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In this paper, the field analysis of a triangular corrugated coaxial cavity with misaligned insert is carried out for a megawatt-class sub-terahertz wave gyrotron. In misalignment analysis, both parallel displacement of the insert axis and tilting of the insert axis with the outer resonator axis are considered. Graff’s addition theorem is used for deriving the eigenvalue equation of the coaxial cavity with such a misaligned insert. Mathematical formulations are carried out to include the effect of insert misalignment on the beam coupling coefficient and ohmic wall loading of the outer cavity and insert of the coaxial cavity gyrotron. The effect of structural misalignment of the insert on the operation of 2-MW, 220-GHz coaxial cavity gyrotron is analyzed. It is shown that insert misalignment reduces RF output power as well as increases ohmic wall loading of the insert beyond the prescribed cooling limit. Compared to tilting of the insert, parallel displacement of the insert causes more deterioration in the gyrotron operation. Comparative studies are also performed for a coaxial cavity with rectangular slots on the insert. [ABSTRACT FROM AUTHOR]

We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite and the Laguerre orthogonal polynomials, and propose an algorithm for finding their particular and general solutions in the closed form. This algorithm is especially suitable for the computer-algebra tools, such as Maple. The resonant equations form an essential part of various applications, e.g., of the efficient functional-discrete method for the solution of operator equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for the di-spin algebra and in the solution of square operator equations, such as A2u = f (e.g., of the biharmonic equation). [ABSTRACT FROM AUTHOR]

• Explicit solutions are found for transient natural convection in ducts. • Eigenfunction superposition method is well suited for fully-developed flow. • Rise time to steady state depends on the first eigenvalue of the Helmholtz equation. This paper models the transient natural convection problem for the hydrodynamical and thermal fully-developed open vertical ducts. The powerful method of eigenfunction superposition yields analytic solutions for general duct cross sections. It is found that the induced flow rate approaches a constant in the steady state, but the transient depends on the Prandtl number (Pr). The rise time to reach 95% of steady state is 3 / (Pr λ 1) if Pr < 1, and is 3/λ 1 if Pr > 1, where λ 1 is the lowest eigenvalue of the Helmholtz equation. The theory is then applied to the circular, semi-circular, rectangular and equilateral triangular ducts. For low Pr at small times, the transient induced velocity is larger in the velocity boundary layer near the walls but the maximum velocity is near the corners of the duct. [ABSTRACT FROM AUTHOR]

A direct eigenvalue analysis approach for solving the stability problem of periodically supported beams with multi-mode coupling vibration under general harmonic excitations is developed based on the Floquet theorem, Fourier series and matrix eigenvalue analysis. The transverse periodic supports are considered for improving the parametrically excited stability of beams under longitudinal periodic excitations. The dynamic stability of parametrically excited vibration of the beam with transverse spaced supports under longitudinal harmonic excitations is studied. The partial differential equation of motion of the beam with spaced supports under harmonic excitations is given and converted into ordinary differential equations with time-varying periodic parameters using the Galerkin method, which describe the parametrically excited vibration of the beam with coupled multiple modes. The fundamental solution to the equations is expressed as the product of periodic and exponential components based on the Floquet theorem. The periodic component and periodic parameters are expanded into Fourier series, and the matrix eigenvalue equation is obtained which is used for directly determining the parametrically excited stability. The dynamic stability of parametrically excited vibration of the beam with spaced supports under harmonic excitations is illustrated by numerical results on unstable regions. The influence of the periodic supports and excitation parameters on the parametrically excited stability is explored. The parametrically excited stability of the beam with multi-mode coupling vibration can be improved by the periodic supports. The developed analysis method is applicable to more general period-parametric beams with multi-mode coupling vibration under various harmonic excitations. [ABSTRACT FROM AUTHOR]

The entanglement of two distant quantum dots coupled to metallic waveguide has been investigated theoretically in the presence of the flip-flop interaction with the analytic solutions of eigenvalue equations of the system. High entanglement of two quantum dots could be achieved by adjusting the direct-coupling strength of two quantum dots, the coupling strength of quantum dots with surface plasmon along metallic waveguide, the group velocity of surface plasmon and detuning. The discussed system with the flip-flop interaction provides us with a rich way to realize the quantum device for quantum information processing, such as quantum communication and quantum computation. [ABSTRACT FROM AUTHOR]

