Dimensions of attractors

This thesis studies different types of dimensions of attractors in low dimensional dissipative dynamical systems. Some of them can be calculated by looking directly at the attractor, other by looking at the system, taking or not into account a probability distribution. We give some simple examples to make the ideas clear, but the generalized baker's transformation is taken as a model for such studies. This transformation is used to illustrate some conjectures about typical chaotic attractor. This thesis may be considered as a partial report of the seminal article of Farmer et al (12

Count of rotational symmetric bent Boolean functions

Counting the Boolean functions having specific cryptographic features is an interesting problem in combinatorics and cryptography. Count of bent functions for more than eight variables is unexplored. In this paper, we propose an upper bound for the count of rotational symmetric bent Boolean functions and characterize its truth table representation from the necessary condition of a rotational symmetric bent Boolean function

A Maiorana-McFarland Construction of a GBF on Galois ring

Bent functions shows some vital properties among all combinatorial objects. Its links in combinatorics, cryptography and coding theory attract the scientific community to construct new class of bent functions. Since the entire characterisation of bent functions is still unexplored but several construction on different algebraic structure is in progress. In this paper we proposed a generalized Maiorana-McFarland construction of bent function from Galois ring

Topics in Nonconvex Optimization

Nonconvex Optimization is a multi-disciplinary research field that deals with the characterization and computation of local/global minima/maxima of nonlinear, nonconvex, nonsmooth, discrete and continuous functions. Nonconvex optimization problems are frequently encountered in modeling real world systems for a very broad range of applications including engineering, mathematical economics, management science, financial engineering, and social science. This contributed volume consists of selected contributions from the Advanced Training Programme on Nonconvex Optimization and Its Applications h

Invexity and optimization.

The book presents an overview (and also some new results) on invex and related functions in various types of optimization problems (with single-valued objective functions, with vector-valued objective functions, and static as well as dynamic problems). The content of the book is structured as follows. Chapter 1 is an introduction, where the authors give the basic notions of convexity and generalized convexity in the finite-dimensional real space $\Bbb{R}^n$Rn. Chapter 2 is devoted to a discussion on the definition and meaning of invexity in the differentiable case. A comparison of invexity with other concepts of generalized convexity is also given. Chapter 3 deals with $\eta$η-pseudolinear ($f$f and $-f$−f both pseudo-invex) functions and the links between invexity and generalized monotonicity. Chapter 4 is concerned with invexity for non-smooth functions; in particular, with the Lipschitzian case using Clarke's theory of generalized gradients. Chapter 5 discusses invexity in nonlinear programming and examines the relevant role of invexity to duality theory. In Chapter 6, the authors consider a multiobjective optimization problem involving invex functions and present several duality results for multiobjective programming problems. Chapter 7 is devoted to some variational and control optimization problems. Finally, in Chapter 8, several applications of invexity to special optimization problems are presented (problems with quadratic functions, fractional programming problems, non-differentiable programs, etc.)

Multiobjective Fractional Programming Involving Generalized Semilocally V-Type I-Preinvex and Related Functions

We study a nonlinear multiple objective fractional programming with inequality constraints where each component of functions occurring in the problem is considered semidifferentiable along its own direction instead of the same direction. New Fritz John type necessary and Karush-Kuhn-Tucker type necessary and sufficient efficiency conditions are obtained for a feasible point to be weakly efficient or efficient. Furthermore, a general Mond-Weir dual is formulated and weak and strong duality results are proved using concepts of generalized semilocally V-type I-preinvex functions. This contribution extends earlier results of Pred

Multiobjective Fractional Programming Involving Generalized Semilocally V-Type I-Preinvex and Related Functions

We study a nonlinear multiple objective fractional programming with inequality constraints where each component of functions occurring in the problem is considered semidifferentiable along its own direction instead of the same direction. New Fritz John type necessary and Karush-Kuhn-Tucker type necessary and sufficient efficiency conditions are obtained for a feasible point to be weakly efficient or efficient. Furthermore, a general Mond-Weir dual is formulated and weak and strong duality results are proved using concepts of generalized semilocally V-type I-preinvex functions. This contribution extends earlier results of Preda (2003), Mishra et al. (2005), Niculescu (2007), and Mishra and Rautela (2009), and generalizes results obtained in the literature on this topic

Introduction to unconstrained optimization with R

This book discusses unconstrained optimization with R — a free, open-source computing environment, which works on several platforms, including Windows, Linux, and macOS. The book highlights methods such as the steepest descent method, Newton method, conjugate direction method, conjugate gradient methods, quasi-Newton methods, rank one correction formula, DFP method, BFGS method and their algorithms, convergence analysis, and proofs. Each method is accompanied by worked examples and R scripts. To help readers apply these methods in real-world situations, the book features a set of exercises at the end of each chapter. Primarily intended for graduate students of applied mathematics, operations research and statistics, it is also useful for students of mathematics, engineering, management, economics, and agriculture