构造奇次同心幻方的一种方法

幻方是古老的数学游戏,经过几个世纪的发展形成了很多有趣的构造方法.利用行列式的性质和变换得到了构造奇数阶同心幻方原基的一种方法,利用这种方法和排列组合都能得到任意奇数阶幻方的多种形式.     

A Formula for the Dedekind ξ-Function of an Imaginary Quadratic Field

In this paper a formula is given for the Dedekind ξ-function, which is similar to that for the Riemann ξ-function. As an application, a lower bound is obtained for the value of the Dedekind ξ-function at the central point.     

Negative asymptotic topological dimension, a new condensate, and their relation to the quantized Zipf law

We introduce the notion of weight for the asymptotic topological dimension. Planck’s formula for black-body radiation is refined. We introduce the notion of negative asymptotic topological dimension (of hole dimension). The condensate in the hole dimension is applied to the quantized Zipf law for frequency dictionaries (obtained earlier by the author).     

Complex-valued Ray-Singer torsion

In the spirit of Ray and Singer we define a complex-valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex-valued Reidemeister torsion, including its phase. We establish this conjecture in some non-trivial situations.     

Principal Γ-cone for a tree

For any tree Γ, we introduce Γ-cones consisting of chambers and enumerate the number of chambers contained in two particular (called principal) Γ-cones. The problem is equivalent to the combinatorial problem of the enumeration of linear extensions of two bipartite orderings on a tree Γ. We characterize the principal Γ-cones among other Γ-cones by the strict maximality of the number of their chambers, and give a formula for this maximal (called principal) number by a finite sum of hook length formulae. We explain the formula through the simplicial block decomposition of principal Γ-cones. The results have their origin and application in the study of the topology related to Coxeter groups and Artin groups.     

A generalized Ihara zeta function formula for simple graphs with bounded degree

We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from vertex-transitive graphs.     

Mapping properties of the zero-balanced hypergeometric functions

In this paper, the order of convexity of $z_2{F}_1(a,b;c;z)$ is given under certain conditions on the positive real parameters a, b and c. We show also that the image domains of the unit disc $\mathbb{D}$ under some shifted zero-balanced hypergeometric functions $z_2{F}_1(a,b;a+b;z)$ are convex and bounded by two horizontal lines. This solves a problem raised by Ponnusamy and Vuorinen in [10].