Orthogonality Relations Research Articles

A density property for tensor products of gradients of harmonic functions and applications

We show that linear combinations of tensor products of k gradients of harmonic functions, with k at least three, are dense in C(Ω‾), for any bounded domain Ω in dimension 3 or higher. The bulk of the argument consists in showing that any smooth compactly supported k-tensor that is L2-orthogonal to all such products must be zero. This is done by using a Gaussian quasi-mode based construction of harmonic functions in the orthogonality relation. We then demonstrate the usefulness of this result by using it to prove uniqueness in the inverse boundary value problem for a coupled quasilinear elliptic system. The paper ends with a discussion of the corresponding property for products of two gradients of harmonic functions, and the connection of this property with the linearized anisotropic Calderón problem.

Fourier Transform of the Orthogonal Polynomials on the Unit Ball and Continuous Hahn Polynomials

Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval’s identity, very recently, some examples of orthogonal systems of this type have been introduced and orthogonality relations have been discussed. In the present paper, this method is applied for multivariate orthogonal polynomials on the unit ball. The Fourier transform of these orthogonal polynomials on the unit ball is obtained. By Parseval’s identity, a new family of multivariate orthogonal functions is introduced. The results are expressed in terms of the continuous Hahn polynomials.

Quadratic-Phase Wave-Packet Transform in L2(R)

Wavelet transform is a powerful tool for analysing the problems arising in harmonic analysis, signal and image processing, sampling, filtering, and so on. However, they seem to be inadequate for representing those signals whose energy is not well concentrated in the frequency domain. In pursuit of representations of such signals, we propose a novel time-frequency transform coined as quadratic-phase wave packet transform in L2(R). The proposed transform is aimed at rectifying the conventional wavelet transform by employing a quadratic-phase Fourier transform with extra degrees of freedom. Besides the formulation of all the fundamental results, including the orthogonality relation, reconstruction formula and the characterization of range, we also derive a direct relationship between the well-known Wigner-Ville distribution and the proposed transform. In addition, we study the quadratic-phase wave-packet transform in the framework of almost periodic functions. Finally, we extend the scope of the present work by investigating the composition of quadratic-phase wave packet transforms.

Certain New Development to the Orthogonal Binary Relations

In this study, inspired by the concept of B-metric-like space (BMLS), we introduce the concept of orthogonal B-metric-like space (OBMLS) via a hybrid pair of operators. Additionally, we establish the concept of orthogonal dynamic system (ODS) as a generalization of the dynamic system (DS), which improves the existing results for analysies such as those presented here. By applying this, some new refinements of the F⊥-Suzuki-type (F⊥-ST) fixed-point results are presented. These include some tangible instances, and applications in the field of nonlinear analysis are given to highlight the usability and validity of the theoretical results.

Vector Bessel beams: General classification and scattering simulations

Apart from a lot of fundamental interest, vector Bessel beams are widely used in optical manipulation, material processing, and imaging. However, the existing description of such beams remains fragmentary, especially when their scattering by small particles is considered. We propose a new general classification of all existing vortex Bessel beam types in an isotropic medium based on the superposition of transverse Hertz vector potentials. This theoretical framework contains duality and coordinate rotations as elementary matrix operations and naturally describes all relations between various beam types. This leads to various bases for Bessel beams and uncovers the novel beam type with circularly symmetric energy density. We also discuss quadratic functionals of the fields (such as the energy density and Poynting vector) and derive orthogonality relations between various beam types. Altogether, it provides a comprehensive reference of all properties of Bessel beams that may be relevant for applications. Next we generalize the formalism of the Mueller scattering matrices to arbitrary Bessel beams accounting for their vorticity. Finally, we implement these beams in the ADDA code - an open-source parallel implementation of the discrete dipole approximation. This enables easy and efficient simulation of Bessel-beams scattering by particles with arbitrary shape and internal structure.

