Functions Of Eigenvalues Research Articles

Parabolic Regularity of Spectral Functions

This paper is devoted to the study of the second-order variational analysis of spectral functions. It is well-known that spectral functions can be expressed as a composite function of symmetric functions and eigenvalue functions. We establish several second-order properties of spectral functions when their associated symmetric functions enjoy these properties. Our main attention is given to characterize parabolic regularity for this class of functions. It was observed recently that parabolic regularity can play a central rule in ensuring the validity of important second-order variational properties, such as twice epi-differentiability. We demonstrates that for convex spectral functions, their parabolic regularity amounts to that of their symmetric functions. As an important consequence, we calculate the second subderivative of convex spectral functions, which allows us to establish second-order optimality conditions for a class of matrix optimization problems. Funding: The research of A. Mohammadi is funded by a postdoctoral fellowship from Georgetown University. E. Sarabi is partially supported by the U.S. National Science Foundation [Grant DMS 2108546].

Points of inflection of special eigenvalue functions as indicators of stiffness maxima/minima of proportionally loaded structures

The stiffness of a proportionally loaded structure may continuously increase or decrease. As a special exception, it may be constant. On the other hand, an initially stiffening (softening) structure may turn into a softening (stiffening) structure. At the load level of such a change the stiffness of the structure attains an extreme value. The task of this work is to present mathematical conditions for these load levels. Lack of them represents a void in the pertinent literature. The practical significance of the aforementioned changes is the one of indicators of the mechanical behavior to be expected after their occurrence. The second and the last author have recently presented a condition for the load level at which the stiffness of a proportionally loaded structure becomes a minimum value. It is given as d2(ℜ(χ1(λ)))/dλ2=0, representing the condition for a point of inflection of the real part of a complex eigenvalue function χ1(λ), where λ denotes a dimensionless load parameter. The underlying linear eigenvalue problem has two indefinite coefficient matrices, which is a necessary condition for complex regions of eigenvalue functions. These matrices are established with hybrid elements, available in a commercial finite element program. In the present work, d2(χ1(λ))/dλ2=0 is shown to be the condition for the load level at which the stiffness of a proportionally loaded structure attains a maximum value. The eigenvalue function concerned has no complex region. It is also shown that the displacement elements, which are the basis for their extension to the employed hybrid elements, are unable to indicate the load level at an extreme value of the stiffness.

A simplified analytical method for lateral dynamic responses of a transmission tower due to rockfall impact

Transmission tower structures support high-voltage power lines that carry electricity over long distance and rockfall is one of critical disasters during its safe operation. This paper presented a simplified analytical methodology for lateral dynamic responses of a transmission tower structure due to rockfall impact. At first, the lateral dynamic displacement of a lattice transmission tower structure can be represented by a second-order partial differential equation and half sine wave was used for rockfall impact. Then, the solution can be approximated by a set of specified shape functions multiplied by time-dependent generalized coordinates. And the partial differential equation is discretized into a set of single degree of freedom system. And then the shape function can be determined by solved an eigenvalue function and the fundamental frequency of a transmission tower can was derived based on the energy method and combination synthesis method. Finally, the lateral dynamic displacements can be approximately obtained. A numerical study of a transmission tower was conducted. Parametric study of the effect of impact location height, impact duration, peak impact force, as well as the distribution of cross-arms on dynamic responses were also carried out. And the results show that the discrepancy between the analytical and the computed of fundamental frequency is less than 3%, the error of dynamic displacement is within 10%, and the fundamental frequency of the structure decreases with the increase of the tower top additional mass ratio.

Interaction of the DKP particle with the Coulomb-type potential and Aharonov–Bohm potential under the cosmic rainbow gravity

The Duffin–Kemmer–Petiau (DKP) particle with spin 0 interacts with the Aharonov–Bohm (AB) magnetic vector potential and scalar Coulomb-type potential in the cosmic string space–times under the framework of rainbow gravity (RG). By using Bethe–Ansatz method, we obtain the energy eigenvalue and approximate solution to the wave function of the DKP particle with spin 0. We select two sets of rainbow functions and analyze their influence on the energy eigenvalue and wave functions. We find that the energy eigenvalue is determined by the parameters of the rainbow function. It further shows that the rainbow function affects the properties of the space–time where the DKP particle located, and also affects the distribution probability of DKP particles in the space.

