"Provides a clear and comprehensive overview of the fundamental theories, numerical methods, and iterative processes encountered in difference calculus. Explores classical problems such as orthological polynomials, the Euclidean algorithm, roots of polynomials, and well-conditioning."

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; andmethods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.As a result, the book represents a blend of new methods in general computational analysis,and specific, but also generic, techniques for study of systems theory ant its particularbranches, such as optimal filtering and information compression.- Best operator approximation,- Non-Lagrange interpolation,- Generic Karhunen-Loeve transform- Generalised low-rank matrix approximation- Optimal data compression- Optimal nonlinear filtering

With the development of the scanning tunneling microscope, nanoscience became an important discipline. Single atoms could be manipulated in a controlled manner, and it became possible to change matter at its 'ultimate' level; it is the level on which the properties of matter emerge. This possibility enables to construct and to produce devices, materials, etc. with very small sizes and completely new properties. That opens up new perspectives for technology and is in particular relevant in connection with nano-engineering.Nanosystems are unimaginably small and very fast. No doubt, this is an important characteristic. But there is another feature, possibly more relevant, in connection with nanoscience and nanotechnology. The essential point here is that we work at the 'ultimate level'. This is the smallest level at which the properties of our world emerge, at which functional matter can exist. In particular, at this level biological individuality comes into existence. This situation can be expressed in absolute terms: This is not only the strongest material ever made, this is the strongest material it will ever be possible to make (D Ratner and M Ratner, Nanotechnology and Homeland Security). This is a very general statement. All aspects of matter are concerned here. Through the variation of the composition various forms of matter emerge with different items.Nanosystems are usually small, but they offer nevertheless the possibility to vary the structure of atomic (molecular) ensembles, creating a diversity of new material-specific properties. A large variety of experimental possibilities come into play and flexible theoretical tools are needed at the basic level. This is reflected in the different disciplines: In nanoscience and nanotechnology we have various directions: Materials science, functional nanomaterials, nanoparticles, food chemistry, medicine with brain research, quantum and molecular computing, bioinformatics, magnetic nanostructures, nano-optics, nano-electronics, etc.The properties of matter, which are involved within these nanodisciplines, are ultimate in character, i.e., their characteristic properties come into existence at this level. The book is organized in this respect.

Integrating both classical and modern treatments of difference equations, this book contains the most updated and comprehensive material on stability, Z-transform, discrete control theory, asymptotic theory, continued fractions and orthogonal polynomials. While the presentation is simple enough for use by advanced undergraduates and beginning graduates in mathematics, engineering science, and economics, it will also be a useful reference for scientists and engineers interested in discrete mathematical models. The text covers a large set of applications in a variety of disciplines, including neural networks, feedback control, Markov chains, trade models, heat transfer, propagation of plants, epidemic models and host-parasitoid systems, with each section rounded off by an extensive and highly selected set of exercises.

This collection of carefully refereed and edited papers were originally presented at the Fourth International Conference on Difference Equations held in Poznan, Poland. Contributions were from a diverse group of researchers from several countries and featured discussions on the theory of difference equations, open problems and conjectures, as well

Celebrating the work of Professor W. Kuperberg, this reference explores packing and covering theory, tilings, combinatorial and computational geometry, and convexity, featuring an extensive collection of problems compiled at the Discrete Geometry Special Session of the American Mathematical Society in New Orleans, Louisiana. Discrete Geometry analy

A must-read for mathematicians, scientists and engineers who want to understand difference equations and discrete dynamics Contains the most complete and comprehenive analysis of the stability of one-dimensional maps or first order difference equations. Has an extensive number of applications in a variety of fields from neural network to host-parasitoid systems. Includes chapters on continued fractions, orthogonal polynomials and asymptotics. Lucid and transparent writing style

The methods of functional analysis have helped solve diverse real-world problems in optimization, modeling, analysis, numerical approximation, and computer simulation. Applied Functional Analysis presents functional analysis results surfacing repeatedly in scientific and technological applications and presides over the most current analytical and numerical methods in infinite-dimensional spaces. This reference highlights critical studies in projection theorem, Riesz representation theorem, and properties of operators in Hilbert space and covers special classes of optimization problems. Supported by 2200 display equations, this guide incorporates hundreds of up-to-date citations.

Revised and updated, this second edition of Walter Gautschi's successful Numerical Analysis explores computational methods for problems arising in the areas of classical analysis, approximation theory, and ordinary differential equations, among others. Topics included in the book are presented with a view toward stressing basic principles and maintaining simplicity and teachability as far as possible, while subjects requiring a higher level of technicality are referenced in detailed bibliographic notes at the end of each chapter. Readers are thus given the guidance and opportunity to pursue advanced modern topics in more depth. Along with updated references, new biographical notes, and enhanced notational clarity, this second edition includes the expansion of an already large collection of exercises and assignments, both the kind that deal with theoretical and practical aspects of the subject and those requiring machine computation and the use of mathematical software. Perhaps most notably, the edition also comes with a complete solutions manual, carefully developed and polished by the author, which will serve as an exceptionally valuable resource for instructors.