The emergence of Western science: the systematization of natural knowledge in the ancient world, the transmission of the classical legacy to the Latin West, and the revolt from classical thought during the scientific revolution. Examines scientific concepts in light of their cultural and historical contexts.

Recent results in cryptography and interactive proofs. Lectures by instructor, invited speakers, and students. Alternate years. The topics covered in this course include interactive proofs, zero-knowledge proofs, zero-knowledge proofs of knowledge, non-interactive zero-knowledge proofs, secure protocols, two-party secure computation, multiparty secure computation, and chosen-ciphertext security.

Differential calculus in one and several dimensions. Java applets and spreadsheet assignments. Vector algebra in 3D, vector- valued functions, gradient, divergence and curl, Taylor series, numerical methods and applications. Given in the first half of the first term. However, those wishing credit for 18.013A only, must attend the entire semester. Prerequisites: a year of high school calculus or the equivalent, with a score of 4 or 5 on the AB, or the AB portion of the BC, Calculus test, or an equivalent score on a standard international exam, or a passing grade on the first half of the 18.01 Advanced Standing exam.

Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression.

Continues 18.100, in the direction of manifolds and global analysis. Differentiable maps, inverse and implicit function theorems, n-dimensional Riemann integral, change of variables in multiple integrals, manifolds, differential forms, n-dimensional version of Stokes' theorem. 18.901 helpful but not required.

This course analyzes combinatorial problems and methods for their solution. Topics include: enumeration, generating functions, recurrence relations, construction of bijections, introduction to graph theory, network algorithms, and extremal combinatorics.

The course consists of a sampling of topics from algebraic combinatorics. The topics include the matrix-tree theorem and other applications of linear algebra, applications of commutative and exterior algebra to counting faces of simplicial complexes, and applications of algebra to tilings.

Introduction to discrete and computational geometry. Topics covered: planar graphs, geometric graphs, the theory of crossings, extremal graph theory, arrangements of curves and points in the plane (mainly pseudolines and pseudocircles), problems involving distances, Gallai-Sylvester-type problems, Davenport-Schinzel sequences. Emphasis on teaching methods in combinatorial geometry. Many results presented are recent, and include open problems.

This graduate-level course focuses on one-dimensional nonparametric statistics developed mainly from around 1945 and deals with order statistics and ranks, allowing very general distributions. For multidimensional nonparametric statistics, an early approach was to choose a fixed coordinate system and work with order statistics and ranks in each coordinate. A more modern method, to be followed in this course, is to look for rotationally or affine invariant procedures. These can be based on empirical processes as in computer learning theory. Robustness, which developed mainly from around 1964, provides methods that are resistant to errors or outliers in the data, which can be arbitrarily large. Nonparametric methods tend to be robust.

In this second term of Algebraic Topology, the topics covered include fibrations, homotopy groups, the Hurewicz theorem, vector bundles, characteristic classes, cobordism, and possible further topics at the discretion of the instructor.

This course is a self-contained introduction to statistics with economic applications. Elements of probability theory, sampling theory, statistical estimation, regression analysis, and hypothesis testing. It uses elementary econometrics and other applications of statistical tools to economic data. It also provides a solid foundation in probability and statistics for economists and other social scientists.

This course treats various methods to design and analyze datastructures and algorithms for a wide range of problems. The most important new datastructure treated is the graph, and the general methods introduced are: greedy algorithms, divide and conquer, dynamic programming and network flow algorithms. These general methods are explained by a number of concrete examples, such as simple scheduling algorithms, Dijkstra, Ford-Fulkerson, minimum spanning tree, closest-pair-of-points, knapsack, and Bellman-Ford. Throughout this course there is significant attention to proving the correctness of the discussed algorithms. All material for this course is in English. The recorded lectures, however, are in Dutch.

This course covers the basic algebra and technological tools used in the social, physical and life sciences to analyze quantitative information. The emphasis is on real world, open-ended problems that involve reading, writing, calculating, synthesizing, and clearly reporting results. Topics include descriptive statistics, linear, and exponential models. Technology used in the course includes computers (spreadsheets, Internet) and graphing calculators.

Modelling is about understanding the nature: our world, ourselves and our work. Everything that we observe has a cause (typically several) and has the effect thereof. The heart of modelling lies in identifying, understanding and quantifying these cause-and-effect relationships.

A model can be treated as a (selective) representation of a system. We create the model by defining a mapping from the system space to the model space, thus we can map system state and behaviour to model state and behaviour. By defining the inverse mapping, we may map results from the study of the model back to the system. In this course, using an overarching modelling paradigm, students will become familiar with several instances of modelling, e.g., mechanics, thermal dynamics, fluid mechanics, etc.

This free online textbook is a one semester course in basic analysis. These were my lecture notes for teaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in fall 2009. The course is a first course in mathematical analysis aimed at students who do not necessarily wish to continue a graduate study in mathematics. A Sample Darboux sums prerequisite for the course is a basic proof course. The course does not cover topics such as metric spaces, which a more advanced course would. It should be possible to use these notes for a beginning of a more advanced course, but further material should be added.

t is increasingly clear that the shapes of reality – whether of the natural world, or of the built environment – are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself ‘tick’, and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside.

The Essence of Mathematics consists of a sequence of 270 problems – with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline.

The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works.

Learning and Understanding Mathematical Concepts in the Areas of Water Distribution and Water Treatment

Intermediate Microeconomics is a comprehensive microeconomic theory text that uses real world policy questions to motivate and illustrate the material in each chapter. Intermediate Microeconomics is an approachable yet rigorous textbook that covers the entire scope of traditional microeconomic theory and includes two mathematical approaches, allowing instructors to teach the material with or without calculus. With real-world policy topics as an entree into each subject, Intermediate Microeconomics will help students engage with the material and facilitate learning not only the concepts, but their importance and application as well.

The inspiration for this text grew out of a simple question that emerged over a number of years of teaching math to Middle School, High School and College students.

Practically speaking, what is the origin of a particular polynomial?

So much time is spent analyzing, factoring, simplifying and graphing polynomials that it is easy to lose sight of the fact that polynomials have a wealth of practical uses. Exploring the techniques of interpolating data allows us to view the development and birth of a polynomial. This text is focused on laying a foundation for understanding and applying several common forms of polynomial interpolation. The principal goals of the text are:

1). Breakdown the process of developing polynomials to demonstrate and give the student a feel for the process and meaning of developing estimates of the trend (s) a collection of data may represent.

2).Introduce basic matrix algebra to assist students with understanding the process without getting bogged down in purely manual calculations. Some manual calculations have been included, however, to assist with understanding the concept.

3).Assist students in building a basic foundation allowing them to add additional techniques, of which there are many, not covered in this text.

Prealgebra is designed to meet scope and sequence requirements for a one-semester prealgebra course. The book’s organization makes it easy to adapt to a variety of course syllabi. The text introduces the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics.

Table of Contents
1 Whole Numbers
2 The Language of Algebra
3 Integers
4 Fractions
5 Decimals
6 Percents
7 The Properties of Real Numbers
8 Solving Linear Equations
9 Math Models and Geometry
10 Polynomials
11 Graphs

Access also available here: https://openstax.org/details/books/prealgebra