Orthonormal Research Articles

Generative pulsar timing analysis

A new Bayesian method for the analysis of folded pulsar timing data is presented that allows for the simultaneous evaluation of evolution in the pulse profile in either frequency or time, along with the timing model and additional stochastic processes such as red spin noise, or dispersion measure variations. We model the pulse profiles using `shapelets' - a complete ortho-normal set of basis functions that allow us to recreate any physical profile shape. Any evolution in the profiles can then be described as either an arbitrary number of independent profiles, or using some functional form. We perform simulations to compare this approach with established methods for pulsar timing analysis, and to demonstrate model selection between different evolutionary scenarios using the Bayesian evidence. %s The simplicity of our method allows for many possible extensions, such as including models for correlated noise in the pulse profile, or broadening of the pulse profiles due to scattering. As such, while it is a marked departure from standard pulsar timing analysis methods, it has clear applications for both new and current datasets, such as those from the European Pulsar Timing Array (EPTA) and International Pulsar Timing Array (IPTA).

Gau–Wu numbers of nonnegative matrices

For any n-by-n matrix A, we consider the maximum number k=k(A) of orthonormal vectors xj∈Cn such that the scalar products 〈Axj,xj〉 lie on the boundary ∂W(A) of the numerical range W(A). This number is called the Gau–Wu number of the matrix A. If A is an n-by-n (n≥2) nonnegative matrix with the permutationally irreducible real part of the form[0A100⋱⋱Am−100],where m≥3 and the diagonal zeros are zero square matrices, then k(A) has an upper bound m−1. In addition, we also obtain necessary and sufficient conditions for k(A)=m−1 for such a matrix A. Another class of nonnegative matrices we study is the doubly stochastic ones. We prove that the value of k(A) is equal to 3 for any 3-by-3 doubly stochastic matrix A. For any 4-by-4 doubly stochastic matrix, we also determine its numerical range, which is then applied to find its Gau–Wu numbers. Furthermore, a lower bound of the Gau–Wu number k(A) is also found for a general n-by-n (n≥5) doubly stochastic matrix A via the possible shapes of W(A).

Phase space localization of orthonormal sequences in [formula omitted

The aim of this paper is to prove a quantitative extension of Shapiro’s result on the time–frequency concentration of orthonormal sequences in Lα2(R+). More precisely, we prove that, if {φn}n=0+∞ is an orthonormal sequence in Lα2(R+), then for all N≥0∑n=0N(‖xφn‖Lα22+‖ξHα(φn)‖Lα22)≥2(N+1)(N+1+α), and the equality is attained for the sequence of Laguerre functions. Particularly if the elements of an orthonormal sequence and their Fourier–Bessel transforms (or Hankel transforms) have uniformly bounded dispersions then the sequence is finite.Moreover we prove the following strong uncertainty principle for bases for Lα2(R+), that is if {φn}n=0+∞ is an orthonormal basis for Lα2(R+) and s>0, then supn(‖xsφn‖Lα22‖ξsHα(φn)‖Lα22)=+∞.

C-Numerical radius isometries on matrix algebras and triangular matrix algebras

Let c=(c1,…,cn)t∈Rn and Mn be the set of n×n complex matrices. For any A∈Mn, define the c-numerical range and the c-numerical radius of A byWc(A)={∑i=1nci〈Axi,xi〉:{x1,…,xn}is an orthonormal set in Cn} andwc(A)=max⁡{|z|:z∈Wc(A)}, respectively. Let Tn be the set of n×n upper triangular matrices. When wc(⋅) is a norm on Mn, mappings T on Mn (or Tn) satisfyingwc(T(A)−T(B))=wc(A−B) for all A, B are characterized. As an intermediate step, we also characterize additive c-numerical range preservers on Mn (or Tn).

