Balanced diagonals in frequency squares

We say that a diagonal in an array is $\lambda$-balanced if each entry occurs $\lambda$ times. Let $L$ be a frequency square of type $F(n;\lambda )$; that is, an $n\times n$ array in which each entry from $\{1,2,\dots ,m=n∕\lambda \}$ occurs $\lambda$ times per row and $\lambda$ times per column. We show that if $m?3$, $L$ contains a $\lambda$-balanced diagonal, with only one exception up to equivalence when $m=2$. We give partial results for $m?4$ and suggest a generalization of Ryser’s conjecture, that every Latin square of odd order has a transversal. Our method relies on first identifying a small substructure with the frequency square that facilitates the task of locating a balanced diagonal in the entire array.     

On Banach spaces with Kasparov and Yu's Property (H)

A new geometric property of Banach spaces recently introduced by G. Kasparov and G. Yu, called Property (H), has important applications to the strong Novikov conjecture and the coarse Novikov conjecture, yet is far from being well understood. In this paper, we investigate various uniformly continuous maps on unit spheres of Banach spaces and prove that all separable Banach lattices with nontrivial cotype have Property (H).     

An Intrinsic Rigidity Theorem for Closed Minimal Hypersurfaces in S5 with Constant Nonnegative Scalar Curvature

Let M⁴ be a closed minimal hypersurface in \(\mathbb{S}^5\) with constant nonnegative scalar curvature. Denote by f₃ the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f₃ and g are constant, then M⁴ is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M⁴. This result provides another piece of supporting evidence to the Chern conjecture.     

Connectivity keeping stars or double-stars in 2-connected graphs

In Mader (2010), Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta \left(G\right)\ge ?\frac{3}{2}k?+m?1$ contains a subtree $T^{\prime }$ isomorphic to $T$ such that $G?V\left(T^{\prime }\right)$ is $k$-connected. In the same paper, Mader proved that the conjecture is true when $T$ is a path. Diwan and Tholiya (2009) verified the conjecture when $k=1$. In this paper, we will prove that Mader’s conjecture is true when $T$ is a star or double-star and $k=2$.     

Remarks on torsionfreeness and its applications

In this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre’s condition (S_n). As its applications, we give a characterization of Cohen-Macaulay rings R such that R_𝔭 is Gorenstein for all prime ideals 𝔭 of height less than n, and we will give a partial answer of Tachikawa conjecture and Auslander-Reiten conjecture.     

Involutive automorphism of symmetric groups

Let (𝔖_n,S) be a Coxeter system of the symmetric group, we show that the set of automorphisms of 𝔖_n which are involutions and leave S stable is a finite group of order less than 3.     

Lifting of polynomial symplectomorphisms and deformation quantization

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the a?ne space and is closely related to several major open problems in algebraic geometry and ring theory.

This paper is a continuation of the study [19 Kanel Belov, A., Razavinia, F., Zhang, W. (2017). Bergman’s centralizer theorem and quantization. Commun. Algebra 1–7.[Taylor &; Francis Online] , [Google Scholar]].