Orthogonal Operators Research Articles

Adaptive Sampling for Signals Associated with the Special Affine Fourier Transform

Adaptive sampling is inspired by the working principle of biological neurons, which converts the amplitude information of the received signal into a time sequence through a time encoding machine (TEM). The article mainly studies uniform sampling and adaptive sampling of non-bandlimited signals in two kinds of function spaces associated with the special affine Fourier transform (SAFT). Firstly, based on the stability characterization of the basis functions, we prove the existence of orthogonal projection operators in function spaces associated with the canonical convolution and the SAFT-convolution, respectively. Secondly, uniform sampling theorems are established for two kinds of signals living in the proposed function spaces. Finally, adaptive sampling schemes based on the crossing TEM and the integrate-and-fire TEM are studied. Moreover, the corresponding iterative reconstruction algorithms with exponential convergence are provided.

Development and application of a power Bonferroni mean operator based on the foreign fiber content grade evaluation method

Cubic q-rung orthogonal fuzzy sets (C q-ROFSs) are a sophisticated mathematical tool used to handle complex evaluation information in multi-attribute decision-making problems. In specific decision-making problems, the power Bonferroni mean (PBM) operator can reflect the correlation between different attributes and mitigate the impact of extreme evaluation information, thereby providing more practical value. This paper focuses on expanding the PBM operator into the C q-ROFS environment and deriving new PBM operators: the cubic q-rung orthogonal power Bonferroni averaging operator and weight cubic q-rung orthogonal PBM operator. The proposed operator shows strong flexibility and stability in the cubic q-rung orthogonal fuzzy environment. In the absence of weight information, there is a dearth of literature addressing the acceptable advantage and decision stability in the C q-ROFS environment; considering the regret behavior of decision information, a VIKOR method based on regret theory is proposed. The proposed method aggregates information using the proposed operator, determines the scheme and weights at two levels of attributes, and constructs a relative proximity decision matrix. Then, the VIKOR method calculates the group utility value and individual regret value based on the regret perception value to rank the alternatives. Finally, the method is applied to evaluate the cotton foreign fiber content, and its stability and effectiveness are verified through sensitivity analysis and comparison with existing methods.

Orthogonal operators: extension to hyperfine structure and equivalent p- and f-electrons

Orthogonal operators: extension to hyperfine structure and equivalent p- and f-electrons

Application of Clifford Algebra on Group Theory

The orthogonal operators defined as similarity transformations on Euclidean space E can also be considered as group actions on the Clifford Algebra. In this paper, we investigate the finite subgroup of Euclidian space E of Geometric Algebra over a finite dimension vector space E. The hierarchy of the finite subgroups of Clifford Algebra C(E) is depicted through the lattice structure and we discussed the group action of these subgroups on the vector space E. Further, we shall address the number of non-trivial finite subgroups, Normal subgroups, and subnormal series of the subgroup of Clifford Algebra C(E) constructed over the vector space E by performing group action over the subgroup of Clifford Algebra C(E).

Wavelengths and Energy Levels of the Upper Levels of Singly Ionized Nickel (Ni ii) from 3d 8(3 F)5f to 3d 8(3 F)9s

Using high-resolution spectra of Ni ii recorded using Fourier transform (FT) spectroscopy of continuous, nickel–helium hollow cathode discharge sources in the region 143–5555 nm (1800–70,000 cm−1, the analysis of 1016 Ni ii lines confirmed and optimized 206 previously reported energy levels of the (3 F) parent term, from 3d 8(3 F)5f to 3d 8(3 F)9s, lying between 122,060 and 138,563 cm−1. The uncertainties of these levels have been improved by at least an order of magnitude compared with their previously reported values. With the increased resolution and spectral range of the FT measurements, compared to previously published grating spectra, we were able to extend our analysis to identify and establish 33 new energy levels of Ni ii, which are reported here for the first time. Eigenvector compositions of all revised and newly established energy levels were calculated using the orthogonal operator method. In addition, an improved ionization energy of 146,541.35 ± 0.15 cm−1 for Ni ii, using highly excited levels of the 3d 8(3 F)5g, 3d 8(3 F)6g, and 3d 8(3 F)6h configurations, has been derived.

