Convexifiable functions in integral calculus

A function is said to be convexifiable if it becomes convex after adding to it a strictly convex quadratic term. In this paper we extend some of the basic integral properties of convex functions to Lipschitz continuously differentiable functions on real line. In particular, we give estimates of the mean value, a "nonstandard" form of Jensen\u27s inequality, and an explicit representation of analytic functions. It is also outlined how one can use convexification to study ordinary differential equations

Union Catalogue of Digitised Documents in India

A lot of digitisation projects are in progress in different parts of India. But many of these efforts are individual and lack a holistic view. The article stresses the need to have coordinated digitisation activities and a union catalogue of the digitised contents in India

An ecofeminist study of Flora Nwapa’s ‘Efuru’

This paper identified the great contribution of Flora Nwapa’s Efuru to ideas of ecological consciousness, and environmental protection, using theory that interlaces ecocriticism cum feminist criticism. The methodology therefore involves the conversation or ideas on the images of women and nature in ‘Efuru’, the association between the oppression of women and exploitation of nature by male chauvinist, thereby enslaving the female and nature in the commercial market value. From an ecofeminist perspective, this paper discovered that Flora Nwapa inculcates her novel with a theme of feminine and natural liberation from domination and violence. Flora Nwapa foresees the establishment of symbiosis, in which there is no male oppression or environment exploitation.Keywords: ecological conscience, male oppression, ecofeminism, domination, interconnectednes

Union Catalogue of Digitised Documents in India

A lot of digitisation projects are in progress in different parts of India. But many of these efforts are individual and lack a holistic view. The article stresses the need to have coordinated digitisation activities and a union catalogue of the digitised contents in India

Characterizing fixed points

A set of sufficient conditions which guarantee the existence of a point x⋆ such that f(x⋆) = x⋆ is called a "fixed point theorem". Many such theorems are named after well-known mathematicians and economists. Fixed point theorems are among most useful ones in applied mathematics, especially in economics and game theory. Particularly important theorem in these areas is Kakutani\u27s fixed point theorem which ensures existence of fixed point for point-to-set mappings, e.g., [2, 3, 4]. John Nash developed and applied Kakutani\u27s ideas to prove the existence of (what became known as) "Nash equilibrium" for finite games with mixed strategies for any number of players. This work earned him a Nobel Prize in Economics that he shared with two mathematicians. Nash\u27s life was dramatized in the movie "Beautiful Mind" in 2001. In this paper, we approach the system f(x) = x differently. Instead of studying existence of its solutions our objective is to determine conditions which are both necessary and sufficient that an arbitrary point x⋆ is a fixed point, i.e., that it satisfies f(x⋆) = x⋆. The existence of solutions for continuous function f of the single variable is easy to establish using the Intermediate Value Theorem of Calculus. However, characterizing fixed points x⋆, i.e., providing answers to the question of finding both necessary and sufficient conditions for an arbitrary given x⋆ to satisfy f(x⋆) = x⋆, is not simple even for functions of the single variable. It is possible that constructive answers do not exist. Our objective is to find them. Our work may require some less familiar tools. One of these might be the "quadratic envelope characterization of zero-derivative point" recalled in the next section. The results are taken from the author\u27s current research project "Studying the Essence of Fixed Points". They are believed to be original. The author has received several feedbacks on the preliminary report and on parts of the project which can be seen on Internet [9]

ON THE DERIVATIVE OF SMOOTH MEANINGFUL FUNCTIONS

The derivative of a function f in n variables at a point x* is one of the most important tools in mathematical modelling. If this object exists, it is represented by the row n-tuple f(x*) = [∂f/∂xi(x*)] called the gradient of f at x*, abbreviated: “the gradient”. The evaluation of f(x*) is usually done in two stages, first by calculating the n partials and then their values at x = x*. In this talk we give an alternative approach. We show that one can characterize the gradient without differentiation! The idea is to fix an arbitrary row n-tuple G and answer the following question: What is a necessary and sufficient condition such that G is the gradient of a given f at a given x*? The answer is given after adjusting the quadratic envelope property introduced in [3]. We work with smooth, i.e., continuously differentiable, functions with a Lipschitz derivative on a compact convex set with a non-empty interior. Working with this class of functions is not a serious restriction. In fact, loosely speaking, “almost all” smooth meaningful functions used in modelling of real life situations are expected to have a bounded “acceleration” hence they belong to this class. In particular, the class contains all twice differentiable functions [1]. An important property of the functions from this class is that every f can be represented as the difference of some convex function and a convex quadratic function. This decomposition was used in [3] to characterize the zero derivative points. There we obtained reformulations and augmentations of some well known classic results on optimality such as Fermats extreme value theorem (known from high school) and the Lagrange multiplier theorem from calculus [2, 3]. In this talk we extend the results on zero derivative points to characterize the relation G = f(x*), where G is an arbitrary n-tuple. Some special cases: If G = O, we recover the results on zero derivative points. For functions of a single variable on I = [a, b], the choice G = [f(b) – f(a)]/(b – a) yields characterizations of points c where the instantaneous and average rates of change coincide [4], etc. The celebrated mean value theorem [2] claims that at least one such point c exists but it does not characterize it. These ideas are illustrated by examples and a photograph of an overpass in Beijing. A successful implementation of the new approach requires familiarity with the basic theory of infinite sequences. [1] Floudas, C. A. and C. E. Gounaris: An overview of advances in global optimization during 2003-2008,” a chapter in the book Lectures on Global Optimization (P. M. Pardalos and T. F. Coleman, editors), Fields Institute Communications, v. 55 (2009) 105-154. [2] Neralić, L. and B. Šego, B.: Matematika, Element, Zagreb, 2009. [3] Characterizing zero-derivative points, J. Global Optimization 46 (2010) 155-161. (Published on line: 2 July 2009.) [4] On the behaviour of functions around zero-derivative points, Int. J. Optimization: Theory, Methods and Applications 1 (2009) 329-340

The fundamental theorem of calculus for Lipschitz functions

Every smooth function in several variables with a Lipschitz derivative, when considered on a compact convex set, is the difference of a convex function and a convex quadratic function. We use this result to decompose anti - derivatives of continuous Lipschitz functions and augment the fundamental theorem of calculus. The augmentation makes it possible to convexify and monotonize ordinary differential equations and obtain possibly new results for integrals of scalar functions and for line integrals. The result is also used in linear algebra where new bounds for the determinant and the spectral radius of symmetric matrices are obtained

Modelling with twice continuously differentiable functions

Many real life situations can be described using twice continuously differentiable functions over convex sets with interior points. Such functions have an interesting separation property: At every interior point of the set they separate particular classes of quadratic convex functions from classes of quadratic concave functions. Using this property we introduce new characterizations of the derivative and its zero points. The results are applied to the study of sensitivity of the Cobb-Douglas production function. They are also used to describe the least squares solutions in linear and nonlinear regression