The explicit series solution of SIR and SIS epidemic models

In this paper the SIR and SIS epidemic models in biology are solved by means of an analytic technique for nonlinear problems, namely the homotopy analysis method (HAM). Both of the SIR and SIS models are described by coupled nonlinear differential equations. A one-parameter family of explicit series solutions are obtained for both models. This parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models. Our analytic results agree well with the numerical ones. This analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology.     

Analyticity of the central path at the boundary point in semidefinite programming

In this paper we study the limiting behavior of the central path for semidefinite programming (SDP). We show that the central path is an analytic function of the barrier parameter even at the limit point, provided that the semidefinite program has a strictly complementary solution. A consequence of this property is that the derivatives – of any order – of the central path have finite limits as the barrier parameter goes to zero.     

精确解析法的子结构算法

文[1]提出精确解析法,用以求解任意变系数常微分方程,并利用初参数算法给出一个解的解析表达式.但利用初参数算法,对某一类问题,如长柱壳弯曲和振动等,它们的解将难以在计算机上得到.本文通过非均匀轴对称长圆柱壳弯曲问题,给出精确解析法的子结构算法,它能够计算初参数算法在计算机上不能解决的问题.问题最后和初参数算法一样能归结为求解一个低阶代数方程组.文末给出算例,表明本文算法的正确性,并和初参数算法作了比较.     

New algebraic approaches to classical boundary layer problems

Classical non-steady boundary layer equations are fundamental nonlinear partial differential equations in the boundary layer theory of fluid dynamics. In this paper, we introduce various schemes with multiple parameter functions to solve these equations and obtain many families of new explicit exact solutions with multiple parameter functions. Moreover, symmetry transformations are used to simplify our arguments. The technique of moving frame is applied in the three-dimensional case in order to capture the rotational properties of the fluid. In particular, we obtain a family of solutions singular on any moving surface, which may be used to study turbulence. Many other solutions are analytic related to trigonometric and hyperbolic functions, which reflect various wave characteristics of the fluid. Our solutions may also help engineers to develop more effective algorithms to find physical numeric solutions to practical models.     

Learning curve parameter estimation beyond traditional statistics

Due to decreasing order quantities, increasing product variety and fluctuating production orders, manufacturing companies have been encountering an increased occurrence of repetitive learning-forgetting phenomenon. In this paper, deterministic methods for the learning curve parameter estimation from the limited production data available from the unstable production environment are studied. Two main learning curve models: cumulative average (Wright) and unit (Crawford) were considered and several different mathematically proven methods were proposed for the parameter estimation. The calculation results illustrated that learning curve parameters can be unequivocally estimated from the limited production data (single random sample) by using deterministic methods for both of the learning curve models, although more accurate estimation was provided by the cumulative average model based methods. Newly proposed methods enable sufficiently accurate parameter estimation from the limited production data where traditional statistical parameter estimation methods cannot be applied.     

Vacation models in discrete time

A class of single server vacation queues which have single arrivals and non-batch service is considered in discrete time. It is shown that provided the interarrival, service, vacation, and server operational times can be cast with Markov-based representation then this class of vacation model can be studied as a matrix–geometric or a matrix-product problem – both in the matrix–analytic family – thereby allowing us to use well established results from Neuts (1981). Most importantly it is shown that using discrete time approach to study some vacation models is more appropriate and makes the models much more algorithmically tractable. An example is a vacation model in which the server visits the queue for a limited duration. The paper focuses mainly on single arrival and single unit service systems which result in quasi-birth-and-death processes. The results presented in this paper are applicable to all this class of vacation queues provided the interarrival, service, vacation, and operational times can be represented by a finite state Markov chain.An erratum to this article can be found at     

D-Optimal Designs for Trigonometric Regression Models on a Partial Circle

In the common trigonometric regression model we investigate the D-optimal design problem, where the design space is a partial circle. It is demonstrated that the structure of the optimal design depends only on the length of the design space and that the support points (and weights) are analytic functions of this parameter. By means of a Taylor expansion we provide a recursive algorithm such that the D-optimal designs for Fourier regression models on a partial circle can be determined in all cases. In the linear and quadratic case the D-optimal design can be determined explicitly.     

Decision analysis models in reinsurance

Decision analysis models are developed and illustrated for the reinsurance (risk transfer) decisions made by insurance companies. Decision analytic models were found to be useful tools both for structuring multistage reinsurance decisions and for comparing alternative options. The insurer is faced with many possible choices involving reinsurance type and extent, and an expected utility model provided insight both as a screening device and as an evaluation criterion. Decision analytic models appeared to be superior to other approaches such as mean/variance and risk of ruin models both because of their flexibility and their more comprehensive treatment the important elements of the decision, namely the complete claims distribution, the cost of reinsurance and the insurer's risk attitude.     

A ranking method based on a full-inefficient frontier

Since in evaluating by traditional data envelopment analysis (DEA) models many decision making units (DMUs) are classified as efficient, a large number of methods for fully ranking both efficient and inefficient DMUs have been proposed. In this paper a ranking method is suggested which basically differs from previous methods but its models are similar to traditional DEA models such as BCC, additive model, etc. In this ranking method, DMUs are compared against an full-inefficient frontier, which will be defined in this paper. Based on this point of view many models can be designed, and we mention a radial and a slacks-based one out of them. This method can be used to rank all DMUs to get analytic information about the system, and also to rank only efficient DMUs to discriminate between them.     