Orthogonal Polynomials and Operator Convergence in Hilbert Spaces: Norm-Attainability, Uniform Boundedness, and Compactness

This research paper investigates the convergence properties of operators constructed from orthogonal polynomials in the context of Hilbert spaces. The study establishes norm-attainability and explores the uniform boundedness of these operators, extending the analysis to include complex-valued orthogonal polynomials. Additionally, the paper uncovers connections between operator compactness and the convergence behaviors of orthogonal polynomial operators, revealing how sequences of these operators converge weakly to both identity and zero operators. These results advance our understanding of the intricate interplay betweenalgebraic and analytical properties in Hilbert spaces, contributing to fields such as functional analysis and approximation theory. The research sheds new light on the fundamental connections underlying the behavior of operators defined by orthogonal polynomials in diverse Hilbert space settings.

A Hilbertian projection method for constrained level set-based topology optimisation

We present an extension of the projection method proposed by Challis et al. (Int J Solids Struct 45(14–15):4130–4146, 2008) for constrained level set-based topology optimisation that harnesses the Hilbertian velocity extension-regularisation framework. Our Hilbertian projection method chooses a normal velocity for the level set function as a linear combination of (1) an orthogonal projection operator applied to the extended optimisation objective shape sensitivity and (2) a weighted sum of orthogonal basis functions for the extended constraint shape sensitivities. This combination aims for the best possible first-order improvement of the optimisation objective in addition to first-order improvement of the constraints. Our formulation utilising basis orthogonalisation naturally handles linearly dependent constraint shape sensitivities. Furthermore, use of the Hilbertian extension-regularisation framework ensures that the resulting normal velocity is extended away from the boundary and enriched with additional regularity. Our approach is generally applicable to any topology optimisation problem to be solved in the level set framework. We consider several benchmark constrained microstructure optimisation problems and demonstrate that our method is effective with little-to-no parameter tuning. We also find that our method performs well when compared to a Hilbertian sequential linear programming method.

Blur Invariants for Image Recognition

Blur is an image degradation that makes object recognition challenging. Restoration approaches solve this problem via image deblurring, deep learning methods rely on the augmentation of training sets. Invariants with respect to blur offer an alternative way of describing and recognising blurred images without any deblurring and data augmentation. In this paper, we present an original theory of blur invariants. Unlike all previous attempts, the new theory requires no prior knowledge of the blur type. The invariants are constructed in the Fourier domain by means of orthogonal projection operators and moment expansion is used for efficient and stable computation. Applying a general substitution rule, combined invariants to blur and spatial transformations are easy to construct and use. Experimental comparison to Convolutional Neural Networks shows the advantages of the proposed theory.

New Ritz wavelengths and transition probabilities for parity-forbidden, singly ionized nickel [Ni ii] lines of astrophysical interest

ABSTRACT We report accurate Ritz wavelengths for parity-forbidden [Ni ii] transitions, derived from energy levels determined using high-resolution Fourier transform spectroscopy. Transitions between the 18 lowest Ni ii energy levels of even-parity produced Ritz wavelengths for 126 parity-forbidden lines. Uncertainties for the Ritz wavelengths derived in this work are up to two orders of magnitude lower than previously published values. Transition probabilities were calculated using the semi-empirical orthogonal operator method, with uncertainties ranging from approximately 1 per cent for strong M1 lines and up to 10 per cent for weak E2 lines. Accurate forbidden line wavelengths and transition probabilities, particularly for lines in the infrared, are important in the analyses of low-density astrophysical plasmas, such as supernova remnants, planetary nebulae, and active galactic nuclei.

