Orthonormal Research Articles

Theory of complete orthonormal sets of relativistic tensor wave functions and Slater tensor orbitals of particles with arbitrary spin in position, momentum and four-dimensional spaces

Using the complete orthonormal basis sets of nonrelativistic and quasirelativistic orbitals introduced by the author in previous papers for particles with arbitrary spin the new analytical relations for the 2 ( 2 s + 1 ) -component relativistic tensor wave functions and Slater tensor orbitals in position, momentum and four-dimensional spaces are derived, where s = 1 / 2 , 1 , 3 / 2 , 2 , … . The relativistic tensor function sets are expressed through the corresponding nonrelativistic and quasirelativistic orbitals. The analytical formulas for overlap integrals over relativistic Slater tensor orbitals in position space are also derived.

Augmented message-matrix approach to deterministic dense-coding theory

A useful method for deriving analytical results applicable to the standard two-party deterministic dense-coding protocol is introduced and illustrated. In this protocol, communication of $K$ perfectly distinguishable messages is attainable via $K$ selected local unitary operations performed on one qudit from a pair of entangled qudits of equal dimension $d$ in a pure state $|\ensuremath{\psi}⟩$ with largest Schmidt coefficient $\sqrt{{\ensuremath{\lambda}}_{0}}$. The method takes advantage of the fact that the $K$ message states, together with ${d}^{2}\ensuremath{-}K$ augmenting orthonormal state vectors, yield a unitary matrix, thereby implying properties of the $K$ message states which otherwise are not readily recognized. Employing this augmented message matrix, we produce simple proofs of previously established results including (i) ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/K$, (ii) ${\ensuremath{\lambda}}_{0}<d/K$ when $K=d+1$, and (iii) the impossibility of finding a $|\ensuremath{\psi}⟩$ that can enable transmission of $K={d}^{2}\ensuremath{-}1$ messages but not ${d}^{2}$. Additional results obtained using the method include proofs that the ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/K$ bound is reduced to at least (i) ${\ensuremath{\lambda}}_{0}\ensuremath{\le}(1/2)[1+\sqrt{(d\ensuremath{-}2)/(d+2)}]$ when $K=d+1$ and (ii) ${\ensuremath{\lambda}}_{0}\ensuremath{\le}(K\ensuremath{-}m)/(2K\ensuremath{-}m\ensuremath{-}d)$ whenever $(d+1)\ensuremath{\le}K\ensuremath{\le}2d$ and the selected local unitaries include the first $m$ non-negative integral powers of the shift operator $X$.

Rotating stellar core-collapse waveform decomposition: a principal component analysis approach

This paper introduces the use of principal component analysis (PCA) as a method to decompose the catalogues of gravitational waveforms to produce a set of orthonormal basis vectors. We apply PCA to a set of gravitational waveforms produced by rotating stellar core-collapse simulations and compare its basis vectors with those obtained through Gram–Schmidt decomposition. The comparison is made using the match parameter which quantifies how well each waveform is reconstructed by a set of basis vectors. The performance of the two methods is found to be comparable with 14 Gram–Schmidt basis vectors and 12 principal components required if we require all waveforms in the catalogue to be reconstructed with a match of 0.9 or better. Additionally, we observe that the chosen set of waveforms has very similar features, and a match of at least 0.7 can be obtained by decomposing only waveforms generated from simulations with A = 2. We discuss the implications of this observation and the advantages of eigen-decomposing waveform catalogues with PCA.

GENERALIZED FOURIER-WIENER FUNCTION SPACE TRANSFORMS

In this paper, we define generalized Fourier-Hermite functionals on a function space <TEX>$C_{a,b}[0,\;T]$</TEX> to obtain a complete orthonormal set in <TEX>$L_2(C_{a,b}[0,\;T])$</TEX> where <TEX>$C_{a,b}[0,\;T]$</TEX> is a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in <TEX>$L_2(C_{a,b}[0,\;T])$</TEX> has a generalized Fourier-Wiener function space transform <TEX>${\cal{F}}_{\sqrt{2},i}(F)$</TEX> also belonging to <TEX>$L_2(C_{a,b}[0,\;T])$</TEX>.

