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This collection of papers ranges over several areas of research, including matrix theory, group theory, functional analysis, Morse theory, global analysis, and probability theory.

Frederick Mosteller has inspired numerous statisticians and other scientists by his creative approach to statistics and its applications. This volume brings together 40 of his most original and influential papers, capturing the variety and depth of his writings. The editors hope to share these with a new generation of researchers, so that they can build upon his insights and efforts. This volume of selected papers is a companion to the earlier volume A Statistical Model: Frederick Mosteller's Contributions to Statistics, Science, and Public Policy, edited by Stephen E. Fienberg, David C. Hoaglin, William H. Kruskal, and Judith M. Tanur (Springer-Verlag, 1990), and to Mosteller's forthcoming autobiography, which will also be published by Springer-Verlag. It includes a biography and a comprehensive bibliography of Mosteller's books, papers, and other writings. Stephen E. Fienberg is Maurice Falk University Professor of Statistics and Social Science, in the Departments of Statistics and Machine Learning at Carnegie Mellon University, Pittsburgh, PA. David C. Hoaglin is Principal Scientist at Abt Associates Inc., Cambridge, MA.

This book presents the refereed proceedings of the 15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing that was held in Linz, Austria, and organized by the Johannes Kepler University Linz and the Austrian Academy of Sciences, in July 2022. These biennial conferences are major events for Monte Carlo and quasi-Monte Carlo researchers. The proceedings include articles based on invited lectures as well as carefully selected contributed papers on all theoretical aspects and applications of Monte Carlo and quasi-Monte Carlo methods. Offering information on the latest developments in these highly active areas, this book is an excellent reference resource for theoreticians and practitioners interested in solving high-dimensional computational problems, in particular arising in finance, statistics and computer graphics.

This book presents a collection of selected papers that represent the current variety of research on the teaching and learning of probability. The respective chapters address a diverse range of theoretical, empirical and practical aspects underpinning the teaching and learning of probability, curricular issues, probabilistic reasoning, misconceptions and biases, as well as their pedagogical implications. These chapters are divided into THREE main sections, dealing with: TEACHING PROBABILITY, STUDENTS'REASONING AND LEARNING AND EDUCATION OF TEACHERS.In brief, the papers presented here include research dealing with teachers and students at different levels and ages (from primary school to university) and address epistemological and curricular analysis, as well as the role of technology, simulations, language and visualisation in teaching and learning probability. As such, it offers essential information for teachers, researchers and curricular designers alike.

This book includes the joint proceedings of the International Conference on Particle Systems and PDEs VI, VII and VIII. Particle Systems and PDEs VI was held in Nice, France, in November/December 2017, Particle Systems and PDEs VII was held in Palermo, Italy, in November 2018, and Particle Systems and PDEs VIII was held in Lisbon, Portugal, in December 2019. Most of the papers are dealing with mathematical problems motivated by different applications in physics, engineering, economics, chemistry and biology. They illustrate methods and topics in the study of particle systems and PDEs and their relation. The book is recommended to probabilists, analysts and to those mathematicians in general, whose work focuses on topics in mathematical physics, stochastic processes and differential equations, as well as to those physicists who work in statistical mechanics and kinetic theory.

This book fills an important gap in studies on D. D. Kosambi. For the first time, the mathematical work of Kosambi is described, collected and presented in a manner that is accessible to non-mathematicians as well. A number of his papers that are difficult to obtain in these areas are made available here. In addition, there are essays by Kosambi that have not been published earlier as well as some of his lesser known works. Each of the twenty four papers is prefaced by a commentary on the significance of the work, and where possible, extracts from technical reviews by other mathematicians.

​This book presents the refereed proceedings of the 13th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing that was held at the University of Rennes, France, and organized by Inria, in July 2018. These biennial conferences are major events for Monte Carlo and quasi-Monte Carlo researchers. The proceedings include articles based on invited lectures as well as carefully selected contributed papers on all theoretical aspects and applications of Monte Carlo and quasi-Monte Carlo methods. Offering information on the latest developments in these very active areas, this book is an excellent reference resource for theoreticians and practitioners interested in solving high-dimensional computational problems, arising, in particular, in finance, statistics and computer graphics.

This volume gathers selected peer-reviewed papers presented at the XXVI International Joint Conference on Industrial Engineering and Operations Management (IJCIEOM), held on July 8-11, 2020 in Rio de Janeiro, Brazil. The respective chapters address a range of timely topics in industrial engineering, including operations and process management, global operations, managerial economics, data science and stochastic optimization, logistics and supply chain management, quality management, product development, strategy and organizational engineering, knowledge and information management, work and human factors, sustainability, production engineering education, healthcare operations management, disaster management, and more. These topics broadly involve fields like operations, manufacturing, industrial and production engineering, and management. Given its scope, the book offers a valuable resource for those engaged in optimization research, operations research, and practitioners alike.