Weighted Golub-Kahan-Lanczos bidiagonalization algorithm(wGKLu) is used to solving the linear response eigenproblem. In this paper, we present an improvement to wGKLu based on the shift-and-invert strategy. Due to the interior eigenproblem being transformed to the exterior eigenproblem, our new algorithm saves lots of calculus. Numerical examples illustrates the behaviors. [ABSTRACT FROM AUTHOR]

We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer G-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a 2 × 2 Riemann–Hilbert problem, and use this representation to obtain the so-called large gap asymptotics. [ABSTRACT FROM AUTHOR]

Using an expanded algebraic formalism with the inclusion of inverse operators, we construct raised and decreased coherent states for a set of exactly solvable quantum confined systems. We assume in this procedure both the ladder-operator and the displacement-operator methods, showing the equivalence between the two approaches. For each coherent state defined, we present its expansion in the Hilbert eigenstate space H e s , eigenvalue equation, overcompleteness relation, as well as other intrinsic properties. Whenever possible, we present an interpretation based on nonlinear deformation models for these new forms of coherent states. We evaluate the relevance of the new coherent states in quantum entanglement and squeezing by taking, as an example, the case of a coupled system. [ABSTRACT FROM AUTHOR]

We start by introducing a new class of structured matrix polynomials, namely, the class of MA-structured matrix polynomials, to provide a common framework for many classes of structured matrix polynomials that are important in applications: the classes of (skew-)symmetric, (anti-) palindromic, and alternating matrix polynomials. Then, we introduce the families of MA-structured strong block minimal bases pencils and of MAstructured block Kronecker pencils, which are particular examples of block minimal bases pencils recently introduced by Dopico, Lawrence, Pérez and Van Dooren, and show that any MA-structured odd-degree matrix polynomial can be strongly linearized via an MA-structured block Kronecker pencil. Finally, for the classes of (skew-)symmetric, (anti-)palindromic, and alternating odd-degree matrix polynomials, the MA-structured framework allows us to perform a global and structured backward stability analysis of complete structured polynomial eigenproblems, regular or singular, solved by applying to a MA-structured block Kronecker pencil a structurally backward stable algorithm that computes its complete eigenstructure, like the palindromic-QR algorithm or the structured versions of the staircase algorithm. This analysis allows us to identify those MA-structured block Kronecker pencils that yield a computed complete eigenstructure which is the exact one of a slightly perturbed structured matrix polynomial. These pencils include (modulo permutations) the well-known block-tridiagonal and block-anti-tridiagonal structure-preserving linearizations. Our analysis incorporates structure to the recent (unstructured) backward error analysis performed for block Kronecker linearizations by Dopico, Lawrence, P#233;erez and Van Dooren, and share with it its key features, namely, it is a rigorous analysis valid for finite perturbations, i.e., it is not a first order analysis, it provides precise bounds, and it is valid simultaneously for a large class of structure-preserving strong linearizations. [ABSTRACT FROM AUTHOR]

In this paper, we study a peculiar model for the scalar field. We add the cuscuton term in a standard model and investigate how this inclusion modifies the usual behavior of kinks. We find the first order equations and calculate the energy density and the total energy of the system. Also, we investigate the linear stability of the model, which is governed by a Sturm-Liouville eigenvalue equation that can be transformed in an equation of the Shcrödinger type. The model is also investigated in the braneworld scenario, where a first order formalism is also obtained and the linear stability is investigated. Surprisingly, the solutions are independent from the cuscuton term. We also show that this modification allows for the presence of analytical results in the variables that describe the transformed Schrödinger-like equation. [ABSTRACT FROM AUTHOR]

The generalized model for plasma, thermal, and elastic waves under dual phase lag model have been applied to determine the carrier density, the displacement, the temperature, and the stresses in a semiconductor medium. Using Laplace transform and the eigenvalue approach methodology, the solutions of all variables have been obtained analytically. A semiconducting material like as silicon was considered. The results were graphically represented to show the different between the dual phase model, Lord and Shulman's theory and the classical dynamical coupled theory. [ABSTRACT FROM AUTHOR]

We tackle with the Dirac equation in the presence of the general molecular potential (GMP). The spin symmetric solution of the relativistic wave equation is obtained by considering the Pekeris-type approximation scheme to deal with the centrifugal term, and in order to solve second-order differential equation, the asymptotic iteration method (AIM) is used. The closed form of the energy eigenvalue equation is found out for any values of the angular momentum quantum number. We calculate the relativistic vibrational bound state energies of 51Δg state of Na2 molecule and compare them with the Rydberg–Klein–Rees (RKR) data. We show that relativistic vibrational energies of this molecule, which are found in the spin symmetric case, are more convenient with experimental RKR data than non-relativistic vibrational energies. We also obtain normalization constant by considering inductive approach and investigate the radial eigenfunctions and probability density functions corresponding to different eigenvalues graphically for the Na2(51Δg) molecule. [ABSTRACT FROM AUTHOR]

