Complete Orthogonal Set Research Articles

Solvable random-matrix ensemble with a logarithmic weakly confining potential.

This work identifies a solvable (in the sense that spectral correlation functions can be expressed in terms of orthogonal polynomials), rotationally invariant random matrix ensemble with a logarithmic weakly confining potential. The ensemble, which can be interpreted as a transformed Jacobi ensemble, is in the thermodynamic limit characterized by a Lorentzian eigenvalue density. It is shown that spectral correlation functions can be expressed in terms of the nonclassical Gegenbauer polynomials C_{n}^{(-1/2)}(x) with n≥2, which have been proven to form a complete orthogonal set with respect to the proper weight function. A procedure to sample matrices from the ensemble is outlined and used to provide a numerical verification for some of the analytical results. This ensemble is pointed out to potentially have applications in quantum many-body physics.

Demonstration of 10-channel mode- and polarization-division multiplexed free-space optical transmission with successive interference cancellation DSP.

We experimentally demonstrate 10-channel mode-division multiplexed free-space optical transmission with five spatial modes, each carrying 19.6925-Gbaud dual-polarization quadrature phase shift keying signals. Strong inter-mode cross talk is observed in our commercially available photonic lantern based system when using a complete orthogonal mode set as independent channels. A successive interference cancellation based multiple-input multiple-output digital signal processing (DSP) algorithm is first applied to mitigate the inter-mode cross talk in mode-division multiplexed systems. The DSP also supports unequal transmit and receive channel numbers to further improve the cross talk resiliency. Compared to the conventional minimum mean square error DSP, the required optical signal-to-noise ratio of the successive interference cancellation DSP is decreased by approximately 5 dB at the hard-decision forward error correction limit. As a result, this system demonstrates a record-high independent channel number of 10 and spectral efficiency of 13.7 b/s/Hz in mode-division multiplexed free-space optical systems.

Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models

The fractional Brownian motion (FBM) is a common model for long and short-range dependent phenomena that appears in different fields, including physics, biology, and finance. In the current work, a new spectral technique named the fractional Wiener Hermite Expansion (FWHE) is developed to analyze stochastic models with FBM. The technique has a theoretical background in the literature and proof of convergence. A new complete orthogonal Hermite basis set is developed. Calculus derivations and statistical analysis are performed to handle the mixed multi-dimensional fractional and/or integer-order integrals that appear in the analysis. Formulas for the mean and variance are deduced and are found to be based on fractional integrals. Using the developed expansion with the statistical properties of the basis functionals will help to reduce the stochastic model to equivalent deterministic fractional models that can be analyzed numerically or analytically with the well-known techniques. A numerical algorithm is developed to be used in case there is no available analytical solution. The numerical algorithm is compared with the fractional Euler-Maruyama (EM) technique to verify the results. In comparison to sampling based techniques, FWHE provides an efficient analytical or numerical alternative. The applicability of FWHE is demonstrated by solving different examples with additive and multiplicative FBM.

Electromagnetic plane wave diffraction by a cylindrical arc with edges: H-polarized case

An accurate hybrid method (numerical-analytical method) for the diffraction of H-polarized electromagnetic plane wave by perfectly electric conducting cylindrical bodies containing edges and a longitudinal slit aperture is proposed. This method is the combination of the Method of Moment and semi-inversion method. The current density function is expressed as the Chebyshev polynomials forming a complete orthogonal set of basis functions. Then, the initial problem is reduced to a system of linear algebraic equations. After inversion, the unknown coefficients are obtained. Then, near and far-field distributions, radar cross-sections are obtained. The resonances are observed for different values of the aperture size, radius of the arc, and the results are compared with previous outcomes.