This book discusses recent developments and contemporary research in mathematics, statistics and their applications in computing. All contributing authors are eminent academicians, scientists, researchers and scholars in their respective fields, hailing from around the world. This is the second conference on mathematics and computing organized at Haldia Institute of Technology, India. The conference has emerged as a powerful forum, offering researchers a venue to discuss, interact and collaborate, and stimulating the advancement of mathematics and its applications in computer science. The book will allow aspiring researchers to update their knowledge of cryptography, algebra, frame theory, optimizations, stochastic processes, compressive sensing, functional analysis, complex variables, etc. Educating future consumers, users, producers, developers and researchers in mathematics and computing is a challenging task and essential to the development of modern society. Hence, mathematics and its applications in computing are of vital importance to a broad range of communities, including mathematicians and computing professionals across different educational levels and disciplines.In current research, modeling and simulation, making decisions under uncertainty and pattern recognition have become very common. Professionals across different educational levels and disciplines need exposure to advances in mathematics and computing. In this context, this book presents research papers on applicable areas of current interest. It also includes papers in which experts summarize research findings, such as signal processing and analysis and low-rank-matrix approximation for solving large systems, which will emerge as powerful tools for further research. These new advances and cutting-edge research in the fields of mathematics and their applications to computing are of paramount importance for young researchers.

The book, belonging to the series “Studies in Theoretical and Applied Statistics– Selected Papers from the Statistical Societies”, presents a peer-reviewed selection of contributions on relevant topics organized by the editors on the occasion of the SIS 2013 Statistical Conference'Advances in Latent Variables. Methods, Models and Applications', held at the Department of Economics and Management of the University of Brescia from June 19 to 21, 2013.The focus of the book is on advances in statistical methods for analyses with latent variables. In fact, in recent years, there has been increasing interest in this broad research area from both a theoretical and an applied point of view, as the statistical latent variable approach allows the effective modeling of complex real-life phenomena in a wide range of research fields.A major goal of the volume is to bring together articles written by statisticians from different research fields, which present different approaches and experiences related to the analysis of unobservable variables and the study of the relationships between them.

This book was originally compiled for a course I taught at the University of Rochester in the fall of 1991, and is intended to give advanced graduate students in statistics an introduction to Edgeworth and saddlepoint approximations, and related techniques. Many other authors have also written monographs on this sub­ ject, and so this work is narrowly focused on two areas not recently discussed in theoretical text books. These areas are, first, a rigorous consideration of Edgeworth and saddlepoint expansion limit theorems, and second, a survey of the more recent developments in the field. In presenting expansion limit theorems I have drawn heavily on notation of McCullagh (1987) and on the theorems presented by Feller (1971) on Edgeworth expansions. For saddlepoint notation and results I relied most heavily on the many papers of Daniels, and a review paper by Reid (1988). Throughout this book I have tried to maintain consistent notation and to present theorems in such a way as to make a few theoretical results useful in as many contexts aS possible. This was not only in order to present as many results with as few proofs as possible, but more importantly to show the interconnections between the various facets of asymptotic theory. Special attention is paid to regularity conditions. The reasons they are needed and the parts they play in the proofs are both highlighted.

This book presents a selection of papers presented to the Second Inter­ national Symposium on Semi-Markov Models: Theory and Applications held in Compiegne (France) in December 1998. This international meeting had the same aim as the first one held in Brussels in 1984 : to make, fourteen years later, the state of the art in the field of semi-Markov processes and their applications, bring together researchers in this field and also to stimulate fruitful discussions. The set of the subjects of the papers presented in Compiegne has a lot of similarities with the preceding Symposium; this shows that the main fields of semi-Markov processes are now well established particularly for basic applications in Reliability and Maintenance, Biomedicine, Queue­ ing, Control processes and production. A growing field is the one of insurance and finance but this is not really a surprising fact as the problem of pricing derivative products represents now a crucial problem in economics and finance. For example, stochastic models can be applied to financial and insur­ ance models as we have to evaluate the uncertainty of the future market behavior in order, firstly, to propose different measures for important risks such as the interest risk, the risk of default or the risk of catas­ trophe and secondly, to describe how to act in order to optimize the situation in time. Recently, the concept of VaR (Value at Risk) was'discovered'in portfolio theory enlarging so the fundamental model of Markowitz.

In response to a growing interest in Total Least Squares (TLS) and Errors-In-Variables (EIV) modeling by researchers and practitioners, well-known experts from several disciplines were invited to prepare an overview paper and present it at the third international workshop on TLS and EIV modeling held in Leuven, Belgium, August 27-29, 2001. These invited papers, representing two-thirds of the book, together with a selection of other presented contributions yield a complete overview of the main scientific achievements since 1996 in TLS and Errors-In-Variables modeling. In this way, the book nicely completes two earlier books on TLS (SIAM 1991 and 1997). Not only computational issues, but also statistical, numerical, algebraic properties are described, as well as many new generalizations and applications. Being aware of the growing interest in these techniques, it is a strong belief that this book will aid and stimulate users to apply the new techniques and models correctly to their own practical problems.

