Orthonormality Relations Research Articles

Numerical Solutions of High-Order Differential Equations with Polynomial Coefficients Using a Bernstein Polynomial Basis

The paper presents a novel method that allows one to establish numerical solutions of linear and nonlinear ordinary differential equations—with polynomial coefficients—that contain any finite products of the unknown functions and/or their general derivatives. The presented algorithm provides numerical solutions of these differential equations subject to initial or boundary conditions. This algorithm proposes the desired solution in terms of B-polynomials (Bernstein polynomial basis) and then uses the orthonormal relation of B-polynomials with its weighted dual basis with respect to the Jacobi weight function to construct a linear/nonlinear system in the unknown expansion coefficients which can be solved using a suitable solver. The properties of B-polynomials provide greater flexibility in which to impose the initial or boundary conditions at the end points of the interval [0, R] and enable us to obtain exactly and explicitly some of the unknown expansion coefficients in the form of a suggested numerical solution. Consequently, the presented algorithm leads to a linear or nonlinear algebraic system in the unknown expansion coefficients that has a simpler form than that was obtained by the other algorithms. So that, this procedure is a powerful tool that we may utilize to overcome the difficulties associated with boundary and initial value problems with less computational effort than the other techniques. An accepted agreement is obtained between the exact and approximate solutions for the given examples. The error analysis was also studied, and the obtained numerical results clarified the validity of the theoretical results.

Quasinormal mode theory for nanoscale electromagnetism informed by quantum surface response

We report a self-consistent quasinormal mode theory for nanometer scale electromagnetism where the possible nonlocal and quantum effects are treated through quantum surface responses. With Feibelman's frequency-dependent \textit{d} parameters to describe the quantum surface responses, we formulate the source-free Maxwell's equations into a generalized linear eigenvalue problem to define the quasinormal modes. We then construct an orthonormal relation for the modes and consequently unlock the powerful toolbox of modal analysis. The orthonormal relation is validated by the reconstruction of the full numerical results through modal contributions. Significant changes in the landscape of the modes are observed due to the incorporation of the quantum surface responses for a number of nanostructures. Our semi-analytical modal analysis enables transparent physical interpretation of the spontaneous emission enhancement of a dipolar emitter as well as the near-field and far-field responses of planewave excitations in the nanostructures.

ON ORTHOGONALIZATION OF BOUBAKER POLYNOMIALS

In this work, we explore some unknown properties of the Boubaker polynomials. The orthogonalization of the Boubaker polynomials has not been discussed in the literature. Since most of the application areas of such polynomial sequences demand orthogonal polynomials, the orthogonality of the Boubaker polynomials will help extend its theareas of application. We investigate orthogonality of classical Boubaker polynomials using Sturm-Liouville form and then apply the Gram-Schmidt orthogonalization process to develop modified Boubaker polynomials which are also orthogonal. Some classical properties, like orthogonality and orthonormality relation and zeros, of the modified Boubaker polynomials, have been proved. The contributions from this study have an impact on the further application of modified Boubaker polynomials to not only the cases where classical polynomials could be used but also in cases where the classical ones could not be used due to orthogonality issue.

Numerical Solutions of Korteweg-de Vries and Korteweg-de Vries-Burger’s Equations in a Bernstein Polynomial Basis

An algorithm for approximating numerical solution to Korteweg-de Vries (KdV) and Korteweg-de Vries-Burger’s (KdVB) equations in B-polynomials (Bernstein polynomial basis) is introduced. The algorithm expands the desired solution in terms of B-polynomials over a closed interval [a, b], and then makes use of the orthonormal relation of B-polynomials with its dual basis to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. Numerical results obtained using the current algorithm are compared with existing analytical results. Excellent agreement is found between the exact and approximate solutions.

Superradiant Phenomena for Spinor Fields in Rotating Black Hole Geometry

We derive the results (i) the ortho-normal and completeness relations for normal modes and (ii) non-existence of zero mode for spinor fields in rotating black hole geometry. From these results, we show that superradiant phenomena for spinor fields should be type 2: positive momentum on the horizon (pH > 0) and negative frequency at infinity (ω < 0).