Sound focusing in inhomogeneous waveguides

A method for analytically studying sound focusing in inhomogeneous waveguides is presented. From the viewpoint of acquiring the maximum acoustic pressure at an arbitrary position with normalized energy flux injection, optimal incident waves can be derived based on the multimodal admittance method. The method involves two steps. The first step is to expand the wave solution onto a complete orthogonal basis set so that the Helmholtz equation can be transformed into two sets of first-order coupled differential equations in the modal domain. The second step is to solve the coupled equations numerically by introducing admittance matrices and propagators, which can be used to derive reflection matrices and transmission matrices. Using the multimodal admittance method, one can circumvent the contamination caused by exponentially diverging evanescent modes and acquire stable wave solutions. Then the mapping between the acoustic pressure at an arbitrary position and that of the incident wave can be constructed, and this mapping changes the problem of wave focusing into solving the extrema of inner products in Hilbert space. The optimal incident waves that generate wave focusing at an arbitrary position can be readily computed together with the corresponding wave solutions. In this paper, we study the sound focusing in waveguides with varying cross-sections, scatterers and sound-speed profiles. The results show that the optimal incident waves will take full advantage of wave scattering caused by the boundaries and inhomogeneities during propagation to achieve the maximum pressure at foci, leading to good single-point and multi-point sound focusing performance. In addition, we find when injecting the spatially sampled optimal incident waves or the optimal incident waves with random perturbations, the resultant wave focusing phenomena will be still apparent. The focusing behaviors are highly robust to the perturbations of the moduli of the incident waves and slightly less robust to that of the arguments of the incident waves. Our method is also available for analyzing wave focusing in other kinds of inhomogeneous waveguides. We believe that our research can provide guidance on designing acoustic lenses or metamaterials to focus sound waves in complex media, and can offer inspiration in wave communications, imagings and non-destructive testing.

History states of systems and operators

We discuss some fundamental properties of discrete system-time history states. Such states arise for a quantum reference clock of finite dimension and lead to a unitary evolution of system states when satisfying a static discrete Wheeler-DeWitt-type equation. We consider the general case where system-clock pairs can interact, analyzing first their different representations and showing there is always a special clock basis for which the evolution for a given initial state can be described by a constant Hamiltonian $H$. It is also shown, however, that when the evolution operators form a complete orthogonal set, the history state is maximally entangled for any initial state, as opposed to the case of a constant $H$, and can be generated with a simple two-clock setting. We then examine the quadratic system-time entanglement entropy, providing an analytic evaluation and showing it satisfies strict upper and lower bounds determined by the energy spread and the geodesic evolution connecting the initial and final states. We finally show that the unitary operator that generates the history state can itself be considered as an operator history state, whose quadratic entanglement entropy determines its entangling power. Simple measurements on the clock enable to efficiently determine overlaps between system states and also evolution operators at any two times.

Harmonic oscillator Wigner function extension to exceptional polynomials

In this paper, we construct isospectral Hamiltonians without shape invariant potentials for a harmonic oscillator Wigner function on a real line. In this case, we actually remove the ground state of the second Hamiltonian, which forms a special case, $$m=0$$ , of an exceptional Laguerre differential equation with solutions $$\{L_n^{-2}\}_{n=2}^\infty $$ as eigenfunctions form a complete orthogonal set in the Hilbert space.

The spectral analysis of three families of exceptional Laguerre polynomials

The Bochner Classification Theorem (1929) characterizes the polynomial sequences {pn}n=0∞, with degpn=n that simultaneously form a complete set of eigenstates for a second-order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders. Indeed, up to a complex linear change of variable, only the classical Hermite, Laguerre, and Jacobi polynomials, with certain restrictions on the polynomial parameters, satisfy these conditions. In 2009, Gómez-Ullate, Kamran, and Milson found that for sequences {pn}n=1∞, degpn=n (without the constant polynomial), the only such sequences satisfying these conditions are the exceptionalX1-Laguerre and X1-Jacobi polynomials. Subsequently, during the past five years, several mathematicians and physicists have discovered and studied other exceptional orthogonal polynomials {pn}n∈N0⧵A, where A is a finite subset of the non-negative integers N0 and where degpn=n for all n∈N0⧵A. We call such a sequence an exceptional polynomial sequence of codimension |A|, where the latter denotes the cardinality of A. All exceptional sequences with a non singular weight, found to date, have the remarkable feature that they form a complete orthogonal set in their natural Hilbert space setting.Among the exceptional sets already known are two types of exceptional Laguerre polynomials, called the Type I and Type II exceptional Laguerre polynomials, each omitting m polynomials. In this paper, we briefly discuss these polynomials and construct the self-adjoint operators generated by their corresponding second-order differential expressions in the appropriate Hilbert spaces. In addition, we present a novel derivation of the Type III family of exceptional Laguerre polynomials along with a detailed disquisition of its properties. We include several representations of these polynomials, orthogonality, norms, completeness, the location of their local extrema and roots, root asymptotics, as well as a complete spectral study of the second-order Type III exceptional Laguerre differential expression.

