On the Sum of Differentiable Functions

1962 ◽  
Vol 58 (2) ◽  
pp. 225-228
Author(s):  
H. T. Croft
Keyword(s):  
Differentiable Function ◽  
Finite Interval ◽  
Differentiable Functions ◽  
Continuous Differentiable Function

Ryll-Nardzewski has proposed the following problem (New Scottish Book, no. 119). If fn(x) are continuous, differentiable* functions in a closed finite interval, do there always exist constants cn (no cn = 0) (depending on the ), such that converges and is also a continuous differentiable function?

scholarly journals A bridging method for global optimization

1999 ◽  
Vol 41 (1) ◽  
pp. 41-57 ◽  
Author(s):  
Y. Liu ◽  
K. L. Teo
Keyword(s):  
Global Optimization ◽  
Differentiable Function ◽  
Numerical Solutions ◽  
Finite Interval ◽  
Optimization Problems ◽  
Optimal Solutions ◽  
Numerical Examples ◽  
One Dimensional ◽  
Differentiable Functions ◽  
Global Optimal

AbstractIn this paper a bridging method is introduced for numerical solutions of one-dimensional global optimization problems where a continuously differentiable function is to be minimized over a finite interval which can be given either explicitly or by constraints involving continuously differentiable functions. The concept of a bridged function is introduced. Some properties of the bridged function are given. On this basis, several bridging algorithm are developed for the computation of global optimal solutions. The algorithms are demonstrated by solving several numerical examples.

scholarly journals A Function Algebra on Riemann Surfaces

1959 ◽  
Vol 15 ◽  
pp. 1-7 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Differentiable Function ◽  
Riemann Surfaces ◽  
Compact Manifold ◽  
Function Algebra ◽  
Closed Surface ◽  
Conformal Structure ◽  
Riemannian Structure ◽  
Normed Algebra ◽  
Similar Investigation ◽  
Differentiable Functions

In this note, we treat the problem to determine the conformal structure of the closed surface by the structure of the differentiable function algebra as the normed algebra with a certain norm.A similar investigation is found in Myers [1]. He concerns himself with determining the Riemannian structure of the compact manifold using a certain normed algebra of differentiable functions.

scholarly journals A programming problem with an Lp norm in the objective function

1976 ◽  
Vol 19 (3) ◽  
pp. 333-342 ◽  
Author(s):  
Bertram Mond ◽  
Murray Schechter
Keyword(s):  
Objective Function ◽  
Programming Problem ◽  
Differentiable Function ◽  
Duality Theorem ◽  
Quadratic Function ◽  
Inequality Constraints ◽  
Square Root ◽  
Lp Norm ◽  
Differentiable Functions ◽  
Converse Duality Theorem

AbstractWe consider a programming problem in which the objective function is the sum of a differentiable function and the p norm of Sx, where S is a matrix and p > 1. The constraints are inequality constraints defined by differentiable functions. With the aid of a recent transposition theorem of Schechter we get a duality theorem and also a converse duality theorem for this problem. This result generalizes a result of Mond in which the objective function contains the square root of a positive semi-definite quadratic function.

On the Composition of Differentiable Functions

2003 ◽  
Vol 46 (4) ◽  
pp. 481-494 ◽  
Author(s):  
M. Bachir ◽  
G. Lancien
Keyword(s):  
Banach Space ◽  
Variational Principle ◽  
Linear Operator ◽  
Differentiable Function ◽  
General Result ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Schur Property ◽  
Differentiable Functions ◽  
Fréchet Differentiable

AbstractWe prove that a Banach space X has the Schur property if and only if every X-valued weakly differentiable function is Fréchet differentiable. We give a general result on the Fréchet differentiability of f ○ T, where f is a Lipschitz function and T is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm ‖ ‖lip on various spaces of Lipschitz functions.

