An observation on the determinant of a Sylvester-Kac type matrix

Based on a less-known result, we prove a recent conjecture concerning the determinant of a certain Sylvester-Kac type matrix and consider an extension of it

Some formulas for determinants of tridiagonal matrices in terms of finite generalized continued fractions: Formulas for determinants of tridiagonal matrices

In the paper, by virtue of induction and properties of determinants, the authors discover explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of finite generalized continued fractions and apply these formulas to evaluations for determinants of the Sylvester matrix and two Sylvester type matrices.In the paper, by virtue of induction and properties of determinants, the authors discover explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of finite generalized continued fractions and apply these formulas to evaluations for determinants of the Sylvester matrix and two Sylvester type matrices

Explicit spectrum of a circulant-tridiagonal matrix with applications

We consider a circulant-tridiagonal matrix and compute its determinant by using generating function method. Then we explicitly determine its spectrum. Finally we present applications of our results for trigonometric factorizations of the generalized Fibonacci and Lucas sequences

A Survey of Alternating Permutations

This survey of alternating permutations and Euler numbers includes refinements of Euler numbers, other occurrences of Euler numbers, longest alternating subsequences, umbral enumeration of classes of alternating permutations, and the cd-index of the symmetric group.Comment: 32 pages, 7 figure

Accurate and Efficient Expression Evaluation and Linear Algebra

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use the so-called Traditional Model (TM). We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high accuracy algorithm or a proof that none exists. When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as $x+y+z$, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case. Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl

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

Locality and Exceptional Points in Pseudo-Hermitian Physics

Pseudo-Hermitian operators generalize the concept of Hermiticity. Included in this class of operators are the quasi-Hermitian operators, which define a generalization of quantum theory with real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices. An implicit assumption of the tensor product model of locality is that the inner product factorizes with the tensor product. Quasi-Hermitian quantum theory generalizes the tensor product model by modifying the Born rule via a metric operator with nontrivial Schmidt rank. Local observable algebras and expectation values are examined in chapter 5. Observable algebras of two one-dimensional fermionic quasi-Hermitian chains are explicitly constructed. Notably, there can be spatial subsystems with no nontrivial observables. Despite devising a new framework for local quantum theory, I show that expectation values of local quasi-Hermitian observables can be equivalently computed as expectation values of Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory. A perturbative feature present in pseudo-Hermitian curves which has no Hermitian counterpart is the exceptional point, a branch point in the set of eigenvalues. An original finding presented in section 2.6.3 is a correspondence between cusp singularities of algebraic curves and higher-order exceptional points. Eigensystems of one-dimensional lattice models admit closed-form expressions that can be used to explore the new features of non-Hermitian physics. One-dimensional lattice models with a pair of non Hermitian defect potentials with balanced gain and loss, Δ±iγ, are investigated in chapter 3. Conserved quantities and positive-definite metric operators are examined. When the defects are nearest neighbour, the entire spectrum simultaneously becomes complex when γ increases beyond a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the PT-phase transition is dictated by the topological phase of the system. In the thermodynamic limit, PT-symmetry spontaneously breaks in the topologically non-trivial phase due to the presence of edge states. Chiral symmetry and representation theory are utilized in chapter 4 to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators. These intertwining operators include positive-definite metric operators in the quasi-Hermitian case. The PT-phase transition is explicitly determined in a special case

Introduction to Linear Algebra: Models, Methods, and Theory

This book develops linear algebra around matrices. Vector spaces in the abstract are not considered, only vector spaces associated with matrices. This book puts problem solving and an intuitive treatment of theory first, with a proof-oriented approach intended to come in a second course, the same way that calculus is taught. The book\u27s organization is straightforward: Chapter 1 has introductory linear models; Chapter 2 has the basics of matrix algebra; Chapter 3 develops different ways to solve a system of equations; Chapter 4 has applications, and Chapter 5 has vector-space theory associated with matrices and related topics such as pseudoinverses and orthogonalization. Many linear algebra textbooks start immediately with Gaussian elimination, before any matrix algebra. Here we first pose problems in Chapter 1, then develop a mathematical language for representing and recasting the problems in Chapter 2, and then look at ways to solve the problems in Chapter 3-four different solution methods are presented with an analysis of strengths and weaknesses of each.https://commons.library.stonybrook.edu/ams-books/1000/thumbnail.jp

Delayed Hawkes birth-death processes

We introduce a variant of the Hawkes-fed birth-death process, in which the conditional intensity does not increase at arrivals, but at departures from the system. Since arrivals cause excitation after a delay equal to their lifetimes, we call this a delayed Hawkes process. We introduce a general family of models admitting a cluster representation containing the Hawkes, delayed Hawkes and ephemerally self-exciting processes as special cases. For this family of models, as well as their nonlinear extensions, we prove existence, uniqueness and stability. Our family of models satisfies the same FCLT as the classical Hawkes process; however, we describe a scaling limit for the delayed Hawkes process in which sojourn times are stretched out by a factor $\sqrt T$, after which time gets contracted by a factor $T$. This scaling limit highlights the effect of sojourn-time dependence. The cluster representation renders our family of models tractable, allowing for transform characterisation by a fixed-point equation and for an analysis of heavy-tailed asymptotics. In the Markovian case, for a multivariate network of delayed Hawkes birth-death processes, an explicit recursive procedure is presented to calculate the $d$th-order moments analytically. Finally, we compare the delayed Hawkes process to the regular Hawkes process in the stochastic ordering, which enables us to describe stationary distributions and heavy-traffic behaviour.Comment: 38 pages, 1 figur