Topics in Model Theory
This book has two chapters. The first is a modern or contemporary account of stability theory. A focus is on the local (formula-by-formula) theory, treated a little differently from in the author's book Geometric Stability Theory. There is also a survey of general and geometric stability theory, as well as applications to combinatorics (stable regularity lemma) using pseudofinite methods.The second is an introduction to 'continuous logic' or 'continuous model theory, ' drawing on the main texts and papers, but with an independent point of view. This chapter includes some historical background, including some other formalisms for continuous logic and a discussion of hyperimaginaries in classical first order logic.These chapters are based around notes, written by students, from a couple of advanced graduate courses in the University of Notre Dame, in Autumn 2018, and Spring 2021.
Topics in Model Theory
This book has two chapters. The first is a modern or contemporary account of stability theory. A focus is on the local (formula-by-formula) theory, treated a little differently from in the author's book Geometric Stability Theory. There is also a survey of general and geometric stability theory, as well as applications to combinatorics (stable regularity lemma) using pseudofinite methods.The second is an introduction to 'continuous logic' or 'continuous model theory, ' drawing on the main texts and papers, but with an independent point of view. This chapter includes some historical background, including some other formalisms for continuous logic and a discussion of hyperimaginaries in classical first order logic.These chapters are based around notes, written by students, from a couple of advanced graduate courses in the University of Notre Dame, in Autumn 2018, and Spring 2021.
Causal Inference in Pharmaceutical Statistics
Causal Inference in Pharmaceutical Statistics introduces the basic concepts and fundamental methods of causal inference relevant to pharmaceutical statistics. This book covers causal thinking for different types of commonly used study designs in the pharmaceutical industry, including but not limited to randomized controlled clinical trials, longitudinal studies, singlearm clinical trials with external controls, and real-world evidence studies. The book starts with the central questions in drug development and licensing, takes the reader through the basic concepts and methods via different study types and through different stages, and concludes with a roadmap to conduct causal inference in clinical studies. The book is intended for clinical statisticians and epidemiologists working in the pharmaceutical industry. It will also be useful to graduate students in statistics, biostatistics, and data science looking to pursue a career in the pharmaceutical industry.Key Features: Causal inference book for clinical statisticians in the pharmaceutical industry Introductory level on the most important concepts and methods Align with FDA and ICH guidance documents Across different stages of clinical studies: plan, design, conduct, analysis, and interpretation Cover a variety of commonly used study designs
Risk Revealed
Explore the concept of risk through numerous examples and their statistical modeling, traveling from a historical perspective all the way to an up-to-date technical analysis. Written with a wide readership in mind, this book begins with accounts of a selection of major historical disasters, such as the North Sea flood of 1953 and the L'Aquila earthquake. These tales serve to set the scene and to motivate the second part of the book, which describes the mathematical tools required to analyze these events, and how to use them. The focus is on the basic understanding of the mathematical modeling of risk and what types of questions the methods allow one to answer. The text offers a bridge between the world of science and that of everyday experience. It is written to be accessible to readers with only a basic background in mathematics and statistics. Even the more technical discussions are interspersed with historical comments and plentiful examples.
Partitions, Objective Indefiniteness, and Quantum Reality
This book presents a new 'partitional' approach to understanding or interpreting the math of standard quantum mechanics (QM). The thesis is that the mathematics (not the physics) of QM is the Hilbert space version of the math of partitions on a set and, conversely, the math of partitions is a skeletonized set level version of the math of QM. Since at the set level, partitions are the mathematical tool to represent distinctions and indistinctions (or definiteness and indefiniteness), this approach shows how to interpret the key non-classical QM notion of superposition in terms of (objective) indefiniteness between definite alternatives (as opposed to seeing it as the sum of 'waves'). Thus, the book develops a new mathematical, or indeed, logical, approach to the century-old problem of interpreting quantum mechanics, ensure it is of interest to philosophers of science as well as mathematicians and physicists.
The Polls Weren't Wrong
This book will equip readers with the tools to navigate the mismatch of expectations. It is not intended to replace more technical applications of statistics but is accessible to those interested in learning about how poll data should be understood.
Ordinal Data Analysis
This book is a step-by-step data story for analysing ordinal data from start to finish. The book is for researchers, statisticians and scientists who are working with data sets where the response is ordinal. This type of data is common in many disciplines, not just in surveys.
Integrated Computer Technologies in Mechanical Engineering - Synergetic Engineering
The best works were selected after the conference "Integrated Computer Technologies in Mechanical Engineering"-Synergetic Engineering (ICTM). The conference provided technical exchange between the scientific community in the form of keynote addresses, panel discussions, and a special session. In addition, participants were treated to a series of techniques that facilitated collaboration among fellow researchers. Moreover, ICTM 2021 received 110 applications from different countries.
Nonparametric Statistical Methods Using R
This thoroughly updated and expanded second edition covers traditional nonparametric methods and rank-based analyses. Two new chapters covering multivariate analyses and big data have been added. Core classical nonparametrics chapters on one- and two-sample problems have been expanded
Integrated Computer Technologies in Mechanical Engineering - Synergetic Engineering Ⅱ
The best works were selected after the conference titled "Integrated Computer Technologies in Mechanical Engineering"-Synergetic Engineering (ICTM) II. The conference provided technical exchange between the scientific community in the form of keynote addresses, panel discussions, and a special session. In addition, participants were treated to a series of techniques that facilitated collaboration among fellow researchers. Moreover, ICTM 2021 received 110 applications from different countries.
