Trigonometric Limit Research Articles

Surface energy and elementary excitations of the XYZ spin chain with integrable open boundary fields

We study the thermodynamic limit of the anisotropic XYZ spin chain with non-diagonal integrable open boundary conditions. Although the U(1)-symmetry is broken, by using the new parametrization scheme, we exactly obtain the surface energy and the excitation energy of the system, which has solved the difficulty in the inhomogeneous T − Q relation. With the boundary parameters in the regions making the Hamiltonian Hermitian, we have obtained the distribution patterns of the zeros of the eigenvalue of the transfer matrix for the ground state and the excited ones. We find that the surface and excitation energies depend on the parities of sites number N, due to the long-range Neel order in the bulk. The easy-axis and thermodynamic limit for all the regions of boundary parameters are studied. We also obtain the physical quantities in the thermodynamic limit of boundary XXZ model by taking the trigonometric limit.

On extension of the Yang-Baxter equation and the fermionic R-operator

We consider the fermionic R-operator based on Bazhanov-Stroganov's three-parameter elliptic parametrization of the free fermion model, and find the most general solution of the related tetrahedral Zamolodchikov algebra in the trigonometric limit for an arbitrary set of parameters. This allows to construct an extension of the R-operator and the corresponding Yang-Baxter equation, which are of the difference type in one of the spectral parameters.

Elliptic Racah polynomials

Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic Racah polynomials. The well-known $q$-Racah polynomials of Askey and Wilson are recovered as a trigonometric limit.

Higher Order Deformed Elliptic Ruijsenaars Operators

We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide a quantum mechanical description of two kinds of relativistic quantum mechanical particles which can be identified with particles and anti-particles in an underlying quantum field theory. We give direct proofs of the commutativity of our operators and of some other fundamental properties such as kernel function identities. In particular, we give a rigorous proof of the quantum integrability of the deformed Ruijsenaars model.

Eigenfunctions of a Discrete Elliptic Integrable Particle Model with Hyperoctahedral Symmetry

We construct the orthogonal eigenbasis for a discrete elliptic Ruijsenaars type quantum particle Hamiltonian with hyperoctahedral symmetry. In the trigonometric limit the eigenfunctions in question recover a previously studied $q$-Racah type reduction of the Koornwinder-Macdonald polynomials. When the inter-particle interaction degenerates to that of impenetrable bosons, the orthogonal eigenbasis simplifies in terms of generalized Schur polynomials on the spectrum associated with recently found elliptic Racah polynomials.

Elliptic Ruijsenaars Difference Operators on Bounded Partitions

Abstract By means of a truncation condition on the parameters, the elliptic Ruijsenaars difference operators are restricted onto a finite lattice of points encoded by bounded partitions. A corresponding orthogonal basis of joint eigenfunctions is constructed in terms of polynomials on the joint spectrum. In the trigonometric limit, this recovers the diagonalization of the truncated Macdonald difference operators by a finite-dimensional basis of Macdonald polynomials.

ANALYSIS OF ERRORS OF ELEVENTH SCIENCE 1 NATIONAL SENIOR HIGH SCHOOL 1 MIDDLE MALUKU IN RESOLVING TRIGONOMETRY FUNCTION LIMIT PROBLEMS AND REMEDIATION EFFORTS

This study was a qualitative descriptive research which aimed to describe the types of errors made by students of class of eleventh science 1 National Senior School 1 Maluku Tengah in solving the trigonometric function limit questions. Also, to determine the factors that cause students to make these mistakes and discover how remedial affects the error. The subjects of this study were students of eleventh science 1 National Senior School 1 Maluku Tengah. Student error data is obtained from the results of the diagnostic test answers identified into the types of errors. After that, four students were selected to be interviewed. Based on the results of data analysis and research results it can be concluded as follows: (1) The students' mistakes in solving the trigonometric function limit problems were errors using definitions or theorems and solutions that were not rechecked. (2) The factors that cause students done the limit of trigonometric functions task incorrectly are students have not mastered the prerequisite material, which plays an essential role in the process of understanding the concept of limit trigonometric functions. Such as the basic concepts of limit functions and trigonometry, students do not memorize basic trigonometric formulas. So that they find it challenging to work on questions; students do not understand the basic concepts of limit trigonometric functions. Students are confused when determining which part of the problem to change first or students are confused when describing questions. Students do not know what steps to use in solving math problems; the time used teachers in teaching in the classroom are too little so that students do not practice questions during the learning process in class. (3) After comparing the results of the diagnostic test and the student's remedial test, it can be seen that there is an increase in student learning outcomes on the subject of limit trigonometric functions.

Lifting the understanding of trigonometric limits from procedural towards conceptual

The purpose of this paper is to follow the reasoning of high school students when asked to explain the standard trigonometric limit . An observational study was conducted in four different phases in order to investigate if visualization, by means of an interactive technology environment (Geogebra), can contribute in lifting high school students’ understanding from a mere procedural understanding to a combination of conceptual and procedural understanding. The obtained results confirm that the students were able to show a conceptual understanding only after using the digital interactive tool. Through comparing, exploring and self-explaining combined with the use of the interactive tool, the students managed to link different concepts together. The students were able to see and interpret the reason making the angle and relate under certain conditions, thus leading to the standard trigonometric limit .

Thermodynamic limit of the spin- frac{1}{2} XYZ spin chain with the antiperiodic boundary condition

Based on its off-diagonal Bethe ansatz solution, we study the thermodynamic limit of the spin- frac{1}{2} XYZ spin chain with the antiperiodic boundary condition. The key point of our method is that there exist some degenerate points of the crossing parameter ηm,l, at which the associated inhomogeneous T − Q relation becomes a homogeneous one. This makes extrapolating the formulae deriving from the homogeneous one to an arbitrary η with O(N−2) corrections for a large N possible. The ground state energy and elementary excitations of the system are obtained. By taking the trigonometric limit, we also give the results of antiperiodic XXZ spin chain within the gapless region in the thermodynamic limit, which does not have any degenerate points.