Field-Free Type- x Spin-Orbit-Torque Switching by Easy-Axis Engineering

The current-induced type-x spin-orbit-torque (SOT) switching configuration describes the orthogonal relationship between the magnetic easy axis (EA) of a ferromagnetic (FM) layer and the injected spin polarization, $\ensuremath{\sigma}$, from a heavy-metal (HM) layer, which has the potential to eclipse the conventional type-y scenario (EA parallel to $\ensuremath{\sigma}$) at the sub-nanosecond pulse regime. Here, we show that, in HM/FM bilayer heterostructures, current-driven differential planar Hall signals can serve as efficient means to probe type-x SOT switching in a simple measurement fashion. Through this approach, we demonstrate field-free type-x SOT switching in all devices with a canted EA, which are engineered via field-annealing processes. By analyzing the switching-phase diagrams, we further verify that such field-free switching stems from a z-direction effective field contributed to by both the dampinglike torque and the canted EA. Our work indicates that the canted EA, as engineered by a field-annealing process, can give rise to a robust field-free SOT switching and further reduce the critical switching current density.

Dirac cohomology and orthogonality relations for weight modules

Let $\mathfrak{g}$ be a reductive Lie algebra over $\mathbb{C}$. For any simple weight module of $\mathfrak{g}$ with finite-dimensional weight spaces, we show that its Dirac cohomology is vanished unless it is a highest weight module. This completes the calculation of Dirac cohomology for simple weight modules since the Dirac cohomology of simple highest weight modules was carried out in our previous work. We also show that the Dirac index pairing of two weight modules which have infinitesimal characters agrees with their Euler-Poincar\'{e} pairing. The analogue of this result for Harish-Chandra modules is a consequence of the Kazhdan's orthogonality conjecture which was settled by the first named author and Binyong Sun.

Quaternion linear canonicalS-transform and associated uncertainty principles

In this paper, we introduce the notion of quaternion linear canonical S-transform (Q-LCST) which is an extension of the linear canonical S-transform and study the uncertainty principles associated with the Q-LCST. First, we propose the definition of Q-LCST and then study the fundamental properties of [Formula: see text]-LCST including linearity, shift, modulation, orthogonality relation and reconstruction formula. Second, we derive the associated Heisenbergs uncertainty inequality and the corresponding logarithmic version for [Formula: see text]-LCST. Finally, some potential applications of the Q-LCST are introduced.

Energy distribution of bifurcated waveguide structure with planar and non-planar surfaces

The present communication addresses the reflection and transmission analysis of a two-dimensional waveguide structure with planar and non-planar surfaces. In the fields of physics, engineering and applied mathematics, research is being carried out to control or reduce unwanted and harmful noise to an acceptable level. Our aim is to solve the boundary value problem developed to simulate the transmission and reflection of a bifurcated waveguide structure in the channels or ducts that have abrupt changes in the height together with planar and non-planar surfaces. Furthermore, the eigenfunctions are linearly independent and non-orthogonal. To solve non-orthogonal eigenfunctions, a generalized orthogonality relation has been developed. Solution is obtained using mode matching technique (MMT) along with generalized orthogonal relations. In the last section, solution is viewed graphically in the inner and outer ducts or channels for different frequencies and height. The expressions for energy distribution in different duct regions are obtained and sketched subject to the various properties of the waveguide material. The results are well supported for a number of mathematical and physical reasons.

Wave trapping due to composite pile-rock structure coupled with vertical barrier

The wave transformation due to pile-rock porous structure in combination with vertical porous barrier is studied under oblique wave action. The pile-rock breakwaters consists of two rows of closely spaced piles and a rock core between them is effective in dissipating wave energy when compared with traditional rigid breakwaters due to its reduced deadweight of construction materials and additional stability. Three different cases of the vertical barrier configurations such as fully-extended barrier, bottom-standing barrier and surface-piercing barrier placed in front of the pile-rock porous structure are considered for the investigation. The numerical study is performed using the eigenfunction expansion and the associated orthogonal mode-coupling relations considering the continuity of pressure and velocity for the vertical barrier, seaward and leeward structural interfaces. The Darcy’s law is incorporated for the flow through porous media and the porosity factor of the structure is introduced using the complex porous effect parameter. The numerical results for the wave reflection, transmission and dissipation coefficient, wave force on front and rear side of porous structure along with the wave force on the barrier interface are evaluated for different hydraulic characteristics. The analysis is presented for varying structural porosity, angle of incidence, structural thickness, friction factor, length between vertical barrier and porous structure for the three different cconfigurations of vertical barrier. The numerical investigation performed in the present study will be useful for the design and analysis of the composite breakwater system to protect the offshore facility from high waves.