In order to provide more sensitive monitoring results, the time dynamics and steady-state operating conditions should be separately monitored by distinguishing time information from the steady-state counterpart. However, it is a more challenging task for batch processes because they vary from phase to phase presenting multiple steady states and complex dynamic characteristics. To address the above issue, a concurrent monitoring strategy of multiphase steady states and process dynamics is developed for batch processes in this paper. On one hand, multiple local models are constructed to identify a steady derivation from the normal operating condition for different phases. On the other hand, based on the recognition that the process dynamics can be considered to be irrelevant with the steady states, a global model is built to detect the dynamics anomalies by monitoring the time variations. Corresponding to alarms issued by different statistics, different operating statuses are indicated with meaningful physical interpretation and deep process understanding. To illustrate the feasibility and efficacy, the proposed algorithm is applied to the injection molding process, which is a typical multiphase batch process. [ABSTRACT FROM AUTHOR]

In this article we prove that the first eigenvalue of the ∞ {\infty} -Laplacian { min ⁡ { - Δ ∞ ⁢ v , | ∇ ⁡ v | - λ 1 , ∞ ⁢ (Ω) ⁢ v } = 0 in ⁢ Ω , v = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle\min\{-\Delta_{\infty}v,|\nabla v|-\lambda% _{1,\infty}(\Omega)v\}&\displaystyle=0&&\displaystyle\text{in }\Omega,\\ \displaystyle v&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as ℓ ↗ 1 {\ell\nearrow 1} of concave problems of the form { min ⁡ { - Δ ∞ ⁢ v ℓ , | ∇ ⁡ v ℓ | - λ 1 , ∞ ⁢ (Ω) ⁢ v ℓ ℓ } = 0 in ⁢ Ω , v ℓ = 0 on ⁢ ∂ ⁡ Ω. \left\{\begin{aligned} \displaystyle\min\{-\Delta_{\infty}v_{\ell},|\nabla v_{% \ell}|-\lambda_{1,\infty}(\Omega)v_{\ell}^{\ell}\}&\displaystyle=0&&% \displaystyle\text{in }\Omega,\\ \displaystyle v_{\ell}&\displaystyle=0&&\displaystyle\text{on }\partial\Omega.% \end{aligned}\right. In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the sub-homogeneous problems as happens for the usual eigenvalue problem for the p-Laplacian for a fixed 1 < p < ∞ {1

Based on an eigenvalue analysis conducted on the scaled memoryless quasi-Newton updating formulas BFGS and DFP, an adaptive choice for the trust region radius is proposed. Then, using a trust region ratio obtained from a nonmonotone line search strategy, an adaptive nonmonotone trust region algorithm is developed. Under proper conditions, it is briefly shown that the proposed algorithm is globally and locally superlinearly convergent. Numerical experiments are done on a set of unconstrained optimization test problems of the CUTEr collection, using the Dolan-Moré performance profile. They show efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract This work presents an efficient numerical approach based on the combination of the mode matching technique and the finite element method (FEM) to model the sound propagation in silencers containing granular material and to evaluate their acoustic performance through the computation of transmission loss (TL). The methodology takes into account the presence of three-dimensional (3D) waves and the corresponding higher order modes, while reducing the computational expenditure of a full 3D FEM calculation. First, the wavenumbers and transversal pressure modes associated with the silencer cross section are obtained by means of a two-dimensional FEM eigenvalue problem, which allows the consideration of arbitrary transversal geometries and material heterogeneities. The numerical approach considers the possibility of using different filling levels of granular material, giving rise to cross sections with abrupt changes of properties located not only in the usual central perforated passage, but also in the transition between air and material, that involves a significant change in porosity. After solving the eigenvalue problem, the acoustic fields (acoustic pressure and axial velocity) are coupled at geometric discontinuities between ducts through the compatibility conditions to obtain the complete solution of the wave equation and the acoustic performance (TL). The granular material is analysed as a potential alternative to the traditional dissipative silencers incorporating fibrous absorbent materials. Sound propagation in granular materials can be modelled through acoustic equivalent properties, such as complex and frequency dependent density and speed of sound. TL results computed by means of the numerical approach proposed here show good agreement with full 3D FEM calculations and experimental measurements. As expected, the numerical mode matching outperforms the computational expenditure of the full 3D FEM approach. Different configurations have been studied to determine the influence on the TL of several parameters such as the size of the material grains, the filling level of the chamber, the granular material porosity and the geometry of the silencer cross section. Highlights • Numerical mode matching is applied to sound waves in silencers with granular medium. • It allows efficient computation of geometries with arbitrary cross section. • TL curves show good agreement with full 3D FEM results and experimental measurements. • The acoustic influence of granular size and silencer cross section is analysed. [ABSTRACT FROM AUTHOR]

