The Statistical Analysis of Small Data Sets
We live in the era of big data. However, small data sets are still common for ethical, financial, or practical reasons. Small sample sizes can cause researchers to seek out the most powerful methods to analyse their data, but they may also be wary that some methodologies and assumptions may not be appropriate when samples are small. The book offers advice on the statistical analysis of small data sets for various designs and levels of measurement, helping researchers to analyse such data sets, but also to evaluate and interpret others' analyses. The book discusses the potential challenges associated with a small sample, as well as the ways in which these challenges can be mitigated. General topics with strong relevance to small sample sizes such as meta-analysis, sequential and adaptive designs, and multiple testing are introduced. While the focus is on hypothesis tests and confidence intervals, Bayesian analyses are also covered. Code written in the statistical software R is presented to carry out the proposed methods, many of which are not limited to use on small data sets, and the book also discusses approaches to computing the power or the necessary sample size, respectively.
Quadrature Formulae
No detailed description available for "Quadrature Formulae".
Geometrodynamics of Gauge Fields
No detailed description available for "Geometrodynamics of Gauge Fields".
Introduction to Transfer Learning
Transfer learning is one of the most important technologies in the era of artificial intelligence and deep learning. It seeks to leverage existing knowledge by transferring it to another, new domain. Over the years, a number of relevant topics have attracted the interest of the research and application community: transfer learning, pre-training and fine-tuning, domain adaptation, domain generalization, and meta-learning. This book offers a comprehensive tutorial on an overview of transfer learning, introducing new researchers in this area to both classic and more recent algorithms. Most importantly, it takes a "student's" perspective to introduce all the concepts, theories, algorithms, and applications, allowing readers to quickly and easily enter this area. Accompanying the book, detailed code implementations are provided to better illustrate the core ideas of several important algorithms, presenting good examples for practice.
Reverse Dynamic Inequalities on Time Scales
Inequalities lie at the heart of the mathematical analysis which is a major and important branch of mathematics. Throughout the history, many researchers discovered agreat number of inequalities that are very useful in many fields of mathematics. Thestudy of inequalities became very popular in the twentieth century specially afterthe publishing of the pioneering book. In 1988, Stephan Hilger introduced in his PhD thesis a new theory, the theoryof time scales, which builds bridges between continuous and discrete mathematics.The main idea is to prove a result for a dynamic inequality where the domain of theunknown function is a so-called time scale T, which is defined as an arbitrary closedsubset of the real numbers R, to avoid proving results twice, once in the continuouscase which leads to an integral inequality and once again on a discrete case whichleads to a discrete inequality.Recently, the study of dynamic inequalities on time scales has attracted the attention of many mathematicians. One of the applications of this kind of inequalitiesis to study the behavior of solutions of dynamic equations on time scales.
The Statistical Analysis of Small Data Sets
We live in the era of big data. However, small data sets are still common for ethical, financial, or practical reasons. Small sample sizes can cause researchers to seek out the most powerful methods to analyse their data, but they may also be wary that some methodologies and assumptions may not be appropriate when samples are small. The book offers advice on the statistical analysis of small data sets for various designs and levels of measurement, helping researchers to analyse such data sets, but also to evaluate and interpret others' analyses. The book discusses the potential challenges associated with a small sample, as well as the ways in which these challenges can be mitigated. General topics with strong relevance to small sample sizes such as meta-analysis, sequential and adaptive designs, and multiple testing are introduced. While the focus is on hypothesis tests and confidence intervals, Bayesian analyses are also covered. Code written in the statistical software R is presented to carry out the proposed methods, many of which are not limited to use on small data sets, and the book also discusses approaches to computing the power or the necessary sample size, respectively.
