Skip to main content
main-content

Über dieses Buch

This textbook presents a rigorous approach to multivariable calculus in the context of model building and optimization problems. This comprehensive overview is based on lectures given at five SERC Schools from 2008 to 2012 and covers a broad range of topics that will enable readers to understand and create deterministic and nondeterministic models. Researchers, advanced undergraduate, and graduate students in mathematics, statistics, physics, engineering, and biological sciences will find this book to be a valuable resource for finding appropriate models to describe real-life situations.

The first chapter begins with an introduction to fractional calculus moving on to discuss fractional integrals, fractional derivatives, fractional differential equations and their solutions. Multivariable calculus is covered in the second chapter and introduces the fundamentals of multivariable calculus (multivariable functions, limits and continuity, differentiability, directional derivatives and expansions of multivariable functions). Illustrative examples, input-output process, optimal recovery of functions and approximations are given; each section lists an ample number of exercises to heighten understanding of the material. Chapter three discusses deterministic/mathematical and optimization models evolving from differential equations, difference equations, algebraic models, power function models, input-output models and pathway models. Fractional integral and derivative models are examined. Chapter four covers non-deterministic/stochastic models. The random walk model, branching process model, birth and death process model, time series models, and regression type models are examined. The fifth chapter covers optimal design. General linear models from a statistical point of view are introduced; the Gauss–Markov theorem, quadratic forms, and generalized inverses of matrices are covered. Pathway, symmetric, and asymmetric models are covered in chapter six, the concepts are illustrated with graphs.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Essentials of Fractional Calculus

Abstract
In recent decades, the field of fractional calculus has attracted interest of researchers in several areas including mathematics, physics, chemistry, engineering, and even finance and social sciences.
A. M. Mathai, H. J. Haubold

Chapter 2. Multivariable Calculus

Abstract
A vector is a quantity that is determined by both its magnitude and its direction: thus, it is a directed line segment.
A. M. Mathai, H. J. Haubold

Chapter 3. Deterministic Models and Optimization

Abstract
There are various types of models that one can construct for a given set of data. The types of model that is chosen depends upon the type of data for which the model is to be constructed. If the data are coming from a deterministic situation, then there may be already an underlying mathematical formula such as a physical law.
A. M. Mathai, H. J. Haubold

Chapter 4. Non-deterministic Models and Optimization

Abstract
We considered models which were governed by definite mathematical rules or fully deterministic in nature. There was no chance variation involved. But most of the practical situations, mostly in social sciences, economics, commerce, management, etc., as well as many physical phenomena, are non-deterministic in nature. Earthquake at a place cannot be predetermined, but with sophisticated prediction tools, we may be able to predict the occurrence to some extent.
A. M. Mathai, H. J. Haubold

Chapter 5. Optimal Regression Designs

Abstract
The general linear model, discussed in Chapter 4, will be re-examined here in order to bring out some more interesting points and to talk about estimability of linear functions of the parameters, Gauss–Markov setup, and related matters.
A. M. Mathai, H. J. Haubold

Chapter 6. Pathway Models

Abstract
Here, we look at model building in general. First, we will deal with real scalar variable cases. Then generalized models, matrix-variate cases, etc., will be examined. First, let us look at the effect of power transformations and exponentiations on a given model. This will give some ideas about the changes required in a given situation of building models over a given data at hand. When building models for the data in hand, one usually takes a member from a parametric family of functions.
A. M. Mathai, H. J. Haubold

Backmatter

Weitere Informationen

Premium Partner

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

    Bildnachweise