Basic Properties of Non-Stationary Ruijsenaars Functions

For any variable number, a non-stationary Ruijsenaars function was recently introduced as a natural generalization of an explicitly known asymptotically free solution of the trigonometric Ruijsenaars model, and it was conjectured that this non-stationary Ruijsenaars function provides an explicit solution of the elliptic Ruijsenaars model. We present alternative series representations of the non-stationary Ruijsenaars functions, and we prove that these series converge. We also introduce novel difference operators called T which, as we prove in the trigonometric limit and conjecture in the general case, act diagonally on the non-stationary Ruijsenaars functions.

Penyelesaian Soal Limit Fungsi Trigonometri dengan Solusi Cepat pada siswa SMA

The purpose of this service is to help students deal with the shortage of time when working on multiple choice math problems, especially in the trigonometric function limit material. This activity was carried out on April 24, 2019 at SMAN 1 Tamban, Barito kuala, South Kalimantan, attended by 25 students as participants. The implementation of this activity consists of opening and introduction, explanation and application of problem solving, then practice questions and gift giving, as well as closing and group photos. The material for the quick solution of trigonometric function limit problem solving consists of trigonometric limits using the spruck rules, trigonometric limits using the delete rules of "evil" cosines, and trigonometric limits using the "good" cosine change rules. In the midst of community service activities, the service team appealed to students to prioritize understanding of mathematical concepts in solving mathematical problems. Quick solutions are recommended to be used only on multiple choice tests and force students to solve problems in a short time. Overall, this activity went well and smoothly and without significant obstacles.

Multivariate Quadratic Transformations and the Interpolation Kernel

We prove a number of quadratic transformations of elliptic Selberg integrals (conjectured in an earlier paper of the author), as well as studying in depth the "interpolation kernel", an analytic continuation of the author's elliptic interpolation functions which plays a major role in the proof as well as acting as the kernel for a Fourier transform on certain elliptic double affine Hecke algebras (discussed in a later paper). In the process, we give a number of examples of a new approach to proving elliptic hypergeometric integral identities, by reduction to a Zariski dense subset of a formal neighborhood of the trigonometric limit.

Gustafson-Rakha-Type Elliptic Hypergeometric Series

We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.

Trigonometric Derivatives Made Easy

SummaryWe give geometric proofs of the basic trigonometric derivative formulas, avoiding the trigonometric limit formulas required in the usual limit definition proofs.

A new perspective on the integrability of Inozemtsev’s elliptic spin chain

The aim of this paper is studying from an alternative point of view the integrability of the spin chain with long-range elliptic interactions introduced by Inozemtsev. Our analysis relies on some well-established conjectures characterizing the chaotic vs. integrable behavior of a quantum system, formulated in terms of statistical properties of its spectrum. More precisely, we study the distribution of consecutive levels of the (unfolded) spectrum, the power spectrum of the spectral fluctuations, the average degeneracy, and the equivalence to a classical vertex model. Our results are consistent with the general consensus that this model is integrable, and that it is closer in this respect to the Heisenberg chain than to its trigonometric limit (the Haldane–Shastry chain). On the other hand, we present some numerical and analytical evidence showing that the level density of Inozemtsev’s chain is asymptotically Gaussian as the number of spins tends to infinity, as is the case with the Haldane–Shastry chain. We are also able to compute analytically the mean and the standard deviation of the spectrum, showing that their asymptotic behavior coincides with that of the Haldane–Shastry chain.

Is a definition always useful to calculate something?

We calculate the limit of a particular trigonometric functionAfter a question proposed by a student: can you show a particular result, involving a trigonometric limit, using the definition of limit? The answer to this question is the aim of this manuscript. in a case in which direct substitution results in an indeterminacy (zero over zero), in four different ways, showing that the definition is not always the best way to obtain a result.

On non-stationary Lamé equation from WZW model and spin-1/2 XYZ chain

We study the link between WZW model and the spin-1/2 XYZ chain. This is achieved by comparing the second-order differential equations from them. In the former case, the equation is the Ward-Takahashi identity satisfied by one-point toric conformal blocks. In the latter case, it arises from Baxter's TQ relation. We find that the dimension of the representation space w.r.t. the V-valued primary field in these conformal blocks gets mapped to the total number of chain sites. By doing so, Stroganov's "The Importance of being Odd" (cond-mat/0012035) can be consistently understood in terms of WZW model language. We first confirm this correspondence by taking a trigonometric limit of the XYZ chain. That eigenstates of the resultant two-body Sutherland model from Baxter's TQ relation can be obtained by deforming toric conformal blocks supports our proposal.

New integrable systems as a limit of the elliptic SL(N, ℂ) top

We consider the scaling limit of an elliptic top. This limit is a combination of a scaling of the elliptic top variables, an infinite shift of the spectral parameter, and the trigonometric limit. We give general necessary constraints on the scaling of the variables and examples of such a degeneracy. A certain subclass of limit systems is integrable in the Liouville sense, which can also be shown directly.

An Intuitive Proof of the Singular Value Decomposition of a Matrix

SummaryUsing a simple trigonometric limit, we provide an intuitive geometric proof of the Singular Value Decomposition of an arbitrary matrix.

Three-coloring statistical model with domain wall boundary conditions: Trigonometric limit

We consider a nontrivial trigonometric limit of the three-coloring statistical model with the domain wall boundary conditions. In this limit, we solve the previously constructed functional equations and find a new determinant representation for the partial partition functions.