We continue to study the simplest closure conditions for chains of spectral transformations of the Laurent biorthogonal polynomials $\left( \text{LBP} \right)$. It is shown that the 1-1-periodic $q$-closure condition leads to the $\text{LBP}$ introduced by Pastro. We introduce classes of semi-classical and Laguerre-Hahn $\text{LBP}$ associated to generic closure conditions of the chain of spectral transformations.

Formulae for the nilpotency class of the p-sylow subgroups of GL(n, q) where (p, q) = 1 are derived. These formulae are used in author's following paper: “On the other pα qβ theorem of Burnside”.

We prove an (n + l)-linear inequality which generalizes the classical bilinear inequality of Young concerning the LP norm of the convolution of two functions.

The following problem arose in connection with the study of Poincaré polynomials for homogeneous spaces. Let

gi hi positive integers and set d = (h1, h1,…, hk), the greatest common divisors of the exponents in the denominator. Let hi = dri and assume that the ri′s are comprime in pairs, i.e.

Let λ be a certain measure supported on a curve in R3. We prove that if the curve has nonzero curvature and torsion, then λ * L3/2 ⊂ L2.

We study the distribution of the discrete spectrumof the Schrödinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$.

A non-zero module M having a minimal generator set contains a maximal submodule. If M is Artinian and all submodules of M have minimal generator sets then M is Noetherian; it follows that every left Artinian module of a left perfect ring is Noetherian. Every right Noetherian module of a left perfect ring is Artinian. It follows that a module over a left and right perfect ring (in particular, commutative) is Artinian if and only if it is Noetherian. We prove that a local ring is left perfect if and only if each left module has a minimal generator set.

In Connections on a parabolic principal bundle over a curve, $I$ we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizes to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence.

We prove Beurling's theorem for rank 1 Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space.

Aczel has conjectured, [1, p. 448], the possibility of developing a net theory for structures more general than quasigroups. Steps in this direction have been taken by Havel who considers nets associated with multigroupoids [2], The work presented here introduces a generalization of 3-nets and their algebraization which is wide enough to encompass most algebraic structures based on a single binary operation.

We give explicit parametrizations for all parabolic geodesics in 3-dimensional Sasakian space forms.

A friendship graph is a graph in which every two distinct vertices have exactly one common adjacent vertex (called a neighbour). Finite friendship graphs have been characterized by Erdós, Rényi and Sós [2]: Each finite friendship graph Fn which consists of n edge disjoint triangles such that all n>1 triangles have one vertex in common (F1 is a triangle i.e. the complete graph with three vertices). Thus Fn has 2n+1 vertices, 2n of them being of degree two and the remaining one (the common vertex of n triangles if n>1) being of degree 2n.

The purpose of this note is to give an elementary proof of a special case of a theorem suggested by Th. Bang (2; 3) and proved by Lee et al (5; see also 1; 4; 6; 7; 8).

In this paper we establish a close connection between three notions attached to a modular subgroup, namely, the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup, and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

Let [α, β] be a closed interval and C[α, β] be the space of continuous functions on [α, β], For g a function on [α, β] define

Let s be a non-negative function on [α, β]. Let F be an approximating function with parameter space P such that F(A, .)∊ C[α, β] for all A∊P. The Chebyshev problem with weight s is given f ∊ C[α, β], to find a parameter A* ∊ P to minimize e(A) = ||s * (f - F(A, .))|| over A∊P. Such a parameter A* is called best and F(A*,.) is called a best approximation to f.

We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_{q}$ with $d\leqslant q+1$, then there is an $\mathbb{F}_{q}$-line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ when $q=p^{r}$.

It is well-known that in a Hausdorff space, a sequence has at most one limit, but that the converse is not true. The condition that every sequence have at most one limit will be called the semi-Hausdorff condition. We will prove that the semi-Hausdorff condition is strictly stronger than the T1 -axiom and is thus between the T1 and T2 axioms. In this note, we investigate into some properties of the spaces satisfying the semi-Hausdorff condition.

Let LSC(X) denote the set of extended real valued lower semicontinuous functions on a metrizable space X. If f, f1, f2, f3,... is a sequence in LSC(X), we say 〈fn〉 is epi-convergent to f provided the sequence of epigraphs 〈epi fn〉 is Kuratowski- Painlevé convergent to epi f. In this note we address the following question: what conditions on f and/or on X are necessary and sufficient for this mode of convergence to force epigraphical convergence with respect to the stronger Hausdorff metric and Vietoris topologies?

Matrix iterative methods of solving systems of linear algebraic equations for a class of matrices which includes strictly and irreducibly diagonally dominant matrices are considered and a convergence theorem proved.

In an important series of papers ([3], [4], [5]), (see also Rosen and Galovich [1], [2]), D. Goss has developed the arithmetic of cyclotomic function fields. In particular, he has introduced Bernoulli polynomials and proved a non-existence theorem for an analogue to Fermat’s equation for regular “exponent”. For each odd prime p and integer n, l ≤ n ≤ p2-2 we derive a closed form for the nth Bernoulli polynomial. Using this result a computer search for regular quadratic polynomials of the form x2-a was made. For primes less than or equal to 269 regular quadratics exist for p= 3, 5, 7, 13, 31.