Advanced Techniques in Control Charting
The book Advanced Techniques in Control Charting for Process Monitoring offers a timely and insightful look into the contemporary uses of control charts. It delivers a comprehensive analysis of advanced methods, focusing on their impact in optimizing process control, improving quality, and increasing efficiency across multiple industries. Covering topics such as the integration of machine learning algorithms and multivariate control charts, this book presents a diverse range of innovative techniques that are revolutionizing how processes are monitored and managed in modern industrial settings.
Probabilistic Spiking Neuronal Nets
This book provides a self-contained introduction to a new class of stochastic models for systems of spiking neurons. These systems have a large number of interacting components, each one evolving as a stochastic process with a memory of variable length. Several mathematical tools are put to use, such as Markov chains, stochastic chains having memory of variable length, point processes having stochastic intensity, Hawkes processes, random graphs, mean field limits, perfect sampling algorithms, the Context algorithm, and statistical model selection.The book's focus on mathematically tractable objects distinguishes it from other texts on theoretical neuroscience. The biological complexity of neurons is not ignored, but reduced to some of its main features, such as the intrinsic randomness of neuronal dynamics. This reduction in complexity aims at explaining and reproducing statistical regularities and collective phenomena that are observed in experimental data, an approach that leads to mathematically rigorous results. With an emphasis on a constructive and algorithmic point of view, this book is directed towards mathematicians interested in learning about stochastic network models and their neurobiological underpinning, and neuroscientists interested in learning how to build and prove results with mathematical models that relate to actual experimental settings.
Lectures on Lie Algebras
No detailed description available for "Lectures on Lie Algebras".
Dynamics of Circle Mappings
This book explores recent developments in the dynamics of invertible circle maps, a rich and captivating topic in one-dimensional dynamics. It focuses on two main classes of invertible dynamical systems on the circle: global diffeomorphisms and smooth homeomorphisms with critical points. The latter is the book's core, reflecting the authors' recent research interests.Organized into four parts and 14 chapters, the content covers rigid rotations, circle homeomorphisms, and the concept of rotation number in the first part. The second part delves into circle diffeomorphisms, presenting classical results. The third part introduces multicritical circle maps--smooth homeomorphisms of the circle with a finite number of critical points. The fourth and final part centers on renormalization theory, analyzing the fine geometric structure of orbits of multicritical circle maps. Complete proofs for several fundamental results in circle dynamics are provided throughout. The book concludes with a list of open questions.Primarily intended for graduate students and young researchers in dynamical systems, this book is also suitable for mathematicians from other fields with an interest in the subject. Prerequisites include familiarity with the content of a standard graduate course in real analysis, along with some understanding of ergodic theory and dynamical systems. Basic knowledge of complex analysis is needed for specific chapters.
Fuzzy Soft Ternary Γ-Semirings
Fuzzy soft ternary gamma semirings are an advanced mathematical structure that combines elements from fuzzy logic, soft set theory, ternary operations, and gamma semirings. This book provides information on Fuzzy Soft Quasi TΓ-Ideals in Fuzzy Soft TΓ-Semirings, On Fuzzy Soft Interior TΓ-Ideals, Fuzzy Soft Radicals in Fuzzy Soft TΓ-Semirings, Fuzzy Soft TΓ-Homomorphism in TΓ-Semirings. Each topic has been started with introductory part and developed gradually up to the standard form.
Foundations of Quantitative Finance, Book VI
Every finance professional wants and needs a competitive edge. A firm foundation in advanced mathematics can translate into dramatic advantages to professionals willing to obtain it. Many are not-and that is the competitive edge these books offer the astute reader.
