Orthogonal Polynomial Techniques Research Articles

Supersymmetric Virasoro minimal strings

A random matrix model definition of a family of N=1 supersymmetric extensions of the Virasoro minimal string of Collier, Eberhardt, Mühlmann, and Rodriguez is presented. An analysis of the defining string equations shows that the models all naturally have unambiguous nonperturbative completions, which are explicitly supplied by the double-scaled orthogonal polynomial techniques employed. Perturbatively, the multiloop correlation functions of the model define a special supersymmetric class of “quantum volumes,” generalizing the prototype case, some of which are computed. Published by the American Physical Society 2024

A Comparative Analysis of Trends in Area, Production and Productivity of Rapeseed Mustard in India and Uttar Pradesh

This paper analyses the trends in area, production and productivity of Rapeseed Mustard in India as well as Uttar Pradesh. India has the largest area and the highest production of mustard in the world. By using secondary data, a fifty-year time series has been used to apply orthogonal polynomial technique to get trends in area production and productivity. In order to know relative strength of area or productivity in production, Area and Productivity effect has been calculated with the help of simple regression analysis. To break the productivity, plateau the investment in technology up gradation and mechanization in the oilseed production, opening of better market avenues and improving the efficiency in the processing sector shall go a long way in augmenting the overall production and productivity. Key Words: Orthogonal Polynomial, Mechanization, Efficiency, Regression, productivity

Receding contact between an inclusion of generalized shape and a remotely-stressed plate

This paper examines the problem of frictionless contact and separation of a rigid inclusion from an infinite elastic plate. The inclusion initially fits perfectly into a hole in the plate, and separates from it upon remote stressing. In a significant generalization of the classical work of Wilson and Goree on receding contacts of elliptical inclusions, the present formulation allows the inclusion to be of quite general shape. In particular, superellipses provide a convenient way to describe a family of such generalized inclusion shapes by varying two parameters. A highly accurate conformal mapping scheme based on rational functions and the Adaptive Antoulas–Anderson (AAA) algorithm is used to map the interior of a unit circle to the exterior of these generalized inclusions. Notably, a rational function representation is far more efficient than a polynomial representation. The governing Singular Integro-Differential Equation (SIDE) of the contact is then derived using the conformal map and analytic continuation, and solved using standard orthogonal polynomial techniques. The contact size and transmitted pressure traction are obtained for a wide range of shapes, including superelliptical, curvilinear hexagonal, and hypotrochoidal square inclusions. The role of the remote stress ratio on the contact response is studied, and a phase diagram demarcating various contact regimes is obtained. A comparison of the pressures for selected inclusion shapes using finite element analysis (FEA) shows good agreement with the present (semi-analytical) SIDE results. For shapes like hypotrochoidal squares, subtle changes in the inclusion geometry can cause significantly different peak contact pressures, pointing to the importance of accurate modeling of geometry in conforming–receding contact problems.

Gauss quadrature based finite temperature Lanczos method

The finite temperature Lanczos method (FTLM), which is an exact diagonalization method intensively used in quantum many-body calculations, is formulated in the framework of orthogonal polynomials and Gauss quadrature. The main idea is to reduce finite temperature static and dynamic quantities into weighted summations related to one- and two-dimensional Gauss quadratures. Then lower order Gauss quadrature, which is generated from Lanczos iteration, can be applied to approximate the initial weighted summation. This framework fills the conceptual gap between FTLM and kernel polynomial method, and makes it easy to apply orthogonal polynomial techniques in the FTLM calculation.

