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It is very interesting and difficult to understand the papers of the great mathematician Prof. Harish-Chandra (1923-1983). While reading any one of them the reader is compelled to know the answers of many questions standing on the way. We have tried to understand his paper¹ and found that the Jacobi polynomial appears on the way of solution of wave equation of electron moving in the field of a magnetic pole. This Jacobi polynomial is not in the usual form appearing in mathematical literature. In this paper we have compared the Jacobi polynomial used by Harish-Chandra with its usual form. We have also deduced the explicit form for such polynomials from identity given in his paper¹ . We have also verified the results found by him concerning Jacobi polynomials. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians in with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains , , . Second, we extend Friedlander''s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains Ω, following an approach introduced by Filonov for this type of problems. [Copyright &y& Elsevier]

This paper presents the topological derivative of the first eigenvalue for the free vibration model of plane structures. We conduct a topological asymptotic analysis to account for perturbations in the domain caused by inserting a small inclusion. The paper includes a rigorous derivation of the topological derivative for the eigenvalue problem along with a proof of its existence. Additionally, we provide numerical examples that illustrate the application of the proposed methodology for maximizing the first eigenvalue in plane structures. The results demonstrate that multiple eigenvalues were not encountered. [ABSTRACT FROM AUTHOR]

The weighted fractional Fourier transform (WFRFT) has always been considered a development of the discrete fractional Fourier transform (FRFT). This paper points out that the WFRFT is a discrete FRFT of eigenvalue decomposition, which will change the consistent understanding of the WFRFT. Extended definitions based on the WFRFT have been proposed and widely used in information processing. This paper proposes a unified framework for extended definitions, and existing extended definitions can serve as special cases of this unified framework. In further analysis, we find that the existing extended definitions are deficient. With the help of a unified framework, we systematically analyze the reasons for the deficiencies. This has great guiding significance for the application of the WFRFT and its extended definitions. [ABSTRACT FROM AUTHOR]

This paper concerns with a Lyapunov-type inequality for the Riesz-Caputo fractional boundary value problem with anti-periodic boundary conditions. As an application for the obtained inequality, a lower bound for the eigenvalues of anti-periodic fractional boundary problems of the Riesz-Caputo derivative has been obtained. [ABSTRACT FROM AUTHOR]

In Ref. [1] , the κ -Dirac equation, based on the κ -deformed Poincaré-Hopf algebra, have been studied. In particular, solutions of the κ -Dirac oscillator (DO), in a three-dimensional space, were obtained by deriving the associated radial equations. We point out, however, a miscalculation in treating these equations, which had led to erroneous conclusions, particularly, about the energy eigenvalues and the breaking of their infinite degeneracy by the κ -deformation. By the way, we present a simple alternative method to solve the problem using an algebraic procedure. [ABSTRACT FROM AUTHOR]

In order to explore a non-destructive method for measuring the polymerization degree (DP) of paper insulation in transformer, a new method that based on the optical properties of free fiber particles in transformer oil was studied. The chromatic dispersion images of fibers with different aging degree were obtained by polarizing microscope, and the eigenvalues (r, b, and Mahalanobis distance) of the images were extracted by the RGB (red, blue, and green) tricolor analysis method. Then, the correlation between the three eigenvalues and DP of paper insulation were simulated respectively. The results showed that the color of images changed from blue-purple to orange-yellow gradually with the increase of aging degree. For the three eigenvalues, the relationship between Mahalanobis distance and DP had the best goodness of fit (R2 = 0.98), higher than that of r (0.94) and b (0.94). The mean square error of the relationship between Mahalanobis distance and DP (52.17) was also significantly lower than that of r and b (97.58, 98.05). Therefore, the DP of unknown paper insulation could be calculated by the simulated relationship of Mahalanobis distance and DP. [ABSTRACT FROM AUTHOR]

