The second order extension of the Gibbs state

We present the second order thermodynamics (in the sense of higher order thermodynamics of the first author) on the base of gauge invariant (forH →H + const. transformation) parameterization of second order states, i.e., $$\rho = Z^{ - 1} \exp \left[ { - \beta H - \beta '\left( {H - U} \right)^2 } \right]$$ whereU is determined self-consistently fromU = TrρH. By this formulation, we can give definite physical interpretation to temperature parametersβ andβ′. In particular, the first order temperatureβ is identified with the conventional concept of temperature, and for large systems the second order state gives the same results as the conventional Gibbs state. We show these facts by some general considerations and by concrete calculations for three simple models, ideal gas, a semiharmonic oscillatory chain of hard-core particles, and a 3-dimensional solid with harmonic oscillations. The results are weekly model-dependent.     

On second order efficient robust inference

General strategies for constructing second order efficient robust distances from suitable properties of the residual adjustment functions (RAF) are discussed. Based on those properties families of estimators are constructed using the truncated polynomial, negative exponential and sigmoidal functions as RAFs and their efficiency and robustness properties are investigated. The estimators have full asymptotic efficiency, and are automatically second order efficient. Many of the proposed estimators are competitive or better than the minimum Hellinger distance estimator (MHDE) and minimum negative exponential disparity estimator (MNEDE) under the combined goals of asymptotic efficiency with strong robustness properties. Hence the proposed families give the user the flexibility to choose from a large class of robust second order efficient estimators based upon specific needs.     

A second order monotone upwind scheme

We analyze a special finite difference scheme of upwind type for an ordinary singularly perturbed nonlinear boundary value problem. In particular we prove the uniqueness and monotone dependence upon the right hand sides of the discrete solutions and the second order accuracy in the global domain.     

A second order theory of data types

Summary A specification method for abstract data types based on many-sorted second order logic is presented. In our approach the specification is composed of two parts: the first part specifies a set of objects and a set of fundamental operations; the other part specifies an arbitrary number of augmenting operations. The fundamental operations are called constructors. On the basis of constructors the axioms can be put into schematic forms and the resulting theory is categorical.     

On first and second order box schemes

The box method for discretizing elliptic boundary value problems are discussed. Error estimates of first and second order between the Galerkin solution and the box method solution are proved. A proposal for a new second order box-like scheme is made.     

A second order theory of infinite trees

An axiomatic approach based on second order logic for specifying finite data types has been given in [12]. In the present paper we generalize the approach to infinite trees. Data logic CT of infinite trees is introduced and investigated. CT is based on the data logic T of finite trees. For treating infinite objects we heavily rely upon the notions of Bishop'apos;s constructive mathematics. Thus the results of the present paper are closely related with an intuitionistic type theory of infinite trees.     

A second order method for unconstrained optimization

An algorithm using second derivatives for solving unconstrained optimization problems is presented. In this brief note the descent direction of the algorithm is based on a modification of the Newton direction, while the Armijo rule for choosing the stepsize is used. The rate of convergence of the algorithm is shown to be superlinear. Our computational experience shows that the method performs quite well and our numerical results are presented in Section 4.     

A second order theory of data types

Summary A specification method for abstract data types based on manysorted second order logic is presented. In our approach the specification is composed of two parts: the first part specifies a set of objects and a set of fundamental operations; the other part specifies an arbitrary number of augmenting operations. The fundamental operations are called constructors. On the basis of constructors the axioms can be put into schematic forms and the resulting theory is categorical.     

Optimum second order stationary extrapolated iterative schemes

In this paper the construction of a stationary second order scheme of de Phillis' type, with optimal convergence rates, for the solution of a linear system is presented. Numerical tests have shown that the new scheme compares favorably with the very fast nonstationary second order one proposed by Manteuffel. It is also proved that a class of already known second order Extrapolated Alternating Direction Implicit schemes can be obtained directly by using the methods of this paper.     

