Algorithms for Bohemian Matrices

This thesis develops several algorithms for working with matrices whose entries are multivariate polynomials in a set of parameters. Such parametric linear systems often appear in biology and engineering applications where the parameters represent physical properties of the system. Some computations on parametric matrices, such as the rank and Jordan canonical form, are discontinuous in the parameter values. Understanding where these discontinuities occur provides a greater understanding of the underlying system. Algorithms for computing a complete case discussion of the rank, Zigzag form, and the Jordan canonical form of parametric matrices are presented. These algorithms use the theory of regular chains to provide a unified framework allowing for algebraic or semi-algebraic constraints on the parameters. Corresponding implementations for each algorithm in the Maple computer algebra system are provided. In some applications, all entries may be parameters whose values are limited to finite sets of integers. Such matrices appear in applications such as graph theory where matrix entries are limited to the sets {0, 1}, or {-1, 0, 1}. These types of parametric matrices can be explored using different techniques and exhibit many interesting properties. A family of Bohemian matrices is a set of low to moderate dimension matrices where the entries are independently sampled from a finite set of integers of bounded height. Properties of Bohemian matrices are studied including the distributions of their eigenvalues, symmetries, and integer sequences arising from properties of the families. These sequences provide connections to other areas of mathematics and have been archived in the Characteristic Polynomial Database. A study of two families of structured matrices: upper Hessenberg and upper Hessenberg Toeplitz, and properties of their characteristic polynomials are presented

Upper Hessenberg and Toeplitz Bohemians

We also acknowledge the support of the Ontario Graduate Institution, The National Science & Engineering Research Council of Canada, the University of Alcala, the Rotman Institute of Philosophy, the Ontario Research Centre of Computer Algebra, and Western University. Part of this work was developed while R. M. Corless was visiting the University of Alcala, in the frame of the project Giner de los Rios. J.R. Sendra is member of the Research Group ASYNACS (Ref. CT-CE2019/683).A set of matrices with entries from a fixed finite population P is called “Bohemian”. The mnemonic comes from BOunded HEight Matrix of Integers, BOHEMI, and although the population P need not be solely made up of integers, it frequently is. In this paper we look at Bohemians, specifically those with population {−1,0,+1} and sometimes other populations, for instance {0,1,i,−1,−i}. More, we specialize the matrices to be upper Hessenberg Bohemian. We then study the characteristic polynomials of these matrices, and their height, that is the infinity norm of the vector of monomial basis coefficients. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the number of normal matrices and the number of stable matrices in these families.Agencia Estatal de Investigació

Education or just Creativity: what matters most for economic performance?

There is a large consensus among social researchers on the positive role played by human capital on economic performances. The standard way to measure the human capital endowment is to consider the educational attainments by the resident population, usually the share of people with a university degree. Recently, Florida (2002) suggested a different measure of human capital - the “creative class†- based on the actual occupations of individuals in specific jobs like science, engineering, arts, culture, entertainment. However, the empirical analyses carried out so far overlooked a serious measurement problem concerning the clear identification of the education and creativity components of human capital. The main purpose of this paper is to try to disentangle this issue by proposing a disaggregation of human capital into three non-overlapping categories of creative graduates, bohemians and non creative graduates. By using a spatial econometric framework to account for spatial dependence, we assess the concurrent effect of the human capital indicators on total factor productivity for 257 regions of EU27. Our main results indicate that the highly educated creative group is the most relevant one in explaining production efficiency, while the other two categories - non creative graduates and bohemians - exhibit negligible effects. Moreover, a relevant influence is exerted by technological capital and by the level of tolerance providing robust evidence that an innovative, open, inclusive and culturally diverse environment is becoming more and more crucial for productivity enhancements.

Algebraic linearizations of matrix polynomials

We show how to construct linearizations of matrix polynomials za(z)d0+c0, a(z)b(z), a(z) +b(z)(when deg (b(z))<deg (a(z))), and za(z)d0b(z) +c0from linearizations of the component parts, a(z)and b(z). This allows the extension to matrix polynomials of a new companion matrix construction.Ministerio de Economía y CompetitividadEuropean Regional Development Fund (ERDF)Part of this work was developed while R.M.Corless was visiting the University of Alcalá, in the frame of the project Giner de los Rios. We acknowledge the support of the Ontario Graduate Institution, the National Science & Engineering Research Council of Canada, the University of Alcalá, the Rotman Institute of Philosophy, the Ontario Research Centre of Computer Algebra, and Western Univ

Financial Communication in Romantic Relationships

Previous research indicates that financial disagreements are common among romantic couples. However, little theoretical development has been offered to explain such disagreements. This study integrates several areas of research pertinent to financial conflict, and proposes two typologies to explain couples’ recurrent arguments over finances. The first typology concerns financial attitudes that work together to create a financial style. The second typology concerns financial power in the relationship, which is comprised of contribution to household funds, dominance in financial decision-making, and keeping money separate from one’s partner. Dyadic data was collected from 80 couples to test the typologies. Analyses revealed that some attitude combinations are less conducive to relational harmony than others, particularly for males. Among all respondents, being romantically involved with a liberal spender increased the perception of financial conflict. Additionally, partners who perceived they had the most financial power in the relationship perceived that conflicts over finances were frequent, irresolvable, and predictable

Integer matrix factorisations, superalgebras and the quadratic form obstruction

We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalgebra. We further obtain a formula for the determinant of a square matrix in terms of adjugates of these matrix decompositions, as well as identifying a $\it co-Latin$ symmetry space.Comment: 20 Page