Heterogeneous information network (HIN) embedding aims at learning the low-dimensional representation of nodes while preserving structure and semantics in a HIN. Existing methods mainly focus on static networks, while a real HIN usually evolves over time with the addition (deletion) of multiple types of nodes and edges. Because even a tiny change can influence the whole structure and semantics, the conventional HIN embedding methods need to be retrained to get the updated embeddings, which is time-consuming and unrealistic. In this paper, we investigate the problem of dynamic HIN embedding and propose a novel Dynamic HIN Embedding model (DyHNE) with meta-path based proximity. Specifically, we introduce the meta-path based first- and second-order proximities to preserve structure and semantics in HINs. As the HIN evolves over time, we naturally capture changes with the perturbation of meta-path augmented adjacency matrices. Thereafter, we learn the node embeddings by solving generalized eigenvalue problem effectively and employ eigenvalue perturbation to derive the updated embeddings efficiently without retraining. Experiments show that DyHNE outperforms the state-of-the-arts in terms of effectiveness and efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we first present multiple numerical simulations of the anti-symmetric matrix in the stability criteria for fractional order systems (FOSs). Subsequently, this paper is devoted to the study of the admissibility criteria for descriptor fractional order systems (DFOSs) whose order belongs to (0, 2). The admissibility criteria are provided for DFOSs without eigenvalues on the boundary axes. In addition, a unified admissibility criterion for DFOSs involving the minimal linear matrix inequality (LMI) variable is provided. The results of this paper are all based on LMIs. Finally, numerical examples were provided to validate the accuracy and effectiveness of the conclusions. [ABSTRACT FROM AUTHOR]

This paper covers aspects of quantifying the symmetry of two- and three-dimensional elastic bar-and-joint structures. The concept of symmetry as a quantitative property instead of a binary question of 'yes' or 'no' is widely accepted and thoroughly investigated, for example, in molecular physics but also in engineering sciences, mainly in chemical engineering. Similarly to most of the articles written on this topic, our method is also based on the comparison of specific metrics of the analyzed structure and a reference one, i.e., which possesses the desired (perfect) symmetry. The deviation of the analyzed (imperfect) structure from the reference structure is quantified by one scalar. The novelty in our approach is that we consider not just the relative position of the nodes but also the normal stiffness of the truss members, even for structures with higher-order, i.e., polyhedral symmetries. For both geometric and material properties to be accounted for, the eigenvalues of the stiffness matrix were chosen as metrics. The difficulty lies in finding the reference structure which will be carried out based on energy principles. [ABSTRACT FROM AUTHOR]

Installing multi-camera systems and inertial measurement units (IMUs) in self-driving cars, micro aerial vehicles, and robots is becoming increasingly common. An IMU provides the vertical direction, allowing coordinate frames to be aligned in a common direction. The degrees of freedom (DOFs) of the rotation matrix are reduced from 3 to 1. In this paper, we propose a globally optimal solver to calculate the relative poses and scale of generalized cameras with a known vertical direction. First, the cost function is established to minimize algebraic error in the least-squares sense. Then, the cost function is transformed into two polynomials with only two unknowns. Finally, the eigenvalue method is used to solve the relative rotation angle. The performance of the proposed method is verified on both simulated and KITTI datasets. Experiments show that our method is more accurate than the existing state-of-the-art solver in estimating the relative pose and scale. Compared to the best method among the comparison methods, the method proposed in this paper reduces the rotation matrix error, translation vector error, and scale error by 53%, 67%, and 90%, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation numbers is proven. Here, we note that finding estimates of approximation numbers, as well as extremal subspaces, for a set of solutions to the equation is a task that is certainly important from both a theoretical and a practical point of view. The paper also obtained an upper bound for the eigenvalues. Note that, in this paper, estimates of eigenvalues and approximation numbers for the degenerate third-order partial differential operators are obtained for the first time. [ABSTRACT FROM AUTHOR]