Abstract Curve squeal noise is a severe railway noise problem that can occur when a railway vehicle negotiates a sharp curve. It is usually characterised by a very loud tonal noise and can be very annoying for people in the vicinity. It is generally attributed to friction-induced instability, either due to a falling friction characteristic with increasing sliding velocity or to a mode coupling mechanism which can lead to instability even for a constant friction coefficient. The squeal frequency is usually associated with one or more wheel modes. However, the wheel is coupled dynamically to the track and insufficient attention has been paid in previous research to the role played by the rail dynamic behaviour. In this paper, the effect of the rail dynamics on curve squeal under constant friction conditions is investigated by means of different modelling approaches. The rail is firstly modelled using a waveguide finite element (WFE) model and it is found that the inclusion of the rail dynamics in the model can lead to squeal in some situations where it would otherwise not occur. Various effects are then considered that may introduce additional resonant behaviour into the rail dynamics. These include the effect of the rail cross mobility, rail cross-section deformation, the influence of the periodic support of the rail and reflections between multiple wheels on the rail. The effect of the rail pad stiffness is also explored. However, the results show that all these factors have little influence on the predicted curve squeal instabilities. By means of a reduced model, the main characteristics of the rail dynamics that can result in squeal are then assessed. It is shown that the mass and damping-like behaviour of the infinite rail are at the origin of the instabilities rather than any modal behaviour of the track. Curve squeal may occur for a single wheel mode even if the rail is represented by a damper, which is a close approximation to the vertical mobility of the track at high frequencies. This forms a third possible mechanism for curve squeal in addition to falling friction and wheel mode coupling. [ABSTRACT FROM AUTHOR]

Highlights • Functionally graded material properties of beams are modelled as random fields. • A generalized eigenvalue equation is derived for stochastic free vibration analysis. • Effective surrogate models are developed based on the polynomial chaos expansion. • Several examples are presented to demonstrate the engineering applications. Abstract The functionally graded material (FGM) has a potential to replace ordinary ones in engineering reality due to its superior thermal and dynamical characteristics. In this regard, the paper presents an effective approach for uncertain natural frequency analysis of composite beams with axially varying material properties. Rather than simply assuming the material model as a deterministic function, we further extend the FGM property as a random field, which is able to account for spatial variability in laboratory observations and in-field data. Due to the axially varying input uncertainty, natural frequencies of the stochastically FGM (S-FGM) beam become random variables. To this end, the Karhunen–Loève expansion is first introduced to represent the composite material random field as the summation of a finite number of random variables. Then, a generalized eigenvalue function is derived for stochastic natural frequency analysis of the composite beam. Once the mechanistic model is available, the brutal Monte-Carlo simulation (MCS) similar to the design of experiment can be used to estimate statistical characteristics of the uncertain natural frequency response. To alleviate the computational cost of the MCS method, a generalized polynomial chaos expansion model developed based on a rather small number of training samples is used to mimic the true natural frequency function. Case studies have demonstrated the effectiveness of the proposed approach for uncertain natural frequency analysis of functionally graded material beams with axially varying stochastic properties. [ABSTRACT FROM AUTHOR]

We propose a unifying approach to the problem of measuring the inconsistency of judgments. More precisely, we define a general framework to allow several well-known inconsistency indices to be expressed as special cases of this new formulation. We consider inconsistency indices as aggregations of 'local', i.e. triple-based, inconsistencies. We show that few reasonable assumptions guarantee a set of good properties for the obtained general inconsistency index. Under this representation, we prove a property of Pareto efficiency and show that OWA functions and t-conorms are suitable aggregation functions of local inconsistencies. We argue that the flexibility of this proposal allows tuning of the index. For example, by using different types of OWA functions, the analyst can obtain the desired balance between an averaging behavior and a 'largest inconsistency-focused' behavior. [ABSTRACT FROM AUTHOR]