Vector-Valued Nonuniform Wavelet Packets

ABSTRACTWe have introduced vector-valued nonuniform multiresolution analysis(VNUMRA) in [12] where the associated subspace V0 of has, an orthonormal basis, a collection of translates of a vector valued function ϕ of the form {ϕ(x−λ)}λ∈Λ where Λ = {0,r∕N}+2ℤ, N≥1 is an integer and r is an odd integer with 1≤r≤2N−1 such that r and N are relatively prime and ℤ is the set of all integers and the corresponding dilation factor is 2N. The notion of vector-valued nonuniform wavelet packets is introduced in this paper and their various properties are investigated. The orthonormal basis of is constructed from these wavelet packets and the orthonormal decomposition relation is also given.

Selberg\u2019s orthonormality conjecture and joint universality of L-functions

In the paper we introduce a new method how to use only an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality.

Robust polarization-insensitive strip-slot waveguide mode converter based on symmetric multimode interference.

Strip-slot waveguide mode converters for TE0 have been widely investigated. Here we demonstrate a polarization-insensitive converter numerically and experimentally. The polarization-insensitive performance is achieved by matching the optical field distribution of the 2-fold image of the Multimode Interference (MMI) and the TE0 (TM0) mode of a slot waveguide. The working principle for this MMI-based mode converter is thoroughly analyzed with the quantitatively evaluated optical field overlap ratio that is theoretically derived from the orthonormal relation of eigenmodes. Based on the analysis, the MMI-based polarization-insensitive converters are then simulated and fabricated. The simulation and measurement results indicate that the proposed scheme is a robust design since it is not only polarization-insensitive but also wavelength-insensitive and fabrication-tolerant. Moreover, the mode converter is as small as 1.22 μm × 4 μm while the measured conversion efficiencies are 95.9% for TE0 and 96.6% for TM0. All these excellent properties make the proposed mode converter an ideal solution for coupling light between strip and slot waveguides when both TE and TM polarizations are considered.

Fisher information of Markovian decay modes

We introduce the Fisher information in the basis of decay modes of Markovian dynamics, arguing that it encodes important information about the behavior of nonequilibrium systems. In particular we generalize an orthonormality relation between decay eigenmodes of detailed balanced systems to normal generators that commute with their time-reversal. Viewing such modes as tangent vectors to the manifold of statistical distributions, we relate the result to the choice of a coordinate patch that makes the Fisher-Rao metric Euclidean at the steady distribution, realizing a sort of statistical equivalence principle. We then classify nonequilibrium systems according to their spectrum, showing that a degenerate Fisher matrix is the signature of the insurgence of a class of dynamical phase transitions between nonequilibrium regimes, characterized by level crossing and power-law decay in time of suitable order parameters. An important consequence is that normal systems cannot manifest critical behavior. Finally, we study the Fisher matrix of systems with time-scale separation.

Solutions of 2nd-order linear differential equations subject to Dirichlet boundary conditions in a Bernstein polynomial basis

An algorithm for approximating solutions to 2nd-order linear differential equations with polynomial coefficients in B -polynomials (Bernstein polynomial basis) subject to Dirichlet conditions is introduced. The algorithm expands the desired solution in terms of B -polynomials over a closed interval [0, 1] and then makes use of the orthonormal relation of B -polynomials with its dual basis to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of B -polynomials, and the procedure is much simpler compared to orthogonal polynomials for solving differential equations. The current procedure is implemented to solve five linear equations and one first-order nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations.