Exact first moments of the RV coefficient by invariant orthogonal integration

The RV coefficient measures the similarity between two multivariate configurations, and its significance testing has attracted various proposals in the last decades. We present a new approach, the invariant orthogonal integration, permitting to obtain the exact first four moments of the RV coefficient under the null hypothesis.Our proposal can be applied to any multivariate setting endowed with Euclidean distances between the observations. It also covers the weighted setting of observations of unequal importance, where the exchangeability assumption, justifying the usual permutation tests, breaks down.The proposed RV moments express as simple functions of the kernel eigenvalues occurring in the weighted multidimensional scaling of the two configurations (spectral effective dimensionality, spectral skewness and spectral excess kurtosis). The expressions for the third and fourth moments seem original, and explain the marked asymmetry and kurtosis of the RV coefficient. They permit to test the significance of the RV coefficient by Cornish–Fisher cumulant expansion, beyond the normal approximation, as illustrated on a small dataset.The first three moments can be obtained by elementary means, but computing the fourth moment requires a more sophisticated apparatus, the Weingarten calculus for orthogonal groups.

Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box

We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the discrete Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows, and then use it to give a partial proof of the conjecture. For the one dimensional case, we give a complete proof by means of Green function bounds.

Pólya’s conjecture for Euclidean balls

The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. Pólya’s conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya’s conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya’s conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk.

In-situ measurement of the heat generation of LiFePO4 and Li(Ni1/3Co1/3Mn1/3)O2 batteries

The heat flow behavior of lithium-ion batteries during normal operation is an important consideration for safety. Yet, there are few studies on this subject. In current work, the heat flow of Li(Ni1/3Co1/3Mn1/3)O2 (NCM) and LiFePO4 (LFP) batteries during a discharge and charge cycle at different ambient temperatures and discharge rates has been tested for the first time using a high precision isothermal microcalorimeter (TAM IV). The agreement of the measured heat flow over several repeated cycles is excellent, with a spike in heat flow occurring at the end of discharge. Low temperatures have a greater effect on the heat flow of the battery than high temperatures. NCM batteries have greater thermal stability than LFP batteries during heating and cooling, as well as when the discharge rate increases. Meanwhile, the heat flow spike of NCM batteries has a good exponential dependence on temperature and discharge rate. At a certain temperature, a model of the heat flow eigenvalue function, which is only dependent on the discharge rate, is developed. There is a strong correlation between the heat flow behavior and the electrical behavior of LFP and NCM batteries. The research provides a practical basis for the application, monitoring, and optimization of lithium-ion batteries.

Topological effects on non-relativistic eigenvalue solutions under AB-flux field with pseudoharmonic- and Mie-type potentials

In this paper, we investigate the quantum dynamics of a non-relativistic particle confined by the Aharonov–Bohm quantum flux field with pseudoharmonic-type potential in the background of topological defect produced by a point-like global monopole. We solve the radial Schrödinger equation analytically and determine the exact eigenvalue solution of the quantum system. Afterwards, we consider a Mie-type potential in the quantum system and solve the radial equation analytically and obtain the eigenvalue solution. We analyze the effects of the topological defect and the quantum flux with these potentials on the energy eigenvalue and wave function of the non-relativistic particles. In fact, it is shown that the energy levels and wave functions are influenced by the topological defect shifted the result compared to the flat space results. In addition, the quantum flux field also shifted the eigenvalue solutions and an analogue of the Aharonov–Bohm effect for bound-states is observed. Finally, we utilize these eigenvalue solutions to some known diatomic molecular potential models and presented the energy eigenvalue and wave function.

