Quadratic Operator Research Articles

Anharmonicity-induced excited-state quantum phase transition in the symmetric phase of the two-dimensional limit of the vibron model

In most cases, excited state quantum phase transitions can be associated with the existence of critical points (local extrema or saddle points) in a system's classical limit energy functional. However, an excited-state quantum phase transition might also stem from the lowering of the asymptotic energy of the corresponding energy functional. One such example occurs in the 2D limit of the vibron model, once an anharmonic term in the form of a quadratic bosonic number operator is added to the Hamiltonian. The study of this case in the broken-symmetry phase was presented in Phys. Rev. A. 81 050101 (2010). In the present work, we delve further into the nature of this excited-state quantum phase transition and we characterize it in the, previously overlooked, symmetric phase of the model making use of quantities such as the effective frequency, the expected value of the quantum number operator, the participation ratio, the density of states, and the quantum fidelity susceptibility. In addition to this, we extend the usage of the quasilinearity parameter, introduced in molecular physics, to characterize the phases in the spectrum of the anharmonic 2D limit of the vibron model and a down-to-earth analysis has been included with the characterization of the critical energies for the linear isomers HCN/HNC.

SPECTRAL INEQUALITIES FOR COMBINATIONS OF HERMITE FUNCTIONS AND NULL-CONTROLLABILITY FOR EVOLUTION EQUATIONS ENJOYING GELFAND–SHILOV SMOOTHING EFFECTS

AbstractThis work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost linearly at infinity, and provide quantitative estimates, with respect to the energy level of the Hermite functions seen as eigenfunctions of the harmonic oscillator, for the constants appearing in these spectral estimates. These spectral inequalities allow us to derive the null-controllability in any positive time for evolution equations enjoying specific regularizing effects. More precisely, for a given index $\frac {1}{2} \leq \mu <1$ , we deduce sufficient geometric conditions on control subsets to ensure the null-controllability of evolution equations enjoying regularizing effects in the symmetric Gelfand–Shilov space $S^{\mu }_{\mu }(\mathbb {R}^{n})$ . These results apply in particular to derive the null-controllability in any positive time for evolution equations associated to certain classes of hypoelliptic non-self-adjoint quadratic operators, or to fractional harmonic oscillators.

An adaptive state transition algorithm with local enhancement for global optimization

State transition algorithm (STA) is an efficient and powerful metaheuristic method for solving global optimization problems, and it has been successfully applied in many engineering fields in the past few years. However, the basic STA has weak local search capability and shows slow convergence rate and low convergence accuracy in the later search stage. In view of the above shortcomings, an adaptive state transition algorithm (ASTA) with local enhancement is proposed in this paper. Firstly, the order of using state transformation operators and the optimal parameters of the operators are considered in each iteration of ASTA, and a statistical method is employed to adaptively select the optimal transformation operator and the parameter values of the optimal operator to speed up the search process. Then, an adaptive call strategy is adopted to determine its convergence to the neighborhood of the optimal solution and to decide whether to perform the quasi-Newton operator for local enhancement. Finally, the degree to which the current solution is close to the optimal solution is judged by the information of historical solutions, and an analytical solution is quickly obtained by calling the quadratic interpolation operator. The effectiveness of the proposed ASTA is checked, through a comparison with other metaheuristic methods, on 15 benchmark functions and several real-world optimization problems. Experimental results show that ASTA has a stronger search capability than the basic STA, STA variants, and some state-of-the-art metaheuristic methods.

Modified Algorithm Based on Quadratic Correlation Time Delay Estimation

In the passive positioning system, the traditional generalized cross-correlation algorithm estimates the effect under the influence of noise and reverberation, while the delay estimation algorithm after homomorphic filtering reduces the anti-reverb ability of the cross-correlation algorithm, but also reduces the cross-correlation of the signal, resulting in the decrease in the noise resistance of the cross-correlation algorithm. Based on this, the quadratic cross-correlation operation of the full-pass component of the signal is performed after homomorphic filtering to improve its noise immunity. In addition, the Introduction of Hilbert Difference sharpens the secondary cross-correlation peaks to make peak detection that reflects the time delay more accurate. Experimental simulation shows that the proposed method can effectively suppress the influence of noise and reverberation in non-stationary speech signals, and improve the performance of delay estimation.

Fixed Points of Quadratic Operators Defined on a Three-Dimensional Simplex of Idempotent Measures

Fixed Points of Quadratic Operators Defined on a Three-Dimensional Simplex of Idempotent Measures

Quadrature operator eigenstates and energy eigenfunctions of f-deformed oscillators

Quadrature operator eigenstates and energy eigenfunctions of f-deformed oscillators

A Hybrid Approach of Spotted Hyena Optimization Integrated with Quadratic Approximation for Training Wavelet Neural Network.

