Negative Eigenvalues Research Articles

On discretizing sea-ice dynamics on triangular meshes using vertex, cell or edge velocities

Discretization of the equations of Viscous Plastic and Elastic Viscous Plastic (EVP) sea ice dynamics on triangular meshes can be done by placing discrete velocities at vertices, cells or edges. Since there are more cells and edges than vertices, the cell- and edge-based discretizations simulate more linear kinematic features at the same mesh than the vertex discretization. However, the discretizations based on cell and edge velocities may suffer from non-trivial kernels in the strain rate or stress divergence operators and need either special strain rate computations for cell velocities, or stabilization for edge velocities. An elementary Fourier analysis clarifies how kernels are removed, and also shows that cell and edge velocity placement leads to spurious branches of stress divergence operator with large negative eigenvalues. Although spurious branches correspond to fast decay and are not expected to distort sea ice dynamics, they demand either smaller internal time steps or higher stability parameters in explicit EVP-like methods.

Isostables for Stochastic Oscillators.

Thomas and Lindner [P. J. Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014).PRLTAO0031-900710.1103/PhysRevLett.113.254101], defined an asymptotic phase for stochastic oscillators as the angle in the complex plane made by the eigenfunction, having a complex eigenvalue with a least negative real part, of the backward Kolmogorov (or stochastic Koopman) operator. We complete the phase-amplitude description of noisy oscillators by defining the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue. Our results suggest a framework for stochastic limit cycle dynamics that encompasses noise-induced oscillations.

ON THE SPECTRUM OF THE TWO-PARTICLE SCHRÖDINGER OPERATOR WITH POINT POTENTIAL: ONE DIMENTIONAL CASE

In the paper a one-dimensional two-particle quantum system interacted by two identical point interactions is considered. The corresponding Schr\"{o}\-dinger operator (energy operator) $h_\varepsilon$ depending on $\varepsilon,$ is constructed as a self-adjoint extension of the symmetric Laplace operator. The main results of the work are based to the study of the operator $h_\varepsilon.$ First the essential spectrum is described. The existence of unique negative eigenvalue of the Schr\"{o}dinger operator is proved. Further, this eigenvalue and corresponding eigenfunction are found.

Microwave Studies of Three Chiral Ensembles in Chains of Coupled Dielectric Resonators

Random matrix theory has proven very successful in the understanding of the spectra of chaotic systems. Depending on symmetry with respect to time reversal and the presence or absence of a spin 1/2 there are three ensembles, the Gaussian orthogonal (GOE), Gaussian unitary (GUE), and Gaussian symplectic (GSE) one. With a further particle-antiparticle symmetry the chiral variants of these ensembles, the chiral orthogonal, unitary, and symplectic ensembles (the BDI, AIII, and CII in Cartan's notation) appear which are the main point of interest in this paper. Following a recently published work on chiral random matrix ensembles and their experimental realizations (Rehemanjiang et al., PRL 124, 116801 (2020)), this is achieved by using dielectric cylinders placed between two parallel aluminium plates. These cylinders act as microwave resonators which are used to create tight-binding chains of finite length up to N=5. The different ensembles are achieved by using different types of couplings: For the orthogonal case spatial proximity is used, for the unitary case microwave circulators are used, and for the symplectic case a combination of circulators and cables is used to create the necessary symmetry. In all cases the predicted repulsion behavior between positive and negative eigenvalues for energies close to zero are verified by a comparison with theory taking the finite size of the systems into account. We will show that the difference to the expected universal behavior is given by logarithmic corrections only. These corrections stem from the Hamiltonians having zero entries in their off-diagonal blocks.

Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue

We say that a bounded domain $\Omega$ is optimal for the first positive curl eigenvalue $\mu_1(\Omega)$ if $\mu_1(\Omega)\leq \mu_1(\Omega')$ for any domain $\Omega'$ with the same volume. In spite of the fact that $\mu_1(\Omega)$ is uniformly lower bounded in terms of the volume, in this paper we prove that there are no axisymmetric optimal (and even locally minimizing) domains with $C^{2,\alpha}$ boundary that satisfies a mild technical assumption. As a particular case, this rules out the existence of $C^{2,\alpha}$ optimal axisymmetric domains with a convex section. An analogous result holds in the case of the first negative curl eigenvalue.

