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Introduction to finite element analysis and design
Finite Element Method (FEM) is one of the numerical methods of solving differential equations that describe many engineering problems. This text covers the basic theory of FEM and includes appendices on each of the main FEA programs as reference.
Book
Introduction to Recognition and Deciphering of Patterns
Introduction to Recognition and Deciphering of Patterns aims to get STEM and non-STEM students acquainted with different patterns, as well as where and when specific patterns arise. In addition, the book seeks to get students to learn how to recognize patterns and distinguish the similarities and differences between them.
Patterns emerge on an everyday basis, such as weather patterns, traffic patterns, behavioral patterns, geometric patterns, linguistic patterns, structural patterns, digital patterns, etc. Recognizing patterns and studying their unique traits is essential for the development and enhancement of our intuitive skills and in strengthening our analytical skills. Mathematicians often apply patterns to get acquainted with new concepts, but this is a technique that can be applied across many disciplines.
Throughout this book we will encounter assorted patterns that emerge from various geometrical configurations of squares, circles, right triangles and equilateral triangles that either repeat at the same scale or at different scales. The book will also focus on describing linear patterns, geometric patterns, alternating patterns, piecewise patterns, summation-type patterns and factorial-type patterns analytically. Deciphering the details of these distinct patterns will lead to the proof by induction method. Furthermore, the book will render properties of Pascal's triangle and provide supplemental practice in deciphering specific patterns and verifying them.
The book will adjourn with first-order recursive relations: describing sequences as recursive relations, obtaining the general solution by solving an initial value problem and determining the periodic traits.
eBook
Solutions manual to accompany Finite mathematics: models and applications
A solutions manual to accompany Finite Mathematics: Models and Applications In order to emphasize the main concepts of each chapter, Finite Mathematics: Models and Applications features plentiful pedagogical elements throughout such as special exercises, end notes, hints, select solutions, biographies of key mathematicians, boxed key principles, a glossary of important terms and topics, and an overview of use of technology. The book encourages the modeling of linear programs and their solutions and uses common computer software programs such as LINDO. In addition to extensive chapters on probability and statistics, principles and applications of matrices are included as well as topics for enrichment such as the Monte Carlo method, game theory, kinship matrices, and dynamic programming. Supplemented with online instructional support materials, the book features coverage including: Algebra Skills Mathematics of Finance Matrix Algebra Geometric Solutions Simplex Methods Application Models Set and Probability Relationships Random Variables and Probability Distributions Markov Chains Mathematical Statistics Enrichment in Finite Mathematics.
eBook
Generalized Network Design Problems
Combinatorial optimization is a fascinating topic. Combinatorial optimization problems arise in a wide variety of important fields such as transportation, telecommunications, computer networking, location, planning, distribution problems, etc. Important and significant results have been obtained on the theory, algorithms and applications over the last few decades. In combinatorial optimization, many network design problems can be generalized in a natural way by considering a related problem on a clustered graph, where the original problem's feasibility constraints are expressed in terms of the clusters, i.e., node sets instead of individual nodes. This class of problems is usually referred to as generalized network design problems (GNDPs) or generalized combinatorial optimization problems. The express purpose of this monograph is to describe a series of mathematical models, methods, propositions, algorithms developed in the last years on generalized network design problems in a unified manner. The book consists of seven chapters, where in addition to an introductory chapter, the following generalized network design problems are formulated and examined: the generalized minimum spanning tree problem, the generalized traveling salesman problem, the railway traveling salesman problem, the generalized vehicle routing problem, the generalized fixed-charge network design problem and the generalized minimum vertex-biconnected network problem. The book will be useful for researchers, practitioners, and graduate students in operations research, optimization, applied mathematics and computer science. Due to the substantial practical importance of some presented problems, researchers in other areas will find this book useful, too.
eBook
Introduction to the explicit finite element method for nonlinear transient dynamics
\"This is the first book to specifically address the explicit finite element method for nonlinear transient dynamics. This book aids readers in mastering the explicit finite element method as well as programming a code without extensively reading the more general finite element books. This book consists of 12 chapters within four sections including: the variation principles and formulation of the explicit finite element method for nonlinear transient dynamics; the finite element technology with 4-node and 3-node Reissner-Mindlin plate bending elements, the 8-node solid elements, etc.; plasticity and nonlinear material models; and contact algorithms and other kinematic constraint conditions. Each chapter contains a list of carefully chosen references intended to help readers to further explore the related subjects\"--
eBook
Finite Mathematics as the Most General (Fundamental) Mathematics
The purpose of this paper is to explain at the simplest possible level why finite mathematics based on a finite ring of characteristic p is more general (fundamental) than standard mathematics. The belief of most mathematicians and physicists that standard mathematics is the most fundamental arose for historical reasons. However, simple mathematical arguments show that standard mathematics (involving the concept of infinities) is a degenerate case of finite mathematics in the formal limit p→∞; standard mathematics arises from finite mathematics in the degenerate case when operations modulo a number are discarded. Quantum theory based on a finite ring of characteristic p is more general than standard quantum theory because the latter is a degenerate case of the former in the formal limit p→∞.
Journal Article
Main Problems in Constructing Quantum Theory Based on Finite Mathematics
As shown in our publications, quantum theory based on a finite ring of characteristic p (FQT) is more general than standard quantum theory (SQT) because the latter is a degenerate case of the former in the formal limit p→∞. One of the main differences between SQT and FQT is the following. In SQT, elementary objects are described by irreducible representations (IRs) of a symmetry algebra in which energies are either only positive or only negative and there are no IRs where there are states with different signs of energy. In the first case, objects are called particles, and in the second antiparticles. As a consequence, in SQT it is possible to introduce conserved quantum numbers (electric charge, baryon number, etc.) so that particles and antiparticles differ in the signs of these numbers. However, in FQT, all IRs necessarily contain states with both signs of energy. The symmetry in FQT is higher than the symmetry in SQT because one IR in FQT splits into two IRs in SQT with positive and negative energies at p→∞. Consequently, most fundamental quantum theory will not contain the concepts of particle–antiparticle and additive quantum numbers. These concepts are only good approximations at present since at this stage of the universe the value p is very large but it was not so large at earlier stages. The above properties of IRs in SQT and FQT have been discussed in our publications with detailed technical proofs. The purpose of this paper is to consider models where these properties can be derived in a much simpler way.
Journal Article
A defense of Isaacson’s thesis, or how to make sense of the boundaries of finite mathematics
Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (
PA
) is
the
theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can be overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some
PA
-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of
PA
. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious
PA
-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below
ε
0
. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below
ε
0
are finitary objects despite being infinite.
Journal Article
Discussion of foundation of mathematics and quantum theory
Following the results of our recently published book [F. Lev,
, Springer, 2020, ISBN 978-3-030-61101-9], we discuss different aspects of classical and finite mathematics and explain why finite mathematics based on a finite ring of characteristic
is more general (fundamental) than classical mathematics: the former does not have foundational problems, and the latter is a special degenerate case of the former in the formal limit
. In particular, quantum theory based on a finite ring of characteristic
is more general than standard quantum theory because the latter is a special degenerate case of the former in the formal limit
Journal Article