Showing papers on "Biorthogonal system published in 2014"

Biorthogonal quantum mechanics

Abstract: The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.

Quasimodal expansion of electromagnetic fields in open two-dimensional structures

Abstract: A quasimodal expansion method (QMEM) is developed to model and understand the scattering properties of arbitrary shaped two-dimensional open structures. In contrast with the bounded case which has only a discrete spectrum (real in the lossless media case), open resonators show a continuous spectrum composed of radiation modes and may also be characterized by resonances associated to complex eigenvalues (quasimodes). The use of a complex change of coordinates to build perfectly matched layers allows the numerical computation of those quasimodes and of approximate radiation modes. Unfortunately, the transformed operator at stake is no longer self-adjoint, and classical modal expansion fails. To cope with this issue, we consider an adjoint eigenvalue problem whose eigenvectors are biorthogonal to the eigenvectors of the initial problem. The scattered field is expanded on this complete set of modes leading to a reduced order model of the initial problem. The different contributions of the eigenmodes to the scattered field unambiguously appears through the modal coefficients, allowing us to analyze how a given mode is excited when changing incidence parameters. This gives physical insights to the spectral properties of different open structures such as nanoparticles and diffraction gratings. Moreover, the QMEM proves to be extremely efficient for the computation of local density of states.

Eigenvalue statistics for product complex Wishart matrices

Abstract: The eigenvalue statistics for complex Wishart matrices , where is equal to the product of r complex Gaussian matrices, and the inverse of s complex Gaussian matrices, are considered. In the case r = s the exact form of the global density is computed. The averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function . For finite N the eigenvalue probability density function is computed, and is shown to be an example of a biorthogonal ensemble. A double contour integral form of the corresponding correlation kernel is derived, which allows the hard edge scaled limit to be computed. The limiting kernel is given in terms of certain Meijer G-functions, and is identical to that found in the recent work of Kuijlaars and Zhang in the case s = 0. Properties of the kernel and corresponding correlation functions are discussed.

Cauchy–Laguerre Two-Matrix Model and the Meijer-G Random Point Field

Abstract: We apply the general theory of Cauchy biorthogonal polynomials developed in Bertola et al. (Commun Math Phys 287(3):983–1014, 2009) and Bertola et al. (J Approx Th 162(4):832–867, 2010) to the case associated with Laguerre measures. In particular, we obtain explicit formulae in terms of Meijer-G functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the Cauchy two-matrix model associated with them. The central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists, and is given by a new determinantal two-level random point field, the Meijer-G random field. We conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of Laguerre weights. We express the joint distributions of the smallest eigenvalues in terms of suitable Fredholm determinants and evaluate them numerically. We also show that in a suitable limit, the Meijer-G random field converges to the Bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the Laguerre ensemble.

Biorthogonal ensembles with two-particle interactions

Abstract: We investigate determinantal point processes on [0, +∞) of the form We prove that the biorthogonal polynomials associated with such models satisfy a recurrence relation and a Christoffel–Darboux formula if $ hetainmathbb Q$ , and that they can be characterized in terms of 1 × 2 vector-valued Riemann–Hilbert problems, which exhibit some non-standard properties. In addition, we obtain expressions for the equilibrium measure associated with our model if w(λ) = e−nV (λ) in the one-cut case with and without a hard edge

Numerical solution for a class of fractional convection–diffusion equations using the flatlet oblique multiwavelets

Abstract: This paper is concerned with the construction of biorthogonal multiwavelet basis in the unit interval to form a biorthogonal flatlet multiwavelet system. Next a method to calculate integer and frac...

Triangular random matrices and biorthogonal ensembles

Abstract: We study the singular values of certain triangular random matrices. When their elements are i.i.d. standard complex Gaussian random variables, the squares of the singular values form a biorthogonal ensemble, and with an appropriate change in the distribution of the diagonal elements, they give the biorthogonal Laguerre ensemble. For triangular Wigner matrices, we give alternative proofs for the convergence of the empirical distribution of the appropriately scaled squares of the singular eigenvalues to a distribution with support $[0, e]$, as well as for the almost sure convergence of the rescaled largest singular eigenvalue to $\sqrt{e}$ under the additional assumption of mean zero and finite fourth moment for the law of the matrix elements.

Development of real-time MHD markers based on biorthogonal decomposition of signals from Mirnov coils

Abstract: The biorthogonal decomposition analysis of signals from an array of Mirnov coils is able to provide the spatial structure and the temporal evolution of magnetohydrodynamic (MHD) instabilities in a tokamak. Such analysis can be adapted to a data acquisition and elaboration system suitable for fast real time applications such as instability detection and disruption precursory markers computation. This paper deals with the description of this technique as applied to the Frascati Tokamak Upgrade (FTU).