Finding a Hamiltonian cycle by finding the global minimizer of a linearly constrained problem

It has been shown that a global minimizer of a smooth determinant of a matrix function corresponds to the largest cycle of a graph. When it exists, this is a Hamiltonian cycle. Finding global minimizers even of a smooth function is a challenge. The difficulty is often exacerbated by the existence of many global minimizers. One may think this would help, but in the case of Hamiltonian cycles the ratio of the number of global minimizers to the number of local minimizers is typically astronomically small. There are various equivalent forms of the problem and here we report on two. Although the focus is on finding Hamiltonian cycles, and this has an interest in and of itself, this is just a proxy for a class of problems that have discrete variables. The solution of relaxations of these problems is typically at a degenerate vertex, and in the neighborhood of the solution the Hessian is indefinite. The form of the Hamiltonian cycle problem we address has the virtue of being an ideal test problem for algorithms designed for discrete nonlinear problems in general. It is easy to generate problems of varying size and varying character, and they have the advantage of being able to determine if a global solution has been found. A feature of many discrete problems is that there are many solutions. For example, in the frequency assignment problem any permutation of a solution is also a solution. A consequence is that a common characteristic of the relaxed problems is that they have large numbers of global minimizers and even larger numbers of both local minimizers, and saddle points whose reduced Hessian has only a single negative eigenvalue. Efficient algorithms that seek to find global minimizers for this type of problem are described. Results using BONMIN, a solver for nonlinear problems with continuous and discrete variables, are also included.

Multi-pulse edge-localized states on quantum graphs

We construct the edge-localized stationary states of the nonlinear Schrödinger equation on a general quantum graph in the limit of large mass. Compared to the previous works, we include arbitrary multi-pulse positive states which approach asymptotically a composition of N solitons, each sitting on a bounded (pendant, looping, or internal) edge. We give sufficient conditions on the edge lengths of the graph under which such states exist in the limit of large mass. In addition, we compute the precise Morse index (the number of negative eigenvalues in the corresponding linearized operator) for these multi-pulse states. If N solitons of the edge-localized state reside on the pendant and looping edges, we prove that the Morse index is exactly N. The technical novelty of this work is achieved by avoiding elliptic functions (and related exponentially small scalings) and closing the existence arguments in terms of the Dirichlet-to-Neumann maps for relevant parts of the given graph. We illustrate the general results with three examples of the flower, dumbbell, and single-interval graphs.

Vibrational Analysis of Paraelectric–Ferroelectric Transition of LiNbO3: An Ab-Initio Quantum Mechanical Treatment

The phase transitions between paraelectric (PE) and ferroelectric (FE) isomorph phases of LiNbO3 have been investigated quantum mechanically by using a Gaussian-type basis set, the B3LYP hybrid functional and the CRYSTAL17 code. The structural, electronic and vibrational properties of the two phases are analyzed. The vibrational frequencies evaluated at the Γ point indicate that the paraelectric phase is unstable, with a complex saddle point with four negative eigenvalues. The energy scan of the A2u mode at −215 cm−1 (i215) shows a dumbbell potential with two symmetric minima. The isotopic substitution, performed on the Li and Nb atoms, allows interpretation of the nontrivial mechanism of the phase transition. The ferroelectric phase is more stable than the paraelectric one by 0.32 eV.

Instantaneous Normal Modes Reveal Structural Signatures for the Herschel-Bulkley Rheology in Sheared Glasses

The Herschel-Bulkley law, a universal constitutive relation, has been empirically known to be applicable to a vast range of soft materials, including sheared glasses. Although the Herschel-Bulkley law has attracted public attention, its structural origin has remained an open question. In this Letter, by means of atomistic simulation of binary Lennard-Jones glasses, we report that the instantaneous normal modes with negative eigenvalues, or so-called imaginary modes, serve as the structural signatures for the Herschel-Bulkley rheology in sheared glasses.

Stability of smooth periodic travelling waves in the Camassa–Holm equation

AbstractWe solve the open problem of spectral stability of smooth periodic waves in the Camassa–Holm equation. The key to obtaining this result is that the periodic waves of the Camassa–Holm equation can be characterized by an alternative Hamiltonian structure, different from the standard formulation common to the Korteweg‐de Vries equation. The standard formulation has the disadvantage that the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We prove that the nonstandard formulation has the advantage that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region where the smooth periodic waves exist.