Math Mammoth Factors & Factoring
Math Mammoth Factors & Factoring is a worktext that covers typical number theory topics for upper elementary school: divisibility rules, finding factors of a given number, prime numbers, prime factorization, the greatest common factor, and the least common multiple. These topics are usually covered in grades 4-6.The book begins with 4th grade topics, starting out with the concept of divisibility and the common divisibility rules. The lesson Prime Numbers brings out the concept of a prime number as one that is only divisible by 1 and by itself. Armed with the knowledge about divisibility, students now learn how to find factors of a given two-digit number in the lesson Finding Factors.The following lesson, Primes and Finding Factors, is a review lesson about factors, primes, and divisibility rules, originally intended for beginning of 5th grade. You can use it as a review of the first four lessons of this book, or for students that have some knowledge of these topics already.The next two lessons delve into prime factorization using a factor tree. The first lesson on the topic only uses two-digit numbers. In the latter, students are introduced the sieve of Eratosthenes and they factorize three-digit and larger numbers.The last topics in the book are the greatest common factor (GCF), the least common multiple (LCM), and factoring sums, intended mainly for 6th grade. Factoring sums means writing a sum such as 42 + 18 as 6(7 + 3). We first find a common factor of 42 and 18 (which is 6), and use that to factorize the sum. The lesson only deals with numeric examples, but it is actually preparing students for algebra, where the same process is done with variables; for example x2 + 2x is factored as x(x + 2).The book ends in a review lesson. Answers are appended.
Math Mammoth Division 2
Math Mammoth Division 2 is a continuation from the Math Mammoth Division 1. It focuses on the topics of long division, remainder, problem solving, average, divisibility, and factors. The book is most suitable for fourth grade.We start out by reviewing basic division facts by single-digit numbers (such as 24 繩 4 or 56 繩 7). After that, we study terminology of division and dividing numbers by whole tens and hundreds (such as 400 繩 20). Next students practice the order of operations-this time with division as one of the operations.Then we study the concept of remainder, preparing students for the upcoming lessons on long division. At first, the concept of remainder is presented visually. Soon, students solve simple division problems with a remainder, written with the long division symbol (or long division "corner", as I like to call it).Next comes a set of lessons intended to teach long division in several small steps. We start with divisions where each of the digits in the dividend (thousands, hundreds, tens, and ones) can be divided evenly by the divisor (for example, 3096 繩 3). As the next step, there is a remainder in the ones. Then, the divisions have a remainder in the tens. Finally, there is a remainder in the hundreds and in the thousands, and this completes the step-by-step learning process for long division. The lessons also include lots of word problems to solve.After long division, we study the concept of average, which is a nice application of division, and problems that involve finding a fractional part of a quantity using division. For example, we can find 3/4 of a number by first finding 1/4 (dividing by 4) and then multiplying the result by 3. Students get help from visual bar models to solve the problems.The last section deals with elementary number theory. We study basic divisibility rules (though not all of them), prime numbers, and finding all factors of a given two-digit number.
Nonlinear Second Order Elliptic Equations
This book focuses on the following three topics in the theory of boundary value problems of nonlinear second order elliptic partial differential equations and systems: (i) eigenvalue problem, (ii) upper and lower solutions method, (iii) topological degree method, and deals with the existence of solutions, more specifically non-constant positive solutions, as well as the uniqueness, stability and asymptotic behavior of such solutions.While not all-encompassing, these topics represent major approaches to the theory of partial differential equations and systems, and should be of significant interest to graduate students and researchers. Two appendices have been included to provide a good gauge of the prerequisites for this book and make it reasonably self-contained.A notable strength of the book is that it contains a large number of substantial examples. Exercises for the reader are also included. Therefore, this book is suitable as a textbook for graduate students who havealready had an introductory course on PDE and some familiarity with functional analysis and nonlinear functional analysis, and as a reference for researchers.
Fundamentals of Order and Rank Statistics
This book is devoted to the fundamentals of order and rank statistics. Primarily focusing on theoretical properties, it also discusses practical aspects, including interesting applications of step and impulse functions for the distribution of random variables, magnitudes, and signs. New concepts are introduced, such as independent and semi-identically distributed random vectors and probability density-mass functions. This book also presents an investigation of relative magnitudes in order statistics, and of correlation coefficients among signs, ranks, and magnitudes. The basic concepts are described in clear terms, and step-by-step details are provided for most of the presented mathematical results. The exposition is accompanied by numerous examples and more than 100 exercises, for which a complete solution manual is available. Providing a useful reference, and requiring only a basic understanding of probability and random variables, the book will appeal to a wide readership.
Recent Trends in AI Enabled Technologies
This book constitutes the refereed proceedings of the First International Conference on Recent Trends in AI Enabled Technologies, ThinkAI 2023, which took place in Hyderabad, India, in December 2023. The 7 full papers presented in these proceedings were carefully reviewed and selected from 51 submissions. The conference focuses on on up to date topics and recent trends in artificial intelligence and related technologies.
