The Mathematical Structure of Stable Physical Systems


Book Description

This book is an introduction to the simple math patterns used to describe fundamental, stable spectral-orbital physical systems (represented as discrete hyperbolic shapes), the containment set has many-dimensions, and these dimensions possess macroscopic geometric properties (which are also discrete hyperbolic shapes). Thus, it is a description which transcends the idea of materialism (ie it is higher-dimensional), and it can also be used to model a life-form as a unified, high-dimension, geometric construct, which generates its own energy, and which has a natural structure for memory, where this construct is made in relation to the main property of the description being, in fact, the spectral properties of both material systems and of the metric-spaces which contain the material systems, where material is simply a lower dimension metric-space, and where both material-components and metric-spaces are in resonance with the containing space. Partial differential equations are defined on the many metric-spaces of this description, but their main function is to act on either the, usually, unimportant free-material components (to most often cause non-linear dynamics) or to perturb the orbits of the, quite often condensed, material trapped by (or within) the stable orbits of a very stable hyperbolic metric-space shape.




A Framework for K-12 Science Education


Book Description

Science, engineering, and technology permeate nearly every facet of modern life and hold the key to solving many of humanity's most pressing current and future challenges. The United States' position in the global economy is declining, in part because U.S. workers lack fundamental knowledge in these fields. To address the critical issues of U.S. competitiveness and to better prepare the workforce, A Framework for K-12 Science Education proposes a new approach to K-12 science education that will capture students' interest and provide them with the necessary foundational knowledge in the field. A Framework for K-12 Science Education outlines a broad set of expectations for students in science and engineering in grades K-12. These expectations will inform the development of new standards for K-12 science education and, subsequently, revisions to curriculum, instruction, assessment, and professional development for educators. This book identifies three dimensions that convey the core ideas and practices around which science and engineering education in these grades should be built. These three dimensions are: crosscutting concepts that unify the study of science through their common application across science and engineering; scientific and engineering practices; and disciplinary core ideas in the physical sciences, life sciences, and earth and space sciences and for engineering, technology, and the applications of science. The overarching goal is for all high school graduates to have sufficient knowledge of science and engineering to engage in public discussions on science-related issues, be careful consumers of scientific and technical information, and enter the careers of their choice. A Framework for K-12 Science Education is the first step in a process that can inform state-level decisions and achieve a research-grounded basis for improving science instruction and learning across the country. The book will guide standards developers, teachers, curriculum designers, assessment developers, state and district science administrators, and educators who teach science in informal environments.




Stability of Collisionless Stellar Systems


Book Description

Recent advances in our understanding of instabilities in galactic type systems have led to an unravelling of some of the mysteries of what determines the form galaxies take. This book focuses on the mathematical development of the subject, assuming no prior knowledge of it, with a strong emphasis on the underlying physical interpretation. This framework is used to discuss the most relevant instabilities which are believed to be closely involved in the way galaxies are formed, in a model independent manner. The relevant observed properties of galaxies that may be used to establish the role of these physical mechanisms are discussed. The book also includes a chapter discussing numerical simulation techniques, with attention paid to their limitations and to recent advances in this approach. It is demonstrated that recent developments in computer hardware enable a detailed comparison of simulations with analysis. Thus the simulations extend our physical understanding beyond the limitations of the analysis. The book is intended for use by postgraduate students and researchers in the areas of cosmology, extragalactic astronomy and dynamics.




Differential Dynamical Systems, Revised Edition


Book Description

Differential equations are the basis for models of any physical systems that exhibit smooth change. This book combines much of the material found in a traditional course on ordinary differential equations with an introduction to the more modern theory of dynamical systems. Applications of this theory to physics, biology, chemistry, and engineering are shown through examples in such areas as population modeling, fluid dynamics, electronics, and mechanics. Differential Dynamical Systems begins with coverage of linear systems, including matrix algebra; the focus then shifts to foundational material on nonlinear differential equations, making heavy use of the contraction-mapping theorem. Subsequent chapters deal specifically with dynamical systems concepts?flow, stability, invariant manifolds, the phase plane, bifurcation, chaos, and Hamiltonian dynamics. This new edition contains several important updates and revisions throughout the book. Throughout the book, the author includes exercises to help students develop an analytical and geometrical understanding of dynamics. Many of the exercises and examples are based on applications and some involve computation; an appendix offers simple codes written in Maple, Mathematica, and MATLAB software to give students practice with computation applied to dynamical systems problems.







