Pao-Lu Hsu (Xu, Bao-lu): The Grandparent of Probability and Statistics in China

The years 1910-1911 are auspicious years in Chinese mathematics with the births of Pao-Lu Hsu, Luo-Keng Hua and Shiing-Shen Chern. These three began the development of modern mathematics in China: Hsu in probability and statistics, Hua in number theory, and Chern in differential geometry. We here review some facts about the life of P.-L. Hsu which have been uncovered recently, and then discuss some of his contributions. We have drawn heavily on three papers in the 1979 Annals of Statistics (volume 7, pages 467-483) by T. W. Anderson, K. L. Chung and E. L. Lehmann, as well as an article by Jiang Ze-Han and Duan Xue-Fu in Hsu's collected papers.Comment: Published in at http://dx.doi.org/10.1214/12-STS387 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spin - or, actually: Spin and Quantum Statistics

The history of the discovery of electron spin and the Pauli principle and the mathematics of spin and quantum statistics are reviewed. Pauli's theory of the spinning electron and some of its many applications in mathematics and physics are considered in more detail. The role of the fact that the tree-level gyromagnetic factor of the electron has the value g = 2 in an analysis of stability (and instability) of matter in arbitrary external magnetic fields is highlighted. Radiative corrections and precision measurements of g are reviewed. The general connection between spin and statistics, the CPT theorem and the theory of braid statistics are described.Comment: 50 pages, no figures, seminar on "spin

On Nonlocal Modified Gravity and Cosmology

Despite many nice properties and numerous achievements, general relativity is not a complete theory. One of actual approaches towards more complete theory of gravity is its nonlocal modification. We present here a brief review of nonlocal gravity with its cosmological solutions. In particular, we pay special attention to two nonlocal models and their nonsingular bounce solutions for the cosmic scale factor.Comment: 11 pages, Published in Springer Proceedings in Mathematics & Statistics 111 (2014) 251-26

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

Signed group orthogonal designs and their applications

Craigen introduced and studied {\it signed group Hadamard matrices} extensively in \cite{Craigenthesis, Craigen}. Livinskyi \cite{Ivan}, following Craigen's lead, studied and provided a better estimate for the asymptotic existence of signed group Hadamard matrices and consequently improved the asymptotic existence of Hadamard matrices. In this paper, we introduce and study signed group orthogonal designs. The main results include a method for finding signed group orthogonal designs for any $k$-tuple of positive integer and then an application to obtain orthogonal designs from signed group orthogonal designs, namely, for any $k$-tuple $\big(u_1, u_2, ..., u_{k}\big)$ of positive integers, we show that there is an integer $N=N(u_1, u_2, ..., u_k)$ such that for each $n\ge N$, a full orthogonal design (no zero entries) of type $\big(2^nu_1,2^nu_2,...,2^nu_{k}\big)$ exists . This is an alternative approach to the results obtained in \cite{EK}.Comment: 16 pages, To appear in Algebraic Design Theory and Hadamard Matrices (ADTHM), Springer Proceeding in Mathematics and Statistics. Editor: Charles Colbourn. Springer Proceeding in Mathematics and Statistics (PROMS), 201

R.A.Fisher, design theory, and the Indian connection

Design Theory, a branch of mathematics, was born out of the experimental statistics research of the population geneticist R. A. Fisher and of Indian mathematical statisticians in the 1930s. The field combines elements of combinatorics, finite projective geometries, Latin squares, and a variety of further mathematical structures, brought together in surprising ways. This essay will present these structures and ideas as well as how the field came together, in itself an interesting story.Comment: 11 pages, 3 figure

On distribution formulas for complex and ℓ\ell-adic polylogarithms

We study an $\ell$-adic Galois analogue of the distribution formulas for polylogarithms with special emphasis on path dependency and arithmetic behaviors. As a goal, we obtain a notion of certain universal Kummer-Heisenberg measures that enable interpolating the $\ell$-adic polylogarithmic distribution relations for all degrees.Comment: This article has appeared in the proceedings volume "Periods in Quantum Field Theory and Arithmetic" (J.~Burgos Gil, K.~Ebrahimi-Fard, H.~Gangl eds), [Conference proceedings ICMAT-MZV 2014] Springer Proceedings in Mathematics \& Statistics {\bf 314} (2020), pp.593--61