ABSTRACT. An n×n matrix M is called a fooling-set matrix of size n if its diagonal entries are nonzero and Mk,lMl,k = 0 for every k 6= l. Dietzfelbinger, Hromkovič, and Schnitger (1996) showed that n ≤ (rkM)2 , regardless of over which field the rank is computed, and asked whether the exponent on rkM can be improved. We settle this question. In characteristic zero, we construct an infinite fami...

We consider (max,+)-algebra products of random matrices, which arise from performance evaluation of acyclic fork-join queueing networks. A new algebraic technique to examine properties of the product and investigate its limiting behaviour is proposed based on an extension of the standard matrix (max,+)-algebra by endowing it with the ordinary matrix addition as an external operation. As an appl...

A transfer-matrix simulation scheme for the three-dimensional (d=3) bond percolation is presented. Our scheme is based on Novotny's transfer-matrix formalism, which enables us to consider arbitrary (integral) number of sites N constituting a unit of the transfer-matrix slice even for d=3. Such an arbitrariness allows us to perform systematic finite-size-scaling analysis of the criticality at th...

The non-abelian tensor product of groups which has its origins in algebraic K-theory as well as inhomotopy theory, was introduced by Brown and Loday in 1987. Group theoretical aspects of non-abelian tensor products have been studied extensively. In particular, some studies focused on the relationship between the exponent of a group and exponent of its tensor square. On the other hand, com...

For DNA sequences of various species we construct the Google matrix [Formula: see text] of Markov transitions between nearby words composed of several letters. The statistical distribution of matrix elements of this matrix is shown to be described by a power law with the exponent being close to those of outgoing links in such scale-free networks as the World Wide Web (WWW). At the same time the...

In 2003 Cohn and Umans introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets S, T and U of a group G satisfying the Triple Product Property (TPP) as a means to bound the exponent ω of the matrix multiplication. We show that S, T and U may be be assumed to contain the identity and be otherwise disjoint. We also give a much shorter proof of the u...

We consider the problem of detecting (testing) Gaussian stochastic sequences (signals) with imprecisely known means and covariance matrices. An alternative is independent identically distributed zero-mean random variables unit variances. For a given false alarm (1st-kind error) probability, quality minimax detection by best miss probability (2nd-kind error probability) exponent over growing obs...

Given a square, nonsingular matrix of univariate polynomials F ∈ K[x]n×n over a field K, we give a deterministic algorithm for finding the determinant of F. The complexity of the algorithm is O ̃(nωs) field operations where s is the average column degree or the average row degree of F. Here O ̃ notation is Big-O with log factors omitted and ω is the exponent of matrix multiplication.

One of the phenomena that influences significantly the performance of low-density parity-check codes is known as trapping sets. An (a, b) elementary trapping set, or simply an ETS where a is the size and b is the number of degree-one check nodes and b a < 1, causes high decoding failure rate and exert a strong influence on the error floor. In this paper, we provide sufficient conditions for exp...

We show that the ultradifferentiable-like classes of smooth functions introduced and studied by S. Pilipović, N. Teofanov F. Tomić are special cases general framework spaces ultradifferentiable defined in terms weight matrices sense A. Rainer third author. study “beyond geometric growth factors” a sequence an exponent sequence, prove these new types admit matrix representation transfer known re...