On Reducing a Resource Allocation Problem to a Single One-Dimensional Minimization of a Differentiable Convex Function

By very elementary arguments on Lagrangian duality it is shown that the classical resource allocation problem can be reduced to a single one-dimensional minimization of a differentiable convex function. An optimality condition is given that can be used for testing optimality of some proposed heuristic solutions.     

The derivative of a continuous nonconstant function

Using elementary ideas and techniques, we prove (Theorem 2) that for a nonconstant differentiable function , a < b, the set S(f) = {x ε [a, b] : f'(x) ≠ 0} cannot be negligible. This result remains valid if f fails to be differentiable on a countable subset of [a, b].     

Monotone transformations and differential properties of functions

The class of all real functions of a single variable which become everywhere differentiable after a certain homeomorphic transformation of coordinate axis is described. Moreover, various examples about differential properties of functions are given (in particular, an elementary construction of a nonconstant continuously differentiable real function of two variables, every value of which is critical-the example of Whitney, is given).Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 859–871, December, 1977.The authors express gratitude to A. L. Brudno, P. L. Ul'yanov, and V. A. Skvortsov for discussion of the results given here.     

A graphical method to solve a family of allocation problems

This paper discusses the classical resource allocation problem. By very elementary arguments on Lagrangian duality it is shown that this problem can be reduced to a single one-dimensional maximization of a differentiable concave function. Moreover, a simple graphical method is developed and applied to a family of well-known problems from the literature.     

可微cantor型函数的构造

可微Ｃａｎｔｏｒ型函数的存在性仅是从理论上给出证明，本文给出一个具体的构造，以使我们可以更好了解这种函数的本质．     

A class of practically nowhere differentiable functions on the complex plane

This note could find classroom use in an introductory course on complex analysis. Using some of the most significant theorems from complex analysis, the main result provides a simple method for transforming many elementary functions (defined on the complex plane) into everywhere continuous functions that are differentiable only on a nowhere dense set. Accordingly, such continuous functions are termed ‘practically nowhere differentiable’. The twofold pedagogical value of this method is that (1) students can readily generate examples of everywhere continuous, practically nowhere differentiable functions that do not require any direct appeal to infinite series, and (2) the often dynamical difference between the behaviour of functions of a complex variable and functions of a real variable is showcased.     

The role of Billingsley dimensions in computing fractal dimensions on Cantor-like spaces

We consider a Cantor-like set as a geometric projection of a Bernoulli process. P. Billingsley (1960) and C. Dai and S.J. Taylor (1994) introduced dimension-like indices in the probability space of a stochastic process. Under suitable regularity conditions we find closed formulae linking the Hausdorff, box and packing metric dimensions of the subsets of the Cantor-like set, to the corresponding Billingsley dimensions associated with a suitable Gibbs measure. In particular, these formulae imply that computing dimensions in a number of well-known fractal spaces boils down to computing dimensions in the unit interval endowed with a suitable metric. We use these results to generalize density theorems in Cantor-like spaces. We also give some examples to illustrate the application of our results.

     

A proof of Yomdin-Gromov’s Algebraic Lemma

Following the analysis of differentiable mappings of Y. Yomdin, M. Gromov has stated a very elegant “Algebraic Lemma” which says that the “differentiable size” of an algebraic subset may be bounded only in terms of its dimension, degree and diameter, regardless of the size and specific values of the underlying coefficients. We give a complete and elementary proof of Gromov’s result.     

Indefinite integrals of special functions from hybrid equations

ABSTRACT

Elementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of these equations. If the function also obeys a conventional differential equation, information from this equation can be introduced into the elementary equations to give blended linear equations which are here called hybrid equations. Integration theorems are derived for these hybrid equations and several universal integrals are also derived. The paper presents integrals derived with these methods for cylinder functions, associated Legendre functions, and the Gegenbauer, Chebyshev, Hermite, Jacobi and Laguerre orthogonal polynomials. All the results presented have been checked using Mathematica.     

一类非光滑广义分式规划的Kuhn-Tucker型最优性必要条件

考虑一类非线性不等式约束的非光滑minimax分式规划问题;目标函数中的分子是可微函数与凸函数之和形式而分母是可微函数与凸函数之差形式,且约束函数是可微的.在Arrow- Hurwicz-Uzawa约束品性下,给出了这类规划的最优解的Kuhn-Tucker型必要条件.所得结果改进和推广了已有文献中的相应结果.     

