pp 147-158
A Variant of Davenport's Constant
Let 𝑝 be a prime number. Let 𝐺 be a finite abelian 𝑝-group of exponent 𝑛 (written additively) and 𝐴 be a non-empty subset of $]n[:=\{1,2,\ldots,n\}$ such that elements of 𝐴 are incongruent modulo 𝑝 and non-zero modulo 𝑝. Let $k ≥ D(G)/|A|$ be any integer where 𝐷(𝐺) denotes the well-known Davenport’s constant. In this article, we prove that for any sequence $g_1,g_2,\ldots,g_k$ (not necessarily distinct) in 𝐺, one can always extract a subsequence $g_{i_1},g_{i_2},\ldots,g_{i_l}$ with $1 ≤ l ≤ k$ such that
$$\sum\limits_{j=1}^l a_j g_{i_j}=0 \text{in} G,$$
where $a_j\in A$ for all 𝑗. We provide examples where this bound cannot be improved. Furthermore, for the cyclic groups, we prove some sharp results in this direction. In the last section, we explore the relation between this problem and a similar problem with prescribed length. The proof of Theorem 1 uses group-algebra techniques, while for the other theorems, we use elementary number theory techniques.
pp 159-165
Reducing System of Parameters and the Cohen–Macaulay Property
Let 𝑅 be a local ring and let $(x_1,\ldots,x_r)$ be part of a system of parameters of a finitely generated 𝑅-module 𝑀, where $r < \dim_R M$. We will show that if $(y_1,\ldots,y_r)$ is part of a reducing system of parameters of 𝑀 with $(y_1,\ldots,y_r)M=(x_1,\ldots,x_r)M$ then $(x_1,\ldots,x_r)$ is already reducing. Moreover, there is such a part of a reducing system of parameters of 𝑀 iff for all primes $P\in \mathrm{Supp} M \cap V_R(x_1,\ldots,x_r)$ with $\dim_R R/P = \dim_R M-r$ the localization $M_P$ of 𝑀 at 𝑃 is an 𝑟-dimensional Cohen–Macaulay module over $R_P$.
Furthermore, we will show that 𝑀 is a Cohen–Macaulay module iff $y_d$ is a non zero divisor on $M/(y_1,\ldots,y_{d-1})M$, where $(y_1,\ldots,y_d)$ is a reducing system of parameters of $M(d:=\dim_R M)$.
pp 167-175
Derivations and Generating Degrees in the Ring of Arithmetical Functions
Alexandru Zaharescu Mohammad Zaki
In this paper we study a family of derivations in the ring of arithmetical functions of several variables over an integral domain, and compute the generating degrees of the ring of arithmetical functions over the kernel of these derivations.
pp 177-183
Ergodic Theory of Amenable Semigroup Actions
In this paper, among other things, we state and prove the mean ergodic theorem for amenable semigroup algebras.
pp 185-196
Weighted Composition Operators on Weighted Bergman Spaces of Bounded Symmetric Domains
Sanjay Kumar Kanwar Jatinder Singh
In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator $W_{\varphi,\psi}$ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten 𝑝-class weighted composition operators.
pp 197-203
Some Further Remarks on Good Sets
We show that in 𝑛-fold cartesian product, 𝑛 ≥ 4, a related component need not be a full component. We also prove that when 𝑛 ≥ 4, uniform boundedness of lengths of geodesics is not a necessary condition for boundedness of solutions of (1) for bounded function 𝑓.
pp 205-211
Borel Hierarchies in Infinite Products of Polish Spaces
Let 𝐻 be a product of countably infinite number of copies of an uncountable Polish space 𝑋. Let $𝛴_𝜉(\overline{𝛴}_𝜉)$ be the class of Borel sets of additive class 𝜉 for the product of copies of the discrete topology on 𝑋 (the Polish topology on 𝑋), and let $\mathcal{B}=\cup_{𝜉 < 𝜔_1}\overline{𝛴}_𝜉$. We prove in the Lévy-Solovay model that
$$\overline{𝛴}_𝜉 = 𝛴_𝜉 \cap\mathcal{B}$$
for 1 ≤ 𝜉 < 𝜔_{1}.
pp 213-218
Law of Iterated Logarithm for NA Sequences with Non-Identical Distributions
Based on a law of the iterated logarithm for independent random variables sequences, an iterated logarithm theorem for NA sequences with non-identical distributions is obtained. The proof is based on a Kolmogrov-type exponential inequality.
pp 219-231
A Polycycle and Limit Cycles in a Non-Differentiable Predator-Prey Model
For a non-differentiable predator-prey model, we establish conditions for the existence of a heteroclinic orbit which is part of one contractive polycycle and for some values of the parameters, we prove that the heteroclinic orbit is broken and generates a stable limit cycle. In addition, in the parameter space, we prove that there exists a curve such that the unique singularity in the realistic quadrant of the predator-prey model is a weak focus of order two and by Hopf bifurcations we can have at most two small amplitude limit cycles.
pp 233-257
Infinite Dimensional Differential Games with Hybrid Controls
A two-person zero-sum infinite dimensional differential game of infinite duration with discounted payoff involving hybrid controls is studied. The minimizing player is allowed to take continuous, switching and impulse controls whereas the maximizing player is allowed to take continuous and switching controls. By taking strategies in the sense of Elliott–Kalton, we prove the existence of value and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities.
pp 259-265
Solutions for a Class of Iterated Singular Equations
Some fundamental solutions of radial type for a class of iterated elliptic singular equations including the iterated Euler equation are given.
pp 267-281
Fractional Extensions of some Boundary Value Problems in Oil Strata
In the present paper, we solve three boundary value problems related to the temperature field in oil strata - the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional generalization of the incomplete lumped formulation in the radial case. By using the Caputo differintegral operator and the Laplace transform, the solutions are obtained in integral forms where the integrand is expressed in terms of the convolution of some auxiliary functions of Wright function type. A generalization of the Laplace transform convolution theorem, known as Efros’ theorem is widely used.
pp 283-285
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.