arXiv subjects

R. S. Vieira

Publications and source records attributed to R. S. Vieira.

14 recordsLinked to original sources

arXiv2020

The solutions of the Yang-Baxter equation for the $(n+1)(2n+1)$-vertex models through a differential approach

The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the $R$ matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which two systems of polynomial equations are obtained in place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nevertheless, the remaining unknowns can be found by solving a few number of simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work we considered the Yang-Baxter equation for the $(n+1)(2n+1)$-vertex models with a generalization based on the $A_n$ symmetry. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way.

nlin.SI↗

arXiv2020

How to count the number of zeros that a polynomial has on the unit circle?

The classical problem of counting the number of real zeros of a real polynomial was solved a long time ago by Sturm. The analogous problem of counting the number of zeros that a polynomial has on the unit circle is, however, still an open problem. In this paper, we show that the second problem can be reduced to the first one through the use of a suitable pair of Möbius transformations - often called Cayley transformations - that have the property of mapping the unit circle onto the real line and vice versa. Although the method applies to arbitrary complex polynomials, we discuss in detail several classes of polynomials with symmetric zeros as, for instance, the cases of self-conjugate, self-adjoint, self-inversive, self-reciprocal and skew-reciprocal polynomials. Finally, an application of this method to Salem polynomials and to polynomials with small Mahler measure is also discussed.

math.CV↗

arXiv2019

Fifteen-vertex models with non-symmetric $R$ matrices

In this work, we employ the algebraic-differential method recently developed by the author to solve the Yang-Baxter equation for arbitrary fifteen-vertex models satisfying the ice-rule. We show that there are four different families of such regular $R$ matrices containing several free-parameters. The corresponding reflection $K$ matrices, solutions of the boundary Yang-Baxter equation, were also found and classified. We found that there are three different families of regular $K$ matrices, regardless of what $R$ matrix we choose.

nlin.SI↗

arXiv2019

Orthogonal polynomials and Möbius transformations

Given an orthogonal polynomial sequence on the real line, another sequence of polynomials can be found by composing these polynomials with a general Möbius transformation. In this work, we study the properties of such Möbius-transformed polynomials. We show that they satisfy an orthogonality relation in given curve of the complex plane with respect to a varying weight function and that they also enjoy several properties common to the orthogonal polynomial sequences on the real line --- e.g. a three-term recurrence relation, Christoffel-Darboux type identities, their zeros are simple, lie on the support of orthogonality and have the interlacing property, etc. Moreover, we also show that the Möbius-transformed polynomials obtained from classical orthogonal polynomials also satisfy a second-order differential equation, a Rodrigues' type formula and generating functions. As an application, we show that Hermite, Laguerre, Jacobi, Bessel and Romanovski polynomials are all related to each other by a suitable Möbius transformation. New orthogonality relations for Bessel and Romanovski polynomials are also presented.

math.CV↗

arXiv2019

Polynomials with symmetric zeros

Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real self-reciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.

math.CV↗

arXiv2018

Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the $R$ matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which two systems of polynomial equations are obtained in place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the $R$ matrix can be fixed through them. Nonetheless, the remaining unknowns can be found by solving a few number of simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work we considered the Yang-Baxter equation for two-state systems, up to the eight-vertex model. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way and also to completely classify its regular solutions.

nlin.SI↗

arXiv2018

Covariant theory of gravitation in the framework of special relativity

In this work, we study the magnetic effects of gravity in the framework of special relativity. Imposing covariance of the gravitational force with respect to the Lorentz transformations, we show from a thought experiment that a magnetic-like force must be present whenever two or more bodies are in motion. The exact expression for this gravitomagnetic force is then derived purely from special relativity and the consequences of such a covariant theory are developed. For instance, we show that the gravitomagnetic fields satisfy a system of differential equations similar to the Maxwell equations of electrodynamics. This implies that the gravitational waves spread out with the speed of light in a flat spacetime, which is in agreement with the recent results concerning the gravitational waves detection. We also propose that the vector potential can be associated with the interaction momentum in the same way as the scalar potential is usually associated with the interaction energy. Other topics are also discussed, for example, the transformation laws for the fields, the energy and momentum stored in the gravitomagnetic fields, the invariance of the gravitational mass and so on. We remark that is not our intention here to propose an alternative theory of gravitation but, rather, only a first approximation for the gravitational phenomena, so that it can be applied whenever the gravitational force can be regarded as an ordinary effective force field and special relativity can be used with safety. To make this point clear we present briefly a comparison between our approach and that based on the (linearized) Einstein's theory. Finally, we remark that although we have assumed nothing from the electromagnetic theory, we found that gravity and electricity share many properties in common - these similarities, in fact, are just a requirement of special relativity that must apply to any physically (...)