Eigenvalue estimates for magnetic Schrodinger operators in a waveguide

We present an upper estimate for the number of negative eigenvalues below the essential spectrum for the magnetic Schrodinger operator with Aharonov-Bohm magnetic field in a strip.

Semi-Implicit Time Integration with Hessian Eigenvalue Corrections for a Larger Time Step in Molecular Dynamics Simulations.

In molecular dynamics simulations, the limited time step size has been a barrier to simulating long-time behaviors. Implicit time integration methods allow markedly larger time steps than the standard explicit time method, although they have major drawbacks such as overheads solving linear systems and instability of Newton iterations. To overcome these issues, we propose a semi-implicit time integration scheme, the semi-implicit Hessian correction (SimHec) scheme, for overdamped Langevin dynamics. The method focuses on the Hessian matrices of bonded and nonbonded interactions, where components with large negative Hessian eigenvalues are cut off in the linear approximation of momentum equations to avoid instability. The narrow band Hessian matrix enables an efficient parallelized linear solution with an overlapping approximation. We tested SimHec for the interdomain fluctuations in adenylate kinase and the powerstroke transition of myosin II using a coarse-grained protein model. SimHec reproduced the same dynamics as the explicit method, although the transition dynamics tended to be accelerated and fluctuations in bonded potentials were slightly reduced. These deviations were corrected using a hybrid method, SimHec-H, which adds explicit time steps after the semi-implicit time step. The proposed scheme allowed us to use time steps 50-200 times larger than those in explicit time integration, which resulted in a speedup factor of 7-30 taking the overhead into account.

On the existence, uniqueness, and stability of periodic waves for the fractional Benjamin–Bona–Mahony equation

AbstractThe existence, uniqueness, and stability of periodic traveling waves for the fractional Benjamin–Bona–Mahony equation is considered. In our approach, we give sufficient conditions to prove a uniqueness result for the single‐lobe solution obtained by a constrained minimization problem. The spectral stability is then shown by determining that the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum and mass at the periodic wave has no negative eigenvalues. We propose the Petviashvili's method to investigate the spectral stability of the periodic waves for the fractional Benjamin–Bona–Mahony equation, numerically. Some remarks concerning the orbital stability of periodic traveling waves are also presented.

Explaining the specific heat of liquids based on instantaneous normal modes

The successful prediction of the specific heat of solids is a milestone in the kinetic theory of matter due to Debye. No such success, however, has ever been obtained for the specific heat of liquids, which has remained a mystery for over a century. A theory of specific heat of liquids is derived here using a recently proposed analytical form of the vibrational density of states of liquids, which takes into account saddle points in the liquid energy landscape via the so-called instantaneous normal modes (INMs), corresponding to negative eigenvalues (imaginary frequencies) of the Hessian matrix. The theory is able to explain the typical monotonic decrease in specific heat with temperature observed in liquids in terms of the average INM excitation lifetime decreasing with T (in accordance with the Arrehnius law) and provides an excellent single-parameter fitting to several sets of experimental data for atomic and molecular liquids. It also correlates the height of the liquid energy barrier with the slope of the specific heat in the function of temperature in accordance with the available data. These findings demonstrate that the specific heat of liquids is controlled by the instantaneous normal modes, i.e., by localized unstable (exponentially decaying) vibrational excitations and provide the missing connection among anharmonicity, saddle points in the energy landscape, and the thermodynamics of liquids.

Design of high-gain observers based on sampled measurements via the interval arithmetic

The main objective of this paper is to implement a high-gain “practical” observer with the interval arithmetic in the case of sampled measurements. In order to pursue this objective, two closed-form expressions are proposed to compute certified bounds on the solution of a single-input linear system, with distinct, real, and negative eigenvalues, when both the initial condition and the control input lie in given intervals. Furthermore, it is shown how such formulas can be extended in the case of sampled input measurements. These results are used to design an interval high-gain observer for non-linear systems in observer normal form. In particular, it is shown how to implement a high-gain “practical” observer with the interval arithmetic, both in continuous-time and in discrete-time, so to compute guaranteed interval estimates of the state of the system.