On the second order topological asymptotic expansion

The topological derivative provides the sensitivity of a given shape functional with respect to an infinitesimal (non smooth) domain perturbation at an arbitrary point of the domain. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal perturbations. However, for practical applications, we need to insert perturbations of finite size. Therefore, we consider one more term in the expansion which is defined as the second order topological derivative. In order to present these ideas, in this work we calculate first as well as second order topological derivatives for the total potential energy associated to the Laplace’s equation, when the domain is perturbed with a hole. Furthermore, we also study the effects of different boundary conditions on the hole: Neumann and Dirichlet (both homogeneous). In the Neumann’s case, the second order topological derivative depends explicitly on higher-order gradients of the state solution and also implicitly on the point where the hole is nucleated through the solution of an auxiliary problem. On the other hand, in the Dirichlet’s case, the first order topological derivative depends explicitly on the state solution as well as implicitly through the solution of an auxiliary problem, and the second order topological derivative depends only explicitly on the solution associated to the original problem. Finally, we present two simple examples showing the influence of both terms in the second order topological asymptotic expansion for each case of boundary condition on the hole.     

A fourth order multiderivative method for linear second order boundary value problems

Fourth order methods are developed and analysed for the numerical solution of linear second order boundary value problems.

The methods are developed by replacing the exponential terms in a three-point recurrence relation by Padé approximants.

The derivations of second order and sixth order methods from the recurrence relation are outlined briefly.

One method is tested on two problems from the literature, one of which is mildly nonlinear.     

Successive linearizations of second order multidimensional time-invariant systems

The present paper discusses a linearization method for second order multidimensional time-invariant systems. The method approximates locally the nonlinear vector field around the equilibrium, where the solution starting from a given initial state near the equilibrium is approximated by a linear solution. This linearization is usually called local trajectory-based linearization. The approximation is computed using an iterative method, which consists of successive approximations in the least square sense. Using a numerical example, it is shown that the linearized solutions exhibit good agreement with the nonlinear solutions.     

Stability properties of second order delay integro-differential equations

A basic theorem on the behavior of solutions of scalar linear second order delay integro-differential equations is established. As a consequence of this theorem, a stability criterion is obtained.     

Finite-time output stabilization with second order sliding modes

In this note, a class of discontinuous feedback laws that switch over branches of parabolas in the auxiliary state plane is analyzed. Conditions are provided under which controllers belonging to this class are second order sliding-mode algorithms: they ensure uniform global finite-time output stability for uncertain systems of relative degree two.     

基于二阶统计量的盲信号分离

研究讨论了一种只利用信号的二阶统计量来分离盲源信号的方法。该方法先估计传感器信号的相关矩阵,通过对其进行特征值分解,可以得到信号的主特征值和特征向量。同时可通过奇异值分解来得到预白化后的时滞信号的奇异值,利用这些信息来估计源信号。仿真结果表明:该方法计算工作量少,对多个盲混合信号有良好的分离性能。     

Component statistical analysis of second order hidden periodicities

The component method is applied to define estimators of the periods for Gaussian periodically correlated random processes (mathematical model of stochastic oscillations). The properties of these period estimators are obtained using some small parameter method and the rate of convergence is shown to be optimal. Specific results for the simplest models of periodically correlated process are presented. Finally the method is illustrated with a simulated sequence and a real life vibration signal.     

Eigenvalue problems of second order impulsive differential equations

This paper is concerned with an eigenvalue problem for second order differential equations with impulse. The existence of a countably infinite set of eigenvalues and eigenfunctions is proved.     

一类三次Bent函数的二阶非线性度*

研究了形如yTr(x5)+Tr(x3)的三次Bent函数,通过研究其导数的非线性度的下界,得到了该函数的二阶非线性度的下界,将所得结果与Carlet的结果进行了比较,结果表明,该函数的二阶非线性度大于Carlet给出的下界。