Loaded differential operators: Convergence of spectral expansions

Abstract: We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a broad class of loaded even-order differential operators defined on a finite interval. These expansions are compared with the Fourier trigonometric series expansions of the same functions in an integral metric on any interior compact set of the main interval or on the entire interval. We obtain estimates for the equiconvergence rate of these expansions.

Four-Dimensional Compact Manifolds with Nonnegative Biorthogonal Curvature

Abstract: The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian manifolds with nonnegative biorthogonal curvature. In particular, we obtain a partial answer to Yau Conjecture on pinching theorem for 4-dimensional compact manifolds.

A new approach to the design of biorthogonal triplet half-band filter banks using generalized half-band polynomials

Abstract: This paper presents a novel approach to design a class of biorthogonal triplet half-band filter banks based on the generalized half-band polynomials. The filter banks are designed with the help of three-step lifting scheme (using three kernels). The generalized half-band polynomial is used to construct these three kernels by imposing the number of zeros at $$z=-1$$ . The maximum number of zeros imposed for the three kernels is half of the order of half-band polynomial ( $$K/2$$ for $$K$$ order polynomial). The three kernels give a set of constraints on the coefficients of half-band polynomial by imposing the zeros. In addition to structural perfect reconstruction and linear phase, the proposed filter banks provide better frequency selectivity, more similarity between analysis and synthesis filters (measure of near-orthogonality), and good time–frequency localization. The proposed technique offers more flexibility in the design of filters using two degrees of freedom. Some examples have been presented to illustrate the method.

Affine fractal functions as bases of continuous functions

Abstract: The objective of the present paper is the study of affine transformations of the plane, which provide self-affine curves as attractors. The properties of these curves depend decisively of the coefficients of the system of affnities involved. The corresponding functions are continuous on a compact interval. If the scale factors are properly chosen one can define Schauder bases of C[a, b] composed by affine fractal functions close to polygonals. They can be chosen bounded. The basis constants and the biorthogonal sequence of coefficient functionals are studied.

The use of the Biorthogonal Decomposition for the identification of zonal flows at TJ-II

Abstract: This work addresses the identification of zonal flows in fusion plasmas. Zonal flows are large scale phenomena, hence multipoint measurements taken at remote locations are required for their identification. Given such data, the Biorthogonal Decomposition (or Singular Value Decomposition) is capable of extracting the globally correlated component of the multipoint fluctuations. By using a novel quadrature technique based on the Hilbert transform, propagating global modes (such as MHD modes) can be distinguished from the non-propagating, synchronous (zonal flow-like) global component. The combination of these techniques with further information such as the spectrogram and the spatial structure then allows an unambiguous identification of the zonal flow component of the fluctuations. The technique is tested using gyro-kinetic simulations. The first unambiguous identification of a zonal flow at the TJ-II stellarator is presented, based on multipoint Langmuir probe measurements.

Non-Hermitian model for asymmetric tunneling

Abstract: We present a simple non-Hermitian model to describe the phenomenon of asymmetric tunneling between two energy-degenerate sites coupled by a non-reciprocal interaction without dissipation. The system was described using a biorthogonal family of energy eigenvectors, the dynamics of the system was determined by the Schrodinger equation, and unitarity was effectively restored by proper normalization of the state vectors. The results show that the tunneling rates are indeed asymmetrical in this model, leading to an equilibrium that displays unequal occupation of the degenerate systems even in the absence of external interactions.

On Rotations as Spin Matrix Polynomials

Abstract: Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an alternate derivation. The central factorial numbers play key roles in both derivations.

Performance Analysis of Steganography based on 5-Wavelet Families by 4 Levels - DWT

Abstract: Steganography is the art and science of writing hidden messages in such a way that no one, excluding the sender and deliberated recipient, suspects the message existence , a form of security through obscurity. Steganography techniques can be utilized to images, a video file or an audio file. Typically, steganography is written in characters including hash marking, but its usage within images is also usual. At any rate, steganography secures from pirating copyrighted materials as well as aiding in unauthorized viewing. In this paper, performance analysis of image steganography based on 2 level and 4 level DWT associated to colored images is done. Performance analysis of steganography is done for selecting better wavelet for application by analyzing Peak Signal Noise Ratio(PSNR) of different wavelet families like Haar, Daubechies, Biorthogonal, Reverse Biorthogonal & Meyer wavelet(dmey) on result oriented basis using Matlab environment.