If M=(M,∇) is an affine surface, let Q(M):=ker(H+1m-1ρs) be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let M~=(M,∇~) be another affine structure on M which is strongly projectively flat. We show that Q(M)=Q(M~) if and only if ∇=∇~ and that Q(M) is linearly equivalent to Q(M~) if and only if M is linearly equivalent to M~. We use these observations to classify the flat Type~A connections up to linear equivalence, to classify the Type~A connections where the Ricci tensor has rank 1 up to linear equivalence, and to study the moduli spaces of Type~A connections where the Ricci tensor is non-degenerate up to affine equivalence. [ABSTRACT FROM AUTHOR]

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive orthants and Lorentz cone are given. [ABSTRACT FROM AUTHOR]

Abstract This paper considers Hopf bifurcation of complex network with two independent delays. By analyzing the eigenvalue equations, the local stability of the system is studied. Taking delay as parameter, the change of system stability with time is studied and the emergence of inherent bifurcation is given. By changing the value of the delay, the bifurcation of a given system can be controlled. Numerical simulation results confirm the validity of the results found. [ABSTRACT FROM AUTHOR]

Abstract The inverse spectral problem for the interior transmission eigenvalue problem with the unit time is studied by given spectral data. The authors show that the refractive index on the whole interval can be uniquely determined by parts of its transmission eigenvalues together with the partial information on the refractive index. In particular, we pose and solve a new type of inverse spectral problems involving the interior transmission eigenvalue problem with complex transmission eigenvalues except for at most finite real eigenvalues. [ABSTRACT FROM AUTHOR]

The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential is Gaussian, and prove three results: localisation at strong disorder, a Wegner-type estimate on the mean density of eigenvalues, and a Minami-type estimate on the probability of having multiple eigenvalues in a short interval. The last two results are proved in the more general setting of deformed block-Gaussian matrices, which includes a class of Gaussian band matrices as a special case. Emphasis is placed on the dependence of the bounds on the number of orbitals. As an additional application, we improve the upper bound on the localisation length for one-dimensional Gaussian band matrices. [ABSTRACT FROM AUTHOR]

We address the reduction to compact band forms, via unitary similarity transformations, for the solution of symmetric eigenvalue problems and the computation of the singular value decomposition (SVD). Concretely, in the first case, we revisit the reduction to symmetric band form, while, for the second case, we propose a similar alternative, which transforms the original matrix to (unsymmetric) band form, replacing the conventional reduction method that produces a triangular-band output. In both cases, we describe algorithmic variants of the standard Level 3 Basic Linear Algebra Subroutines (BLAS)-based procedures, enhanced with look-ahead, to overcome the performance bottleneck imposed by the panel factorization. Furthermore, our solutions employ an algorithmic block size that differs from the target bandwidth, illustrating the important performance benefits of this decision. Finally, we show that our alternative compact band form for the SVD is key to introduce an effective look-ahead strategy into the corresponding reduction procedure. [ABSTRACT FROM AUTHOR]

We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions' systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem. [ABSTRACT FROM AUTHOR]

Reservoir computing (RC) is a machine‐learning paradigm that is capable to process empirical time series data. This paradigm is based on a neural network with a fixed hidden layer having a high‐dimensional state space, called a reservoir. Reservoirs including time delays are considered to be good candidates for practical applications because they make hardware realization of the high‐dimensional reservoirs simple. Performance of the well‐trained RCs depends both on dynamical properties of attractors of the reservoirs and tasks they solve. Therefore, in the conventional monostable RCs, there arise task‐wise optimization problems of the reservoirs, which have been solved based on trial and error approaches. In this study, we analyzed the relationship between the dynamical properties of the time‐delay reservoir and the performance in terms of the spectra of the delayed dynamical systems, which might facilitate the development of the unified systematic optimization techniques for the time‐delay reservoirs. In addition, we propose a novel RC framework that performs well on distinct tasks without the task‐wise optimization using bistable reservoir dynamics, which can reduce complicated hardware management of the reservoirs. [ABSTRACT FROM AUTHOR]

