• Volume 123, Issue 2

May 2013,   pages  151-302

• Bell Numbers, Determinants and Series

In this article, we study Bell numbers and Uppuluri Carpenter numbers. We obtain various expressions and relations between them. These include polynomial recurrences and expressions as determinants of certain matrices of binomial coefficients.

• The Smallest Randić Index for Trees

The general Randic index Rα(G) is the sum of the weight d(u)d(v)α over all edges uv of a graph G, where α is a real number and d(u) is the degree of the vertex u of G. In this paper, for any real number α=0, the first three minimum general Randic indices among trees are determined, and the corresponding extremal trees are characterized.

• A New Generalization of Hardy-Berndt Sums

In this paper, we construct a new generalization of Hardy–Berndt sums which are explicit extensions of Hardy–Berndt sums. We express these sums in terms of Dedekind sums $s_r(h,k:x,y|𝜆)$ with 𝑥=𝑦=0 and obtain corresponding reciprocity formulas.

• On Partial Sums of a Spectral Analogue of the Möbius Function

Sankaranarayanan and Sengupta introduced the function 𝜇*(𝑛) corresponding to the Möbius function. This is defined by the coefficients of the Dirichlet series $1/L_f(s)$, where $L_f(s)$ denotes the 𝐿-function attached to an even Maaß cusp form 𝑓. We will examine partial sums of 𝜇*(𝑛). The main result is $𝛴_{n≤ x} 𝜇*(𝑛)=O(x \exp(-A\sqrt{\log x}))$, where 𝐴 is a positive constant. It seems to be the corresponding prime number theorem.

• Distribution of Residues and Primitive Roots

Given an integer 𝑁 ≥ 3, we shall prove that for all primes $p≥(N-2)^2 4^N$, there exists 𝑥 in $(\mathbb{Z}/p\mathbb{Z})^∗$ such that $x,x+1,\ldots,x+N-1$ are all squares (respectively, non-squares) modulo 𝑝. Similarly, for an integer $N≥ 2$, we prove that for all primes $p≥ \exp(2^{5.54N})$, there exists an element $x\in(\mathbb{Z}/p\mathbb{Z})^∗$ such that $x,x+1,\ldots,x+N-1$ are all generators of $(\mathbb{Z}/p\mathbb{Z})^∗$.

• On Rationality of Moduli Spaces of Vector Bundles on Real Hirzebruch Surfaces

Let 𝑋 be a real form of a Hirzebruch surface. Let $M_H(r,c_1,c_2)$ be the moduli space of vector bundles on 𝑋. Under some numerical conditions on $r,c_1$ and $c_2$, we identify those $M_H(r,c_1,c_2)$ that are rational.

• Characteristic Classes for $GO(2n)$ in étale Cohomology

Let $GO(2n)$ be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic $≠ 2$. We determine the smooth-étale cohomology ring with $\mathbb{F}_2$ coefficients of the algebraic stack $BGO(2n)$. In the topological category, Holla and Nitsure determined the singular cohomology ring of the classifying space $BGO(2n)$ of the complex Lie group $GO(2n)$ in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are:

• an extension to étale cohomology of an idea of Totaro, originally used in the context of Chow groups, which allows us to approximate the classifying stack by quasi projective schemes; and
• construction of a Gysin sequence for the $\mathbb{G}_m$-fibration $BO(2n)→ BGO(2n)$ of algebraic stacks.
• A Note on 2-Nilpotence of Finite Groups

Ballester-Bolinches and Guo showed that a finite group 𝐺 is 2-nilpotent if 𝐺 satisfies: (1) a Sylow 2-subgroup 𝑃 of 𝐺 is quaternion-free and (2) $𝛺_1(P\cap G')≤ Z(P)$ and $N_G(P)$ is 2-nilpotent. In this paper, it is obtained that 𝐺 is a non-2-nilpotent group of order 16𝑞 for an odd prime 𝑞 satisfying (1) a Sylow 2-subgroup 𝑃 of 𝐺 is not quaternion-free and (2) $𝛺_1(P\cap G')≤ Z(P)$ and $N_G(P)$ is 2-nilpotent if and only if 𝑞=3 and $G≅ GL_2(3)$.

• Conjugacy Class Sizes and Solvability of Finite Groups

Let 𝐺 be a finite group and 𝐺* be the set of primary, biprimary and triprimary elements of 𝐺. We prove that if the conjugacy class sizes of 𝐺* are {1,𝑚,𝑛,𝑚𝑛} with positive coprime integers 𝑚 and 𝑛,then 𝐺 is solvable. This extends a recent result of Kong (Manatsh. Math. 168(2)(2012) 267–271).

• Notes on Discrete Subgroups of Möbius Transformations

Jørgensen’s inequality gives a necessary condition for a nonelementary two generator subgroup of $SL(2,\mathbb{C})$ to be discrete. By embedding $SL(2,\mathbb{C})$ into $Û(1,1;\mathbb{H})$, we obtain a new type of Jørgensen’s inequality, which is in terms of the coefficients of involved isometries. We provide an example to show that this result gives an improvement over the classical Jørgensen’s inequality.

• A Note on the Fuglede-Putnam Theorem

We prove the following generalization of the Fuglede–Puntam theorem. Let 𝑁 be an unbounded normal operator in the Hilbert space, and let 𝐴 be an unbounded self-adjoint operator such that $D(N)\subseteq D(A)$. Then, $AN\subseteq N^∗A\Rightarrow AN^∗\subseteq NA$.

• Multiple Decomposability of Probabilities on Contractible Locally Compact Groups

Operator decomposable probabilities on vector spaces – generalizing (semi-)stable and self-decomposable laws – are well known. Here we are concerned with multiple decomposability on locally compact groups. In fact, as it turned out that contraction properties play an essential role, throughout we concentrate on contractible locally compact groups.

• Asymptotic Distribution of Products of Sums of Independent Random Variables

In the paper we consider the asymptotic distribution of products of weighted sums of independent random variables.

• Series Solutions and a Perturbation Formula for the Extended Rayleigh Problem of Hydrodynamic Stability

We generalize Tollmien’s solutions of the Rayleigh problem of hydrodynamic stability to the case of arbitrary channel cross sections, known as the extended Rayleigh problem. We prove the existence of a neutrally stable eigensolution with wave number $k_s>0$; it is also shown that instability is possible only for $0 < k < k_s$ and not for $k>k_s$. Then we generalize the Tollmien–Lin perturbation formula for the behavior of $c_i$, the imaginary part of the phase velocity as the wave number $k→ k_s$ − to the extended Rayleigh problem and subsequently, we use this formula to demonstrate the instability of a particular shear flow.

• # Proceedings – Mathematical Sciences

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