Orthogonal Families Research Articles

On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on R that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval [0,1]. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues.In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a spatio-spectral limiting operator (SSLO) on L2(Rd), which is an alternating product of projection operators associated to given spatial and frequency domains in Rd. If one of the domains is a hypercube, and the other domain is convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval [0,1].To prove our results, we design an orthonormal system of wave packets in L2(Rd) that are highly concentrated in the spatial and frequency domains. We show that these wave packets are “approximate eigenfunctions” of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for L2[0,1], and we lift the basis to higher dimensions using a tensor product.

Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form

Orthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of C^ by vertical and horizontal trajectories of a non-vanishing quadratic differential. Yet the identification of these trajectories with real quartics in CP2 is subtle. Here, we give an efficient, geometric argument in the course of updating the classical theory of confocal families in the modern language of quadratic differentials and the Edwards normal form for elliptic curves. In particular, we define a parameterized Edwards transformation, providing explicit birational equivalence between each curve in a confocal family and a fixed curve in normal form.

Norm-Attainable Polynomials: Characterizations and Properties in Orthogonal Polynomial Families

Norm-attainable polynomials play a crucial role in various mathematical contexts. This research paper investigates the characterizations and properties of norm-attainable polynomials in different orthogonal polynomial families. We explore their behavior, convexity, and positive definiteness. Additionally, we establish their norm-attainability in specific intervals for various weight functions. The findings contribute to a deeper understanding of norm-attainable polynomials and their applications in approximation theory and mathematical modeling.

The effect of tectonic subsidence on the evolution of depocenters and development of the Mundaú sub-basin, Brazilian Equatorial margin

Located on the Brazilian Equatorial Margin, the Mundaú Sub-Basin is the only offshore hydrocarbon-producing region of the Ceará Basin. The origin and evolution of the Mundaú Sub-basin are a consequence of the rupture of the Pangea supercontinent during the Early Cretaceous, which led to the opening and development of the Equatorial Atlantic Ocean. This area has been considered a new exploration frontier, promoting research to be developed in recent years. However, the interactions between transtensive events and spatial-temporal distributions of tectonic subsidence have been poorly studied. To cover this gap, fifty-seven wells and approximately 5800 km of 2D seismic lines were analyzed to understand these events. From these data, stratigraphic and structural interpretations were performed throughout the sub-basin, enabling the construction of 43 pseudo-wells. Using descompactation of the wells and pseudo-wells using the backstripping method, it was possible to reconstruct seven tectonic phases for the Sub-Basin, which correspond to the rift sequence (RP01, RP02, and RP03), the transition sequence (BP01), and the drift sequence (DP01, DP02, and DP03). Four sets of faults of three zones of the three zones (WZ, CZ and EZ) proposed for the Mundaú Sub-Basin were also analyzed. The faults are distributed in two orthogonal families, formed in two distinct events. Older R and P faults, respectively trending NWW-SEE and NNESSW, formed during RP01 (first event). Younger R and P faults, respectively trending NNW-SSE and NEE-SWW, formed during RP02 (second event). Tectonic reconstruction revealed the diachronic formation of two depocenters, one in the West Zone (WZ) during RP01, and the other in the Central Zone (CZ) during RP02. Despite the intense tectonic activity in WZ and CZ, the East Zone (EZ) showed low subsidence rates and little development during the entire rift phase. In turn, BP01 showed faster subsidence rates when compared to RP03, indicating tectonic influence even during the breakup phase. Finally, the drift phase showed the lowest subsidence rates, spatially distributed in an opposite pattern to the rift phase. From these results, a rotation of the Principal Displacement Zone (PDZ) was identified, with initial parallel motion vectors, forming the WZ depocenter and generating the first fault family during RP01; after RP02, motion vectors were oblique, forming the second depocenter in CZ and generating the second fault family. Significant thermal control associated with magmatic events was identified during the drift phase. This study also describes the main structures formed along with the overlap of parallel and oblique PDZ displacement.

