Orthogonality Relations Research Articles

New proofs of theorems on q-orthogonal functions

In this paper, we establish a [Formula: see text]-integral formula by using the orthogonality relation, and also provide a new proof of the [Formula: see text]-orthogonality relation for the continuous [Formula: see text]-ultraspherical polynomials. A new [Formula: see text]-beta integral with five parameters is evaluated.

Biquaternion linear canonical stockwell transform and associated uncertainty principles

The intersection of geometric methods and signal processing has led to significant advancements in various fields, including physics, engineering, and mathematics. In recent years, the biquaternion domain, a generalization of complex numbers with a geometric interpretation, has emerged as a promising tool for signal analysis. This paper introduces the Biquaternion Linear Canonical Stockwell Transform (BiQLCST), a novel fusion of advanced mathematical frameworks that offers enhanced signal analysis capabilities and unveils new uncertainty principles, bridging the gap between complex transformations and uncertainty analysis in high-dimensional spaces. First, we established the various fundamental properties, including linearity, shift, modulation, parity, orthogonality relation, reconstruction formula and Plancherel’s theorem. Heisenberg uncertainty principle associated with Biquaternion Linear Canonical Stockwell Transform (BiQLCST) is also established. Toward the end, some potential applications of the BiQLCST are presented.

Two twinning operators when imposing nonlinear modulation in short‐time Fourier transform

Two types of short‐time Fourier transforms involving respectively nonlinear modulation and nonlinear translation are designed in this note. Some important properties such as twin in time‐frequency domain and orthogonality relations are investigated. Moreover, Lieb type inequalities and uncertainty principle are established.

Fluorescence scattering under the Stokes–Mueller formalism determines the orientational distribution of fluorophores and the nature of optically active biological proteins

Abstract In the current paper, we represent intrinsic fluorescence anisotropies as four-dimensional normalized Stokes vectors defined by the maximum excitation and emission in the fluorescence process with respect to linear, linear-45 and circular polarizations of light. Depending upon the transition moments for absorption/excitation and emission of fluorophores, eight types of these Stokes vectors can be realized from the Mueller fluorescence matrix of the system. These Stokes vectors probe the orientational distribution of fluorophores and predict the nature of optically active biological proteins, whether laevorotatory or dextrorotatory. The orthogonality relation between the Stokes vectors corresponding to the excitation and emission processes of fluorescence connects the molecular ground and excited states of biological and non-biological systems.

Explicit A Posteriori Error Representation for Variational Problems and Application to TV-Minimization

AbstractIn this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix–Raviart and the Raviart–Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation –initially based on continuous arguments only– practicable from a numerical point of view. In addition, we provide a generalized Marini formula that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin–Osher–Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.

Waveguide finite element modelling for broadband vibration analysis of tyres including rotation and prestress

In the framework of tyre/road noise prediction, our aim is to introduce a waveguide finite element (WFE) method for broadband vibration analysis of rotating inflated structures of arbitrary cross-section. Approximating the geometry as perfectly circular, the finite element discretization is restricted to the two-dimensional cross-section, enabling fast and high-frequency computations. Tyres are particularly challenging to model due to various complicating effects, often approximated or neglected in WFE models : rotation, inflation, structural anisotropy, heterogenetity, damping among others. Our formulation uses Lagrangian perturbations of displacement in Eulerian coordinates, providing a concise way to express equilibrium equations in vibrating media initially moving, prestressed and subjected to fixed point forces (typically, contact excitations in rotating tyres). The method comprises three steps: computing the steady rolling state (including rotation, inflation and centrifugal force), determining the undamped vibration modes superimposed onto the steady state (solving an eigenvalue problem of quadratic type owing to Coriolis acceleration), and obtaining the forced response of the damped structure through specific orthogonality relationships (considering a specific form of damping). The WFE method is compared with a rotating shell analytical model from existing literature and is subsequently applied to a rotating tyre with heterogeneous cross-section up to 4kHz.

