Symmetric Orthogonalization Research Articles

Birkhoff-James orthogonality and the simultaneous zero inclusion property

A normed space X has the simultaneous zero inclusion (S0I) property if, for every invertible bounded linear operator T on X, 0 lies in the spatial numerical range of T if and only if 0 lies in the spatial numerical range of T−1. It is shown that the only absolute symmetric norms on C2 with the S0I property are scalar multiples of the Euclidean norm. However, the S0I property does not characterize inner products in a slightly wider class of norms—for example, there is a family of absolute polygonal norms on C2 that have the S0I property. There are two keys: (1) The S0I property is a weak version of symmetric Birkhoff-James orthogonality. (2) Complex spaces of dimension two are constructed, based on Radon curves, in which Birkhoff-James orthogonality is symmetric.

Coherence, superposition, and Löwdin symmetric orthogonalization

The notions of coherence and superposition are conceptually the same; however, an important distinction exists between their resource-theoretic formulations. Namely, while basis states are orthogonal in the resource theory of coherence, they are not necessarily orthogonal in the resource theory of superposition. Owing to the nonorthogonality, the manipulation and characterization of superposition states require significant efforts. Here, we demonstrate that the Löwdin symmetric orthogonalization (LSO) method offers a useful means for characterizing pure superposition states. The principal property of LSO is that the structure and symmetry of the original nonorthogonal basis states are preserved to the greatest extent possible, which prompts us to study the role of LSO in identifying the hierarchical relations of resource states. Notably, we reveal that the maximally coherent states turn into the states with maximal superposition with the help of LSO: in other words, they are equivalent under the action of symmetric orthogonalization. Our results facilitate further connections between coherence and superposition, where LSO is the main tool.

Applications of the Spectral Theory of Chemical Bonding to Simple Hydrocarbons.

The finite-basis, pair formulation of the Spectral Theory of chemical bonding is briefly surveyed. Solutions of the Born-Oppenheimer polyatomic Hamiltonian that are totally antisymmetric in electron exchange are obtained from diagonalization of an aggregate matrix built up from conventional diatomic solutions to atom-localized problems. A succession of transformations of the bases of the underlying matrices and the unique character of symmetric orthogonalization in producing the archived matrices calculated "once-of-all" in the pairwise-antisymmetrized basis are described. Application is made to molecules containing hydrogens and a single carbon atom. Results in conventional orbital bases are given and compared to experimental and high-level theoretical results. Chemical valence is shown to be respected and subtle angular effects in polyatomic contexts are reproduced. Means of reducing the size of the atomic-state basis and improving the fidelity of the diatomic descriptions for fixed basis size, so as to enable application to larger polyatomic molecules, are outlined along with future initiatives and prospects.

Dyadic diagonalization of positive definite band matrices and efficient [formula omitted]-spline orthogonalization

A dyadic algorithm for diagonalizing an arbitrary positive definite band matrix, referred to as a band Gramian, is obtained to efficiently orthogonalize the B-splines. The algorithm can be also used as a fast inversion method for a band Gramian characterized by remarkable sparsity of the diagonalizing matrix. There are two versions of the algorithm: the first one is more efficient and is applicable to a Toeplitz band Gramian while the second one is more general, works with any Gramian matrix, but is more computationally intensive. In the context of the B-splines, these two cases result in new symmetric orthogonalization procedures and correspond to equally and arbitrarily spaced knots, respectively. In the algorithm, the sparsity of a band Gramian is utilized to produce a natural dyadic net of orthogonal splines, rather than a sequence of them. Such a net is thus naturally referred to as a splinet. The splinets exploit “near-orthogonalization” of the B-splines and feature locality expressed through a small size of the total support set and computational efficiency that is a result of a small number of inner product evaluations needed for their construction. These and other efficiencies are formally quantified by upper bounds and asymptotic rates with respect to the number of splines in a splinet. An additional assessment is provided through numerical experiments. They suggest that the theoretical bounds are rather conservative and the method is even more efficient than the bounds indicate. The dyadic net-like structures and the locality bear some resemblance to wavelets but in fact, the splinets are fundamentally different because they do not aim at capturing the resolution scales. The orthogonalization method together with efficient spline algebra and calculus has been implemented in R-package Splinets available on CRAN.

Reliable transition properties from excited-state mean-field calculations.

