Volume 113, Issue 3
August 2003, pages 213-353
pp 213-221
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.
pp 223-241
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.
pp 243-250
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 $|\overline{N}, q_n|_s$.
pp 251-279
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.
pp 281-291
Equivalence of Quotient Hilbert Modules
Ronald G Douglas Gadadhar Misra
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.
pp 293-319
Nonlinear Second-Order Multivalued Boundary Value Problems
Leszek Gasiński Nikolaos S Papageorgiou
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.
pp 321-332
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.
pp 333-352
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.
pp 353-353 Erratum
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