A digital model of coupled oscillators

A new model of coupled oscillators is proposed and investigated. All phase variables and parameters are integer-valued. The model is shown to exhibit two types of motions, those which involve periodic phase differences, and those which involve drift. Traditional dynamical concepts such as stability, bifurcation and chaos are examined for this class of integer-valued systems. Numerical results are presented for systems of two and three oscillators. This work has application in digital technology.     

A multicriteria model for risk sorting of natural gas pipelines based on ELECTRE TRI integrating Utility Theory

This paper proposes a multicriteria model for assessing risk in natural gas pipelines, and for classifying sections of pipeline into risk categories. The model integrates Utility Theory and the ELECTRE TRI method. It aims to help transmission and distribution companies, when engaged in risk management and decision-making, to consider the multiple dimensions of risk that may arise from pipeline accidents. Pipeline hazard scenarios are presented, and it is argued that the assessment of risk in natural gas pipelines should not be based solely on probabilities of human fatalities, but should involve a wider perspective that simultaneously takes into consideration the human, environmental and financial dimensions of impacts of pipeline accidents. Finally, in order to verify the effectiveness of the model set out, a numerical application based on a real case study is presented.     

Geometric polyhedral models for nanotubes comprising hexagonal lattices

Two new models for the geometric structure of nanotubes comprising hexagonal lattices are described. The existing models for nanotubes typically involve rolled up planar sheets and ignore discrepancies due to curvature. The first of the models presented here assumes that all atomic locations are equidistant from the tube axis which applies for single species nanotubes such as carbon nanotubes. This model assumes that all bond angles and all bond lengths are equal in the cylindrical state, and that all atoms are equidistant from the tube axis, and from these three assumptions, expressions are given for the major geometric parameters. The second model extends this notion to tubes where all the atomic locations are not equidistant from the tube axis, which may be employed to model nanotubes comprising two chemical species that bond into a hexagonal lattice such as boron nitride nanotubes. In the second model, all bond lengths are taken to be equal and the atoms of the same species are taken to be equidistant from the tube axis, and the nanotube is assumed to comprise two species and thus there may be two radii. Fundamental to both models is the determination of a solution of a transcendental equation. Here we present a new formal Lagrange expansion of the solution. Previously given asymptotic series expansions of the exact formulae for both models lead to the conventional expressions as the leading order term. Although the correction terms are typically small, knowledge of the precise structure may be critical to comprehending many nanoscale phenomena. The new models also give rise to an expression for the wall thickness, an important geometric parameter for which at present no reliable information is available.     

Conditional least squares and copulae in claims reserving for a single line of business

One of the main goals in non-life insurance is to estimate the claims reserve distribution. A generalized time series model, that allows for modeling the conditional mean and variance of the claim amounts, is proposed for the claims development. On contrary to the classical stochastic reserving techniques, the number of model parameters does not depend on the number of development periods, which leads to a more precise forecasting.Moreover, the time series innovations for the consecutive claims are not considered to be independent anymore. Conditional least squares are used to estimate model parameters and consistency of these estimates is proved. The copula approach is used for modeling the dependence structure, which improves the precision of the reserve distribution estimate as well.Real data examples are provided as an illustration of the potential benefits of the presented approach.     

State space models on special manifolds

This paper concerns modeling time series observations in state space forms considered on the Stiefel and Grassmann manifolds. We develop a state space model relating the time series observations to a sequence of unobserved state or parameter matrices assuming the matrix Langevin noise processes on the Stiefel manifolds. We show a Bayes method for estimating the state matrices by the posterior modes. We consider a further extended state space model where two sequences of unobserved state matrices are involved. A simple state space model on the Grassmann manifolds with matrix Langevin noise processes is also investigated.     

Bootstrapping the empirical distribution of a linear process

The validity of the moving block bootstrap for the empirical distribution of a short memory causal linear process is established under simple conditions that do not involve mixing or association. Sufficient conditions can be expressed in terms of the existence of moments of the innovations and summability of the coefficients of the linear model. Applications to one and two sample tests are discussed.     