$\mathrm{BC}_{2}$-Type Multivariable Matrix Functions and Matrix Spherical Functions

Matrix spherical functions associated to the compact symmetric pair (\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(m))) , m\geq 2 , having a reduced root system of type \mathrm{BC}_{2} , are studied. We consider an irreducible K -representation (\pi,V) arising from the \mathrm{U}(2) -part of K , and the induced representation \mathrm{Ind}_{K}^{G} \pi splits multiplicity-free. The corresponding spherical functions, i.e. \Phi \colon G \to \mathrm{End}(V) satisfying \Phi(k_{1}gk_{2})=\pi(k_{1})\Phi(g)\pi(k_{2}) for all g\in G , k_{1},k_{2}\in K , are studied by examining certain leading terms which involve hypergeometric functions. This is done explicitly using the action of the radial part of the Casimir operator on these functions and their leading terms. To suitably grouped matrix spherical functions we associate two-variable matrix orthogonal polynomials giving a matrix analogue of Koornwinder’s 1970s two-variable orthogonal polynomials, which are Heckman–Opdam polynomials for \mathrm{BC}_{2} . In particular, we find explicit orthogonality relations with the matrix polynomials being eigenfunctions to an explicit second-order matrix partial differential operator. The scalar part of the matrix weight is less general than Koornwinder’s weight.

Electromagnetic wave scattering in plasma beam-driven waveguides under strong magnetic fields

This study analyzes the scattering of electromagnetic waves in a cold and uniform plasma-filled waveguide driven by an intense relativistic plasma beam under a strong magnetic field. The strong interaction of plasma with electromagnetic waves enables its potential use in different types of waveguides. The Helmholtz equation governs the boundary value problem, which is solved by incorporating the mode matching technique. Invoking the boundary and matching conditions and the derived orthogonality and dispersion relations in this scheme gives an exact solution to the scattering problem. The numerical results shed light on the occurrence of reflection and transmission and flow of power. The power flux is plotted against angular frequency and various duct configurations. The solution is completely validated through the benefit of analytical and numerical results. The investigation of this structure reveals not only its mathematical, but also its physical features.

Examining relations between sea-level anomalies in the Niño 3.4 region and terrestrial hydroclimatic conditions in China

As a key driver of global climate variability, El Niño–Southern Oscillation (ENSO) significantly impacts the regional hydrological cycle. Similar to the recognized ENSO monitoring index (i.e., sea surface temperature anomaly), a potential advancement in the usage of sea level anomaly (SLA) in the equatorial Pacific is expected to enhance our knowledge of ENSO and its influence on regional terrestrial hydroclimatic conditions (THCs). In this study, we employed wavelet coherence and multi-channel singular spectral analysis to investigate the ability of satellite altimetry-based SLA, and its two major components (i.e., steric SLA (SSL) and non-steric SLA (NSSL)) averaged over Niño 3.4 region, to identify ENSO-related signals. Subsequently, we explored their spatial association with THCs at interannual scales in China using a cross-correlation method. We found that the time series of sea surface temperature anomaly index (SSTaI), SLA index (SLAI), and SSL index (SSLI) displayed high consistency. SSTaI was found to present higher correlations to SLAI (0.91) and SSLI (0.88), compared to NSSL index (NSSLI; 0.40). Furthermore, the high (low) wavelet coherence and in-phase (orthogonal) relationship between SSTaI and SLAI/SSLI (NSSLI) was also revealed. This indicated that SLAI and SSLI may be good complementary indices to SSTaI in observing ENSO-related signals. Similar to the temporal association of SSTaI and SLAI/SSLI, their spatial correlation pattern with THCs also displayed consistency. The time lag characterizing the relationship between SSTaI/SLAI/SSLI and THCs was also consistent, despite some regional differences. In summary, our results indicated that the combined analyses of the time lag determined between SSTaI/SLAI/SSLI and THCs can eventually complement the time lag analysis based solely on SSTaI. This, in turn, provides a more comprehensive understanding of ENSO’s influence on THCs in China.

Point evaluation in Paley–Wiener spaces

We study the norm of point evaluation at the origin in the Paley–Wiener space PWp for 0 < p < ∞, i.e., we search for the smallest positive constant C, called Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document}, such that the inequality |f(0)|p≤C∥f∥pp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert f(0)\\vert ^{p}\\leq C \\Vert f\\Vert _{p}^{p}$$\\end{document} holds for every f in PWp. We present evidence and prove several results supporting the following monotonicity conjecture: The function p↦Cp/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\mapsto {\\mathscr{C}}_{p}/p$$\\end{document} is strictly decreasing on the half-line (0, ∞). Our main result implies that Cp<p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} < p/2$$\\end{document} for 2 < p < ∞, and we verify numerically that Cp>p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} > p/2$$\\end{document} for 1 ≤ p < 2. We also estimate the asymptotic behavior of Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as p → ∞ and as p → 0+. Our approach is based on expressing Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as the solution of an extremal problem. Extremal functions exist for all 0 < p < ∞; they are real entire functions with only real zeros, and the extremal functions are known to be unique for 1 ≤ p < ∞. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau–Pollak–Slepian operator of time-frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range 1 < p < ∞, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} and the extremal functions.

