Orthonormal Research Articles

Characterization of infrasonic wind noise using the Discrete Orthonormal Stockwell Transform

The Discrete Orthonormal Stockwell Transform (DOST) (Stockwell, 2007) projects a finite-length, periodically sampled signal onto a set of orthonormal basis vectors that are localized in time and scale such that all basis functions at each scale are represented by a finite number of Discrete Fourier Transform (DFT) basis functions. It is based on the Stockwell Transform (Stockwell et al., 1996), sometimes referred to as the S-transform, which is applicable to continuously defined waveforms. Furthermore, the DFT basis functions corresponding to each DOST scale are non-overlapping. The shorter DOST scales correspond to higher-frequency DFT basis functions and greater bandwidths than the longer DOST scales. Thus, the DOST is a type of time-frequency transform and is suitable for non-stationary time-series analysis. Additionally, the DOST of an N-point record can be calculated efficiently in order N log N operations (Wang and Orchard, 2009). Several applications of the S-transform and the DOST have been documented including beamforming with array data (Collar and Frazier, 2022; Frazier, 2022). In this presentation, the DOST used is to examine the statistics of data measured with infrasound sensors as a function of the DOST scales, adaptively characterize the amplitude distributions of the DOST coefficients at each scale and perform near-optimal detection of transient signals. Results corresponding to infrasound data recorded under various conditions are presented.

Common canonical variate analysis (CCVA) based modeling and monitoring for multimode processes

Multiple operating modes are common in industrial processes due to feed stock alterations, product specifications, working environment changes and so on. Although different modes show different behaviors, some underlying process characteristics may stay invariable as mode changes, which reveal the essence information of the process. In this work, the issue of multimode process monitoring is studied with subspace separation, in which each mode is divided into the common subspace and specific subspace. A modified canonical variate analysis (CVA), termed as common CVA (CCVA), is put forward to extract the mode-common features based on joint approximate diagonalization method. The concatenation of all Hankel matrices is analyzed to find a common orthonormal set eigenvectors by minimization of joint diagonality criterion. Then, the remaining part of each mode is regarded as local specific subspace, which provides more representative information in each different mode. CVA algorithm is applied to build multiple local models based on mode-specific information. Two case studies, a numerical example and Tennessee Eastman (TE) process, are provided to validate the feasibility and effectiveness of the proposed method in monitoring abnormal operation for multimode processes.

Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems

Dynamical vectors characterizing instability and applicable as ensemble perturbations for prediction with geophysical fluid dynamical models are analysed. The relationships between covariant Lyapunov vectors (CLVs), orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors and finite-time normal modes (FTNMs) are examined for periodic and aperiodic systems. In the phase-space of FTNM coefficients, SVs are shown to equate with unit norm FTNMs at critical times. In the long-time limit, when SVs approach OLVs, the Oseledec theorem and the relationships between OLVs and CLVs are used to connect CLVs to FTNMs in this phase-space. The covariant properties of both the CLVs, and the FTNMs, together with their phase-space independence, and the norm independence of global Lyapunov exponents and FTNM growth rates, are used to establish their asymptotic convergence. Conditions on the dynamical systems for the validity of these results, particularly ergodicity, boundedness and non-singular FTNM characteristic matrix and propagator, are documented. The findings are deduced for systems with nondegenerate OLVs, and, as well, with degenerate Lyapunov spectrum as is the rule in the presence of waves such as Rossby waves. Efficient numerical methods for the calculation of leading CLVs are proposed. Norm independent finite-time versions of the Kolmogorov-Sinai entropy production and Kaplan-Yorke dimension are presented.