The Doubly-Pareto Uniform Distribution with Applications in Uncertainty Analysis and Econometrics

In this paper we present a novel four parameter continuous univariate distribution that can be motivated from at least two approaches. The first one views the distribution as a generalization of the uniform one that allows for uncertainty specification at the vicinity of its bounds (gradually) represented via two Pareto tails. The second one is that of an asymmetric heavy-tailed, peaked distribution with an unbounded domain with the property that the location of the mode is not uniquely determined but rather is described by a uniform range. Properties of the distribution are described and a maximum likelihood estimation (MLE) procedure for the mode location and the Pareto tails parameters is presented. The procedure is illustrated by means of an i.i.d. sample of standardized log-differences of quarterly monthly US certificate deposit interest rates for the period 1964–2004. The sample is constructed utilizing the Auto-Regressive Conditional Heteroscedastic (ARCH) time series model devised by the Nobel Laureate R.F. Engle (1982).     

Estimation of the Extreme Flow Distributions by Stochastic Models

The t-year event is a commonly used characteristic to describe the extreme flood peak in hydrological designs. The annual maximum series (AMS) and the partial duration series (PDS) are two basic approaches in flood analyses. In this paper, we first derive the distribution of the maximum extreme or the joint distribution of two or more maximum extremes from historical records based on a stochastic model, and then estimate statistical characteristics, including the t-year event, from the distribution. In addition to the two classical approaches (AMS and PDS), two additional approaches are proposed for estimating the unknown parameters in this paper. The first one uses two or more annual maximums (MAMS) as the sample to estimate the distribution of the maximum extremes. The second one uses multi-variate shock model to estimate the distribution of the maximum extremes for a multi-modal streamflow. The distribution of the extreme streamflow and the associated characteristics in the Bird Creek in Avant, Oklahoma, in the St. Johns River in Deland, Florida, and in the West Walker River in Coleville, California are estimated by using the stochastic model. To investigate further the performance of the estimation, the stochastic models based on AMS, MAMS and PDS related are also applied to the simulated data. The results show that the stochastic model and the related methods are reliable.     

半无穷大裂纹端部粘聚力分析

准脆性材料裂纹端部断裂过程区粘聚力是导致非线性断裂特性的重要原因,根据准脆性材料的断裂特性,对存在粘聚力分布的半无穷大裂纹力学分析模型,由变形叠加原理得到以该粘聚应力分布为未知函数的积分方程,通过对积分方程的分析推证,得到了该分布函数解的数学结构和级数型表达式;提出了由实际裂纹张开位移,确定裂纹端部粘聚力分布函数的两种方法:其一由连续的裂纹张开位移通过积分变换求解未知函数级数展开项的系数,其二是由离散的裂纹张开位移数据通过最小二乘法确定该函数;推导出了相应方法求解未知量的代数方程,并且给出了适当的算例和讨论。     

On a Sharp Interface Model for Phase Transitions in Steel

Steel is one of the most widely used materials in the world with a broad spectrum of properties. The microstructure and the distribution of the different phases are of great importance, since they each possess different properties. A sharp interface model for the austenite-ferrite phase transition is presented. Mechanical effects due to eigenstrains resulting from the different densities of the phases are taken into account. The governing PDEs in each phase are a diffusion equation for the carbon concentration and the balance of momentum. Across the free interface, separating the two phases, the physical quantities may have discontinuities, which are controlled by jump conditions. Consistency of the model with the 2nd law of thermodynamics is shown. Numerical simulations for these types of free boundary problems are quite complex and involve appropriate methods to determine the interface position. One possibility to circumvent the explicit determination of the free boundary is the use of regularization techniques in form of phase field methods, where the interface is tracked implicitly. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of periodic orbits of nonlinear oscillators by means of piecewise-linearization methods

