Huaiqing Zuo

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function ψ(x, y) which is the number of positive integers ≤ x and free of prime factors > y. Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional (n ≥ 3) real right-angled simplices. In this paper, we prove this Number Theoretic Conjecture for n = 5. As an application, we give a sharp estimate of Dickman-De Bruijn function ψ(x, y) for 5 ≤ y < 13.

Lojasiewicz inequality for weighted homogeneous polynomial with isolated singularity

Let be a real continuous function on an interval, and consider the operator function defined for Hermitian operators . We will show that if is increasing w.r.t. the operator order, then for the operator function is convex. Let and be functions defined on an interval . Suppose is non-decreasing and is increasing. Then we will define the continuous kernel function by , which is a generalization of the Lowner kernel function. We will see that it is positive definite if and only if whenever for Hermitian operators , and we give a method to construct a large number of infinitely divisible kernel functions.

Notes on classification of toric surface codes of dimension 5

This is an addendum to the beautiful paper by Little and Schwarz (Appl Algebra Eng Commun Comput 18:349–367, 2007) in which one case of toric surface codes of dimension 5 was missing in their classification result of toric surface codes of dimension less than 6. Our main purpose is to fill the gap of this paper. We find that our new code

On classification of toric surface codes of low dimension

{C_{P_5^{(7)}}}

Coordinate-free characterization of homogeneous polynomials with isolated singularities

enjoys more symmetry, and it has more codewords of minimum distance in general. However, over some special fields

Complete characterization of isolated homogeneous hypersurface singularities

{\mathbb{F}_{2^m}}