Spotted hyena optimization (SHO) is one of the newly evolved swarm-based metaheuristic optimization methods based on the social life cycle of hyenas. In recent times SHO is being applied to various engineering applications as well as to solve real-life complications. In this paper, we have hybridized SHO with quadratic approximation operator (QAO), termed as QASHO. The proposed QASHO has been scrutinized to enhance the exploitation ability, aiming to achieve global optimum, as QAO performs better within the local confinement region. Furthermore, the proposed approach shows improved strength in terms of escaping from the local minima trap, as in each iteration we discard some of the worst individuals by some suitable ones. To validate the proficiency of the proposed QASHO approach, 28 standard problems have been preferred in connection with IEEE-CEC-2017. The outcome observed from the suggested method has also equated with contemporary metaheuristic approaches. To prove the statistical significance, a nonparametric test has also been accomplished. Additionally as a real-life application, the suggested approach QASHO has utilized to train wavelet higher-order neural networks (HONN) by choosing datasets from the UCI store. The above correlations reveal that QASHO can deal with complex optimization tasks.

Tomography in loop quantum cosmology

We analyze the tomographic representation for the Friedmann–Robertson–Walker (FRW) model within the Loop Quantum Cosmology framework. We focus on the Wigner quasi-probability distributions associated with Gaussian and Schrödinger cat states, and then, by applying a Radon integral transform for those Wigner functions, we are able to obtain the symplectic tomograms which define measurable probability distributions that fully characterize the quantum model of our interest. By appropriately introducing the quantum dispersion for a rotated and squeezed quadrature operator in terms of the position and momentum, we efficiently interpret the properties of such tomograms, being consequent with Heisenberg’s uncertainty principle. We also obtain, by means of the dual tomographic symbols, the expectation value for the volume operator, which coincides with the values reported in the literature. We expect that our findings result interesting as the introduced tomographic representation may be further benefited from the well-developed measure techniques in the areas of Quantum optics and Quantum information theory.

On nonhomogeneous geometric quadratic stochastic operators

n this paper, we construct a nonhomogeneous geometric quadratic stochastic operator generated by 2-partition $\xi$ on countable state space $X=\mathbb{Z}^{*}$. The limiting behavior of such operator is studied. We have proved that such operator possesses the regular property.

A framework for fitting quadratic-bilinear systems with applications to models of electrical circuits

We propose a method for fitting quadratic-bilinear models from data. Although the dynamics characterizing the original model consist of general analytic nonlinearities, we rely on lifting techniques for equivalently embedding the original model into the quadratic-bilinear structure. Here, data are given by generalized transfer function values that can be sampled from the time-domain steady-state response. This method is an extension of methods that perform bilinear, or quadratic inference, separately. It is based on first identifying the underline minimal linear model with the interpolatory method known as the Loewner framework, and then on inferring the best supplementing nonlinear (quadratic and bilinear) operators, by solving an optimization problem. The proposed data-driven method finds applications in engineering serving the scopes of robust simulation, design, and control. Model examples of electrical circuits with nonlinear components (diodes) were used to test the method's performance.

Бесконечномерные бистохастические квадратичные операторы в пространстве li

The article is devoted to bistochastic quadratic operators. The concept of a bistochastic quadratic operator is introduced and the property of such operators is studied. A necessary and sufficient condition for bistochasticity is given. An analogue of Birkhoff s theorem is studied for the class of bistochastic quadratic operators. A sufficient condition for the extremity of bistochastic quadratic operators is obtained, and a necessary condition is also obtained for small dimensions. The concept of bistochastic quadratic operators is generalized, and the concept of a bistochastic operator of arbitrary order is introduced. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.

The Moore-Penrose inverse of accretive operators with application to quadratic operator pencils

We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.

Differential quadrature technique for frequencies of the coupled circular arch–arch beam bridge system

This study is appointed to discover the dynamic behavior related to one of the most applicable systems used in engineering, namely coupled beam bridge system. For this reason, the system, which is constructed by two circular arch beams and an elastic layer between them, is vibrationally investigated here. In addition, the geometrical strategy is hired to modify the essential position associated with the situation of the two beams lying on each other. Moreover, the first-order shear deformation theory (FSDT) and the cylindrical panel scheme are employed to discover the fundamental features of the relationships linked to the system. Moreover, by engaging Hamilton’s principle and part-by-part integration (or Green–Gauss theory), the governing equations related to the motion of the system are achieved. Furthermore, the robust semi-analytical technique, renowned by the generalized differential quadrature operator is executed to split up the motion equations linked to the system. Additionally, to confirm the offered configuration, some benchmarks are chosen and solved. Finally, the influence of the diverse boundary conditions, geometrical characteristics, and different values related to the elastic mid-layer stiffness on the natural frequencies of the coupled circular arch–arch beam bridge system are determined.

