Critical value computation for H∞-Riccati difference equations

In designing finite horizon discrete time H_∞ controllers, the associated H_∞-Riccati difference equations must be solved. But the Riccati equation has a non-negative solution only when γ⁻² is small enough. So it is important to get the upper bound of the parameter, i.e., the critical value that ensures the existence of the solution to the Riccati equation. The solution sequence of the Riccati difference equation can be constructed by the conjoined basis of an associated linear Hamiltonian difference system. Based on this expression and the Hamiltonian difference system eigenvalue theorems, the equivalence between the critical value and the first order eigenvalue of the linear Hamiltonian difference system is presented. Since the critical value is also shown to be the fundamental eigenvalue of a generalized Rayleigh quotient, an extended form of Wittrick-Williams algorithm is presented to search this value.     

Linearization of homogeneous Riccati equations

An autonomous system of n coupled homogeneous Riccati equations of the projective type linearizes to its linear part by means of a map which is analytic for almost all initial data. The explicit form of the map is presented.     

Multi-objective optimization in quantum parameter estimation

We investigate quantum parameter estimation based on linear and Kerr-type nonlinear controls in an open quantum system, and consider the dissipation rate as an unknown parameter. We show that while the precision of parameter estimation is improved,it usually introduces a significant deformation to the system state. Moreover, we propose a multi-objective model to optimize the two conflicting objectives:(1) maximizing the Fisher information, improving the parameter estimation precision, and(2)minimizing the deformation of the system state, which maintains its fidelity. Finally, simulations of a simplified ε-constrained model demonstrate the feasibility of the Hamiltonian control in improving the precision of the quantum parameter estimation.     

On the non-extensive nature of the isothermal depolarization relaxation currents in cement mortars

The study of isothermal depolarization current relaxation during the cement hardening process follows a generalized exponential function, which explicitly introduces hierarchically constrained dynamics and macroscopic interactions. The interactions are associated to the non-extensive entropy parameter q, which varies during hardening evolution. Combining ideas of Levy and Tsallis statistics we argue that a Levy walk type mechanism can organize the geometry of the cement mortar heterogeneous system. The estimation of non-extensive q parameter safe concludes that cement's hardening is a subextensive process.     

Weighted Fisher informations, their derivation and use

Fisher information has been used to derive many laws of physics, including its differential equations. In many of these Fisher-based derivations the information in the time measurement is required to be negative. Yet by its expression in the standard derivation of the Cramer-Rao (CR) inequality, a negative Fisher seems impossible. To the contrary, we show that in fact the standard CR derivation allows the Fisher to be meaningfully regarded as either negative or positive. The mathematics allow it, and the choice to be made depends upon the physical parameter whose information level is sought. For example, the rules of special relativity require an imaginary time parameter ict, , with c the speed of light and t the time. The squared time parameter is then negative, requiring (as shown) the Fisher information to be negative as well. Further generalizations of the CR derivation are also made, which allow the estimation of general parameters using arbitrarily weighted mean-squared error criteria. The weights are found to define a family of correspondingly weighted Fisher informations. Finally, a condition is found for achieving efficient estimation in the presence of any weight function.     

The Three-Step Master Equation: Class of Parametric Stationary Solutions

We examine the three-step master equation from the standpoint of the general solution of the associated discrete Riccati equation. We report by this means stationary master solutions depending on a free constant parameter, denoted by D, that should be negative in order to assure the positivity of the solution. These solutions correspond to different discrete Markov processes characterized by the value of D, which is related to specific renormalizations of the transition rates of the chain of states.     

A Constructive Approach to the Soliton Solutions of Integrable Quadrilateral Lattice Equations

Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Bäcklund transformation, and the method of solving a Riccati map by exploiting two known particular solutions. This leads to an expression for the N-soliton-type solutions of a generic equation within this class. As a particular instance we give an explicit N-soliton solution for the primary model, which is Adler’s lattice equation (or Q4).     

Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series

The detection of long range dependence (LRD) is an important task in time series analysis. LRD is often summarized by the well-known Hurst parameter (or exponent) H∈[0,1], which can be estimated by a number of methods. Some of these techniques are designed to be applied to signals behaving as a stationary fractional Gaussian noise (fGn), whereas others imply that the analyzed time series behave as a non-stationary fractional Brownian motion (fBm). Moreover, some estimators do not yield the Hurst parameter but indexes related to H and ranging outside the unit interval. Therefore, the fGn or fBm nature of the studied time series has to be preliminarily analyzed before applying any estimation method, and the relationships between H and the indexes resulting from the analyses have to be taken into account to obtain coherent results. Since fGn-like series represent the increments of fBm-like processes and both the signals are characterized by the same H value by definition, estimators designed for fGn-like series can be applied to fBm-like sequences after preventive differentiation, and conversely estimators designed for fBm-like processes can be applied to fGn-like series after preventive integration. The signal characterization is particularly important when H is estimated on financial time series because the returns represent the first difference of price time series, which are often assumed to behave like self-affine sequences. The analysis of simulated fGn and fBm time series shows that all the considered methods yield comparable H values when properly applied. The reanalysis of several market price time series already studied in the literature points out that a correct application of the estimators (supported by a preventive signal classification) yields homogeneous H values allowing for a useful cross-validation of results reported in different works. Moreover, some conclusions reported in the literature about the anti-persistence of some financial series are shown to be incorrect because of the inappropriate application of the estimation methods.     

Shooter localization and bullet trajectory, caliber, and speed estimation based on detected firing sounds

Shooter localization and estimation of bullet trajectory, caliber and speed have become essential tasks for example in peacekeeping and police assignments. A novel approach for such estimation and localization is presented in this paper, as a numerical estimation method is applied to the problem. Both simulated and recorded gunshot data are considered, as a known bullet shock wave model and detected firing sounds are utilized in creating a likelihood function corresponding to different bullet states. For this, a state-space model of the underlying dynamic system is developed, and a well-known optimization algorithm is used to find the global maximum of the evaluated function. Two different criteria are used to measure the likelihood values, namely the Generalized Cross Correlation (GCC) and the Mean-Squared Error (MSE). The achieved localization and estimation results are accurate and applicable when considering the usability of the method against hostile snipers. The shooter position and bullet state estimation errors vary between 2% and 10%, depending on the estimated parameter at stake.     

Analytic presentation of a solution of the Schrödinger equation

High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into the nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.     

Recognition of gravitation impulses on a correlated non-Gaussian noise background

We consider the problem of gravitation-wave impulse recognition against the background of correlated noise produced by resonance gravitational antennas. A new algorithm for the joint processing of data recorded by spatially separated gravitational detectors is obtained. The algorithm is an alternative to the coincidence scheme, which is traditionally used in gravitation-wave experiments. It is based on the principles, well known in statistical radiophysics, of joint detection and parameter estimation in quasi-deterministic signals.     

贝叶斯频率估计中频率的先验分布对有色噪声作用的影响

在贝叶斯参数估计理论框架下,研究了被测参数的先验分布对有色噪声的抑制作用.选择一个受1/f~α型谱密度有色噪声影响的自旋1/2量子比特作为量子探测系统来估计一个磁场强度的大小,利用贝叶斯代价函数的动力学演化来评判估计的精度,重点研究先验概率分布对噪声非高斯性的限制作用.研究发现:当先验概率的不确定度比较大时,有色噪声的非高斯性对频率估计精度的影响比较小;当先验概率的不确定度比较小时,有色噪声的非高斯性对频率估计精度的影响比较大.     

Three superintegrable two-dimensional oscillators: Superintegrability,nonlinearity, and curvature

The superintegrability of three different two-dimensional oscillators is studied: (i) a nonlinear oscillator dependent on a parameter λ (two-dimensional version of the oscillator of Lakshmanan and Mathews), (ii) a nonlinear oscillator related to the Riccati equation, and (iii) the standard harmonic oscillator on constant curvature spaces. They can be considered as nonlinear deformations, or curvature-dependent versions, of the linear harmonic oscillator. The text was submitted by the authors in English.     

