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Within the field of elastic metamaterials, topological metamaterials have recently received much attention due to their ability to host topologically robust edge states. Introducing local resonators to these metamaterials also opens the door for many applications such as energy harvesting and reconfigurable metamaterials. However, the interactions between phenomena from local resonance and modulation patterning are currently unknown. This work fills that gap by studying multiple cases of spatially modulated metamaterials with local resonators to reveal the mechanisms behind bandgap formation. Their dispersion relations are determined analytically for infinite chains and validated numerically using eigenvalue analysis. The inverse method is used to determine the imaginary wavenumber components from which each bandgap is characterized by its formation mechanism. The topological nature of the bandgaps is also explored through calculating the Chern number and integrated density of states. The band structures are obtained for various sources of modulation as well as multiple resonator parameters to illustrate how both local resonance and modulation patterning interact together to influence the band structure. Other unique features of these metamaterials are further demonstrated through the mode shapes obtained using the eigenvectors. The results reveal a complex band structure that is highly tunable, and the observations given here can be used to guide designers in choosing resonator parameters and patterning to fit a variety of applications. [ABSTRACT FROM AUTHOR]

Interference is widely regarded as a foundational attribute of quantum mechanics. However, for a given experimental arrangement, interference can either contribute or not contribute to the outcome depending upon the basis in which it is measured. This observation is both foundational and particularly relevant to coherent control of molecular processes, an approach based upon quantum interference. Here, we address this issue and its relevance to controlling molecular processes via the "coherent control scattering (CCS) matrix," a formalism that allows for an analysis of modifications in an interference structure resulting from a change of basis. This analysis reveals that the change in the interference structure can be attributed to the non-commutativity of the transformation matrix with the CCS matrix and the non-orthogonality of the transformation. Additionally, minimal interference is shown to be associated with the CCS eigenbasis and that the Fourier transform of the eigenvectors of the CCS matrix provides the maximal interference and hence the best coherent control. The change of controllability through a change of basis is illustrated with an example of 85Rb+ 85Rb scattering. In addition, the developed formalism is applied to explain recent experimental results on He + D2 inelastic scattering demonstrating the presence or absence of interference depending on the basis. [ABSTRACT FROM AUTHOR]

This document is an erratum for a journal article that corrects errors in the implementation of equations. It provides corrected values for eigenvectors and perturbative correction terms. The document also presents data on triplet exciton transfer couplings in anthracene dimers and singlet fission couplings for pentacene dimers. The corrected tables compare couplings for different compounds and calculation methods, and provide excitation energies and weights of basis states for pentacene dimers. This information can be useful for researchers studying singlet fission and its applications. The correction was provided by Soichiro Nishio and Yuki Kurashige. [Extracted from the article]

We derive the explicit expression of the three self-energies that one encounters in many-body perturbation theory: the well-known GW self-energy, as well as the particle–particle and electron–hole T-matrix self-energies. Each of these can be easily computed via the eigenvalues and eigenvectors of a different random-phase approximation linear eigenvalue problem that completely defines their corresponding response function. For illustrative and comparative purposes, we report the principal ionization potentials of a set of small molecules computed at each level of theory. The performance of these schemes on strongly correlated systems (B2 and C2) is also discussed. [ABSTRACT FROM AUTHOR]

Inhomogeneous damping distribution leads to the occurrence of complex modes of structures. Complex modes' vibration and acoustic radiation characteristics are different from real modes. This paper studies the complex modal characteristics of inhomogeneously damped plates for forced vibration. We used the state space method to obtain the complex eigenvalues and eigenvectors of the vibration system and calculated the relationship between modal coordinates and frequency under unit force. The results show that the absolute values of the modal coordinates of two conjugate complex eigenvectors are different, and the structure's vibration is dominated by the eigenvectors corresponding to eigenvalues with positive imaginary parts near natural frequencies. When investigating the structural sound radiation in the vibration mode space, each mode's acoustic radiation is not independent but coupled with each other. The influence of the coupling between complex modes on the calculation results of the acoustic radiation power of the plate is studied in this paper. The results show that the coupling effect between modes cannot be ignored when calculating the acoustic radiation power based on complex modes. Finally, the error of calculating acoustic power using the decoupling approximation instead of the complex mode method is analyzed. [ABSTRACT FROM AUTHOR]