The Philosophy of Penelope Maddy
This volume features more than 20 essays that explore the work of one of the most important contemporary philosophers of mathematics. It will help readers to better appreciate this significant and prolific philosopher. Within philosophy of mathematics, Penelope Maddy initially advocated realism. She then went on to advance naturalism. Both of her positions became very influential in the field, along with her other work in the philosophy of logic. The contributors comment on and otherwise engage with Maddy's work. They also weigh in on the state of set theory and its philosophy, the philosophy and history of logic, naturalism, skepticism, and the myriad other areas to which Maddy left her mark. Overall, coverage traces her influence on these various ideas over the years. It will also help readers to better understand how philosophers working at the forefront of these areas see these concepts today. These essays will be essential reading for the wide group of philosophers working in these different areas as well as graduate students studying philosophy of mathematics and logic and the other related issues to which Maddy has contributed. The volume will also appeal to logicians and set theorists in general, as well as to philosophers working in analytic philosophy more widely, as well as to those working in the history of philosophy.
Enveloping Algebras
No detailed description available for "Enveloping Algebras".
Measure and Integral
This classic text provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then introducing a more general treatment based on abstract notions characterized by axioms and with less geometric content. Packed with new exercises and material, the long-awaited s
Introduction to Mathematical Logic
This bestselling, classic textbook continues to provide a complete one-semester introduction to mathematical logic. The sixth edition incorporates recent work on G繹del's second incompleteness theorem as well as an appendix on consistency proofs for first-order arithmetic. It also offers historical perspectives and many new exercises of varying d
Topological Vector Spaces
With many new concrete examples and historical notes, Topological Vector Spaces, Second Edition provides one of the most thorough and up-to-date treatments of the Hahn-Banach theorem. This edition explores the theorem's connection with the axiom of choice, discusses the uniqueness of Hahn-Banach extensions, and includes an entirely new chapter on vector-valued Hahn-Banach theorems. It also considers different approaches to the Banach-Stone theorem as well as variations of the theorem.The book covers locally convex spaces; barreled, bornological, and webbed spaces; and reflexivity. It traces the development of various theorems from their earliest beginnings to present day, providing historical notes to place the results in context. The authors also chronicle the lives of key mathematicians, including Stefan Banach and Eduard Helly. Suitable for both beginners and experienced researchers, this book contains an abundance of examples, exercises of varying levels of difficulty with many hints, and an extensive bibliography and index.
Computational Mathematics
Computational Mathematics: Models, Methods, and Analysis with MATLAB(R) and MPI is a unique book covering the concepts and techniques at the core of computational science. The author delivers a hands-on introduction to nonlinear, 2D, and 3D models; nonrectangular domains; systems of partial differential equations; and large algebraic problems requiring high-performance computing. The book shows how to apply a model, select a numerical method, implement computer simulations, and assess the ensuing results.Providing a wealth of MATLAB, Fortran, and C++ code online for download, the Second Edition of this very popular text: Includes a new chapter with two sections on the finite element method, two sections on shallow water waves, and two sections on the driven cavity problem Introduces multiprocessor/multicore computers, parallel MATLAB, and message passing interface (MPI) in the chapter on high-performance computing Updates and adds code and documentation Computational Mathematics: Models, Methods, and Analysis with MATLAB(R) and MPI, Second Edition is an ideal textbook for an undergraduate course taught to mathematics, computer science, and engineering students. By using code in practical ways, students take their first steps toward more sophisticated numerical modeling.
Statistical Methods in Biology
Written in simple language with relevant examples, this illustrative introductory book presents best practices in experimental design and simple data analysis. Taking a practical and intuitive approach, it only uses mathematical formulae to formalize the methods where necessary and appropriate. The text features extended discussions of examples
Introduction to General and Generalized Linear Models
Providing a flexible framework for data analysis and model building, this text focuses on the statistical methods and models that can help predict the expected value of an outcome, dependent, or response variable. It offers a sound introduction to general and generalized linear models using the popular and powerful likelihood techniques. The aut
Linear Algebra
This book is a comprehensive guide to Linear Algebra and covers all the fundamental topics such as vector spaces, linear independence, basis, linear transformations, matrices, determinants, inner products, eigenvectors, bilinear forms, and canonical forms. It also introduces concepts such as fields, rings, group homomorphism, and binary operations early on, which gives students a solid foundation to understand the rest of the material. Unlike other books on Linear Algebra that are either too theory-oriented with fewer solved examples or too problem-oriented with less good quality theory, this book strikes a balance between the two. It provides easy-to-follow theorem proofs and a considerable number of worked examples with various levels of difficulty. The fundamentals of the subject are explained in a methodical and straightforward way. This book is aimed at undergraduate and graduate students of Mathematics and Engineering Mathematics who are studying Linear Algebra. It is also a useful resource for students preparing for exams in higher education competitions such as NET, GATE, lectureships, etc. The book includes some of the most recent and challenging questions from these exams.
