Recurrent relations for certain determinants

This note presents recurrent relations for certain determinants, e.g., connected with Hermite–Padé approximations of the second kind, for certain generalized hypergeometric functions.     

Counting theorems for finitep-groups

Dedicated to George Glauberman on occasion of his 50^th birthday     

Minimal number of idempotent generators for certain algebras

It is known [KRS] that for each finitely generated Banach algebra there exists a numberN such that for eachn>N the matrix algebras can be generated by three idempotents. In this paper we show that the same statement is true for direct sums and , where is a finitely generated free algebra, i.e. polynomials in several non-commuting variables. These results are new even for algebras because the numberN we obtain here improves known estimates (see for example [R]). We show that the algebra can be generated by two idempotents if and only ifn _j =2 for eachj and is singly generated. Also we give an example of a free singly generated algebra for which can not be generated by two idempotents. But% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacuWFSeIqgaacaaaa!409A!\[{\tilde {\cal B}}\] can be generated by three idempotents for each singly generated free algebra .     

Duality relations for certain single server queues

In this paper, relations are given between the joint distribution of several variables in a GI/G/1 queue and the joint distribution of variables associated with the busy cycle in the dual queue, that is in the queue which results from the original when the interarrival times and the service times are interchanged. It is assumed that the primal queue has the preemptive-resume last-come-first-served queue discipline while the dual queue may have any queue discipline which is work conserving. These relations generalize a result given recently for M/G/1 and GI/M/1 queues.     

Power automorphisms of finitep-groups

For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp ² possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.     

Two-generator Frattini subgroups of finitep-groups

This paper deals with nonabelianp-groupsT (p a prime andp>2) which are either metacyclic or Redei. These groups are classified into those which are Frattini subgroups of a finitep-groupG and those which are not. Finally, it is shown that a nonabelian two-generator group of orderp ⁿ (n>4) which is the Frattini subgroup of ap-group must be metacyclic. This work is contained in the author’s dissertation.     

Minimal sets of generators for the relation ideals of certain monomial curves

LetO be a monomial curve in the affine algebraice-space over a fieldK andP be the relation ideal ofO. ifO is defined by a sequence ofe positive integers somee-1 of which form an arithmetic sequence then we construct a minimal set of generators forP and write an explicit formula for μ(P).     

On certain integral relations and their applications

The aim of this paper is to establish certain integral relations involving the H-function of two variables and employ these to evaluate certain double integrals involving the products of two H-function of two variables of generalised arguments. The results established in this paper are of a very general nature and are capable of yielding a number of results as particular cases.     

Inclusion relations between certain sets of matrices

In this paper we study inclusion relations between the following four classes of matrices: normal matrices, matrices with equal spectral radius and spectral norm, matrices whose numerical range coincides with the convex polygon spanned by their eigenvalues, and matrices with equal numerical and spectral radii.     

Extension of certain types of generating relations

This paper gives a new expansion formula which essentially involves a double sum. As a consequence of our main result, eq. (6), various types of generating relations are seen to emerge and relevance with some known results is pointed out briefly. The usefulness of our main result is also indicated by considering its application to a probabilistic method for a lattice path enumeration problem.     

On the dynamics of certain recurrence relations

In recent analyses [3, 4] the remarkable AGM continued fraction of Ramanujan—denoted ${\cal R}_1$ (a, b)—was proven to converge for almost all complex parameter pairs (a, b). It was conjectured that ${\cal R}_1$ diverges if and only if (0≠ a = be ^{i φ} with cos ²φ ≠ 1) or (a ² = b ²? (?∞, 0)). In the present treatment we resolve this conjecture to the positive, thus establishing the precise convergence domain for ${\cal R}_1$ . This is accomplished by analyzing, using various special functions, the dynamics of sequences such as (t _n ) satisfying a recurrence $$ t_n = (t_{n-1} + (n-1) \kappa_{n-1}t_{n-2})/n, $$ where κ _n ? a ², b ² as n be even, odd respectively. As a byproduct, we are able to give, in some cases, exact expressions for the n-th convergent to the fraction ${\cal R}_1$ , thus establishing some precise convergence rates. It is of interest that this final resolution of convergence depends on rather intricate theorems for complex-matrix products, which theorems evidently being extensible to more general continued fractions.