Turing patterns formation may depend on variation of the initial conditions. In this work, the dependent mechanism of initial distributions on pattern formation for the coupled discrete diffusion system is discussed, some asymptotic behavior between pattern formation and initial value determined by the corresponding eigenvectors of the eigenvalue for discrete Laplace operator can be given. To control the coupled discrete diffusion system for pattern selection, an adaptive adjustment mechanism is applied, and some theoretical generalizations are obtained. Then, the effectiveness of the theory analysis is validated though simulation results. [ABSTRACT FROM AUTHOR]

In this paper, we develop an eigenvector-assisted estimation framework for a collection of signal-plus-noise matrix models arising in high-dimensional statistics and many applications. The framework is built upon a novel asymptotically unbiased estimating equation using the leading eigenvectors of the data matrix. However, the estimator obtained by directly solving the estimating equation could be numerically unstable in practice and lacks robustness against model misspecification. We propose to use the quasi-posterior distribution by exponentiating a criterion function whose maximizer coincides with the estimating equation estimator. The proposed framework can incorporate heteroskedastic variance information, but does not require the complete specification of the sampling distribution and is also robust to the potential misspecification of the distribution of the noise matrix. Computationally, the quasi-posterior distribution can be obtained via a Markov chain Monte Carlo sampler, which exhibits superior numerical stability over some of the existing optimization-based estimators and is straightforward for uncertainty quantification. Under mild regularity conditions, we establish the large sample properties of the quasi-posterior distributions. In particular, the quasi-posterior credible sets have the correct frequentist nominal coverage probability provided that the criterion function is carefully selected. The validity and usefulness of the proposed framework are demonstrated through the analysis of synthetic datasets and the real-world ENZYMES network datasets. [ABSTRACT FROM AUTHOR]

This study investigates distributed convex optimisation that accounts for the inequality constraints over unbalanced directed graphs. In distributed optimisation, agents exchange information on a network to obtain an optimal solution when they only know their own cost function. We propose a distributed primal-dual subgradient method based on a row stochastic weight matrix that is associated with a communication network. In the proposed method, the normalised left eigenvector of the weight matrix is estimated by a consensus algorithm. Then, the subgradient of the Lagrange function is scaled by the estimated values of the left eigenvector to compensate for the imbalance of the information flow in the network. We show that the pair of estimations for the primal and dual problems converge to an optimal primal-dual solution. We also show the relation between the convergence rate of the proposed algorithm and the step-size rule. A numerical example confirms the validity of the proposed method. [ABSTRACT FROM AUTHOR]

An innovative method of obtaining the weights for a Finite Impulse Response (FIR) notch filter using the Eigen-approach is proposed. The Eigen-approach was used to identify the frequencies of the source signals by determining the eigenvector corresponding to the minimum eigenvalue of the autocorrelation matrix. The FIR notch filter was constructed by shifting the root corresponding to the desired signal, which filtered out all other undesired signals. The obtained results indicate that using the Eigen-approach to generate a selective FIR notch filter is a viable solution. [ABSTRACT FROM AUTHOR]

Except for the empty graph, we show that the orthogonal matrix X of the adjacency matrix A determines that adjacency matrix completely, but not always uniquely. The proof relies on interesting properties of the Hadamard product Ξ = X ∘ X. As a consequence of the theory, we show that irregular co-eigenvector graphs exist only if the number of nodes N ≥ 6. Co-eigenvector graphs possess the same orthogonal eigenvector matrix X , but different eigenvalues of the adjacency matrix. Co-eigenvector graphs are the dual of co-spectral graphs, that share all eigenvalues of the adjacency matrix, but possess a different orthogonal eigenvector matrix. We deduce general properties of co-eigenvector graph and start to enumerate all co-eigenvector graphs on N = 6 and N = 7 nodes. Finally, we list many open problems. [ABSTRACT FROM AUTHOR]

Despite the huge importance that the centrality metrics have in understanding the topology of a network, too little is known about the effects that small alterations in the topology of the input graph induce in the norm of the vector that stores the node centralities. If so, then it could be possible to avoid re-calculating the vector of centrality metrics if some minimal changes occur in the network topology, which would allow for significant computational savings. Hence, after formalising the notion of centrality, three of the most basic metrics were herein considered (i.e., Degree, Eigenvector, and Katz centrality). To perform the simulations, two probabilistic failure models were used to describe alterations in network topology: Uniform (i.e., all nodes can be independently deleted from the network with a fixed probability) and Best Connected (i.e., the probability a node is removed depends on its degree). Our analysis suggests that, in the case of degree, small variations in the topology of the input graph determine small variations in Degree centrality, independently of the topological features of the input graph; conversely, both Eigenvector and Katz centralities can be extremely sensitive to changes in the topology of the input graph. In other words, if the input graph has some specific features, even small changes in the topology of the input graph can have catastrophic effects on the Eigenvector or Katz centrality. [ABSTRACT FROM AUTHOR]

