The Spectral Structure of TN Matrices

2011 ◽  
Author(s):  
Shaun M. Fallat ◽  
Charles R. Johnson
Keyword(s):  
Special Class ◽  
Spectral Structure ◽  
Nonnegative Matrices ◽  
Eigenvalues And Eigenvectors ◽  
Real Numbers ◽  
Sign Patterns ◽  
Oscillatory Case

This chapter highlights the spectral structure of TN matrices. By “spectral structure” this chapter refers to facts about the eigenvalues and eigenvectors of matrices in a particular class. In view of the well-known Perron-Frobenius theory that describes the spectral structure of general entrywise nonnegative matrices, it is not surprising that TN matrices have an important and interesting special spectral structure. Nonetheless, that spectral structure is remarkably strong, identifying TN matrices as a very special class among nonnegative matrices. All eigenvalues are nonnegative real numbers, and the sign patterns of the entries of the eigenvectors are quite special as well. This spectral structure is most apparent in the case of IITN matrices (that is, the classical oscillatory case originally described by Gantmacher and Krein).

scholarly journals AN INTEGRATION APPROACH TO THE TOEPLITZ SQUARE PEG PROBLEM

2017 ◽  
Vol 5 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
Finite Sets ◽  
Real Numbers ◽  
Integration Approach ◽  
Sign Patterns ◽  
Lipschitz Constants ◽  
Lipschitz Graphs ◽  
Periodic Version ◽  
Rule Out

The ‘square peg problem’ or ‘inscribed square problem’ of Toeplitz asks if every simple closed curve in the plane inscribes a (nondegenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ‘homological’ nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper, we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs $f$, $g:[t_{0},t_{1}]\rightarrow \mathbb{R}$ that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums $y_{1}+y_{2}+y_{3}$ for $y_{1},y_{2},y_{3}$ ranging in finite sets of real numbers.

scholarly journals On the infinite Borwein product raised to a positive real power

2021 ◽  
Author(s):  
Michael J. Schlosser ◽  
Nian Hong Zhou
Keyword(s):  
Asymptotic Formula ◽  
Circle Method ◽  
Sign Pattern ◽  
Real Numbers ◽  
Positive Real ◽  
Infinite Products ◽  
Real Power ◽  
Sign Patterns ◽  
Divisibility Properties ◽  
Made In

AbstractIn this paper, we study properties of the coefficients appearing in the q-series expansion of $$\prod _{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta $$ ∏ n ≥ 1 [ ( 1 - q n ) / ( 1 - q pn ) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power $$\delta $$ δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For $$p=3$$ p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $$\delta $$ δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $$p=3$$ p = 3 case.

scholarly journals On the inverse eigenvalue problem of symmetric nonnegative matrices

2019 ◽  
Vol 14 (1) ◽  
pp. 11-19
Author(s):  
A. M. Nazari ◽  
A. Mashayekhi ◽  
A. Nezami
Keyword(s):  
Eigenvalue Problem ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Nonnegative Matrices ◽  
Real Numbers ◽  
Inverse Eigenvalue ◽  
The Given

AbstractIn this paper, at first for a given set of real numbers with only one positive number, and in continue for a given set of real numbers in special conditions, we construct a symmetric nonnegative matrix such that the given set is its spectrum.

HENSTOCK–KURZWEIL TYPE INTEGRAL BASED ON GENERALIZED g-SEMIRING

2002 ◽  
Vol 10 (supp01) ◽  
pp. 89-104
Author(s):  
IVANA ŠTAJNER-PAPUGA
Keyword(s):  
Generating Function ◽  
Special Class ◽  
Real Numbers ◽  
Positive Real ◽  
Type Integral ◽  
Pseudo Inverse

We shall consider special class of generalized semiring based on generalized pseudo-operations of the following form: x ⊕ y = g(-1)(εg(x) + g(y)), x ⊙ y = g(-1)(g(x)γg(y)), where ε and γ are arbitrary but fixed positive real numbers, g is a positive strictly monotone generating function and g(-1) is its pseudo-inverse. Using this pseudo-operations, corresponding pseudo-measure and the Henstock–Kurzweil type integral will be introduced and investigated.

scholarly journals Minimum Rank of Matrices Described by a Graph or Pattern over the Rational, Real and Complex Numbers

10.37236/749 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Avi Berman ◽  
Shmuel Friedland ◽  
Leslie Hogben ◽  
Uriel G. Rothblum ◽  
Bryan Shader
Keyword(s):  
Computational Complexity ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Real Numbers ◽  
Minimum Rank ◽  
The Real ◽  
Sign Patterns ◽  
Symmetric Patterns

We use a technique based on matroids to construct two nonzero patterns $Z_1$ and $Z_2$ such that the minimum rank of matrices described by $Z_1$ is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by $Z_2$ is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture by Arav, Hall, Koyucu, Li and Rao about rational realization of minimum rank of sign patterns. Using $Z_1$ and $Z_2$, we construct symmetric patterns, equivalent to graphs $G_1$ and $G_2$, with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.

scholarly journals Brauer's theorem and nonnegative matrices with prescribed diagonal entries

2019 ◽  
Vol 35 ◽  
pp. 53-64 ◽  
Author(s):  
Ricardo Soto ◽  
Ana Julio ◽  
Macarena Collao
Keyword(s):  
Inverse Problem ◽  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Simple Condition ◽  
Inverse Spectral Problems ◽  
Real Numbers ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

The problem of the existence and construction of nonnegative matrices with prescribed eigenvalues and diagonal entries is an important inverse problem, interesting by itself, but also necessary to apply a perturbation result, which has played an important role in the study of certain nonnegative inverse spectral problems. A number of partial results about the problem have been published by several authors, mainly by H. \v{S}migoc. In this paper, the relevance of a Brauer's result, and its implication for the nonnegative inverse eigenvalue problem with prescribed diagonal entries is emphasized. As a consequence, given a list of complex numbers of \v{S}migoc type, or a list $\Lambda = \left\{\lambda _{1},\ldots ,\lambda _{n} \right \}$ with $\operatorname{Re}\lambda _{i}\leq 0,$ $\lambda _{1}\geq -\sum\limits_{i=2}^{n}\lambda _{i}$, and $\left\{-\sum\limits_{i=2}^{n}\lambda _{i},\lambda _{2},\ldots ,\lambda _{n} \right\}$ being realizable; and given a list of nonnegative real numbers $% \Gamma = \left\{\gamma _{1},\ldots ,\gamma _{n} \right\}$, the remarkably simple condition $\gamma _{1}+\cdots +\gamma _{n} = \lambda _{1}+\cdots +\lambda _{n}$ is necessary and sufficient for the existence and construction of a realizing matrix with diagonal entries $\Gamma .$ Conditions for more general lists of complex numbers are also given.

From Christoffel Words to Markoff Numbers

2018 ◽  
Author(s):  
Christophe Reutenauer
Keyword(s):  
Free Group ◽  
Continued Fractions ◽  
Quadratic Forms ◽  
Special Class ◽  
Rational Numbers ◽  
Real Numbers ◽  
Inner Automorphisms ◽  
Sturmian Words ◽  
Markoff Numbers ◽  
Lyndon Words

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.

scholarly journals Sign patterns of inverse doubly-nonnegative matrices

2021 ◽  
Vol 610 ◽  
pp. 480-487
Author(s):  
Sandip Roy ◽  
Mengran Xue
Keyword(s):  
Nonnegative Matrices ◽  
Sign Patterns
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