Narrow Results

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms,

It is well known that the full matrix ring over a skew-field is a simple ring. We generalize this theorem to the case of semirings. We characterize the case when the matrix semiring $M_n(S)$, of all $n\times n$ matrices over a semiring S, is congruence-simple, provided that either S has a multiplicatively absorbing element or S is commutative and additively cancellative.

In this paper, we study representations of $G_n^3$-like groups. The group $G_n^3$ itself appeared in works of the third named author on non-Reidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of $G_n^3$-like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on the $G_n^3$-like group structure.

In an article by Hamiache (IJGT, 2001) an axiomatization of the Shapley value has been proposed. Three axioms were called on, inessential game, continuity and associated consistency. This present article proposes a new proof, based on elementary linear algebra. Games are represented by vectors. Associated games are the results of matrix operations. The eigenvalues of the involved matrices are computed and it is shown that they are diagonalizable. The present contribution offers a powerful tool allowing further generalizations of the Shapley value, which were difficult to consider on the basis of the previous proof.

In this paper, we survey the main known constructions of Ferrers diagram rank-metric codes, and establish new results on a related conjecture by Etzion and Silberstein. We also give a sharp lower bound on the dimension of linear rank-metric anticodes with a given profile. Combining our results with the multilevel construction, we produce examples of subspace codes with the largest known cardinality for the given parameters. We also apply results from algebraic geometry to the study of the analogous problem over an algebraically closed field, proving that the bound by Etzion and Silberstein can be improved in this case, and providing a sharp bound for full-rank matrices.

We prove that the zero-divisor graph of a direct product of matrices over finite zero-divisor free semirings uniquely determines the sizes of matrices and cardinalities of semirings in question. We also give an example that the semirings themselves are not necessarily uniquely determined.

Let $S$ be an additively idempotent semiring and $M_n(S)$ be the semiring of all $n\times n$ matrices over $S$. We characterize the conditions of when the semiring $M_n(S)$ is congruence-simple provided that the semiring $S$ is either commutative or finite. We also give a characterization of when the semiring $M_n(S)$ is subdirectly irreducible for $S$ being almost integral (i.e. $xy+yx+x=x$ for all $x,y\in S$). In particular, we provide this characterization for the semirings $S$ derived from the pseudo MV-algebras.

Every real $m\times n$ matrix $A$ can be factored in three ways that arise from the steps of elimination: a lower triangular/upper triangular factorization $PA=LU$, a column-row factorization $A=CR$, and a triple factorization $A=C{W}^{-1}B$. The column-row factorization provides both a constructive proof that the row rank $r$ of $A$ equals the column rank, and a formula for the pseudoinverse $A^{+}$ not based on the singular value decomposition. In the triple factorization, $C$ and $B$ contain the first $r$ independent columns of $A$ and the first $r$ independent rows of $A$; $W$ is the invertible submatrix of $A$ where $B$ meets $C$. An alternative to the traditional elimination method, using slide steps in place of the usual swap steps, identifies the first $r$ independent rows of $A$.

For a nilpotent group $G$ without $\pi$-torsion, and $x$, $y\in G$, if $x^n=y^n$ for a $\pi$-number $n$, then $x=y$; if $x^m{y}^n=y^n{x}^m$ for $\pi$-numbers $m$, $n$, then $xy=yx$. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let $m$ be a given positive integer and $A$ a complex square matrix satisfying that (i) all eigenvalues of $A$ are nonnegative, and (ii) $\mathrm{rank}{A}^2=\mathrm{rank}A$; then $A$ has a unique $m$-th root $X$ with $\mathrm{rank}{X}^2=\mathrm{rank}X$, all eigenvalues of $X$ are nonnegative, and moreover there is a polynomial $f(\lambda )$ with $X=f(A)$. In addition, let $A$ and $B$ be complex $n\times n$ matrices with all eigenvalues nonnegative, and $\mathrm{rank}{A}^2=\mathrm{rank}A$, $\mathrm{rank}{B}^2=\mathrm{rank}B$; then (i) $A=B$ when $A^r=B^r$ for some positive integer $r$, and (ii) $AB=BA$ when $A^s{B}^t=B^t{A}^s$ for two positive integers $s$ and $t$.

In this paper, we get several new results on permutation polynomials over finite fields. First, by using the linear translator, we construct permutation polynomials of the forms and . These forms generalize the results obtained by Kyureghyan in 2011. Consequently, we characterize permutation polynomials of the form , which extends a theorem of Charpin and Kyureghyan obtained in 2009.

In the area of response-adaptive design based on randomized urn models, Professor Bai's research has been focused on providing a mathematically rigorous of generalized Friedman's urn model. In a series of papers, matrix recursions and martingale theory were introduced to study randomized urn models. Based on these techniques, some fundamental questions were answered.

We show that for n × n nonsingular matrices A₁, A₂, …, A_k (k ≥ 2) over a commutative principal ideal domain R with relatively coprime determinants there exist invertible n × n matrices U, V₁, …, V_k over R such that UA_iV_i = S_Ai are the Smith normal forms of the matrices A_i.

The objects of investigation are the most widespread Nature Systems on the Land - River Basins, their geodynamics and geoinformatics. For the purpose we worked out the Virtual Nature Systems (VNS) as computer doubles of River Basins and of their components in view to research their dynamics and evolution under continual influence of external factors, in analogy with the Actual Nature Systems (ANS).