A spectral study of the second-order exceptional [formula omitted]-Jacobi differential expression and a related non-classical Jacobi differential expression

The exceptional X1-Jacobi differential expression is a second-order ordinary differential expression with rational coefficients; it was discovered by Gómez-Ullate, Kamran and Milson in 2009. In their work, they showed that there is a sequence of polynomial eigenfunctions {Pˆn(α,β)}n=1∞ called the exceptional X1-Jacobi polynomials. There is no exceptional X1-Jacobi polynomial of degree zero. These polynomials form a complete orthogonal set in the weighted Hilbert space L2((−1,1);wˆα,β), where wˆα,β is a positive rational weight function related to the classical Jacobi weight. Among other conditions placed on the parameters α and β, it is required that α,β>0. In this paper, we develop the spectral theory of this expression in L2((−1,1);wˆα,β). We also consider the spectral analysis of the ‘extreme’ non-exceptional case, namely when α=0. In this case, the polynomial solutions are the non-classical Jacobi polynomials {Pn(−2,β)}n=2∞. We study the corresponding Jacobi differential expression in several Hilbert spaces, including their natural L2 setting and a certain Sobolev space S where the full sequence {Pn(−2,β)}n=0∞ is studied and a careful spectral analysis of the Jacobi expression is carried out.

Restricted Schurs and correlators for SO(N ) and Sp(N )

In a recent work, restricted Schur polynomials have been argued to form a complete orthogonal set of gauge invariant operators for the 1/4-BPS sector of free N = 4 super Yang-Mills theory with an SO(N) gauge group. In this work, we extend these results to the theory with an Sp(N) gauge group. Using these operators, we develop techniques to compute correlation functions of any multi-trace operators with two scalar fields exactly in the free theory limit for both SO(N) and Sp(N).

Trapped waves on jet currents: asymptotic modal approach

AbstractAn asymptotic theory of surface waves trapped on vertically uniform jet currents is developed as a first step towards a systematic description of wave dynamics on oceanic jet currents. It has been shown that in a linear setting an asymptotic separation of vertical and horizontal variables, which underpins the modal description of the wave field on currents, is possible if either the current velocity is small compared to the wave celerity or the current width is large compared to the wavelength along the current. The scheme developed enables us to obtain solutions as an asymptotic series with any desired accuracy. The initially three-dimensional problem is reduced to solving one-dimensional equations with the lateral and vertical dependence being prescribed by the corresponding modal structure. To leading order in current magnitude to wave celerity, the boundary value problem specifying the modes and eigenvalues reduces to classical Sturm–Liouville type based upon the one-dimensional stationary Schrödinger equation. The modes, both trapped and ‘passing-through’, form a complete orthogonal set. This makes the modal description of waves on currents a mathematically attractive alternative to the approaches currently adopted. Properties of trapped eigenmodes and their dispersion relations are examined both for broad currents of arbitrary magnitude, where the modes are not orthogonal, and for weak currents, where the modes are orthogonal. Several model profiles for which nice analytical solutions of the leading-order boundary value problem are known were used to get an insight. The asymptotic solutions proved not only to capture qualitative behaviour well but also to provide a good quantitative description even for unrealistically strong and narrow currents. The results are discussed for various oceanic currents, with particular attention paid to the Agulhas Current, for which specific estimates were derived. For typical dominant wind waves and swell, all oceanic-jet-type currents are weak and, correspondingly, the developed asymptotic scheme based upon one-dimensional stationary Schrödinger equation for modes applies.

A New Class of Orthogonal Spline Moments and Its Application

This paper introduces a new set of orthogonal moment Functions-franklin Moments (FMs). The kernel functions of FMs is Franklin set, which is a class of complete orthogonal spline function set of degree 1. The implementation of FMs does not involve any numerical approximation and has a rather low computation complexity, since the basis set has the advantages of lower order. These properties make FMs superior to the conventional orthogonal moments such as Legendre moments and Zernike moments, in terms of the image reconstruction. Our simulation results also show that FMs have a better feature representation capability.

