Mathematical rigor

This book introduces a variety of problem statements in classical optimal control, in optimal estimation and filtering, and in optimal control problems with nonscalarvalued performance criteria. Many example problems are solved completely in the body of the text. All chapterend exercises are sketched in the appendix. The theoretical part of the book is based on the calculus of variations, so the exposition is very transparent and requires little mathematical rigor.
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Solve Two of the Toughest Problems When Preparing for the Stockbroker’s Exam Those wishing to become licensed as stockbrokers must pass the series 7 examination. This exam, known officially as the General Securities Registered Representative Examination, is very rigorous. Traditionally, students without a financial background have a difficult time with the mathematical calculations peculiar to the world of stocks, bonds, and options. Many are also relatively unfa miliar with proper use of the calculator and thus are dou bly hampered in their efforts to become registered.
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As states and local school districts implement more rigorous assessment and accountability systems, teachers often face long lists of mathematics topics or learning expectations to address at each grade level, with many topics repeating from year to year. Lacking clear, consistent priorities and focus, teachers stretch to find the time to present important mathematical topics effectively and in depth.
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Continuing interest in the subject of reliability and the heretofore unavailability of our book Mathematical Theory of Reliability have encouraged publication of this SIAM Classics edition. We have not revised the original version, although much has transpired since its original publication in 1965. Although many contemporary reliability books are now available, few provide as mathematically rigorous a treatment of the required probability background as this one.
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We have attempted to explain the concepts which have been used and developed to model the stochastic dynamics of natural and biological systems. While the theory of stochastic differential equations and stochastic processes provide an attractive framework with an intuitive appeal to many problems with naturally induced variations, the solutions to such models are an active area of research, which is in its infancy. Therefore, this book should provide a large number of areas to research further.
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Lecture Mathematics 53  Lecture 5.2 provides knowledge of the integrals yielding logarithmic and exponential functions. This chapter presents the following content: Integrals of f(x) = 1/x and of the other circular functions, integrals of exponential functions, the natural logarithmic function: A rigorous approach.
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This book is intended for a rigorous introductory Ph.D. level course in econometrics, or for use in a field course in econometric theory. It is based on lecture notes that I have developed during the period 19972003 for the first semester econometrics course “Introduction to Econometrics” in the core of the Ph.D. program in economics at the Pennsylvania State University. Initially these lecture notes were written as a companion to Gallant’s (1997) textbook, but have been developed gradually into an alternative textbook.
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Our primary objective herein is not to determine how approximate calculations introduce errors into situations with accurate hypotheses, but instead to study how rigorous calculations transmit errors due to inaccurate parameters or hypotheses. Unlike quantities represented by entire numbers, the continuous quantities generated from physics, economics or engineering sciences, as represented by one or several real numbers, are compromised by errors.
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Although Continuum Mechanics belongs to a traditional topic, the research in this field has never been stopped. The goal of this book is to introduce the latest progress in the fundamental aspects and the applications in various engineering areas. The first three chapters are on the fundamentals of Continuum Mechanics. Chapter 1 introduces the Spencer Operator and presents the applications of this useful operator in solving Continuum Mechanics problems. The authors extend the ideas for tackling general Mathematical Physics problems. Chapter 2 is on Transversality Condition.
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The emergence of a new paradigm in science offers vast perspectives for future investigations, as well as providing fresh insight into existing areas of knowledge, discovering hitherto unknown relations between them. We can observe this kind of process in connection with the appearance of the concept of solitons [465].
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Journal of Mathematical Neuroscience (2011) 1:1 DOI 10.1186/2190856711 RESEARCH Open Access Stability of the stationary solutions of neural ﬁeld equations with propagation delays Romain Veltz · Olivier Faugeras Received: 22 October 2010 / Accepted: 3 May 2011 / Published online: 3 May 2011 © 2011 Veltz, Faugeras; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License Abstract In this paper, we consider neural ﬁeld equations with spacedependent delays.
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This paper introduces a rigorous computerassisted procedure for analyzing hyperbolic 3manifolds. This procedure is used to complete the proof of several longstanding rigidity conjectures in 3manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3manifold. Theorem 0.1. Let N be a closed hyperbolic 3manifold.
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The ILO gender audit has added impetus to the process of integrating gender equality into the ILO by setting in motion an institutional learning mechanism. The challenge now is to rigorously monitor the implementation of its recommendations. The ILO constituents and the UN system are increasingly interested in conducting gender audits in order to accelerate gender mainstreaming within their own organizations. ILO is responding by modifying and revising the audit methodology to meet their speciﬁ c needs.
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Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements.
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Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system).
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Mathematicians seek out patterns[9][10] and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
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One of our objectives is to develop rigorously the concepts of limit, continuity, differen tiability, and integrability, which you have seen in calculus. To do this requires a better understanding of the real numbers than is provided in calculus. The purpose of this section is to develop this understanding. Since the utility of the concepts introduced here will not become apparent until we are well into the study of limits and continuity, you should re serve judgment on their value until they are applied. As this occurs, you should reread the applicable parts of this section.
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Aristotle defined mathematics as "the science of quantity", and this definition prevailed until the 18th century.[21] Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clearcut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[22] Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics.
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This book presents a mathematical development of a recent approach to the modeling and simulation of turbulent flows based on methods for the approximate solution of inverse problems. The resulting Approximate Deconvolution Models or ADMs have some advantages (as well as some disadvantages) over more commonly used turbulence models: ADMs are supported by a mathematically rigorous theoretical foundation. ADMs are a family of models of increasing accuracy O(δ2N+2), where δ is the averaging (or filter) radius...
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The Dictionary of Classical and Theoretical Mathematics, one volume of the Comprehensive Dictionary of Mathematics, includes entries from the fields of geometry, logic, number theory, set theory, and topology. The authors who contributed their work to this volume are professional mathematicians, active in both teaching and research. The goal in writing this dictionary has been to define each term rigorously, not to author a large and comprehensive survey text in mathematics.
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