One-dimensional Matrices Research Articles

An Auto-calibrated Measurement System for One-dimensional Matrices of Impedimetric Sensors

An Auto-calibrated Measurement System for One-dimensional Matrices of Impedimetric Sensors

Optimal Adjacent Output Phase Difference Assignments in One-Dimensional Parallel Switching Matrices With Four Beams

Optimal Adjacent Output Phase Difference Assignments in One-Dimensional Parallel Switching Matrices With Four Beams

A transfer matrix for the input impedance of weakly tapered, dissipative cones as of wind instruments (L).

A formula for the local acoustical admittance in a conical waveguide with viscous and thermal losses given by Nederveen [(1969). Acoustical Aspects of Woodwind Instruments (Frits Knuf, Amsterdam)] is rewritten as an impedance transmission matrix. Based on a self-consistent approximation for the cone, it differs from other one-dimensional transmission matrices used in musical acoustics, which implicitly include the loss model of a cylinder. The resonance frequencies of air columns calculated with this transmission matrix are in better agreement with more comprehensive models. Even for long cones with a slight taper, there is no need to discretize along the axis.

Editorial: Tensor numerical methods and their application in scientific computing and data science

Editorial: Tensor numerical methods and their application in scientific computing and data science

Three-dimensional shape measurement for the steep surface using DMD camera

Three-Dimensional shape measurement has been confronted with the ambiguity on the steep surface. To address the problem, compressed sensing theory is employed to reconstruct phase shifting images in DMD camera. Specially, every CCD pixel in the region of interest is aligned to N × N DMD mirrors to construct DMD camera. The one-dimensional measurement matrices are chosen to collect the measured values on CCD pixel according to the directional judgement of original gradient image. Due to the enhancement of the spatial sampling frequency and the noise robustness, the reconstructed sinusoidal stripe images are utilized to obtain the three-dimensional model of the steep surface accurately. We measure the planes with various slopes to discuss the measurement capability. The comparative experiments show that our proposed method can correct the deformed phase and repair the defect on the steep surface.

Special assignments of adjacent output phase differences per input ports in generalized one-dimensional parallel switching matrices

Special assignments of adjacent output phase differences per input ports in generalized one-dimensional parallel switching matrices

Generalized One-Dimensional Parallel Switching Matrices With an Arbitrary Number of Beams

This paper introduces a novel family of one-dimensional switching matrices, also known as beamforming networks (BFNs), which can be sized for an arbitrary number of beams. It also describes the associated numerical design method, supported by several design examples. The concept is further validated with a full-wave model of a generic 5 × 5 switching matrix designed at 76 GHz. The proposed matrices are obtained by cascading units consisting of a two-way coupler and a phase shifter in a parallel configuration, providing theoretically lossless operation as a direct consequence of the orthogonal BFN topology and equalized path lengths, expected to benefit the frequency bandwidth. The design method is general, also including the configurations equivalent to Butler matrices when the number of beams is a power of two. The obtained planar matrix designs combine some benefits of the well-known Butler and Nolen matrices, namely the parallel topology and the arbitrary number of beams. The minimum number of cascaded units required by the proposed topology, also corresponding to the number of layers, is found to be <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> is the number of beams, which is less than the number of cascaded units for the corresponding “parallel” Nolen matrix, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> larger than three, thus providing a reduction in the number of layers of up to 38% for an 8-beam matrix. This reduction is close to the theoretical upper boundary of 50%, reached for large values of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> . This significant reduction is expected to have a positive impact on RF performance, and in particular insertion losses, as well as amplitude and phase dispersion over the operating band. Interestingly, the proposed switching matrix configuration reduces to a standard Butler matrix when the number of beams is equal to 4, and demonstrates a reduction in the number of layers of 20% over the corresponding planar Butler matrix for a design with 8 beams. The proposed matrix configuration is a promising solution for the design of low-cost single-layer beam-switching matrices.