An approximate method based on piecewise linearization is developed for the determination of periodic orbits of nonlinear oscillators. The method is based on Taylor series expansions, provides piecewise analytical solutions in three-point intervals which are continuous everywhere and explicit three-point difference equations which are P-stable and have an infinite interval of periodicity. It is shown that the method presented here reduces to the well-known Störmer technique, is second-order accurate, and yields, upon applying Taylor series expansion and a Padé approximation, another P-stable technique whenever the Jacobian is different from zero. The method is generalized for single degree-of-freedom problems that contain the velocity, and (approximate) analytical solutions are presented. Finally, by introducing the inverse of a vector and the vector product and quotient, and using Taylor series expansions and a Padé approximation, the method has been generalized to multiple degree-of-freedom problems and results in explicit three-point finite difference equations which only involve vector multiplications.     

Asymptotic results for conditional measures of association of a random sum

Asymptotic results are obtained for several conditional measures of association. The chosen random variables are the first two order statistics and the total sum within a random sum. Many of the results have confirmed the “one-jump” property of the risk model. Non-trivial limits are obtained when the dependence among the first two order statistics is considered. Our results help in understanding the extreme behaviour of well-known reinsurance treaties that involve only few large claims. Interestingly, the Pearson product-moment correlation coefficient between the first two order statistics provides an alternative procedure to estimate the tail index of the underlying distribution.     

IN-SEASON FISHERY MANAGEMENT: A BAYESIAN MODEL

A Bayesian model is presented for optimizing harvest rates on an uncertain resource stock during the course of a fishing season. Pre-season stock status information, in the form of a “prior” probability distribution, is updated using new data obtained through the operation of the fishery, and harvest rates are chosen to achieve a balance between conservation concerns and fishing interests. A series of fishery scenarios are considered, determined by the stock size distribution and the timing distribution; the uncertainty in the fish stock is seen to have a rather complex influence on optimal harvest rates. The model is applied to a specific example, the Skeena River sockeye salmon fishery.     

Distribution free tests for comparing trends in Poisson series

Tests are presented for comparing trends in the rate of occurence of events for two Poisson series. The tests are based on a product model which is similar to the one proposed by Cox [2]. The model allows arbitrary trends in the individual series. Although the density function of the observable variables does not belong to the exponential family, the method often used for exponential families is shown to be applicable for constructing the tests.     

A degradation path-dependent approach for remaining useful life estimation with an exact and closed-form solution

Remaining useful life (RUL) estimation is regarded as one of the most central components in prognostics and health management (PHM). Accurate RUL estimation can enable failure prevention in a more controllable manner in that effective maintenance can be executed in appropriate time to correct impending faults. In this paper we consider the problem of estimating the RUL from observed degradation data for a general system. A degradation path-dependent approach for RUL estimation is presented through the combination of Bayesian updating and expectation maximization (EM) algorithm. The use of both Bayesian updating and EM algorithm to update the model parameters and RUL distribution at the time obtaining a newly observed data is a novel contribution of this paper, which makes the estimated RUL depend on the observed degradation data history. As two specific cases, a linear degradation model and an exponential-based degradation model are considered to illustrate the implementation of our presented approach. A major contribution under these two special cases is that our approach can obtain an exact and closed-form RUL distribution respectively, and the moment of the obtained RUL distribution from our presented approach exists. This contrasts sharply with the approximated results obtained in the literature for the same cases. To our knowledge, the RUL estimation approach presented in this paper for the two special cases is the only one that can provide an exact and closed-form RUL distribution utilizing the monitoring history. Finally, numerical examples for RUL estimation and a practical case study for condition-based replacement decision making with comparison to a previously reported approach are provided to substantiate the superiority of the proposed model.     