Modeling, Analysis, and Optimization Issues for Large Space Structures


Book Description

This document contains the proceedings of the Air Force/NASA Workshop on Modeling, Analysis, and Optimization Issues for Large Space Structures held in Williamsburg, Virginia, May 13-14 1982. The theme of the workshop was modeling, analysis, and optimization of large space structures, including structure control interaction. Speakers were drawn primarily from industry, with participation from universities and government. The workshop was organized into three sessions: mathematical modeling, analysis methodology, and optimization for controllability. Results of the workshop were discussed in a final session. The workshop presentations ranged over many topics in large space structures, including structure-control interaction, structural and structural dynamics modeling, thermal analysis, testing, design, and optimization. The interdisciplinary area of structure-control interaction, which is a challenge to analysts, designers, and test engineers, was clearly emphasized. Not addressed in the workshop was the important subject of structural deployment.




Multiparameter Stability Theory with Mechanical Applications


Book Description

This book deals with fundamental problems, concepts, and methods of multiparameter stability theory with applications in mechanics. It presents recent achievements and knowledge of bifurcation theory, sensitivity analysis of stability characteristics, general aspects of nonconservative stability problems, analysis of singularities of boundaries for the stability domains, stability analysis of multiparameter linear periodic systems, and optimization of structures under stability constraints. Systems with finite degrees of freedom and with continuous models are both considered. The book combines mathematical foundation with interesting classical and modern mechanical problems.A number of mechanical problems illustrating how bifurcations and singularities change the behavior of systems and lead to new physical phenomena are discussed. Among these problems, the authors consider systems of rotating bodies, tubes conveying fluid, elastic columns under the action of periodic and follower forces, optimization problems for conservative systems, etc. The methods presented are constructive and easy to implement in computer programs.This book is addressed to graduate students, academics, researchers, and practitioners in aerospace, naval, civil, and mechanical engineering. No special background is needed; just a basic knowledge of mathematics and mechanics.




Theory of Stability of Continuous Elastic Structures


Book Description

Theory of Stability of Continuous Elastic Structures presents an applied mathematical treatment of the stability of civil engineering structures. The book's modern and rigorous approach makes it especially useful as a text in advanced engineering courses and an invaluable reference for engineers.




Model Validation


Book Description

Validation is a central issue to future model design in environmental science. This book is the first to provide a critical appraisal of today's validation needs, capabilities, and required changes in philosophy. It takes examples from four different scales: hillslope and river channel, catchment, regional, and global. This timely book offers unique, multifaceted coverage of model validation in hydrological science today. Topics covered include calibration procedures, data assimilation, scaling, critical future need in validation, and evidence of field data. * State-of-the-art research book on an important new topic * End-of-section discussion chapters written by leading international researchers




Mathematics for Stability and Optimization of Economic Systems


Book Description

Economic Theory and Mathematical Economics: Mathematics for Stability and Optimization of Economic Systems provides information pertinent to the stability aspects and optimization methods relevant to various economic systems. This book presents relevant mathematical theorems sufficient to develop important economic systems, including Leontief input–output systems, Keynesian dynamic models, the Ramsey optimal accumulation systems, and von Neumann expanding economic systems. Organized into two parts encompassing nine chapters, this book begins with an overview of useful theorems on matrices, eigenvalue problems, and matrices with dominant diagonals and P-matrices. This text then explores the linear transformations on vector spaces. Other chapters consider the Hawkins–Simon theorem concerning non-negative linear systems. This book discusses as well the dual linear relations and optimization methods applicable to inequality economic systems. The final chapter deals with powerful optimal control method for dynamical systems. This book is a valuable resource for mathematicians, economists, research workers, and graduate students.