The quantum phase estimation algorithm has been utilized by a variety of quantum algorithms, most notably Shor's algorithm, to obtain information regarding the period of a function that is appropriately encoded into a unitary operator. In many cases, it is desired to estimate whether a specific state exhibits a certain pattern quickly. In this paper, we exhibit a methodology based on the QPE algorithm to identify certain patterns. In particular, starting from a properly encoded state, we demonstrate how to construct unitary operators whose eigenvectors correspond to states with proper diagonals. QPE will then output an eigenphase equal to zero with certainty for these states, thereby identifying this class of matrices. For partial matches, our algorithm, based on the tolerance threshold used, will show areas of high similarity, and it will outperform classical ones when the number of mismatches defined by the tolerance is significantly low when compared to the size of the diagonal. [ABSTRACT FROM AUTHOR]

This paper discusses continuous-time quantum walks and asymptotic state transfer in graphs with an involution. By providing quantitative bounds on the components of the eigenvectors of the Hamiltonian, it provides an approach to achieving high-fidelity state transfer by strategically selecting energy potentials based on the maximum degrees of the graphs. The study also involves an analysis of the time necessary for quantum transfer to occur. [ABSTRACT FROM AUTHOR]

The existence of real symmetric interval matrices is a natural phenomenon that can be found in a variety of real-life scenarios. In this particular feature, we develop the principal axis/spectral theorem for real symmetric interval matrices. This theorem paves the way for exciting developments in interval matrix theory, particularly concerning issues such as approximations, dealing with inexact or vague data, coping with the unavailability of data, and managing errors in measurements. To demonstrate the practical utility of our approach, we delve into a concrete real-world example involving vibration analysis within an uncertain environment. This research opens up new horizons for the effective utilization of interval matrices in addressing challenges arising from uncertainty and imprecision in diverse fields. [ABSTRACT FROM AUTHOR]

We derive asymptotic estimates for the projection of the vorticity onto principal directions of material stretching in 3D flows. In flows with pointwise bounded vorticity, these estimates predict vorticity alignment with Lyapunov vectors along trajectories with positive Lyapunov exponents. Specifically, we find that in inviscid flows with conservative body forces, the vorticity exactly aligns with the intersection of the planes orthogonal to the dominant forward and backward Lyapunov vectors along trajectories with positive Lyapunov exponent. Furthermore, we derive asymptotic estimates for the vorticity alignment with the intermediate eigenvector of the rate-of-strain tensor for viscous flows under general forcing. We illustrate these results on explicit solutions of Euler's equation and on direct numerical simulations of homogeneous isotropic turbulence. [ABSTRACT FROM AUTHOR]

Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. We review some known results obtained in the last 10 years, unify several approaches used to achieve such bounds, and extend their scope to a large class of linear and nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue—much in the spirit of earlier results by Donsker–Varadhan and Bañuelos–Carrol—as well as upper bounds on heat kernels. Our methods solely rely on order properties of operators: We devote special attention to the case where the relevant operators enjoy various forms of elliptic or parabolic maximum principle. Additionally, we illustrate our findings with several examples, including p$p$‐Laplacians on domains and graphs as well as Schrödinger operators with magnetic and electric potential, also by means of elementary numerical experiments. [ABSTRACT FROM AUTHOR]

Partition problems in graphs are extremely important in applications, as shown in the Data Science and Machine Learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue a (G) of the Laplacian matrix L G of the graph G. This problem corresponds to the minimization of a quadratic form associated with L G , under certain constraints involving the ℓ 2 -norm. We introduce and investigate a similar problem, but using the ℓ 1 -norm to measure distances. This leads to a new parameter b (G) as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for b (G) for trees. We also comment on an ℓ ∞ -norm version of the problem. [ABSTRACT FROM AUTHOR]