Free vibration characteristics in uniform beams with several lumped attachments are an important problem in engineering applications that have to deal with mounting different equipment (e.g. motors, oscillators or engines) on a structural beam. In order to solve the lack of a generalized automatic procedure, this investigation presents a simple solving approach based on analytical means applied to a secular frequency equation for obtaining the natural frequencies of an arbitrarily supported single-span, or multi-span Euler–Bernoulli beam carrying any combination of miscellaneous attachments. The approach is obtained by solving a characteristic polynomial equation using a classical method for computing the roots of a polynomial. Interestingly, if the number of elements is greater than one, a pole-zero cancellation is needed, but it does not require manual interventions such as initial values and iteration. The mathematical approach is validated with bibliographic references and evaluated for accuracy and computational effectiveness. A good agreement is observed with relative error values practically negligible mostly ranging between 10−3 and 10−9 in the first five natural frequencies, which confirms the validity of the presented approach in this paper. The MatLab code that has been developed with the solving approach is freely available as a supplementary material to this paper. Additionally, a MatLab graphical user interface has also been developed in this work which allows to obtain the eigenvalues of a simply supported Euler–Bernoulli beam carrying an undetermined number of lumped elements. The graphical user interface is also available for download, along with help facilities to be run in a Windows operating system and detailed instructions to reproduce the case studies presented here. The proposed scheme (and also the MatLab graphical user interface) is very easy to code, and can be slightly modified to accommodate beams with arbitrary supports. [ABSTRACT FROM AUTHOR]

Contemporary quantum computers open up novel possibilities for tackling intricate problems, encompassing quantum system modeling and solving partial differential equations (PDEs). This paper explores the optimization of quantum algorithms aimed at resolving PDEs, presenting a significant challenge within the realm of computational science. The work delves into the application of the Variational Quantum Eigensolver (VQE) for addressing equations such as Poisson’s equation. It employs a Hamiltonian constructed using a modified Feynman-Kitaev formalism for a VQE, which represents a quantum system and encapsulates information pertaining to the classical system. By optimizing the parameters of the quantum circuit that implements this Hamiltonian, it becomes feasible to achieve minimization, which corresponds to the solution of the original classical system. The modification optimizes quantum circuits by minimizing the cost function associated with the VQE. The efficacy of this approach is demonstrated through the illustrative example of solving the Poisson equation. The prospects for its application to the integration of more generalized PDEs are discussed in detail. This study provides an in-depth analysis of the potential advantages of quantum algorithms in the domain of numerical solutions for the Poisson equation and emphasizes the significance of continued research in this direction. By leveraging quantum computing capabilities, the development of more efficient methodologies for solving these equations is possible, which could significantly transform current computational practices. The findings of this work underscore not only the practical advantages but also the transformative potential of quantum computing in addressing complex PDEs. Moreover, the results obtained highlight the critical need for ongoing research to refine these techniques and extend their applicability to a broader class of PDEs, ultimately paving the way for advancements in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

In this paper, a novel weighted block sparse method based on the signal subspace is proposed to realize the Direction-of-Arrival (DOA) estimation under unknown mutual coupling in the uniform linear array. Firstly, the signal subspace is obtained by decomposing the eigenvalues of the sampling covariance matrix. Then, a block sparse model is established based on the deformation of the product of the mutual coupling matrix and the steering vector. Secondly, a suitable set of weighted coefficients is calculated to enhance sparsity. Finally, the optimization problem is transformed into a second-order cone (SOC) problem and solved. Compared with other algorithms, the simulation results of this paper have better performance on DOA accuracy estimation. [ABSTRACT FROM AUTHOR]

In the field of radar target detection, the conventional approach is to employ the range profile energy accumulation method for detecting extended targets. However, this method becomes ineffective when dealing with non-stationary and non-uniform radar clutter scenarios, as well as long-distance targets with weak radar cross sections (RCSs). In such cases, the signal-to-noise ratio (SNR) of the target echo is severely degraded, rendering the energy accumulation detection algorithm unreliable. To address this issue, this paper presents a new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. This method utilizes a detection model that incorporates observed data and employs the likelihood ratio test (LRT) theory to derive the maximum eigenvalue detector at low SNR. Specifically, the detector constructs a matrix using a sliding window block with the available data and then computes the maximum eigenvalue of the covariance matrix. Subsequently, the maximum eigenvalue matrix is transformed into a one-dimensional eigenvalue image, enabling extended target detection through analogy with the energy accumulation detection method. Furthermore, this paper analyzes the proposed extended target detection method from both theoretical and experimental perspectives, validating it through field-measured data. The results obtained from the measured data demonstrate that the method effectively enhances the SNR in low SNR conditions, thereby improving target detection performance. Additionally, the method exhibits robustness across different scattering center targets. [ABSTRACT FROM AUTHOR]

