Showing papers on "Biorthogonal system published in 2016"
TL;DR: In this paper, a global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator is developed.
Abstract: In this paper, we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. There are no assumptions on the regularity of the boundary which is allowed to have arbitrary singularities. We give applications of the developed analysis to obtain a priori estimates for solutions of boundary value problems that are elliptic within the constructed calculus.
78 citations
TL;DR: In this article, fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles are studied for which the underlying biorthyogonal family of families is known.
Abstract: We study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles We study those biorthogonal ensembles for which the underlying biorthogonal family s
75 citations
Book•
30 Jul 2016
TL;DR: In this paper, the basic linear structure and biorthogonal systems are used for smoothness, smooth approximation, and smoothness approximation problems in nonlinear geometry and linear geometry.
Abstract: Preface.- Basic linear structure.- Basic linear geometry.- Biorthogonal systems.- Smoothness, smooth approximation.- Nonlinear geometry.- Some more nonseparable problems.- Some applications.- Bibliography.- List of concepts and problems.- Symbol index.- Subject index.
48 citations
07 Nov 2016
TL;DR: In this article, the relation between the joint densities of the singular values and the eigenvalues for complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant) is investigated.
Abstract: We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint densities of the singular values and the eigenvalues for complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that each of these joint densities determines the other one. Moreover, we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore, we show how to generalize the relation between the singular value and eigenvalue statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.
46 citations
TL;DR: In this paper, the authors investigated Gabor frames on locally compact abelian groups with time-frequency shifts along non-separable, closed subgroups of the phase space, where the necessary conditions are given in terms of the size of the subgroup.
42 citations
TL;DR: The various performance metrics like Ratio of Edge pixels to size of image (REPS), peak signal to noise ratio (PSNR) and computation time are compared for various wavelets for edge detection and biorthogonal wavelet bior1.3 performs well in detecting the edges with better quality.
31 citations
TL;DR: In this paper, the singular values of the product of two coupled rectangular random matrices were studied as a determinantal point process, and exact formulae for the correlation kernel were derived in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis.
Abstract: We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel–Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.
30 citations
TL;DR: In this article, the authors introduce a general theory of regular biorthogonal sequences and its physical operators and show that there exists a non-singular positive self-adjoint operator Tf in H defined by an orthonormal basis (ONB) f ≡ {fn} in H such that ϕn = Tffn and ψn=Tf−1fn, n = 0, 1, etc.
Abstract: In this paper, we introduce a general theory of regular biorthogonal sequences and its physical operators. Biorthogonal sequences {ϕn} and {ψn} in a Hilbert space H are said to be regular if Span {ϕn} and Span {ψn} are dense in H. The first purpose is to show that there exists a non-singular positive self-adjoint operator Tf in H defined by an orthonormal basis (ONB) f ≡ {fn} in H such that ϕn = Tffn and ψn=Tf−1fn, n = 0, 1, …, and such an ONB f is unique. The second purpose is to define and study the lowering operators Af and Bf†, the raising operators Bf and Af†, and the number operators Nf and Nf† determined by the non-singular positive self-adjoint operator Tf. These operators connect with quasi-Hermitian quantum mechanics and its relatives. This paper clarifies and simplifies the mathematical structure of this framework and minimizes the required assumptions.
18 citations
TL;DR: The mathematical underpinnings of the biorthogonal von Neumann method for quantum mechanical simulations (PvB) are described and a detailed discussion of the important issue of nonorthogonal projection onto subspaces of biorThogonal bases is presented.
Abstract: We describe the mathematical underpinnings of the biorthogonal von Neumann method for quantum mechanical simulations (PvB). In particular, we present a detailed discussion of the important issue of nonorthogonal projection onto subspaces of biorthogonal bases, and how this differs from orthogonal projection. We present various representations of the Schrodinger equation in the reduced basis and discuss their relative merits. We conclude with illustrative examples and a discussion of the outlook and challenges ahead for the PvB representation.
17 citations
TL;DR: In this article, a non-self-adjoint bosonic Hamiltonian H possessing real eigenvalues is investigated and it is shown that the operator can be diagonalized by making use of pseudo-bosonic operators.
Abstract: A non-self-adjoint bosonic Hamiltonian H possessing real eigenvalues is investigated. It is shown that the operator can be diagonalized by making use of pseudo-bosonic operators. The biorthogonal sets of eigenvectors for the Hamiltonian and its adjoint are explicitly constructed. The positive definite operator which connects both sets of eigenvectors is also given. The dynamics of the model is briefly analyzed.