Integral Transforms of Functionals in L 2(C a,b [0,T

In this paper we use generalized Fourier-Hermite functionals to obtain a complete orthonormal set in L2(Ca,b[0,T]) where Ca,b[0,T] is a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in L2(Ca,b[0,T]) has an integral transform ℱγ,βF also belonging to L2(Ca,b[0,T]).

Structures and harmonic vibrational frequencies for excited states of diatomic molecules with CCSD(R12) and CCSD(F12) models

The equation-of-motion coupled-cluster method for excited states with the singles-and-doubles model (CCSD) has been implemented for ansatz 2 of the explicitly correlated CCSD(R12) and CCSD(F12) methods as part of the program package Dalton. In this model, an orthonormal complementary auxiliary basis set is used for the resolution-of-identity approximation in order to calculate the three-electron integrals needed for CCSD(R12) and CCSD(F12). The additional CCSD(R12) or CCSD(F12) terms introduced within ansatz 2, which are not present in ansatz 1, are derived and discussed with regard to the extra costs needed for their computation. As a first application the basis set convergence of equilibrium bond lengths and harmonic vibrational frequencies has been investigated for some singlet excited states of the diatomic molecules N(2), CO, BF, and BH. The calculated CCSD(F12) results show that the average absolute deviations of the bond lengths and frequencies from the basis set limits are below 0.1 pm and 5 cm(-1) as well as 0.05 pm and 1 cm(-1) for the triple- and quadruple-zeta basis sets, respectively. These deviations are shown to largely arise from the SCF basis set incompleteness errors.

The G-Scheme: A framework for multi-scale adaptive model reduction

The numerical solution of mathematical models for reaction systems in general, and reacting flows in particular, is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics. However, the dynamics can develop very-slow and very-fast time scales separated by a range of active time scales. An opportunity to reduce the complexity of the problem arises when the fast/active and slow/active time scales gaps becomes large. We propose a numerical technique consisting of an algorithmic framework, named the G-Scheme, to achieve multi-scale adaptive model reduction along-with the integration of the differential equations (DEs). The method is directly applicable to initial-value ODEs and (by using the method of lines) PDEs. We assume that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. The G-Scheme introduces locally a curvilinear frame of reference, defined by a set of orthonormal basis vectors with corresponding coordinates, attached to this decomposition. The evolution of the curvilinear coordinates associated with the active subspace is described by non-stiff DEs, whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem. Adjusting the active DEs dynamically during the time integration is the most significant feature of the G-Scheme, since the numerical integration is accomplished by solving a number of DEs typically much smaller than the dimension of the original problem, with corresponding saving in computational work. To demonstrate the effectiveness of the G-Scheme, we present results from illustrative as well as from relevant problems.

A frame-invariant formulation of Fung elasticity

Fung elasticity refers to the hyperelasticity constitutive relation proposed by Fung and co-workers for describing the pseudo-elastic behavior of biological soft tissues undergoing finite deformation. A frame-invariant formulation of Fung elasticity is provided for material symmetries ranging from orthotropy to isotropy, which uses Lamé-like material constants. In the orthotropic case, three orthonormal vectors are used to define mutually orthogonal planes of symmetry and associated texture tensors. The strain energy density is then formulated as an isotropic function of the Lagrangian strain and texture tensors. The cases of isotropy and transverse isotropy are derived from the orthotropic case. Formulations are provided for both material and spatial frames. These formulations are suitable for implementation into finite element codes. It is also shown that the strain energy function can be naturally uncoupled into a dilatational and a distortional part, to facilitate the computational implementation of incompressibility.

Krylov-Proportionate Adaptive Filtering Techniques Not Limited to Sparse Systems

This paper proposes a novel adaptive filtering scheme named the Krylov-proportionate normalized least-mean-square (KPNLMS) algorithm. KPNLMS exploits the benefits (i.e., fast convergence for sparse unknown systems) of the proportionate NLMS algorithm, but its applications are not limited to sparse unknown systems. A set of orthonormal basis vectors is generated from a certain Krylov sequence. It is proven that the unknown system is sparse with respect to the basis vectors in case of fairly uncorrelated input data. Different adaptation gain is allocated to a coefficient of each basis vector, and the gain is roughly proportional to the absolute value of the corresponding coefficient of the current estimate. KPNLMS enjoys i) fast convergence, ii) linear complexity per iteration, and iii) no use of any a priori information. Numerical examples demonstrate significant advantages of the proposed scheme over the reduced-rank method based on the multistage Wiener filter (MWF) and the transform-domain adaptive filter (TDAF) both in noisy and silent situations.