Weak linear representation of M-estimaton in GLMs with dependent errors

2016 ◽  
Vol 17 (05) ◽  
pp. 1750034 ◽  
Author(s):  
Zhen Zeng ◽  
Hongchang Hu
Keyword(s):  
Convex Function ◽  
Asymptotic Normality ◽  
Differentiable Function ◽  
Generalized Linear Models ◽  
Linear Models ◽  
Linear Representation ◽  
M Estimator ◽  
Continuous Differentiable Function ◽  
M Estimation

In this paper, we consider the generalized linear models (GLMs) [Formula: see text] where [Formula: see text] is a continuous differentiable function, [Formula: see text] are dependent errors. We obtain the M-estimator [Formula: see text] of [Formula: see text] from the following equation: [Formula: see text] where [Formula: see text] is assumed to be a convex function. We also show the linear representation and asymptotic normality of the estimator, which extend the correspondingly results of Wu et al. (M-estimation of linear models with dependent errors, Ann. Statist. 2007) to GLMs.

scholarly journals Fractional Integral Inequalities for Strongly h -Preinvex Functions for a kth Order Differentiable Functions

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1448 ◽  
Author(s):  
Saima Rashid ◽  
Muhammad Amer Latif ◽  
Zakia Hammouch ◽  
Yu-Ming Chu
Keyword(s):  
Integral Operator ◽  
Differentiable Function ◽  
Real World ◽  
Fractional Integral ◽  
Higher Order ◽  
Fractional Integrals ◽  
Differentiable Functions ◽  
Real World Applications ◽  
Mathematical Problems ◽  
Preinvex Functions

The objective of this paper is to derive Hermite-Hadamard type inequalities for several higher order strongly h -preinvex functions via Riemann-Liouville fractional integrals. These results are the generalizations of the several known classes of preinvex functions. An identity associated with k-times differentiable function has been established involving Riemann-Liouville fractional integral operator. A number of new results can be deduced as consequences for the suitable choices of the parameters h and σ . Our outcomes with these new generalizations have the abilities to be implemented for the evaluation of many mathematical problems related to real world applications.

scholarly journals Minimal approximate Hessians for continuously Gâteaux differentiable functions

2004 ◽  
Vol 45 (3) ◽  
pp. 349-359
Author(s):  
Hongxu Li ◽  
Falun Huang
Keyword(s):  
Optimality Conditions ◽  
Differentiable Function ◽  
Sufficient Optimality Conditions ◽  
Differentiable Functions ◽  
Open Questions ◽  
Necessary And Sufficient ◽  
Approximate Hessian

AbstractIn this paper, we investigate minimal (weak) approximate Hessians, and completely answer the open questions raised by V. Jeyakumar and X. Q. Yang. As applications, we first give a generalised Taylor's expansion in terms of a minimal weak approximate Hessian. Then we characterise the convexity of a continuously Gâteaux differentiable function. Finally some necessary and sufficient optimality conditions are presented.

scholarly journals Using Determinants for Determining Extreme Values of a Function of One Variable, Two Variables and Three Variables

2021 ◽  
Vol 07 (12) ◽  
Author(s):  
Pham Ngoc Thinh ◽  
Keyword(s):  
High School ◽  
High School Students ◽  
Differentiable Function ◽  
Extreme Values ◽  
Sufficient Conditions ◽  
Difficult Problem ◽  
School Students ◽  
High School Math ◽  
Differentiable Functions ◽  
Necessary And Sufficient

Finding the maximum and minimum values of a function is essential in high school math. However, Vietnamese high school students have only been taught how to find the extreme values of a function of 1 variable. Seeing the extreme values of a function of 2 and 3 variables is a difficult problem for students. Using the determinants, our aim in this paper is to show the necessary and sufficient conditions for a continuous and differentiable function (1 variable, two variables, and three variables) to reach its maximum over a specified domain. Furthermore, our method can be used to find the extremes of n-variable differentiable functions.

On Asymmetrical Derivates of Non-Differentiable Functions

1968 ◽  
Vol 20 ◽  
pp. 135-143 ◽  
Author(s):  
K. M. Garg
Keyword(s):  
Continuous Function ◽  
Differentiable Function ◽  
Differentiable Functions ◽  
Linear Interval ◽  
Image Position ◽  
Infinite Derivative

Let ƒ(x) be a non-differentiable function, i.e. a realvalued continuous function denned on a linear interval which has nowhere a finite or infinite derivative. We shall say that ƒ(x) has symmetrical derivates at a point x if the four Dini derivates of ƒ(x) at x satisfy the relationsand otherwise we shall say that ƒ(x) has asymmetrical dérivâtes at x.

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