codes whose Tanner graphs have known girth. For a prime power q and m/spl ges/2, Lazebnik and Ustimenko construct a q-regular bipartite graph D(m,q) on 2q/sup m/ vertices, which has girth at least 2/spl lceil/m/2/spl rceil/+4. We regard these graphs as Tanner graphs of binary codes LU(m,q). We can determine the dimension and minimum weight of LU(2,q), and show that the weight of its minimum stopping set is at least q+2 for q odd and exactly q+2 for q even. We know that D(2,q) has girth 6 and diameter 4, whereas D(3,q) has girth 8 and diameter 6. We prove that for an odd prime p, LU(3,p) is a [p/sup 3/,k] code with k/spl ges/(p/sup 3/-2p/sup 2/+3p-2)/2. We show that the minimum weight and the weight of the minimum stopping set of LU(3,q) are at least 2q and they are exactly 2q for many LU(3,q) codes. We find some interesting LDPC codes by our partial row construction. We also give simulation results for some of our codes.