The present volume contains a selection of papers presented at the Eighth Conference of the International Federation of Classification Societies (IFCS) which was held in Cracow, Poland, July 16-19, 2002. All originally submitted papers were subject to a reviewing process by two independent referees, a procedure which resulted in the selection of the 53 articles presented in this volume. These articles relate to theoretical investigations as well as to practical applications and cover a wide range of topics in the broad domain of classifi­ cation, data analysis and related methods. If we try to classify the wealth of problems, methods and approaches into some representative (partially over­ lapping) groups, we find in particular the following areas: • Clustering • Cluster validation • Discrimination • Multivariate data analysis • Statistical methods • Symbolic data analysis • Consensus trees and phylogeny • Regression trees • Neural networks and genetic algorithms • Applications in economics, medicine, biology, and psychology. Given the international orientation of IFCS conferences and the leading role of IFCS in the scientific world of classification, clustering and data anal­ ysis, this volume collects a representative selection of current research and modern applications in this field and serves as an up-to-date information source for statisticians, data analysts, data mining specialists and computer scientists.

In 1992, when Paul Erdos was awarded a Doctor Honoris Causa by Charles University in Prague, a small conference was held, bringing together a distin­ guished group of researchers with interests spanning a variety of fields related to Erdos'own work. At that gathering, the idea occurred to several of us that it might be quite appropriate at this point in Erdos'career to solicit a col­ lection of articles illustrating various aspects of Erdos'mathematical life and work. The response to our solicitation was immediate and overwhelming, and these volumes are the result. Regarding the organization, we found it convenient to arrange the papers into six chapters, each mirroring Erdos'holistic approach to mathematics. Our goal was not merely a (random) collection of papers but rather a thor­ oughly edited volume composed in large part by articles explicitly solicited to illustrate interesting aspects of Erdos and his life and work. Each chap­ ter includes an introduction which often presents a sample of related ErdOs'problems'in his own words'. All these (sometimes lengthy) introductions were written jointly by editors. We wish to thank the nearly 70 contributors for their outstanding efforts (and their patience). In particular, we are grateful to Bela Bollobas for his extensive documentation of Paul Erdos'early years and mathematical high points (in the first part of this volume); our other authors are acknowledged in their respective chapters. We also want to thank A. Bondy, G. Hahn, I.

This IMA Volume in Mathematics and its Applications TIME SERIES ANALYSIS AND APPLICATIONS TO GEOPHYSICAL SYSTEMS contains papers presented at a very successful workshop on the same title. The event which was held on November 12-15, 2001 was an integral part of the IMA 2001-2002 annual program on'Mathematics in the Geosciences.'We would like to thank David R. Brillinger (Department of Statistics, Uni­ versity of California, Berkeley), Enders Anthony Robinson (Department of Earth and Environmental Engineering, Columbia University), and Fred­ eric Paik Schoenberg (Department of Statistics, University of California, Los Angeles) for their superb role as workshop organizers and editors of the proceedings. We are also grateful to Robert H. Shumway (Department of Statistics, University of California, Davis) for his help in organizing the four-day event. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA v PREFACE This volume contains a collection of papers that were presented dur­ ing the Workshop on Time Series Analysis and Applications to Geophysical Systems at the Institute for Mathematics and its Applications (IMA) at the University of Minnesota from November 12-15, 2001. This was part of the IMA Thematic Year on Mathematics in the Geosciences, and was the last in a series of four Workshops during the Fall Quarter dedicated to Dynamical Systems and Ergodic Theory.

This volume focuses on innovative approaches and recent developments in clustering, analysis of data and models, and applications: The first part of the book covers a broad range of innovations in the area of clustering, from algorithmic innovations for graph clustering to new visualization and evaluation techniques. The second part addresses new developments in data and decision analysis (conjoint analysis, non-additive utility functions, analysis of asymmetric relationships, and regularization techniques). The third part is devoted to the application of innovative data analysis methods in the life-sciences, the social sciences and in engineering. All contributions in this volume are revised and extended versions of selected papers presented in the German/Japanese Workshops at Karlsruhe (2010) and Kyoto (2012).

This book focuses mainly on fractional Brownian fields and their extensions. It has been used to teach graduate students at Grenoble and Toulouse's Universities. It is as self-contained as possible and contains numerous exercises, with solutions in an appendix. After a foreword by Stéphane Jaffard, a long first chapter is devoted to classical results from stochastic fields and fractal analysis. A central notion throughout this book is self-similarity, which is dealt with in a second chapter with a particular emphasis on the celebrated Gaussian self-similar fields, called fractional Brownian fields after Mandelbrot and Van Ness's seminal paper. Fundamental properties of fractional Brownian fields are then stated and proved. The second central notion of this book is the so-called local asymptotic self-similarity (in short lass), which is a local version of self-similarity, defined in the third chapter. A lengthy study is devoted to lass fields with finite variance. Among these lass fields, we find both Gaussian fields and non-Gaussian fields, called Lévy fields. The Lévy fields can be viewed as bridges between fractional Brownian fields and stable self-similar fields. A further key issue concerns the identification of fractional parameters. This is the raison d'être of the statistics chapter, where generalized quadratic variations methods are mainly used for estimating fractional parameters. Last but not least, the simulation is addressed in the last chapter. Unlike the previous issues, the simulation of fractional fields is still an area of ongoing research. The algorithms presented in this chapter are efficient but do not claim to close the debate.