The transport of charged particles or photons in a scattering medium can be modelled with a Boltzmann equation. The mathematical treatment for scattering in such scenarios is often simplified if evaluated in a frame where the scattering centres are, on average, at rest. It is common therefore to use a mixed coordinate system, wherein space and time are measured in a fixed inertial frame, while momenta are measured in a 'co-moving' frame. To facilitate analytical and numerical solutions, the momentum dependence of the phase-space density may be expanded as a series of spherical harmonics, typically truncated at low order. A method for deriving the system of equations for the expansion coefficients of the spherical harmonics to arbitrary order is presented in the limit of isotropic, small-angle scattering. The method of derivation takes advantage of operators acting on the space of spherical harmonics. The matrix representations of these operators are employed to compute the system of equations. The computation of matrix representations is detailed and subsequently simplified with the aid of rotations of the coordinate system. The eigenvalues and eigenvectors of the matrix representations are investigated to prepare the application of standard numerical techniques, e.g. the finite volume method or the discontinuous Galerkin method, to solve the system. [ABSTRACT FROM AUTHOR]

The branch point singularities in the Riemann surface of the parameter space are known as the exceptional points (EPs), at which two or more eigenvalues and their associated eigenvectors simultaneously coalesce. The abrupt bifurcation property around an EP shows a strong spectral response to external perturbations, and hence, is exploited as a new approach to ultrasensitive sensors. Recently, an intriguing proposal for implementing second‐order EPs in optical resonators with external perturbations has shown the superiority of non‐Hermitian degeneracies in the enhancement of sensing. Of particular importance is further improving the sensitivity to even greater extents. To this end, a novel physical mechanism is proposed, namely introducing extra resonances of external Rayleigh scatterers to accomplish strong‐coupling augmentation in sensing. The results, grounded by both theoretical coupled‐mode‐theory calculations and experimental spectral‐domain measurements, show that the EP‐based sensor working at the coherent resonance equips simultaneously substantial perturbation strength and azimuth sensitivity even subjected to extremely weak perturbations. This work paves the way to a new class of highly functional tunable sensors for applications in optics, microwave and acoustics. [ABSTRACT FROM AUTHOR]

This work examines the time complexity of quantum search algorithms on combinatorial t-designs with multiple marked elements using the continuous-time quantum walk. Through a detailed exploration of t-designs and their incidence matrices, we identify a subset of bipartite graphs that are conducive to success compared to random-walk-based search algorithms. These graphs have adjacency matrices with eigenvalues and eigenvectors that can be determined algebraically and are also suitable for analysis in the multiple-marked vertex scenario. We show that the continuous-time quantum walk on certain symmetric t-designs achieves an optimal running time of O n , where n is the number of points and blocks, even when accounting for an arbitrary number of marked elements. Upon examining two primary configurations of marked elements distributions, we observe that the success probability is consistently o(1), but it approaches 1 asymptotically in certain scenarios. [ABSTRACT FROM AUTHOR]

Time history response derivatives with respect to the design variables are frequently used in optimization design, damage detection, structural control, etc. This paper proposes a substructuring method for efficient calculation of higher order time history response derivatives of large-scale structures. First, the global structure is disassembled into several small substructures and the substructural displacement is projected onto the range space of a few substructural master eigenvectors. Afterwards, the derivative of substructural master eigenvectors of a few substructures containing the design variables are assembled to form the reduced first, second, and higher order sensitivity equations with a size of the number of master eigenvectors. The equivalent eigenvector which relates the slave and master eigenvectors is derived to compensate for the inertial effect of discarded slave eigenvectors. Finally, the first, second, and higher order time history response derivatives of global large-scale structure are efficiently solved from the reduced sensitivity equations by using Newmark- β method. A numerical one-bay plane frame and a numerical highway bridge are applied to verify the accuracy and efficiency of proposed substructuring method. [ABSTRACT FROM AUTHOR]

Currently, magnetic gradient tensor-based localization methods face challenges such as significant errors in geomagnetic field estimation, susceptibility to local optima in optimization algorithms, and inefficient performance. In addressing these issues, this article propose a two-point localization method under the constraint of overlaying geometric invariants. This method initially establishes the relationship between the target position and the magnetic gradient tensor by substituting an intermediate variable for the magnetic moment. Exploiting the property of the eigenvector corresponding to the minimum absolute eigenvalue being perpendicular to the target position vector, this constraint is superimposed to formulate a nonlinear system of equations of the target's position. In the process of determining the target position, the Nara method is employed for obtaining the initial values, followed by the utilization of the Levenberg–Marquardt algorithm to derive a precise solution. Experimental validation through both simulations and experiments confirms the effectiveness of the proposed method. The results demonstrate its capability to overcome the challenges faced by a single-point localization method in the presence of some errors in geomagnetic field estimation. In comparison to traditional two-point localization methods, the proposed method exhibits the highest precision. The localization outcomes under different noise conditions underscore the robust noise resistance and resilience of the proposed method. [ABSTRACT FROM AUTHOR]

