Orthonormal Research Articles

Direct mathematical method for real-time ischemic episodes detection from electrocardiograms using the discrete Hermite transform

A real-time automated identification technique is developed for the detection of ischemic episodes in long-term electrocardiographic (ECG) signals using mathematical expansions involving the Discrete Dilated Hermite Transform. The Discrete Hermite functions could be viewed as a set of orthogonal vectors that resemble a finite Fourier series. They are generated easily as eigenvectors of a symmetric tridiagonal matrix that commutes with the centered Fourier matrix. The Discrete Hermite Transform (DHmT) values are computed from a simple dot product between an individual ECG complex extracted from the European Society of Cardiology (ESC) ST-T database and the corresponding discrete Hermite function. These values are found to contain information about the ECG shape, highlighting changes between ST segment and T wave alterations which are the features of ischemic episodes. This information from the discrete Hermite transform, based on an orthonormal set of n-dimensional digital Hermite functions that serve as shape-identification functions, can be used to identify ischemic episodes from the ECG. The performance measures resulting from applying this method to detect ischemic episodes were Sensitivity 87 %, Specificity 86 %, and positive predictive accuracy 81 %. The computer time to analyze one heartbeat for ischemia with this method is 0.031 s on a standard PC.

Quantum-Mechanical Explanation of Young’s double slit Experiment through Simulation of the Photon Coordinate Wave Function

Background: In the rapidly developing areas of photonics associated with quantum teleportation of photons, quantum cryptography, and quantum computing, ideas are becoming increasingly relevant about the certain localized states of single or several entangled photons moving in space and time. As is known, in quantum mechanics, localized states of particles can only be described using wave functions in coordinate representation – the corresponding wave packets. Aims: To explain Young's one- and two-photon experiments using coordinate 6-component quantum mechanical and 1-component quasi-classical photon wave functions. Methods: The article outlines the method developed in authors' previous work in constructing the 6-component coordinate wave function of free photons in the form of the wave packet (an integral over the entire momentum space and sum over two possible values +1 and -1 of the helicity). The basis functions are the circularly polarized monochromatic waves, corresponding to certain values of the momentum, helicity, and photon energy operators and forming a complete orthonormal set of generalized eigenfunctions of these operators. This photon wave function (PWF) is normalized to the unit probability of finding the photon anywhere in space. Both the basis functions and the coordinate PWF satisfy a Schrödinger-type equation derived from the Maxwell-Lorentz equations written in quantum mechanical form first introduced by Majorana. Then the results of modeling and consideration of the space-time evolution of the wave packet with Gaussian momentum distribution corresponding to the directed photon emission for 80 fs are briefly presented. Further, in view of the possibility of constructing the coordinate PWF, the basic formula for the waveparticle duality of photons and particles is formulated. It is emphasized that the photon is fundamentally different from “ordinary” particles, since, according to the authors, the photon is the result of the propagation of a certain complex spin wave in a physical vacuum, the details of which can only be considered simultaneously with the study of the structure of the physical vacuum at Planckian distances. Results: Young's one- and two-photon experiments are explained using various options for modeling 6-component coordinate PWFs, including approximate functions corresponding to spherically symmetric and electric dipole photon radiation. Conclusion: This explanation is illustrated using specific examples of radiation and, at the same time, is compared with numerical simulations for spherically symmetric radiation and the use of 1-component quasi-classical PWFs.

The sectional curvature of the infinite dimensional manifold of Hölder equilibrium probabilities