Sergei Kuznetsov is one of the top experts on measure valued branching processes (also known as “superprocesses”) and their connection to nonlinear partial differential operators. His research interests range from stochastic processes and partial differential equations to mathematical statistics, time series analysis and statistical software; he has over 90 papers published in international research journals. His most well known contribution to probability theory is the'Kuznetsov-measure.'A conference honoring his 60th birthday has been organized at Boulder, Colorado in the summer of 2010, with the participation of Sergei Kuznetsov's mentor and major co-author, Eugene Dynkin. The conference focused on topics related to superprocesses, branching diffusions and nonlinear partial differential equations. In particular, connections to the so-called “Kuznetsov-measure” were emphasized. Leading experts in the field as well as young researchers contributed to the conference. The meeting was organized by J. Englander and B. Rider (U. of Colorado).

Copulas are mathematical objects that fully capture the dependence structure among random variables and hence offer great flexibility in building multivariate stochastic models. Since their introduction in the early 1950s, copulas have gained considerable popularity in several fields of applied mathematics, especially finance and insurance. Today, copulas represent a well-recognized tool for market and credit models, aggregation of risks, and portfolio selection. Historically, the Gaussian copula model has been one of the most common models in credit risk. However, the recent financial crisis has underlined its limitations and drawbacks. In fact, despite their simplicity, Gaussian copula models severely underestimate the risk of the occurrence of joint extreme events. Recent theoretical investigations have put new tools for detecting and estimating dependence and risk (like tail dependence, time-varying models, etc) in the spotlight. All such investigations need to be further developed and promoted, a goal this book pursues. The book includes surveys that provide an up-to-date account of essential aspects of copula models in quantitative finance, as well as the extended versions of talks selected from papers presented at the workshop in Cracow.

Paul Erdös was one of the most influential mathematicians of the twentieth century, whose work in number theory, combinatorics, set theory, analysis, and other branches of mathematics has determined the development of large areas of these fields. In 1999, a conference was organized to survey his work, his contributions to mathematics, and the far-reaching impact of his work on many branches of mathematics. On the 100th anniversary of his birth, this volume undertakes the almost impossible task to describe the ways in which problems raised by him and topics initiated by him (indeed, whole branches of mathematics) continue to flourish. Written by outstanding researchers in these areas, these papers include extensive surveys of classical results as well as of new developments.

This book brings together papers of well-known specialists in game theory and adjacent problems. It presents the basic results in dynamic games, stochastic games, applications of game theoretical methods in ecology and economics and methodological aspects of game theory. Special attention was given to problems in dynamic games under partial information and to the development of numerical methods for high-dimensional games (there is an increasing focus on this arena as recent theory is leading to solution methods for problems which were heretofore quite intractable). The interdisciplinary aspects touched upon were related to dynamical systems via replicator dynamics, with probability (measure-valued processes), statistical mechanics (kinetic equation, non-equilibrium behaviour) and max-plus (tropical or idempotent) mathematics. In this volume, we publish the review of Martin Shubik (The Present and Future of Game Theory) and the contributions presenting extending versions of the talks given at the workshop'Game Theory for Finance, Social and Biological Sciences', held in Warwick 14-17 of April 2010. The conference was exclusively marked by a rare natural event: the eruption of an Icelandic volcano Eyjafjallajökull, which blocked the functioning of most of the airlines, turning the return trip into an adventurous enterprise for many participants.

Professor Kiyosi Ito is well known as the creator of the modern theory of stochastic analysis. Although Ito first proposed his theory, now known as Ito's stochastic analysis or Ito's stochastic calculus, about fifty years ago, its value in both pure and applied mathematics is becoming greater and greater. For almost all modern theories at the forefront of probability and related fields, Ito's analysis is indispensable as an essential instrument, and it will remain so in the future. For example, a basic formula, called the Ito formula, is well known and widely used in fields as diverse as physics and economics. This volume contains 27 papers written by world-renowned probability theorists. Their subjects vary widely and they present new results and ideas in the fields where stochastic analysis plays an important role. Also included are several expository articles by well-known experts surveying recent developments. Not only mathematicians but also physicists, biologists, economists and researchers in other fields who are interested in the effectiveness of stochastic theory will find valuable suggestions for their research. In addition, students who are beginning their study and research in stochastic analysis and related fields will find instructive and useful guidance here. This volume is dedicated to Professor Ito on the occasion of his eightieth birthday as a token of deep appreciation for his great achievements and contributions. An introduction to and commentary on the scientific works of Professor Ito are also included.

Since the early eighties, Ali Süleyman Üstünel has been one of the main contributors to the field of Malliavin calculus. In a workshop held in Paris, June 2010 several prominent researchers gave exciting talks in honor of his 60th birthday. The present volume includes scientific contributions from this workshop. Probability theory is first and foremost aimed at solving real-life problems containing randomness. Markov processes are one of the key tools for modeling that plays a vital part concerning such problems. Contributions on inventory control, mutation-selection in genetics and public-private partnerships illustrate several applications in this volume. Stochastic differential equations, be they partial or ordinary, also play a key role in stochastic modeling. Two of the contributions analyze examples that share a focus on probabilistic tools, namely stochastic analysis and stochastic calculus. Three other papers are devoted more to the theoretical development of these aspects. The volume addresses graduate students and researchers interested in stochastic analysis and its applications.

