On matrix equations and

With the help of the concept of Kronecker map, an explicit solution for the matrix equation X−AXF=C is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation is also studied. An explicit solution for this matrix equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.     

On nonatomic submeasures on . II

It is shown that if two submeasures on that are lim sup's of sequences of measures have the same zero sets, and one is nonatomic, so is the other, and they are (ε-δ)-equivalent. Moreover, if a submeasure η is the lim sup of a sequence of lower semi-continuous (lsc) submeasures and is 0-dominated by the so-called core γ^• of an lsc submeasure γ, then η is also (ε-δ)-dominated by γ^•. And if a submeasure η of this type is 0-dominated by a nonatomic submeasure, then it is nonatomic as well.     

Busy period analysis of queue: Lattice path approach

In this paper, we find the busy period density of queues in explicit computational form, through lattice path (LP) approach. Both the arrival and service time distributions are approximated by 2-phase Cox distribution C₂, which has a Markovian property enabling us to use LP combinatorics. Since any distribution with rational Laplace–Stieltjes transform (LST) and square coefficient of variation (CV²) lying in [1/2,∞) can be approximated by a C₂([M. Agarwal, K. Sen, B. Borkakaty, Busy period density of queueing system C₃/M/1, Journal of Combinatorics, Information and Systems Sciences 31 (1–4) (2006) 127–161]), the results obtained would be applicable to a very wide class of distributions occurring in real life.     

On the nonlinear matrix equation

Based on fixed point theorems for monotone and mixed monotone operators in a normal cone, we prove that the nonlinear matrix equation always has a unique positive definite solution. A conjecture which is proposed in [X.G. Liu, H. Gao, On the positive definite solutions of the matrix equation X^s±A^TX^-tA=I_n, Linear Algebra Appl. 368 (2003) 83–97] is solved. Multi-step stationary iterative method is proposed to compute the unique positive definite solution. Numerical examples show that this iterative method is feasible and effective.     

Minimal cyclic random motion in and hyper-Bessel functions

We obtain the explicit distribution of the position of a particle performing a cyclic, minimal, random motion with constant velocity c in . The n+1 possible directions of motion as well as the support of the distribution form a regular hyperpolyhedron (the first one having constant sides and the other expanding with time t), the geometrical features of which are here investigated.The distribution is obtained by using order statistics and is expressed in terms of hyper-Bessel functions of order n+1. These distributions are proved to be connected with (n+1)th order p.d.e. which can be reduced to Bessel equations of higher order.Some properties of the distributions obtained are examined. This research has been inspired by a conjecture formulated in Orsingher and Sommella [E. Orsingher, A.M. Sommella, A cyclic random motion in R³ with four directions and finite velocity, Stochastics Stochastics Rep. 76 (2) (2004) 113–133] which is here proved to be false.     

The operator equation

The solution of the linear operator equation:A^n-1X+A^n-2XB++AXB^n-2+XB^n-1=Y is given by if the spectra of A and B are in the sector {z:z≠0,-π/n<argz<π/n}.     

Trace formulae for irreducible polynomials over with minimal order roots in

The extension field where q is a prime divisor of (P−1), has a unique structure. This paper describes this unique structure and uses it to derive formulas relating the trace values for elements in . These formulas can be refined for certain elements to produce a formula for the trace.     

Varieties of modules for

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety , that is the set of pairs of (n×n)-matrices (A,B) such that A²=B²=[A,B]=0, is equidimensional. can be identified with the ‘variety of n-dimensional modules’ for , or equivalently, for k[X,Y]/(X²,Y²). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic >2. We also prove that if e²=0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of as a direct sum of indecomposable components of varieties of -modules.     

Construction of nonseparable dual -wavelet frames in

Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are s₁×s₁ and s₂×s₂ dilation matrices with a₁(|det(M)|-1) and (2a₂+1)(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in are constructed, where s=s₁+s₂, and with Θ and M^-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.     

Compactness in

This paper is concerned with compactness for some topologies on the collection of bounded linear operators on Banach spaces. New versions of the Eberlein–Šmulian theorem and Day's lemma in the collection are established. Also we obtain a partial solution of the dual problem for the quasi approximation property, that is, it is shown that for a Banach space X if X^** is separable and X^* has the quasi approximation property, then X has the quasi approximation property.     

Characterizations of spaces in the unit ball of

Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Q_p and Q_p,0 on the unit ball of are defined as

and

In this paper, we give some derivative-free, mixture and oscillation characterizations for Q_p and Q_p,0 spaces in the unit ball of .     

Stickelberger elements for -extensions of function fields

We study Stickelberger elements associated to -extensions over global function fields of characteristic p>0 and show that they are in some sense generically irreducible in the Iwasawa algebras.     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

On the recursive sequence

Our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation

where B, C and α, β, γ are positive, k {1, 2, 3, … }, and the initial conditions x_−2k+1, … , x₋₁, x₀ are positive real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients. Our concentration is on invariant intervals, the character of semicycles, and the boundedness of the above mentioned equation. Our final comments are about informative examples.     

Coboundaries in

Let T be an ergodic automorphism of a probability space, f a bounded measurable function, . It is shown that the property that the probabilities μ(|S_n(f)|>n) are of order n^−p roughly corresponds to the existence of an approximation in L^∞ of f by functions (coboundaries) g−g○T, gL^p. Similarly, the probabilities μ(|S_n(f)|>n) are exponentially small iff f can be approximated by coboundaries g−g○T where g have finite exponential moments.

Résumé

Soit T un automorphisme ergodique d'un espace probabilisé, f une fonction bornée mesurable et . Une correspondance est établie entre l'existence de l'estimation des probabilités μ(|S_n(f)|>n) d'ordre n^−p et l'existence de l'approximation dans L^∞ de la fonction f par des cobords g−g○T où g est “presque” dans L^p. De manière similaire, les probabilités μ(|S_n(f)|>n) sont d'ordre e^−cn, pour un certain c>0, n=1,2… , si et seulement si f admet une approximation dans L^∞ par des cobords g−g○T avec g ayant des moments exponentiels.     

Weyl sums in with digital restrictions

Let be a finite field and consider the polynomial ring . Let . A function , where G is a group, is called strongly Q-additive, if f(AQ+B)=f(A)+f(B) holds for all polynomials with degB<degQ. We estimate Weyl sums in restricted by Q-additive functions. In particular, for a certain character E we study sums of the form

where is a polynomial with coefficients contained in the field of formal Laurent series over and the range of P is restricted by conditions on f_i(P), where f_i (1ir) are Q_i-additive functions. Adopting an idea of Gel'fond such sums can be rewritten as sums of the form

with . Sums of this shape are treated by applying the kth iterate of the Weyl–van der Corput inequality and studying higher correlations of the functions f_i. With these Weyl sum estimates we show uniform distribution results.     

On Rado numbers for

For all integers m3 and all natural numbers a₁,a₂,…,a_m−1, let n=R(a₁,a₂,…,a_m−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to

a₁x₁+a₂x₂++a_m−1x_m−1=x_m.