Investigating nearshore spatial and temporal trends in nutrient concentrations along an urban northern shoreline, Lake Ontario

The nearshore zone of the Laurentian Great Lakes is of significant ecological importance, providing critical biogeochemical processes supporting spatially diverse habitats for a variety of species. Extreme plant and algal growth are common due to excessive anthropogenic eutrophication, with exacerbations from dreissenid establishment and lake mixing (e.g., coastal upwelling events and nearshore-offshore water exchanges), resulting in nuisance algal blooms across the nearshore area adjacent to Western Durham, Ontario, Canada. Thus, our main goal was to characterize the trends in essential nutrients (i.e., phosphorus and nitrogen) within the nearshore by applying general additive models on irregular time-series from 2011 to 2022 and to identify plausible contributing factors using exploratory principal component analysis. Increasing trends for total phosphorus were observed in surface (0.5 m below the surface) and benthic (0.5 m above the bottom) waters despite decreases in point source loading to the area. Most notably, the local water pollution control plant (WPCP) outfall did not seem to drive lake phosphorus concentrations across the study area, depicted by an orthogonal relationship within the principal component analysis. While the WPCP is a point source to the nearshore, it does not appear to be the primary driver of temporal lake phosphorus trends. Our results suggest that weather, inertial forcings and lake hydrodynamics in combination with traditional point-sources across Western Durham are contributing to the increasing total phosphorus trends observed as the peak period of wave spectra, wave height, alongshore wind speeds, watershed loadings, and total phosphorus concentrations within Lake Ontario were positively associated.

Macdonald polynomials for super-partitions

We discuss generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables: usual pk variables are accompanied by anti-commuting Grassmann variables θk. Starting from recently defined super-Schur polynomials and exploiting orthogonality relations with triangular decompositions we are able to fully determine super-Macdonald polynomials. These polynomials have similar properties to canonical Macdonald polynomials – they respect two different orderings in the set of (super)-Young diagrams simultaneously.

Consequences of the μ−τ reflection symmetry for leptogenesis in a seesaw model with diagonal Dirac neutrino mass matrix

In this paper, we have studied the consequences of the μ−τ reflection symmetry for leptogenesis in the type-I seesaw model with diagonal Dirac neutrino mass matrix. We have first obtained the phenomenologically allowed values of the model parameters, which show that there may exist zero or equal entries in the Majorana mass matrix for the right-handed neutrinos, and then studied their predictions for three right-handed neutrino masses, which show that there may exist two nearly degenerate right-handed neutrinos. Then, we have studied the consequences of the model for leptogenesis. Due to the μ−τ reflection symmetry, leptogenesis can only work in the two-flavor regime. Furthermore, leptogenesis cannot work for the particular case of r=1. Accordingly, for some benchmark values of r≠1, we have given the constraints of leptogenesis for relevant parameters. Furthermore, we have investigated the possibilities of leptogenesis being induced by the renormalization group evolution effects for two particular scenarios. For the particular case of r=1, the renormalization group evolution effects will break the orthogonality relations among different columns of MD and consequently induce leptogenesis to work. For the low-scale resonant leptogenesis scenario which is realized for nearly degenerate right-handed neutrinos, the renormalization group evolution effects can break the μ−τ reflection symmetry and consequently induce leptogenesis to work. Published by the American Physical Society 2024

RVE determination and developement of an anisotropic elastic model for auxetic sheet metal