This volume contains the majority of the papers presented at the 5th Inter­ national Workshop on Model-Oriented Data Analysis held in June 1998. This series started in March 1987 with a meeting on the Wartburg, Eisenach (Germany). The next three meetings were in 1990 (St Kyrik monastery, Bulgaria), 1992 (Petrodvorets, StPetersburg, Russia) and 1995 (Spetses, Greece). The main purpose of these workshops was to bring together lead­ ing scientists from'Eastern'and'Western'Europe for the exchange of ideas in theoretical and applied statistics, with special emphasis on experimen­ tal design. Now that the separation between East and West has become less rigid, this dialogue has, in principle, become much easier. However, providing opportunities for this dialogue is as vital as ever. MODA meetings are known for their friendly atmosphere, leading to fruitful discussions and collaboration, especially between young and senior scien­ tists. Indeed, many long term collaborations were initiated during these events. This intellectually stimulating atmosphere is achieved by limiting the number of participants to around eighty, by the choice of location so that participants can live as a community, and, of course, through the care­ ful selection of scientific direction made by the Programme Committee.

The first Pannonian Symposium on Mathematical Statistics was held at Bad Tatzmannsdorf (Burgenland/Austria) from September 16th to 21st, 1979. The aim of it was to furthe~ and intensify scientific cooperation in the Pannonian area, which, in a broad sense, can be understood to cover Hungary, the eastern part of Austria, Czechoslovakia, and parts of Poland, ¥ugoslavia and Romania. The location of centers of research in mathematical statistics and probability theory in this territory has been a good reason for the geographical limitation of this meeting. About 70 researchers attended this symposium, and 49 lectures were delivered; a considerable part of the presented papers is collected in this volume. Beside the lectures, vigorous informal discussions among the participants took place, so that many problems were raised and possible ways of solutions were attacked. We take the opportunity to thank Dr. U. Dieter (Graz), Dr. F. Konecny (Wien), Dr. W. Krieger (G8ttingen) and Dr. E. Neuwirth (Wien) for their valuable help in the refereeing work for this volume. The Pannonian Symposium could not have taken place without the support of several institutions: The Austrian Ministry for Research and Science, the State government of Burgenland, the Community Bad Tatzmannsdorf, the Kurbad Tatzmannsdorf AG, the Austrian Society for Information Science and Statistics, IBM Austria, Volksbank Oberwart, Erste Osterreichische Spar-Casse and Spielbanken AG Austria. The Austrian Academy of Sciences iv made possible the participation in the Symposium for several mathematicians. We express our gratitude to all these institutions for their generous help.

A collection of 20 refereed research or review papers presented at a six-day seminar in Switzerland. The contributions focus on stochastic analysis, its applications to the engineering sciences, and stochastic methods in financial models, which was the subject of a minisymposium.

The interaction between mathematicians and statisticians reveals to be an effective approach for dealing with actuarial, insurance and financial problems, both in an academic and in an operative perspective. The international conference MAF 2008, held at the University Ca'Foscari of Venezia (Italy) in 2008, had precisely this purpose, and the collection here published gathers a selection of about the one hundred papers presented at the conference and successively referred and reviewed to this aim. They cover a wide variety of subjects in actuarial, insurance and financial fields, all treated in light of the successful cooperation between the two quantitative approaches.

This book provides an introductory overview of random variables and their transformations. The authors approach the topic with statistics students in mind, along with researchers in various fields who are interested in data analysis. The book begins with by defining and explaining mathematical expectation. The authors then discuss transformations of random variables, including distribution functions and special functions. The book also covers joint probability distribution and its applications. The authors have updated and expanded upon their writing on these topics, which they originally covered in their previous book, Statistics for Scientists and Engineers.

This book presents a groundbreaking new mathematical paradigm and system (including inventing a new number) that enables a new approach to advancing the foundations of science and technology. The new approach allows connecting subjective and objective points of view that do not rely solely on the classical deductive approach. There is no need to fear paradoxical situations, because it's precisely in them wherein lies the key to a new understanding that is more connected to life itself. This new model also serves to advance the theory of consciousness. Life is more colorful and diverse than just the binary colors of black and white. For this purpose, the authors propose a new type of number called Soft Number. The basic idea is to distinguish between the different multiples of the number zero. Soft Logic is a continuous extension of the idea of giving different meanings to the zero. The authors create a tangible model of an infinitely small world. This seemingly inconceivable world is important for understanding the essence of the connection between the external world of reality and the inner world of human beings and offers a new language to express it. This book systematically and axiomatically describes the mathematical development of Soft Logic and Soft Numbers and points to possible directions for scientific and technological applications. This book is a valuable resource for researchers working in mathematics, physics, biology, engineering, computer science and artificial intelligence as well as advanced-level students majoring in computer science, engineering and mathematics. Practitioners, Data Scientists, AI developers and readers interested in a scientific understanding of the concept of consciousness will also want to purchase this book.