High-Order Method with Moving Frames to Compute the Covariant Derivatives of Vectors on General 2D Curved Surfaces

The covariant derivative is a generalization of differentiating vectors. The Euclidean derivative is a special case of the covariant derivative in Euclidean space. The covariant derivative gathers broad attention, particularly when computing vector derivatives on curved surfaces and volumes in various applications. Covariant derivatives have been computed using the metric tensor from the analytically known curved axes. However, deriving the global axis for the domain has been mathematically and computationally challenging for an arbitrary two-dimensional (2D) surface. Consequently, computing the covariant derivative has been difficult or even impossible. A novel high-order numerical scheme is proposed for computing the covariant derivative on any 2D curved surface. A set of orthonormal vectors, known as moving frames, expand vectors to compute accurately covariant derivatives on 2D curved surfaces. The proposed scheme does not require the construction of curved axes for the metric tensor or the Christoffel symbols. The connectivity given by the Christoffel symbols is equivalently provided by the attitude matrix of orthonormal moving frames. Consequently, the proposed scheme can be extended to the general 2D curved surface. As an application, the Helmholtz‐Hodge decomposition is considered for a realistic atrium and a bunny.

New approach to the generalized cross product. Orthonormal sets and Cramer's rule

New approach to the generalized cross product. Orthonormal sets and Cramer's rule

Conditional wiener integral associated with Gaussian processes and applications

Let C0[0,T] denote the one-parameter Wiener space and let C?0[0,T] be the Cameron-Martin space in C0[0,T]. Given a function k in C?0[0,T], define a stochastic process Zk : C0[0,T] ? [0, T] ? R by Zk(x, t) = R t 0 Dk(s)dx(s), where Dk ? d/dt k. Let a random vector XG,k : C0[0,T] ? Rn be given by XG,k(x) = ((g1,Zk(x,?))~,..., (gn,Zk(x,?))~), where G = {g1, ..., gn} is an orthonormal set with respect to the weighted inner product induced by the function k on the space C?0[0,T], and (g,Zk(x,?))~ denotes the Paley-Wiener-Zygmund stochastic integral. In this paper, using the reproducing kernel property of the Cameron-Martin space, we establish a very general evaluation formula for expressing conditional generalized Wiener integrals, E(F(Zk(x,?)) |XG,k(x) = ??), associated with the Gaussian processes Zk. As an application, we establish a translation theorem for the conditional Wiener integral and then use it to obtain various conditional Wiener integration formulas on C0[0,T].

Embedding snowflakes of Carnot groups into bounded dimensional Euclidean spaces with optimal distortion

We show that for any Carnot group $G$ there exists a natural number $D_G$ such that for any $0<\varepsilon<1/2$ the metric space $(G,d_G^{1-\varepsilon})$ admits a bi-Lipschitz embedding into $\mathbb{R}^{D_G}$ with distortion $O_G(\varepsilon^{-1/2})$. This is done by building on the approach of T. Tao (2018), who established the above assertion when $G$ is the Heisenberg group using a new variant of the Nash--Moser iteration scheme combined with a new extension theorem for orthonormal vector fields. Beyond the need to overcome several technical issues that arise in the more general setting of Carnot groups, a key point where our proof departs from that of Tao is in the proof of the orthonormal vector field extension theorem, where we incorporate the Lov\'{a}sz local lemma and the concentration of measure phenomenon on the sphere in place of Tao's use of a quantitative homotopy argument.

Reconstructing a rotor from initial and final frames using characteristic multivectors: With applications in orthogonal transformations

If an initial frame of vectors is related to a final frame of vectors by, in geometric algebra (GA) terms, a rotor, or in linear algebra terms, an orthogonal transformation, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or orthogonal matrix representing rotation, given knowledge of initial and transformed points.In this paper, we discuss methods in the literature for recovering such rotors and then outline a GA method, which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of characteristic multivectors as discussed in the book by Hestenes and Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create fractional transformations and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a ‘best fit’ rotor between these sets and compare our results with other methods.