Estimation and experimental test of the sound-absorption coefficient of a pin-holder structure (Case of sound waves incidence upon the side surfaces of a group of cylinders)

In this study, we conducted theoretical analyses and experiments related to the acoustic characteristics of the situation where sound waves are incident upon the side surfaces of a group of cylinders forming a pin-holder structure. The sound-absorption coefficient, entering its clearance between cylinders through the geometrical dimension of the clearance or the physical property of gas, was calculated. In the analytical model, the gap part of the pin-holder structure was divided into elements and approximated as a gap surrounded by two parallel planes. The characteristic impedance and propagation constant of the approximate gap were obtained and treated as one-dimensional transfer matrices; the sound-absorption coefficient was then calculated using the transfer-matrix method. The calculated value was compared to that obtained in an experiment with a sample prepared using a 3D printer; the sound-absorption coefficient was measured using a 2-microphone impedance-measuring tube. We attempted to make a simple yet accurate estimation of sound-absorption coefficient using these procedures. Our theoretical values displayed a similar tendency to that obtained by experiment.

One-dimensional Block Matching Algorithm in Video Image Motion Estimation

<p indent=0mm>Motion estimation is a basic problem in the fields of video image compression and video image restoration. The traditional block matching methods have good search quality, but the search speed is not fast enough. Aiming at the shortcomings of the search speed in the traditional block matching methods, we proposed a fast one-dimensional block matching motion estimation algorithm. Firstly, the motion vector is orthogonally decomposed, and the two-dimensional matching block is reduced by a special weight coefficient matrix to obtain two sets of one-dimensional feature matrices. Then the one-dimensional three step search method is selected as the search strategy. The sum of absolute differences criterion is used as the matching criterion. The two sets of one-dimensional feature matrices are used to search for the two components of the matching motion vector. Finally the two components are formed into a complete motion vector. The results of multiple sets of comparative experiments show that the search speed of motion estimation is significantly improved while the algorithm is guaranteed to quantitatively evaluate the PSNR. The higher the video definition and the larger the pixel size of the matching block, the better the algorithm can improve the search speed of motion estimation.

The density of states of 1D random band matrices via a supersymmetric transfer operator

Recently,M. and T. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral properties of non-self-adjoint operators in the semiclassical regime. The method is applied to a transfer operator constructed from the supersymmetric integral representation for the density of states. We present a simpler proof of a slightly upgraded version of the semicircle law, which requires only standard semiclassical arguments and some peculiar elementary computations. The simplification is due to the use of supersymmetry, which manifests itself in the commutation between the transfer operator and a family of transformations of superspace, and was applied earlier in the context of band matrices by Constantinescu. Other versions of this supersymmetry have been a crucial ingredient in the study of the localization–delocalization transition by theoretical physicists.

Correlations in Perturbed Dual-Unitary Circuits: Efficient Path-Integral Formula

Interacting many-body systems with explicitly accessible spatio-temporal correlation functions are extremely rare, especially in the absence of integrability. Recently, we identified a remarkable class of such systems and termed them dual-unitary quantum circuits. These are brick-wall type local quantum circuits whose dynamics are unitary in both time and space. For these systems the spatio-temporal correlation functions are non-trivial only at the edge of the causal light cone and can be computed in terms of one-dimensional transfer matrices. Dual-unitarity, however, requires fine-tuning and the degree of generality of the observed dynamical features remained unclear. Here we address this question by introducing arbitrary perturbations of the local gates. Considering fixed perturbations, we prove that for a particular class of unperturbed elementary dual-unitary gates the correlation functions are still expressed in terms of one-dimensional transfer matrices. These matrices, however, are now contracted over generic paths connecting the origin to a fixed endpoint inside the causal light cone. The correlation function is given as a sum over all such paths. Our statement is rigorous in the "dilute limit", where only a small fraction of the gates is perturbed, and in the presence of random longitudinal fields, but we provide theoretical arguments and stringent numerical checks supporting its validity even in the clean case and when all gates are perturbed. As a byproduct, in the case of random longitudinal fields -- which turns out to be equivalent to certain classical Markov chains -- we find four types of non-dual-unitary(and non-integrable) interacting many-body systems where the correlation functions are exactly given by the path-sum formula.

