Related papers: Generalized tournament matrices with the same prin…
We prove that, with high probability, in every $2$-edge-colouring of the random tournament on $n$ vertices there is a monochromatic copy of every oriented tree of order $O (n / \sqrt{\log n})$. This generalises a result of the first, third…
Crucial to an Evolutionary Algorithm's performance is its selection scheme. We mathematically investigate the relation between polynomial rank and probabilistic tournament methods which are (respectively) generalisations of the popular…
According to the classification scheme of the generalized random matrix ensembles, we present various kinds of concrete examples of the generalized ensemble, and derive their joint density functions in an unified way by one simple formula…
In this article we present a unification of the theory of algebraic, singular, topological and directional transition matrices by introducing the (generalized) transition matrix which encompasses each of the previous four. Some transition…
We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random…
The ensemble of antagonistic matrices is introduced and studied. In antagonistic matrices the entries $\mathcal A_{i,j}$ and $\mathcal A_{j,i}$ are real and have opposite signs, or are both zero, and the diagonal is zero. This…
We study basic properties of the generalized ideal transforms $D_I(M, N)$ and the set of associated primes of the modules $R^iD_I(M,N).$
While dealing with the nontrivial task of classifying Mueller matrices, of special interest is the study of the degenerate Mueller matrices (matrices with vanishing determinant, for which the law of multiplication holds, but there exists no…
Given a tournament T=(V,A), a subset X of $V$ is an interval of T provided that for every a, b in X and x\in V-X, (a,x) in A if and only if (b,x) in A. For example, $\emptyset$, {x}(x in V) and V are intervals of T, called trivial…
We determine the inducibility of all tournaments with at most $4$ vertices together with the extremal constructions. The $4$-vertex tournament containing an oriented $C_3$ and one source vertex has a particularly interesting extremal…
We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of…
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce…
Generalized majority-minority (GMM) operations are introduced as a common generalization of near unanimity operations and Mal'tsev operations on finite sets. We show that every instance of the constraint satisfaction problem (CSP), where…
A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…
A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general…
Let $X = (x_1,\dots,x_n)$ be a vector of distinct positive integers. The $n \times n$ matrix $S = S(X) := (\gcd(x_i,x_j))_{i,j=1}^n$, where $\gcd(x_i,x_j)$ denotes the greatest common divisor of $x_i$ and $x_j$, is called the greatest…
We give a description of the minimal primes of the ideal generated by the 2 x 2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m x m minors of an m x n generic matrix when the…
The random matrix uniformly distributed over the set of all m-by-n matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called k-good random…
We consider the manipulability of tournament rules which map the results of $\binom{n}{2}$ pairwise matches and select a winner. Prior work designs simple tournament rules such that no pair of teams can manipulate the outcome of their match…
For positive integers $n,n'$, we give a combinatorial characterization for the set of quadratic inequalities on minors that are valid for all $n\times n'$ totally nonnegative matrices. This is obtained as a consequence from our earlier…