MATRIX is Australia's international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-2 weeks in duration. This book is a scientific record of the 24 programs held at MATRIX in 2021-2022, including tandem workshops with Mathematisches Forschungsinstitut Oberwolfach (MFO), with Research Institute for Mathematical Sciences Kyoto University (RIMS), and with Sydney Mathematical Research Institute (SMRI).

This new edition presents the essential topics in probability and statistics from a rigorous standpoint. Any discipline involving randomness, including medicine, engineering, and any area of scientific research, must have a way of analyzing or even predicting the outcomes of an experiment. The authors focus on the tools for doing so in a thorough, yet introductory way. After providing an overview of the basics of probability, the authors cover essential topics such as confidence intervals, hypothesis testing, and linear regression. These subjects are presented in a one semester format, suitable for engineers, scientists, and STEM students with a solid understanding of calculus. There are problems and exercises included in each chapter allowing readers to practice the applications of the concepts.

This book provides a concise yet rigorous introduction to probability theory. Among the possible approaches to the subject, the most modern approach based on measure theory has been chosen: although it requires a higher degree of mathematical abstraction and sophistication, it is essential to provide the foundations for the study of more advanced topics such as stochastic processes, stochastic differential calculus and statistical inference. The text originated from the teaching experience in probability and applied mathematics courses within the mathematics degree program at the University of Bologna; it is suitable for second- or third-year students in mathematics, physics, or other natural sciences, assuming multidimensional differential and integral calculus as a prerequisite. The four chapters cover the following topics: measures and probability spaces; random variables; sequences of random variables and limit theorems; and expectation and conditional distribution. The text includes a collection of solved exercises.

This IMA Volume in Mathematics and its Applications TOPICS IN STOCHASTIC ANALYSIS AND NONPARAMETRIC ESTIMATION contains papers that were presented at the IMA Participating Institution conference on'Asymptotic Analysis in Stochastic Processes, Nonparamet­ ric Estimation, and Related Problems'held on September 15-17, 2006 at Wayne State University. The conference, which was one of approximately ten selected each year for partial support by the IMA through its affiliates program, was dedicated to Professor Rafail Z. Khasminskii on the occasion th of his 75 birthday, in recognition of his profound contributions to the field of stochastic processes and nonparametric estimation theory. We are grateful to the participants and, especially, to the conference organizers, for making the event so successful. Pao-Liu Chow, Boris Mor­ dukhovich, and George Yin of the Department of Mathematics at Wayne State University did a superb job organizing this first-rate event and in editing these proceedings. We take this opportunity to thank the Nation al Science Foundation for its support of the IMA.

This book presents a comprehensive overview of Structural Equation Modeling and how it can be applied to address research issues in different disciplines. The authors employ a ‘simple to complex'approach. The book reviews topics such as variance, covariance, correlation, multiple regression, mediation, moderation, path analysis, and confirmatory factor analysis. The authors then discuss the initial steps for performing structural equation modeling, including model specification, model identification, model estimation, model testing, and model modification. The book includes an introduction to the IBM SPSS and IBM SPSS Amos software. The authors the explain how this software can be utilized for developing measurement, structural models, and SEM models. The book provides conceptual clarity in understanding the models and discusses practical approaches to solving them. The authors also highlight how these techniques can be applied to various disciplines, including psychology, education,sociology, business, medicine, political science, and biological sciences.

This book tells the story of the probability integral, the approaches to analyzing it throughout history, and the many areas of science where it arises. The so-called probability integral, the integral over the real line of a Gaussian function, occurs ubiquitously in mathematics, physics, engineering and probability theory. Stubbornly resistant to the undergraduate toolkit for handling integrals, calculating its value and investigating its properties occupied such mathematical luminaries as De Moivre, Laplace, Poisson, and Liouville. This book introduces the probability integral, puts it into a historical context, and describes the different approaches throughout history to evaluate and analyze it. The author also takes entertaining diversions into areas of math, science, and engineering where the probability integral arises: as well as being indispensable to probability theory and statistics, it also shows up naturally in thermodynamics and signal processing. Designed to be accessible to anyone at the undergraduate level and above, this book will appeal to anyone interested in integration techniques, as well as historians of math, science, and statistics.

This book explores and demonstrates how geometric tools can be used in data analysis. Beginning with a systematic exposition of the mathematical prerequisites, covering topics ranging from category theory to algebraic topology, Riemannian geometry, operator theory and network analysis, it goes on to describe and analyze some of the most important machine learning techniques for dimension reduction, including the different types of manifold learning and kernel methods. It also develops a new notion of curvature of generalized metric spaces, based on the notion of hyperconvexity, which can be used for the topological representation of geometric information.In recent years there has been a fascinating development: concepts and methods originally created in the context of research in pure mathematics, and in particular in geometry, have become powerful tools in machine learning for the analysis of data. The underlying reason for this is that data are typically equipped with somekind of notion of distance, quantifying the differences between data points. Of course, to be successfully applied, the geometric tools usually need to be redefined, generalized, or extended appropriately.Primarily aimed at mathematicians seeking an overview of the geometric concepts and methods that are useful for data analysis, the book will also be of interest to researchers in machine learning and data analysis who want to see a systematic mathematical foundation of the methods that they use.