Thermal buckling and post-buckling analyses of rotating Timoshenko microbeams reinforced with graphene platelet

A comprehensive study is established to investigate on the thermal buckling instability and the subsequent post-critical deflection of a rotating nanocomposite microbeam reinforced with graphene platelet. The main scope of the current research is shedding light on the effectiveness of reinforcing a rotating microbeam with graphene platelet subjected to a constant temperature development. The Timoshenko beam theory comes along the von-Karman strain displacement relations to extract the governing equations of motion in accordance with the modified couple stress theory. The modified Halpin–Tsai micromechanical model is implemented to determine the practical elasticity modulus of the nanocomposite beam. The Ritz technique is accompanied by the Chebyshev orthonormal polynomial set to discretize the weak form of the nonlinear equations of motion. The Newton–Raphson technique is applied to the discretized static nonlinear equations of motion to compute the static deformation due to the centrifugal force. On the basis of two effective algorithms the nonlinear equations of motion are attacked to find out the critical buckling temperature change as well as the succeeding post-critical deflection. It is observed that adjoining the graphene phase in the form of an X-pattern promotes the static strength of a rotating microbeam against the buckling instability. However, in contrast, for stationary microbeams and microbeams that rotate with a velocity below a threshold speed adjoining the graphene platelet reinforcement in the form of an O-Pattern not only does not strengthen the microbeam but also weakens it against the buckling instability. Moreover, the buckling modeshape of rotating simply-clamped microbeams is more impressed by the graphene phase.

THE NOTES ON ENERGY AND ELECTROMAGNETIC FIELD VECTORS IN THE NULL CONE Q2

In this paper, we characterize the directional derivatives in accordance with the asymptotic orthonormal frame {x,α,y} in Q^2. Also, we express the extended Serret-Frenet relations by using cone Frenet formulas and we explain the geometrical understanding of energy on each asymptotic orthonormal vector fields in null cone. Furthermore, we express the bending elastic energy function for the same particle according to curve x(s,ξ,η) and we finalize our results by providing energy variation sketches according to directional derivatives for different cases. Additionally, we explain a geometrical interpretation of the energy for unit vector fields and we express Maxwell’s equations for the electric and magnetic field vectors in null cone 3-space.

Generalized multimode scattering parameters and the applications in antenna and radio frequency component designs

This article applies the generalized multimode S-parameters to multiport antennas and radio frequency (RF) components for CAD applications. Consider the special case of impedance matching which could significantly simplify the multimode representation on port and remove the requirement on calculating port impedance. N single-ended ports of a network are regarded as one generalized multimode port, supporting N orthonormal modes represented by N orthonormal basis vectors, respectively. By doing this, in antenna and RF component designs, one can only focus on specific modes interested, while ignore others. A multiport network's characteristics could be represented and understood in a much simpler manner because fewer but more intuitive physical quantities are considered. Therefore, this can significantly simplify the analysis, design, and optimization of multiport networks. For better illustration, several examples, including branch-line coupler, mixed-mode case, sequential rotation feeding antenna, and monopulse feed are analyzed and designed. Complicated analyses in network design using the single-ended S-parameters, including matching, amplitude and phase unbalance, could be converted into much simpler ones from the generalized multimode perspective.

Visualizing Laguerre polynomials as a complete orthonormal set for the inner product space ℙ n

This linear algebra note offers teaching and learning ideas in the treatment of the orthonormal Laguerre polynomials as vectors of the inner product space with the associated improper integral inner product in a DGS (Dynamic Geometry Software)-MATLAB facilitated learning environment. The proposed learning module demonstrates creative ways of integrating DGS-MATLAB technology into various properties of the integral inner products of Laguerre polynomials along with other notions inherent in the inner product space, such as Triangle Inequality, Pythagorean Theorem, Parallelogram Law, Orthogonality and Orthonormality, Coordinates Relative to an Orthonormal Basis, and Parseval’s Identity.