The Relay Race to Infinity
Why were most historically important mathematicians wealthy? Why were they often lawyers and many had pastors for fathers? Why were original results sometimes discovered by two mathematicians independently within a short time of each other? Why did the Italian Fibonacci, speak Arabic?It all began a couple of years ago, when one of the authors started to write short biographies of important historical mathematicians for the teaching journal Australian Primary Mathematics Classroom. It was felt that teachers generally knew very little about the way the subject developed or the people who developed it. And it was felt that historical knowledge would help them see how the subject progressed and enable them to fit in with the historical episodes that would be of interest to students.Clearly, the book that developed contains mathematics up to the 17th century, but we are keen to set the subject in those times, to try to give short biographies of the people involved, as well as provide a perspective of the events that led up to the times and led up to the mathematics. Importantly, it is shown that the maths enterprise was not undertaken by a small few, but worked like a relay race. One or a few might take up an idea and develop it, but it often gets only so far. Later, others would take up the idea, the baton, and the relay race to find results continues.
Algebraic Topology
The aim of the textbook is two-fold: first to serve as an introductory graduate course in Algebraic Topology and then to provide an application-oriented presentation of some fundamental concepts in Algebraic Topology to the fixed point theory. A simple approach based on point-set Topology is used throughout to introduce many standard constructions of fundamental and homological groups of surfaces and topological spaces. The approach does not rely on Homological Algebra. The constructions of some spaces using the quotient spaces such as the join, the suspension, and the adjunction spaces are developed in the setting of Topology only. The computations of the fundamental and homological groups of many surfaces and topological spaces occupy large parts of the book (sphere, torus, projective space, Mobius band, Klein bottle, manifolds, adjunctions spaces). Borsuk's theory of retracts which is intimately related to the problem of the extendability of continuous functions is developed in details. This theory together with the homotopy theory, the lifting and covering maps may serve as additional course material for students involved in General Topology. The book comprises 280 detailed worked examples, 320 exercises (with hints or references), 80 illustrative figures, and more than 80 commutative diagrams to make it more oriented towards applications (maps between spheres, Borsuk-Ulam Theory, Fixed Point Theorems, ...) As applications, the book offers some existence results on the solvability of some nonlinear differential equations subject to initial or boundary conditions. The book is suitable for students primarily enrolled in Algebraic Topology, General Topology, Homological Algebra, Differential Topology, Differential Geometry, and Topological Geometry. It is also useful for advanced undergraduate students who aspire to grasp easily some new concepts in Algebraic Topology and Applications. The textbook is practical both as a teaching and research document for Bachelor, Master students, and first-year PhD students since it is accessible to any reader with a modest understanding of topological spaces. The book aspires to fill a gap in the existing literature by providing a research and teaching document which investigates both the theory and the applications of Algebraic Topology in an accessible way without missing the main results of the topics covered.