Palm Print Recognition using Zernike Moments

ABSTRACT This paper proposes an automated system for recognizing palmprints for biometric identification of individuals. Complex Zernike moments are constructed using a set of complex polynomials which form a complete orthogonal basis set defined on the unit disc. Palmprint images are projected onto the basis set resulting in a set of complex signals. The magnitude of the complex value is computed and a scalar value is derived from it by computing the mean of the vector elements. Classification is done by subtracting the test samples from the mean of the training set. The data set consists of 80 images divided into 4 classes. Accuracy obtained is comparable to the best results reported in literature General Terms Pattern Recognition, Computer Vision Keywords Zernike moment, Palmprint recognition, Texture classification 1. INTRODUCTION Biometrics refers to automatic recognition of individuals based on their physiological and behavioral characteristics, which are nowadays used extensively for personal identification worldwide. Different physical traits like face, iris, fingerprint, palmprint, retina etc. fall under the perview of biometrics. Palmprint recognition involves identifying an individual by the principal lines, wrinkles, ridges on the surface of the palm. The basis for using palmprints lies in the fact that no two individuals have exactly the same palmprint pattern, moreover palmprints remain more or less stable throughout the lifetime and are easily obtainable using standard imaging techniques. Challenges in palmprint recognition are related to building a reliable data model from randomly oriented irregular lines that enable high amount of accuracies in security based systems and applications. This paper presents an efficient algorithm for palm print recognition by utilizing complex Zernike moments. The organization of the paper is as follows: section 2 provides an overview of the related works, section 3 outlines the proposed approach, section 4 details the experimentations done and results obtained, section 5 analyses the current work vis-a-vis contemporary works, section 6 brings up the overall conclusions and future scopes.

Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters $$\alpha =\beta =-1$$

In this paper, we consider the second-order differential expression $$\begin{aligned} \ell [y](x)=(1-x^{2})(-(y^{\prime }(x))^{\prime }+k(1-x^{2})^{-1} y(x))\quad (x\in (-1,1)). \end{aligned}$$ This is the Jacobi differential expression with nonclassical parameters $$\alpha =\beta =-1$$ in contrast to the classical case when $$\alpha ,\beta >-1$$ . For fixed $$k\ge 0$$ and appropriate values of the spectral parameter $$\lambda ,$$ the equation $$\ell [y]=\lambda y$$ has, as in the classical case, a sequence of (Jacobi) polynomial solutions $$\{P_{n}^{(-1,-1)} \}_{n=0}^{\infty }.$$ These Jacobi polynomial solutions of degree $$\ge 2$$ form a complete orthogonal set in the Hilbert space $$L^{2}((-1,1);(1-x^{2})^{-1})$$ . Unlike the classical situation, every polynomial of degree one is a solution of this eigenvalue equation. Kwon and Littlejohn showed that, by careful selection of this first-degree solution, the set of polynomial solutions of degree $$\ge 0$$ are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a self-adjoint operator $$T$$ , generated by $$\ell [\cdot ],$$ in this Sobolev space that has these Jacobi polynomials as a complete orthogonal set of eigenfunctions. The classical Glazman–Krein–Naimark theory is essential in helping to construct $$T$$ in this Sobolev space as is the left-definite theory developed by Littlejohn and Wellman.

Weak-lensing statistics from the Coyote Universe

Analyzing future weak lensing data sets from KIDS, DES, LSST, Euclid, WFIRST requires precise predictions for the weak lensing measures. In this paper we present a weak lensing prediction code based on the Coyote Universe emulator. The Coyote Universe emulator predicts the (non-linear) power spectrum of density fluctuations (P_delta) to high accuracy for k \in [0.002;3.4] h/Mpc within the redshift interval z \in [0;1], outside this regime we extend P_delta using a modified Halofit code. This pipeline is used to calculate various second-order cosmic shear statistics, e.g., shear power spectrum, shear-shear correlation function, ring statistics and COSEBIs (Complete Orthogonal Set of EB-mode Integrals), and we examine how the upper limit in k (and z) to which P_delta is known, impacts on these statistics. For example, we find that k_max~8 h/Mpc causes a bias in the shear power spectrum at l~4000 that is comparable to the statistical errors (intrinsic shape-noise and cosmic variance) of a DES-like survey, whereas for LSST-like errors k_max~15 h/Mpc is needed to limit the bias at l~4000. For the most recently developed second-order shear statistics, the COSEBIs, we find that 9 modes can be calculated accurately knowing P_delta to k_max=10 h/Mpc. The COSEBIs allow for an EB-mode decomposition using a shear-shear correlation function measured over a finite range, thereby avoiding any EB-mode mixing due to finite survey size. We perform a detailed study in a 5-dimensional parameter space in order to examine whether all cosmological information is captured by these 9 modes with the result that already 7-8 modes are sufficient.