16 citations
TL;DR: In this article, a method of constructing a regular biorthogonal pair based on the commutation rule is presented. But the assumption that the pseudo-bosons coincide with the definition of the regular pair is challenged.
Abstract: The first purpose of this paper is to show a method of constructing a regular biorthogonal pair based on the commutation rule: ab − ba = I for a pair of operators a and b acting on a Hilbert space H with inner product (⋅| ⋅ ). Here, sequences {ϕn} and {ψn} in a Hilbert space H are biorthogonal if (ϕn|ψm) = δnm, n, m = 0, 1, …, and they are regular if both Dϕ ≡ Span{ϕn} and Dψ ≡ Span{ψn} are dense in H. Indeed, the assumptions to construct the regular biorthogonal pair coincide with the definition of pseudo-bosons as originally given in F. Bagarello [“Pseudobosons, Riesz bases, and coherent states,” J. Math. Phys. 51, 023531 (2010)]. Furthermore, we study the connections between the pseudo-bosonic operators a, b, a†, b† and the pseudo-bosonic operators defined by a regular biorthogonal pair ({ϕn}, {ψn}) and an ONB e of H in H. Inoue [“General theory of regular biorthogonal pairs and its physical applications,” e-print arXiv:math-ph/1604.01967]. The second purpose is to define and study the notion of D-p...
TL;DR: In this article, a space curved surface shape reconstruction algorithm is proposed for shape perception and reconstruction of flexible plate structure, based on biorthogonal strain data measured by optimal distribution.
Abstract: A space curved surface shape reconstruction algorithm is proposed for shape perception and reconstruction of flexible plate structure. First, biorthogonal strain data measured by optimal distribute...
TL;DR: In this article, Inoue and Takakura introduced general theories of semi-regular biorthogonal pairs, generalized Riesz bases, and its physical applications, and showed that the same result holds true if the pair is only semi regular by using operators Tϕ,e, Te, ϕ, Tψ,e and Te,ψ defined by an orthonormal basis e in H and a biorhogonal pair ({ϕn, {ψn}).
Abstract: In this paper we introduce general theories of semi-regular biorthogonal pairs, generalized Riesz bases and its physical applications. Here we deal with biorthogonal sequences {ϕn} and {ψn} in a Hilbert space H , with domains D ( ϕ ) = { x ∈ H ; ∑ k = 0 ∞ ( x | ϕ k ) 2 < ∞ } and D ( ψ ) = { x ∈ H ; ∑ k = 0 ∞ ( x | ψ k ) 2 < ∞ } and linear spans Dϕ ≡ Span{ϕn} and Dψ ≡ Span{ψn}. A biorthogonal pair ({ϕn}, {ψn}) is called regular if both Dϕ and Dψ are dense in H , and it is called semi-regular if either Dϕ and D(ϕ) or Dψ and D(ψ) are dense in H . In a previous paper [H. Inoue, J. Math. Phys. 57, 083511 (2016)], we have shown that if ({ϕn}, {ψn}) is a regular biorthogonal pair then both {ϕn} and {ψn} are generalized Riesz bases defined in the work of Inoue and Takakura [J. Math. Phys. 57, 083505 (2016)]. Here we shall show that the same result holds true if the pair is only semi-regular by using operators Tϕ,e, Te,ϕ, Tψ,e, and Te,ψ defined by an orthonormal basis e in H and a biorthogonal pair ({ϕn}, {ψn}). F...
TL;DR: Numerical examples and comparison are included to show that the proposed approach is more flexible in making trade-off between the spectral selectivity and reconstruction error over the existing method.
Abstract: Narang and Ortega have constructed a two-channel biorthogonal graph filter bank with compact support. The design method does not consider the spectral response of the kernels. In this letter, we employ optimization approach to design the spectral kernels. The analysis and synthesis kernels are, respectively, optimized with constrained optimization problems, in which the reconstruction error and spectral selectivity are controlled simultaneously. The optimization problems are semidefinite programming (SDP), which can be solved effectively. Numerical examples and comparison are included to show that the proposed approach is more flexible in making trade-off between the spectral selectivity and reconstruction error over the existing method.
TL;DR: In this article, it was shown that for rational values of ε > 0, there is an equivalent vector equilibrium problem that describes the large n behavior of the Muttalib{Borodin biorthogonal ensemble.
Abstract: The Muttalib{Borodin biorthogonal ensemble is a joint density function for n particles on the positive real line that depends on a parameter . There is an equilibrium problem that describes the large n behavior. We show that for rational values of there is an equivalent vector equilibrium problem.