The Relay Race to Infinity
Why were most historically important mathematicians wealthy? Why were they often lawyers and many had pastors for fathers? Why were original results sometimes discovered by two mathematicians independently within a short time of each other? Why did the Italian Fibonacci, speak Arabic?It all began a couple of years ago, when one of the authors started to write short biographies of important historical mathematicians for the teaching journal Australian Primary Mathematics Classroom. It was felt that teachers generally knew very little about the way the subject developed or the people who developed it. And it was felt that historical knowledge would help them see how the subject progressed and enable them to fit in with the historical episodes that would be of interest to students.Clearly, the book that developed contains mathematics up to the 17th century, but we are keen to set the subject in those times, to try to give short biographies of the people involved, as well as provide a perspective of the events that led up to the times and led up to the mathematics. Importantly, it is shown that the maths enterprise was not undertaken by a small few, but worked like a relay race. One or a few might take up an idea and develop it, but it often gets only so far. Later, others would take up the idea, the baton, and the relay race to find results continues.
A Stitch in Line
This book provides readers with instructions for creating hitomezashi items with minimum outlay. The reader is guided through the practical steps involved in creating each design, and then the mathematics which underpins it is explained in a friendly, accessible way.
Computational mathematics in R Programming
Computational mathematics in R programming harnesses the power of R's extensive libraries and tools to tackle complex numerical analysis, statistical computations, and data visualization tasks. This versatile programming language is widely adopted in a range of scientific and engineering disciplines for executing matrix operations, optimization techniques, and solving differential equations. Whether applied in data science, physics, economics, or engineering, R provides a robust platform for performing sophisticated mathematical computations, enabling researchers and professionals to analyze data efficiently, model real-world systems, and gain deeper insights from their work. Its user-friendly interface, combined with powerful computational capabilities, makes it an invaluable resource for anyone engaged in computational mathematics.
Theory of Nonlinear Operators
No detailed description available for "Theory of Nonlinear Operators".
Invertibility and Asymptotics of Toeplitz Matrices
No detailed description available for "Invertibility and Asymptotics of Toeplitz Matrices".
Singularities and Low Dimensional Topology
The special semester 'Singularities and low dimensional topology' in the Spring of 2023 at the Erdős Center (Budapest) brought together algebraic geometers and topologists to discuss and explore the strong connection between surface singularities and topological properties of three- and four-dimensional manifolds. The semester featured a Winter School (with four lecture series) and several focused weeks. This volume contains the notes of the lecture series of the Winter School and some of the lecture notes from the focused weeks. Topics covered in this collection range from algebraic geometry of complex curves, lattice homology of curve and surface singularities to novel results in smooth four-dimensional topology and grid homology, and to Seiberg-Witten homotopy theory and 'spacification' of knot invariants. Some of these topics are already well-documented in the literature, and the lectures aim to provide a new perspective and fresh connections. Other topics are rather new and have been covered only in research papers. We hope that this volume will be useful not only for advanced graduate students and early-stage researchers, but also for the more experienced geometers and topologists who want to be informed about the latest developments in the field.
The (wrong) paths of tourism
This study deals with the socio-environmental issues related to the implementation and paving of the GO-239 road, called Estrada-parque Prefeito Divaldo Rinco, which links Alto Para穩so de Goi獺s to Colinas do Sul, in the Chapada dos Veadeiros, in Goi獺s. The road borders and also leads to the Chapada dos Veadeiros National Park (PNCV) and its route presents a scenic beauty typical of the region, showing the high-altitude Cerrado all around. The objectives are to describe the condition of the road and to show what people think about it. This is a qualitative, exploratory study that used direct observation of the road and interviews with residents and institutional agents (PNCV, Municipal Tourism Office) to capture their perceptions of the road's importance and issues surrounding its construction. It was concluded that the road still lacks the conditions that characterise it as a parkway, such as a management plan, but that it has great tourist potential due to the landscape that surrounds it.
Geometric mathematics in the slab cistern
This work came about through research in the rural community of S穩tio Barra do Japi in the municipality of Japi-RN. It shows the step-by-step mathematics involved in building a cement slab cistern. However, the aim was to bring together the mathematical concepts involved in carrying out this work. The aim is for the community's math teacher to be able to integrate his lessons with the daily reality of these people, bringing the school closer to the daily lives of the community's residents.
