On certain classes of matrix norms

Some characterizations of certain classes oi submultiplicative and/or general matrix norms are given.     

Congruences for certain binomial sums

We exploit the properties of Legendre polynomials defined by the contour integral ${{\rm{P}}_n}(z) = {(2{\rm{\pi i}})^{ - 1}}\oint {{{(1 - 2tz + {t^2})}^{ - 1/2}}{t^{ - n - 1}}{\rm{d}}t} $ , where the contour encloses the origin and is traversed in the counterclockwise direction, to obtain congruences of certain sums of central binomial coefficients. More explicitly, by comparing various expressions of the values of Legendre polynomials, it can be proved that for any positive integer r, a prime p ?5 and n = rp ² ? 1, we have $\sum\limits_{k = 0}^{\left\lfloor {n/2} \right\rfloor } {(_k^{2k} ) \equiv 0,1} $ or ?1 (mod p ²), depending on the value of r (mod 6).     

Some inequalities for certain power sums

Acta Mathematica Hungarica -     

Non-cancellation for certain classes of groups

For a certain class of groups, which are semidirect products arising from an action of a finite rank free abelian group on another group, we study cancellation of the infinite cyclic group in isomorphic direct products. As an application we obtain a sufficient condition for triviality of the genus of certain nilpotent groups.     

Combinatorial sums and finite differences

We present a new approach to evaluating combinatorial sums by using finite differences. Let and be sequences with the property that Δb_k=a_k for k?0. Let , and let . We derive expressions for g_n in terms of h_n and for h_n in terms of g_n. We then extend our approach to handle binomial sums of the form , , and , as well as sums involving unsigned and signed Stirling numbers of the first kind, and . For each type of sum we illustrate our methods by deriving an expression for the power sum, with a_k=k^m, and the harmonic number sum, with a_k=H_k=1+1/2+?+1/k. Then we generalize our approach to a class of numbers satisfying a particular type of recurrence relation. This class includes the binomial coefficients and the unsigned Stirling numbers of the first kind.     

Singular value inequalities for matrix sums and minors

This paper gives new proofs for certain inequalities previously established by the author involving sums of singular values of matrices A, B, C = A + B, and also sums of singular values of A, B, and C when A, B are complementary submatrices of C. Some new facts concerning these inequalities are also included.     

Lattice theory for certain classes of rings

Conclusion We observe that every Boolean ring is aP ₁-ring, where each element is its own minimal idempotent duplicator; a Boolean ring with unity is also aP ₂-ring, where each element is the maximal idempotent annihilator of its complement; and thus,P ₁-rings andP ₂-rings are generalisations of a Boolean ring in terms of its lattice structure. Then, it is only natural that these two classes of rings have such close affiliations with the lattice structure of a Boolean ring.     

A hybrid mean value related to certain hardy sums and Kloosterman sums

The main purpose of this paper is using the mean value formula of Dirichlet L-functions and the analytic methods to study a hybrid mean value problem related to certain Hardy sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.