TL;DR: In this paper, it was shown that a 4-dimensional compact manifold with harmonic Weyl tensor must be either locally conformally flat or isometric to a complex projective space, provided that the biorthogonal curvature satisfies a suitable pinching condition.
Abstract: We prove that a 4-dimensional compact manifold \(M^4\) with harmonic Weyl tensor must be either locally conformally flat or isometric to a complex projective space \(\mathbb {CP}^2,\) provided that the biorthogonal (sectional) curvature satisfies a suitable pinching condition. In particular, we improve the pinching constants considered by some preceding works on a rigidity result for 4-dimensional compact manifolds.
TL;DR: In this article, a method incorporating biorthogonal orbital optimization, symmetry projection, and double-occupancy screening with a non-unitary similarity transformation generated by the Gutzwiller factor is presented.
Abstract: We present a method incorporating biorthogonal orbital-optimization, symmetry projection, and double-occupancy screening with a non-unitary similarity transformation generated by the Gutzwiller factor [Formula: see text], and apply it to the Hubbard model. Energies are calculated with mean-field computational scaling with high-quality results comparable to coupled cluster singles and doubles. This builds on previous work performing similarity transformations with more general, two-body Jastrow-style correlators. The theory is tested on 2D lattices ranging from small systems into the thermodynamic limit and is compared to available reference data.
TL;DR: In this article, the authors define the notion of semi-regular biorthogonal pairs, which is a generalization of regular biorhogonal pair, and show that if $(\{ \phi{n}, \{ \psi_{n} \})$ is a semirefinite pair, then it is a generalized Riesz base.
Abstract: In this paper we define the notion of semi-regular biorthogonal pairs what is a generalization of regular biorthogonal pairs in Ref. \cite{hiroshi1} and show that if $(\{ \phi_{n} \} , \{ \psi_{n} \})$ is a semi-regular biorthogonal pair, then $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. This result improves the results of Ref. \cite{h-t, hiroshi1, h-t2} in the regular case.
TL;DR: In this paper, a wavelet transform is constructed from B-spline scaling functions defined on a grid of non-equispaced knots, and the scaling coefficients are derived from the observations.
Abstract: This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen–Daubechies–Feauveau wavelets. The new construction is based on the factorization of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replaci...
TL;DR: The relation between the joint density of the singular values and the eigenvalues of complex random matrices which are bi-unitarily invariant is investigated in this article, where it is shown that one of these joint densities determines the other one.
Abstract: We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore we show how to generalize the relation between the eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.
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TL;DR: In this article, the authors discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space and discuss properties of convolution and give a number of examples.
Abstract: In this note we discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space. We develop the associated biorthogonal Fourier analysis and the theory of distributions, discuss properties of convolutions and give a number of examples.
TL;DR: In this paper, the authors give a characterization of the existence of an enriched linear finite element approximation based on biorthogonal systems and show that the enriched element exists if and only if a certain generalized trapezoidal type cubature formula has a nonzero approximation error.
Abstract: The present paper is intended to give a characterization of the existence of an enriched linear finite element approximation based on biorthogonal systems. It is shown that the enriched element exists if and only if a certain multivariate generalized trapezoidal type cubature formula has a nonzero approximation error. Furthermore, for such an enriched element we derive simple explicit formulas for the basis functions, and show that the approximation error can be written as the sum of the error of the (non-enriched) element plus a perturbation that depends on the enrichment function. Finally, we estimate the approximation error in L2 norm. We also give an alternative approach to estimate the approximation error, which relies on an appropriate use of the Poincare inequality.
Journal Article•
TL;DR: In this article, a new numerical method for solving fractional optimal control problems (FOCPs) is presented based upon biorthogonal cubic Hermite spline multiwavelets approximations.
Abstract: In this paper, a new numerical method for solving fractional optimal control problems (FOCPs) is presented. The fractional derivative in the dynamic system is described in the Caputo sense. The method is based upon biorthogonal cubic Hermite spline multiwavelets approximations. The properties of biorthogonal multiwavelets are first given. The operational matrix of fractional Riemann-Lioville integration and multiplication are then utilized to reduce the given optimization problem to the system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain algebraic equations. Illustrative examples are provided to confirm the applicability of the new method.
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TL;DR: In this paper, the authors study biorthogonal functions related to basic hypergeometric integrals with coupled continuous and discrete components, which appear as superconformal indices for three-dimensional quantum field theories and also in the context of solvable lattice models.