Twelvefold symmetric quasicrystallography from the latticesF4,B6andE6

One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter–Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors inn-dimensional Euclidean space which lead to suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter–Weyl group is identified to determine the symmetry of the quasicrystal structure. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter–Weyl groupsWa(F4),Wa(B6) andWa(E6). These groups share the same Coxeter numberh= 12 with different Coxeter exponents. The dihedral subgroupD12of the Coxeter groups can be obtained by defining two generatorsR1andR2as the products of generators of the Coxeter–Weyl groups. The reflection generatorsR1andR2operate in the Coxeter planes where the Coxeter elementR1R2of the Coxeter–Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the latticesWa(F4) andWa(B6) are compatible with some experimental results.

Machine-Learning Methods for Earthquake Ground Motion Analysis and Simulation

This paper presents a novel method of data-based probabilistic seismic hazard analysis (PSHA) and ground motion simulation, verified using previously recorded strong-motion data and machine-learning techniques. The procedure consists of three parts: (1) selection of an orthonormal set of basis vectors called eigenquakes to represent characteristic earthquake records; (2) estimation of response spectra for the anticipated level of shaking for a scenario earthquake at a site using Gaussian process regression; and (3) optimal combination of the eigenquakes to generate time series of ground acceleration consistent with the response spectral ordinates obtained in the second part. The paper discusses the benefits of applying such machine-learning methods to strong-motion databases for PSHA and ground motion simulation, particularly in large urban areas where dense instrumentation is available or expected. The effectiveness of the proposed methodology is exhibited using four scenario examples for downtown Los Angeles. Advantages, disadvantages, and future research needs for this machine-learning approach to PSHA are discussed.

Combined open shell Hartree-Fock theory of atomic and molecular systems constructed from noncharged scalar particles

By the use of condition of relativistic covariance, Dirac group theory, Clifford algebra and complete orthonormal sets of w ða � Þ -self-frictional exponential type orbitals (w ða � Þ -SFETOs) introduced by the author in standard convention, the Hartree-Fock (HF) theory is suggested for multideterminantal single configuration states with any number of open shells of atoms and molecules constructed from the Standard Model-Fermi (SM-F) particles with m 6 0;s ¼ 0; and e ¼ 0 defined in the Standard Model of particle physics. It is shown that the origin of stability of these systems is the quantum damping or self-frictional forces produced by the SM-F particle itself. As an application, we have presented the periodic table for the SM-F atomic elements using Pauli principle of spinless noncharged identical SM-F particles. V C 2014 Physics Essays Publication .( http://dx.doi.org/10.4006/0836-1398-27.3.351) Resume´: Par l'utilisation de la condition de covariance relativiste, theorie des groupes de Dirac, algebre de Clifford et des ensembles complets de orthonormeesw ða � Þ -auto-friction orbitales de type exponentielle (w ða � Þ

Adaptive Clutter Nulling Approach for Heterogeneous Environments

Radar processors may suffer from performance loss when the heavy clutter is not sufficiently suppressed in a heterogeneous environment. In order to achieve the clutter suppression and further improve the detection performance, the clutter nulling method is widely addressed in radar systems, especially for the low-rank clutter in space---time adaptive processing, where the rank of the clutter covariance matrix is smaller than the length of test data vector. For the ubiquitous clutter-plus-noise environment in practice, where it is assumed as the superposition of the white Gaussian noise and the low-rank compound-Gaussian clutter without the accurately prior information of the texture, this paper develops a clutter nulling approach, whose kernel and emphasis are to obtain the maximum-likelihood estimation of the orthonormal basis vectors of the clutter subspace. Precisely in processing, the proposed clutter nulling method is mainly derived with the application of the Lagrange multiplier method and adaptively implemented using iteration method with the training data. Finally, the results on numerical data validate the advantages of the proposed nulling approach, in comparison with the existing method.