基于EGB2分布族的GAS-EGARCH模型与VaR预测

GARCH族模型是刻画资产收益率的常用工具,在风险度量领域具有广泛应用。为了更有效地描述收益率的偏斜厚尾等特征,越来越多学者对GARCH族模型的条件分布形式进行了研究。但是仅对GARCH模型条件分布进行修正是不够的,还需要对模型本身的函数形式进行修正。基于得分函数的时变参数建模思想近年来受到广泛关注,本文借助这一思想对EGARCH模型中对数标准差进行时变波动建模,并利用EGB2分布族作为模型的条件分布,进而建立GAS-EGARCH-EGB2模型。以我国10只中证行业指数为研究对象考察GAS-EGARCH-EGB2模型的风险预测效果,GAS-EGARCH-EGB2模型样本外VaR预测表现普遍优于ACM-EGARCH-EGB2模型。     

An analytical solution for two and three dimensional nonlinear Burgers' equation

This paper derives analytical solutions for the two dimensional and the three dimensional Burgers' equation. The two-dimensional and three-dimensional Burgers' equation are defined in a square and a cubic space domain, respectively, and a particular set of boundary and initial conditions is considered. The analytical solution for the two dimensional Burgers' equation is given by the quotient of two infinite series which involve Bessel, exponential, and trigonometric functions. The analytical solution for the three dimensional Burgers' equation is given by the quotient of two infinite series which involve hypergeometric, exponential, trigonometric and power functions. For both cases, the solutions can describe shock wave phenomena for large Reynolds numbers (R_e ≥ 100), which is useful for testing numerical methods.     

Convex upper and lower bounds for present value functions

In this paper we present an efficient methodology for approximating the distribution function of the net present value of a series of cash‐flows, when discounting is presented by a stochastic differential equation as in the Vasicek model and in the Ho–Lee model. Upper and lower bounds in convexity order are obtained. The high accuracy of the method is illustrated for cash‐flows for which no analytical results are available. Copyright © 2001 John Wiley & Sons, Ltd.     

Modeling fat tails in stock returns: a multivariate stable-GARCH approach

In this paper a new multivariate volatility model is proposed. It combines the appealing properties of the stable Paretian distribution to model the heavy tails with the GARCH model to capture the volatility clustering. Returns on assets are assumed to follow a sub-Gaussian distribution, which is a particular multivariate stable distribution. In this way the characteristic function of the fitted returns has a tractable expression and the density function can be recovered by numerical methods. A multivariate GARCH structure is then adopted to model the covariance matrix of the Gaussian vectors underlying the sub-Gaussian system. The model is applied to a bivariate series of daily U.S. stock returns. Value-at-risk for long and short positions is computed and compared with the one obtained using the multivariate normal and the multivariate Student’s t distribution. Finally, exploiting the recent developments in the vast dimensional time-varying covariances modeling, possible feasible extensions of our model to higher dimensions are suggested and an illustrative example using the Dow Jones index components is presented.     

A Levin-type algorithm for accelerating the convergence of Fourier series

A nonlinear sequence transformation is presented which is able to accelerate the convergence of Fourier series. It is tailored to be exact for a certain model sequence. As in the case of the Levin transformation and other transformations of Levin-type, in this model sequence the partial sum of the series is written as the sum of the limit (or antilimit) and a certain remainder, i.e., it is of Levin-type. The remainder is assumed to be the product of a remainder estimate and the sum of the first terms oftwo Poincaré-type expansions which are premultiplied by two different phase factors. This occurrence of two phase factors is the essential difference to the Levin transformation. The model sequence for the new transformation may also be regarded as a special case of a model sequence based on several remainder estimates leading to the generalized Richardson extrapolation process introduced by Sidi. An algorithm for the recursive computation of the new transformation is presented. This algorithm can be implemented using only two one-dimensional arrays. It is proved that the sequence transformation is exact for Fourier series of geometric type which have coefficients proportional to the powers of a numberq, |q|<1. It is shown that under certain conditions the algorithm indeed accelerates convergence, and the order of the convergence is estimated. Finally, numerical test data are presented which show that in many cases the new sequence transformation is more powerful than Wynn's epsilon algorithm if the remainder estimates are properly chosen. However, it should be noted that in the vicinity of singularities of the Fourier series the new sequence transformation shows a larger tendency to numerical instability than the epsilon algorithm.