• Volume 122, Issue 2

May 2012,   pages  153-311

• 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.

• On Non-Frattini Chief Factors and Solvability of Finite Groups

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.

• Permutation Groups with Bounded Movement having Maximum Orbits

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.

• Split Malcev Algebras

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.

• 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.

• On the Uniqueness of Meromorphic Functions that Share Three or Two Finite Sets on Annuli

• 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.

• 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.

• On the Extrema of Dirichlet's First Eigenvalue of a Family of Punctured Regular Polygons in Two Dimensional Space Forms

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 $\wp 1$.

• Asymptotic Behavior of Stochastic Two-Dimensional Navier-Stokes Equations with Delays

The paper proves the 𝐿2-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.

• 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.

• # Proceedings – Mathematical Sciences

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