Triangular arrays are a unifying thread throughout various areas of discrete mathematics such as number theory and combinatorics. They can be used to sharpen a variety of mathematical skills and tools, such as pattern recognition, conjecturing, proof-techniques, and problem-solving techniques.
While a good deal of research exists concerning triangular arrays and their applications, the information is scattered in various journals and is inaccessible to many mathematicians. This is the first text that will collect and organize the information and present it in a clear and comprehensive introduction to the topic. An invaluable resource book, it gives a historical introduction to Pascal's triangle and covers application topics such as binomial coefficients, figurate numbers, Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers, graph theory, Fibonomial and tribinomial coefficients and Fibonacci and Lucas polynomials, amongst others. The book also features the historical development of triangular arrays, including short biographies of prominent mathematicians, along with the name and affiliation of every discoverer and year of discovery. The book is intended for mathematicians as well as computer scientists, math and science teachers, advanced high school students, and those with mathematical curiosity and maturity.

This extended and revised second edition is intended for engineering students and researchers working with finite element methods in structural and mechanical analysis. Discussing numerical structural analysis from first mechanical and mathematical principles, it establishes the central role of influence functions (Green's functions) in finite element analysis, reanalysis, sensitivity analysis, parameter identification and in optimization, with a particular focus on computational aspects and questions of accuracy. It also presents a one-click reanalysis, a new technique that allows instantaneous modifications to a structure to be made by clicking on single elements. Lastly, the book features four programs that can be downloaded for the solution of the Poisson equation, 2-D elasticity, plate-bending problems and planar frames.

This book provides a basic grounding in the use of probability to model random financial phenomena of uncertainty, and is targeted at an advanced undergraduate and graduate level. It should appeal to finance students looking for a firm theoretical guide to the deep end of derivatives and investments. Bankers and finance professionals in the fields of investments, derivatives, and risk management should also find the book useful in bringing probability and finance together. The book contains applications of both discrete time theory and continuous time mathematics, and is extensive in scope. Distribution theory, conditional probability, and conditional expectation are covered comprehensively, and applications to modeling state space securities under market equilibrium are made. Martingale is studied, leading to consideration of equivalent martingale measures, fundamental theorems of asset pricing, change of numeraire and discounting, risk-adjusted and forward-neutral measures, minimal and maximal prices of contingent claims, Markovian models, and the existence of martingale measures preserving the Markov property. Discrete stochastic calculus and multiperiod models leading to no-arbitrage pricing of contingent claims are also to be found in this book, as well as the theory of Markov Chains and appropriate applications in credit modeling. Measure-theoretic probability, moments, characteristic functions, inequalities, and central limit theorems are examined. The theory of risk aversion and utility, and ideas of risk premia are considered. Other application topics include optimal consumption and investment problems and interest rate theory.

This authoritative book presents recent research results on nonlinear problems with lack of compactness. The topics covered include several nonlinear problems in the Euclidean setting as well as variational problems on manifolds. The combination of deep techniques in nonlinear analysis with applications to a variety of problems make this work an essential source of information for researchers and graduate students working in analysis and PDE's.

This book covers two major classes of mixed effects models, linear mixed models and generalized linear mixed models. It presents an up-to-date account of theory and methods in analysis of these models as well as their applications in various fields. The book offers a systematic approach to inference about non-Gaussian linear mixed models. Furthermore, it includes recently developed methods, such as mixed model diagnostics, mixed model selection, and jackknife method in the context of mixed models. The book is aimed at students, researchers and other practitioners who are interested in using mixed models for statistical data analysis.

This book provides a guided approach to the geostatistical modelling of compositional spatial data. These data are data in proportions, percentages or concentrations distributed in space which exhibit spatial correlation. The book can be divided into four blocks. The first block sets the framework and provides some background on compositional data analysis. Block two introduces compositional exploratory tools for both non-spatial and spatial aspects. Block three covers all necessary facets of multivariate spatial prediction for compositional data: variogram modelling, cokriging and validation. Finally, block four details strategies for simulation of compositional data, including transformations to multivariate normality, Gaussian cosimulation, multipoint simulation of compositional data, and common postprocessing techniques, valid for both Gaussian and multipoint methods. All methods are illustrated via applications to two types of data sets: one a large-scale geochemical survey, comprised of a full suite of geochemical variables, and the other from a mining context, where only the elements of greatest importance are considered. R codes are included for all aspects of the methodology, encapsulated in the R package "gmGeostats", available in CRAN.

