The Dickson polynomial Dn, (x, a) of degree n is defined by denotes the greatest integer function. In particular, we define D0 (x, a) = 2 for all real x and a. By using Dickson polynomials we present new types of generalized Stirling numbers of the first and second kinds. Some basic properties of these numbers and a combinatorial application to the enumeration of functions on finite sets in terms of their range values is also given.

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaces

where H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

We prove that, for operator spaces V and W, the operator space V** ⊗hW** can be completely isometrically embedded into (V ⊗hW)**, ⊗h being the Haagerup tensor product. We also show that, for exact operator spaces V and W, a jointly completely bounded bilinear form on V × W can be extended uniquely to a separately w*-continuous jointly completely bounded bilinear form on V× W**. This paves the way to obtaining a canonical embedding of into with a continuous inverse, where is the operator space projective tensor product. Further, for C*-algebras A and B, we study the (closed) ideal structure of which, in particular, determines the lattice of closed ideals of completely.

In a recent paper Professor A. C. Aitken directed attention to a little known type of inverse factorial series, which he calls inverse central factorial series. He showed that this type of series is a very powerful means of asymptotic representation of functions of a certain type, and concluded his note with the-remark that there is here scope for considerable investigation. As a modest contribution to this investigation, the asymptotic theory, foreshadowed in the last paragraph of Aitken's paper, will be given here: it is hoped to give the convergence theory of inverse central factorial series later.

If R is a ring with identity and E is the semigroup of identity preserving ring endomorphisms of R then the fixed ring of E is denned by for each φɛE}. R is said to be rigid if RE = R.Equivalently, R is rigid if and only if

The sampling theorem, often referred to as the Shannon or Whittaker-Kotel'nikov- Shannon sampling theorem, is of considerable importance in many fields, including communication engineering, electronics, control theory and data processing, and has appeared frequently in various forms in engineering literature (a comprehensive account of its numerous extensions and applications is given in [3]). The result states that a band-limited signal, i.e. a real function f of the form

where w>0, is under reasonable conditions on the even function F, determined by its values on the sampling set (l/2w)ℤ and can be reconstructed from the samples f(k/2w), k∈ℤ, by the series

It is characteristic for the development of many processes that in certain moments they change their state in jumps. Systems with impulse effect provide an adequate mathematical model of such processes. The investigation of these systems begins with the paper of Millman and Myshkis [7] and afterwards the number of publications dedicated to this problem rapidly increases.

Let M(u), H(u) be the maximal operator and Hilbert transform along the parabola (t, ut2). For U ⊂ (0, ∞) we consider Lp estimates for the maximal functions sup u∈U|M(u)f| and sup u∈U|H(u)f|, when 1 < p ≤ 2. The parabolas can be replaced by more general non-flat homogeneous curves.

For positive integral n and positive real x1, … xn let

Where

and let

The main purpose of this paper is to obtain an explicit Capelli identity relating skew-symmetric matrices under the action of the general linear group $GL_N$. In particular, we give an explicit formula for the skew Capelli element in terms of the trace of powers of a matrix defined by the standard infinitesimal generators of $GL_N$.

AMS 2000 Mathematics subject classification: Primary 17B35. Secondary 17B10; 15A15; 20G05

The problem considered in this paper is that of finding conditions on a range space such that the closed-graph theorem holds for linear mappings from a class of linear topological spaces. The concept of a -space, which is a result of this investigation, is meaningful for commutative topological groups but we limit our consideration in this paper to linear topological spaces. On restricting ourselves to locally convex linear topological spaces, we see that the notion of a -space is an extension of the powerful idea of a B-complete space.

In this paper we first obtain elementary solutions of the integral equations

and

Using these solutions we then define operators of fractional integration. These operators may be regarded as a generalisation of the operators of fractional integration introduced by Sneddon (1) as a modification of Erdé1yi–Kober operators. In fact Erdélyi–Kober–Sneddon operators may be obtained by multiplying both sides of the equations by α-1½β and considering the limiting case α→0. We employ these operators to find a generalisation of the Mellin transform.

In the present paper, which is a direct sequel of our paper [14] joint with Roozbeh Hazrat, we prove an unrelativized version of the standard commutator formula in the setting of Chevalley groups. Namely, let Φ be a reduced irreducible root system of rank ≥ 2, let R be a commutative ring and let I,J be two ideals of R. We consider subgroups of the Chevalley group G(Φ, R) of type Φ over R. The unrelativized elementary subgroup E(Φ, I) of level I is generated (as a group) by the elementary unipotents xα(ξ), α ∈ Φ, ξ ∈ I, of level I. Obviously, in general, E(Φ, I) has no chance to be normal in E(Φ, R); its normal closure in the absolute elementary subgroup E(Φ, R) is denoted by E(Φ, R, I). The main results of [14] implied that the commutator [E(Φ, I), E(Φ, J)] is in fact normal in E(Φ, R). In the present paper we prove an unexpected result, that in fact [E(Φ, I), E(Φ, J)] = [E(Φ, R, I), E(Φ, R, J)]. It follows that the standard commutator formula also holds in the unrelativized form, namely [E(Φ, I), C(Φ, R, J)] = [E(Φ, I), E(Φ, J)], where C(Φ, R, I) is the full congruence subgroup of level I. In particular, E(Φ, I) is normal in C(Φ, R, I).

The class of non-metacyclic finite soluble groups known to have 2-generator 2-relation presentations is small. Classes of such groups are given in (3), (4), (8) and (9). Some subclasses of the groups discussed in (1) and (2) also provide examples, while a class of finite nilpotent 2-generator 2-relation groups is given by Macdonald in (7).

The term Geometrography is new to mathematical science, and it may be defined, in the words of its inventor, as “the art of geometrical constructions.”

We study the biharmonic equation Δ2u = u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn, n ≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

We establish a defect relation for algebraically non-degenerate meromorphic maps over generalized p-parabolic manifolds that intersect hypersurfaces in smooth projective algebraic varieties, extending certain results of H. Cartan, L. Ahlfors, W. Stoll, M. Ru, P. M. Wong and Philip P. W. Wong and others.

Proper ample monoids are described by means of a certain category acted upon on both sides by a cancellative monoid. Making use of this characterization, we show that every ample monoid $S$ has a proper ample cover, which can be taken to be finite whenever $S$ is finite.

We establish Bohr inequalities for operator-valued functions, which can be viewed as analogues of a couple of interesting results from scalar-valued settings. Some results of this paper are motivated by the classical flavour of Bohr inequality, while others are based on a generalized concept of the Bohr radius problem.

Let A be an abelian surface and let |D| be a polarization of type (1,3) on A. If (A,|D|) is not a product of elliptic curves, such a polarization induces a finite morphism Q: A →p2C of degree 6. In this paper we describe the branch locus of Q when A is bielliptic in thesense of K. Hulek and S. H. Weintraub (see [13]), generalizing the results proved by Ch. Birkenhake and H. Lange in [4].