Abstract Here we consider the discrete time dynamics described by a transformation $T:M \to M$ , where T is either the action of shift $T=\sigma$ on the symbolic space $M=\{1,2, \ldots,d\}^{\mathbb{N}}$ , or, T describes the action of a d to 1 expanding transformation $T:S^1 \to S^1$ of class $C^{1+\alpha}$ (for example $x \to T(x) =\mathrm{d} x $ (mod 1)), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of Hölder equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a certain normalized Hölder potential A denote by $\mu_A \in \mathcal{N}$ the associated equilibrium probability. The set of tangent vectors X (functions $X: M \to \mathbb{R}$ ) to the manifold $\mathcal{N}$ at the point µA (a subspace of the Hilbert space $L^2(\mu_A)$ ) coincides with the kernel of the Ruelle operator for the normalized potential A. The Riemannian norm $|X|=|X|_A$ of the vector X, which is tangent to $\mathcal{N}$ at the point µA, is described via the asymptotic variance, that is, satisfies $ |X|^2 = \langle X, X \rangle = \lim_{n \to \infty} \frac{1}{n} \int (\sum_{i=0}^{n-1} X\circ T^i )^2 \,\mathrm{d} \mu_A$ . Consider an orthonormal basis Xi, $i \in \mathbb{N}$ , for the tangent space at µA. For any two orthonormal vectors X and Y on the basis, the curvature $K(X,Y)$ is \begin{equation*}K(X,Y) = \frac{1}{4}[ \sum_{i=1}^\infty (\int X Y X_i \,\mathrm{d} \mu_A)^2 - \sum_{i=1}^\infty \int X^2 X_i \,\mathrm{d} \mu_A \int Y^2 X_i \,\mathrm{d} \mu_A ].\end{equation*} When the equilibrium probabilities µA is the set of invariant Markov probabilities on $\{0,1\}^{\mathbb{N}}\subset \mathcal{N}$ , introducing an orthonormal basis $\hat{a}_y$ , indexed by finite words y, we show explicit expressions for $K(\hat{a}_x,\hat{a}_z)$ , which is a finite sum. These values can be positive or negative depending on A and the words x and z. Words $x,z$ with large length can eventually produce large negative curvature $K(\hat{a}_x,\hat{a}_z)$ . If $x, z$ do not begin with the same letter, then $K(\hat{a}_x,\hat{a}_z)=0$ .

Numerical investigation of the Klein-Gordon and sine-Gordon equations using the Christov expansion

Abstract The numerical interactions of solitary waves given by the Klein-Gordon and sine-Gordon equations are demonstrated. To do that we apply the “Christov expansion method”, which is a Galerking type expansion. The Christov expansion is based on the Christov functions which are functions that form a complete orthonormal set of functions on L 2(−∞, ∞) that allow us to expand derivatives, nonlinear products back into the same basis. Here we show how can implement this method to solitary wave equations that form kink or anti-kink type of solutions. We also show how can speed up the algorithms by rewriting the expressions for the nonlinear terms more efficiently. To this end we need to mention that the numerical integration is performed using a Crank–Nicolson type scheme. Our results are in a very good agreement with other published works.

On semi-orthogonal matrices with row vectors of equal lengths

When does a rectangular matrix with an orthonormal set of column vectors have row vectors of equal lengths? The column spaces of such matrices are multidimensional generalizations of the projection plane used in isometric perspective. We show that in the absence of unexpected linear relations, any rectangular matrix can be row-scaled so that if we were to orthonormalize the column vectors, the row vectors would attain equal lengths in the process. We use Grassmann coordinates to reduce the question into an instance of the famous matrix scaling problem, and with the help of existing theory introduce simple numerical solutions.

Derivation, characterization, and application of complete orthonormal sequences for representing general three-dimensional states of residual stress

Residual stresses are self-equilibrated stresses on unloaded bodies. Owing to their complex origins, it is useful to develop functions that can be linearly combined to represent any sufficiently regular residual stress field. In this work, we develop orthonormal sequences that span the set of all square-integrable residual stress fields on a given three-dimensional region. These sequences are obtained by extremizing the most general quadratic, positive-definite functional of the stress gradient on the set of all sufficiently regular residual stress fields subject to a prescribed normalization condition; each such functional yields a sequence. For the special case where the sixth-order coefficient tensor in the functional is homogeneous and isotropic and the fourth-order coefficient tensor in the normalization condition is proportional to the identity tensor, we obtain a three-parameter subfamily of sequences. Upon a suitable parameter normalization, we find that the viable parameter space corresponds to a semi-infinite strip. For a further specialized spherically symmetric case, we obtain analytical expressions for the sequences and the associated Lagrange multipliers. Remarkably, these sequences change little across the entire parameter strip. To illustrate the applicability of our theoretical findings, we employ three such spherically symmetric sequences to accurately approximate two standard residual stress fields. Our work opens avenues for future exploration into the implications of different sequences, achieved by altering both the spatial distribution and the material symmetry class of the coefficient tensors, toward specific objectives.