On the study of faster-than-Nyquist multicarrier signaling based on frame theory

Abstract: Multicarrier transmissions are classically based on undercomplete or exact Weyl-Heisenberg Riesz (biorthogonal or orthogonal) bases implemented thanks to oversampled filter-banks. This can be seen as a transmission below the Nyquist rate. However, when overcomplete Weyl-Heisenberg frames are used, we obtain a “faster-than-Nyquist” (FTN) system and it is theoretically impossible to recover exactly transmitted symbols using a linear receiver. Various studies have shown the interest of this high density signaling scheme as well as practical implementations based on trellis and/or iterative decoding. Nevertheless, there is still a lack of theoretical justifications with regard to pulse design in the FTN case. In this paper, we consider a linear transceiver operating over an additive white Gaussian noise channel. Using the frame theory and simulation results, we show that the mean squared error (MSE) is minimized when tight frames are used.

Universality conjecture and results for a model of several coupled positive-definite matrices

Abstract: The paper contains two main parts: in the first part, we analyze the general case of $p\geq 2$ matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a $(p+1)\times (p+1)$ matrix valued solution of a Riemann-Hilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multi-level random determinantal point fields which include the Bessel$_ u$ universality for $1$-level and the Meijer-$G$ universality for $2$-level. Our analysis uses the Deift-Zhou nonlinear steepest descent method and the explicit construction of a $(p+1)\times (p+1)$ origin parametrix in terms of Meijer G-functions. The solution of the full Riemann-Hilbert problem is derived rigorously only for $p=3$ but the general framework of the proof can be extended to the Cauchy chain of arbitrary length $p$.

Method for constructing biorthogonal wavelet filter bank

Abstract: The invention discloses a method for constructing a biorthogonal wavelet filter bank. The method includes the following steps: determining the length N of a decomposition low-pass filter h(n) and the length M of a reconstruction low-pass filter (please see the specifications) according to the requirements of wavelet functions to be constructed, wherein the N and the M are even numbers; determining the orders of vanishing moments of dual wavelet generating functions (please see the specifications) to be constructed; calculating the biorthogonal complete reconstruction filter bank according to a formula set to obtain a set of coefficients of the biorthogonal complete reconstruction filter bank; substituting the results into a dual-scale equation to construct a set of biorthogonal wavelet bases (please see the specifications), decomposing a decomposition high-pass filter corresponding to the wavelet function (please see the specifications), and reconstructing a reconstruction high-pass filter corresponding to the wavelet function (please see the specifications). Compared with common db-series wavelet filter banks and common bior-series wavelet filter banks, heart sound signals are processed through the biorthogonal wavelet filter bank constructed with the method, and the better noise reduction effect, the accurate heart sound classified information and the smaller reconstruction error rate can be obtained.

Spectral representation of transient signals

Abstract: Signal processing techniques exploiting natural and efficient representations of a class of signals with an underlying parametric model have been extensively studied and successfully applied across many disciplines. In this paper, we focus attention to the representation of one such class, i.e. transient structured signals. The class of transient signals in particular often results in computationally ill-conditioned problems which are further degraded by the presence of noise. We develop the Discrete Transient Transform, a biorthogonal transform to a basis parameterized by decay rate, along with algorithms for its implementation which mitigate these numerical issues and enable a spectral approach to parameter identification, estimation, and modeling for signals with transient behavior. The three algorithms developed have varying degrees of numerical robustness for generating the biorthogonal transient basis. Issues pertaining to transient spectral leakage and resolution are characterized and discussed in the context of an example related to Vandermonde system inversion.

Underwater Images Enhancement using Multi-Wavelet Transform and Median Filter

Abstract: Autonomous underwater vehicles (AUV) are usually equipped with vision sensors. However, the underwater images captured by AUV often suffer from effects such as diffusion, scatter and caustics. So image enhancement methods are necessary to increase visual quality. A Median filter de-noising approach based on multi-wavelet transform was proposed to remove the impulse noise viewed as random noise from the blurred underwater image. Biorthogonal muti-wavelet has two scaling functions that may generate different multiresolution analysis, so it was chosen as the basic wavelet for underwater image two-layer decomposition and reconstruction. On this basis, the blurred underwater image was decomposed and reconstructed adopting Biorthogonal and the Median filter was applied for removing the impulse noise form the decomposition images of each layer. Four indexes were involved to evaluate the performance of de-noising. The results show that the proposed approach provides superior results compared to other de-noising method. DOI : http://dx.doi.org/10.11591/telkomnika.v12i3.4505 Full Text: PDF