This article deals with the development of an elastic tetragonal model for the 2D auxetic rotating units structures in the framework of orthogonal transformations. The existing anisotropy in the structure was first determined by numerical simulations on the samples with different pattern orientation angles. A suitable representative volume element (RVE), which correctly represents the mechanical properties of the whole structure both in macroscale and in microscale, was then proposed by implementation of the kinematic periodic boundary conditions. In the next step, with the help of the orthogonal transformations relations, an anisotropic elastic model was developed, which correctly reflects the present tetragonal symmetry in the structure. Finally, the model parameters were identified and validated with the help of the corresponding experiments

Optimizing the noise control in a two-layer conduit

In the modern world, noise pollution is a major concern due to its prevalence. This work focuses on optimizing noise control in a two-layer conduit. A conduit comprises an inlet and double-outlet zones (DOZ). The upper walls of the DOZ are lined with perforated absorbent material while the lower walls are layered with fibrous. Mathematically, the physical problem is formulated with a field wave equation together with rigid and impedance boundary conditions in the respective zones. Such governing boundary value problem (BVP) leads to the Sturm-Liouville (SL) category in which standard orthogonality relations (OR) are indispensable. The system of the linear equations of the BVP is acquired with semi-analytical Mode-Matching (MM) approach by implementing the continuity conditions of sound pressures and velocities at the matching junction with the aid of OR. These systems are truncated and solved numerically with computation learning to obtain the reflected and transmitted modal amplitudes in respective zones. Due to the lining's perforated upper walls, fibrous lower walls, and reversal in the DOZ, the analysis of the reflected and transmitted powers versus frequencies is significantly observed and is shown in graphical findings. The algebraic derivation is validated by satisfying the conservation law of power flux and matching continuity conditions for impedance, perforated, and fibrous lined boundaries.

Multivariate Meixner polynomials related to holomorphic discrete series representations of [formula omitted]

We show that Griffiths’ multivariate Meixner polynomials occur as matrix coefficients of holomorphic discrete series representations of the group SU(1,d). Using this interpretation we derive several fundamental properties of the multivariate Meixner polynomials, such as orthogonality relations and difference equations. Furthermore, we also show that matrix coefficients for specific group elements lead to degenerate versions of the multivariate Meixner polynomials and their properties.

A novel wavelet transform in the quaternion quadratic-phase domain

In this paper, we propose a novel integral transform coined as quaternion quadratic-phase wavelet transform (QQPWLT) by invoking the elegant convolution structure associated with the quaternion quadratic-phase Fourier transform. First, we explore some mathematical properties of the QQPWLT, including the orthogonality relation, inversion formula, reproducing kernel and some notable inequalities. Second, we study Heisenberg’s uncertainty principles and the logarithmic uncertainty principle associated with the quadratic-phase wavelet transform in quaternion domain. We culminate our investigation by presenting some illustrative examples.

PROJECTIVE CHARACTER VALUES ON REAL AND RATIONAL ELEMENTS

Abstract Let $\alpha $ be a complex-valued $2$ -cocycle of a finite group G with $\alpha $ chosen so that the $\alpha $ -characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also $\alpha $ -regular are characterised by the values that the irreducible $\alpha $ -characters of G take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of $G;$ however, the initial choice of $\alpha $ from its cohomology class is not unique in general and it is shown the results can vary for a different choice.

Uncertainty principles for windowed coupled fractional Fourier transform

The windowed coupled fractional Fourier transform was recently proposed in the literature. It may be considered as a generalized version of the windowed fractional Fourier transform. In this study, we first present various basic properties of the windowed coupled fractional Fourier transform including linearity, shifting, modulation, parity, orthogonality relation, and inversion formula. Further, the close relation of the windowed coupled fractional Fourier transform with the two‐dimensional Fourier transform and the windowed fractional Fourier transform is studied. By combining the properties and relation, we derive several versions of the uncertainty inequalities related to the windowed coupled fractional Fourier transform.

Linear canonical Fourier–Bessel wavelet transform: properties and inequalities

The purpose of this paper is to introduce and study the linear canonical Fourier–Bessel wavelet transform. We prove an orthogonality relation, inversion formula and some inequality for linear canonical Fourier–Bessel wavelet transform. We first present a direct relationship between the linear canonical Bessel wavelet transform and ordinary Bessel wavelet transform. Based on this relation, we provide an alternative proof of the orthogonality relation for the linear canonical Bessel wavelet transform. Some of its essential properties are also studied in detail. Finally, we explicitly derive several versions of inequalities associated with the linear canonical Bessel wavelet transform.