Abstract Fluidelastic instability is a key issue in steam generator tube arrays subjected in two-phase flow. Lots of experimental analyses were conducted on fluidelastic instability of a tube array subjected to air-water cross-flow. However, there is seldom theoretical analysis to calculate the critical velocity. Therefore, considering the effect of two-phase flow, the fluidelastic instability of a rotated triangular tube array was studied in this paper. A mathematical model of a tube array with unsteady fluid force model was set up. A program based on the model was written, and an experiment was carried out to verify the correctness of the program. The results of this program were in good agreement with the experimental data. Using this program, the critical velocity of fluid-elastic instability considering the effect of two-phase flow was determined by the eigenvalue analysis. This paper investigated the critical velocities of fluid-elastic instability for void fraction ranging from 0% to 90% with five tube natural frequencies, respectively. The results show that void fraction and tube natural frequency are the key factors in fluidelastic instability, which have an obvious effect on the critical velocity of fluidelastic instability. Highlights • A mathematical model of a tube array with unsteady fluid force model was set up. • A program based on the model was written, and an experiment was conducted to verify the correctness of the program. • The critical velocity of fluid-elastic instability considering two-phase flow was determined by the eigenvalue analysis. [ABSTRACT FROM AUTHOR]

Abstract A geometrically accurate infinitesimal-rotation planar beam element is developed and implemented in this study. The performance of the element, which is suited for developing reduced-order models for both structural and multibody systems (MBS), is evaluated using an eigenvalue analysis. Unlike conventional infinitesimal-rotation finite elements (FE), the new element is compatible with the computer-aided design (CAD) B-spline and NURBS (N on- U niform R ational B - S pline) representations and allows for a straightforward linear transformation of CAD solid models to FE analysis meshes. The absolute nodal coordinate formulation (ANCF) elements, which are related to B-splines and NURBS by linear mapping, are used as the basis for developing the planar beam element. The new element has a shape function matrix expressed in terms of geometric coefficients obtained using the ANCF position vector gradients in the reference configuration. The change in the position vector gradients is written in terms of infinitesimal rotation coordinates using a velocity transformation that defines constant element mass and stiffness matrices. Using this approach, initially straight and curved configurations can be modeled using the same displacement field. The eigenvalue analysis is used to evaluate the element performance and examine the effect of shear locking on the predicted frequencies. Several elastic force formulations are used to evaluate the convergence characteristics, including the direct displacement method (DDM), general continuum mechanics (GCM) approach, elastic line (EL) approach, and strain split method (SSM). The element performance is compared with the conventional Euler-Bernoulli and Timoshenko elements as well as the analytical solutions. [ABSTRACT FROM AUTHOR]

The recently introduced by us, two- and three-parameter (p, q)- and (p, q, μ)-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite η (N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of "almost free" (but essentially nonlinear) Hamiltonian. [ABSTRACT FROM AUTHOR]

Abstract We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on L p -spaces (affine isometric, and more generally (2 − 2 ϵ) 1 / 2 p -uniformly Lipschitz) with p varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal p for which L p -fixed point properties hold and the conformal dimension of the boundary. In the Gromov density model, we prove that for every p 0 ∈ [ 2 , ∞) for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on L p -spaces that are (2 − 2 ϵ) 1 / 2 p -uniformly Lipschitz, and this for every p ∈ [ 2 , p 0 ]. To accomplish these goals we find new bounds on the first eigenvalue of the p -Laplacian on random graphs, using methods adapted from Kahn and Szemerédi's approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to L p -spaces previous results for Kazhdan's Property (T) established by Żuk and Ballmann–Świa̧tkowski. [ABSTRACT FROM AUTHOR]

We present a method for reducing the generalized eigenvalue problem Ax = Bxλwith banded hermitian matrices A, B, and B positive definite to an equivalent standard eigenvalue problem Cy = yλ, such that C again is banded. Our method combines ideas of an algorithm proposed by Crawford in 1973 and LAPACK's reduction routines _{SY,HE}GST and retains their respective advantages, namely, being able to rely on matrix-matrix operations (Crawford) and to handle split factorizations and different bandwidths bA and bB (LAPACK). In addition, it includes two algorithmic parameters (block size, nb, and width of blocked orthogonal transformations, w) that can be adjusted to optimize performance. We also present a heuristic for automatically determining suitable values for these parameters. Numerical experiments confirm the efficiency of our new method. [ABSTRACT FROM AUTHOR]