Non-Hermitian Model for Asymmetrical Tunneling

Abstract: We present a simple non-hermitian model to describe the phenomenon of asymmetric tunneling between two energy-degenerate sites coupled by a non-reciprocal interaction without dissipation. The system was described using a biorthogonal family of energy eigenvectors, the dynamics of the system was determined by the Schrodinger equation, and unitarity was effectively restored by proper normalization of the state vectors. The results show that the tunneling rates are indeed asymmetrical in this model, leading to an equilibrium that displays unequal occupation of the degenerate systems even in the absence of external interactions.

Constructions of Reproducing Kernel Banach Spaces via Generalized Mercer Kernels

Abstract: This article mainly studies with the constructions of reproducing kernel Banach spaces (RKBSs) that are the generalization of reproducing kernel Hilbert spaces (RKHSs). Firstly, we verify many advance properties of the general RKBSs such as density, continuity, implicit representation, imbedding, compactness, and representer theorems for learning methods. Next, we develop a new concept of generalized Mercer kernels to construct the $p$-norm RKBSs for $1\leq p\leq\infty$. The $p$-norm RKBSs preserve the same simple format as the Mercer representation of RKHSs. Moreover, the p-norm RKBSs are isometrically equivalent to the standard p-norm spaces of countable sequences; hence the $p$-norm RKBSs possess more abundant geometrical structures than RKHSs including sparsity. To be more precise, the suitable countable expansion terms of the generalized Mercer kernels can be used to represent the pairs of Schauder bases and biorthogonal systems of the $p$-norm RKBSs such that the generalized Mercer kernels become the reproducing kernels of the $p$-norm RKBSs. The theory of the generalized Mercer kernels also cover many well-known kernels, for instance, min kernels, Gaussian kernels, and power series kernels. Finally, we propose to solve the support vector machines in the $p$-norm RKBSs, which are to minimize the regularized empirical risks over the $p$-norm RKBSs. We show that the infinite-dimensional support vector machines in the $p$-norm RKBSs can be equivalently transferred into the finite dimensional convex optimization problems such that we can obtain the finite dimensional representations of their support vector machine solutions for practical applications and computer programs. In particular, we verify that some typical support vector machines in the $1$-norm RKBSs are equivalent to the classical $1$-norm sparse regressions.

Introduction to a Gram-Schmidt-type biorthogonalization method

Abstract: The aim of this expository/pedagogical paper is to describe a Gram-Schmidt biorthogonalization method in such a way that it can be used as an introduction to the subject for undergraduate presentation. The task of biorthogonalization naturally arises when the scalar product of vectors formed are linear combinations of two sets of linearly independent vectors, as the case may be. If one wants the scalar product to have the usual form, the two sets of basis vectors should be biorthogonal. If they are not, the question of biorthogonalization arises. New is the detailed description of the biorthogonalzation method for teaching purposes as well as the comparison of this method with Schmidt's orthogonalization method in the case when the sets of linearly independent vectors are identical.

Electronic transition dipole moment: A semi‐biorthogonal approach within valence universal coupled cluster framework

Abstract: Electronic dipole strengths (square of transition moments) and oscillator strengths are evaluated for various transitions, arising from the ground state to a few valence excited states. Parallel to two other methods of calculating the dipole strength within the Fock-space multireference coupled cluster framework, a new semi-biorthogonal approach is formulated and implemented in this article. This semi-biorthogonal approach can evaluate dipole strengths at a lower computational effort than the biorthogonal approach without compromising on the accuracy. This new method is compared and tested against the previously developed expectation value and biorthogonal approach for various molecular transitions. © 2014 Wiley Periodicals, Inc.

MPEG-4 single grade encoding method and device based on all phase position biorthogonal transform

Abstract: The invention relates to an MPEG-4 single grade encoding method and a device based on all phase position biorthogonal transform, and aims at solving the technical problems that a conventional MPEG-4 video compression method has a remarkable blocking effect at a reconstructed video block boundary, a quantization table is complex, a larger storage space is required and a great deal of calculation is required when the compression rate is changed The method comprises encoding a frame I and encoding a frame P, wherein in the step of encoding the frame I, the all phase position biorthogonal transform is adopted to replace conventional two-dimensional discrete cosine transform, and all transform coefficients are uniformly quantified The method and the device are widely applied to the technical field of video compression