Delta-self-consistent field (ΔSCF) theory is a conceptually simple and computationally inexpensive method for finding excited states. Using the maximum overlap method to guide optimization of the excited state, ΔSCF has been shown to predict excitation energies with a level of accuracy that is competitive with, and sometimes better than, that of time-dependent density functional theory. Here, we benchmark ΔSCF on a larger set of molecules than has previously been considered, and, in particular, we examine the performance of ΔSCF in predicting transition dipole moments, the essential quantity for spectral intensities. A potential downfall for ΔSCF transition dipoles is origin dependence induced by the nonorthogonality of ΔSCF ground and excited states. We propose and test a simple correction for this problem, based on symmetric orthogonalization of the states, and demonstrate its use on bacteriochlorophyll structures sampled from the photosynthetic antenna in purple bacteria.

Compensation for In-Phase/Quadrature Phase Mismatch in Coherent Free-Space Optical QPSK Communication Systems

The Gram–Schmidt orthogonalization procedure (GSOP) and Löwdin symmetric orthogonalization procedure (SYOP) are the two mainstream algorithms for the compensation of phase mismatch in an imperfect optical 90° hybrid. In this paper, we put forward an algorithm switching orthogonalization procedure (ASOP) according to the quality of in-phase and quadrature signals based on the Q value of the eye diagram with less computation. If the quality of the in-phase and quadrature signals has a significant difference, we use the GSOP and select the signal branch with better quality as the initial reference vector for orthogonalization. If they are of about the same quality, then we use the SYOP. We present computer simulations for a coherent free-space optical (FSO) quadrature phase-shift keying (QPSK) communication system and demonstrate the system improvement that can be achieved using the ASOP. Finally, we also show that the proposed ASOP scheme can contribute to the frequency offset and phase estimation of the FSO system in the environment of atmospheric turbulence.

RIEMANN ZETA FUNCTION, BERNOULLI POLYNOMIALS AND THE BEST CONSTANT OF SOBOLEV INEQUALITY

Green function for periodic boundary value problem of 2M-th order or- dinary differential equation is found by symmetric orthogonalization method under a suitable solvability condition. As an application, the best constants and the best functions of the Sobolev inequalities in a certain series of Hilbert spaces are found and expressed by means of the well-known Bernoulli polynomials. This result has clarified the variational meaning of the special values ζ(2M )( M =1 , 2, 3,···) of Riemann zeta function ζ(z).

Multiple constrained minimum variance beamformer (MCMV) performance in connectivity analyses

Functional brain connectivity is increasingly being seen as critical for cognition, perception and motor control. Magnetoencephalography and electroencephalography are modalities that offer noninvasive mapping of electrophysiological interactions among brain regions, yet suffer from signal leakage and signal cancellation when estimating brain activity. This leads to biased connectivity values which complicate interpretation. In this study, we test the hypothesis that a Multiple Constrained Minimum Variance beamformer (MCMV) outperforms the more traditional Linearly Constrained Minimum Variance beamformer (LCMV) for estimation of electrophysiological connectivity. To this end, MCMV and LCMV performance is compared in task related analyses with both simulated data and human MEG recordings of visual steady state signals, and in resting state analyses with simulated data and human MEG data of 89 subjects. In task related scenarios connectivity was estimated using coherence and phase locking values, whereas envelope correlations were used for the resting state data. We also introduce a novel Augmented Pairwise MCMV (APW-MCMV) approach for signal leakage suppression in resting state analyses and assess its performance against LCMV and more conventional MCMV approaches. We demonstrate that with MCMV effects of signal mixing and coherent source cancellation are greatly reduced in both task related and resting state conditions, while in contrast to other approaches 0- and short time lag interactions are preserved. In addition, we demonstrate that in resting state analyses, APW-MCMV strongly reduces spurious connections while better controlling for false negatives compared to more conservative measures such as symmetrical orthogonalization.

Existence of special rainbow triangles in weak geometries

Abstract We show that, in any ordered plane with a symmetric orthogonality relation which allows for a meaningful definition of acute and obtuse angles, in which all points are colored with three colors, such that each color is used at least once, there must exist both an acute triangle whose vertices have all three colors and an obtuse triangle with the same property. We also show that, in both a geometry endowed with an orthogonality relation, in which there is a reflection in every line, in which all right angles are bisectable, which satisfies Bachmann’s Lotschnittaxiom (the perpendiculars raised on the sides of a right angle intersect), and in plane absolute geometry, in which all points are colored with three colors, such that each color is used at least once, there exists a right triangle with all vertices of different colors.

Big data analysis of ab Initio molecular integrals in the neglect of diatomic differential overlap approximation.