Generalized finite element method enrichment functions for curved singularities in 3D fracture mechanics problems

This paper presents a study of generalized enrichment functions for 3D curved crack fronts. Two coordinate systems used in the definition of singular curved crack front enrichment functions are analyzed. In the first one, a set of Cartesian coordinate systems defined along the crack front is used. In the second case, the geometry of the crack front is approximated by a set of curvilinear coordinate systems. A description of the computation of derivatives of enrichment functions and curvilinear base vectors is presented. The coordinate systems are automatically defined using geometrical information provided by an explicit representation of the crack surface. A detailed procedure to accurately evaluate the surface normal, conormal and tangent vectors along curvilinear crack fronts in explicit crack surface representations is also presented. An accurate and robust definition of orthonormal vectors along crack fronts is crucial for the proper definition of enrichment functions. Numerical experiments illustrate the accuracy and robustness of the proposed approaches.

A Multivariate Orthonormal Vector Fitting based estimation technique

This paper modifies a recent robust parametric macromodeling technique called Multivariate Orthonormal Vector Fitting (MOVF), to handle noisy data in an output error estimation framework. The new method provides accurate and compact rational parametric macromodels based on measurements in the frequency domain. The performance of the multivariate method is shown on simulation as well as on real measurements.

Two-dimensional nonradiating currents for imaging systems: theoretical development and preliminary assessment

An analytical orthonormal set of basis functions for 2D nonradiating current is derived. Preliminary numerical investigation shows the capabilities of this set of currents in enhancing the solution to an inverse scattering problem.

Interior Controllability of a Broad Class of Reaction Diffusion Equations

We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spaces Z = L2(Ω) given by z′ = −Az + 1ωu(t), t ∈ [0, τ], where Ω is a domain in ℝn, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u ∈ L2(0, t1; L2(Ω)) and A : D(A) ⊂ Z → Z is an unbounded linear operator with the following spectral decomposition: 〈z, ϕj,k〉ϕj,k. The eigenvalues 0 &lt; λ1 &lt; λ2 &lt; ⋯&lt;⋯λn → ∞ of A have finite multiplicity γj equal to the dimension of the corresponding eigenspace, and {ϕj,k} is a complete orthonormal set of eigenvectors of A. The operator −A generates a strongly continuous semigroup {T(t)} given by 〈z, ϕj,k〉ϕj,k. Our result can be applied to the nD heat equation, the Ornstein‐Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.

One-Range Addition Theorems in Terms of Ψα -ETOs for STOs and Coulomb–Yukawa Like Correlated Interaction Potentials of Integer and Noninteger Indices

The expansion formulas in terms of complete orthonormal sets of Ψα-exponential type orbitals are established for the Slater type orbitals and Coulomb–Yukawa-like correlated interaction potentials of integer and noninteger indices. These relations are used in obtaining their unsymmetrical and symmetrical one-range addition theorems. The final results are especially useful in the calculations of multicentre multielectron integrals occurring when Hartree–Fock–Roothaan and explicitly correlated methods are employed.

Zinssensitivitäten börsennotierter deutscher Finanzdienstleister: Eine empirische Untersuchung

Zusammenfassung/Summary Zinssensitivitaten borsennotierter deutscher Finanzdienstleister: Eine empirische Untersuchung Dieser Beitrag untersucht die Zinssensitivitat borsennotierter Finanzdienstleister am deutschen Kapitalmarkt anhand des Grundansatzes von Stone (1974). Hiernach werden die Aktienrenditen durch einen Markt- und einen Zinsfaktor erklart, wobei in entsprechenden Untersuchungen regelmasig unterschiedliche Varianten insbesondere hinsichtlich der Konstruktion des Zinsfaktors verwendet werden. Im Fokus der empirischen Analyse steht der Vergleich der Ergebnisse zentraler Ausgestaltungsformen dieses 2-Faktor-Regressionsmodells auf Basis eines einheitlichen Datensatzes fur deutsche Finanzdienstleister uber den Zeitraum von 1973 bis 2003. Wesentlichen Einfluss auf die Ergebnisse haben die Orthogonalisierungsvarianten bezuglich der unabhangigen Variablen, die Laufzeit des Zinsfaktors und die Steigung der Zinsstruktur im betrachteten Untersuchungszeitraum. Alternative Konstruktionen des Zinsfaktors wi...