Statistical Machine Learning for Engineering with Applications
Mathematical Modelling
This book investigates human-machine systems through the use of case studies such as crankshaft maintenance, liner piston maintenance, and biodiesel blend performance.
Modeling and Simulation of Environmental Systems
This book presents an overview of modeling and simulation of environmental systems via diverse research problems and pertinent case studies, covering air pollution modeling, sustainable water resources modeling, IoT based applications in environmental systems, and future algorithms and conceptual frameworks in environmental systems.
A Stitch in Line
This book provides readers with instructions for creating hitomezashi items with minimum outlay. The reader is guided through the practical steps involved in creating each design, and then the mathematics which underpins it is explained in a friendly, accessible way.
Everything Is Predictable
A "fascinating, witty, and perspective-shifting" (Oliver Burkeman, New York Times bestselling author) tour of Bayes's theorem and its global impact on modern life from the acclaimed science writer and author of The Rationalist's Guide to the Galaxy.At its simplest, Bayes's theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. But in Everything Is Predictable, Tom Chivers lays out how it affects every aspect of our lives. He explains why highly accurate screening tests can lead to false positives and how a failure to account for it in court has put innocent people in jail. A cornerstone of rational thought, many argue that Bayes's theorem is a description of almost everything. But who was the man who lent his name to this theorem? How did an 18th-century Presbyterian minister and amateur mathematician uncover a theorem that would affect fields as diverse as medicine, law, and artificial intelligence? "Witty, lively, and best of all, extremely nerdy" (Tim Harford, author of The Undercover Economist), Everything Is Predictable is an entertaining and accessible illustration of how a single compelling idea can have far reaching consequences.
Spline Approch for Solution of Fluid Flow Problems
The book on the numerical solution of Magnetohydrodynamic (MHD) fluid flow problems using the spline collocation method provides a comprehensive exploration of how spline functions can be applied to solve complex MHD equations. It covers the fundamentals of MHD, where fluid dynamics and electromagnetism intersect, and introduces spline collocation as an efficient numerical technique for approximating solutions. The book details the method's application in various MHD problems, emphasizing its accuracy and computational efficiency. Through examples and case studies, the text illustrates how the spline collocation method can effectively handle boundary layers, singularities, and other challenges typical in MHD fluid flow analysis.
Evolutionary Computation
Evolutionary computation offers powerful problem-solving methodologies inspired by models of natural genetics and evolutionary processes. Potential applications are wide-ranging and include problems related to combinatorial optimization, numerical optimization, multi-objective optimization, and others, as well as specific applications of these problems in diverse domains, such as engineering, design, medicine, robotics, science, etc. Techniques from evolutionary computation often lend themselves well to parallel and distributed implementations and are often more effective in dealing with challenging problem characteristics such as non-linearity and high dimensionality than alternative approaches. This Special Issue brings together recent advances in the theory and application of evolutionary computation.
A First Course on Orthogonal Polynomials
This book provides an introduction to orthogonal polynomials and special functions aimed at graduate students studying these topics for the first time. A large part of its content is essentially inspired by the works of Pascal Maroni on the so-called algebraic theory of orthogonal polynomials.
Theory of Distributions
The Theory of Distributions, also known as the theory of generalized functions, is a mathematical analysis framework extending the classical notion of functions and derivatives. Introduced by Laurent Schwartz in the mid-20th century, this theory addresses limitations in traditional function analysis, particularly in handling functions that are not differentiable in the classical sense. Distributions generalize functions by defining them as continuous linear functional on a space of test functions, typically smooth and rapidly decreasing functions. This approach facilitates the rigorous treatment of objects like the Dirac delta function, which is pivotal in various applications across physics and engineering. Distributions are integral in solving differential equations, especially in contexts where classical solutions may not exist or be easily identifiable. They enable the extension of operations such as differentiation and convolution to a broader class of objects, thereby providing a robust toolkit for addressing problems in partial differential equations, signal processing, and quantum mechanics.