A comprehensive survey of undergraduate mathematics, compressing four years of study into one robust overview.In The Math You Need, Thomas Mack provides a singular, comprehensive survey of undergraduate mathematics, compressing four years of math curricula into one volume. Without sacrificing rigor, this book provides a go-to resource for the essentials that any academic or professional needs. Each chapter is followed by numerous exercises to provide the reader an opportunity to practice what they learned. The Math You Need is distinguished in its use of the Bourbaki style—the gold standard for concision and an approach that mathematicians will find of particular interest. As ambitious as it is compact, this text embraces mathematical abstraction throughout, avoiding ad hoc computations in favor of general results.Covering nine areas—group theory, commutative algebra, linear algebra, topology, real analysis, complex analysis, number theory, probability, and statistics—this thorough and highly effective overview of the undergraduate curriculum will prove to be invaluable to students and instructors alike.

This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications. The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples.After a review of some background material, the reader is introduced to semigroup theory, including the Hille–Yosida Theorem, used to construct continuous parameter Markov processes. Illustrated with examples, it is a cornerstone of Feller's seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes, and processes with independent increments, or Lévy processes. The greater part of the book is devoted to Itô's fascinating theory of stochastic differential equations, and to the study of asymptotic properties of diffusions in all dimensions, such as explosion, transience, recurrence, existence of steady states, and the speed of convergence to equilibrium. A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes. Among Special Topics chapters, two study anomalous diffusions: one on skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.

This textbook introduces the reader to Machine Learning (ML) applications in Earth Sciences. In detail, it starts by describing the basics of machine learning and its potentials in Earth Sciences to solve geological problems. It describes the main Python tools devoted to ML, the typical workflow of ML applications in Earth Sciences, and proceeds with reporting how ML algorithms work. The book provides many examples of ML application to Earth Sciences problems in many fields, such as the clustering and dimensionality reduction in petro-volcanological studies, the clustering of multi-spectral data, well-log data facies classification, and machine learning regression in petrology. Also, the book introduces the basics of parallel computing and how to scale ML models in the cloud. The book is devoted to Earth Scientists, at any level, from students to academics and professionals.

This book facilitates both the theoretical background and applications of fuzzy, intuitionistic fuzzy and rough, fuzzy rough sets in the area of data science. This book provides various individual, soft computing, optimization and hybridization techniques of fuzzy and intuitionistic fuzzy sets with rough sets and their applications including data handling and that of type-2 fuzzy systems. Machine learning techniques are effectively implemented to solve a diversity of problems in pattern recognition, data mining and bioinformatics. To handle different nature of problems, including uncertainty, the book highlights the theory and recent developments on uncertainty, fuzzy systems, feature extraction, text categorization, multiscale modeling, soft computing, machine learning, deep learning, SMOTE, data handling, decision making, Diophantine fuzzy soft set, data envelopment analysis, centrally measures, social networks, Volterra–Fredholm integro-differential equation, Caputo fractional derivative, interval optimization, decision making, classification problems. This book is predominantly envisioned for researchers and students of data science, medical scientists and professional engineers.

This book develops limit theorems for a natural class of long range random walks on finitely generated torsion free nilpotent groups. The limits in these limit theorems are Lévy processes on some simply connected nilpotent Lie groups. Both the limit Lévy process and the limit Lie group carrying this process are determined by and depend on the law of the original random walk. The book offers the first systematic study of such limit theorems involving stable-like random walks and stable limit Lévy processes in the context of (non-commutative) nilpotent groups.

This book presents the latest developments in the theory of gradient flows in random walk spaces. A broad framework is established for a wide variety of partial differential equations on nonlocal models and weighted graphs. Within this framework, specific gradient flows that are studied include the heat flow, the total variational flow, and evolution problems of Leray-Lions type with different types of boundary conditions. With many timely applications, this book will serve as an invaluable addition to the literature in this active area of research.Variational and Diffusion Problems in Random Walk Spaces will be of interest to researchers at the interface between analysis, geometry, and probability, as well as to graduate students interested in exploring these areas.

The first edition of the Encyclopedia of Complexity and Systems Science (ECSS, 2009) presented a comprehensive overview of granular computing (GrC) broadly divided into several categories: Granular computing from rough set theory, Granular Computing in Database Theory, Granular Computing in Social Networks, Granular Computing and Fuzzy Set Theory, Grid/Cloud Computing, as well as general issues in granular computing. In 2011, the formal theory of GrC was established, providing an adequate infrastructure to support revolutionary new approaches to computer/data science, including the challenges presented by so-called big data. For this volume of ECSS, Second Edition, many entries have been updated to capture these new developments, together with new chapters on such topics as data clustering, outliers in data mining, qualitative fuzzy sets, and information flow analysis for security applications. Granulations can be seen as a natural and ancient methodology deeply rooted in the human mind. Many daily'things'are routinely granulated into sub'things': The topography of earth is granulated into hills, plateaus, etc., space and time are granulated into infinitesimal granules, and a circle is granulated into polygons of infinitesimal sides. Such granules led to the invention of calculus, topology and non-standard analysis. Formalization of general granulation was difficult but, as shown in this volume, great progress has been made in combing discrete and continuous mathematics under one roof for a broad range of applications in data science.

