Ke-Pao Lin

Counting the Number of Integral Points in General

Recently there has been tremendous interest in counting the number of integral points in n-dimensional tetrahedra with non-integral vertices due to its applications in primality testing and factoring in number theory and in singularities theory. The purpose of this note is to formulate a conjecture on sharp upper estimate of the number of integral points in n-dimensional tetrahedra with non-integral vertices. We show that this conjecture is true for low dimensional cases as well as in the case of homogeneous n-dimensional tetrahedra. We also show that the Bernoulli polynomials play a role in this counting. Received by the editors September 5, 2001; revised October 1, 2002. The second author was supported in part by NSA. Ze-Jiang Professor of East China Normal University. AMS subject classification: 11B75, 11H06, 11P21, 11Y99. c ©Canadian Mathematical Society 2003. 229

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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.

-Dimensional Tetrahedra and Bernoulli Polynomials

It is well known that the Kolmogorov equation is a fundamental equation in applied science, especially in electrical engineering. We present two closed form solutions to the Kolmogorov equation which plays an essential role in nonlinear filtering.

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

Counting the number of integral points in n-dimensional tetrahedra with non-integral vertice is an important problem. It has applications in primality testing and factoring in number theory and interesting applications in geometry and singularity theory. We proposed GLY conjecture on sharp upper estimate of the number of integral points in n-dimensional tetrahedra with non-integral vertice in 2003. But GLY conjecture claim that the n dimensional (n ≥ 3) real right-angled simplice with vertices whose distance to the origin are at least n - 1. A natural problem is how to form a new sharp estimate without the minimal distance assumption. In this paper, we formulate a Number Theoretic Conjecture which is a direct correspondence of the Yau Geometry conjecture. This paper gives hope to prove the new conjecture in general. As an application, we give a sharp estimate of Dickman-De Bruijn function Ψ(x, y) for y ＜ 11.

Recent advance in computing Kolmogorov equation arising from nonlinear filtering

The celebrated work of Yau and Yau [1] solved the nonlinear filtering problem in theory in the following manner. They reduced the problem of solving the Duncan-Mortensen-Zakai equation in real-time to the off-time solution of a Kolmogorov type equation. For the Yau filtering system, this Kolmogorov equation can be transformed as the Schrödinger equation. In this paper, we shall describe the fundamental solution of this Schrödinger equation with quartic potential.