Abstract: We study biorthogonal functions related to basic hypergeometric integrals with coupled continuous and discrete components. Such integrals appear as superconformal indices for three-dimensional quantum field theories and also in the context of solvable lattice models. We obtain explicit biorthogonal systems given by products of two of Rahman's biorthogonal rational 10W9-functions or their degenerate cases. We also give new bilateral extensions of the Jackson and q-Saalschutz summation formulas and new continuous and discrete biorthogonality measures for Rahman's functions.
TL;DR: Numerical results and comparison are included to show the proposed algorithm can lead to biorthogonal graph filter banks with improved performance.
Abstract: In this study, the lifting scheme is first employed to design two-channel biorthogonal graph filter bank. The biorthogonal condition is parameterised by imposing a single-level lifting structure on the analysis and synthesis graph kernels. Based on the parametric structure, the two kernels are separately optimised by constrained quadratic programming. The obtained two-channel biorthogonal graph filter banks are of structurally perfect reconstruction. Numerical results and comparison are included to show the proposed algorithm can lead to biorthogonal graph filter banks with improved performance.
TL;DR: In this paper, a system of biorthogonal polynomials with respect to a complex valued measure supported on the unit circle is considered and all the terms with bounds are explicitly given for the remainder of an asymptotic formula given by R. Askey for this system.
TL;DR: In this article, the authors discuss some facts on generalized Gibbs states and their related KMS-like conditions and propose some extended versions of the Heisenberg algebraic dynamics, deducing some of their properties that are useful for our purposes.
Abstract: Motivated by the growing interest in PT-quantum mechanics, in this paper we discuss some facts on generalized Gibbs states and their related KMS-like conditions. To achieve this, we first consider some useful connections between similar (Hamiltonian) operators and we propose some extended versions of the Heisenberg algebraic dynamics, deducing some of their properties that are useful for our purposes.
01 Jan 2016
TL;DR: A new algorithm for the watermarking of digital images is proposed by cascading of two frequency domain transform techniques, i.e., biorthogonal wavelet transforms (BWT) and discrete cosine transform (DCT).
Abstract: Modern communication technology facilitates easy transmission of multimedia content. But if copyright protection of multimedia data, prevention of illegal access, and rights of intellectual property are considered, so this field needs more attention of researchers. It is a very simple process to make a copy of the multimedia data, alter it, and then put it back for business profits. Digital watermarking techniques provide a solution for this issue, by embedding some information, which can be further used to claim the ownership of multimedia data. In this paper, a new algorithm for the watermarking of digital images is proposed by cascading of two frequency domain transform techniques, i.e., biorthogonal wavelet transforms (BWT) and discrete cosine transform (DCT). Proposed technique takes the approximation component of the biorthogonal transform of cover image and then applies DCT to embed watermark. Embedding of watermark data is done in middle frequency component by comparison-based correlation technique. Also, this technique has been analyzed and compared with the existing ones by applying various image attacks and subsequently measuring the results and proved to be fairly robust.
TL;DR: The novel approach taken in this paper is to work with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system thus ensuring that the scheme is numerically efficient, and the formulation is stable.
Abstract: The thin plate spline method is a widely used data fitting technique as it has the ability to smooth noisy data. Here we consider a mixed finite element discretisation of the thin plate spline. By using mixed finite elements the formulation can be defined in-terms of relatively simple stencils, thus resulting in a system that is sparse and whose size only depends linearly on the number of finite element nodes. The mixed formulation is obtained by introducing the gradient of the corresponding function as an additional unknown. The novel approach taken in this paper is to work with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system thus ensuring that the scheme is numerically efficient, and the formulation is stable. Some numerical results are presented to demonstrate the performance of our approach. A preconditioned conjugate gradient method is an efficient solver for the arising linear system of equations.
TL;DR: The prior-art mesh-to-mesh spatial field projection is extended to the time domain and a space-time error norm is defined between a given field distribution and the target field to be determined and minimize it using the Galerkin method.
Abstract: In this paper, field projection methods are used to couple finite-element analysis carried out on different meshes and different temporal discretization bases. The prior-art mesh-to-mesh spatial field projection is extended to the time domain in this paper. We first define a space-time error norm between a given field distribution and the target field to be determined, and then minimize it using the Galerkin method. Biorthogonal test functions are also introduced into the projection process to replace inner product matrices with diagonal matrices and reduce the computation cost in terms of memory as well as the calculation time required.