Let G be a connected general hypergraph of order n with rank r. The unique positive eigenvector x with -n i=1 x r i = 1 corresponding to the spectral radius ρ(G) is called the principal eigenvector of G. In this paper, the relation between each entry of the principal eigenvector of G and the vertex degree associated with this entry is presented. And some bounds for the extreme entries of the principal eigenvector are obtained. As applications, we give some bounds of the spectral radius of G. [ABSTRACT FROM AUTHOR]

As an extension of Sylvester's matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between left and right eigenvectors are derived. Two determinants of the matrices constructed by the left and right eigenvectors are evaluated in closed form. [ABSTRACT FROM AUTHOR]

In this article, we construct free centroid hom-associative algebras and free centroid hom-Lie algebras. We also construct some other relatively free centroid hom-associative algebras by applying the Gröbner–Shirshov basis theory for (unital) centroid hom-associative algebras. Finally, we prove that the "Poincaré–Birkhoff–Witt theorem" holds for certain type of centroid hom-Lie algebras over a field of characteristic 0, namely, every centroid hom-Lie algebra such that the eigenvectors of the map β linearly generates the whole algebra can be embedded into its universal enveloping centroid hom-associative algebra, and the linear basis of the universal enveloping algebra does not depend on the multiplication table of the centroid hom-Lie algebra under consideration. [ABSTRACT FROM AUTHOR]

Two types of spatial heterogeneity can exist simultaneously: continuous variations across an entire space and significant changes that occur only in specific spatial units. Moreover, each of these can act across multiple spatial scales. To effectively detect both continuous and discrete spatial heterogeneity across different scales, this study proposes a novel approach that combines the random effects eigenvector spatially filtering‐based spatially varying coefficient (RE‐ESF‐SVC) model and the generalized lasso (GL) technique. Additionally, a restricted maximum likelihood estimation (REML)‐based two‐step iterative algorithm is developed for parameter estimation. Simulation experiments and an empirical application using rental price data confirm the ability of the proposed model to identify multiscale continuous and discrete spatial heterogeneity. [ABSTRACT FROM AUTHOR]

Analyzing nodal domains is a way to discern the structure of eigenvectors of operators on a graph. We give a new definition extending the concept of nodal domains to arbitrary signed graphs, and therefore to arbitrary symmetric matrices. We show that for an arbitrary symmetric matrix, a positive fraction of eigenbases satisfy a generalized version of known nodal bounds for un-signed (that is classical) graphs. We do this through an explicit decomposition. Moreover, we show that with high probability, the number of nodal domains of a bulk eigenvector of the adjacency matrix of a signed Erdős-Rényi graph is |$\Omega (n/\log n)$| and |$o(n)$|⁠. [ABSTRACT FROM AUTHOR]

Exceptional points (EPs) are non‐Hermitian degeneracies or branch points where eigenvalues and their corresponding eigenvectors coalesce. Due to the complex non‐trivial topology of Riemann surfaces associated with non‐Hermitian Hamiltonians, the dynamical encirclement or proximity of EPs in parameter space has been shown to lead to topological mode conversions and some novel physical phenomena. In fact, degeneracies can also form continuous line geometries, which are called exceptional lines (ELs). The problem is whether the state transfer around the ELs can show different characteristics from the EPs, which is less explored. Here, novel properties of quantum state transfer around the ELs based on a quantum walk platform are explored. It is found that the evolutionary state around the ELs is independent of the initial state and evolution direction, and the transfer of quantum state is more efficient than the case around the EPs. Furthermore, based on such a property, an entangled state generation insensitive to the incident state is realized experimentally. The work opens up a new way for the application of non‐Hermitian physics in the field of quantum information. [ABSTRACT FROM AUTHOR]

We analyze the recently introduced family of preconditioners in [15] for the stage equations of implicit Runge-Kutta methods for two stage methods. We give explicit formulas for the eigenvalues and eigenvectors of the preconditioned systems for a general method and use these to give explicit convergence estimates of preconditioned GMRES for some common choices of the implicit Runge-Kutta methods. This analysis also allows us to qualitatively predict and explain the main observed features of the GMRES convergence behavior, not only bound it. We illustrate our analysis with numerical experiments. We also consider the direction of numerical optimization for improving the preconditioners performance , as suggested in [15]. We consider two different ways – both distinct to the one introduced in [15] – and numerically optimize these, using the explicit bounds obtained beforehand. [ABSTRACT FROM AUTHOR]

