pp 153-162
Farey Sequences and Resistor Networks
In this article, we employ the Farey sequence and Fibonacci numbers to establish strict upper and lower bounds for the order of the set of equivalent resistances for a circuit constructed from 𝑛 equal resistors combined in series and in parallel. The method is applicable for networks involving bridge and non-planar circuits.
pp 163-173
On Non-Frattini Chief Factors and Solvability of Finite Groups
Jianjun Liu Xiuyun Guo Qianlu Li
A subgroup 𝐻 of a group 𝐺 is said to be a semi $CAP^∗$-subgroup of 𝐺 if there is a chief series $1 = G_0$ < $G_1$ <$\cdots$ < $G_m = G$ of 𝐺 such that for every non-Frattini chief factor $G_i/G_{i-1},H$ either covers $G_i/G_{i-1}$ or avoids $G_i/G_{i-1}$. In this paper, some sufficient conditions for a normal subgroup of a finite group to be solvable are given based on the assumption that some maximal subgroups are semi $CAP^∗$-subgroups.
pp 175-179
Permutation Groups with Bounded Movement having Maximum Orbits
Mehdi Alaeiyan Behnam Razzaghmaneshi
Let 𝐺 be a permutation group on a set 𝛺 with no fixed points in 𝛺 and let 𝑚 be a positive integer. If no element of 𝐺 moves any subset of 𝛺 by more than 𝑚 points (that is, $|𝛤^g\backslash𝛤|≤ m$ for every $𝛤\subseteq𝛺$ and $g\in G$), and also if each 𝐺-orbit has size greater than 2, then the number 𝑡 of 𝐺-orbits in 𝛺 is at most $\frac{1}{2}(3m-1)$. Moreover, the equality holds if and only if 𝐺 is an elementary abelian 3-group.
pp 181-187
Antonio J Calderón Martín Manuel Forero Piulestán José M Sánchez Delgado
We study the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero. We show that any such algebras 𝑀 is of the form $M=\mathcal{U}+\sum_jI_j$ with $\mathcal{U}$ a subspace of the abelian Malcev subalgebra 𝐻 and any $I_j$ a well described ideal of 𝑀 satisfying $[I_j, I_k]=0$ if 𝑗 ≠ 𝑘. Under certain conditions, the simplicity of 𝑀 is characterized and it is shown that 𝑀 is the direct sum of a semisimple split Lie algebra and a direct sum of simple non-Lie Malcev algebras.
pp 189-202
Discretization of Continuous Frame
In this paper we consider the notion of continuous frame of subspaces and define a new concept of continuous frame, entitled continuous atomic resolution of identity, for arbitrary Hilbert space $\mathcal{H}$ which has a countable reconstruction formula. Among the other results, we characterize the relationship between this new concept and other known continuous frames. Finally, we state and prove the assertions of the stability of perturbation in this concept.
pp 203-220
On the Uniqueness of Meromorphic Functions that Share Three or Two Finite Sets on Annuli
The purpose of this article is to investigate the uniqueness of meromorphic functions that share three or two finite sets on annuli.
pp 221-242
On Cohomology Theory for Topological Groups
We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of the identity. We show that if 𝐺 and 𝐴 are locally compact and second countable, then the second cohomology group based on locally continuous measurable cochains as above parametrizes the collection of locally split extensions of 𝐺 by 𝐴.
pp 243-255
Absolutely Continuous Spectrum and Spectral Transition for some Continuous Random Operators
In this paper we consider two classes of random Hamiltonians on $L^2(\mathbb{R}^d)$: one that imitates the lattice case and the other a Schrödinger operator with non-decaying, non-sparse potential both of which exhibit a.c. spectrum. In the former case we also know the existence of dense pure point spectrum for some disorder thus exhibiting spectral transition valid for the Bethe lattice and expected for the Anderson model in higher dimension.
pp 257-281
Let $\wp 1,\wp 0$ be two regular polygons of 𝑛 sides in a space form $M^2(𝜅)$ of constant curvature 𝜅=0,1 or -1 such that $\wp 0\subset\wp 1$ and having the same center of mass. Suppose $\wp 0$ is circumscribed by a circle 𝐶 contained in $\wp 1$. We fix $\wp 1$ and vary $\wp 0$ by rotating it in 𝐶 about its center of mass. Put $𝛺 =(\wp 1\backslash\wp 0)^0$, the interior of $\wp 1\backslash\wp 0$ in $M^2(𝜅)$. It is shown that the first Dirichlet’s eigenvalue 𝜆 1(𝛺) attains extremum when the axes of symmetry of $\wp 0$ coincide with those of $\wp 1$.
pp 283-295
Asymptotic Behavior of Stochastic Two-Dimensional Navier-Stokes Equations with Delays
The paper proves the 𝐿^{2}-exponential stability of weak solutions of two-dimensional stochastic Navier–Stokes equations in the presence of delays. The results extend some of the existing results.
pp 297-311
On the Limit Distribution of Lower Extreme Generalized Order Statistics
In a wide subclass of generalized order statistics $(gOs)$, which contains most of the known and important models of ordered random variables, weak convergence of lower extremes are developed. A recent result of extreme value theory of $m-gOs$ (as well as the classical extreme value theory of ordinary order statistics) yields three types of limit distributions that are possible in case of linear normalization. In this paper a similar classification of limit distributions holds for extreme $gOs$, where the parameters $𝛾_j,j=1,\ldots,n$, are assumed to be pairwise different. Two illustrative examples are given to demonstrate the practical importance for some of the obtained results.
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.