Introduction to the Geometrical Foundations of General Relativity
This book is intended for physics students (undergraduate/transition to graduate) who want to prepare for lectures on general relativity. Some knowledge of linear algebra and analysis are required. This textbook starts with basic topics such as vector space and vectors (chapter 1), dual space and covectors (chapter 2), tensors (chapter 3), etc.. Great importance is always attached to the clarity of the explanations and derivations of the topics. 31 figures support these intentions. General relativity is in its deeper sense a geometric theory. Therefore, the emphasis in this textbook has been placed on understanding space in its geometric configuration. Space is a component of the representation of the physical real. And it is thus the stage on which the physical processes and procedures show themselves. Calculation tasks have been deliberately omitted. The focus is on understanding a topic. For this purpose, many examples and detailed extra introductions have been made. For exercises, there are enough examples in the relevant textbooks that can be used to deepen a topic. This book serves as a good basis for mastering tasks. If you have studied the book thoroughly, you will be prepared to start working on the physics of gravitation as described by general relativity.
The Power of Probability
"The Power of Probability: Making Sense of Uncertainty" is a comprehensive exploration of probability theory and probabilistic reasoning. From its historical origins to its practical applications in diverse domains, the book delves into the fundamental concepts, methodologies, and implications of understanding uncertainty. Through engaging discussions and practical examples, readers are empowered to embrace probabilistic thinking, make informed decisions, and navigate complexities in an uncertain world.
From Concepts to Code
The breadth of problems that can be solved with data science is astonishing, and this book provides the required tools and skills to a broad audience. The necessary background in computer science, mathematics, and statistics is provided in an approachable manner.
Dynamic Network Flows with Adaptive Route Choice Based on Current Information
In this book Lukas Graf studies dynamic network flows which are a model for individual car traffic in road networks. It is assumed that drivers choose their routes based on information about the current state of the network in such a way as to selfishly minimize their own arrival time at their destination. Whilst on their journey the drivers adapt their current route choices based on the changing state of the network. A dynamic flow wherein every (infinitesimally small) flow particle behaves in this way is then called an instantaneous dynamic equilibrium. After giving a mathematically precise definition of this equilibrium concept the author shows existence of those equilibrium flows, studies their computational complexity and derives bounds on their quality.
Mathematical Structures
This textbook is intended to be accessible to any second-year undergraduate in mathematics who has attended courses on basic real analysis and linear algebra. It is meant to help students to appreciate the diverse specialized mathematics courses offered at their universities. Special emphasis is on similarities between mathematical fields and ways to compare them. The organizing principle is the concept of a mathematical structure which plays an important role in all areas of mathematics. The mathematical content used to explain the structural ideas covers in particular material that is typically taught in algebra and geometry courses. The discussion of ways to compare mathematical fields also provides introductions to categories and sheaves, whose ever-increasing role in modern mathematics suggests a more prominent role in teaching. The book is the English translation of the second edition of "Mathematische Strukturen" (Springer, 2024) written in German. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.
Math-Ish
From Stanford professor, author of Limitless Mind, youcubed.org founder, and leading expert in the field of mathematics education Jo Boaler comes a groundbreaking guide to finding joy and understanding by adopting a diverse approach to learning math."Every once in a while, someone revolutionizes an approach to a difficult subject and changes it forever. That is what Jo Boaler has done for math. Fresh, smart, and inclusive, Jo Boaler's strategy eschews the one-size-fits-a-few approach and instead allows math to be seen and solved by everyone. A huge achievement. Math-ish is the only math book I've ever enjoyed reading in my entire life. Honestly." -Bonnie Garmus, author of Lessons in ChemistryMathematics is a fundamental part of life, yet every one of us has a unique relationship with learning and understanding the subject. Working with numbers may inspire confidence in our abilities or provoke anxiety and trepidation. Stanford researcher, mathematics education professor, and the leading expert on math learning Dr. Jo Boaler argues that our differences are the key to unlocking our greatest mathematics potential.In Math-ish, Boaler shares new neuroscientific research on how embracing the concept of "math-ish"--a theory of mathematics as it exists in the real world--changes the way we think about mathematics, data, and ourselves. When we can see the value of diversity among people and multi-faceted approaches to learning math, we are free to truly flourish. Utilizing the latest research on math education, Jo guides us through seven principles that can radically reframe our relationship with the subject: - The power of mindset on learning- Utilizing a visual approach to math- The impact of physical movement and communication on understanding- Understanding the value of an "ish" perspective - in mathematics and beyond- The importance of connected and flexible knowledge- New data on diverse teaching modes that work with different learning styles, not against them- The value of diversity in learning mathematics--and beyondWhen mathematics is approached more broadly, inclusively, and with a greater sense of wonder and play--when we value the different ways people see, approach, and understand it--we empower ourselves and gain a beneficial understanding of its value in our lives.
