Invariant functions of the elements of the powers of a square matrix

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

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The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-2006-018-7 ◽

2006 ◽

Vol 49 (2) ◽

pp. 170-184

Author(s):

Richard Atkins

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Shortest Paths ◽

Equivalence Problem ◽

Second Order ◽

System Of Differential Equations ◽

Lagrange Equations ◽

Invariant Functions ◽

Underlying Geometry ◽

The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

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Algebraic conditions and the sparsity of spectrally arbitrary patterns

Special Matrices ◽

10.1515/spma-2020-0136 ◽

2021 ◽

Vol 9 (1) ◽

pp. 257-274

Author(s):

Louis Deaett ◽

Colin Garnett

Keyword(s):

Major Goal ◽

Algebraic Condition ◽

Open Question ◽

Algebraic Tool ◽

General Method ◽

Square Matrix ◽

Spectrally Arbitrary

Abstract Given a square matrix A, replacing each of its nonzero entries with the symbol * gives its zero-nonzero pattern. Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A. A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2n. An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for ℝ, and that there are essentially only two possible counterexamples over ℂ. Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over ℂ. A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.

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Of lions and Yakshis

Semantic Web ◽

10.3233/sw-200417 ◽

2020 ◽

pp. 1-21

Author(s):

Franziska Pannach ◽

Caroline Sporleder ◽

Wolfgang May ◽

Aravind Krishnan ◽

Anusharani Sewchurran

Keyword(s):

Knowledge Base ◽

Case Studies ◽

Narrative Structure ◽

Function Class ◽

Special Focus ◽

Cultural Backgrounds ◽

Class Hierarchy ◽

Invariant Functions ◽

Narrative Functions ◽

Query System

Vladimir Propp’s theory Morphology of the Folktale identifies 31 invariant functions, subfunctions, and seven classes of folktale characters to describe the narrative structure of the Russian magic tale. Since it was first published in 1928, Propp’s approach has been used on various folktales of different cultural backgrounds. ProppOntology models Propp’s theory by describing narrative functions using a combination of a function class hierarchy and characteristic relationships between the Dramatis Personae for each function. A special focus lies on the restrictions Propp defined regarding which Dramatis Personae fulfill a certain function. This paper investigates how an ontology can assist traditional Humanities research in examining how well Propp’s theory fits for folktales outside of the Russian–European folktale culture. For this purpose, a lightweight query system has been implemented. To determine how well both the annotation schema and the query system works, twenty African tales and fifteen tales from the Kerala region in India were annotated. The system is evaluated by examining two case studies regarding the representation of characters and the use of Proppian functions in African and Indian tales. The findings are in line with traditional analogous Humanities research. This project shows how carefully modelled ontologies can be utilized as a knowledge base for comparative folklore research.

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Analysis of nonlinear dynamics by square matrix method

Physical Review Accelerators and Beams ◽

10.1103/physrevaccelbeams.20.034001 ◽

2017 ◽

Vol 20 (3) ◽

Cited By ~ 3

Author(s):

Li Hua Yu

Keyword(s):

Nonlinear Dynamics ◽

Matrix Method ◽

Square Matrix

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On equicontinuous factors of flows on locally path-connected compact spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.104 ◽

2020 ◽

pp. 1-18

Author(s):

NIKOLAI EDEKO

Keyword(s):

Lie Group ◽

Metric Space ◽

Betti Number ◽

Alternative Proof ◽

Simple Consequence ◽

Compact Metric Space ◽

Invariant Functions ◽

Compact Spaces ◽

Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

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A Schur-Cohn theorem for matrix polynomials

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500004806 ◽

1990 ◽

Vol 33 (3) ◽

pp. 337-366 ◽

Cited By ~ 6

Author(s):

Harry Dym ◽

Nicholas Young

Keyword(s):

Unit Circle ◽

Matrix Polynomial ◽

Hermitian Matrix ◽

Krein Space ◽

Gram Matrix ◽

Natural Basis ◽

Test Matrix ◽

Matrix Polynomials ◽

Rational Vector ◽

Square Matrix

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

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