Although eigenvectors belong to the core of linear algebra, relatively few closed-form expressions exist, which we bundle and discuss here. A particular goal is their interpretation for graph-related matrices, such as the adjacency matrix of an undirected, possibly weighted graph. [ABSTRACT FROM AUTHOR]

In this paper we consider the Gabriel form of cross-validation (CV) and we investigate how to estimate the optimum rank for lower rank approximations of any dataset that can be written in matrix form, with particular application in multivariate analysis and in the analysis of multienvironment trials. The literature related to the method suggests that it can produce overfitting and poor-quality predictions, characteristics that result in overestimation of the rank. Because of this, it is proposed to change the rank selection criterion, testing thirteen statistics both in the original method and in four proposed extensions that seek to solve the above problems. A comparison is made with two gold standard methods for CV through a simulation study and through the analysis of seventeen real datasets, two of which are general multivariate and fifteen are from experiments with genotype-by-environment interaction. It is concluded that from a predictive point of view, the highest accuracy in estimating the rank is obtained by using a regularized singular value decomposition. [ABSTRACT FROM AUTHOR]

We adapt the Lanchester combat model to represent conflict between vampires and humans. It is assumed that vampires attack humans during the hours of darkness whilst humans attack vampires during the hours of light. The right-hand side of the differential equation model therefore depends upon the hour of the day. A key insight is that to answer the question ‘who wins’ it is not required to ‘stitch together’ the solutions to the differential equation over many days. Rather, the number of combatants surviving at the end of each day can be cast as a difference equation in terms of the numbers surviving at the end of the previous day. Two models are investigated and from the solutions the boundary delimiting the parameter regions of victory and defeat is found. Where appropriate we introduce discussion points, both mathematical and modelling. The mathematical techniques used include finding the eigenvalues and eigenvectors of a matrix, solving linear differential and discrete systems, and considering the physical meaning of the model and questioning its assumptions. Thus a variety of problem-solving skills are developed within the context of a model where students can question the underlying assumptions and propose, and investigate, models of their own. [ABSTRACT FROM AUTHOR]

Let M be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra G , where G is gl (m | n) , osp (2 m | 2 n) or spo (2 m | 2 n). We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for G on M can be obtained from the singular eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. As a consequence, the (super) Gaudin Hamiltonians for G are diagonalizable on the space spanned by singular vectors of M and hence on M. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra so (2 n) and the symplectic Lie algebra sp (2 n) , on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules. [ABSTRACT FROM AUTHOR]

In this paper, a Toeplitz construction method based on eigenvalues and eigenvectors is proposed to combine with traditional denoising algorithms, including fractional low-order moment (FLOM), phased fractional low-order moment (PFLOM), and correntropy-based correlation (CRCO) methods. It can improve the direction of arrival (DOA) estimation of signals in impulsive noise. Firstly, the algorithm performs eigenvalue decomposition on the received covariance matrix to obtain eigenvectors and eigenvalues, and then the Toeplitz matrix is created according to the eigenvectors corresponding to its eigenvalues. Secondly, the spatial averaging method is used to obtain an unbiased estimate of the Toeplitz matrix, which is then weighted and added based on the corresponding eigenvalues. Next, the noise subspace of the Toeplitz matrix is reconstructed to obtain the one that has less angle information. Finally, the DOA of the coherent signal is estimated using the Multiple Signal Classification (MUSIC) algorithm. The improved method based on the Toeplitz matrix can not only suppress the effect of impulsive noise but can also solve the problem of aperture loss due to its decoherence. A series of simulations have shown that they have better performances than other algorithms. [ABSTRACT FROM AUTHOR]

Quasiparticle self‐consistent many‐body perturbation theory (MBPT) methods that update both eigenvalues and eigenvectors can calculate the excited‐state properties of molecular systems without depending on the choice of starting points. However, those methods are computationally intensive even on modern multi‐core central processing units (CPUs) and thus typically limited to small systems. Many‐core accelerators such as graphics processing units (GPUs) may be able to boost the performance of those methods without losing accuracy, making starting‐point‐independent MBPT methods applicable to large systems. Here, we GPU accelerate MOLGW, a Gaussian‐based MBPT code for molecules, with open accelerators (OpenACC) and achieve speedups of up to 9.7×$$ 9.7\times $$ over 32 open multi‐processing (OpenMP) CPU threads. [ABSTRACT FROM AUTHOR]