New Set of Non-separable Orthogonal Invariant Moments for Image Recognition

It is known that the rotation, scaling and translation invariant property of image moments has a high significance in image recognition. For this reason, the seven invariant moments presented by Hu are widely used in the field of image analysis. These moments are of finite order; therefore, they do not comprise a complete set of image descriptors. For this reason, we introduce in this paper another series of invariant moments of infinite order, which are based on normalized central moments. The non-orthogonal property of these moments causes the redundancy of information. To overcome this problem, we propose a new construction technique of non-separable orthogonal polynomials in two variables based on a recurrence formula and we present a new set of orthogonal moments, which are invariant to translation, scaling and rotation. The presented approaches are tested in several well-known computer vision datasets including moment’s invariability, image retrieval and classification of objects, this latter based on fuzzy K-means clustering algorithm. The performance of these invariant moments for classification and image retrieval is compared with some recent invariant moments such as invariants of multi-channel orthogonal radial-substituted Chebyshev moments, invariants of quaternion radial-substituted Chebyshev moments, invariants of rotational moments in Radon space and Legendre–Fourier moments in Radon space. The experimental results made using four databases of images, namely Columbia Object Image Library (COIL-20) database, MPEG7-CE shape database, COIL-100 database and ORL database, show that our orthogonal invariant moments have done better than the other descriptors tested.

Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

We investigate a one-parameter family of Coulomb gases in two dimensions, which are confined to an ellipse due to a hard wall constraint, and are subject to an additional external potential. At inverse temperature we can use the technique of planar orthogonal polynomials, borrowed from random matrix theory, to explicitly determine all k-point correlation functions for a fixed number of particles N. These are given by the determinant of the kernel of the corresponding orthogonal polynomials, which in our case are the Gegenbauer polynomials, or a subset of the asymmetric Jacobi polynomials, depending on the choice of external potential, as shown in a companion paper recently published by three of the authors. In the rotationally invariant case, when the ellipse becomes the unit disc, our findings agree with that of the ensemble of truncated unitary random matrices. The thermodynamical large-N limit is investigated in the local scaling regime in the bulk and at the edge of the spectrum at weak and strong non-Hermiticity. We find new universality classes in these limits and recover the sine- and Bessel-kernel in the Hermitian limit. The limiting global correlation functions of particles in the interior of the ellipse are more difficult to obtain but found in the special cases corresponding to the Chebyshev polynomials.

Real-Space Approach for the Electronic Calculation of Twisted Bilayer Graphene Using the Orthogonal Polynomial Technique

We discuss technical issues involving the implementation of a computational method for the electronic structure of material systems of arbitrary atomic arrangement. The method is based on the analysis of time evolution of electron states in the real lattice space. The Chebyshev polynomials of the first kind are used to approximate the time evolution operator. We demonstrate that the developed method is powerful and efficient since the computational scaling law is linear. We invoked the method to study the electronic properties of special twisted bilayer graphene whose atomic structure is quasi-crystalline. We show the density of states of an electron in this graphene system as well as the variation of the associated time auto-correlation function. We find the fluctuation of electron density on the lattice nodes forming a typical pattern closely related to the typical atomic pattern of the quasi-crystalline bilayer graphene configuration.

Asymptotics for the Partition Function in Two-Cut Random Matrix Models

We obtain large N asymptotics for the Hermitian random matrix partition function \[Z_N(V)=\int_{\mathbb R^N}\prod_{i<j}(x_i-x_j)^2 \prod_{j=1}^N e^{-N V(x_j)}dx_j,\] in the case where the external potential $V$ is a polynomials such that the random matrix eigenvalues accumulate on two disjoint intervals (the two-cut case). We compute leading and sub-leading terms in the asymptotic expansion for $\log Z_N(V)$, up to terms that are small as $N$ goes to infinity. Our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential $V$. Afterwards, we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two-cut potentials $V$. The asymptotic expansion of $\log Z_N(V)$ as $N$ goes to infinity contains terms that depend analytically on the potential $V$ and that have already appeared in the literature. In addition our method allows to compute the $V$-independent terms of the asymptotic expansion of $\log Z_N(V)$ which, to the best of our knowledge, had not appeared before in the literature. We use rigorous orthogonal polynomial and Riemann-Hilbert techniques which had so far been successful to compute asymptotics for the partition function only in the one-cut case.

Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite $$n$$ n Gaussian Unitary Ensembles

In this paper we study the gap probability problem in the Gaussian unitary ensembles of \(n\) by \(n\) matrices : The probability that the interval \(J := (-a,a)\) is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke, and Forrester and Witte on this subject, it has been shown that two Painlevé type differential equations arise in this context. The first is the Jimbo–Miwa–Okomoto \(\sigma \)-form and the second is a particular Painlevé IV. Using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted as \(\sigma _n(a)\), \(R_n(a)\) and \(r_n(a)\). We show that each one satisfies a second order Painlevé type differential equation as well as a discrete Painlevé type equation. In particular, in addition to providing an elementary derivation of the aforementioned \(\sigma \)-form and Painlevé IV we are able to show that the quantity \(r_n(a)\) satisfies a particular case of Chazy’s second degree second order differential equation. For the discrete equations we show that the quantity \(r_n(a)\) satisfies a particular form of the modified discrete Painlevé II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities \(R_n(a)\) and \(\sigma _n(a)\).

Universality in Gaussian random normal matrices

We prove that for Gaussian random normal matrices the correlation function has universal behavior. Using the technique of orthogonal polynomials and identities similar to the Christoffel-Darboux formula, we find that in the limit, as the dimension of the matrix tends to infinity, the density of eigenvalues converges to a constant inside of an ellipse and to zero outside. The convergence holds locally uniformly. At the boundary, in scaled coordinates holding the distance between eigenvalues constant, we show that the density is falling off to zero like the complementary error function. The convergence is uniform on the boundary. Further we give an explicit expression for the the correlation function.

The Dirac–Coulomb problem: a mathematical revisit

We obtain a symmetric tridiagonal matrix representation of the Dirac–Coulomb operator in a suitable complete square integrable basis. Orthogonal polynomial techniques along with the Darboux method are used to obtain the bound states energy spectrum, the relativistic scattering amplitudes and phase shifts from the asymptotic behavior of the polynomial solutions associated with the resulting three-term recursion relation.

Rank 1 real Wishart spiked model

AbstractIn this paper, we consider N‐dimensional real Wishart matrices Y in the class \input amssym $W_{\Bbb R} (\Sigma ,M)$ in which all but one eigenvalue of Σ is 1. Let the nontrivial eigenvalue of Σ be 1+τ; then as N, M → ∞, with M/N → γ2 finite and nonzero, the eigenvalue distribution of Y will converge into the Marchenko‐Pastur distribution inside a bulk region. When τ increases from 0, one starts to see a stray eigenvalue of Y outside of the support of the Marchenko‐Pastur density. As this stray eigenvalue leaves the bulk region, a phase transition will occur in the largest eigenvalue distribution of the Wishart matrix.In this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs. We will first establish the results that are valid for all N and M and will use them to carry out the asymptotic analysis. In particular, we have derived a contour integral formula for the Harish‐Chandra Itzykson‐Zuber integral $\int_{O(N)} {e^{{\rm tr}(XgYg^{\rm T} )} } g^{\rm T} dg$ when X and Y are real symmetric and Y is a rank 1 matrix. This allows us to write down a Fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques. As a result, we obtain an integral formula for the largest eigenvalue distribution in the large‐ N limit characterized by Painlevé transcendents. The approach used in this paper is very different from a recent paper by Bloemenal and Virág, in which the largest eigenvalue distribution was obtained using a stochastic operator method. In particular, the Painlevé formula for the largest eigenvalue distribution obtained in this paper is new. © 2012 Wiley Periodicals, Inc.

직교다항식을 이용한 자동차 압축기용 가변 사판의 구조최적설계

The variable-swash-plate compressor has recently been adopted as a vehicle compressor to improve fuel efficiency. The rotation torque in the variable-swash-plate compressor and the pressure-affected piston have a great influence on the swash-plate design and deformation. This paper suggests the optimal configuration design by using Chebyshev orthogonal polynomial and optimization techniques. The orthogonal array (OA) and analysis of variance (ANOVA) techniques and response surface optimization, are employed to determine the main effects and their optimal design variables. According to the optimal design, we confirm an effective design variable in swash plate and explain the optimal solution, the usefulness for satisfying the constraints of maximum stress and deformation.