A continuum model based on Rayleigh dissipation functions to describe a Coulomb-type constitutive law for internal friction in woven fabrics

A continuum model intended to provide predictions for the response of a woven fabric that includes the effects of friction between fibers is proposed. Specifically, we consider a macroscopic formulation in which the fabric weave is composed of two orthogonal families of continuously distributed yarns. The elastic behavior of the planar fabric is characterized by a second-gradient formulation, incorporating the capacity of the fibers to resist a bending deformation. Particular care is devoted to modeling the action of preventing fiber overlapping through a potential energy barrier. The frictional sliding effect of warp threads interwoven with the weft yarns is introduced through a Rayleigh dissipative function that can be appropriately shaped to consider a Coulomb-type law. Spinning friction of yarns belonging to different families also is conceived when a relative rotation between fibers is present to generalize the dissipation phenomenon involved in the considered sheet. Numerical simulations of the proposed model are provided and discussed.

On spectral structure and spectral eigenvalue problems for a class of self similar spectral measure with product form

Let μ be a Borel probability measure with compact support on . We say that μ is a spectral measure if there exists , called a spectrum of μ, such that forms an orthonormal basis for L 2(μ). In this paper, we study the structure of spectra for a class of self-similar spectral measure μ R,B with product form on . We first give a partially characterize for E Λ to be a maximal orthogonal family in L 2(μ R,B ) by using the notion of maximal tree mapping. Based on this, we give a sufficient condition for a maximal orthogonal family E Λ (which corresponds to a maximal tree mapping) to be an orthonormal basis of L 2(μ R,B ). Moreover, we completely settle two types of spectral eigenvalue problems for μ R,B . Precisely, on the first case, for the model spectrum (simplest spectrum) of μ R,B , we characterize all possible real numbers t such that tΛ is also a spectrum of μ R,B . On the other case, we characterize all possible real numbers t such that there exists a countable set Λ such that Λ and tΛ are both spectra of μ R,B .

A general algorithmic scheme for combinatorial decompositions with application to modular decompositions of hypergraphs

We study here algorithmic aspects of modular decomposition of hypergraphs. In the literature one can find three different definitions of modules: standard modules [25], k-subset modules [6], and Courcelle's modules [12]. Using fundamental tools from combinatorial decompositions such as partitive and orthogonal families, we directly derive a linear time algorithm for Courcelle's decomposition. We then introduce a general algorithmic tool for weakly partitive families and apply it for the three definitions of modules to derive polynomial time algorithms. For standard modules, we give an O(n3⋅l) time algorithm (where n is the number of vertices and l the sum of the size of the edges). For k-subset modules, we obtain an O(n3⋅l) runtime. This is an improvement from the best known algorithm for k-subset modular decomposition, which was not polynomial w.r.t. n and m, and is in O(n3k−5) time [6] where k denotes the maximal size of an edge, and m the number of hyperedges. Finally we focus on applications of orthogonality to modular decompositions of tournaments, simplifying the algorithm from [24]. The question of designing a O(n⋅l) time algorithm for the standard modular decomposition of hypergraphs remains open.

On the eigenvalues of matrices with common Gershgorin regions

This paper is a study of the eigenvalues of a complex square matrix with one variable nondiagonal entry expressed in polar form. Changing the angle of the variable entry while leaving the radius fixed generates an algebraic curve; as does the process of fixing an angle and varying the radius. The authors refer to these two curves as eigenvalue orbits and eigenvalue trajectories, respectively. Eigenvalue orbits and trajectories are orthogonal families of curves, and eigenvalue orbits are sets of eigenvalues from matrices with identical Gershgorin regions. Algebraic and geometric properties of both types of curves are examined. Features such as poles, singularities, and foci are discussed.