Unified Treatment of One-Range Addition Theorems for Complete Orthonormal Sets of Generalized Exponential-Type Orbitals and Noninteger n Slater Functions

Abstract By the use of L(pl*)-generalized Laguerre polynomials (L(pl*)-GLPs) introduced by the author, the combined formulas for the one- and two-center one-range addition theorems of complete orthonormal sets of φ (pl*)-generalized exponential-type orbitals (φ (pl*)-GETOs) and χ-noninteger n Slater type orbitals (χ-NISTOs) in terms of χ-integer STOs (χ-ISTOs) are suggested, where pl* = 2l + 2 − α* and α* are the integer (α* = α, −∞ < α ≤ 2) or noninteger (α* ≠ α, −∞ < α* < 3) self-frictional quantum numbers. The series expansion coefficients of these theorems are expressed through the overlap integrals over χ-NISTOs. As an application, the Combined Hartree–Fock–Roothaan (CHFR) total energy values for the ground states of some atoms obtained in the minimal basis set approximation are presented.

Different wavelengths selection methods for identification of early blight on tomato leaves by using hyperspectral imaging technique

Identification of early blight on tomato leaves by using hyperspectral imaging technique based on different effective wavelengths selection methods (successive projections algorithm, SPA; x-loading weights, x-LW; gram-schmidt orthogonaliza-tion, GSO) was studied in the present paper. Hyperspectral images of seventy healthy and seventy infected tomato leaves were obtained by hyperspectral imaging system across the wavelength range of 380-1023 nm. Reflectance of all pixels in region of interest (ROI) was extracted by ENVI 4. 7 software. Least squares-support vector machine (LS-SVM) model was established based on the full spectral wavelengths. It obtained an excellent result with the highest identification accuracy (100%) in both calibration and prediction sets. Then, EW-LS-SVM and EW-LDA models were established based on the selected wavelengths suggested by SPA, x-LW and GSO, respectively. The results showed that all of the EW-LS-SVM and EW-LDA models performed well with the identification accuracy of 100% in EW-LS-SVM model and 100%, 100% and 97. 83% in EW-LDA model, respectively. Moreover, the number of input wavelengths of SPA-LS-SVM, x-LW-LS-SVM and GSO-LS-SVM models were four (492, 550, 633 and 680 nm), three (631, 719 and 747 nm) and two (533 and 657 nm), respectively. Fewer input variables were beneficial for the development of identification instrument. It demonstrated that it is feasible to identify early blight on tomato leaves by using hyperspectral imaging, and SPA, x-LW and GSO were effective wavelengths selection methods.

From finite line graphs to infinite derived signed graphs

Let X be a subset of {±α±β:α,β∈Bandα≠β} where B is an orthonormal set in an inner product space over R, such that x∈X⇒−x∉X. Then the signed graph which is defined as described below is called a derived signed graph: its vertex set is X; two vertices x, y are joined by a positive (negative) edge when 〈x,y〉 is positive (negative); when 〈x,y〉=0, x, y are not joined. Let D denote the family of all derived signed graphs—the order of a member of D may be infinite. (The family of all generalized line graphs—line graphs belong to this family—is a subfamily of D.) Let M be the class of all minimal nonderivable signed graphs. [M includes the 31 (finite) minimal nongeneralized line graphs computed by various methods in the literature.] In this article, we characterize D, determine M and classify the family of all signed graphs S for which, the following holds: for each finite subset X of V(S), the least eigenvalue of S[X] is at least −2. The third result substantially generalizes the well known result (Cameron et al. (1976) [1]) on classifying the family of all finite (signed) graphs with least eigenvalues ⩾−2.

Novel Tridiagonal Commuting Matrices for Types I, IV, V, VIII DCT and DST Matrices

In this letter, we first propose new nearly tridiagonal commuting matrices of discrete Fourier transform (DFT) matrix and generalized DFT (GDFT) matrix. Then, using the block diagonalizations technique of circular-centrosymmetric and centrosymmetric matrices, we derive novel tridiagonal commuting matrices for each discrete cosine transform (DCT) and discrete sine transform (DST) matrices of the types I, IV, V, and VIII from the commuting matrices of the DFT and GDFT based on the relationships in matrix forms among DFT, GDFT, and various types of DCT and DST. Moreover, the novel tridiagonal commuting matrices of various types of DCT and DST do not have multiple eigenvalues. Last, with these novel commuting matrices, we can easily determine an orthonormal set of Hermite-like eigenvectors for each of their corresponding DCT or DST matrix.