Most modern semiempirical quantum-chemical (SQC) methods are based on the neglect of diatomic differential overlap (NDDO) approximation to ab initio molecular integrals. Here, we check the validity of this approximation by computing all relevant integrals for 32 typical organic molecules using Gaussian-type orbitals and various basis sets (from valence-only minimal to all-electron triple-ζ basis sets) covering in total more than 15.6 million one-electron (1-e) and 10.3 billion two-electron (2-e) integrals. The integrals are calculated in the nonorthogonal atomic basis and then transformed by symmetric orthogonalization to the Löwdin basis. In the case of the 1-e integrals, we find strong orthogonalization effects that need to be included in SQC models, for example, by strategies such as those adopted in the available OMx methods. For the valence-only minimal basis, we confirm that the 2-e Coulomb integrals in the Löwdin basis are quantitatively close to their counterparts in the atomic basis and that the 2-e exchange integrals can be safely neglected in line with the NDDO approximation. For larger all-electron basis sets, there are strong multishell orthogonalization effects that lead to more irregular patterns in the transformed 2-e integrals and thus cast doubt on the validity of the NDDO approximation for extended basis sets. Focusing on the valence-only minimal basis, we find that some of the NDDO-neglected integrals are reduced but remain sizable after the transformation to the Löwdin basis; this is true for the two-center 2-e hybrid integrals, the three-center 1-e nuclear attraction integrals, and the corresponding three-center 2-e hybrid integrals. We consider a scheme with a valence-only minimal basis that includes such terms as a possible strategy to go beyond the NDDO integral approximation in attempts to improve SQC methods. © 2018 Wiley Periodicals, Inc.

Symmetric orthogonality and non-expansive projections in metric spaces

In this paper known results of symmetric orthogonality, as introduced by G. Birkhoff, and non-expansive nearest point projections are extended from the linear to the metric setting. If the space has non-positive curvature in the sense Busemann then it is shown that those concepts are actually equivalent. In the end it is shown that every space having non-positive curvature in the sense of Busemann is a CAT(0)-space provided that its tangent cones are uniquely geodesic and their nearest point projections onto convex sets are non-expansive.

LEFT SYMMETRIC POINTS FOR BIRKHOFF ORTHOGONALITY IN THE PREDUALS OF VON NEUMANN ALGEBRAS

In this paper, we give a complete description of left symmetric points for Birkhoff orthogonality in the preduals of von Neumann algebras. As a consequence, except for $\mathbb{C}$, $\ell _{\infty }^{2}$ and $M_{2}(\mathbb{C})$, there are no von Neumann algebras whose preduals have nonzero left symmetric points for Birkhoff orthogonality.

Analytic derivative couplings and first-principles exciton/phonon coupling constants for an ab initio Frenkel-Davydov exciton model: Theory, implementation, and application to compute triplet exciton mobility parameters for crystalline tetracene.

Recently, we introduced an ab initio version of the Frenkel-Davydov exciton model for computing excited-state properties of molecular crystals and aggregates. Within this model, supersystem excited states are approximated as linear combinations of excitations localized on molecular sites, and the electronic Hamiltonian is constructed and diagonalized in a direct-product basis of non-orthogonal configuration state functions computed for isolated fragments. Here, we derive and implement analytic derivative couplings for this model, including nuclear derivatives of the natural transition orbital and symmetric orthogonalization transformations that are part of the approximation. Nuclear derivatives of the exciton Hamiltonian's matrix elements, required in order to compute the nonadiabatic couplings, are equivalent to the "Holstein" and "Peierls" exciton/phonon couplings that are widely discussed in the context of model Hamiltonians for energy and charge transport in organic photovoltaics. As an example, we compute the couplings that modulate triplet exciton transport in crystalline tetracene, which is relevant in the context of carrier diffusion following singlet exciton fission.

Fermionic Quantum Orthogonalizations I

We generalize classical orthogonalization procedures from real linear algebra to the setting of fermionic quantum (FQ) operations. In the case of the Gram-Schmidt orthogonalization procedure, the generalization is easy. This, however, helps to obtain general information regarding FQ operations, and to generalize the symmetric orthogonalization procedure.

Bonding Analysis of TM(cAAC)2 (TM = Cu, Ag, and Au) and the Importance of Reference State

A recent analysis of the bonding in transition metal (TM) complexes with cyclic aminoalkyl carbene (cAAC) ligands, TM(cAAC)2 (TM = Cu, Ag, and Au), purports to show that metal–ligand bonding involves the TM in the excited 2P state and that TM(pπ) → (cAAC)2 backdonation is not properly recognized in NBO analysis because of biases against participation of np functions in transition metal bonding. The questions of TM np orbital involvement in bonding and the possible biases in the NBO occupancy-weighted symmetric orthogonalization procedure have been examined by performing NBO analyses in two ways: (1) single Lewis structure (loc) analysis with TM np orbitals treated as valence (NBOs) or nonvalence (NBOx) and (2) direct comparison of a two-configuration resonance model (res/NBOs) treatment with a single configuration model using the expanded valency (loc/NBOx) treatment. The principal bonding picture that emerges from NBO analysis features a TM cation with two “non-innocent” cAAC ligands that are each reduce...

