In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

Modifications to the Feynman propagation functions Sp and Dp are suggested in order to deal with field theoretic problems involving the presence in the initial and the final states, of free fermions or bosons, having all values of momentum (and energy) within a certain specified range.

We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group \[\mathbb{H} = \mathbb{C} \times \mathbb{R}\], endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection \[\pi (E)\] (projection along the center of \[\mathbb{H}\]) almost surely equals \[\min \{ 2,{\dim _\operatorname{H} }(E)\} \] and that \[\pi (E)\] has non-empty interior if \[{\dim _{\text{H}}}(E) > 2\]. As a corollary, this allows us to determine the Hausdorff dimension of E with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension \[{\dim _{\text{H}}}(E)\].

We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere \[{{\text{S}}^3}\] endowed with the visual metric d obtained by identifying \[{{\text{S}}^3}\] with the boundary of the complex hyperbolic plane. In \[{{\text{S}}^3}\], we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in \[{{\text{S}}^3}\] satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.

Let N > 1 and let AN be the polydisc algebra, i.e. the algebra of all continuous functions on the closed polydisc δ¯N ⊂ N, analytic on the open polydisc δN, with sup norm. Call a closed set F ⊂ δ¯N a peak interpolation set for AN if given any f ε C(F), f ≠ 0, there is an extension f ε AN of f such that ¦f˜(z)¦ < ‖ f ‖ (z ε δ¯N - F); call F a norm preserving interpolation set for AN if given any f ε C(F) there is an extension f˜ ε AN of f such that ‖f˜‖ = ‖f‖. The paper gives a complete description of norm preserving interpolation sets for AN in terms of peak interpolation sets for AM, M ≤ N.

In hypergeometry loci defined by parametric equations

have been frequently discussed from various points of view. Among other properties these loci possess two systems of generating spaces obtained by keeping the α's (or β's) fixed while the β's (or α's) vary, the simplest example being the generators of a quadric in S3. Another method of obtaining loci having these properties is by taking the flat spaces through the sets of a linear series of sets of points on an elliptic curve and the flat spaces through the sets of the residual series (the curve being supposed normal. In this way we shall obtain a configuration of ∞PSP's and ∞qSq's, the dual of which gives a locus included among those represented by (1).

An explicit formula is derived for exp (iβJz) as a finite sum of irreducible tensor components. With this formula, a technique is developed to obtain the matrix elements of exp (iβJy).

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we mean

and let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfying

For such kernels, the iterated kernels,

are well-defined (1), as are the higher order traces

Carleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinant

the latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).

The thermal boundary layer in the converging flow between non-parallel plane walls is studied. Analytical solutions of the boundary-layer equations are derived and the heat transfer across the wall is obtained from these solutions.

In this paper comparisons are made between the class of continuous functions of generalized bounded variation and the class of continuous functions with graphs having σ-finite length i.e. linear measure. An investigationof the differentiability and approximate differentiability of such functions discloses the fact that the latter class is considerably more extensivethan the former one. The following definitions will be needed:

(1) A function f is said to be of bounded variation (VB) on a set E if

where the supremum is taken over all sequences {[ai, bi]} of non-overlapping intervals with endpoints in E.

Further consideration is given to properties of the Stokes phenomenon associated with a class of nth order differential equations with transition points of order m (a rational fraction) at the origin. The systematic vanishing of the Stokes multipliers leads to the discovery of sets of equations that are transformable from certain specific values of m to particular lower values of m. A series of theorems in the theory of congruences is first proved, from which the main transformation theorem is then deduced. The theory is ifiustrated by various numerical examples.

We study the iteration of transcendental self-maps of $\mathcal{C}^*:\=\mathcal{C}\{0}$, that is, holomorphic functions $\fnof:\mathcal{C}^*:\rarr\mathcal{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0},\infin$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathcal{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

The forced torsional motion of a semi-infinite isotropic right-circular elastic cylinder is studied in this paper, the vibrations being excited by a harmonically oscillating rigid disc, securely attached to the plane face of the cylinder. Two distinctsituations are considered, namely, when the curved surface of the cylinder is clamped and when it is stress free. In both cases the problem is reduced to the solution of a Fredholm integral equation of the second kind which can be solved iteratively when the frequency of oscillation is small and the cylinder radius large. Using the iterative solutions, approximate expressions are deduced for the distribution of shear stress under the disc and the driving couple required. In an appendix a discussion is given of a mathematically similar problem in the theory of the swirling flow of a perfect fluid.

1. Introduction. In a recent paper the author (2) has derived a number of integrals with the help of results ((5), p. 343)

where α, β, α′, β′ are parameters such that

and

whereK has the same meaning as above.

Let A be a finite set of integers and FA(x) = ∑a∈A exp(2πiax) be its exponential sum. McGehee, Pigno and Smith and Konyagin have independently proved that ∥FA∥1 ≥ c log|A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L1-norm of exponential sums of sets in the d-dimensional grid d. We show that ∥FA∥1 is considerably larger than log|A| when A ⊂ d has multidimensional structure. We furthermore prove similar lower bounds for sets in , which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno and Smith and Konyagin.

Cromwell and Marar present an analysis of semi-regular (generic) surfaces with a single triple point and connected self-intersection set. Six of their surfaces are the projective plane, including Boy's surface and Steiner's surface. We build on their work by incorporating twists similar to that of Apery's immersion of the projective plane and show that with a few additional surfaces, all such generic maps of the projective plane are now identified.

We extend to a wider class of rearrangement invariant spaces a result of S. Guerre concerning the extraction of almost symmetric sequences from a given weakly null sequence in Lp.

A condition of frequency domain type on the vector polynomial G(z) is obtained which is both necessary and sufficient for the Hurwitzian set of G(z) to include the special Hurwitzian set , where E(z) = col (1, z, …, zn). This result is extended to Hurwitzian sets arising from scalar delay-differential equations. Analogous conditions are also given for each of the inclusions where denotes the Schur set of G(z).