Spin current and internal Zeeman field in spin-orbit coupled rings

We investigate the one-dimensional quantum ring constructed by the spin-orbit coupled material, in which the quantum spin-Hall Bernevig-Zhang (BZ) Hamiltonian and Rashba-Dirac (RD) type spin-orbit coupling are taken into account (called RD-BZ Hamiltonian in this paper). It is known that the curvature of the ring generates an out-of-plane effective magnetic field, acting as an internal Zeeman field. We find that only the BZ coupling can change the strength of the internal Zeeman field, which enables us to detect the effect of the internal Zeeman field. Furthermore, we find that the total angular momentum is conserved in the RD-BZ Hamiltonian, and the energy eigenvalue and wave function must be modified to fit the conserved quantity, which are ignored in the previous studies. The conductance without leads is discussed. Different from the previous results, we find that the conductance behaves like a beat phenomenon resulting from the interplay between the magnetic flux and Aharonov-Casher (AC) phase, and thus, it can oscillate without passing through the insulating state in some regimes of magnetic flux or AC phase. Importantly, we find that the conductance with integer and half-integer magnetic flux provides us a method to measure the AC phase. The cancellation of the internal Zeeman field due to the BZ coupling can be detected by using specific fractional magnetic flux. In the ring with nonvanishing RD and BZ couplings, the conductance can exhibit a quasiplateau near the small RD coupling. The increase in the strength of BZ coupling would result in wider quasiplateau in conductance, which implies that the ring could remain insulating state (or conducting state) regardless of the small change in RD coupling. The thermal average of spin and charge currents are calculated at low temperatures without impurities. Importantly, we find that the persistent spin and charge currents as a function of the magnetic flux exhibit nodelike lines, which amazingly are perpendicular to each other. As a consequence, the persistent spin current could be nonzero even when the charge current vanishes at nonzero magnetic flux. The result suggests a pure spin current in quantum rings, and its direction can be reversed by changing the magnetic flux. The increase in the BZ coupling would exhibit the plateaulike pure spin current.

Integral formulas for a foliated sub-Riemannian manifold

We apply the notion of foliation to a nonholonomic manifold, which was introduced for the geometric interpretation of constrained systems in mechanics. We prove a series of integral formulas for a foliated sub-Riemannian manifold, that is, a Riemannian manifold equipped with a distribution $${{\mathscr {D}}}$$ and a foliation $${{\mathscr {F}}}$$ whose tangent bundle is a subbundle of $${{\mathscr {D}}}$$ . Our integral formulas generalize some results for a foliated Riemannian manifold and involve the shape operators of $${\mathscr {F}}$$ with respect to normals in $${\mathscr {D}}$$ , the curvature tensor of induced connection on $${\mathscr {D}}$$ and arbitrary functions depending on elementary symmetric functions of eigenvalues of the shape operators. For a special choice of these functions, integral formulas with the Newton transformations of the shape operators of $${\mathscr {F}}$$ are obtained. Application to a foliated sub-Riemannian manifold with restrictions on the curvature and extrinsic geometry of $${\mathscr {F}}$$ and also to codimension-one foliations are given.

Study the Ground State Perturbed Eigenfunction with Modified Order Parameters of Hydrogen Atom

Studying the eigen value function of a perturbed hydrogen atom in its ground state with various orders is the goal of this investigation (1st, 2nd, 3rd, 4th, 5th, 6th and 7th). The new equation is created using the theoretic way to investigate the nature of eigen function. At various perturbed order parameters, it is discovered that the nature rapidly decreases with distance from the Bohr’s radius. Additionally, the nature with perturbed order parameters was discovered to be linear at various separations beyond Bohr’s radius. This demonstrates that while a large amount of energy is needed to perturbed a system close to the Bohr’s radius, very little energy is needed to perturbed a system farther away from the Bohr’s radius.

Elasticity-based locally-exact homogenization theory for three-phase composites considering the morphological effect of carbon fibers

Elasticity-based locally-exact homogenization theory (LEHT) is further extended in this paper to investigate the mechanical behaviors of three-phase composites with orthotropic interlayer as well as circumferentially orthotropic (CO) and radially orthotropic (RO) carbon fibers that are determined by their morphological basal plane. Taking advantage of the Fourier series representations for displacements of each phase, in-plane and out-of-plane exact solutions are obtained for orthotropic media through rigorous mathematical derivations and the internal eigenvalue functions are directly solved within rectangular and hexagonal representative unit cells (RUCs) subjected to axial normal, transverse loading and axial shear loading, respectively. The balanced variational principle is applied to enforce periodical boundary conditions which ensures rapid convergence of displacement fields. The solutions of LEHT are verified with far-field Eshelby problems with good agreement and the morphological effects of fiber are investigated by comparing CO/RO properties and equivalent transversely isotropic (TI) parameters obtained from replacement scheme. Homogenized elastic moduli of RUCs with orthotropic interlayer are predicted through homogenization constitutive relations and compared to special extension of the composite cylinders approach and numerical results based on the asymptotic expansion homogenization method to demonstrate excellent correlation. Finally, the effects of orthotropic interlayer have been demonstrated and local stress distributions are obtained from a “top-to-down” procedure, illustrating constituents’ interactions from microscope perspective and identifying the possible crack initiations/propagations within microstructures.