High-order angular response beamformer for vector sensors

A linear processing scheme for computing higher-order angular response modes of a vector sensor is described. Examples of modal response beampatterns are presented. The response modes form (in principle) a complete, orthonormal set that can be transformed into steerable sets of one or more directive beams. The linear processing scheme facilitates calibration of vector sensor measurement systems. The angular resolution that can be achieved with the new processing scheme is predicted to be (155/ N m ) degrees, where N m is the highest order of computed response mode, for the higher orders. The number of higher-order response modes appears to be limited only by the computational power available.

Simultaneous estimation of aerosol cloud concentration and spectral backscatter from multiple-wavelength lidar data

We present a sequential algorithm for estimating both concentration dependence on range and time and backscatter coefficient spectral dependence of optically thin localized atmospheric aerosols using data from rapidly tuned lidar. The range dependence of the aerosol is modeled as an expansion of the concentration in an orthonormal basis set whose coefficients carry the time dependence. Two estimators are run in parallel: a Kalman filter for the concentration range and time dependence and a maximum-likelihood estimator for the aerosol backscatter wavelength and time dependence. These two estimators exchange information continuously over the data-processing stream. The state model parameters of the Kalman filter are also estimated sequentially together with the concentration and backscatter. Lidar data collected prior to the aerosol release are used to estimate the ambient lidar return. The approach is illustrated on atmospheric backscatter long-wave infrared (CO2) lidar data.

Theory of complete orthonormal relativistic vector wave function sets and Slater type relativistic vector orbitals in coordinate, momentum and four-dimensional spaces

The new analytical relations for the relativistic vector wave functions and Slater type relativistic vector orbitals in coordinate, momentum and four-dimensional spaces are derived using the properties of quasirelativistic vector spherical harmonics introduced by the author in previous paper (I.I. Guseinov, J. Math. Chem., 44, 197 (2008)) and complete orthonormal scalar basis sets of nonrelativistic ψ α -exponential type orbitals (ψ α -ETO), \({\phi ^{\alpha}}\)-momentum space orbitals (\({\phi ^{\alpha}}\)-MSO) and z α -hyperspherical harmonics (z α -HSH). The 6-component relativistic vector wave functions obtained are complete without the inclusion of the continuum. The relativistic vector wave function sets and Slater type relativistic vector orbitals are expressed through the corresponding quasirelativistic vector wave functions and Slater vector orbitals, respectively. The analytical formulas are also derived for overlap integrals over Slater type relativistic vector orbitals with the same screening constants in coordinate space.

Unified treatment of nonrelativistic and quasirelativistic atomic integrals over complete orthonormal sets of $${\Psi^{\alpha}}$$ -exponential type orbitals

The new one- and two-electron nonrelativistic and quasirelativistic basic functions are introduced. The general analytical relations in terms of basic functions suggested are derived for the non- and quasi-relativistic atomic integrals over complete orthonormal sets of \({\Psi ^{\alpha}}\) -exponential type orbitals introduced by the author, where α = 1,0, − 1, − 2, . . . The relationships obtained are valid for the arbitrary values of quantum numbers and screening constants of orbitals.

Application of Complete Orthonormal Sets of Ψα-Exponential-Type Orbitals to Accurate Ground and Excited States Calculations of One-Electron Diatomic Molecules Using Single-Zeta Approximation

The applicability of the complete orthonormal sets of ψα-exponential-type orbitals introduced by one of the authors to the study of electronic structure of one electron diatomic molecules is demonstrated using single-zeta approximation. As an example of application, the calculations have been performed for α, π and δ states of one electron homo- and hetero-nuclear diatomic molecules H+2 and HeH2+, respectively. The calculation results are presented. The values for these molecules obtained in eight-digit accuracy are close to the results of solution presented in literature.