arXiv subjects

Xiao-Nan Lu

Publications and source records attributed to Xiao-Nan Lu.

5 recordsLinked to original sources

arXiv2026

Geometric designs and Hilbert-Kamke equations of degree five for classical orthogonal polynomials

In this paper we elucidate the advantage of examining the connections between Hilbert-Kamke equations and geometric designs, or Chebyshev-type quadrature, for classical orthogonal polynomials. We first establish that if a $5$-design with $6$ rational points for a symmetric classical measure is parametrized by rational functions, then the corresponding measure should be the Chebyshev measure $(1-t^2)^{-1/2}dt/π$ on $(-1,1)$. Our proof is based on the collaboration of a certain polynomial identity and some advanced techniques on the computation of the genus of a certain irreducible curve. Next, we prove a necessary and sufficient condition for the existence of rational $5$-designs for the Chebyshev measure. Moreover, as one of our main theorems, we construct an infinite family of ideal solutions for the Prouhet-Tarry-Escott (PTE) problem by utilizing rational $5$-designs for the Chebyshev measure, and then establish that, up to affine equivalence over $\mathbb{Q}$, such ideal solutions are included in the famous parametric solutions found by Borwein (2002).

math.NT↗

arXiv2025

Completely (Quasi-)Uniform Nested Boolean Steiner Quadruple Systems

Nested Steiner quadruple systems are designs derived from Steiner quadruple systems (SQSs) by partitioning each block into pairs. A nested SQS is completely uniform if every possible pair appears with equal multiplicity, and completely quasi-uniform if every pair appears with multiplicities that differ by at most one. An explicit construction on the Boolean SQS of order $2^m$ is presented, producing a nested SQS$(2^m)$ that is completely uniform when $m$ is odd and completely quasi-uniform when $m$ is even for each integer $m \ge 3$ . These results resolve two open problems posed by Chee et al. (2025). The notion of completely uniform pairings is further generalized for $t$-designs with $t \ge 2$. As an application, completely uniform nested $2$-$(2^m,4,3)$ designs give rise to fractional repetition codes with zero skip cost, requiring fewer storage nodes than constructions based on SQSs. In addition, small examples are provided for non-Boolean orders, establishing the existence of completely uniform nested SQS$(v)$ for all $v \le 50$.

math.CO↗

arXiv2024

Efficient pooling designs and screening performance in group testing for two type defectives

Group testing is utilized in the case when we want to find a few defectives among large amount of items. Testing n items one by one requires n tests, but if the ratio of defectives is small, group testing is an efficient way to reduce the number of tests. Many research have been developed for group testing for a single type of defectives. In this paper, we consider the case where two types of defective A and B exist. For two types of defectives, we develop a belief propagation algorithm to compute marginal posterior probability of defectives. Furthermore, we construct several kinds of collections of pools in order to test for A and B. And by utilizing our belief propagation algorithm, we evaluate the performance of group testing by conducting simulations.

stat.CO↗

arXiv2022

A remark on statistics for detecting laboratory effects in ORDANOVA

The present study defines a new statistic for detecting laboratory effects in the analysis of ordinal variation (ORDANOVA). The ORDANOVA is an analysis method similar to one-way analysis of variance for analysing ordinal data obtained from interlaboratory comparison studies. In this paper, we present an approximate continuous distribution for the new statistic for the case of an arbitrary number of ordinal levels, and we demonstrate that $alpha$-percentiles of the distribution are suitable criteria for conducting statistical tests. In addition, a real example involving data from an interlaboratory comparison study is analysed using the proposed statistic.

stat.AP↗

arXiv2019

Further Results on Existentially Closed Graphs Arising from Block Designs

A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets $A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design with block set $\cal B$, its block intersection graph is the graph whose vertex set is $\cal B$ and two vertices (blocks) are adjacent if they have non-empty intersection. In this paper, we investigate the block intersection graphs of pairwise balanced designs, and propose a sufficient condition for such graphs to be $2$-e.c. In particular, we study the $λ$-fold triple systems with $λ\ge 2$ and determine for which parameters their block intersection graphs are $1$- or $2$-e.c. Moreover, for Steiner quadruple systems, the block intersection graphs and their analogue called $\{1\}$-block intersection graphs are investigated, and the necessary and sufficient conditions for such graphs to be $2$-e.c. are established.

math.CO↗