This book enables readers to understand, model, and predict complex dynamical systems using new methods with stochastic tools. The author presents a unique combination of qualitative and quantitative modeling skills, novel efficient computational methods, rigorous mathematical theory, as well as physical intuitions and thinking. An emphasis is placed on the balance between computational efficiency and modeling accuracy, providing readers with ideas to build useful models in practice. Successful modeling of complex systems requires a comprehensive use of qualitative and quantitative modeling approaches, novel efficient computational methods, physical intuitions and thinking, as well as rigorous mathematical theories. As such, mathematical tools for understanding, modeling, and predicting complex dynamical systems using various suitable stochastic tools are presented. Both theoretical and numerical approaches are included, allowing readers to choose suitable methods in different practical situations. The author provides practical examples and motivations when introducing various mathematical and stochastic tools and merges mathematics, statistics, information theory, computational science, and data science. In addition, the author discusses how to choose and apply suitable mathematical tools to several disciplines including pure and applied mathematics, physics, engineering, neural science, material science, climate and atmosphere, ocean science, and many others. Readers will not only learn detailed techniques for stochastic modeling and prediction, but will develop their intuition as well. Important topics in modeling and prediction including extreme events, high-dimensional systems, and multiscale features are discussed.

This book is devoted to the study of multivariate discrete q-distributions, which is greatly facilitated by existing multivariate q-sequences and q-functions. Classical multivariate discrete distributions are defined on a sequence of independent and identically distributed Bernoulli trials, with either being a success of a certain rank (level) or a failure. The author relaxes the assumption that the probability of success of a trial is constant by assuming that it varies geometrically with the number of trials and/or the number of successes. The latter is advantageous in the sense that it permits incorporating the experience gained from the previous trials and/or successes, which leads to multivariate discrete q-distributions. Furthermore, q-multinomial and negative q-multinomial formulae are obtained. Next, the book addresses q-multinomial and negative q-multinomial distributions of the first and second kind. The author also examines multiple q-Polya urn model, multivariate q-Polya and inverse q-Polya distributions. Presents definitions and theorems that highlight key concepts and worked examples to illustrate the various applicationsContains numerous exercises at varying levels of difficulty that consolidate the presented concepts and resultsIncludes hints and answers to all exercises via the appendix and is supplemented with an Instructor's Solution Manual

The Encyclopedia of Mathematical Geosciences is a complete and authoritative reference work. It provides concise explanation on each term that is related to Mathematical Geosciences. Over 300 international scientists, each expert in their specialties, have written around 350 separate articles on different topics of mathematical geosciences including contributions on Artificial Intelligence, Big Data, Compositional Data Analysis, Geomathematics, Geostatistics, Geographical Information Science, Mathematical Morphology, Mathematical Petrology, Multifractals, Multiple Point Statistics, Spatial Data Science, Spatial Statistics, and Stochastic Process Modeling. Each topic incorporates cross-referencing to related articles, and also has its own reference list to lead the reader to essential articles within the published literature. The entries are arranged alphabetically, for easy access, and the subject and author indices are comprehensive and extensive.

This volume is an original collection of articles by 44 leading mathematicians on the theme of the future of the discipline. The contributions range from musings on the future of specific fields, to analyses of the history of the discipline, to discussions of open problems and conjectures, including first solutions of unresolved problems. Interestingly, the topics do not cover all of mathematics, but only those deemed most worthy to reflect on for future generations. These topics encompass the most active parts of pure and applied mathematics, including algebraic geometry, probability, logic, optimization, finance, topology, partial differential equations, category theory, number theory, differential geometry, dynamical systems, artificial intelligence, theory of groups, mathematical physics and statistics.

This book introduces undergraduate students of engineering and science to applied mathematics essential to the study of many problems. Topics are differential equations, power series, Laplace transforms, matrices and determinants, vector analysis, partial differential equations, complex variables, and numerical methods. Approximately, 160 examples and 1000 homework problems aid students in their study. This book presents mathematical topics using derivations rather than theorems and proofs. This textbook is uniquely qualified to apply mathematics to physical applications (spring-mass systems, electrical circuits, conduction, diffusion, etc.), in a manner that is efficient and understandable. This book is written to support a mathematics course after differential equations, to permit several topics to be covered in one semester, and to make the material comprehensible to undergraduates. An Instructor Solutions Manual, and also a Student Solutions Manual that provides solutions to select problems, is available.^

The Strunk and White of statistics team up to help the average person navigate the numbers in the news Drawing on their hugely popular BBC Radio 4 show More or Less, journalist Michael Blastland and internationally known economist Andrew Dilnot delight, amuse, and convert American mathphobes by showing how our everyday experiences make sense of numbers. The radical premise of The Numbers Game is to show how much we already know and give practical ways to use our knowledge to become cannier consumers of the media. If you've ever wondered what'average'really means, whether the scare stories about cancer risk should convince you to change your behavior, or whether a story you read in the paper is biased (and how), you need this book. Blastland and Dilnot show how to survive and thrive on the torrent of numbers that pours through everyday life.