The book is intended as a textbook providing a deliberately simple introduction to finite element methods in a way that should be readily understandable to engineers, both students and practicing professionals. Only the very simplest elements are considered, mainly two-dimensional three-noded constant strain triangles, with simple linear variation of the relevant variables. Chapters of the book deal with structural problems (beams), classification of a broad range of engineering into harmonic and biharmonic types, finite element analysis of harmonic problems, finite element analysis of biharmonic problems (plane stress and plane strain). Full FORTRAN programs are listed and explained in detail, and a range of practical problems solved in the text. Despite being somewhat unfashionable for general programming purposes, the FORTRAN language remains very widely used in engineering. The programs listed, which were originally developed for use on mainframe computers, have been thoroughly updated for use on desktops and laptops. Unlike the first edition, the new edition has Problems (with Solutions) at the end of each chapter. Electronic copies of the programs should be freely available for download from the internet.

Inverse problems such as imaging or parameter identification deal with the recovery of unknown quantities from indirect observations, connected via a model describing the underlying context. While traditionally inverse problems are formulated and investigated in a static setting, we observe a significant increase of interest in time-dependence in a growing number of important applications over the last few years. Here, time-dependence affects a) the unknown function to be recovered and / or b) the observed data and / or c) the underlying process. Challenging applications in the field of imaging and parameter identification are techniques such as photoacoustic tomography, elastography, dynamic computerized or emission tomography, dynamic magnetic resonance imaging, super-resolution in image sequences and videos, health monitoring of elastic structures, optical flow problems or magnetic particle imaging to name only a few. Such problems demand for innovation concerning their mathematical description and analysis as well as computational approaches for their solution.

This monograph introduces breakthrough control algorithms for partial differential equation models with moving boundaries, the study of which is known as the Stefan problem. The algorithms can be used to improve the performance of various processes with phase changes, such as additive manufacturing. Using the authors' innovative design solutions, readers will also be equipped to apply estimation algorithms for real-world phase change dynamics, from polar ice to lithium-ion batteries. A historical treatment of the Stefan problem opens the book, situating readers in the larger context of the area. Following this, the chapters are organized into two parts. The first presents the design method and analysis of the boundary control and estimation algorithms. Part two then explores a number of applications, such as 3D printing via screw extrusion and laser sintering, and also discusses the experimental verifications conducted. A number of open problems and provided as well, offering readers multiple paths to explore in future research. Materials Phase Change PDE Control & Estimation is ideal for researchers and graduate students working on control and dynamical systems, and particularly those studying partial differential equations and moving boundaries. It will also appeal to industrial engineers and graduate students in engineering who are interested in this area.

This book first presents an overview of the history of a national character survey by the Institute of Statistical Mathematics that has been conducted for more than 65 years. The Japanese National Character Survey, launched in 1953, is a rare longitudinal survey in the world of survey research based on rigorous statistical sampling theory, motivating other countries to launch similar longitudinal surveys, including the General Social Survey (GSS), the Allgemeine Bevoelkerungsumfrage der Sozialwissenschaften (ALLBUS, German General Social Survey (GGSS)), Eurobarometer, and others. Since the early 1970s, the Japanese survey has been extended as a cross-national survey for more advanced research of the Japanese national character in a comparative context. Second, the book explains the paradigm of cross-national studies called the Cultural Manifold Analysis (CULMAN), developed in the longitudinal and cross-national surveys, with practical examples of analysis. This explanation will help helps a wide range of readers to better understand the cross-national comparative surveys of attitudes, opinion, and social values as basic information for evidence-based policymaking and research.

Indicators are more and more applied to describe and analyze complex systems. Typical examples: Innovation potential of nations, child-well being, Environmental health, poverty, chemical pollution, corruption of nations. The task is: How can a system of indicators be defined in order to fulfill the above expectations. One possibility is the application of the mathematical theory of partial order, especially when the indicator system shall be used for ranking purposes.

