Duality Unary Convex Functions Programming

Most nonliner programming problems consist of functions which are sums of unary convex functions of linear fuctions.In this paper,we derive the duality forms of the unary convex optimization,and these technuques are applied to the geometric programming and minimum discriminaiton information problems.     

Duality of Unary Convex Functions Programming

Most nonliner programming problems consist of functions which are sums of unary,convexfunctions of linear fuctions.In this paper.we derive the duality forms of the unary oonvex optimization,and these technuqucs are applied to the geometric programming and minimum discrimination informationproblems.     

关于概率论中的分段函数问题

分段函数在概率论中有着广泛的应用.通过对几个概率问题的研究,探讨针对分段函数如何合理分段或分区域进行积分问题,体现分段函数在概率论中的重要性.     

Rigid Unary Functions and the Axiom of Choice

We shall investigate certain statements concerning the rigidity of unary functions which have connections with (weak) forms of the axiom of choice.     

Least Squares Smoothing of Univariate Data to achieve Piecewise Monotonicity

If measurements of a univariate function include uncorrelatederrors, then it is usual for the first-order divided differencesof the measurements to show far more sign changes than the correspondingdifferences of the underlying function. Therefore we addressthe problem of making the least sum of squares change to thedata so that the piecewise linear interpolant to the smootheddata is composed of at most k monotonic sections, k being aprescribed positive integer. The positions of the joins of thesections are integer variables whose optimal values are determinedautomatically, which is a combinatorial problem that can haveO(n^k) local minima, where n is the number of data. Fortunatelywe find that a dynamic programming procedure can calculate theglobal minimum of the sum of squares in at most O(n² + kn logn) computer operations. Further, the complexity reduces to onlyO(n) when k = 1 or k = 2, this result being well known in themonotonic case (k = 1). Algorithms that achieve these efficienciesare described. They perform well in practice, but a discussionof complexity suggests that there is still room for improvementwhen k 3.     

Bessel Functions: Monotonicity and Bounds

Monotonicity with respect to the order v of the magnitude ofgeneral Bessel functions C_v(x) = aJ_v(x)+bY_v(x) at positive stationarypoints of associated functions is derived. In particular, themagnitude of C_v at its positive stationary points is strictlydecreasing in v for all positive v. It follows that sup_x|J_v(x)|strictly decreases from 1 to 0 as v increases from 0 to . Themagnitude of x^1/2C_v(x) at its positive stationary points isstrictly increasing in v. It follows that sup_x|x^1/2J_v(x)| equals2/ for 0 v 1/2 and strictly increases to as v increases from1/2 to . It is shown that v^1/3sup_x|J_v(x)| strictly increases from 0 tob = 0.674885... as v increases from 0 to . Hence for all positivev and real x, where b is the best possible such constant. Furthermore, forall positive v and real x, where c = 0.7857468704... is the best possible such constant. Additionally, errors in work by Abramowitz and Stegun and byWatson are pointed out.     

The Maximal Closed Classes of Unary Functions in p-Valued Logic

In many-valued logic the decision of functional completeness is a basic and important problem, and the thorough solution to this problem depends on determining all maximal closed sets in the set of many-valued logic functions. It includes three famous problems, i.e., to determine all maximal closed sets in the set of the total, of the partial and of the unary many-valued logic functions, respectively. The first two problems have been completely solved ([1], [2], [8]), and the solution to the third problem boils down to determining all maximal subgroups in the k-degree symmetric group S_k, which is an open problem in the finite group theory. In this paper, all maximal closed sets in the set of unary p-valued logic functions are determined, where p is a prime. Mathematics Subject Classification: 03B50, 20B35.     

Differential Inclusions and Monotonicity Conditions for Nonsmooth Lyapunov Functions

In this paper we address the problem of characterizing the infinitesimal properties of functions which are nonincreasing along all the trajectories of a differential inclusion. In particular, we extend the condition based on the proximal gradient to the case of semicontinuous functions and Lipschitz continuous differential inclusions. Moreover, we show that the same criterion applies also in the case of Lipschitz continuous functions and continuous differential inclusions.     

Piecewise Constant Roughly Convex Functions

This paper investigates some kinds of roughly convex functions, namely functions having one of the following properties: -convexity (in the sense of Klötzler and Hartwig), -convexity and midpoint -convexity (in the sense of Hu, Klee, and Larman), -convexity and midpoint -convexity (in the sense of Phu). Some weaker but equivalent conditions for these kinds of roughly convex functions are stated. In particular, piecewise constant functions satisfying f(x) = f([x]) are considered, where [x] denotes the integer part of the real number x. These functions appear in numerical calculation, when an original function g is replaced by f(x):=g([x]) because of discretization. In the present paper, we answer the question of when and in what sense such a function f is roughly convex.     

Monotonicity Properties of Determinants of Special Functions

We prove the absolute monotonicity or complete monotonicity of some determinant functions whose entries involve

Fourier Expansions of Piecewise Smooth Functions

Although Fourier series or integrals of piecewise smooth functions may be slowly convergent, sometimes it is possible to accelerate their speed of convergence by adding and subtracting suitable combination of known functions.     

Convergence of Lagrange Interpolation Polynomials for Piecewise Smooth Functions

The present paper first establishes a decomposition result for f(x)∈ C ^r C ^r+1. By using this decomposition we thus can obtain an estimate of ∣f(x) - L _n (f,x)∣ which reflects the influence of the position of the x's and ω(f ^(r+1),δ)_j, j = 0,1,...,s, on the error of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Iterative Characterizations of Computable Unary Functions: A General Method

Iterative characterizations of computable unary functions are useful patterns for the definition of programming languages based on iterative constructs. The features of such a characterization depend on the pairing producing it: this paper offers an infinite class of pairings involving very nice features.     

A Property of Piecewise Smooth Functions

Piecewise smooth equations are increasingly important in the numerical treatment of complementarity problems and models of equilibrium. This note brings out a property of the functions that enter such equations, for instance through penalty expressions.     

两个分段函数卷积的数值算法

本文推广了文［１］中卷积的数值计算方法，使之适用于更广的范围，并用Ｃ语言编程举例说明．     

Melnikov Functions for a Class of Piecewise Hamiltonian Systems

This paper is concerned with the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. By constructing a Poincar\''{e} map, we obtain explicit expressions of the first, second and third order Melnikov functions. In addition, we apply their expressions to give upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.     

关于一类函数单调性的再讨论

针对函数F(x)=x∫0(x-ct)f(t)dt的单调性,通过反例说明某文献的相关论述存在错误,并给出命题,全面讨论此类函数在各种情况下的单调性.     

Sub differential Monotonicity as Characterization of Convex Functions

We prove that a lower semicontinuous function defined on a Banach space is convex if and only if its subdifferential ismonotone.     

Convex Hull Algorithms for Piecewise Linear-Quadratic Functions in Computational Convex Analysis

Computing the convex envelope is a core operation in nonsmooth analysis that bridges the convex with the nonconvex world. Although efficient algorithms to compute fundamental transforms of convex analysis have been proposed over the years, they are limited to convex functions until an efficient algorithm becomes available to compute the convex envelope of a piecewise linear-quadratic function (of one variable) efficiently. We present two such algorithms, one based on maximum and conjugate computation that is easy to implement but has quadratic time complexity, and another based on direct computation that requires more work to implement but has optimal (linear time) complexity. We prove their time (and space) complexity, and compare their performances.