Quadratic non-stochastic operators: examples of splitted chaos

There is one-to-one correspondence between quadratic operators (mapping \({\mathbb {R}}^m\) to itself) and cubic matrices. It is known that any quadratic operator corresponding to a stochastic (in a fixed sense) cubic matrix preserves the standard simplex. In this paper we find conditions on the (non-stochastic) cubic matrix ensuring that corresponding quadratic operator preserves simplex. Moreover, we construct several quadratic non-stochastic operators which generate chaotic dynamical systems on the simplex. These chaotic behaviors are splitted meaning that the simplex is partitioned into uncountably many invariant (with respect to quadratic operator) subsets and the restriction of the dynamical system on each invariant set is chaos in the sense of Devaney.

Normal modes of a waveguide as eigenvectors of a self-adjoint operator pencil

A waveguide with a constant, simply connected section S is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section S, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.

Volterra-type Quadratic Stochastic Operators with a Homogeneous Tournament

As is known [1], each quadratic stochastic operator of Volterra type acting on a finitedimensional simplex defines a certain tournament, the properties of which make it possible to study the asymptotic behavior of the trajectories of this Volterra operator. In this paper, we introduce the concept of a homogeneous tournament and study the dynamic properties of Volterra operators corresponding to homogeneous tournaments in the simplex S4.

Estimation of Gaussian random displacement using non-Gaussian states

In continuous-variable quantum information processing, quantum error correction of Gaussian errors requires simultaneous estimation of both quadrature components of displacements on phase space. However, quadrature operators $x$ and $p$ are non-commutative conjugate observables, whose simultaneous measurement is prohibited by the uncertainty principle. Gottesman-Kitaev-Preskill (GKP) error correction deals with this problem using complex non-Gaussian states called GKP states. On the other hand, simultaneous estimation of displacement using experimentally feasible non-Gaussian states has not been well studied. In this paper, we consider a multi-parameter estimation problem of displacements assuming an isotropic Gaussian prior distribution and allowing post-selection of measurement outcomes. We derive a lower bound for the estimation error when only Gaussian operations are used, and show that even simple non-Gaussian states such as single-photon states can beat this bound. Based on Ghosh's bound, we also obtain a lower bound for the estimation error when the maximum photon number of the input state is given. Our results reveal the role of non-Gaussianity in the estimation of displacements, and pave the way toward the error correction of Gaussian errors using experimentally feasible non-Gaussian states.

A non-linear discrete-time dynamical system related to epidemic SISI model

Abstract We consider SISI epidemic model with discrete-time. The crucial point of this model is that an individual can be infected twice. This non-linear evolution operator depends on seven parameters and we assume that the population size under consideration is constant, so death rate is the same with birth rate per unit time. Reducing to quadratic stochastic operator (QSO) we study the dynamical system of the SISI model.

Determination of the Range of Magnetic Interactions from the Relations between Magnon Eigenvalues at High-Symmetry κ PointsSupported by the National Natural Science Foundation of China (Grant Nos. 11834006, 12004170, and 12104215), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20200326), and the Excellent Programme in Nanjing University. Xiangang Wan also acknowledges the support from the Tencent Foundation through

Magnetic exchange interactions (MEIs) define networks of coupled magnetic moments and lead to a surprisingly rich variety of their magnetic properties. Typically MEIs can be estimated by fitting experimental results. Unfortunately, how many MEIs need to be included in the fitting process for a material is unclear a priori, which limits the results obtained by these conventional methods. Based on linear spin-wave theory but without performing matrix diagonalization, we show that for a general quadratic spin Hamiltonian, there is a simple relation between the Fourier transform of MEIs and the sum of square of magnon energies (SSME). We further show that according to the real-space distance range within which MEIs are considered relevant, one can obtain the corresponding relationships between SSME in momentum space. By directly utilizing these characteristics and the experimental magnon energies at only a few high-symmetry k points in the Brillouin zone, one can obtain strong constraints about the range of exchange path beyond which MEIs can be safely neglected. Our methodology is also generally applicable for other Hamiltonian with quadratic Fermi or Boson operators.

Description of the Set of Strictly Regular Quadratic Bistochastic Operators and Examples

The present paper focuses on the dynamical systems of the quadratic bistochastic operators (QBO) on the standard simplex. In the paper, we show the character of connection of the dynamical systems of a bistochastic operator with the dynamical systems of the extreme bistochastic operators. In addition, we prove that almost all quadratic bistochastic operators are strictly regular and give a description of the strictly regular quadratic bistochastic operators in the convex polytope QBO. Furthermore, the density of the set of strictly regular QBO in the set of QBO is proved and nontrivial examples of strictly regular bistochastic operators are given.