Elliptic Racah polynomials

Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic Racah polynomials. The well-known $q$-Racah polynomials of Askey and Wilson are recovered as a trigonometric limit.

Octonion short‐time linear canonical transform

We investigate the octonion short‐time linear canonical transform (OCSTLCT) in this paper. First, we propose the new definition of the OCSTLCT, and then several important properties of newly defined OCSTLCT, such as bounded, shift, modulation, time‐frequency shift, inversion formula, and orthogonality relation, are derived based on the spectral representation of the octonion linear canonical transform (OCLCT). Second, by the Heisenberg uncertainty principle for the OCLCT and the orthogonality relation property for the OCSTLCT, the Heisenberg uncertainty principle for the OCSTLCT is established. Finally, we give an example of the OCSTLCT.

Decoupling the electronic and geometric effects of Pt catalysts in selective hydrogenation reaction

Decoupling the electronic and geometric effects has been a long cherished goal for heterogeneous catalysis due to their tangled relationship. Here, a novel orthogonal decomposition method is firstly proposed to settle this issue in p-chloronitrobenzene hydrogenation reaction on size- and shape-controlled Pt nanoparticles (NPs) carried on various supports. Results suggest Fermi levels of catalysts can be modulated by supports with varied work function (Wf). And the selectivity on Pt NPs of similar size and shape is linearly related with the Wf of support. Optimized Fermi levels of the catalysts with large Wf weaken the ability of Pt NPs to fill valence electrons into the antibonding orbital of C–Cl bond, finally suppressing the hydrodehalogenation side reaction. Foremost, the geometric effect is firstly spun off through orthogonal relation based on series of linear relationships over various sizes of Pt NPs reflecting the electronic effect. Moreover, separable nested double coordinate system is established to quantitatively evaluate the two effects.

Degenerate r-Whitney numbers and degenerate r-Dowling polynomials via boson operators

Dowling showed that the Whitney numbers of the first kind and of the second kind satisfy Stirling number-like relations. Recently, Kim-Kim introduced the degenerate r-Whitney numbers of the first kind and of the second kind, as degenerate versions and further generalizations of the Whitney numbers of both kinds. The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. In this paper, it is noted that the normal ordering of a certain quantity involving the number operator is expressed in terms of the degenerate r-Whitney numbers of the second kind. We derive some properties, recurrence relations, orthogonality relations and several identities on those numbers from such normal ordering. In addition, we consider the degenerate r-Dowling polynomials as a natural extension of the degenerate r-Whitney numbers of the second kind and investigate their properties.

A Scalable Deep Network for Graph Clustering via Personalized PageRank

Recently, many models based on the combination of graph convolutional networks and deep learning have attracted extensive attention for their superior performance in graph clustering tasks. However, the existing models have the following limitations: (1) Existing models are limited by the calculation method of graph convolution, and their computational cost will increase exponentially as the graph scale grows. (2) Stacking too many convolutional layers causes the over-smoothing issue and neglects the local graph structure. (3) Expanding the range of the neighborhood and the model depth together is difficult due to the orthogonal relationship between them. Inspired by personalized pagerank and auto-encoder, we conduct the node-wise graph clustering task in the undirected simple graph as the research direction and propose a Scalable Deep Network (SDN) for graph clustering via personalized pagerank. Specifically, we utilize the combination of multi-layer perceptrons and linear propagation layer based on personalized pagerank as the backbone network (i.e., the Quasi-GNN module) and employ a DNN module for auto-encoder to learn different dimensions embeddings. After that, SDN combines the two embeddings correspondingly; then, it utilizes a dual self-supervised module to constrain the training of the embedding and clustering process. Our proposed Quasi-GNN module reduces the computational costs of traditional GNN models in a decoupled approach and solves the orthogonal relationship between the model depth and the neighborhood range. Meanwhile, it also alleviates the degraded clustering effect caused by the over-smoothing issue. We conducted experiments on five widely used graph datasets. The experimental results demonstrate that our model achieves state-of-the-art performance.