Math Mammoth Multiplication 2
Math Mammoth Multiplication 2 is a worktext focusing on multi-digit multiplication and related topics. It best suits fourth grade mathematics.The first lessons briefly review the concept of multiplication and the multiplication tables. Next, students encounter equations in disguise - presented with shapes on both sides of a pan balance - in the lesson Scales Puzzles. This lesson is intended to be fun and motivational.Then, the focus shifts to multi-digit multiplication (also called multiplication algorithm or multiplying in columns). We start out by learning to multiply numbers by multiples of ten and hundred (for example, 20 ? 4 or 500 ? 6). After this is mastered, students learn the very important concept of multiplying in parts, or partial products. This means that, for example, we multiply 4 ? 63 in two parts: first we multiply 4 ? 60 = 240 and 4 ? 3 = 12, and lastly the results are added: 240 + 12 = 252.This principle underlies all other multiplication algorithms, so it is important to master. We don't want children to "blindly" memorize the multiplication algorithm without understanding what is going on with it. The partial products algorithm (multiplying in parts) also ties in with an area model, and it is very important that students see the connection between this visual model and the procedure.The book contains two lessons about multiplying in columns the "easy way". This "easy way" is a simplified form of the traditional multiplication algorithm, based on partial products. You may skip these two lessons at your discretion. The method taught in those lessons is most useful for students who may have trouble with the traditional form of the algorithm. This method is also helpful in cementing the student's understanding of the partial products method.The traditional, or standard, form of multiplication algorithm is taught next, and is hopefully fairly easy, with the partial products as a foundation.Students also study estimation, the order of operations, and multiplying with money. There are numerous word problems in all of the lessons. Students are encouraged to write number sentences for the word problems-essentially learning to show their work and their thinking process.The lesson So Many of the Same Thing has to do with proportional reasoning. The idea is really simple, and prepares students for learning ratios and proportions in middle school.The last major topic in the book is multiplying two-digit numbers by two-digit numbers. Again, we first study partial products and tie that in with an area model. The lesson Multiplying in Parts: Another Way is optional. Lastly, the book teaches the standard algorithm for two-digit by two-digit multiplication. Students will practice multiplication with more digits in the book Mammoth Multiplication & Division 3.
Combinatorics Problems and Solutions
This book will help you learn combinatorics in the most effective way possible - through problem solving. It contains 263 combinatorics problems with detailed solutions. Combinatorics is the part of mathematics that involves counting. It is therefore an essential part of anyone's mathematical toolkit. The applications of combinatorics include probability, cryptography, error correcting, games, music and visual art. In this new edition we have expanded the introductory section by more than twice the original size, and the number of problems has grown by over 30%. There are new sections on the pigeon hole principle and integer partitions with accompanying problems. Many of the new problems are application oriented. There are also new combinatorial geometry problems. Someone with no prior exposure to combinatorics will find enough introductory material to quickly get a grasp of what combinatorics is all about and acquire the confidence to start tackling problems.
Shrinkage for Stabilizing the Detection of Changepoints in Covariances for High-Dimensional Data
Diploma Thesis from the year 2012 in the subject Mathematics, grade: 1,0, University of Kaiserslautern (Fakult瓣t f羹r Mathematik), language: English, abstract: In mathematical statistics, detecting changes in parameters of real-life data series, known as change-point problems, is crucial. Originating in quality control during the 1950s, these problems have widespread applications today, spanning fields like economics, finance, medicine, and geology. In finance, fluctuations in asset returns can violate assumptions of constant variance, leading to inaccurate forecasts. Chapter 2 briefly discusses the univariate case, focusing on detecting changes in mean and variance parameters over time. The Cumulative Sums (CUSUM) test statistics, derived from likelihood ratios, serve as change-point estimators. However, their asymptotic distribution complexity and slow convergence limit applicability to small sample sizes. Nevertheless, asymptotic quantiles help determine if changes have occurred. Chapter 3 extends this analysis to the multivariate case, specifically addressing changes in covariance matrices. Estimating the covariance matrix, particularly in scenarios with many variables and few observations, poses challenges. Shrinkage estimators, like the Ledoit-Wolf (LW) estimator, offer improvements over sample covariance matrices, especially in small sample sizes. The Rao-Blackwell theorem leads to the development of the Rao-Blackwellized Ledoit-Wolf (RBLW) estimator, enhancing performance under Gaussian assumptions. A simulation study in Chapter 5 demonstrates the effectiveness of using these shrinkage estimators in detecting change-points, resulting in improved test power and accuracy. However, due to the absence of an asymptotic distribution for the test statistics, quantiles must be obtained through simulation.
General Combination Theorem and Selected Combinations
Document from the year 2024 in the subject Mathematics - Algebra, language: English, abstract: So for combinations are discussed with different theorems in algebra. In this chapter I apply assembly analysis to get the theorems easy and memorable. After assembly analysis applied there becomes a lot of new theorems and all the theorems get a new face by summation methods. We have a full idea about combination. It indicates the outcome of arandom experiment. That is combination is the selection of M different components taken V at a time what is called usually a random experiment where order is not taken into account and repetitions are not allowed.
General Permutation Theorem and Selected Permutations
Document in the subject Mathematics - Algebra, language: English, abstract: We have permutations are discussed with different theorems in algebra. In this chapter I apply B system analysis to get the theorems easy and memorable. After B system analysis applied there becomes a lot of new theorems and all the theorems get a new face by summation methods. One usual theorem described with summation method and face new looks.
Actuarial Loss Models
This book covers part of the learning outcomes of the Fundamentals of Actuarial Mathematics (FAM) exam and the Advanced Short-Term Actuarial Mathematics (ASTAM) exam administered by the Society of Actuaries. It can be used by students and practitioners who prepare for actuarial exams.
Analyzing Repartition Numbers of Parent Partitions. Theorems and their Implications
Research Paper (postgraduate) from the year 2024 in the subject Mathematics - Algebra, language: English, abstract: This paper delves into the intricate realm of partition theory, specifically focusing on the repartition numbers of parent partitions. Through a systematic development of six theorems, insights into the distribution of repartitions, delineating distinct patterns for two separate parent partitions, are unveiled. Theorems are structured to elucidate the occurrences of repartitions based on the number of components involved, providing a comprehensive understanding of the repartition phenomena. The findings contribute to a deeper comprehension of partition theory and its implications across various domains.