Sobolev inequality of free boundary value problem for ( − 1 ) M ( d / d x ) 2 M

The Green function of the free boundary value problem for $(-1)^{M}(d/dx)^{2M}$ is found by using Whipple’s formula. The Green function is constructed through so-called symmetric orthogonalization method under a suitable solvability condition. Its Green function is a reproducing kernel for a suitable set of Hilbert space and an inner product. By using the fact, we compute the best constant ( $M=1,2,3,4,5$ ) and a family of the best functions for a Sobolev inequality. It is possible for us to expect the best constant of the Sobolev inequality, but the proof has not been completed for $M\geq6$ in the present paper.

Cholesky-Like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms

It is well known that orthogonalization of column vectors in a rectangular matrix $B$ with respect to the bilinear form induced by a nonsingular symmetric indefinite matrix $A$ can be eventually seen as its factorization $B=QR$ that is equivalent to the Cholesky-like factorization in the form $B^TAB=R^T \Omega R$, where $R$ is upper triangular and $\Omega$ is a signature matrix. Under the assumption of nonzero principal minors of the matrix $M=B^T A B$ we give bounds for the conditioning of the triangular factor $R$ in terms of extremal singular values of $M$ and of only those principal submatrices of $M$ where there is a change of sign in $\Omega$. Using these results we study the numerical behavior of two types of orthogonalization schemes and we give the worst-case bounds for quantities computed in finite precision arithmetic. In particular, we analyze the implementation based on the Cholesky-like factorization of $M$ and the Gram--Schmidt process with respect to the bilinear form induced by the matrix...

Spread-Out Overlapping Sources by Independent Component Analysis for Location Positioning

The problem of spread-out overlapping sensors and sources of electromagnetic fields is discussed, and the way to distinguished them having same amplitude, phase, and frequency is presented. The case of sources separated by small distance ${\rm d}\ll\lambda$ is studied. To do so, the method of independent component analysis is applied. In this work an algorithm based on maximizing the negentropy and symmetric orthogonalization is used. A measurement setup is proposed to construct the database for the algorithm. Two half wavelength dipoles, at 1.8 GHz, are considered as sources. A high gain broadband antenna is used to receive the incoming signals. The correlation coefficient is used to evaluate how much the estimated signals using the independent component analysis are similar to the originals signals. Correlation coefficient high as 0.98 is achieved for sources placed $\lambda/10$ apart. The results obtained by independent component analysis are promising considering the complexity of the measurement setup and the proximity of sources.

On the Convergence of Symmetrically Orthogonalized Bounded Component Analysis Algorithms for Uncorrelated Source Separation

Bounded Component Analysis (BCA) has recently been introduced as an alternative linear decomposition scheme. In this approach the boundedness property of sources is exploited to replace the usual independence assumption with a weaker assumption, which enables development of methods to separate both independent and dependent components from their mixtures. This paper positions total output range minimization based blind source separation approach as a BCA method for the separation of uncorrelated sources. It is shown that the global minimizers of the corresponding optimization problem are the perfect separators. Furthermore, a stationary point analysis for the corresponding algorithms based on symmetrical orthogonalization is provided. The main result of this analysis is that the range minimization based parallel BCA algorithm and the kurtosis maximization based Independent Component Analysis algorithm have related set of identified stationary points.

Mayer’s orthogonalization: relation to the Gram-Schmidt and Löwdin’s symmetrical scheme

A method introduced by Mayer (Theor Chem Acc 104:163, 2000) for generating an orthogonal set of basis vectors, perpendicular to an arbitrary start vector, is examined. The procedure provides the complementary vectors in closed form, expressed with the components of the start vector. Mayer’s method belongs to the family of orthogonalization schemes, which keep an arbitrary vector intact without introducing any non-physical sequence-dependence. It is shown that Mayer’s orthogonalization is recovered by performing a two-step combination of the Gram-Schmidt and Lowdin’s symmetrical orthogonalization. Processor time requirement of constructing Mayer’s orthonormal set is proportional to ∼N2, in contrast to the rough ∼N3 CPU requirement of performing either a full Gram-Schmidt or Lowdin’s symmetrical orthogonalization. Utility of Mayer’s orthogonalization is demonstrated on an electronic structure application using perturbation theory to improve multiconfigurational wavefunctions.