ОДНО- ТА ДВОРОЗМІРНІ ДІЙСНІ КОМПОЗИЦІЙНІ МАТРИЦІ

The definition of compositional matrices (compomatrixes) is given, mathematical objects are defined that can be elements of compomatrixes. Shown are the requirements for indexing one-, two-, and three-dimensional compomatrixes and their purpose. The symbol designation of the real compromomats is established and it is indicated that they are intended for the analytical formalization of the description of geometric figures. It is indicated that the need to introduce the concept of “composite matrices” is caused by the nature of the formation of geometric figures (GF). It is determined what is the unification of the GF and why it is needed in compositional geometric modeling. The rules for the formation of a point compomatrix and a parametric compomatrix are provided, and their conventions are defined. It is proved that compomatrixes are used for geometric modeling of objects, each point of which is K-valued, that is, defined by the k-coordinates of the parameter space. It has been established that the number of elements and the form of their recording in the compatrix are in complete accordance with the number of points and their location on the original GF. A representation of the geometric model of the initial HF is provided, in a complimentary form, and examples of its creation are given for one- and two-dimensional compatatrices. The zero and single compomatrixes and their notation are defined. Also shown are the formations and designations of the numeric compomatrix. Recording rules and notation for calculated (coordinate) compomatrixes are provided. The sequence of transition from the compatrix form of a GF to a point polynomial, which is an interpolant of this GF, is shown. A compatrix is ​​proposed for the interpolant, which is written in expanded form, and the sequence of its separation into geometric and parametric components in the form of the corresponding compomatrix is ​​shown. It is proved that the pointwise interpolant compomatrix is ​​composite, and the parametric is combinative. The analysis of the parametric compatrix for point polynomials is carried out, its trace and determinant are given importance, the features of the transposed compomatrix to the original parametric are indicated. The main feature of the transposed parametric compomatrix is ​​that it is equal to the original compomatrix, which allows the mobile simplex method to be used without restrictions for the geometric modeling method. Keywords: one-dimensional and two-dimensional composite matrices, compomatic equations, point polynomial.

Characteristic Polynomials for Random Band Matrices Near the Threshold

The paper continues previous works which study the behavior of second correlation function of characteristic polynomials of the special case of $n\times n$ one-dimensional Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $J=(-W^2\triangle+1)^{-1}$. Applying the transfer matrix approach, we study the case when the bandwidth $W$ is proportional to the threshold $\sqrt{n}$

Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality

Consider $N\times N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W \geq N^{3/4+\varepsilon}$ for any $\varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the following results. (i) The semicircle law holds up to the scale $N^{-1+\varepsilon}$ for any $\varepsilon>0$. (ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning their ${\rm L}^\infty$ norms are all simultaneously bounded by $N^{-\frac{1}{2}+\varepsilon}$ (after normalization in ${\rm L}^2$) with overwhelming probability, for any $\varepsilon>0$. (iv )Quantum unique ergodicity holds, in the sense that the local ${\rm L}^2$ mass of eigenvectors becomes equidistributed with overwhelming probability. We extend the mean-field reduction method \cite{BouErdYauYin2017}, which required $W=\Omega(N)$, to the current setting $W \ge N^{3/4+\varepsilon}$. Two new ideas are: (1) A new estimate on the generalized resolvent of band matrices when $W \geq N^{3/4+\varepsilon}$. Its proof, along with an improved fluctuation average estimate, will be presented in parts 2 and 3 of this series \cite {BouYanYauYin2018,YanYin2018}. (2) A strong (high probability) version of the quantum unique ergodicity property of random matrices. For its proof, we construct perfect matching observables of eigenvector overlaps and show they satisfying the eigenvector moment flow equation \cite{BouYau2017} under the matrix Brownian motions.

Scalable tensor-product preconditioners for high-order finite-element methods: Scalar equations

We present a tensor-product-based preconditioner for high-order discontinuous-Galerkin (DG) discretizations. The preconditioner is based on approximating the block diagonal of the Jacobian matrix corresponding to element-wise coupling with the sum of tensor products of small one-dimensional matrices. The preconditioner is obtained through an algebraic procedure which minimizes the error between the tensor-product approximation and the exact elemental block Jacobian.Inverting the full elemental block Jacobian requires O(Nd) memory storage and O(N2d) operations, while applying its inverse requires O(Nd) operations per degree of freedom, where N is the solution order and d is the dimension of the problem. Thus, traditional block preconditioners become impractical with increasing dimension, d, and order, N. The cost of forming, storing, or applying the current tensor-product-based preconditioner scales linearly with solution order (O(N)) per degree of freedom for arbitrary number of dimensions. Furthermore, the tensor-product-based preconditioner recovers the exact block-Jacobi preconditioner in the case of constant-coefficient scalar problems on right parallelepiped elements. Numerical results demonstrate the effectiveness of the preconditioner for solving 4D (3D-space+time) DG discretizations of scalar advection-diffusion problems up to 32nd order.