A Noisy Channel Tolerant Image Encryption Scheme

In this paper, we present an image encryption scheme that has the capability to tolerate noisy effects of a wireless channel. This means if the encrypted image data is corrupted by channel noise up to a certain level, correct decryption is possible with some distortion. The proposed image encryption scheme relies on some very interesting properties of orthogonal matrices containing columns that form a set of orthonormal basis vectors. Besides being tolerant to noisy channels, the proposed scheme also provides good security against well-known cryptographic attacks as demonstrated in this paper by a number of experimental results and security analysis.

Graph-theoretic quantum system modelling for neuronal microtubules as hierarchical clustered quantum Hopfield networks

Graph-theoretic quantum system modelling (GTQSM) is facilitated by considering the fundamental unit of quantum computation and information, viz. a quantum bit or qubit as a basic building block. Unit directional vectors “ket 0” and “ket 1” constitute two distinct fundamental quantum across variable orthonormal basis vectors, for the Hilbert space, specifying the direction of propagation of information, or computation data, while complementary fundamental quantum through, or flow rate, variables specify probability parameters, or amplitudes, as surrogates for scalar quantum information measure (von Neumann entropy). This paper applies GTQSM in continuum of protein heterodimer tubulin molecules of self-assembling polymers, viz. microtubules in the brain as a holistic system of interacting components representing hierarchical clustered quantum Hopfield network, hQHN, of networks. The quantum input/output ports of the constituent elemental interaction components, or processes, of tunnelling interactions and Coulombic bidirectional interactions are in cascade and parallel interconnections with each other, while the classical output ports of all elemental components are interconnected in parallel to accumulate micro-energy functions generated in the system as Hamiltonian, or Lyapunov, energy function. The paper presents an insight, otherwise difficult to gain, for the complex system of systems represented by clustered quantum Hopfield network, hQHN, through the application of GTQSM construct.

On Fitting a Surface

This article deals with the problem of fitting the surface f=g (x) h(y) to the set of points (x/sub i/,y/sub j/,f/j). Functions g(x) and h(y) are supposed to be expressible in terms of orthonormal sets of functions. The desired coefficients of these functions are determined as characteristic vectors corresponding to the largest characteristic root of two materials having common characteristic roots.

Global series solutions of nonlinear differential equations with shocks using Walsh functions

An orthonormal basis set composed of Walsh functions is used for deriving global solutions (valid over the entire domain) to nonlinear differential equations that include discontinuities. Function ...

Optimizing Low-Frequency Mode Stirring Performance Using Principal Component Analysis

We formulate and perform principal component analysis (PCA) of mechanically stirred fields, based on tuner sweep data collected across a frequency band at a single location of a sensor inside a reverberation chamber. Both covariance- and correlation-based PCA in undermoded and overmoded regime are performed and intercompared. The nonstationarity of the stir performance as a function of the angular position of the stirrer is demonstrated. It is shown that this nonuniformity can be quantified and exploited to select a set of optimal stir angles. The rotated principal components are found to be interpretable as energy stirred by specific angular sectors of the stirrer and are related to the correlation structure of the data. The analysis leads to the concept of eigen-stirrings (stir modes), which form an orthonormal set of empirical basis functions for expanding stir data.

A Setting for Generalized Orthogonalization

Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica; Argentina

Orthonormal Gas-Chromatography Sets of Extra Virgin Olive Oil

Starting from gas-chromatograms of extra virgin olive oil samples, choosing the most significant peaks, authors try to define a standard ortho-normal base to be deposited for quality valuation.

基于方形域内标准正交矢量多项式的波前重建

基于方形域内标准正交矢量多项式的波前重建