This paper explores the effectiveness of different similarity measures for characterizing vertices and edges in rough graphs, which were introduced to handle imprecise and uncertain information. The authors examine traditional similarity measures like the Jaccard index, Dice coefficient, and overlap measure in this context. Additionally, a new ζ-labeling similarity measure for rough graphs is proposed. The main goal is to perform a comparative analysis evaluating the performance of these diverse similarity measures when applied to rough graphs. Furthermore, the paper computes the energy of rough graphs, defined as the sum of absolute eigenvalues, to demonstrate the superior potency of the proposed ζ-labeling measure compared to the other similarity measures considered. Overall, this work aims to advance techniques for assessing similarity in rough graphs, which have applications in dealing with vague and imprecise data. [ABSTRACT FROM AUTHOR]

In this paper, we give a simple sufficient condition for the eigenvalue-separation properties of real tridiagonal matrices T. This result is much more than the statement that the pertinent eigenvalues are distinct. Its derivation is based on recurrence formulae satisfied by the polynomials made up by the minors of the characteristic polynomial det(xE - T) that are proven to form a Sturm sequence. This is a new result, and it proves the simple spectrum property of a symmetric tridiagonal matrix studied in a Grünbaum paper. Two numerical examples underpin the theoretical findings. The style of the paper is expository in order to address a large readership. [ABSTRACT FROM AUTHOR]

In this paper, we shall study the spectral properties of the non-selfadjoint operator in the space ... generated by the Sturm-Liouville differential equation ... with the integral type boundary condition ... and the non-standard weight function ρ (x) = -1 where ... . There are an enormous number of papers considering the positive values of ρ (x) for both continuous and discontinuous cases. The structure of the weight function affects the analytical properties and representations of the solutions of the equation. Differently from the classical literature, we used the hyperbolic type representations of the fundamental solutions of the equation to obtain the spectrum of the operator. Moreover, the conditions for the finiteness of the eigenvalues and spectral singularities were presented. Hence, besides generalizing the recent results, Naimark's and Pavlov's conditions were adopted for the negative weight function case. [ABSTRACT FROM AUTHOR]

This communication is to correct the sensitivity analysis formula of frequencies for composite structures in the paper [1]. It is further pointed out that even though the sensitivity analysis formula of frequencies has been corrected, the first derivatives of frequencies with respect to design variables cannot be achieved by direct differentiating the equations of eigenvalues and eigenvectors because the first derivatives of vibration modes with respect to design variables are still unknown. The readers are referred to the analytical solution of the sensitivity analysis of frequencies and modes for composite structures in the literature [2], which is derived using linear space theory. [ABSTRACT FROM AUTHOR]

Ice-shelf forced vibration modelling is performed using a full 3-D finite-difference elastic model, which also takes into account sub-ice seawater flow. The ocean flow in the cavity is described by the wave equation; therefore, ice-shelf flexures result from hydrostatic pressure perturbations in sub-ice seawater layer. Numerical experiments have been carried out for idealized rectangular and trapezoidal ice-shelf geometries. The ice-plate vibrations are modelled for harmonic ingoing pressure perturbations and for high-frequency spectra of the ocean swells. The spectra show distinct resonance peaks, which demonstrate the ability to model a resonant-like motion in the suitable conditions of forcing. The spectra and ice-shelf deformations obtained by the developed full 3-D model are compared with the spectra and the deformations modelled by the thin-plate Holdsworth and Glynn model (1978). The main resonance peaks and ice-shelf deformations in the corresponding modes, derived by the full 3-D model, are in agreement with the peaks and deformations obtained by the Holdsworth and Glynn model. The relative deviation between the eigenvalues (periodicities) in the two compared models is about 10 %. In addition, the full model allows observation of 3-D effects, for instance, the vertical distribution of the stress components in the plate. In particular, the full model reveals an increase in shear stress, which is neglected in the thin-plate approximation, from the terminus towards the grounding zone with a maximum at the grounding line in the case of the considered high-frequency forcing. Thus, the high-frequency forcing can reinforce the tidal impact on the ice-shelf grounding zone causing an ice fracture therein. [ABSTRACT FROM AUTHOR]