Commuting Hermitian matrices may be simultaneously diagonalized by a common unitary matrix. However, the numerical aspects are delicate. We revisit a previously rejected numerical approach in a new algorithm called 'do-one-then-do-the-other'. One of two input matrices is diagonalized by a unitary similarity, and then the computed eigenvectors are applied to the other input matrix. Additional passes are applied as necessary to resolve invariant subspaces associated with repeated eigenvalues and eigenvalue clusters. The algorithm is derived by first developing a spectral divide-and-conquer method and then allowing the method to break the spectrum into, not just two invariant subspaces, but as many as safely possible. Most computational work is delegated to a black-box eigenvalue solver, which can be tailored to specific computer architectures. The overall running time is a small multiple of a single eigenvalue-eigenvector computation, even on difficult problems with tightly clustered eigenvalues. The article concludes with applications to a structured eigenvalue problem and a highly sensitive eigenvector computation. [ABSTRACT FROM AUTHOR]

We develop the Euclidean time method of the variational quantum eigensolver for solving the generalized eigenvalue equation A | ϕ n ⟩ = λ n B | ϕ n ⟩ , where A and B are hermitian operators, and | ϕ n ⟩ and λ n are called the eigenvector and the corresponding eigenvalue of this equation, respectively. For the purpose, we modify the usual Euclidean time formalism, which was developed for solving the time-independent Schrödinger equation. We apply our formalism to three numerical examples for test. It is shown that our formalism works very well in all numerical examples. We also apply our formalism to the hydrogen atom and compute the electric polarizability. It turns out that our result is slightly less than that of the perturbation method. [ABSTRACT FROM AUTHOR]

Conversational recommender systems use natural language conversations to elicit user preferences and recommend items proactively. Existing methods based on graph neural networks have been proven to be effective in exploiting knowledge graphs. However, node positions are often treated as constants, which leads to the neglect of graph connectivity due to fuzzy processing. In addition, although the transformer has significant advantages in understanding the text, its secondary computational complexity may be incapable when dealing with long texts. In order to solve these problems, we propose an additive positional conversational recommender model called APCR. This model converts the pair product of transformer into a linear operation, and uses the Laplacian eigenvector to build a location graph. The extended graph neural network captures the topology structure of the location knowledge graph. Specifically, we design an encoder based on additive attention to break through the bottleneck of long text. Furthermore, we develop a recommendation model based on a positional graph neural network to match items with dialogue context, thereby capturing the graph topology. Extensive experiments on the REDIAL dataset show significant improvements in our proposed model over the state-of-the-art methods in recommendation and dialogue generation evaluations. [ABSTRACT FROM AUTHOR]

We establish an almost sure convergence theorem of the stochastic approximation process of Oja for estimating eigenvectors of the Q-symmetric expectation of a random matrix, under a correlation model between the incoming random matrices. This theorem generalizes previous theorems and extends them to the case where the metric Q is unknown and estimated online in parallel. We apply it to streaming principal component analysis of a random vector Z, when a mini-batch of observations of Z is used at each step or all the observations up to the current step. We deal with the case of streaming generalized canonical correlation analysis, with a metric estimated online in parallel. [ABSTRACT FROM AUTHOR]

Principal component analysis is a versatile tool to reduce dimensionality which has wide applications in statistics and machine learning. It is particularly useful for modeling data in high-dimensional scenarios where the number of variables p is comparable to, or much larger than the sample size n. Despite an extensive literature on this topic, researchers have focused on modeling static principal eigenvectors, which are not suitable for stochastic processes that are dynamic in nature. To characterize the change in the entire course of high-dimensional data collection, we propose a unified framework to directly estimate dynamic eigenvectors of covariance matrices. Specifically, we formulate an optimization problem by combining the local linear smoothing and regularization penalty together with the orthogonality constraint, which can be effectively solved by manifold optimization algorithms. We show that our method is suitable for high-dimensional data observed under both common and irregular designs, and theoretical properties of the estimators are investigated under l q (0 ≤ q ≤ 1) sparsity. Extensive experiments demonstrate the effectiveness of the proposed method in both simulated and real data examples. [ABSTRACT FROM AUTHOR]