Targeted optimization in small-scale atomic structure calculations: application to Au I

The lack of reliable atomic data can be a severe limitation in astrophysical modelling, in particular of events such as kilonovae that require information on all neutron-capture elements across a wide range of ionization stages. Notably, the presence of non-orthonormalities between electron orbitals representing configurations that are close in energy can introduce significant inaccuracies in computed energies and transition probabilities. Here, we propose an explicit targeted optimization (TO) method that can effectively circumvent this concern while retaining an orthonormal orbital basis set. We illustrate this method within the framework of small-scale atomic structure models of Au I, using the Grasp2018 multiconfigurational Dirac–Hartree–Fock atomic structure code. By comparing to conventional optimization schemes we show how a TO approach improves the energy level positioning and ordering. TO also leads to better agreement with experimental data for the strongest E1 transitions. This illustrates how small-scale models can be significantly improved with minor computational costs if orbital non-orthonormalities are considered carefully. These results should prove useful to multi-element atomic structure calculations in, for example, astrophysical opacity applications involving neutron-capture elements.

Post-processing for Bayesian analysis of reduced rank regression models with orthonormality restrictions

Orthonormality constraints are common in reduced rank models. They imply that matrix-variate parameters are given as orthonormal column vectors. However, these orthonormality restrictions do not provide identification for all parameters. For this setup, we show how the remaining identification issue can be handled in a Bayesian analysis via post-processing the sampling output according to an appropriately specified loss function. This extends the possibilities for Bayesian inference in reduced rank regression models with a part of the parameter space restricted to the Stiefel manifold. Besides inference, we also discuss model selection in terms of posterior predictive assessment. We illustrate the proposed approach with a simulation study and an empirical application.

Orthonormal eigenfunction expansions for sixth-order boundary value problems

Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. To solve such problems, we first derive a complete set of odd and even orthonormal eigenfunctions — resembling trigonometric sines and cosines, as well as the so-called “beam” functions. These functions intrinsically satisfy boundary conditions (BCs) of relevance to thin-film flows, since they are the solutions of a self-adjoint sixth-order Sturm–Liouville BVP with the same BCs. Next, we propose a Galerkin spectral approach for sixth-order problems; namely the sought function as well as all its derivatives and terms appearing in the differential equation are expanded into an infinite series with respect to the derived complete orthonormal (CON) set of eigenfunctions. The unknown coefficients in the series expansion are determined by solving the algebraic system derived by taking successive inner products with each member of the CON set of eigenfunctions. The proposed method and its convergence are demonstrated by solving two model sixth-order BVPs.

Christov expansion method for nonlocal nonlinear evolution equations

Christov functions are a complete orthonormal set of functions on L 2(-∞,∞) that allow us to expand derivatives, nonlinear products, and nonlocal (integro-differential) terms back into the same basis. These properties are beneficial when solving nonlinear evolution equations using Galerkin spectral methods. In this work, we demonstrate such a “Christov expansion method” for the Benjamin–Ono (BO) equation. In the BO equation, the dispersion term is nonlocal, given by the Hilbert transform of the second spatial derivative of the unknown function. The Hilbert transform of the Christov functions can be computed using complex integration and Cauchy’s residue theorem to obtain simple relations. Then, a Galerkin spectral expansion can be used to the solve the BO equation. Time integration is performed using a Crank–Nicolson-type scheme. Importantly, the Christov expansion method yields a banded matrix for the spatial discretization, even though the spatial terms are nonlocal. To demonstrate the approach and its implementation, we perform numerical experiments showing the steady propagation of single and the overtaking interaction of multiple BO solitary waves.