Abstract When vector-valued observations are of high dimension N relative to the sample size T , it is common to employ a linear factor model in order to estimate the underlying covariance structure or to further understand the relationship between coordinates. Asymptotic analyses of such models often consider the case in which both N and T tend jointly to infinity. Within this framework, we derive weak convergence results for processes of partial sample estimates of the largest eigenvalues of the sample covariance matrix. It is shown that if the effect of the factors is sufficiently strong, then the processes associated with the largest eigenvalues have Gaussian limits under general conditions on the divergence rates of N and T , and the underlying observations. If the common factors are "weak", then N must grow much more slowly in relation to T in order for the largest eigenvalue processes to have a Gaussian limit. We apply these results to develop general tests for structural stability of linear factor models that are based on measuring the fluctuations in the largest eigenvalues throughout the sample, which we investigate further by means of a Monte Carlo simulation study and an application to US treasury yield curve data. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a method to calculate modal displacement, stress, and strain energy participation in shell finite-element eigen-buckling solutions of thin-walled structural members using Generalized Beam Theory (GBT). The method provides quantitative information that can be used to interpret coupled buckling in structural designs. A finite-element (FE) eigen buckling solution is transformed to a GBT solution, and equivalent GBT modal amplitudes representing the FE solution are retrieved. The modal displacement field, stress tensor and strain energy are retrieved using GBT modal amplitude field and applying GBT constitutive relationships between strain and stress. Theory and examples are provided. Highlights • Presents a method obtaining buckling mode participation of displacements, stresses, strain energy of thin-walled members. • Deepens our understanding of coupled buckling. • Facilitates strength prediction of cold-formed members per AISI S100 or other codes who employ Direct Strength Method. • The method transfers a shell finite-element Eigen buckling solution into a Generalized Beam Theory solution. [ABSTRACT FROM AUTHOR]

Abstract We approximate the spectra of a class of 2 n -order differential operators using isogeometric analysis in mixed formulations. This class includes a wide range of differential operators such as those arising in elliptic, biharmonic, Cahn–Hilliard, Swift–Hohenberg, and phase-field crystal equations. The spectra of the differential operators are approximated by solving differential eigenvalue problems in mixed formulations, which require auxiliary parameters. The mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2 p where p is the order of the underlying B-spline space. We improve this order to be 2 p + 2 by applying optimally-blended quadrature rules developed in Puzyrev et al. (2017), Caloet al. (0000) and this order is an optimum in the view of dispersion error. We also compare these results with the mixed finite elements and show numerically that the mixed isogeometric analysis leads to significantly better spectral approximations. [ABSTRACT FROM AUTHOR]

The main goal of this study is investigating the interfacial instability in the shear density stratified flow in non-Boussinesq regime using linear stability theory. In the current study, the pseudospectral collocation method employed Chebyshev polynomials is applied to solve two coupled eigenvalue equations. Using the linear stability analysis in the temporal framework, the effects of various parameters on the flow instability have been studied. Obtained results in the present paper are showing that increasing the bed slope, the flow becomes more unstable; also at R = 1, Kelvin–Helmholtz and Holmboe waves appear. Furthermore, Holmboe waves were not observed only at θ = 0. This study shows that at R ≠ 1, in addition to observing Kelvin–Helmholtz and Holmboe waves with higher growth rates, by increasing the bed slope, the growth rate and the number of Kelvin–Helmholtz modes increase. With an improved understanding of the instability mechanisms and features with including the non-Boussinesq effects, one can confirm some of the previous experimental results and offer new indications to observations that have not been fully explained. In designing laboratory experiments to observe Holmboe waves and estimating their wavelengths and phase speeds the results of present paper are also could be useful. [ABSTRACT FROM AUTHOR]

Abstract The Induced Dimension Reduction method (IDR(s)) (Sonneveld and van Gijzen, 2008) is a short-recurrences Krylov method to solve systems of linear equations. In this work, we accelerate this method using spectral information. We construct a Hessenberg relation from the IDR(s) residual recurrences formulas, from which we approximate the eigenvalues and eigenvectors. Using the Ritz values, we propose a self-contained variant of the Ritz-IDR(s) method (Simoncini and Szyld, 2010) for solving a system of linear equations. In addition, the Ritz vectors are used to speed-up IDR(s) for the solution of sequence of systems of linear equations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the power method of the right eigenvalue problem of a quaternion matrix A. If A is Hermitian, we propose the power method that is a direct generalization of that of complex Hermitian matrix. When A is non-Hermitian, by applying the properties of quaternion right eigenvalues, we propose the power method for computing the standard right eigenvalue with the maximum norm and the associated eigenvector. We also briefly discuss the inverse power method and shift inverse power method for the both cases. The real structure-preserving algorithm of the power method in the two cases are also proposed, and numerical examples are provided to illustrate the efficiency of the proposed power method and inverse power method. [ABSTRACT FROM AUTHOR]