This paper focused on the stability analysis of the Dirichlet-to-Neumann (DN) map for the fractional diffusion equation with a reaction coefficient q . The main result provided a Hölder-type stability estimate for the map, which was formulated in terms of the Dirichlet eigenvalues and normal derivatives of eigenfunctions of the operator A q := − Δ + q . This paper focused on the stability analysis of the Dirichlet-to-Neumann (DN) map for the fractional diffusion equation with a reaction coefficient . The main result provided a Hölder-type stability estimate for the map, which was formulated in terms of the Dirichlet eigenvalues and normal derivatives of eigenfunctions of the operator . [ABSTRACT FROM AUTHOR]

This paper focuses on the reversibility of multidimensional linear cellular automata with an intermediate boundary condition. We begin by addressing the matrix representation of these automata, and the question of reversibility boils down to the invertibility of this matrix representation. We introduce a decomposition method that factorizes the matrix representation into a Kronecker sum of significantly smaller matrices. The invertibility of the matrix hinges on determining whether zero can be expressed as the sum of eigenvalues of these smaller matrices, which happen to be tridiagonal Toeplitz matrices. Notably, each of these smaller matrices represents a one-dimensional cellular automaton. Leveraging the rich body of research on the eigenvalue problem of Toeplitz matrices, our result provides an efficient algorithm for addressing the reversibility problem. As an application, we show that there is no reversible nontrivial linear cellular automaton over Z 2 . This paper focuses on the reversibility of multidimensional linear cellular automata with an intermediate boundary condition. We begin by addressing the matrix representation of these automata, and the question of reversibility boils down to the invertibility of this matrix representation. We introduce a decomposition method that factorizes the matrix representation into a Kronecker sum of significantly smaller matrices. The invertibility of the matrix hinges on determining whether zero can be expressed as the sum of eigenvalues of these smaller matrices, which happen to be tridiagonal Toeplitz matrices. Notably, each of these smaller matrices represents a one-dimensional cellular automaton. Leveraging the rich body of research on the eigenvalue problem of Toeplitz matrices, our result provides an efficient algorithm for addressing the reversibility problem. As an application, we show that there is no reversible nontrivial linear cellular automaton over . [ABSTRACT FROM AUTHOR]

For the adjacency matrix A(G) and diagonal matrix of the vertex degrees D(G) of a simple graph G, the A(G) matrix is the convex combinations of D(G) and A(G), and is defined as A(G) = D(G)+(1)A(G), for 0 n be the eigenvalues of A(G) (which we call -adjacency eigenvalues of the graph G). The generalized adjacency energy also called -adjacency energy of the graph G is defined as EA (G) = is the average vertex degree, m is the size and n is the order of G. The -adjacency energy of a graph G merges the theory of energy (adjacency energy) and the signless Laplacian energy, as EA0 (G) = E (G) and 2E A 12 (G) = QE(G), where E (G) is the energy and QE(G) is the signless Laplacian energy of G. In this paper, we obtain some new upper and lower bounds for the generalized adjacency energy of a graph, in terms of different graph parameters like the vertex covering number, the Zagreb index, the number of edges, the number of vertices, etc. We characterize the extremal graphs attained these bounds. [ABSTRACT FROM AUTHOR]

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes n ≤ 10 . We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks. [ABSTRACT FROM AUTHOR]

In this paper, new Sturmian comparison results were obtained for linear and nonlinear hyperbolic equations on a rectangular prism. The results obtained for linear equations extended those given by Kreith [Sturmian theorems on hyperbolic equations, Proc. Amer. Math. Soc., 22 (1969), 277-281] in which the Sturmian comparison theorem for linear equations was obtained on a rectangular region in the plane. For the purpose of verification, an application was described using an eigenvalue problem. [ABSTRACT FROM AUTHOR]

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics and has wide applications. In this paper, for the biharmonic eigenvalue problem, based on the work of Gudi [Numer. Methods Partial Differ. Equ., 27 (2011), 315-328], we study the a posteriori error estimates of the approximate eigenpairs obtained by the Ciarlet-Raviart mixed finite element method. We prove the reliability and efficiency of the error estimator of the approximate eigenfunction and analyze the reliability of the error estimator of the approximate eigenvalues. We also implement the adaptive calculation and exhibit the numerical experiments which show that our method is efficient and can get an approximate solution with high accuracy. [ABSTRACT FROM AUTHOR]