Joint k-numerical ranges of operators

Let \(\mathcal{H} \) be a complex Hilbert space and \(\mathcal{B}(\mathcal{H}) \) the algebra of all bounded linear operators on \( \mathcal{H} \). For a positive integer k less than the dimension of \(\mathcal{H} \) and \({\bf A}=(A_1,...,AM)\in\mathcal{B}(\mathcal{H})^{m} \), the joint k-numerical range \(W_k(\bf A) \) is the set of \((\alpha_1,..,\alpha_m)\in \mathbb{C^m}\) such that \(\alpha_{i}={\Sigma^{k}_{j=1}} \langle A_{i}x{j},x{j }\rangle \) for an orthonormal set \(\{x_{1},...,x_{k}\} \) in \(\mathcal{H} \). The geometrical properties of \( W_k(\bf A) \) and their relations with the algebraic properties of \(\{A_{1},...,A_{m}\} \) are investigated in this paper. First, conditions for\(W_k(\bf A) \) to be convex are studied, and some results for finite dimensional operators are extended to the infinite dimensional case. An example of A is constructed such that \(W_k(\bf A) \) is not convex, but \(W_r(\bf A) \) is convex for all positive integer r not equal to k. Second, descriptions are given for the closure of \(W_k(\bf A) \) and the closure of conv\(W_k(\bf A) \) in terms of the joint essential numerical range of A for infinite dimensional operators\(A_{1},...,A_{m} \) . These lead to characterizations of \(W_k(\bf A) \)Wk(A) or conv\(W_k(\bf A) \) to be closed. Moreover, it is shown that conv \(W_k(\bf A) \) is closed whenever \(W_{k+1}(\bf A) \) or conv\(W_{k+1}(\bf A) \) is. These results are used to study the connection between the geometric properties of \( W_k(\bf A) \) and algebraic properties of \(A_{1},...,A_{m} \). For instance, \(W_k(\bf A) \) is a polyhedral set, i.e., the convex hull of a finite set, if and only if \(A_{1},...,A_{k} \) have a common reducing subspace V of finite dimension such that the compression of \(A_{1},...,A_{m} \) on the subspace V are diagonal operators \( D_{1},...,D_{m} \) and \( W_k(A)=W_{k}(D_1,...,D_m) \). Characterization is also given to A such that the closure of \(W_k(\bf A) \) is polyhedral. For finite rank operators the following two condition are equivalent: (1) \(A_{1},...,A_{m} \) is a commuting family of normal operators. (2) \(W_k(A)=W_{k}(A_1,...,A_m) \) is polyhedral for every positive integer k less than dim\(\mathcal{H} \). However, the two conditions are not equivalent for compact operators. Characterizations are given for compact operators \( A_{1},...,A_{m} \) satisfying (1) and (2), respectively. Results are also obtained for general non-compact operators. Open problems and future research topics are presented.

Oscillating charge currents of one-dimensional Hubbard model in electric field

The time evolution properties of charge current for the one-dimensional Hubbard model in an electric field have been studied in a rigorous manner. We find that there is a complete and orthonormal set of time-evolution states for which the charge current can only keep zero or oscillate constantly, differing from the possible picture of damped or over-damped Bloch oscillations due to strong correlations. It is also found that, associated with these states, there is a set of constant phase factors, which are uniquely determined and are very useful on discussing the long-time evolution behaviors of the system.

A non-trivial PT-symmetric continuum Hamiltonian and its eigenstates and eigenvalues

In this paper, a non-trivial system governed by a continuum PT-symmetric Hamiltonian is discussed. We show that this Hamiltonian is iso-spectral to the simple harmonic oscillator. We find its eigenfunctions and the path in the complex plane along which these functions form an orthonormal set. We also find the hidden symmetry operator, C, for this system. All calculations are performed analytically and without approximation.

Introduction and propagation properties of circular lorentz-bessel-gaussian beams

In this paper, circular Lorentz–Bessel–Gaussian beams (CLBGBs) are introduced as a novel member of the Lorentz-Gaussian beams family. The analytical expression of the propagation of these beams passing through a paraxial optical ABCD system is derived. The generalized Huygens-Fresnel diffraction integral of the form Collins’s formula and the expansion of the Lorentz distribution in terms of the complete orthonormal basis set of the Hermite-Gauss modes are used. The influences of the beam-order and Bessel part β, Gaussian and Lorentzian waists, and propagation distance z on the propagation of CLBGBs are investigated. Some numerical simulation results are done. The beams family in this work may be useful to the practical applications in free-space optical communications because they have vortex properties with their being experimentally realizable.