physics.gen-ph↗

arXiv2018

The algebraic Bethe Ansatz and combinatorial trees

We present in this paper a comprehensive introduction to the algebraic Bethe Ansatz, taking as examples the six-vertex model with periodic and non-periodic boundary conditions. We propose a diagrammatic representation of the commutation relations used in the algebraic Bethe Ansatz, so that the action of the transfer matrix in the nth excited state gives place to labeled combinatorial trees. The analysis of these combinatorial trees provides in a straightforward way the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations. Several identities between the R-matrix elements can also be derived from the symmetry of these diagrams regarding the permutation of their labels. This combinatorial approach gives some insights about how the algebraic Bethe Ansatz works, which can be valuable for non-experts readers.

math.CO↗

arXiv2017

Boundary Algebraic Bethe Ansatz for a nineteen vertex model with $U_{q}[\mathrm{osp}(2|2)^{(2)}]$ symmetry

The boundary algebraic Bethe Ansatz for a supersymmetric nineteen vertex-model constructed from a three-dimensional representation of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}(2|2)^{(2)}]$ is presented. The eigenvalues and eigenvectors of Sklyanin's transfer matrix, with diagonal reflection $K$-matrices, are calculated and the corresponding Bethe Ansatz equations are obtained.

nlin.SI↗

arXiv2017

Reflection matrices with $U_q[osp^{(2)}(2|2m)]$ symmetry

We propose a classification of the reflection $K$-matrices (solutions of the boundary Yang-Baxter equation) for the $U_{q}[\mathrm{osp}^{\left(2\right)}\left(2|2m\right)]=U_{q}[C^{\left(2\right)}\left(m+1\right)]$ vertex-model. We have found four families of solutions, namely, the complete solutions, in which no elements of the reflection $K$-matrix is null, the block-diagonal solutions, the $X$-shape solutions and the diagonal solutions. We highlight that these diagonal $K$-matrices also hold for the $U_{q}[\mathrm{osp}^{\left(2\right)}\left(2n+2|2m\right)]=U_{q}[D^{\left(2\right)}\left(n+1,m\right)]$ vertex-model.

nlin.SI↗

arXiv2017

Reflection $K$-matrices for a nineteen vertex model with $U_{q}[\mathrm{osp}\left(2|2\right)^{\left(2\right)}]$ symmetry

We derive the solutions of the boundary Yang-Baxter equation associated with a supersymmetric nineteen vertex model constructed from the three-dimensional representation of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}\left(2|2\right)^{\left(2\right)}]\simeq U_{q}[C\left(2\right)^{\left(2\right)}]$. We found three classes of solutions. The type I solution is characterized by three boundary free-parameters and all elements of the corresponding reflection $K$-matrix are different from zero. In the type II solution, the reflection $K$-matrix is even (every element of the $K$-matrix with an odd parity is null) and it has only one boundary free-parameter. Finally, the type III solution corresponds to a diagonal reflection $K$-matrix with two boundary free-parameters.

nlin.SI↗

arXiv2017

Solution of Supplee's submarine paradox through special and general relativity

In 1989 Supplee described an apparent relativistic paradox on which a submarine seems to sink to observers at rest within the ocean, but it rather seems to float in the submarine proper frame. In this letter, we show that the paradox arises from a misuse of the Archimedes principle in the relativistic case. Considering first the special relativity, we show that any relativistic force field can be written in the Lorentz form, so that it can always be decomposed into a \emph{static} (electric-like) and a \emph{dynamic} (magnetic-like) part. These gravitomagnetic effects provide a relativistic formulation of Archimedes principle, from which the paradox is explained. Besides, if the curved spacetime on the vicinity of the Earth is taken into account, we show that the gravitational force exerted by Earth on a moving body must increase with the speed of the body. The submarine paradox is then analyzed again with this speed-dependent gravitational force.

physics.class-ph↗

arXiv2016

On the number of roots of self-inversive polynomials on the complex unit circle

We present a sufficient condition for a self-inversive polynomial to have a fixed number of roots on the complex unit circle. We also prove that these roots are simple when that condition is satisfied. This generalizes the condition found by Lakatos and Losonczi for all the roots of a self-inversive polynomial to lie on the complex unit circle.

math.CV↗

arXiv2015

Where are the roots of the Bethe Ansatz equations?

Changing the variables in the Bethe Ansatz Equations (BAE) for the XXZ six-vertex model we had obtained a coupled system of polynomial equations. This provided a direct link between the BAE deduced from the Algebraic Bethe Ansatz (ABA) and the BAE arising from the Coordinate Bethe Ansatz (CBA). For two magnon states this polynomial system could be decoupled and the solutions given in terms of the roots of some self-inversive polynomials. From theorems concerning the distribution of the roots of self-inversive polynomials we made a thorough analysis of the two magnon states, which allowed us to find the location and multiplicity of the Bethe roots in the complex plane, to discuss the completeness and singularities of Bethe's equations, the ill-founded string-hypothesis concerning the location of their roots, as well as to find an interesting connection between the BAE with Salem's polynomials.

cond-mat.stat-mech↗