Abstract We investigate the following eigenvalue problem { − div (L (x) | ∇ u | p − 2 ∇ u) = λ K (x) | u | p − 2 u in A R 1 R 2 , u = 0 on ∂ A R 1 R 2 , where A R 1 R 2 : = { x ∈ R N : R 1 < | x | < R 2 } (0 < R 1 < R 2 ≤ ∞) , λ > 0 is a parameter, the weights L and K are measurable with L positive a.e. in A R 1 R 2 and K possibly sign-changing in A R 1 R 2 . We prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions. The asymptotic estimates for u (x) and ∇ u (x) as | x | → R 1 + or R 2 − are also investigated. [ABSTRACT FROM AUTHOR]

Instabilities of a two-dimensional quasigeostrophic circular flow around a rigid circular wall (island) with radial offshore bottom slope are studied analytically. The basic flow is composed of two concentric, uniform potential-vorticity (PV) rings with zero net vorticity attached to the island. Linear stability analysis for perturbations in the form of azimuthal modes leads to a transcendental eigenvalue equation. The non-dimensional governing parameters are beta (associated with the steepness of the bottom slope, hence taken to be negative), the PV in the inner ring and the radii of the inner and outer rings. This setting up of the problem allows us to derive analytically the eigenvalue equation. We first analyse this equation for weak slopes to understand the asymptotic first-order corrections to the flat-bottom case. For azimuthal modes 1 and 2, it is found that the conical topographic beta effect stabilizes the counterclockwise flows, but destabilizes clockwise flows. For a clockwise flow, the beta effect gives rise to the mode-1 instability, contrary to the flat-bottom case where this mode is always stable. Moreover, however small the slope steepness (beta) is, it leads to the mode-1 instability in a large region in the parameter space. For steep slopes, the beta term in the PV expression may dominate the relative vorticity term, causing stabilization of the flow, as compared to the flat-bottom case, for both directions of the basic flow. When the flow is counterclockwise and the slope steepness is increased, mode 2 turns out to be entirely stable and modes 3, 4 and 5 enlarge their stability regions. In a clockwise flow, when the slope steepness is increased, mode 1 regains its stability in the entire parameter space, and mode 2 becomes more stable than mode 3. The bifurcation of mode 1 from stability to instability is discussed in terms of the Rossby waves at the contours of discontinuity of the basic PV and outside the uniform-PV rings. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert–Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space. [ABSTRACT FROM AUTHOR]

The article studies the exponential localization of eigenfunctions associated with isolated eigenvalues of Schrödinger operators on infinite metric graphs. We strengthen the result obtained in [3] providing a bound for the rate of exponential localization in terms of the distance between the eigenvalue and the essential spectrum. In particular, if the spectrum is purely discrete, then the eigenfunctions decay super-exponentially. [ABSTRACT FROM AUTHOR]

Motivated by questions in biology, we investigate the stability of equilibria of the dynamical system which arise as critical points of f, under the assumption that is positive semi-definite. It is shown that the condition , where is the smallest eigenvalue of , plays a key role in guaranteeing uniform asymptotic stability and in providing information on the basis of attraction of those equilibria. [ABSTRACT FROM AUTHOR]

We put forward an algorithm for the issue on the partial eigenstructure assignment in a high‐order system, in which some certain eigenpairs, in a given system, can be assigned without changing others. Then a differential equation could be used to model the given systems, so that a multi‐input state feedback control deal with this assignment. Moreover, the algorithm requests the information of a few of the eigenpairs merely. As well some numerical cases are shown to prove the effect of our algorithm. [ABSTRACT FROM AUTHOR]

Abstract Based on a new approximation of a Schur complement matrix and some preconditioning techniques considered in earlier papers, three robust preconditioners are presented for solving the KKT systems arising from discretizing optimal control problem with time-periodic parabolic equation. The eigenvalue bounds of the preconditioned matrices and some new theoretical results are obtained. Moreover, selection of quasi-optimal parameter is studied and an explicit expression of the quasi-optimal parameter is given. Numerical examples are given to illustrate the effectiveness of the three proposed optimized preconditioners. [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices and m edges, and let S k ( G ) be the sum of the k largest Laplacian eigenvalues of G . It was conjectured by Brouwer that S k ( G ) ≤ m + ( k + 1 2 ) holds for 1 ≤ k ≤ n . In this paper, we present several families of graphs for which Brouwer's conjecture holds, which improve some previously known results. We also establish a new upper bound on S k ( G ) for split graphs, which is tight for each k ∈ { 1 , 2 , … , n − 1 } and turns out to be better than that conjectured by Brouwer. [ABSTRACT FROM AUTHOR]