On the optimization of Gegenbauer operational matrix of integration

The theory of Gegenbauer (ultraspherical) polynomial approximation has received considerable attention in recent decades. In particular, the Gegenbauer polynomials have been applied extensively in the resolution of the Gibbs phenomenon, construction of numerical quadratures, solution of ordinary and partial differential equations, integral and integro-differential equations, optimal control problems, etc. To achieve better solution approximations, some methods presented in the literature apply the Gegenbauer operational matrix of integration for approximating the integral operations, and recast many of the aforementioned problems into unconstrained/constrained optimization problems. The Gegenbauer parameter ? associated with the Gegenbauer polynomials is then added as an extra unknown variable to be optimized in the resulting optimization problem as an attempt to optimize its value rather than choosing a random value. This issue is addressed in this article as we prove theoretically that it is invalid. In particular, we provide a solid mathematical proof demonstrating that optimizing the Gegenbauer operational matrix of integration for the solution of various mathematical problems by recasting them into equivalent optimization problems with ? added as an extra optimization variable violates the discrete Gegenbauer orthonormality relation, and may in turn produce false solution approximations.

A new finite-dimensional thermal coherent state and its Wigner distribution function

This paper constructs a new type of finite-dimensional thermal coherent states (FDTCS), which differs from the proceeding thermal coherent state in construction, as realisations of SU(2) Lie algebra. Using the technique of integration within an ordered product of operator, it investigates the orthonormality and completeness relation of the FDTCS. Based on the thermal Wigner operator in the thermal entangled state representation, the Wigner function of the FDTCS is obtained. The nonclassical properties of the FDTCS are discussed in terms of the negativity of its Wigner function.

A new approach to the quantum mechanics of atoms and small molecules

When the Schrodinger equation for a system of N particles interacting through Coulomb forces is expressed in terms of hyperspherical coordinates, it closely resembles the d-dimensional generalization of the hydrogen atom wave equation (where d = 3N). The d-dimensional hydrogenlike wave functions can be found exactly, and a set of these functions, all corresponding to the same energy (but to different charges), fulfills a potential weighted orthonormality relation. When such a set of “generalized Coulomb Sturmians” is used as a basis set for bound states, solution of the resulting secular equations yields a spectrum of parameters k0, which are proportional to the square roots of the binding energies. These parameters also govern the asymptotic behavior of the basis set appropriate for the expansion of the bound states to which they correspond.

Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

A Toeplitz matrix is one in which the matrix elements are constantalong diagonals. The Fisher–Hartwig matrices are much-studied singularmatrices in the Toeplitz family. The matrices are defined for all orders,N. They are parameterizedby two constants, α and β. Their spectrum of eigenvalues has a simple asymptotic form in the limit asN goes to infinity. Here we study the structure of their eigenvalues andeigenvectors in this limiting case. We specialize to the case with realα andβ and0<α<|β|<1, where the behavior is particularly simple.The eigenvalues are labeled by an indexl whichvaries from 0 to N−1. An asymptotic analysis using Wiener–Hopf methods indicates that for largeN, thejth component of thelth eigenvector variesroughly in the fashion lnψjl≈iplj+O(1/N). The lth wavevector, pl, varies as for negative values of β and values of l/(N−1) not too close to zero or one. Correspondingly thelth eigenvalue is given by where a is the Fourier transform (also called the symbol) of the Toeplitz matrix.Note that pl has a small positive imaginary part. For values ofj/N not tooclose to zero or one, this imaginary part acts to produce an eigenfunction which decays exponentiallyas j/N increases. Thus, the eigenfunction appears similar to that of a bound state, attached to a wall atj = 0.Near j = 0 this decay is modified by a set of bumps, probably not universal in character. Forj/N above0.6 the eigenfunction begins to oscillate in magnitude and shows deviations from theexponential behavior.The case of 0<α<β<1 need not be studied separately. It can be obtained from the previous one by a ‘conjugacy’ transformationwhich takes ψj into ψN−j−1. This ‘conjugacy’ produces interesting orthonormality relations for the eigenfunctions.

Normal modes and the no zero mode theorem of scalar fields in BTZ black hole spacetime

The eigenfunctions for normal modes of scalar fields in BTZ black hole spacetime are studied. The orthonormal relations among them are derived. Quantization for scalar fields is done, and particle number, energy and angular momentum are expressed by the creation and annihilation operators. The allowed physical normal mode region is studied on the basis of the no zero mode theorem. Its implications for statistical mechanics is also studied.