Three-dimensional flow field reconstruction in the wake of a confined square cylinder using planar PIV data

A method is presented to perform the three-dimensional reconstruction of the velocity field in laminar flows using planar PIV data. The developed method relies on a convenient, repeated application of two data processing tools developed by two of the authors and called the higher order dynamic mode decomposition and the spatio-temporal Koopman decomposition. The present reconstruction method is an extension of a general method recently presented by three of the authors (Pérez et al., 2020), in which synthetic PIV data (obtained via numerical simulation) along three mutually orthogonal families of planes were used. In this article, instead, actual experimental data are considered, which requires additional data processing tools, developed in the article, to both cope with experimental noise and synchronize the various PIV datasets obtained in different timespans. The method is applied to analyze the wake of a square cylinder confined in a channel, at moderate Reynolds number. Due to limitations of the experimental setup used to obtain data for this configuration, only two families of planar PIV measurements are available that are contained in planes parallel to the sides of the channel. Because of this limitation, only the streamwise velocity component is three-dimensionally reconstructed. The performance of the method is tested in the wake of the confined square cylinder, comparing the reconstructed data with experimental PIV data measured on additional test planes (not used in the reconstruction). The obtained reconstructions of the streamwise velocity are reasonably accurate.

Data-driven edge detectors for seismic data interpretation

We have developed a novel method to assist in seismic interpretation. The algorithm learns data-driven edge detectors for structure enhancement when applied to time slices of 3D poststack seismic data. We obtain the operators by distilling the local and structural information retrieved from patches taken randomly from the input time slices. The filters conform to an orthogonal family that behaves as structure-aware Sobel-like edge detectors, and the user can set their size and number. The results from marine Canada and New Zealand 3D seismic data demonstrate that our algorithm allows the semblance attribute to improve the delineation of subsurface channels. This fact is further supported by testing the method with realistic synthetic 2D and 3D data sets containing channeling and meandering systems. We contrast the results with standard plain Sobel filtering, multidirectional Sobel filters of variable size, and the dip-oriented plane-wave destruction Sobel attribute. Our method gives results that are comparable or superior to those of Sobel-based approaches. In addition, the obtained filters can adapt to the geologic structures present in each time slice, which reduces the number of unwanted artifacts in the final product.

Spin and pseudospin symmetries in radial Dirac equation and exceptional hermite polynomials

We have generalized the solutions of the radial Dirac equation with a tensor potential under spin and pseudospin symmetry limits to exceptional orthogonal Hermite polynomials family. We have obtained new general rational potential models which are the generalization of the nonlinear isotonic potential families and also energy dependent.

Transfer learning based attack detection for wireless communication networks

AbstractWireless communication network attack detection is important in the field of wireless communication network data mining. Transfer learning is a powerful tool to solve the problem of link prediction in unlabeled wireless networks. In the feature mapping process of transfer learning, the loss of source domain information is very serious, which leads to the loss of knowledge in inverse mapping. To obtain more specific and complete information in the source domain, a link prediction method based on the distribution function fitting method of domain orthogonal term family is proposed. The key samples in the source domain are used to construct a family of orthogonal polynomials, and the least square approximation of the distribution function is obtained by using the family of orthogonal polynomials. The feature vector set of the key samples is extracted from the source domain, and the pseudo label is assigned to the target training set, so that the link classifier can complete the supervised link prediction task. In addition, an advanced performance of TLAD is obtained by comparing with other state‐of‐the‐art models in practice.

On the computation of recurrence coefficients for univariate orthogonal polynomials.

Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern computational tools that facilitate evaluation and manipulation of polynomials with respect to the measure, and such tasks are foundational in numerical approximation and quadrature. Although the recurrence coefficients for classical measures are known explicitly, those for nonclassical measures must typically be numerically computed. We survey and review existing approaches for computing these recurrence coefficients for univariate orthogonal polynomial families and propose a novel "predictor-corrector" algorithm for a general class of continuous measures. We combine the predictor-corrector scheme with a stabilized Lanczos procedure for a new hybrid algorithm that computes recurrence coefficients for a fairly wide class of measures that can have both continuous and discrete parts. We evaluate the new algorithms against existing methods in terms of accuracy and efficiency.