Math Mammoth Grade 8 Answer Keys, International Version
This book includes answers for the Math Mammoth Grade 8-A and 8-B student worktexts (international version), for all the chapter tests, the end-of-year test, and for the cumulative revisions.
General Mathematical Theory Dynamic Global Politic Eco
William Jevons (1866 and 1871) established a ground-breaking milestone with 'A General Mathematical Theory of Political Economy' for economic analysis. Jevons' work was praised as the start of the mathematical method in the discipline of economics, which is inherently a subject involved with mathematics and quantities. This book focuses on the most fast-evolving and encompassing area in political economy -- the dynamic global political economy. Under the high level of globalization currently, intertemporal and cross-boundary interactive elements are present in political-economic encounters. Indeed, almost all studies in the political economy may fall into the study of dynamic global political economy. Since the world has changed significantly, new mathematics developed by the authors of this book is used to formulate a general mathematical theory for the dynamic global political economy nowadays. A distinctive feature of the current book is that it combines advanced mathematics, game-theoretic concepts, and economics to develop a general mathematical theory supporting the study of the dynamic global political economy.The book covers mathematical theory for different areas of the dynamic global political economy. In addition, it explicates the application of the mathematical theory in real-world scenarios, including (i) environmental degradation under an uncoordinated interaction scenario, (ii) global climate accords with collaboration and cooperation, (iii) trade network involving the Belt-Road Initiative (BRI) and Build Back Better World (B3W) Initiative, and (iv) random termination of international joint ventures.
Distribution Dependent Stochastic Differential Equations
Corresponding to the link of It繫's stochastic differential equations (SDEs) and linear parabolic equations, distribution dependent SDEs (DDSDEs) characterize nonlinear Fokker-Planck equations. This type of SDEs is named after McKean-Vlasov due to the pioneering work of H P McKean (1966), where an expectation dependent SDE is proposed to characterize nonlinear PDEs for Maxwellian gas. Moreover, by using the propagation of chaos for Kac particle systems, weak solutions of DDSDEs are constructed as weak limits of mean field particle systems when the number of particles goes to infinity, so that DDSDEs are also called mean-field SDEs. To restrict a DDSDE in a domain, we consider the reflection boundary by following the line of A V Skorohod (1961).This book provides a self-contained account on singular SDEs and DDSDEs with or without reflection. It covers well-posedness and regularities for singular stochastic differential equations; well-posedness for singular reflected SDEs; well-posedness of singular DDSDEs; Harnack inequalities and derivative formulas for singular DDSDEs; long time behaviors for DDSDEs; DDSDEs with reflecting boundary; and killed DDSDEs.
Math Mammoth Grade 8 Tests and Cumulative Revisions, International Version
Math Mammoth Grade 8 Tests and Cumulative Revisions (international version) contains the chapter tests, end-of-year test, and cumulative revision worksheets for Math Mammoth Grade 8 curriculum (international version). It does not contain answers; the answer key book may be purchased separately.
Math Mammoth Grade 8-B Worktext, International Version
Math Mammoth Grade 8-B Worktext (International Version) is the second student book for eighth grade mathematics, and part of the Math Mammoth Grade 8 complete curriculum.The book starts with chapter 5, graphing linear equations. Students study slope, and graph and write linear equations in slope-intercept and standard forms. They also learn about parallel and perpendicular lines.In chapter 6, students delve into square roots, cube roots, and irrational numbers and solve equations involving square and cube roots. The rest of the chapter is spent studying the Pythagorean Theorem and its applications.Next, in chapter 7, we focus on systems of linear equations. Students learn the three basic techniques for solving these and also solve a variety of problems that are solved with a pair of linear equations. The last chapter covers statistical topics: scatter plots and two-way tables.