Lipschitz stability of canonical Jordan bases of H-selfadjoint matrices under structure-preserving perturbations

In this paper we study Jordan-structure-preserving perturbations of matrices selfadjoint in the indefinite inner product. The main result of the paper is Lipschitz stability of the corresponding affiliation matrices. The result can be reformulated as Lipschitz stability, under small perturbations, of canonical Jordan bases (i.e., eigenvectors and generalized eigenvectors enjoying a certain flipped orthonormality relation) of matrices selfadjoint in the indefinite inner product. The proof relies upon the analysis of small perturbations of invariant subspaces, where the size of a permutation of an invariant subspace is measured using the concepts of a gap and of a semigap.

Scattering of Surface and Internal Waves by Rectangular Dikes

Abstract The scattering of surface and internal waves by a single dike or a pair of identical dikes in a two-layer fluid is analyzed in two dimensions within the context of linearized theory of water waves. The dikes are approximated as cylinders of rectangular geometry and are placed in a two-layer fluid of finite depth. In the study, both the cases of surface-piercing and bottom-standing dikes are considered. The solution of the associated boundary value problem is derived by a matched eigenfunction expansion method. Because of the flow discontinuity at the interface, the eigenfunctions involved have an integrable singularity at the interface and the orthonormal relation used in the present analysis is a generalization of the classical one corresponding to a single-layer fluid. The reflection coefficients and force amplitudes are computed and analyzed in various cases. The computed results in a two-layer fluid are compared with those existing in the literature for a single-layer fluid. The results obtained by the matched eigenfunction expansion method are compared with that of wide-spacing approximation method, and it is observed that the results from both the methods are in good agreement when the dikes are widely spaced. The general behavior of reflection coefficients for interface-piercing and non-interface-piercing obstacles is found to be different in both cases of surface-piercing and bottom-standing dikes. Moreover, for surface-piercing dikes, the results show the possibility of very large resonant motions between the dikes but with a very narrow bandwidth for the frequency of interest.

Null subspace-based damage detection of structures using vibration measurements

A damage detection method of mechanical system based on subspace identification concepts and statistical process techniques is presented. The aim is to propose a method that is sensitive to small-sized structural damages and suitable for on-line monitoring. Measured time-responses of structures subjected to artificial or environmental vibrations are assembled to form the Hankel matrix, which is further factorised by performing singular-value decomposition to obtain characteristic subspaces. It may be demonstrated that the structural responses are mainly located in the active subspace defined by the first principal components, which is orthonormal to the null subspace defined by the remaining principal components. If no structural damage occurs, the orthonormality relation between the subspaces remains valid with small residues when consecutive data sets are compared, and these residues may be evaluated by the proposed damage indicators. The method is validated using an experimental mock-up of an airplane subjected to different levels of damages simulated. It is also applied in environmental vibration testing of a street lighting device to monitor structural fatigue evolution.

On the computation of scaling coefficients of Daubechies' wavelets

Abstract In the present paper, Daubechies' wavelets and the computation of their scaling coefficients are briefly reviewed. Then a new method of computation is proposed. This method is based on the work [7] concerning a new orthonormality condition and relations among scaling moments, respectively. For filter lengths up to 16, the arising system can be explicitly solved with algebraic methods like Gröbner bases. Its simple structure allows one to find quickly all possible solutions.

Symplectic solution system for reissner plate bending

Based on the Hellinger-Reissner variatonal principle for Reissner plate bending and introducing dual variables, Hamiltonian dual equations for Reissner plate bending were presented. Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem, and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized. So in the symplectic space which consists of the original variables and their dual variables, the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction-vector expansion. All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail, and their physical meanings are showed clearly. The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed. It is showed that the all eigensolutions for zero eigenvalue are basic solutions of the Saint-Venant problem and they form a perfect symplectic subspace for zero eigenvalue. And the eigensolutions for nonzero eigenvalue are covered by the Saint-Venant theorem. The symplectic solution method is not the same as the classical semi-inverse method and breaks through the limit of the traditional semi-inverse solution. The symplectic solution method will have vast application.