IN×N is theN×N identity matrix, q(t) ∈ L(R;R) is T-periodic and satisfies ∫ T  q(t)dt = , A(t) = [aij(t)] is aT-periodic symmetricN×N matrix-valued functionwith aij ∈ L∞(R;R) (∀i, j = , , . . . ,N ), B = [bij] is an antisymmetric N × N constant matrix, u = u(t) ∈ C(R,RN ), H(t,u) ∈ C(R × RN ,R) is T-periodic and Hu(t,u) denotes its gradient with respect to the u variable. In fact, there ...

Let A = {A1, . . . , Ar} be a partition of a set {1, . . . , m} × {1, . . . , n} into r nonempty subsets, and A = (aij) be an m × n matrix. We say that A has a pattern A provided that aij = ai′j′ if and only if (i, j), (i, j) ∈ At for some t ∈ {1, . . . , r}. In this note we study the following function f defined on the set of all m × n matrices M with s distinct entries: f(M ;A) is the smalles...

For a graph G with vertex set V (G) = {v1, v2, . . . , vn}, let S be the covering set of G having the maximum degree over all the minimum covering sets of G. Let NS[v] = {u ∈ S : uv ∈ E(G)} ∪ {v} be the closed neighbourhood of the vertex v with respect to S. We define a square matrix AS(G) = (aij), by aij = 1, if |NS[vi] ∩NS[vj]| ≥ 1, i 6= j and 0, otherwise. The graph G associated with the mat...

Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian H=T(p)+V(q) with kinetic energy T(p)=p2/2 in existing references. When a operator is appropriately adjusted as new operator, it still suitable class of problems H=K(p,q)+V(q) integrable part K(p,q)=?i=1n?j=1naijpipj+?i=1nbipi, where aij=aij(q) and bi=bi(q) are functions coordinates q. The newly...

The Tracy–Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for A=1n(aij)1≤i,j≤n∈Rn×n symmetric (aij)1≤i≤j≤n i.i.d. standard normal, fluctuations of its largest eigenvalue λ1(A) asymptotically described by a real-valued distribution TW1:n2/3(λ1(A)−2)⇒TW1. As it often happens, Gaussianity can be relaxed, ...

Generalized Kac-Moody algebras can be described in two ways: either using generators and relations, or as Lie algebras with an almost positive definite symmetric contravariant bilinear form. Unfortunately it is usually hard to check either of these conditions for any naturally occurring Lie algebra. In this paper we give a third characterization of generalized Kac-Moody algebras which is easier...