Stepwise covariance matrix decomposition for efficient simulation of multivariate large-scale three-dimensional random fields

• A stepwise covariance matrix decomposition method is proposed for multivariate large-scale 3-D random field simulation. • This method is accurate when the 3D correlation function is separable. • The requirement on memory space can be reduced by ten orders of magnitude. • The maximum 3-D random field resolution can be improved from 21 × 21 × 41 to over 501 × 501 × 1001. The spatial variability of geomaterials affects the failure mechanism and reliability of geotechnical structures significantly, and can be modeled rigorously as a three-dimensional (3-D) random field. However, the simulation of multivariate, large-scale and high-resolution 3-D random fields is a challenging task due to extraordinary demands in computational resources. This paper proposes a stepwise covariance matrix decomposition method (CMD) with the aid of separable correlation functions, in which the 3-D random field is generated sequentially along each single dimension with small one-dimensional correlation matrices. The method not only inherits the simplicity of the widely-used general CMD, but also significantly reduces the computational time and required memory space. It only takes a few seconds to construct large-scale and high-resolution 3-D random fields, with the requirement on memory space reduced by more than ten orders of magnitude. The maximum random field resolution is significantly improved from approximately 21 × 21 × 41 using the general CMD to over 501 × 501 × 1001 using the stepwise CMD, which suffices in most engineering applications. The stepwise CMD facilitates 3-D spatial variability modeling in probabilistic site characterization and routine geotechnical reliability analysis.

Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O(p2d) storage and O(p3d) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O(pd+1) storage, O(pd+1) work in two spatial dimensions, and O(pd+2) work in three spatial dimensions. Combined with a matrix-free Newton–Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O(p9) to O(p5) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier–Stokes equations, using polynomials of degree up to p=30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p.

Evaluation of stress intensity factors and T-stress by finite block method: Static and dynamic

In this paper, the Finite Block Method (FBM) for computing the Stress Intensity Factors (SIFs) and the T-stress under static load and transient dynamic load is presented. Based on the one-dimensional first order partial differential matrices derived from the Lagrange series interpolation, the higher order partial differential matrices can be determined directly. By using the mapping technique, a block with arbitrary shape of the boundary is transformed in the polar coordinate (r,θ) into the normalised coordinate (ξ,η)|ξ|⩽1,|η|⩽1. In order to capture the stress intensity factor and the T-stress, the Williams' series of stress function is introduced in the circular core, centred at the crack tip, with the consideration of traction and displacement continuities along the circumference of the circular core. Time dependent problems are analysed by using the Laplace transformation method and the Durbin’s inversion method is used to determine all the physical time dependent variables. However, in the Laplace domain, the Deng's series of stress and displacement has to be used. The accuracy and the convergence of the FBM are demonstrated by four examples. Comparisons have been made with the solutions obtained by the Boundary Collocation Method (BCM) and the Finite Element Method (FEM).

On the Wegner Orbital Model

The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential is Gaussian, and prove three results: localisation at strong disorder, a Wegner-type estimate on the mean density of eigenvalues, and a Minami-type estimate on the probability of having multiple eigenvalues in a short interval. The last two results are proved in the more general setting of deformed block-Gaussian matrices, which includes a class of Gaussian band matrices as a special case. Emphasis is placed on the dependence of the bounds on the number of orbitals. As an additional application, we improve the upper bound on the localisation length for one-dimensional Gaussian band matrices.

Non-Hermitian localization in biological networks.

We explore the spectra and localization properties of the N-site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes. For large N, the spectrum has remarkable symmetries not only with respect to reflections across the real and imaginary axes but also with respect to 90^{∘} rotations, with an unusual anisotropic divergence in the localization length near the origin. When chains with periodic boundary conditions become directed, with a systematic directional bias superimposed on the randomness, a hole centered on the origin opens up in the density-of-states in the complex plane. All states are extended on the rim of this hole, while the localized eigenvalues outside the hole are unchanged. The bias-dependent shape of this hole tracks the bias-independent contours of constant localization length. We treat the large-N limit by a combination of direct numerical diagonalization and using transfer matrices, an approach that allows us to exploit an electrostatic analogy connecting the "charges" embodied in the eigenvalue distribution with the contours of constant localization length. We show that similar results are obtained for more realistic neural networks that obey "Dale's law" (each site is purely excitatory or inhibitory) and conclude with perturbation theory results that describe the limit of large directional bias, when all states are extended. Related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns.