• Volume 121, Issue 3

August 2011,   pages  245-377

• On Diophantine Equations of the Form $(x-a_1)(x-a_2)\ldots(x-a_k)+r=y^n$

Erdős and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation $x(x+1)(x+2)\ldots(x+(m-1))=y^n$ has no solutions in positive integers $x,m,n$ where $m,n>1$ and $y\in Q$. We consider the equation

$$(x-a_1)(x-a_2)\ldots(x-a_k)+r=y^n$$

where $0≤ a_1 < a_2 <\cdots < a_k$ are integers and, with $r\in Q,n≥ 3$ and we prove a finiteness theorem for the number of solutions 𝑥 in $Z,y$ in 𝑄. Following that, we show that, more interestingly, for every nonzero integer 𝑛>2 and for any nonzero integer 𝑟 which is not a perfect 𝑛-th power for which the equation admits solutions, 𝑘 is bounded by an effective bound.

• A Classification of Cubic Symmetric Graphs of Order 16𝑝2

A graph is called symmetric if its automorphism group acts transitively on its arc set. In this paper, we classify all connected cubic symmetric graphs of order 16𝑝2 for each prime 𝑝.

• Deficiently Extremal Gorenstein Algebras

The aim of this article is to study the homological properties of deficiently extremal Gorenstein algebras. We prove that if 𝑅/𝐼 is an odd deficiently extremal Gorenstein algebra with pure minimal free resolution, then the codimension of 𝑅/𝐼 must be odd. As an application, the structure of pure minimal free resolution of a nearly extremal Gorenstein algebra is obtained.

• Existence of Dicritical Divisors Revisited

We characterize the dicriticals of special pencils. We also initiate higher dimensional dicritical theory.

• Test Rank of an Abelian Product of a Free Lie Algebra and a Free Abelian Lie Algebra

Let 𝐹 be a free Lie algebra of rank 𝑛 ≥ 2 and 𝐴 be a free abelian Lie algebra of rank 𝑚 ≥ 2. We prove that the test rank of the abelian product $F× A$ is 𝑚. Morever we compute the test rank of the algebra $F/𝛾 k(F)'$.

• Uniqueness and Zeros of 𝑞-Shift Difference Polynomials

In this paper, we consider the zero distributions of 𝑞-shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to 𝑞-shift difference polynomials. We also investigate the uniqueness problem of 𝑞-shift difference polynomials that share a common value.

• Spherical Means in Annular Regions in the 𝑛-Dimensional Real Hyperbolic Spaces

Let $Z_{r,R}$ be the class of all continuous functions 𝑓 on the annulus $\mathrm{Ann}(r,R)$ in the real hyperbolic space $\mathbb{B}^n$ with spherical means $M_sf(x)=0$, whenever $s>0$ and $x\in\mathbb{B}^n$ are such that the sphere $S_s(x)\subset\mathrm{Ann}(r,R)$ and $B_r(o)\subseteq B_s(x)$. In this article, we give a characterization for functions in $Z_{r,R}$. In the case 𝑅=∞, this result gives a new proof of Helgason’s support theorem for spherical means in the real hyperbolic spaces.

• Composition Operators between Bloch Type Spaces and Zygmund Spaces in the Unit Ball

The boundedness and compactness of composition operators between Bloch type spaces and Zygmund spaces of holomorphic functions in the unit ball are characterized in the paper.

• Existence of Solution of the Pullback Equation Involving Volume Forms

Let $𝛺\subset\mathbb{R}^n$ be a smooth, bounded domain. We study the existence and regularity of diffeomorphisms of 𝛺 satisfying the volume form equation

$$𝜙^*(g)=f,\quad\text{in}\, 𝛺$$

where $f,g\in C^{m,𝛼}(\overline{𝛺};𝛬^n)$ are given positive volume forms.

• Value Functions for Certain Class of Hamilton Jacobi Equations

We consider a class of Hamilton Jacobi equations (in short, HJE) of type

$$u_t+\frac{1}{2}\left(|u_{x_n}|^2+\cdots+|u_{x_{n-1}}|^2\right)+\frac{e^u}{m}|u_{x_n}|^m=0,$$

in $\mathbb{R}^n×\mathbb{R}_+$ and 𝑚>1, with bounded, Lipschitz continuous initial data. We give a Hopf-Lax type representation for the value function and also characterize the set of minimizing paths. It is shown that the minimizing paths in the representation of value function need not be straight lines. Then we consider HJE with Hamiltonian decreasing in 𝑢 of type

$$u_t+H_1(u_{x_1},\ldots,u_{x_i})+e^{-u}H_2(u_{x_{i+1}},\ldots,u_{x_n})=0$$

where $H_1,H_2$ are convex, homogeneous of degree $n,m>1$ respectively and the initial data is bounded, Lipschitz continuous. We prove that there exists a unique viscosity solution for this HJE in Lipschitz continuous class. We also give a representation formula for the value function.

• An almost Sure Central Limit Theorem for the Weight Function Sequences of NA Random Variables

Consider the weight function sequences of NA random variables. This paper proves that the almost sure central limit theorem holds for the weight function sequences of NA random variables. Our results generalize and improve those on the almost sure central limit theorem previously obtained from the i.i.d. case to NA sequences.

• # Proceedings – Mathematical Sciences

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