Derivatives of nonlinear dynamic response are calculated in an exact and efficient manner with respect to material, geometry, mass, and damping parameters. Developments are presented for the Rayleigh and modal damping options that use the updated tangent stiffness. The calculation of exact response sensitivities for those options requires the calculation of derivatives of eigenvalues and eigenvectors; it is also shown that the third-order tensor formed by the derivative of the stiffness matrix with respect to the displacement vector is needed. That tensor, which amends the coefficient matrix of the system of equations that governs the response sensitivities, is nonzero for materials with continuously varying stiffness. The Bouc–Wen material model exhibits that feature and is selected to demonstrate the developments. Correct differentiation and assembly at the material, section, and element levels are highlighted. The results, verified by finite difference, suggest that the sensitivity of the response is influenced by the choice of damping model, strongly for some parameters, particularly when higher modes contribute to the nonlinear structural behavior. [ABSTRACT FROM AUTHOR]

Using a flexible form for ladder operators that incorporates confluent hypergeometric functions, we show how one can determine all of the discrete energy eigenvalues and eigenvectors of the time-independent Schrödinger equation via a single factorization step and the satisfaction of boundary (or normalizability) conditions. This approach determines the bound states of all exactly solvable problems whose wavefunctions can be expressed in terms of confluent hypergeometric functions. It is an alternative that shares aspects of the conventional differential equation approach and Schrödinger's factorization method, but is different from both. We also explain how this approach relates to Natanzon's treatment of the same problem and illustrate how to numerically determine nontrivial potentials that can be solved this way. [ABSTRACT FROM AUTHOR]

Consider the normalized adjacency matrices of random d‐regular graphs on N vertices with fixed degree d⩾3$d\geqslant 3$. We prove that, with probability 1−N−1+ε$1-N^{-1+\varepsilon }$ for any ε>0$\varepsilon >0$, the following two properties hold as N→∞$N \rightarrow \infty$ provided that d⩾3$d\geqslant 3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in N, that is, λ2,|λN|⩽2+N−c$\lambda _2, |\lambda _N|\leqslant 2+N^{-c}$. (ii) All eigenvectors of random d‐regular graphs are completely delocalized. [ABSTRACT FROM AUTHOR]

We study schemes of tensor eigenvectors from an algebraic and geometric viewpoint. We characterize determinantal defining equations of such eigenschemes via linear equations in their coefficients, both in the general and in the symmetric case. We give a geometric necessary condition for a 0-dimensional scheme to be an eigenscheme. [ABSTRACT FROM AUTHOR]

Multifractal analysis is a powerful approach for characterizing ergodic or localized nature of eigenstates in complex quantum systems. In this context, the eigenvectors of random matrices belonging to invariant ensembles naturally serve as models for ergodic states. However, it has been found that the finite-size versions of multifractal dimensions for these eigenvectors converge to unity logarithmically slowly with increasing system size N. In fact, this strong finite-size effect is capable of distinguishing the ergodicity behavior of orthogonal and unitary invariant classes. Motivated by this observation, in this work, we provide semi-analytical expressions for the ensemble-averaged multifractal dimensions associated with eigenvectors in the orthogonal-to-unitary crossover ensemble. Additionally, we explore shifted and scaled variants of multifractal dimensions, which, in contrast to the multifractal dimensions themselves, yield distinct values in the orthogonal and unitary limits as N → ∞ and, therefore, may serve as a convenient measure for studying the crossover. We substantiate our results using Monte Carlo simulations of the underlying crossover random matrix model. We then apply our results to analyze the multifractal dimensions in a quantum kicked rotor, a Sinai billiard system, and a correlated spin-chain model in a random field. The orthogonal-to-unitary crossover in these systems is realized by tuning relevant system parameters, and we find that in the crossover regime, the observed finite-dimension multifractal dimensions can be captured very well with our results. [ABSTRACT FROM AUTHOR]

Pancake-like vortices are often generated by turbulence in geophysical flows. Here, we study the inertia-gravity oscillations that can exist within such geophysical vortices, due to the combined action of rotation and gravity. We consider a fluid enclosed within a triaxial ellipsoid, which is stratified in density with a constant Brunt–Väisälä frequency (using the Boussinesq approximation) and uniformly rotating along a (possibly) tilted axis with respect to gravity. The wave problem is then governed by a mixed hyperbolic-elliptic equation for the velocity. As in the rotating non-stratified case considered by Vantieghem (2014, Proc. R. Soc. A, 470, 20140093. (doi:10.1098/rspa.2014.0093)), we find that the spectrum is pure point in ellipsoids (i.e. only consists of eigenvalues) with polynomial eigenvectors. Then, we characterize the spectrum using numerical computations (obtained with a bespoke Galerkin method) and asymptotic spectral theory. Finally, the results are discussed in light of natural applications (e.g. for Mediterranean eddies or Jupiter's vortices). [ABSTRACT FROM AUTHOR]