Spaces of analytic functions of Hardy-Bloch type

For 0<p≤∞ and 0<q≤∞, the space of Hardy-Bloch type ℬ(p,q) consists of those functionsf which are analytic in the unit diskD such that (1−r)M _p (r,f′)⊂L ^q (dr/(1−r)). We note that ℬ(∞,∞) coincides with the Bloch space ℬ and that ℬ⊂ℬ(p,∞) for allp. Also, the space ℬ(p,p) is the Dirichlet spaceD _p−1 ^p . We prove a number of results on decomposition of spaces with logarithmic weights which allow us to obtain sharp results about the mean growth of the ℬ(p,q). In particular, we prove that iff is an analytic function inD and 2≤p<∞, then the conditionM _p (r,f′)=O((1−r)⁻¹), asr→1, implies that

Subtotally positive and Monge matrices

A real matrix is called k-subtotally positive if the determinants of all its submatrices of order at most k are positive. We show that for an m × n matrix, only mn inequalities determine such class for every k, 1 ? k ? min(m,n). Spectral properties of square k-subtotally positive matrices are studied. Finally, completion problems for 2-subtotally positive matrices and their additive counterpart, the anti-Monge matrices, are investigated. Since totally positive matrices are 2-subtotally positive as well, the presented necessary conditions for this completion problem are also necessary conditions for totally positive matrices.     

On bipartite graphs with small number of laplacian eigenvalues greater than two and three

All connected bipartite graphs with exactly two Laplacian eigenvalues greater than two are determined. Besides, all connected bipartite graphs with exactly one Laplacian eigenvalue greater than three are determined.     

Natural Operators Lifting Vector Fields to Bundles of Weil Contact Elements

Let A be a Weil algebra. The bijection between all natural operators lifting vector fields from m-manifolds to the bundle functor K ^A of Weil contact elements and the subalgebra of fixed elements SA of the Weil algebra A is determined and the bijection between all natural affinors on K ^A and SA is deduced. Furthermore, the rigidity of the functor K ^A is proved. Requisite results about the structure of SA are obtained by a purely algebraic approach, namely the existence of nontrivial SA is discussed.     

Congruence-preserving functions on Stone and Kleene algebras

We investigate local polynomial functions on Stone algebras and on Kleene algebras. We find a generating set for the clone of all local polynomial functions. We also represent local polynomial functions on a given algebra by polynomial functions of some canonical extension of this algebra.     

Lower bounds of the Laplacian graph eigenvalues

In this paper we prove that all positive eigenvalues of the Laplacian of an arbitrary simple graph have some positive lower bounds. For a fixed integer k 1 we call a graph without isolated vertices k-minimal if its kth greatest Laplacian eigenvalue reaches this lower bound. We describe all 1-minimal and 2-minimal graphs and we prove that for every k 3 the path P_k+1 on k + 1 vertices is the unique k-minimal graph.     

Asymptotic behaviour of oscillatory solutions of n-th order differential equations with quasiderivatives

Sufficient conditions are given under which the sequence of the absolute values of all local extremes of y[i], i {0,1,..., n – 2} of solutions of a differential equation with quasiderivatives y ^[n] = f(t, y ^[0],..., y ^[n–1]) is increasing and tends to . The existence of proper, oscillatory and unbounded solutions is proved.     

Classes of Quasi-nearly Subharmonic Functions

It is well known and important that if u ≥ 0 is subharmonic on a domain Ω in ℝ ⁿ and p > 0, then there is a constant C(n,p) ≥ 1 such that for each open ball B(x,r) ⊂ Ω. The definition of a relatively new function class, quasi-nearly subharmonic functions, is based on such a generalized mean value inequality. It is pointed out that the obtained function class is natural. It has important and interesting properties and, at the same time, it is large: In addition to nonnegative subharmonic functions, it includes, among others, Hervé’s nearly subharmonic functions, functions satisfying certain natural growth conditions, especially certain eigenfunctions, polyharmonic functions and generalizations of convex functions. Further, some of the basic properties of quasi-nearly subharmonic functions are stated in a unified form. Moreover, a characterization of quasi-nearly subharmonic functions with the aid of the quasihyperbolic metric and two weighted boundary limit results are given.      

On the Hahn-Banach Extension Property in Hardy and mixed norm spaces on the unit ball

For a nonempty setE of nonnegative integers letH _E ^{p, q, a} andH _E ^p be the closed linear span of