Control design based on multi-index nonlinear control method

Firstly, a nonlinear control system based on the Riemannian manifold is defined from the perspective of holism, and the expression of the system’s state equation in the local coordinate system of the Riemannian manifold is given. The geometric structure of the Riemannian manifold is discussed. The influence of nonlinear systems is studied. The controllability and observability of nonlinear systems is studied. Secondly, using the properties of involute distribution and fully geodesic submanifolds, the local parts of nonlinear systems based on Riemannian manifolds are given. Controllable structured decomposition, local observable structure decomposition and Kalman decomposition. Thirdly, using the properties of orthogonal involute distribution family, incremental involute distribution family and fully geodesic submanifold family, the research is based on Riemannian manifolds. The parallel decoupling problem and cascade decoupling problem of nonlinear control systems on the above, as well as the local disturbance decoupling problem of affine nonlinear control systems.

Classifying Galois groups of an orthogonal family of quartic polynomials

We consider the quartic generalized Laguerre polynomials $L_{4}^{(\alpha)}(x)$ for $\alpha \in \mathbb Q$. It is shown that except $\mathbb Z/4\mathbb Z$, every transitive subgroup of $S_{4}$ appears as the Galois group of $L_{4}^{(\alpha)}(x)$ for infinitely many $\alpha \in \mathbb Q$. A precise characterization of $\alpha\in \mathbb Q$ is obtained for each of these occurrences. Our methods involve the standard use of resolvent cubics and the theory of p-adic Newton polygons. Using these, the Galois group computations are reduced to Diophantine problem of finding integer and rational points on certain curves.

Heat transport in shear-driven flow with axial conduction

The classical Graetz problem of fluid entering a long channel possessing different temperature from that of fluid is extended here to the case where the upper wall of the channel is not stationary. Formulation of the current problem incorporates the simple Couette flow driven by the channel movement and taking into account the axial heat conduction, besides the usual convection. Analytical solutions in the thermally developing regime regarding the temperature distribution as well as the heat transfer rates from both the upper moving and the lower fixed channel walls are then obtained by means of the series expansion method making use of the proper orthogonal family of eigenfunctions. Simple approximate formulae for the rates of heat transfer at the channel entrance valid for large Peclet numbers and at the far downstream locations valid for all Peclet numbers are further derived. More simplistic outcomes are also provided when the axial pressure gradient is assumed to be constant. Results are then discussed from low to high Peclet numbers, which are in compliant with the well-known results available in the literature for the channel flow within stationary walls. On the other hand, the rate of heat transfer from the moving wall is shown to be of high importance when Peclet number increases, which requires special attention in technological applications.

Orthogonal sums in Kreĭn spaces

Infinite orthogonal families of regular subspaces of a Kreĭn space exhibit a wide range of behavior. An elementary method is used to show that conditions on sums of projections produce behavior similar to that of orthogonal closed subspaces of a Hilbert space.

A wavelet filter comparison on multiple datasets for signal compression and denoising

In this paper, we explicitly analyze the performance effects of several orthogonal and bi-orthogonal wavelet families. For each family, we explore the impact of the filter order (length) and the decomposition depth in the multiresolution representation. In particular, two contexts of use are examined: compression and denoising. In both cases, the experiments are carried out on a large dataset of different signal kinds, including various image sets and 1D signals (audio, electrocardiogram and seismic). Results for all the considered wavelets are shown on each dataset. Collectively, the study suggests that a meticulous choice of wavelet parameters significantly alters the performance of the above mentioned tasks. To the best of authors’ knowledge, this work represents the most complete analysis and comparison between wavelet filters. Therefore, it represents a valuable benchmark for future works.

ON A FABRIC OF KISSING CIRCLES

AbstractApplying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. We focus on the curvature inside the individual components of the fabric, which are two orthogonal frames and two orthogonal families of chains. We show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures in each chain are arranged in a quadratic bi-sequence. We also prove a sufficient condition for the fabric to be integral.

Multivariate holomorphic Hermite polynomials

We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multivariate analog of the Ito polynomials. We show how these multivariate polynomials generalize the univariate complex Hermite and Ito polynomials. Generating functions, orthogonality relations, Rodrigues formulas, recurrence and linearization relations, and operator formulas are also derived for these multivariate holomorphic Hermite and Ito polynomials. A Kibble–Slepian formula and a Mehler-type formula for the multivariate Ito polynomials are established.