Exploring Unique Expressions of Positive Integers through Partitions and Theorems
Research Paper (postgraduate) from the year 2024 in the subject Mathematics - Algebra, language: English, abstract: This paper delves into the realm of natural numbers and their expression as sums of other natural numbers, a concept known as partitions. Focusing on partitions originating from a positive integer and comprising positive integers, a systematic analysis is presented. Essential terms are defined to lay the groundwork, followed by the introduction of three key theorems and a consequential corollary. These theorems elucidate the uniqueness of expressions formed through arithmetic addition operations on such partitions, offering valuable insights into the structure and properties of positive integers. This exploration not only contributes to the theory of numbers but also holds implications for various mathematical and computational applications.
Math Mammoth Add & Subtract 2-B
Math Mammoth Add & Subtract 2-B is a continuation to the book Math Mammoth Add & Subtract 2-A. The goal of this book is to study addition and subtraction within 0-100, both mentally and in columns, especially concentrating on regrouping in addition (carrying) and in subtraction (borrowing).Mental math is important because it builds number sense. This book includes many lessons that practice mental math. For example, the child practices adding and subtracting two-digit numbers when one of the numbers is a whole ten (such as 30 + 14 or 66 - 20).The students also learn to solve sums such as 36 + 8 or 45 + 9 with mental math (there is a regrouping), comparing them to the sums 6 + 8 and 5 + 9. Since 6 + 8 fills the first ten and is four more than the next ten, the sum 36 + 8 fills the next whole ten (40), and is four more than that, or 44.At the same time, students also learn to add two-digit numbers with regrouping. This is explained in detail with the help of visual models (base-ten blocks). You are welcome to use actual physical manipulatives, if you prefer. The main concept to understand is that 10 ones are regrouped to form a new ten, and this new ten is written using a little "1" in the tens column.After addition, the lessons focus on column subtraction, initially without regrouping and then with regrouping.The lesson Regrouping practices breaking down a ten into 10 ones. It is crucial that the child understands what happens here. Otherwise, they might end up only memorizing the procedure, and will probably at some point misremember how it was done. If you notice that the child does not understand the concept of regrouping, provide more practice with concrete manipulatives or visual exercises before proceeding.After learning regrouping, students practice mental subtraction in two separate lessons. One of them expounds on several methods for mental subtracting. The other is about Euclid's game-a fun game that also practices subtraction of two-digit numbers.
Math Mammoth Add & Subtract 2-A
Math Mammoth Add & Subtract 2-A deals with two main themes: strategies for adding and subtracting within 0-20; such as adding just one more, a trick with nine and eight, and subtracting using addition;memorizing the basic addition and subtraction facts of single-digit numbers with an answer between 10 and 18.While focusing on addition and subtraction techniques, the lessons also include many word problems.The goal is to memorize the facts, or at least become so fluent with them that an outsider cannot tell if the child remembers the answer or uses some mental math strategy to get the answer.Some children will accomplish this quicker and need less practice, whereas others will take longer. Thus, don't assign all the exercises in the book by default. Use your judgment, and try to match the amount of exercises to your child's need. The ones that don't get assigned can be used later for review. You can also use games to reinforce the facts, and in place of some of the exercises in the book (a list of games is provided in the book).Learning addition and subtraction facts is quite important for later study. For example, regrouping in addition and subtraction (carrying/borrowing) requires the ability to recall the basic facts efficiently and fluently.We will start the book with a few mental math strategies (Add Using "Just One More" and A Trick with Nine and Eight). The lesson Adding within 20 reviews those strategies and gives more practice. These initial lessons don't yet involve actual memorization techniques.The lesson Subtract to Ten explains another basic strategy, and has to do with subtracting in parts. For example, to do 13 - 5, one can subtract 5 in two parts: first do 13 - 3, which equals 10, and then subtract the rest, or 2 more. Memorizing the subtraction facts will be more efficient, but I want children to understand this strategy, because it is useful in many other situations also.Then we review how to complete the next whole ten, which is an important concept. An example of this concept is the question: What number do you add to 23 to get 30? As an equation, we write: 23 + __ = 30.In the next lesson, we study sums that go over ten, doing these sums in two parts. For example, in the sum 9 + 7, the child first completes 10 by adding 9 + 1. Then, the child adds the rest, or 6, to 10. Learning this prepares the child for addition facts where the sum is more than 10.The next lessons, Adding with 9, Adding with 8, Adding with 7, and Adding with 6, provide lots of practice for learning and memorizing the basic addition facts. There are 20 such facts: 9 + 2 till 9 + 9: 8 facts8 + 3 till 8 + 8: 6 facts7 + 4 till 7 + 7: 4 facts6 + 5 till 6 + 6: 2 factsThe last part of the book includes various lessons titled Number Rainbows and Fact Families with.... These give lots of practice and reinforcement for the basic addition and subtraction facts, emphasizing the connection between addition and subtraction as a strategy for subtraction facts.The follow-up to this book, Add & Subtract 2-B, practices adding and subtracting two-digit numbers mentally and in columns (regrouping).
The Eternal Life
Applied Math to Life ResearchUsing the objectiveness of the Mathematical Language to analyze, study and conclude on the spiritual eternal life, the successive lifetimes, life, death and related subjects.A "must not miss" book, quite deep on its subjects but joyfully presented on a youngish style.
Holy Trinity & Inverted Holy Trinity
The Trinities are all over the place!There is the positive Holy Trinity turned upwords... and there is the negative or Inverted Holy Trinity, turned upside down. From a Mathematical perspective. Applied Math to Life, Research. Bridge/Fusion between Mathematical Science & Spirituality.