Responses of SDOF Structures With SPIS-<b>Ⅱ</b> Dampers Under Random Seismic Excitation

A closed-form solution of response of SDOF structure with SPIS-II damper subjected to seismic excitation modeled by Clough-Pezien spectrum is proposed, and the shock absorption performance and influencing factors of that system are studied based on the proposed method. Firstly, the equation of motion of SPIS-II damper is established, and the unified expression of the structural response frequency domain solution such as structural displacement and inertial output force are obtained. Secondly, based on the rational expression decomposition and residue theorem, quadratic quadrature equations of frequency response eigenvalue function and Clough-Pezien spectrum are obtained respectively, and then quadratic quadrature equations of structural response power spectrum are induced. Thirdly, concise closed-form solutions of the 0 ~ 2 order spectral moments of the structural response are acquired. The proposed method and the virtual excitation method are used to analyze a case respectively, which verifies the correctness of the proposed method. Finally, the proposed method is used to analyze the influence of the parameters of inertia capacity on the seismic performance of the structure. The research shows that the response solution obtained by the method in this paper is a closed-form solution, which is better than the virtual excitation method in terms of computational efficiency and computational accuracy; when <i>μ</i>_<i>ω</i>unchanged, the damping performance is enhanced with the increasing of <i>μ</i>_<i>m</i>and <i>μ</i>_<i>ξ</i>; when <i>μ</i>_<i>ω</i>=1, the damping performance is improved to the greatest.

Conditions for minimum stiffness of proportionally loaded structures

During proportional loading of structures a significant re-arrangement of stresses may occur. It may result e.g. in the transition from a softening to a stiffening structure. This may even be the case for constant material stiffness. Since the terms stiffening structure and softening structure are lacking universally accepted definitions, it is not astonishing that the search for conditions for minimum stiffness of proportionally loaded multi degree of freedom structures has so far attracted very little attention. Presentation of such a condition, in the framework of the Finite Element Method (FEM), is the task of this publication. It is based on a linear eigenvalue problem that renders complex eigenvalues and eigenvectors possible. The point of inflection in the real part of the complex section of a specific eigenvalue function is shown to mark the load-level of minimum stiffness of a proportionally loaded structure. Complex eigenvalues require the indefiniteness of both real symmetric coefficient matrices of the underlying linear eigenvalue problem. In the given case, the first one depends on a dimensionless load parameter, whereas the second one is obtained by specialization of the first one for the onset of loading. These matrices are established with the help of a special hybrid finite element, available in a commercial FE program. The proposed condition for minimum stiffness of proportionally loaded structures is verified numerically. The results obtained in a parameter study of the investigated structure are found to agree well with the ones following from an alternative condition for the transition from a softening to a stiffening structure. This condition is based on a mechanically objective arc length in the form of a representative displacement of the structure.

Wind vibration responses of structure with generalized Maxwell model viscoelastic dampers

A new method for analysis of wind vibration response of structures with viscoelastic dampers is presented. Most existing studies focused on the numerical solution, and in this paper, an analytical method based on quadratic orthogonalized power spectral density function (QO-PSDF) was applied to obtain the closed-form solution of the structural dynamic responses with generalized Maxwell model (GMM) viscoelastic damper under Davenport spectral wind excitation. Firstly, the equations in motion of GMM viscoelastic dampers were coupled with the structural dynamic equation, which was decoupled into a linear combination of first-order vibration equations through complex mode method (CMM). Secondly, the frequency response eigenvalue function (FREF) was quadratic orthogonalized, which was combined with power spectral density function (PSDF) of Davenport spectral to obtain a QO-PSDF of the structural dynamic responses, and the exact closed-form solutions of the spectral moments and variances of the dynamic response were successfully acquired using the QO-PSDF. Finally, a 16-storey reinforced concrete building structure equipped with GMM viscoelastic dampers is taken as an example to validate the accuracy and applicability of the closed-form solution through a series of structural responses, including absolute displacements, inter-storey displacements, absolute velocities, inter-storey velocities.