Electromagnetic Spectrum Allocation Method for Multi-Service Irregular Frequency-Using Devices in the Space–Air–Ground Integrated Network

The management and allocation of electromagnetic spectrum resources is the inner driving force of the construction of the space-air-ground integrated network. Existing spectrum allocation methods are difficult to adapt to the scenario where the working bandwidth of multi-service frequency-using devices is irregular and the working priorities are different. In this paper, an orthogonal genetic algorithm based on the idea of mixed niches is proposed to transform the problem of frequency allocation into the optimization problem of minimizing the electromagnetic interference between frequency-using devices in the integrated network. At the same time, a system model is constructed that takes the minimum interference effect of low-priority-to-high-priority devices as the objective function and takes the protection frequency and natural frequency as the constraint conditions. In this paper, we not only introduce the thought of niches to improve the diversity of the population but also use an orthogonal uniform crossover operator to improve the search efficiency. At the same time, we use a standard genetic algorithm and a micro genetic algorithm to optimize the model. The global searchability and local search precision of the proposed algorithm are all improved. Simulation results show that compared with the existing methods, the proposed algorithm has the advantages of fast convergence, strong stability and good optimization effect.

SYMPLECTIC DIRAC OPERATORS FOR LIE ALGEBRAS AND GRADED HECKE ALGEBRAS

The aim of this paper is to define a pair of symplectic Dirac operators (D+, D–) in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of ℤ/2-graded quadratic Lie algebras 𝔤 = 𝔨 + 𝔭 and of graded affine Hecke algebras ℍ. In these contexts, we show analogues of the Parthasarathy’s formula for [D+, D–] and certain generalisations of the Casimir inequality.

Wavelengths and Energy Levels of Singly Ionized Nickel (Ni ii) Measured Using Fourier Transform Spectroscopy

High-resolution spectra of singly ionized nickel (Ni ii) have been recorded using Fourier transform spectroscopy in the region 143–5555 nm (1800–70,000 cm−1) with continuous, nickel–helium hollow cathode discharge sources. An extensive analysis of identified Ni ii lines resulted in the confirmation and revision of 283 previously reported energy levels, from the ground state up to the 3d 8( M L)6s subconfigurations. Typical energy-level uncertainties are a few thousandths of a cm−1, representing at least an order-of-magnitude reduction in uncertainty with respect to previous measurements. Twenty-five new energy levels have now been established and are reported here for the first time. Eigenvector compositions of the energy levels have been calculated using the orthogonal operator method. In total, 159 even and 149 odd energy levels and 1424 classified line wavelengths of Ni ii are reported and will enable more accurate and reliable analyses of Ni ii in astrophysical spectra.

The structure of almost Abelian Lie algebras

An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media — cosmology, crystallography, etc. In differential geometry and theoretical physics, almost Abelian Lie groups have given rise to some of the simplest solvmanifolds on which various geometric structures such as symplectic, Kähler, spin, etc., are currently studied in explicit terms. However, a systematic study of almost Abelian Lie groups and algebras from mathematics perspective has not been carried out yet, and this paper is the first step in addressing this wide and diverse class of groups and algebras. This paper studies the structure and important algebraic properties of almost Abelian Lie algebras of arbitrary dimension over any field of scalars. A classification of almost Abelian Lie algebras is given. All Lie subalgebras and ideals, automorphisms and derivations, Lie orthogonal operators and quadratic Casimir elements are described exactly.

Fusion of implementers for spinors on the circle

We consider the space of odd spinors on the circle, and a decomposition into spinors supported on either the top or on the bottom half of the circle. If an operator preserves this decomposition, and acts on the bottom half in the same way as a second operator acts on the top half, then the fusion of both operators is a third operator acting on the top half like the first, and on the bottom half like the second. Fusion restricts to the Banach-Lie group of restricted orthogonal operators, which supports a central extension of implementers on a Fock space. In this article, we construct a lift of fusion to this central extension. Our construction uses Tomita-Takesaki theory for the Clifford-von Neumann algebras of the decomposed space of spinors. Our motivation is to obtain an operator-algebraic model for the basic central extension of the loop group of the spin group, on which the fusion of implementers induces a fusion product in the sense considered in the context of transgression and string geometry. In upcoming work we will use this model to construct a fusion product on a spinor bundle on the loop space of a string manifold, completing a construction proposed by Stolz and Teichner.