This book presents time synchronization and its essential role as a conduit of optimized networks and as one of the key imperatives of ubiquitous connectivity. The author discusses how, without proper time synchronization, many mission critical infrastructures such as 5G mobile networks, smart grids, data centres CATV, and industrial networks would render in serious performance issues and may be subject to catastrophic failure. The book provides a thorough understanding of time synchronization from fundamental concepts to the application of time synchronization in NextGen mission critical infrastructure. Readers will find information not only on designing the optimized products for mission critical infrastructure but also on building NextGen mission critical infrastructure.

Why do some games seem to be universal while others have a particular connection to the culture of the people playing them? Around the World in 80 Games is about the mathematics of chance, game theory, gamification, gaming strategies and computer games. Traversing the globe, Marcus du Sautoy looks at the genesis of games new and old, explores how to invent a good game and explains the fascination of a popular lockdown game. From the secrets of whist to hopscotch, Scrabble to Wordle. The most simple games endure: board games, card games and dice games have captivated us for centuries and the acclaimed mathematician and author of The Creativity Code (among many others) will once again bring mathematics to the fore with insight and aplomb in Around the World in 80 Games.

Galois theory has such close analogies with the theory of coverings that algebraists use a geometric language to speak of field extensions, while topologists speak of "Galois coverings". This book endeavors to develop these theories in a parallel way, starting with that of coverings, which better allows the reader to make images. The authors chose a plan that emphasizes this parallelism. The intention is to allow to transfer to the algebraic framework of Galois theory the geometric intuition that one can have in the context of coverings. This book is aimed at graduate students and mathematicians curious about a non-exclusively algebraic view of Galois theory.

This book provides an overview of the current advances in artificial intelligence and neural nets. Artificial intelligence (AI) methods have shown great capabilities in modelling, prediction and recognition tasks supporting human-machine interaction. At the same time, the issue of emotion has gained increasing attention due to its relevance in achieving human-like interaction with machines. The real challenge is taking advantage of the emotional characterization of humans' interactions to make computers interfacing with them emotionally and socially credible. The book assesses how and to what extent current sophisticated computational intelligence tools might support the multidisciplinary research on the characterization of appropriate system reactions to human emotions and expressions in interactive scenarios. Discussing the latest recent research trends, innovative approaches and future challenges in AI from interdisciplinary perspectives, it is a valuable resource for researchers and practitioners in academia and industry.

Drawing on published works, correspondence and manuscripts, this book offers the most comprehensive reconstruction of Boscovich's theory within its historical context. It explains the genesis and theoretical as well as epistemological underpinnings in light of the Jesuit tradition to which Boscovich belonged, and contrasts his ideas with those of Newton, Leibniz, and their legacy. Finally, it debates crucial issues in early-modern physical science such as the concept of force, the particle-like structure of matter, the idea of material points and the notion of continuity, and shares novel insights on Boscovich's alleged influence on later developments in physics. With its attempt to reduce all natural forces to one single law, Boscovich's Theory of Natural Philosophy, published in 1758, left a lasting impression on scientists and philosophers of every age regarding the fundamental unity of physical phenomena. The theory argues that every pair of material points is subject to one mutual force - and always the same force - which is their propensity to be mutually attracted or repelled, depending on their distance from one another. Furthermore, the action of this unique force is visualized through a famous diagram that fascinated generations of scientists. But his understanding of key terms of the theory - such as the notion of force involved and the very idea of a material point - is only ostensibly similar to our current conceptual framework. Indeed, it needs to be clarified within the plurality of contexts in which it has emerged rather than being considered in view of later developments. The book is recommended for scholars and students interested in the ideas of the early modern period, especially historians and philosophers of science, mathematicians and physicists with an interest in the history of the discipline, and experts on Jesuit science and philosophy.

This book strikes a healthy balance between theory and applications, ensuring that it doesn't offer a set of tools with no mathematical roots. It is intended as a comprehensive and largely self-contained introduction to probability and statistics for university students from various faculties, with accompanying implementations of some rudimentary statistical techniques in the language R. The content is divided into three basic parts: the first includes elements of probability theory, the second introduces readers to the basics of descriptive and inferential statistics (estimation, hypothesis testing), and the third presents the elements of correlation and linear regression analysis. Thanks to examples showing how to approach real-world problems using statistics, readers will acquire stronger analytical thinking skills, which are essential for analysts and data scientists alike.

