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It is assumed that a sheet of paper performs transverse vibrations as a plate having additional stiffness due to static tension in its plane. The first eigenmodes are determined. The setup for experimental investigation of the nodal lines of the standing waves and for the determination of eigenmodes using projection moire is created for the analysis of unsymmetric loading of the paper. The comparison of the theoretical and experimental results of investigations is performed. The obtained results could be used in the process of design of the elements of the printing device. [ABSTRACT FROM AUTHOR]

In this communication we provide our answers to the comments by Usoskin (2017) on our recent paper (Popova et al, 2017a). We show that Principal Component Analysis (PCA) allows us to derive eigen vectors with eigen values assigned to variance of solar magnetic field waves from full disk solar magnetograms obtained in cycles 21–23 which came in pairs. The current paper (Popova et al, 2017a) adds the second pair of magnetic waves generated by quadruple magnetic sources. This allows us to recover a centennial cycle, in addition to the grand cycle, and to produce a closer fit to the solar and terrestrial activity features in the past millennium. [ABSTRACT FROM AUTHOR]

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]

In this note I examine a number of statement made in Wadee and Yiatros (Z. Angew. Math. Phys. 61:565-577 ) in relation to my paper mentioned in the title. Most of the claims by Drs. Wadee and Yiatros are re-examined here in the proper context and it is shown that the results of the title paper are free of any errors. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which improve the recent results of Tang and He [X.H. Tang, X. He, Lower bounds for generalized eigenvalues of the quasilinear systems, J. Math. Anal. Appl. 385 (2012) 72–85] when 1< p i <2 for i =1,2,…, n. [Copyright &y& Elsevier]

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]

The focus of this paper is on the solution of Maxwell's equations for time-harmonic fields on triangular, possibly non-orthogonal meshes. The method described was first introduced in References 1 and 2 for eigenvalue problems arising in the design of accelerator components and dielectric loaded waveguides. It is based on the well-known Finite Integration Technique (FIT) which is a proven consistent discretization method for the computation of electromagnetic fields. The FIT-discretization on non-orthogonal 2D grids has close relations to the Nédélec elements or edge elements in the Finite Element Method. Revisiting FIT on triangular grids this paper intends to stimulate thorough studies of the latter subject which is well worth further investigations. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we analyze the stability of the problem of performing a rational QZ step with a shift that is an eigenvalue of a given regular pencil H - λ K in unreduced Hessenberg–Hessenberg form. In exact arithmetic, the backward rational QZ step moves the eigenvalue to the top of the pencil, while the rest of the pencil is maintained in Hessenberg–Hessenberg form, which then yields a deflation of the given shift. But in finite-precision the rational QZ step gets "blurred" and precludes the deflation of the given shift at the top of the pencil. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the rational QZ step can be constructed using this eigenvector, so that the exact deflation is also obtained in finite-precision. [ABSTRACT FROM AUTHOR]

We continue the program first initiated by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and develop a modification of the technique introduced in that paper to study the spectral asymptotics, namely the Riesz means and eigenvalue counting functions, of functional difference operators H_0 = \mathcal {F}^{-1} M_{\cosh (\xi)} \mathcal {F} with potentials of the form W(x) = \left \lvert {x} \right \rvert ^pe^{\left \lvert {x} \right \rvert ^\beta } for either \beta = 0 and p > 0 or \beta \in (0, 2] and p \geq 0. We provide a new method for studying general potentials which includes the potentials studied by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and [J. Math. Phys. 60 (2019), p. 103505]. The proof involves dilating the variance of the gaussian defining the coherent state transform in a controlled manner preserving the expected asymptotics. [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]

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]

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, we deal with the global dynamics of a Lotka-Volterra competition-diffusion-advection system with nonlinear boundary conditions, including the existence, nonexistence and global stability of coexistence steady states. We start with the investigation of the principal eigenvalue of linearized system to get the local stability of steady states and then discuss the global dynamics in terms of competition coefficients. [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]

Frequency domain transfer functions of microwave systems are usually calculated either directly in the frequency domain or in the time domain in combination with a Fourier analysis. The computation of loss-free linear devices based on eigenfields (resonant fields) offers an alternative to obtaining broadband frequency characteristics with a high accuracy. Such methods are very effective for problems with multiple narrow resonances as many points in frequency domain have to be calculated to sample the maximas, or time domain computations have to deal with long transients. We describe a method of this type and then an improvement that leads directly to transfer functions, which are very accurate even if only modes in the chosen frequency range are taken into account. Generalized macroscopic parameters are introduced as input and output quantities, which are related to conventional quantities such as discrete currents and voltages, wave parameters, or even beam currents and voltages (as defined in accelerator physics for beams of ultra-relativistic charged particles). When the generalized input/output parameters are causal they are related to each other and to state parameters of the eletromagnetic fields by a differential equation with time independent system matrices. The input/output transfer matrix in the frequency domain is an impedance matrix, which is derived from the system matrices of the discretization as well as from a series expansion using eigensolutions. The number of eigenmodes is typically three times the number of meshpoints, but good approximations can be achieved even with a much smaller set of eigenvalues around the frequency range of interest. Another approach to characterizing the system response is based on the output/input relation described by an admittance matrix. Combining both approaches, the error of the approximations can be estimated and significantly reduced. The application of the method is shown for several examples. Copyright © 1999 John Wiley & Sons, Ltd. [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]

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]

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 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]

We consider two-dimensional, non-relativistic quantum system with asymptotically straight soft waveguide. We show that the local deformation of the symmetric waveguide can lead to the emerging of the embedded eigenvalues living in the continuous spectrum. The main problem of this paper is devoted to the analysis of weak perturbation of the symmetric system. We show that the original embedded eigenvalues turn to the second sheet of the resolvent analytic continuation and constitute resonances. We describe the asymptotics of the real and imaginary components of the complex resonant pole depending on deformation. Finally, we generalize the problem to three dimensional system equipped with a soft layer. [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, 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 this paper, we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the optimal domain. As a possible perspective, we conjecture that this maximal property of the Reuleaux triangle holds for the first eigenvalue of the p -Laplacian for any p ∈ (1 , + ∞) (this paper covers the case p = 1 whereas the case p = + ∞ was already known). [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]

We prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either the Dirichlet energy E f ⁢ (Ω) of the Laplacian in the domain Ω or the first eigenvalue λ 1 ⁢ (Ω) of the Dirichlet-Laplacian. Usually, one considers minimization of such functionals (often with measure constraint), as for example for the famous Saint-Venant and Faber-Krahn inequalities. By adding the convexity constraint (and possibly other natural constraints), we instead consider the rather unusual and difficult question of maximizing these functionals. This paper follows a series of papers by the authors, where the leading idea is that a certain concavity property of the shape functional that is minimized leads optimal shapes to locally saturate their convexity constraint, which geometrically means that they are polygonal. In these previous papers, the leading term in the shape functional was usually the opposite of the perimeter, for which the aforementioned concavity property was rather easy to obtain through computations of its second order shape derivative. By carrying classical shape calculus, a similar concavity property can be observed for the opposite of E f ⁢ (Ω) or λ 1 ⁢ (Ω) when shapes are smooth and convex. The main novelty in the present paper is the proof of a weak convexity property of E f ⁢ (Ω) and λ 1 ⁢ (Ω) among planar convex shapes, namely rather nonsmooth shapes. This involves new computations and estimates of the second order shape derivatives of E f ⁢ (Ω) and λ 1 ⁢ (Ω) interesting for themselves. [ABSTRACT FROM AUTHOR]