Math Mammoth Subtraction 1
Math Mammoth Subtraction 1 is a worktext that covers the concept of subtraction, the relationship between addition and subtraction, and the various meanings of subtraction.In the first lesson, Subtraction Is Taking Away, the child learns the basic meaning of subtraction as taking away objects, and learns to write subtractions from an illustration where some objects are crossed out.In the next lesson, the child counts down to subtract, which ties in subtraction with the number line. This is a transitional strategy to solve subtraction problems, because later students will learn more efficient ways to subtract, but it is important conceptually.The following lesson, Subtraction and Addition in the Same Picture, begins the study of the relationship between addition and subtraction. This concept will span several lessons. This first lesson presents two sets of objects, such as blue and white balls, and the student writes both an addition sentence and a subtraction sentence from this illustration.The lesson When Can You Subtract? concentrates on the idea that some subtractions, such as 4 - 5, are meaningless when you think of taking away. The child also makes subtraction patterns in this lesson.Then we continue studying the connection between addition and subtraction in the lesson Two Subtractions from One Addition. As an example, the child writes both 8 - 3 = 5 and 8 - 5 = 3 from the addition 3 + 5 = 8. This idea ties in with fact families, a concept that is coming up soon.In the lesson Two Parts - One Total, we study word problems that do not involve the idea of taking away but have two parts making up a total. For example, if there are 10 flowers of which some are white and some are red, and seven of them are white, how many are red?Then we study fact families, which means writing two additions and two subtractions using the same three numbers.In the lesson How Many More? students find how many more or how many fewer objects one person has than the other by drawing the objects. This lesson can easily be done with manipulatives if desired.In the very next lesson, "How Many More" Problems and Differences, we continue the theme, this time writing a missing addend addition for problems that ask "how many more." For example, Veronica has 4 marbles and Ann has 6. We write the missing addend sentence 4 + ___ = 6 to find how many more Ann has. In the next lesson the child then learns to write subtraction sentences for such problems.
Reliability
This text provides an elementary introduction to the probabilistic models and statistical methods used by reliability engineers that are applied to a system of components. Probability models include the exponential distribution, Weibull distribution, competing risks, mixtures, accelerated life model, proportional hazards model, and repairable systems models. Statistical methods emphasize determining point and interval estimates for parameters from censored data sets. Applications are drawn from a variety of disciplines. Over 600 exercises make this text appropriate for a class on reliability.
The 2018-2024 Canadian Twin Prime Conjecture Solo Expedition
... one of - if not the most famous proof in all of mathematics - admired for its elegance - simplicity and profundity is Euclid's Proof of Infinite Primes - mathematical rigor currently enjoys an unprecedented historical resurgence over its long history - yet to this author - it seems the strength of the deep-learned analysis and logical arguments used in ancient Greek proof somehow remains missing in contemporary mathematics - to address this - the author does not follow conventional research protocol and abandons research into all prior knowledge on the elusive proof of the long-standing Twin Prime Conjecture - rather - a rare summit attempt is made from first principals alone - equipped with little more than the definition of a twin prime - this Canadian mathematician begins a six year mathematical solo research expedition into unravelling the chaos that is the twin primes - always in search of patterns - the nature of the of twin primes has led several mathematicians to conclude that the exhibited random pattern is chaotic - no discernible pattern exists - under such conditions we cannot predict anything about how twin primes behave nor attempt to count them over increasingly large - infinite domains - lacking a proof akin to Euclid's - that outcome places twin primes outside the harmonious mathematics of the ancient Greeks - a world that we mathematicians have always sought to retain - such incongruity and disharmony is jarring and harsh for any mathematician to accept - after undergoing my transformation into a mathematical alpinist - the expedition strategy was to focus not on the (twin) primes but rather their theoretical complement - the odd composites - it was conjectured by the author that a deeper understanding of these simplistic mathematical entities would indirectly reveal how twin primes are constructed - to summit one of the highest unclimbed mathematical peaks - required new innovative techniques - bold daring thoughts and inexhaustible patience and persistence - if you continue to climb and gain altitude you must - eventually - reach the summit - one painful step at a time - lastly - having reached the summit - by far the most daunting task was to collate and present the argument in such a manner that others would not only understand it but be convinced of the result in the same manner that those who read Euclid's proof are - with elegance - simplicity and profundity ...... this long-awaited Proof of the Twin Prime Conjecture rescues the harmonious nature of mathematics out of chaos ... it was an exhilarating unforgettable intellectual mathematical alpine expedition ... more than once I felt at the brink of physical and mental exhaustion and had to take a brief walk away simply to recover from the task - too often any discussion of the hard intellectual struggles are omitted from the discourse and too often the final result alone is presented as if it was the initial effort and attempt - when the truth involved so much more ...