Case’s Method for Anlı-Güngör Scattering Formula

Case method is a powerful method in solving one-speed neutron transport equation. The method can be applied to one-speed neutron transport problems and pseudo- geometry problems. The method basis on the usage of Case’s eigenfunctions and the orthogonality relations with the certain boundary conditions according to the interested problem. The scattering effects can be investigated via Mika scattering formula and also İnönü’s scattering formula. In this study Case method’s formalism is derived by using Anlı-Güngör scattering formula as an analogue of Mika’s scattering function. This study is about the Case’s eigenfunctions, normalization relation and the orthogonality properties among these eigenfunctions and, moreover; Case’s eigenfunctions and the orthogonality properties must be rewritten according to the studied scattering order as the number of scattering order increase in Anlı-Güngör scattering formula.

Incomplete Metric With The Application For Integral Equations of Volterra Type

In this article, we propose some weaker orthogonal τ , F T − type of contraction mappings in the setting of metric spaces endowed with an orthogonal relation, as well as certain sufficient criteria for the existence of fixed points for this class of mappings. To establish all of our results in the manuscript, we just used Wardowski’s strictly increasing property. Using the aforementioned results as an application, we demonstrate that the Volterra type integral equation has a solution and is stable in the Hyers-Ulam-Rassias-Wright sense.

Generation of Ince-Gaussian Beams Using Azocarbazole Polymer CGH.

Ince–Gaussian beams, defined as a solution to a wave equation in elliptical coordinates, have shown great advantages in applications such as optical communication, optical trapping and optical computation. However, to ingress these applications, a compact and scalable method for generating these beams is required. Here, we present a simple method that satisfies the above requirement, and is capable of generating arbitrary Ince–Gaussian beams and their superposed states through a computer-generated hologram of size 1 mm2, fabricated on an azocarbazole polymer film. Other structural beams that can be derived from the Ince–Gaussian beam were also successfully generated by changing the elliptical parameters of the Ince–Gaussian beam. The orthogonality relations between different Ince–Gaussian modes were investigated in order to verify applicability in an optical communication regime. The complete python source code for computing the Ince–Gaussian beams and their holograms are also provided.

3D hydroelastic modelling of fluid–structure interactions of porous flexible structures

A 3D hydroelastic model subjected to linear wave interaction with horizontal flexible floating and submerged porous structures is developed. The dispersion relation of the aforesaid problem is derived based on Green’s functions approach in water of finite and infinite depths. Three-dimensional Fourier-type expansion formulae for the velocity potentials associated with orthogonal mode-coupling relations are derived in a channel of finite and infinite water depths of finite width. A real physical problem of surface gravity wave interaction with a moored floating and submerged porous plate in finite water depth is analysed by the expansion formula to validate the usefulness of the current method. The numerical accuracy of the present numerical results is checked via convergence test. Further, the effect of different design parameters on the hydroelastic response of floating and a submerged porous plate is studied by analysing the deflection amplitude and structural deformation. It is observed that the present analysis could be useful for a greater insight to design a multi-use flexible floating system.

Transitivity and homogeneity of orthosets and inner-product spaces over subfields of {{mathbb {R}}}

An orthoset (also called an orthogonality space) is a set X equipped with a symmetric and irreflexive binary relation perp , called the orthogonality relation. In quantum physics, orthosets play an elementary role. In particular, a Hilbert space gives rise to an orthoset in a canonical way and can be reconstructed from it. We investigate in this paper the question to which extent real Hilbert spaces can be characterised as orthosets possessing suitable types of symmetries. We establish that orthosets fulfilling a transitivity as well as a certain homogeneity property arise from (anisotropic) Hermitian spaces. Moreover, restricting considerations to divisible automorphisms, we narrow down the possibilities to positive definite quadratic spaces over an ordered field. We eventually show that, under the additional requirement that the action of these automorphisms is quasiprimitive, the scalar field embeds into {{mathbb {R}}}.