Holographic entanglement negativity and replica symmetry breaking

Since the work of Ryu and Takayanagi, deep connections between quantum entanglement and spacetime geometry have been revealed. The negative eigenvalues of the partial transpose of a bipartite density operator is a useful diagnostic of entanglement. In this paper, we discuss the properties of the associated entanglement negativity and its Rényi generalizations in holographic duality. We first review the definition of the Rényi negativities, which contain the familiar logarithmic negativity as a special case. We then study these quantities in the random tensor network model and rigorously derive their large bond dimension asymptotics. Finally, we study entanglement negativity in holographic theories with a gravity dual, where we find that Rényi negativities are often dominated by bulk solutions that break the replica symmetry. From these replica symmetry breaking solutions, we derive general expressions for Rényi negativities and their special limits including the logarithmic negativity. In fixed-area states, these general expressions simplify dramatically and agree precisely with our results in the random tensor network model. This provides a concrete setting for further studying the implications of replica symmetry breaking in holography.

Magneto-nanofluid coolants past heated shrinking/stretching surfaces: Dual solutions and stability analysis

Recent advancement in nanotechnology has provides a veritable platform for the emergence of a better ultrahigh-performance coolant known as nanofluid for many engineering and industrial technologies. In this study, we examine the influence of magnetic field on heat transfer enhancement of nanofluids coolants consisting of Cu-water, Al2O3-water, and Fe3O4-water over a slippery but convectively heated shrinking and stretching surfaces. Based on some realistic assumptions, the nonlinear model differential equations are obtained and numerically tackled using shooting procedure with Runge-Kutta-Fehlberg integration scheme. The existence dual solutions in the specific range of shrinking surface parameters is found. Temporal stability analysis to small disturbances is performed on these dual solutions. It is detected that the upper solution branch is stable, substantially realistic with positive smallest eigenvalues while the lower solution branch is unstable with negative smallest eigenvalues. Influence of numerous emerging parameters on the momentum and thermal boundary layer profiles, skin friction, Nusselt number and heat transfer enhancement rate are depicted graphically and quantitatively discussed.

Theoretical study of energy, inertia and nullity of phenylene and anthracene

Abstract Energy of a molecule plays an important role in physics, chemistry and biology. In mathematics, the concept of energy is used in graph theory to help other subjects such as chemistry and physics. In graph theory, nullity is the number of zeros extracted from the characteristic polynomials obtained from the adjacency matrix, and inertia represents the positive and negative eigenvalues of the adjacency matrix. Energy is the sum of the absolute eigenvalues of its adjacency matrix. In this study, the inertia, nullity and signature of the aforementioned structures have been discussed.

Shrinkage Priors on Complex-Valued Circular- Symmetric Autoregressive Processes

We investigate shrinkage priors on power spectral densities for complex-valued circular-symmetric autoregressive processes. We construct shrinkage predictive power spectral densities, which asymptotically dominate (i) the Bayesian predictive power spectral density based on the Jeffreys prior and (ii) the estimative power spectral density with the maximum likelihood estimator, where the Kullback-Leibler divergence from the true power spectral density to a predictive power spectral density is adopted as a risk. Furthermore, we propose general constructions of objective priors for Kähler parameter spaces by utilizing a positive continuous eigenfunction of the Laplace-Beltrami operator with a negative eigenvalue. We present numerical experiments on a complex-valued stationary autoregressive model of order 1.

Investigation of the effect of DamKohler and Stanton non-dimensional numbers on the stability of continuous stirred tank reactors

The fully mixed continuous stirred tank reactor is an important type of industrial reactors mainly used to produce high volume products such as petrochemicals, detergents, sanitary products and products that are in demand in the market. Knowing the dynamic behavior of chemical reactors is of great importance in setting up, designing, controlling and stopping reactors. In this paper, the effect of non-dimensional numbers Damkohler and Stanton on the stability of a continuous stirred tank reactor in which a first-order exothermic reaction takes place is investigated. First, a mathematical model of the system's dynamic behavior was presented. Then, by simultaneous solving of the equations of mass and energy around the fixed point in MATLAB software, the effect of the mentioned numbers was investigated. The results show that the continuous stirred tank reactor shows different behaviors in different ranges of Damkohler and Stanton numbers. This reactor behaves unstable in small and large ranges of Damkohler and Stanton numbers due to the presence of mixed or positive and negative eigenvalues. The best range for Damkohler and Stanton numbers is close to 1, because in this range the reactor shows stable behavior due to having two negative eigenvalues. In this range, in addition to the stability, the conversion is also 100%. Finaly the ratio of Stanton to DamKohler was investigated as St / Da˃1 and St / Da = 1. If St / Da = 1, the system is in steady state, but in St / Da˃1, the system moves away from steady state.