Truncated optical Bessel modes

An orthonormal set of optical vortex modes is put forward and identified as the polarized truncated optical Bessel (TOB) set, which is endowed with orbital as well as spin angular momentum. Members of this set of modes can be realized once a circular aperture of radius $R$ is placed centrally in the path of an optical Bessel beam of winding number $\ensuremath{\ell}$. For a fixed power input $\mathcal{P}$, the properties of the TOB set, namely, its helicity, energy, linear momentum, and spin and orbital angular momenta, are evaluated, and their main features are explored. The similarities and differences between the properties of the TOB mode set and those of the Laguerre-Gaussian set are pointed out and discussed.

Novel Outlook on the Eigenvalue Problem for the Orbital Angular Momentum Operator

Based on the novel prescription for the power function, (x+iy)m, the new expression for Ψ(x,y|m), the eigenfunction of the operator of the third component of the angular momentum, M^z, is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x+iy)m=ρmeimϕ, are invariant under the rotations at 2π for any, not necessarily integer, m—the eigenvalue of M^z. For any real m the functions Ψ(x,y|m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of M^z. The eigenfunctions and eigenvalues of the angular momentum operator squared, M^2, derived for the two different prescriptions for the square root, (m2)1/2, (m2)1/2=|m| and (m2)1/2=±m, are reported. The normalizable eigenfunctions of M^2 are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the M^2.

A closed local-orbital unified description of DFT and many-body effects

Density functional theory (DFT) is usually formulated in terms of the electron density as a function of position n(r). Here we discuss an alternative formulation of DFT in terms of the orbital occupation numbers {n α } associated with a local-orbital orthonormal basis set {ϕ α }. First, we discuss how the building blocks of DFT, namely the Hohenberg–Kohn theorems, the Levy–Lieb approach and the Kohn–Sham method, can be adapted for a description in terms of {n α }. In particular, the total energy is now a function of {n α }, E[{n α }], and a Kohn–Sham-like Hamiltonian is derived introducing the effects of the electron–electron interactions via effective potentials, . In a second step we consider the Hartree and exchange energies and discuss how to describe them, in the spirit of a DFT approach, in terms of the orbital occupation numbers. In this contribution special attention is paid to the description of the (intra-atomic) correlation energy and corresponding correlation potentials {V corr,α }. For this purpose, a model system is analyzed in detail, whereby an atomic Hamiltonian interacts with the environment via a simplified model; the use of this model allows us to obtain the correlation energy and potentials (in terms of {n α }) for different cases corresponding to low, intermediate and high electron correlations.

Strain-Based Geometrically Nonlinear Beam Formulation for Rigid–Flexible Multibody Dynamic Analysis

Geometrically nonlinear strain-based beam formulation has the potential to analyze flexible slender components in multibody systems efficiently owing to the minimum number of variables and constant stiffness matrix. The objective of this study is to develop a multibody dynamic analysis framework based on the strain-based beam formulation. To this end, we describe the constraint equation using the vector variables of the absolute nodal coordinate formulation that exhibits a velocity-transformation relationship with the strain-based formulation. Subsequently, we divide the Jacobian of the constraint equation into two terms. One term is equivalent to the velocity transformation matrix that is currently implemented in the existing strain-based analysis framework. The other term is a simple constant or linear Jacobian defined based on the orthonormal vectors of the absolute nodal coordinate formulation. This simple Jacobian description integrates the strain-based beam formulation with multibody dynamic theories to handle open- and closed-loop joints. The obtained numerical results validate that the proposed strain-based multibody dynamic analysis method concurs well with and exhibits a better convergence than the conventional flexible multibody dynamic analysis method.