Ring Theory - Proceedings of the Biennial Ohio State-Denison Conference 1992
Math Mammoth Fractions 2
Math Mammoth Fractions 2 continues the study of fraction topics after Math Mammoth Fractions 1. (I sincerely recommend that the student study the Fractions 1 book prior to studying this book, if he has not already done so.)This book is meant for fifth grade, and deals in-depth with the following topics: simplifying; including simplifying before multiplyingmultiplication of fractions (and of mixed numbers)division of fractionsWe start out by simplifying fractions. Since this process is the opposite of making equivalent fractions, studied in Math Mammoth Fractions 1, it should be relatively simple for students to understand. We also use the same visual model, just backwards: This time the pie pieces are joined together instead of split apart.Next comes multiplying a fraction by a whole number. Since this can be solved by repeated addition, it is not a difficult concept at all.Multiplying a fraction by a fraction is first explained as taking a certain part of a fraction, in order to teach the concept. After that, students are shown the usual shortcut for the multiplication of fractions.Then, we find the area of a rectangle with fractional side lengths, and show that the area is the same as it would be found by multiplying the side lengths. Students multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.Simplifying before multiplying is a process that is not absolutely necessary for fifth graders. I have included it here because it prepares students for the same process in future algebra studies and because it makes fraction multiplication easier.Students also multiply mixed numbers, and study how multiplication can be seen as resizing or scaling. This means, for example, that the multiplication (2/3) ? 18 km can be thought of as finding two-thirds of 18 km.Next, we study the division of fractions. The first lesson on the topic shows how fractions can be seen as divisions; in other words, for example, 5/3 is the same as 5 繩 3. This of course gives us a means of dividing whole numbers and getting fractional answers (for example, 20 繩 6 = 3 2/6).The next lesson on the topic explores the thought of solving division problems by multiplying, and then introduces reciprocal numbers and the common shortcut for fraction division.The following two lessons focus on two ways to think about division: equal sharing, and fitting the divisor (measurement division), and how they apply to fraction division. Here is an example of equal sharing: If two people equally share 4/5 of a pizza, how much will each person get? This is represented by the division (4/5) 繩 2 = 2/5.
Math Mammoth Addition 1
Math Mammoth Addition 1 is a worktext (includes both teaching and exercises), dealing with the concept of addition and addition facts within 0-10 (in few occasions numbers between 10 and 20 are used). The book is most suitable for first grade.The book starts out with very easy addition problems within 0-5, using pictures. If the student does not yet know the symbols + and =, you can introduce them first orally. In other words, use blocks or rocks and make addition problems where you use both kinds of wordings: "Three blocks and four blocks makes seven blocks. Three blocks PLUS four blocks EQUALS seven blocks." Play like that until the child can use the words PLUS and EQUALS in his/her own speech. This will make it easier for him/her to use the written symbols.Soon the lessons start including "missing addend" problems. We first use pictures, and gradually get to the abstract 1 + __ = 5 with symbols only. Keep in mind that children may confuse this problem with1 + 5 = __. You can word these problems like this: "1 and how many more makes 5?" You can MODEL them by drawing: First draw 1 ball. Tell the child that we need to have a total of 5 balls. He is to draw more until there are five balls. In the missing addend problem, however many balls the child has to draw is the number that goes on the empty line. So, first there is one ball, then we need to add (draw) some more to make 5. But how many more were drawn?After that, the book contains many lessons called Sums with... whose goal is to help the child memorize addition facts.My approach to memorization is many-fold: Structured drill, such as you see in the lessons Sums with 5, Sums with 6, and so on. This is not random drill, because you will start it by showing the pattern or the structure in the facts. This will help the student to tie the addition facts in with a context and help him understand the facts more on a conceptual level, instead of merely memorizing them at random. In sums with 5, the child learns the number combinations that add up to 5: 0 & 5, 1 & 4, and 2 & 3. This understanding is the basis for the drills.Using addition facts in games, in math problems, everyday life, or anywhere else. Games are especially useful because they help children like mathematics.Random drilling may also be used as a tool among others.Memory helps such as silly mnemonics or writing math facts on a poster and hanging it on the wall. These are not needful for all children.However, keep in mind that children will need LOTS of opportunities to add numbers to actually memorize the facts, so the memorization may not totally occur as your child works through this book. These same addition facts are further studied and used in the Math Mammoth Subtraction 1 book, and in all later math work since they are constantly used. At the latest, your child should memorize addition facts during second grade.
Math Mammoth Grade 8 Review Workbook
Math Mammoth Review Workbook, Grade 8 is intended to give students a thorough review of eighth grade math, following the topics listed in the Common Core Standards.The book has both topical as well as mixed (spiral) review worksheets, and includes both topical tests and a comprehensive end-of-year test. The tests can also be used as review worksheets, instead of tests.Please note: The content for this workbook is taken from Math Mammoth Grade 8 Complete Curriculum (Light Blue Series), so this book is redundant if you already own Math Mammoth grade 8.You can use this workbook for various purposes: for summer math practice, to keep a student from forgetting math skills during other break times, or to prepare students who are going into algebra 1.The topics reviewed in this workbook are: exponentsscientific notationgeometric transformationsvolumeangle relationshipslinear equationsintroduction to functionsgraphing linear equationssystems of linear equationssquare roots and the Pythagorean Theorembivariate dataPlease note this book does not contain lessons or instruction for the topics. It is not intended for initial teaching. It also will not work if the student needs to completely re-study these topics (the student has not learned the topics at all). For that purpose, please consider Math Mammoth Grade 8 Complete Curriculum, which has all the necessary instruction and lessons.