Monte Carlo method and sensitivity estimations

It is shown that, starting from any existing Monte Carlo algorithm for estimation of a physical quantity A, it is possible to implement a simple additional procedure that simultaneously estimates the sensitivity of A to any problem parameter. The corresponding supplementary cost is very low as no additional random sampling is required. The principle is presented on a formal basis and simple radiative transfer examples are used for illustration.     

ON THE PROLONGATION STRUCTURE OF ERNST EQUATION

An SL(2R) ×R¹(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is pfesented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neugebauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbertproblem (HHP), would be covered by this condition.     

Lattice dynamics and the incommensurability of the structures of hexagonal polytypes of diamond

In the density functional theory approximation, we have calculated the phonon dispersions and the densities of vibrational states of the 2H, 3C, 4H, 6H, and 8H hexagonal polytypes of diamond. We have found that a one-dimensional incommensurate modulation of the structure arises along the hexagonal axis, the parameter of which not only exceeds the parameters of basic translations of the crystal lattices of polytypes, but also is not a multiple of them. Based on the estimation of interactions between bilayers of carbon atoms in the structure of polytypes, we have assumed that competing interactions between bilayers are the main mechanism by which incommensurability arises. We have shown that optical measurements of vibrational frequencies in the center of the Brillouin zone of polytypes make it possible to retrieve the dispersion of acoustic branches of cubic diamond.     

Computational issues in parameter estimation and feedback control problems for partial differential equation systems

We consider computational aspects of two problems governed by partial differential equations systems: (i) the computation of approximate Riccati feedback gains for LQR control problems,(ii) the estimation of parameters using methods based on approximate gradients computed via a costate algorithm. Sufficient conditions for convergence of algorithms for these problems are discussed; in both cases, some type of adjoint or costate system convergence is required in addition to the expected system convergence. Examples are discussed that illustrate computational difficulties which may arise if only convergence of the approximate system holds (i.e., in the event costate convergence does not hold).     

Relation between the Grüneisen ratio and the pressure dependence of Poisson's ratio for metals

The Grüneisen ratio of crystalline solids is shown to be dependent on a parameter n whose values are characteristic of each solid, and can be determined by two independent ways: from experimental shock data and from the pressure derivative of Poisson's ratio. The determinations are made for several metals, using data on the pressure derivatives of polycrystalline elastic moduli or of the second order elastic constants measured on single crystals, and giving the pressure derivatives of Poisson's ratio by means of the Voigt-Reuss-Hill averaging procedure. The values of the parameter n deduced from shock data are found to be in good agreement with those deduced from the pressure derivatives of Poisson's ratio. Positive and negative values of parameter n correspond respectively to increasing and decreasing Poisson's ratio with increasing pressure. Discussion of the results is made using the linear and the quadratic relationships between shock velocity and particle velocity. It is shown that shock wave data cannot yield directly an accurate estimation of the derivative of the initial slope of the Hugoniot.     

Experimental Study of High Temperature Fracture Behavior of A286 Superalloy at 650°C

In this paper an experimental work is done for investigation of high temperature fracture properties of A286 superalloy at 650°C. Stress intensity factor K and parameter C* for this superalloy are determined experimentally. For estimation of these parameters, an instrument is developed for investigation of high temperature fracture properties. For estimation of stress intensity factor, compliance method is used. For this purpose four different compact tension specimens are tested and the parameter K is estimated. Creep tests are done for the selected specimens and parameter C* is determined by semiempirical relationships at 650°C. In these tests it is concluded that the specimens are placed near the plane stress condition. Crack growth behavior of this alloy is also studied. High incubation time (600 h) leads to overaging and therefore this alloy after this time showed very ductile creep properties, and fast creep crack growth was the major result of this overaging phenomenon. Finally the obtained results are compared with well-known nonexperimental methods for determination of these parameters. The obtained results showed that the results are in good agreement with each other.