Elementary properties of arcwise connected sets and functions

The purpose of this paper is to investigate some elementary, basic properties of arcwise connected sets and functions. Since these concepts are generalizations of convexity, it is natural to ask if any of the basic properties of convex sets and functions are carried over to these new generalized classes. All the functions involved are considered to be not necessarily differentiable.     

一类不可微广义分式规划的K-T型必要条件

本文对一类在Rn的开子集X上的非线性不等式约束的广义分式规划问题: 目标函数中的分子是可微函数与凸函数之和而分母是可微函数与凸函数之差,且约束函数是可微的,在Abadie约束品性或Calmness约束品性下,给出了最优解的Kuhn-Tucker 型必要条件,所得结果改进和推广了已有文献中的相应结果．     

Star-shaped Differentiable Functions and Star-shaped Differentials

Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.     

Non-monotonic solutions and continuously differentiable solutions of conjugacy equations

In this paper solutions of conjugacy equation φ(f(x))=g(φ(x)) for a strictly decreasing continuous given function f and a continuous given function g (maybe non-monotonic) are constructed by piecewise defining. We determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g. Finally, the recursive algorithm is implemented in MATLAB software and two examples are respectively presented for a non-monotonic solution and a continuously differentiable one.     

A new arc algorithm for unconstrained optimization

The gradient path of a real valued differentiable function is given by the solution of a system of differential equations. For a quadratic function the above equations are linear, resulting in a closed form solution. A quasi-Newton type algorithm for minimizing ann-dimensional differentiable function is presented. Each stage of the algorithm consists of a search along an arc corresponding to some local quadratic approximation of the function being minimized. The algorithm uses a matrix approximating the Hessian in order to represent the arc. This matrix is updated each stage and is stored in its Cholesky product form. This simplifies the representation of the arc and the updating process. Quadratic termination properties of the algorithm are discussed as well as its global convergence for a general continuously differentiable function. Numerical experiments indicating the efficiency of the algorithm are presented.     

Fractal dimensions of fractional integral of continuous functions

In this paper,we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals.Riemann–Liouville integral of a continuous function f(x) of order v(v0) which is written as D~(-v) f(x) has been proved to still be continuous and bounded.Furthermore,upper box dimension of D~(-v) f(x) is no more than 2 and lower box dimension of D~(-v) f(x) is no less than 1.If f(x) is a Lipshciz function,D~(-v) f(x) also is a Lipshciz function.While f(x) is differentiable on [0,1],D~(-v) f(x) is differentiable on [0,1] too.With definition of upper box dimension and further calculation,we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions.If a continuous function f(x) satisfying H?lder condition,upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x).Appeal to auxiliary functions,we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying H?lder condition of order v(v0) is strictly less than 2-v.Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary.Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.     

The topology of stirred fluids

There are simple idealized mathematical models representing the stirring of fluids. The models we consider involve two fluids entering a chamber, with the overflow leaving it. The stirring creates a Cantor-like, but connected, boundary between the fluids that is best described point-set topologically. We prove that in many cases the boundary between the fluids is an indecomposable continuum.     

Expanding maps on Cantor sets and analytic continuation of zeta functions

In this paper, we study a class of Ruelle dynamical zeta functions related to uniformly expanding maps on Cantor sets. We show that under a non-local integrability condition, the zeta function enjoys a non-vanishing analytic continuation in a strip on the left of the line of absolute convergence. Applying these results to Fuchsian Schottky groups and Julia sets yields precise asymptotics of the number of closed geodesics for convex co-compact surfaces and the distribution of periodic points for a family of Cantor-like Julia sets.     

On differentiable exact penalty functions

In this work, we study a differentiable exact penalty function for solving twice continuously differentiable inequality constrained optimization problems. Under certain assumptions on the parameters of the penalty function, we show the equivalence of the stationary points of this function and the Kuhn-Tucker points of the restricted problem as well as their extreme points. Numerical experiments are presented that corroborate the theory, and a rule is given for choosing the parameters of the penalty function.     

A Globally Convergent Algorithm for a Locally Lipschitz Function

In this paper, an algorithm of global convergence is proposed for a locally Lipschitz function, which is strictly differentiable at almost all differentiable points, and several examples are computed on an IBM PC.