On Higher Dimensional Orchard Visibility Problem

Author：Shengning Zhang

Abstract：In this article, we study  P$\mathrm{\acute o}$lya's orchard visibility problem in arbitrary dimension $d$: suppose at every integral point in $\mathbb{R}^d$, centered a small $d$-dimensional ball with radius $r$ (which is considered as a tree at the integral point), given a  $d$-dimensional ball centered at the origin $O$ with radius $R$ (which is considered as the orchard), it asks for the smallest $r$ such that every ray starting from $O$ will hit some tree in the orchard. We give both the upper and the lower bounds of the minimal value of $r$, say $\rho$ in terms of $R$. Moreover, we prove that as $R\rightarrow\infty$, $\rho=\mathcal O(R^{-\frac{1}{d-1}})$.

On higher dimensional orchard problem.pdf

Properties to Determine Inscribed Ellipses of Polygons

Author: Yarong Li

Abstract: In this paper, we extend the result of \cite{1} by calculating some examples in detail, including the inscribed ellipses in triangles, quadrilaterals, and pentagons. We also improve the original proof and reduce the requirements through projective geometry methods in the quadrilateral and pentagon cases. Furthermore, we see the inscribed ellipse problems from the perspective of two projective planes simultaneously, which offers a new way to determine the inscribed ellipses in triangles. Also, we use python to realize the method provided in this paper of drawing inscribed ellipse.

Properties to Determine Inscribed Ellipses of Polygons.pdf

A STUDY ON MÖBIUS FUNCTION AND EULER’S TOTIENT OF ORDER K

Author: Wenhan Zhang and Ziyang Zheng

Abstract: Defined by Tom M. Apostol (see [2]), the Möbius function of order k is a natural generalization of Möbius function which is one of the most important arithmetic functions studied in analytic number theory. In this paper we study some properties of Möbius function of order k, denoted by µk, some of which are analogous to ordinary Möbius function. This involves some summation formulas involving µk. We also use some of them to study “k-free integers”. And from here, we define Euler’s Totient of order k, denoted by φk, and study its properties and relation with µk. We also present asymptotic formula about φk with proof. Furthermore, we study the asymptotic behavior of k-free integers with the help of µk.

A STUDY ON MÖBIUS FUNCTION AND EULER’S TOTIENT OF ORDER K.pdf

Multi-Objective Problem Pareto Front Metamodeling Optimization Using NSGA-II

Author: Anonymous

Abstract: Though neural networks have been applied on approximating Pareto fronts of multi-objective optimization problems using surrogate models, existing works have focused on categorizing optimization genetic algorithms with BPNN surrogate models. However, this paper explores specifically the use of the NSGA-II algorithm with a BPNN surrogate model, with the aid of the ZDT1 test function. The exploration concluded the effectiveness of NSGA-II in approximating the Pareto front with a surrogate model of a multi-objective optimization problem using the IGD value and HV indicator of the obtained Pareto front.

Multi-Objective Problem Pareto Front Metamodeling Optimization.pdf