Fluctuations around periodic BPS-density waves in the Calogero model

The collective field formulation of the Calogero model supports periodic density waves. An important set of such density waves is a two-parameter family of BPS solutions of the equations of motion of the collective field theory. One of these parameters is essentially the average particle density, which determines the period, while the other parameter determines the amplitude. These BPS solutions are sometimes referred to as "small amplitude waves" since they undulate around their mean density, but never vanish. We present complete analysis of quadratic fluctuations around these BPS solutions. The corresponding fluctuation hamiltonian (i.e., the stability operator) is diagonalized in terms of bosonic creation and annihilation operators which correspond to the complete orthogonal set of Bloch-Floquet eigenstates of a related periodic Schr\"odinger hamiltonian, which we derive explicitly. Remarkably, the fluctuation spectrum is independent of the parameter which determines the density wave's amplitude. As a consequence, the sum over zero-point energies of the field-theoretic fluctuation hamiltonian, and its ensuing normal-ordering and regularization, are the same as in the case of fluctuations around constant density background, namely, the ground state. Thus, quadratic fluctuations do not shift the energy density tied with the BPS-density waves studied here, compared to its ground state value. Finally, we also make some brief remarks concerning fluctuations around non-BPS density waves

EXTENSIONS OF GENERALIZED STABLE RINGS

In this paper, we investigate the extensions of generalized stable rings. It is shown that a ring R is a generalized stable ring if and only if R has a complete orthogonal set {e<TEX>$_1$</TEX>, . . . , e<TEX>$_n$</TEX>} of idempotents such that e<TEX>$_1$</TEX>Re<TEX>$_1$</TEX>, . . . , e<TEX>$_n$</TEX>Re<TEX>$_n$</TEX> are generalized stable rings. Also, we prove that a ring R is a generalized stable ring if and only if R[[X]] is a generalized stable ring if and only if T(R,M) is a generalized stable ring.

Quantization of the elastic modes in an isotropic plate

We quantize the elastic modes in a plate. For this, we find a complete orthogonal set of eigenfunctions of the elastic equations and normalize them. These are the phonon modes in the plate and their specific forms and dispersion relations are manifested in low-temperature experiments in ultra-thin membranes.

Brukner-Zeilinger invariant information

We present an alternative interpretation of the notion of invariant information by establishing that it is directly related to the total ordinary variance of a quantum state. Here, “total” means summing the variance over any complete orthogonal set of observables or, equivalently, averaging over a certain sufficiently general ensemble of the observables. This simple, intuitive substratum of the Brukner-Zeilinger invariant information sheds further light on the informational and statistical nature of quantum measurements.

Jacobi–Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression

We develop the left-definite analysis associated with the self-adjoint Jacobi operator A k ( α , β ) , generated from the classical second-order Jacobi differential expression ℓ α , β , k [ y ] ( t ) = 1 w α , β ( t ) ( ( - ( 1 - t ) α + 1 ( 1 + t ) β + 1 y ′ ( t ) ) ′ + k ( 1 - t ) α ( 1 + t ) β y ( t ) ) ( t ∈ ( - 1 , 1 ) ) , in the Hilbert space L α , β 2 ( - 1 , 1 ) ≔ L 2 ( ( - 1 , 1 ) ; w α , β ( t ) ) , where w α , β ( t ) = ( 1 - t ) α ( 1 + t ) β , that has the Jacobi polynomials { P m ( α , β ) } m = 0 ∞ as eigenfunctions; here, α , β > - 1 and k is a fixed, non-negative constant. More specifically, for each n ∈ N , we explicitly determine the unique left-definite Hilbert–Sobolev space W n , k ( α , β ) ( - 1 , 1 ) and the corresponding unique left-definite self-adjoint operator B n , k ( α , β ) in W n , k ( α , β ) ( - 1 , 1 ) associated with the pair ( L α , β 2 ( - 1 , 1 ) , A k ( α , β ) ) . The Jacobi polynomials { P m ( α , β ) } m = 0 ∞ form a complete orthogonal set in each left-definite space W n , k ( α , β ) ( - 1 , 1 ) and are the eigenfunctions of each B n , k ( α , β ) . Moreover, in this paper, we explicitly determine the domain of each B n , k ( α , β ) as well as each integral power of A k ( α , β ) . The key to determining these spaces and operators is in finding the explicit Lagrangian symmetric form of the integral composite powers of ℓ α , β , k [ · ] . In turn, the key to determining these powers is a double sequence of numbers which we introduce in this paper as the Jacobi–Stirling numbers. Some properties of these numbers, which in some ways behave like the classical Stirling numbers of the second kind, are established including a remarkable, and yet somewhat mysterious, identity involving these numbers and the eigenvalues of A k ( α , β ) .