This book is a description of why and how to do Scientific Computing for fundamental models of fluid flow. It contains introduction, motivation, analysis, and algorithms and is closely tied to freely available MATLAB codes that implement the methods described. The focus is on finite element approximation methods and fast iterative solution methods for the consequent linear(ized) systems arising in important problems that model incompressible fluid flow. The problems addressed are the Poisson equation, Convection-Diffusion problem, Stokes problem and Navier-Stokes problem, including new material on time-dependent problems and models of multi-physics. The corresponding iterative algebra based on preconditioned Krylov subspace and multigrid techniques is for symmetric and positive definite, nonsymmetric positive definite, symmetric indefinite and nonsymmetric indefinite matrix systems respectively. For each problem and associated solvers there is a description of how to compute together with theoretical analysis that guides the choice of approaches and describes what happens in practice in the many illustrative numerical results throughout the book (computed with the freely downloadable IFISS software). All of the numerical results should be reproducible by readers who have access to MATLAB and there is considerable scope for experimentation in the "computational laboratory " provided by the software. Developments in the field since the first edition was published have been represented in three new chapters covering optimization with PDE constraints (Chapter 5); solution of unsteady Navier-Stokes equations (Chapter 10); solution of models of buoyancy-driven flow (Chapter 11). Each chapter has many theoretical problems and practical computer exercises that involve the use of the IFISS software. This book is suitable as an introduction to iterative linear solvers or more generally as a model of Scientific Computing at an advanced undergraduate or beginning graduate level.

The nonequilibrium behavior of nanoscopic and biological systems, which are typically strongly fluctuating, is a major focus of current research. Lately, much progress has been made in understanding such systems from a thermodynamic perspective. However, new theoretical challenges emerge when the fluctuating system is additionally subject to time delay, e.g. due to the presence of feedback loops. This thesis advances this young and vibrant research field in several directions. The first main contribution concerns the probabilistic description of time-delayed systems; e.g. by introducing a versatile approximation scheme for nonlinear delay systems. Second, it reveals that delay can induce intriguing thermodynamic properties such as anomalous (reversed) heat flow. More generally, the thesis shows how to treat the thermodynamics of non-Markovian systems by introducing auxiliary variables. It turns out that delayed feedback is inextricably linked to nonreciprocal coupling, information flow, and to net energy input on the fluctuating level.

This book presents the fundamentals of evolutionary game theory and applies them to the analysis of epidemics, which is of paramount importance in the aftermath of the worldwide COVID-19 pandemic. The primary objective of this monograph is to deliver a powerful tool to model and analyze the spread of an infectious disease during a pandemic as well as the human decision dynamics. The book employs a variant of the "vaccination game," in which a mathematical epidemiological model dovetails with evolutionary game theory. From a social physics standpoint, this book introduces an extended concept of the vaccination game starting from the fundamental issues and touching on the newest practical applications. The book first outlines the fundamental basis of evolutionary game theory, in which a two-player and two-strategy game, the so-called 2 x 2 game, and a multi-player game are concisely introduced, and the important issue of how social dilemmas are quantified is highlighted. Subsequently, the book discusses various recent applications of the extended concept of the vaccination game so as to quantitatively evaluate provisions other than vaccination, including practical intermediate protective measures such as mask-wearing, efficiency of quarantine compared with that of isolation policies for suppressing epidemics, efficiency of preemptive versus late vaccination, and optimal subsidy policies for vaccination.

This book contains international perspectives that unifies the themes of strategic management, decision theory, and data science. It contains thought-provoking presentations of case studies backed by adequate analysis adding significance to the discussions. Most of the decision-making models in use do take due advantage of collection and processing of relevant data using appropriate analytics oriented to provide inputs into effective decision-making. The book showcases applications in diverse fields including banking and insurance, portfolio management, inventory analysis, performance assessment of comparable economic agents, managing utilities in a health-care facility, reducing traffic snarls on highways, monitoring achievement of some of the sustainable development goals in a country or state, and similar other areas that showcase policy implications. It holds immense value for researchers as well as professionals responsible for organizational decisions.

This book intends to introduce some recent results on passivity of complex dynamical networks with single weight and multiple weights. The book collects novel research ideas and some definitions in complex dynamical networks, such as passivity, output strict passivity, input strict passivity, finite-time passivity, and multiple weights. Furthermore, the research results previously published in many flagship journals are methodically edited and presented in a unified form. The book is likely to be of interest to university researchers and graduate students in Engineering and Mathematics who wish to study the passivity of complex dynamical networks.