On the distribution of poles of Padé approximants to the Z-transform of complex Gaussian white noise

In the application of Padé methods to signal processing a basic problem is to take into account the effect of measurement noise on the computed approximants. Qualitative deterministic noise models have been proposed which are consistent with experimental results. In this paper the Padé approximants to the Z-transform of a complex Gaussian discrete white noise process are considered. Properties of the condensed density of the Padé poles such as circular symmetry, asymptotic concentration on the unit circle and independence on the noise variance are proved. An analytic model of the condensed density of the Padé poles for all orders of the approximants is also computed. Some Monte Carlo simulations are provided.     

An Infinite Asymptotic Expansion for the Extreme Zeros of the Pollaczek Polynomials

In this paper, we first establish an integral expression for the Pollaczek polynomials P_n ( x ; a , b ) from a generating function. By applying a canonical transformation to the integral and carrying out a detailed analysis of the integrand, we derive a uniform asymptotic expansion for P_n (cosθ; a , b ) in terms of the Airy function and its derivative, in descending powers of n . The uniformity is in an interval next to the turning point , with M being a constant. The coefficients of the expansion are analytic functions of a parameter that depends only on t where , and not on the large parameter n . From the expansion of the polynomials we obtain an asymptotic expansion in powers of n ^−1/3 for the largest zeros. As a special case, a four-term approximation is provided for comparison and illustration. The method used in this paper seems to be applicable to more general situations.     

Adomian decomposition method is a special case of Lyapunov’s artificial small parameter method

The Adomian decomposition method and Lyapunov’s artificial small parameter method are two popular analytic methods for solving nonlinear differential equations. In this short paper, we show that the Adomian decomposition method can be derived from the more general Lyapunov’s artificial small parameter method.     

Arbitrary Initial Term Structure within the CIR Model: A Perturbative Solution

Single‐factor interest rate models with constant coefficients are not consistent with arbitrary initial term structures. An extension which allows both arbitrary initial term structure and analytical tractability has been provided only in the Gaussian case. In this paper, within the context of the HJM methodology, an extension of the CIR model is provided which admits arbitrary initial term structure. It is shown how to calculate bond prices via a perturbative approach, and closed formulas are provided at every order. Since the parameter selected for the expansion is typically estimated to be small, the perturbative approach turns out to be adequate to our purpose. Using results on affine models, the extended CIR model is estimated via maximum likelihood on a time series of daily interest rate yields. Results show that the CIR model has to be rejected with respect to the proposed extension, and it is pointed out that the extended CIR model provides a more flexible characterization of the link between risk neutral and natural probability.     

Moving estimates test with time varying bandwidth

In this paper, we consider the problem of testing for parameter changes in time series models based on a moving estimates (ME) test. It is widely accepted that detecting some changes, for instance, those caused by temporary parameter shifts by the existing cusum test is difficult. A MV test with a fixed bandwidth has been developed to circumvent the defect, but the test still does not perform well under certain conditions. Motivated by this, we propose a MV test with a time varying bandwidth to outperform the original test. In order to illustrate our findings, we have provided simulation results.     

More Higher-Order Efficiency: Concentration Probability

Based on concentration probability of estimators about a true parameter, third-order asymptotic efficiency of the first-order bias-adjusted MLE within the class of first-order bias-adjusted estimators has been well established in a variety of probability models. In this paper we consider the class of second-order bias-adjusted Fisher consistent estimators of a structural parameter vector on the basis of an i.i.d. sample drawn from a curved exponential-type distribution, and study the asymptotic concentration probability, about a true parameter vector, of these estimators up to the fifth-order. In particular, (i) we show that third-order efficient estimators are always fourth-order efficient; (ii) a necessary and sufficient condition for fifth-order efficiency is provided; and finally (iii) the MLE is shown to be fifth-order efficient.     

The practical use of the Euler transformation

The generalised Euler transformation is a powerful transformation of infinite series which can be used, in theory, for the acceleration of convergence and for analytic continuation. When the transformation is applied to a series with rounded coefficients, its behaviour can differ substantially from that predicted theoretically. In general, analytic continuation is impossible in this case. It is still possible, however, to use the transformation for acceleration of convergence, but some changes are necessary in the method of choosing the optimum parameter value.     

Pseudopower expansion of solutions of generalized equations and constrained optimization problems

We show that the solution of a strongly regular generalized equation subject to a scalar perturbation expands in pseudopower series in terms of the perturbation parameter, i.e., the expansion of orderk is the solution of generalized equations expanded to orderk and thus depends itself on the perturbation parameter. In the polyhedral case, this expansion reduces to a usual Taylor expansion. These results are applied to the problem of regular perturbation in constrained optimization. We show that, if the strong regularity condition is satisfied, the property of quadratic growth holds and, at least locally, the solutions of the optimization problem and of the associated optimality system coincide. If, in addition the number of inequality constraints is finite, the solution and the Lagrange multiplier can be expanded in Taylor series. If the data are analytic, the solution and the multiplier are analytic functions of the perturbation parameter.