Basis scaling and double pruning for efficient inference in network-based transfer learning

Network-based transfer learning allows the reuse of deep learning features with limited data, but the resulting models can be unnecessarily large. Although network pruning can improve inference efficiency, existing algorithms usually require fine-tuning that may not be suitable for small datasets. In this paper, using the singular value decomposition, we decompose a convolutional layer into two layers: a convolutional layer with the orthonormal basis vectors as the filters, and a “BasisScalingConv” layer which is responsible for rescaling the features and transforming them back to the original space. As the filters in each decomposed layer are linearly independent, when using the proposed basis scaling factors with the Taylor approximation of importance, pruning can be more effective and fine-tuning individual weights is unnecessary. Furthermore, as the numbers of input and output channels of the original convolutional layer remain unchanged after basis pruning, it is applicable to virtually all architectures and can be combined with existing pruning algorithms for double pruning to further increase the pruning capability. When transferring knowledge from ImageNet pre-trained models to different target domains, with less than 1% reduction in classification accuracies, we can achieve pruning ratios up to 74.6% for CIFAR-10 and 98.9% for MNIST in model parameters.

Uncertainty principles for the short‐time Fourier transform on the lattice

AbstractIn this paper, we study a few versions of the uncertainty principle for the short‐time Fourier transform on the lattice . In particular, we establish the uncertainty principle for orthonormal sequences, Donoho–Stark's uncertainty principle, Benedicks‐type uncertainty principle, Heisenberg‐type uncertainty principle, and local uncertainty inequality for this transform on . Also, we obtain the Heisenberg‐type uncertainty inequality using the ‐entropy of the short‐time Fourier transform on .

Relative acceleration of orthonormal basis vectors for the geometric conduction blocks of the cardiac electric signal propagation on anisotropic curved surfaces

Geometric conduction blocks stop cardiac electric propagation due to the shape or conductivity properties of the domain. The blocks are considered to cause many abnormal cardiac electric propagations, leading to cardiac electrophysiological pathologies, such as cardiac fibrillation and arrhythmia. Locating such multidimensional conduction blocks is challenging, particularly in a complex domain with a complex shape and strong anisotropy, such as the heart. To address this problem, we propose a novel mathematical model of the geometric conduction block using the relative acceleration adopted from space-time physics. An efficient numerical scheme for the mathematical model is also proposed to predict the unidirectional conduction block effectively, even in a complex domain. The relative acceleration in the cardiac electric propagation corresponds to the sink-source relationship between the excited (after repolarization) and excitable (before depolarization) cardiac cells, representing the geometric growth rate of the volume of metric balls. The trajectory is constructed from the wavefront of diffusion-reaction equations by aligning orthonormal basis vectors along the gradient of the action potential. Relative acceleration is computed along the propagational direction from the connection 1-form of the basis vectors. The proposed mathematical model and numerical scheme are applied to demonstrate geometric conduction blocks in two-dimensional (2D) simple curved domains with strong anisotropy.

Optical dipole potential energy of a two-level atom interacting with a tightly focused truncated optical Bessel beam

Truncated optical Bessel (TOB) beams are an orthonormal set of optical vortex beams that possess orbital angular momentum (OAM). It is characterized by a significant longitudinal electric field component at the sub-wavelength scales. The photons of a circularly polarized TOB beam also carry spin angular momentum (SAM) and can interact with a two-level atom, in the far-off resonance regime, giving rise to an optical dipole potential energy that involves a spin–orbit term that stems from the inclusion of the longitudinal field component. When a two-level atom is simultaneously irradiated by two TOB beams, the optical dipole potential energy, in some special cases, is entirely dependent on the OAM–SAM coupling. We explore the cold-atom trapping options enabled by the use of TOB beams, either single ones or combinations thereof. We put emphasis on case in which the OAM–SAM coupling is the dominant contribution. The similarities and differences between the results obtained for the TOB beams and those for the Laguerre–Gaussian beams are also discussed.

