• Volume 113, Issue 3

August 2003,   pages  213-353

• Analogues of Euler and Poisson Summation Formulae

Euler–Maclaurin and Poisson analogues of the summations $\sum_{a < n ≤ b}𝜒(n)f(n), \sum_{a < n ≤ b}d(n) f(n), \sum_{a < n ≤ b}d(n)𝜒(n) f(n)$ have been obtained in a unified manner, where (𝜒(𝑛)) is a periodic complex sequence; 𝑑(𝑛) is the divisor function and 𝑓(𝑥) is a sufficiently smooth function on [𝑎, 𝑏]. We also state a generalised Abel's summation formula, generalised Euler's summation formula and Euler's summation formula in several variables.

• On the Absolute $N_{q_𝛼}$-Summability of 𝑟th Derived Conjugate Series

The object of the present paper is to study the absolute $N_{q𝛼}$-summability of 𝑟th derived conjugate series generalizing a known result.

• Necessary and Sufficient Conditions for Inclusion Relations for Absolute Summability

We obtain a set of necessary and sufficient conditions for $| \overline{N}, p_n|_k$ to imply $|\overline{N}, q_n|_s$ for 1 < 𝑘 ≤ 𝑠 < ∞. Using this result we establish several inclusion theorems as well as conditions for the equivalence of $|\overline{N}, p_n|_k$ and $|\overline{N}, q_n|_s$.

• Subanalytic Bundles and Tubular Neighbourhoods of Zero-Loci

We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector space 𝑃 of sections) on a subanalytic subset 𝑋 of a real analytic manifold 𝑀, and prove that when 𝑀 is compact, there is a Baire subset 𝑈 of sections in 𝑃 whose zero-loci in 𝑋 have tubular neighbourhoods, homeomorphic to the restriction of the given bundle to these zero-loci.

• Equivalence of Quotient Hilbert Modules

Let $\mathcal{M}$ be a Hilbert module of holomorphic functions over a natural function algebra $\mathcal{A}(𝛺)$, where $𝛺 \subseteq \mathbb{C}^m$ is a bounded domain. Let $\mathcal{M}_0 \subseteq \mathcal{M}$ be the submodule of functions vanishing to order 𝑘 on a hypersurface $\mathcal{Z} \subseteq 𝛺$. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modules $\mathcal{Q} = \mathcal{M} \ominus \mathcal{M}_0$. The invariants are given explicitly in the particular case of 𝑘 = 2.

• Nonlinear Second-Order Multivalued Boundary Value Problems

In this paper we study nonlinear second-order differential inclusions involving the ordinary vector 𝑝-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operatory theory and from multivalued analysis, we obtain solutions for both the convex' and nonconvex' problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.

• Probabilistic Representations of Solutions to the Heat Equation

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if 𝜙 is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition 𝜙, is given by the convolution of 𝜙 with the heat kernel (Gaussian density). Our results also extend the probabilistic representation of solutions of the heat equation to initial conditions that are arbitrary tempered distributions.

• Vibrations of Thin Piezoelectric Shallow Shells: Two-Dimensional Approximation

In this paper we consider the eigenvalue problem for piezoelectric shallow shells and we show that, as the thickness of the shell goes to zero, the eigensolutions of the three-dimensional piezoelectric shells converge to the eigensolutions of a two-dimensional eigenvalue problem.

• A Variational Proof for the Existence of a Conformal Metric with Preassigned Negative Gaussian Curvature for Compact Riemann Surfaces of Genus > 1

• # Proceedings – Mathematical Sciences

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