Quasi-Birth-and-Death Processes and Matrix-Valued Orthogonal Polynomials

We consider a matrix-valued spectral decomposition of a family of block-tridiagonal matrices arising as the transition matrices of so-called quasi-birth-and-death processes. This representation is a generalization of the Karlin-McGregor representation for the $n$-step transition probabilities of a birth-and-death process via a system of orthogonal polynomials. At the heart of the representation is a self-adjoint matrix-valued measure associated to the process. We make use of a previously known formula relating the Stieltjes transform of this measure to that of the measure associated to the “0th associated process,” generalizing a theorem of Karlin and McGregor, to compute the Stieltjes transform of the spectral measure for several examples. In addition, we apply matrix-valued orthogonal polynomial techniques to the study of “sin-graphs” and higher-dimensional birth-and-death processes, for which the relevant polynomials are multivariate.

Indentation of a round punch into a rough elastic half-space

We consider the axisymmetric contact problem on the indentation of a round punch with plane base into an elastic half-space. Roughness is taken into account in the framework of the well-known model in which the local surface displacements are proportional to some power of the contact pressures at the same point. To study the problem, we use the theory of nonlinear integral equations of Hammerstein type together with the mathematical technique of orthogonal Legendre polynomials and the contraction mapping principle. We obtain an approximate analytic solution of the problem, numerically analyze the results, and reveal typical laws of variation of the main mechanical variables.

PROBLEMAS Y MÉTODOS COMUNES DEL ANÁLISIS DE EXPERIMENTOS FACTORIALES

In the agronomic research, factorial experiments are frequently carried out. These are tools that make an optimus use of resources; produce more precise estimates of contrasts of treatment means, and make it possible the study of the interactions among factors. The full release of these advantages, however, is not always obtained due to one or several of the following common problems: Poor interpretation of the interaction concept; abuse of multiple comparison procedures; subutilization and even omission of regression techniques, orthogonal contrasts and polynomials, etc. To contribute to solve these problems and thereby to improve the quality of the scientific articles to be published, in this paper several meanings of the interaction concept, including since its etymological aspects until its meaning as a source of variation in the analysis of variance, were analyzed. In addition, cases related with the pertinence, use and advantages of regression, contrasts, and orthogonal contrast and polynomial techniques were analyzed in the context of factorial experiments. To make a more objective presentation, hypothetical cases of factorial experiments were considered. It is hoped that the reader’s conceptual perception and capacity to successfully associate cases of factorial experiments and statistical methods to analyze them properly will be improved and thereby will enable them to increase the quality of the interpretation of the results and their scientific publications.

Bulk Universality and Related Properties of Hermitian Matrix Models

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \ref{t:U.t1}). The proof as our previous proof in \cite{Pa-Sh:97} is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper \cite{BPS:95} on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.

Higher spin fields from a worldline perspective

Higher spin fields in four dimensions, and more generally conformal fields in arbitrary dimensions, can be described by spinning particle models with a gauged SO(N) extended supergravity on the worldline. We consider here the one-loop quantization of these models by studying the corresponding partition function on the one-dimensional torus. After gauge fixing the supergravity multiplet, the partition function reduces to an integral over the corresponding moduli space which is computed using orthogonal polynomial techniques. We obtain a compact formula which gives the number of physical degrees of freedom for all N in all dimensions. As an aside we compute the physical degrees of freedom of the SO(4) = SU(2)xSU(2) model with only a SU(2) factor gauged, which has attracted some interest in the literature.

Vapor−Liquid Equilibrium of Binary Mixtures. 1. Ethanol + 1-Butanol, Ethanol + Octane, 1-Butanol + Octane

The activity coefficients of the binary mixtures ethanol + 1-butanol, ethanol + octane, and 1-butanol + octane were determined at temperatures of (308.15, 313.15, and 318.15) K. The determination of the vapor phase composition at equilibrium was carried out using headspace gas chromatography analysis. Multiple headspace extraction was used to calibrate the headspace gas chromatograph. Comparison of the phase diagrams produced using standard Legendre orthogonal polynomial techniques with phase diagrams from the literature showed good agreement. The composition of the azeotropes were reported, where applicable.

Characteristic polynomials of complex random matrix models

We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their Hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for Hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite- N, where N is the size of the matrix. We give some explicit examples at finite- N for specific weight functions. The characteristic polynomials in the large- N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large- N limit.