The main aim of the book is, naturally, to give students the fundamental notions and instruments in linear algebra. Linearity is the main assumption used in all fieldsof science. It gives a first approximation to any problem under study and is widely used in economics and other social sciences. One may wonder why we decided to write a book in linear algebra despite the fact that there are many excellent books such as [10, 11, 19, 27, 34]? Our reasons can be summarized as follows. First, we try to fit the course to the needs of the students in economics and the students in mathematics and informatics who would like to get more knowledge in economics. Second, we constructed all expositions in the book in such a way to help economics students to learn mathematics and the proof making in mathematics in a convenient and simple manner. Third, since the hours given to this course in economics departments are rather limited, we propose a slightly different way of teaching this course. Namely, we do not try to give all proofs of all theorems presented in the course. Those theorems which are not proved are illustrated via figures and examples, and we illustrated all notions appealing to geometric intuition. Those theorems which are proved are proved in a most accurate way as it is done for the students in mathematics. The main notions are always supported with economic examples. The book provides many exercises referring to pure mathematics and economics. The book consists of eleven chapters and five appendices. Chapter 1 contains the introduction to the course and basic concepts of vector and scalar. Chapter 2 introduces the notions of vectors and matrices, and discusses some core economic examples used throughout the book. Here we begin with the notion of scalar product of two vectors, define matrices and their ranks, consider elementary operations over matrices. Chapter 3 deals with special important matrices – square matrices and their determinants. Chapter 4 introduces inverse matrices. In Chap. 5 we analyze the systems of linear equations, give methods how to solve these systems. Chapter ends with the discussion of homogeneous equations. Chapter 6 discusses more general type of algebraic objects – linear spaces. Here the notion of linear independence of vectors is introduced, which is very important from economic point of view for it defines how diverse is the obtained information. We consider here the isomorphism of linear spaces and the notion of subspace. Chapter 7 deals with important case of linear spaces – the Euclidean ones. We consider the notion of orthogonal bases and use it to construct the idea of projection and, particularly, the least square method widely used in social sciences. In Chapter 8 we consider linear transformations, and all related notions such as an image and kernel of transformation. We also consider linear transformations with respect to different bases. Chapter 9 discusses eigenvalues and eigenvectors. Here we consider self-adjoint transformations, orthogonal transformations, quadratic forms and their geometric representation. Chapter 10 applies the concepts developed before to the linear production model in economics. To this end we use, particularly, Perron–Frobenius Theorem. Chapter 11 deals with the notion of convexity, and so-called separation theorems. We use this instrument to analyse the linear programming problem. We observe during the years of our teaching experience that induction argument creates some difficulties among students. So, we explain this argument in Appendix A. In Appendix B we discuss how to evaluate the determinants. In Appendix C we give a brief introduction to complex numbers, which are important for better understanding the eigenvalues of linear operators. In Appendix D we consider the notion of the pseudoinverse, or generalized inverse matrix, widely used in different economic applications. Each chapter endswith the number of problemswhich allowbetter understanding the issues considered. In Appendix E the answers and hints to solutions to the problems from previous chapters and appendices are given.
Эта книга представляет собой введение в математическую теорию оптимизации. Она подчеркивает теорию конвергенции нелинейных алгоритмов оптимизации и приложений нелинейной оптимизации в комбинаторной оптимизации. Математическая теория оптимизации включает недавние разработки в глобальной конвергенции, гипотезу Пауэлла, полуопределенное программирование и методы релаксации для проектирования приближенного решения задач комбинаторной оптимизации.
This volume consists of chapters written by eminent scientists and engineers from the international community and presents significant advances in several theories, and applications of an interdisciplinary research. These contributions focus on both old and recent developments of Global Optimization Theory, Convex Analysis, Calculus of Variations, and Discrete Mathematics and Geometry, as well as several applications to a large variety of concrete problems, including applications of computers to the study of smoothness and analyticity of functions, applications to epidemiological diffusion, networks, mathematical models of elastic and piezoelectric fields, optimal algorithms, stability of neutral type vector functional differential equations, sampling and rational interpolation for non-band-limited signals, recurrent neural network for convex optimization problems, and experimental design.