Once Upon a Prime
A New York Times Book Review Editors' Choice "Wide-ranging and thoroughly winning." --Jordan Ellenberg, The New York Times Book Review "An absolute joy to read!" --Steven Levitt, New York Times bestselling author of Freakonomics For fans of Seven Brief Lessons in Physics, an exploration of the many ways mathematics can transform our understanding of literature and vice versa, by the first woman to hold England's oldest mathematical chair. We often think of mathematics and literature as polar opposites. But what if, instead, they were fundamentally linked? In her clear, insightful, laugh-out-loud funny debut, Once Upon a Prime, Professor Sarah Hart shows us the myriad connections between math and literature, and how understanding those connections can enhance our enjoyment of both. Did you know, for instance, that Moby-Dick is full of sophisticated geometry? That James Joyce's stream-of-consciousness novels are deliberately checkered with mathematical references? That George Eliot was obsessed with statistics? That Jurassic Park is undergirded by fractal patterns? That Sir Arthur Conan Doyle and Chimamanda Ngozi Adichie wrote mathematician characters? From sonnets to fairytales to experimental French literature, Professor Hart shows how math and literature are complementary parts of the same quest, to understand human life and our place in the universe. As the first woman to hold England's oldest mathematical chair, Professor Hart is the ideal tour guide, taking us on an unforgettable journey through the books we thought we knew, revealing new layers of beauty and wonder. As she promises, you're going to need a bigger bookcase.
Math Mammoth Introduction to Fractions
Math Mammoth Introduction to Fractions contains lessons for fraction arithmetic for grades 1-4. This material does not include division or multiplication of fractions, nor adding unlike fractions, which are topics for 5th and 6th grades. Also, most of the material here is for 3rd and 4th grades, and only a few lessons are meant for grades 1-2.The topics covered are on a simple level, constantly illustrated with visual models, and with small denominators. The presentation avoids spelling out specific rules for manipulating fractions, but instead relies on the usage of pictures on a very concrete level. Children easily confuse the various rules for fraction arithmetic, because there are so many. There is a place for the rules, as shortcuts for ideas that are already understood, but we do not start with them. The goal is to let the big ideas sink in conceptually first, followed by some shortcuts.The topics covered areone half and one fourthconcept of a fractionconcept of a mixed numbercomparing fractionsequivalent fractionsadding and subtracting like fractionsadding and subtracting mixed numbers with like fractional partsadding one fraction that has tenths and another that has hundredths (such as 3/10 + 7/100)multiplying a fraction by a whole numberThe lessons are organized by topic, not by increasing difficulty. For reference, 1st grade students study only the concept of one half and one fourth. In 2nd grade, they study the concept of a fraction and optionally the easiest (first) lesson on comparing fractions. In 3rd grade, students study the concept of a fraction, fractions on a number line, comparing fractions, and equivalent fractions. Then in 4th, they study mixed numbers, comparing fractions, equivalent fractions, adding and subtracting fractions and mixed numbers, and multiplying fractions by whole numbers.The answers are appended.
Math Mammoth Fractions 1
Math Mammoth Fractions 1 is the first book of two that cover all aspects of fraction arithmetic. This book covers the concepts of fraction and mixed number, equivalent fractions, adding and subtracting like and unlike fractions, adding and subtracting mixed numbers, comparing fractions, and measuring in inches. The book Math Mammoth Fractions 2 covers the rest of fraction arithmetic: simplifying fractions and multiplication and division of fractions.Studying fractions involves lots of rules, and many students learn them only mechanically, not really understanding the underlying concepts and principles. Students then end up making lots of mistakes because they confuse the different rules, and either apply the wrong one or apply the right rule but don't remember it quite right. All this can make students even fear fractions.To avoid that, we use the visual model of a pie divided into slices all the way through the book. It is a natural model because a circle can be divided into any number of circle sectors (slices). When students work with this model from lesson to lesson, they will eventually start seeing these pies in their mind. This, in turn, gives them the ability to do many of the easier fraction calculations mentally. It also enables students to really UNDERSTAND these concepts, and not just learn mechanical rules.You are welcome to use manipulatives along with the book; however the visual pie model is probably sufficient for most students in the fifth grade level. I have also included (in the appendix) printable cut-outs for fractions from halves to twelfths.This workbook is most suitable for 5th grade. However, it is not tied to any grade level and does not mention anything about grade levels (except in the introduction), and so it can be used in various settings. It is also very good for remedial teaching or for students with learning challenges because of its heavy usage of visual models.The answers are appended.
Probability and Statistics
Probability and Statistics: Theory and Exercises is a textbook focused on practical examples of probability theory and statistics, with the goal of giving readers a thorough understanding of mathematical relationships in these subjects. The book is designed for basic courses in probability and statistics, and is aimed primarily at non-specialists and beginner level students. The book is divided into 2 sections, respectively. Probability: Includes a primer on set theory, basic probability theory definitions and calculations, combinatorial analysis, random variables and distribution laws Statistics: Covers basic concepts of descriptive statistics Key features - Simple, clear language for easy comprehension of key concepts - Carefully chosen exercises with solutions for self-learning - Over 40 Illustrations for clear explanations - References for further reading and tutorials
Tensor Analysis
Tensor calculus is a prerequisite for many tasks in physics and engineering. This book introduces the symbolic and the index notation side by side and offers easy access to techniques in the field by focusing on algorithms in index notation. It explains the required algebraic tools and contains numerous exercises with answers, making it suitable for self study for students and researchers in areas such as solid mechanics, fluid mechanics, and electrodynamics. ContentsAlgebraic ToolsTensor Analysis in Symbolic Notation and in Cartesian CoordinatesAlgebra of Second Order TensorsTensor Analysis in Curvilinear CoordinatesRepresentation of Tensor FunctionsAppendices: Solutions to the Problems; Cylindrical Coordinates and Spherical Coordinates