Hofstadter-like spectrum and magnetization of artificial graphene constructed with cylindrical and elliptical quantum dots

A comparative study of the electronic and magnetic properties of quasi-two-dimensional electrons in an artificial graphene-like superlattice composed of circular and elliptical quantum dots is presented. A complete orthonormal set of basis functions has been implemented for calculation of the energy dispersions, Hofstadter spectra, density of states and orbital magnetization of the considered systems. Our calculations indicate a topological change in the miniband structure due to the ellipticity of the quantum dots, and non-trivial modifications of the electron energy dispersion surfaces with the change of the magnetic flux per unite cell. The ellipticity of the quantum dots leads to an opening of a gap and considerable modifications of the Hofstadter spectrum. The orbital magnetization is shown to reveal significant oscillations with the change of the magnetic flux. The ellipticity of quantum dots has a qualitative impact on the dependencies of the magnetization on both the magnetic flux and the temperature.

Uncertainty principles for the windowed Opdam–Cherednik transform

In this paper, we study a few versions of the uncertainty principle for the windowed Opdam–Cherednik transform. In particular, we establish the uncertainty principle for orthonormal sequences, Donoho–Stark's uncertainty principle, Benedicks‐type uncertainty principle, Heisenberg‐type uncertainty principle, and local uncertainty inequality for this transform. We also obtain the Heisenberg‐type uncertainty inequality using the ‐entropy of the windowed Opdam–Cherednik transform.

Complete and orthonormal sets of exponential-type orbitals with non-integer quantum numbers

Atomic and molecular orbitals show exponential decrease at long range. Complete orthonormal basis sets for atoms should satisfy this criterion. A number of such bases have been used in physics (e.g. Coulomb Sturmians). The challenge of this work is first adapting Slater type Orbitals for this role, as they are not radially orthogonal. Even more important is their generalization to non-integer quantum numbers that have applications for configuration interaction. This generalization requires the whole apparatus of non-integer calculus that is presented using the Riemann–Liouville approach.

Using deep LSD to build operators in GANs latent space with meaning in real space.

Generative models rely on the idea that data can be represented in terms of latent variables which are uncorrelated by definition. Lack of correlation among the latent variable support is important because it suggests that the latent-space manifold is simpler to understand and manipulate than the real-space representation. Many types of generative model are used in deep learning, e.g., variational autoencoders (VAEs) and generative adversarial networks (GANs). Based on the idea that the latent space behaves like a vector space Radford et al. (2015), we ask whether we can expand the latent space representation of our data elements in terms of an orthonormal basis set. Here we propose a method to build a set of linearly independent vectors in the latent space of a trained GAN, which we call quasi-eigenvectors. These quasi-eigenvectors have two key properties: i) They span the latent space, ii) A set of these quasi-eigenvectors map to each of the labeled features one-to-one. We show that in the case of the MNIST image data set, while the number of dimensions in latent space is large by design, 98% of the data in real space map to a sub-domain of latent space of dimensionality equal to the number of labels. We then show how the quasi-eigenvectors can be used for Latent Spectral Decomposition (LSD). We apply LSD to denoise MNIST images. Finally, using the quasi-eigenvectors, we construct rotation matrices in latent space which map to feature transformations in real space. Overall, from quasi-eigenvectors we gain insight regarding the latent space topology.

TWISTED SHIFT-INVARIANT SYSTEM IN

AbstractWe consider a general twisted shift-invariant system, $V^{t}(\mathcal {A})$ , consisting of twisted translates of countably many generators and study the problem of obtaining a characterization for the system $V^{t}(\mathcal {A})$ to form a frame sequence or a Riesz sequence. We illustrate our theory with some examples. In addition to these results, we study a dual twisted shift-invariant system and also obtain an orthonormal sequence of twisted translates from a given Riesz sequence of twisted translates.

Topological two-body bands in a multiband Hubbard model

In a multiband Hubbard model the self-consistency relations for the two-body bound-state bands are in the form of a nonlinear eigenvalue problem. Assuming that the resultant eigenvectors form an orthonormal set, e.g., in the strong-binding regime, here we reformulate their Berry curvatures and the associated Chern numbers. As an illustration we solve the two-body problem in a Haldane-Hubbard model with attractive on-site interactions and analyze its topological phase diagrams from weak to strong couplings, i.e., by keeping track of the gap closings in between the low-lying two-body bands. The resultant Chern numbers are consistent with the lobe structure of the phase diagrams in the strong-coupling regime.