Special Congruences of Symmetric and Hermitian Matrices and Their Invariants

Let Vn be the arithmetic space of dimension n and let the inner product be introduced in Vn using a symmetric or a skew-symmetric involution M. In the resulting indefinite metric space, one can define the classes of special matrices playing the parts of symmetric, skew-symmetric, and orthogonal operators. Such matrices are said to be M-symmetric, M-skew-symmetric, and Morthogonal, respectively. Invariants of M-orthogonal congruences performed with M-symmetric and M-skew-symmetric matrices are indicated. The Hermitian counterpart of these constructions is also discussed.

Spectral approximation based on a mixed scheme and its error estimates for transmission eigenvalue problems

In this paper, we develop and analyze an efficient spectral method for the transmission eigenvalue problem. By rewriting the original problem into its equivalent fourth order coupled linear eigenvalue problem, a new variational formulation is proposed based on a mixed scheme. With the help of the spectral theory of the compact operator and the approximation property of the orthogonal projection operators in non-uniform weighted Soblev spaces, error estimates for both the eigenvalues and eigenfunctions approximations are established in the d(d=2,3) dimensional setting. Numerical examples are presented to show that the method converges exponentially, and is effective and efficient for computing the eigenvalues.

Core–EP orthogonal operators

The concept of the core orthogonality (CO) was recently introduced for two square matrices A and B of the same size and of index at most 1 as follows: A is core orthogonal to B if and , where is the core inverse of A. Using the core–EP inverse instead of the core inverse, we extend the concept of the CO and present the new concept of the core–EP orthogonality (CEPO) for two Drazin invertible bounded linear Hilbert space operators A and B as follows: A is said to be core–EP orthogonal to B if and , where is the core–EP inverse of A. A number of characterizations for CEPO are presented as well as the relation between the CEPO and core–EP additivity. Applying CEPO, the concept of strong core–EP orthogonality is defined and characterized.

Inframonogenic decomposition of higher‐order Lipschitz functions

Euclidean Clifford analysis has become a well‐established theory of monogenic functions in higher‐dimensional Euclidean space with a variety of applications both inside and outside of mathematics. Noncommutativity of the geometric product in Clifford algebras leads to what are now known as inframonogenic functions, which are characterized by certain elliptic system associated to the orthogonal Dirac operator in . The main question we shall be concerned with is whether or not a higher‐order Lipschitz function on the boundary Γ of a Jordan domain can be decomposed into a sum of the two boundary values of a sectionally inframonogenic function with jump across Γ. To this end, a kind of Cauchy‐type integral and singular integral operator, very specific to the inframonogenic setting, are widely used.

Композиции независимых случайных операторов и связанные с ними дифференциальные уравнения

Iterations of independent random linear operators in the Hilbert space of square integrable functions on a finite dimensional Euclidean space are studied. Random operator under consideration take values in the algebra of operators which is generated by an operators of a shift on a vector of Euclidean space of the argument of a function or the argument of its Fourier image, operators of orthogonal mapping and operators of contraction of argument space. We obtain the conditions sufficient to convergence of a sequence of mean values of compositions of operator valued processes with values in the considered algebra of linear operators to the semigroup describing the diffusion in finite dimensional Euclidean space. Generators of limit semigroups are described.

An efficient spectral-Galerkin method for a new Steklov eigenvalue problem in inverse scattering

<abstract><p>An efficient spectral method is proposed for a new Steklov eigenvalue problem in inverse scattering. Firstly, we establish the weak form and the associated discrete scheme by introducing an appropriate Sobolev space and a corresponding approximation space. Then, according to the Fredholm Alternative, the corresponding operator forms of weak formulation and discrete formulation are derived. After that, the error estimates of approximated eigenvalues and eigenfunctions are proved by using the spectral approximation results of completely continuous operators and the approximation properties of orthogonal projection operators. We also construct an appropriate set of basis functions in the approximation space and derive the matrix form of the discrete scheme based on the tensor product. In addition, we extend the algorithm to the circular domain. Finally, we present plenty of numerical experiments and compare them with some existing numerical methods, which validate that our algorithm is effective and high accuracy.</p></abstract>