The book also contains some review works, which could prove particularly useful for a broader audience of readers in Mathematical and Engineering subjects and especially to graduate students who search for the latest information.
The contributions in this volume have been written by eminent scientists from the international mathematical community and present significant advances in several theories, methods and problems of Mathematical Analysis, Discrete Mathematics, Geometry and their Applications. The chapters focus on both old and recent developments in Functional Analysis, Harmonic Analysis, Complex Analysis, Operator Theory, Combinatorics, Functional Equations, Differential Equations as well as a variety of Applications.
The book also contains some review works, which could prove particularly useful for a broader audience of readers in Mathematical Sciences, and especially to graduate students looking for the latest information.
This book gives a systematic presentation of modern measure theory as it has developed over the past century. It includes material for a standard graduate course, advanced material not covered by the standard course but necessary in order to read research literature in the area, and extensive additional information on the most diverse aspects of measure theory and its connections with other fields. Over 850 exercises with detailed hints or references are given. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided. The subject index includes more than 1000 items. The book is intended for graduate students, instructors of courses in measure and integration theory, and researchers in all fields of mathematics; it may serve as either a textbook, a source for a variety of advanced courses, or a reference work.
The volume addresses major features in empirical social research from methodological and theoretical perspectives. Prominent researchers discuss central problems in empirical social research in a theory-driven way from political science, sociological or social-psychological points of view. These contributions focus on a renewed discussion of foundations together with innovative and open research questions or interdisciplinary research perspectives.
This volume contains two types of papers—a selection of contributions from the “Second International Conference in Network Analysis” held in Nizhny Novgorod on May 7–9, 2012, and papers submitted to an "open call for papers" reflecting the activities of LATNA at the Higher School for Economics.
This volume contains many new results in modeling and powerful algorithmic solutions applied to problems in
- vehicle routing
- single machine scheduling
- modern financial markets
- cell formation in group technology
- brain activities of left- and right-handers
- speeding up algorithms for the maximum clique problem
- analysis and applications of different measures in clustering
The broad range of applications that can be described and analyzed by means of a network brings together researchers, practitioners, and other scientific communities from numerous fields such as Operations Research, Computer Science, Bioinformatics, Medicine, Transportation, Energy, Social Sciences, and more. The contributions not only come from different fields, but also cover a broad range of topics relevant to the theory and practice of network analysis. Researchers, students, and engineers from various disciplines will benefit from the state-of-the-art in models, algorithms, technologies, and techniques including new research directions and open questions.
В этом томе содержится подбор статей "Первой международной конференции по Анализу Сетевых Структур", состоявшейся в Университете Флориды 14-16 декабря 2011 года. Удивительное разнообразие областей, которые используют преимущества сетевого анализа, дает возможность сбора современного материала в единый сборник, что является полезным, но тяжелым занятием. Цель этой книги заключается в преодолении этих трудностей путем сбора основных результатов, полученных участниками и в объединении их в один легкодоступный сборник.
The contributions in this volume cover a broad range of topics including maximum cliques, graph coloring, data mining, brain networks, Steiner forest, logistic and supply chain networks. Network algorithms and their applications to market graphs, manufacturing problems, internet networks and social networks are highlighted. The "Fourth International Conference in Network Analysis," held at the Higher School of Economics, Nizhny Novgorod in May 2014, initiated joint research between scientists, engineers and researchers from academia, industry and government; the major results of conference participants have been reviewed and collected in this Work. Researchers and students in mathematics, economics, statistics, computer science and engineering will find this collection a valuable resource filled with the latest research in network analysis.
This book brings together the latest findings in the area of stochastic analysis and statistics. The individual chapters cover a wide range of topics from limit theorems, Markov processes, nonparametric methods, acturial science, population dynamics, and many others. The volume is dedicated to Valentin Konakov, head of the International Laboratory of Stochastic Analysis and its Applications on the occasion of his 70th birthday. Contributions were prepared by the participants of the international conference of the international conference “Modern problems of stochastic analysis and statistics”, held at the Higher School of Economics in Moscow from May 29 - June 2, 2016. It offers a valuable reference resource for researchers and graduate students interested in modern stochastics.