Seismic responses of structure with six-element generalized model viscoelastic dampers

In this paper, the closed-form solution of seismic response of the structure with six-element generalized model (SGM) viscoelastic damper is analytically investigated based on complex model method (CMM) and pseudo excitation method (PEM), named as CM-PEM. Firstly, the damping characteristic of SGM viscoelastic damper is combined into structural dynamic equation which was decoupled into linear combination of first order equations by CMM. And the closed-form solutions of the structural dynamic responses are obtained as follow. Based on PEM, the power spectral density function (PSDF) of the structure is divided into the product of PSDF of the ground excitation and frequency response eigenvalue function. Through mathematical transformation, PSDF of the structural dynamic responses is quadratically orthogonalized, which can be integrated to obtain the closed-form expression of spectral moments. A 20-storey reinforced concrete structure is taken as an example study for verifying the accuracy and efficiency of the proposed CM-PEM method. A comparatively study between increasing column lateral stiffness and installation of SGM damper is finally conducted, and the results indicate installation of SGM damper shows better energy dissipation capacity.

On soliton solutions and soliton interactions of Kulish–Sklyanin and Hirota–Ohta systems

In this paper we consider a simplest two-dimensional reduction of the remarkable three-dimensional Hirota-Ohta system. The Lax pair of the Hirota-Ohta system was extended to a Lax triad by adding extra third linear equation, whose compatibility conditions with the Lax pair of the Hirota-Ohta imply another remarkable systems: the Kulish-Sklyanin system (KSS) together with its first higher commuting flow, which we can call as vector complex MKdV. This means that any common particular solution of these both two-dimensional integrable systems yields a corresponding particular solution of the three-dimensional Hirota-Ohta system. Using the dressing Zakharov-Shabat method we derive the $N$-soliton solutions of these systems and analyze their interactions, i.e. derive explicitly the shifts of the relative center-of-mass coordinates and the phases as functions of the discrete eigenvalues of the Lax operator. Next we relate to these NLEE system of Hirota--Ohta type and obtain its $N$-soliton solutions

A few results on permittivity variations in electromagnetic cavities

We study the eigenvalues of time-harmonic Maxwell's equations in a cavity upon changes in the electric permittivity ε of the medium. We prove that all the eigenvalues, both simple and multiple, are locally Lipschitz continuous with respect to ε. Next, we show that simple eigenvalues and the symmetric functions of multiple eigenvalues depend real analytically upon ε and we provide an explicit formula for their derivative in ε. As an application of these results, we show that for a generic permittivity all the Maxwell eigenvalues are simple.

Study on the Size and Occurrence Effects of Equivalent Elasticity Modulus of Orthogonal Random Jointed Rock Masses

Under the engineering background of a + 144-m mining platform on an open pit slope of Xinqiao Mining Corporation, the eigenvalue distribution function of joints was acquired based on the massive statistics of field joints. Next, an orthogonal random joint model was constructed using the Monte Carlo method based on the UDEC software platform. Subsequently, the representative elementary volume (REV) size and occurrence effects of the equivalent elasticity modulus of the orthogonal random jointed rock mass were explored through numerical experiments with lots of discrete elements. The results showed that the equivalent elasticity modulus of the orthogonal random jointed rock mass was of evident size effect, and the relationship between the mean equivalent elasticity modulus Eeq and the REV size of the rock mass followed the negative power function distribution. Before the model size reached the REV size, its Eeq was greatly discrete. Under different joint angles, both the standard deviation σf and coefficient of variation δ of the equivalent elasticity modulus of this model presented discrete distribution. As the model size reached the REV size, the mean equivalent elasticity modulus Eeq showed “two-high and one-low” morphological distribution. Eeq was small when the main joint angle was α∈[30°, 40°], but it was large when α = 0° or 90°. δ was taken as 10%, so the REV size of the corresponding Eeq was 10 m × 20 m. When the model size reaches the REV size, the occurrence effect of its equivalent elasticity modulus was no longer significant.