We will develop in some detail algebraic tools such as the theory of Dedekind domains, discrete valuation rings, p-adic numbers, completions, etc., and also some analytic tools such as zeta functions. We will also discuss applications of analytic methods to classical results such as the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions.
This course is designed to give an introduction to the representation theory of finite and compact groups. It will start with the representations of finite groups over the complex numbers. In particular, we will discuss how finite-dimensional representations break up into sums of irreducible representations, Schur's lemma, the group algebra, character theory, induced representation and Frobenius reciprocity. We will give many examples, culminating in the representation theory of the symmetric groups and the remarkable combinatorics associated with it. We will then turn to the representation theory of compact groups, which is formally very similar to the finite group situation. The course will end with a discussion of the representation theory of the unitary groups and its close relationship with the representation theory of the symmetric groups.
Unit 1 Algebra Basics Homework 1 The Real Numbers
This course will serve as an introduction to mapping class groups of surfaces. Mapping class groups are a fundamental object of study in topology, as the automorphism groups of 2-manifolds, but also arise naturally in many other fields, such as complex analysis and algebraic geometry. We will survey some basics such as generators and relations for the mapping class group, subgroups important in 3-manifold theory such as the Torelli group and the handlebody subgroup, and representations of mapping class groups. As time permits, and according to the interests of the class, we will also discuss related groups in geometric group theory, Teichmüller theory, and associated combinatorial structures such as the curve complex, among other topics.
1) Number sets and real functions.Natural numbers, integers, rational numbers, real numbers.Set theory.Functions and Cartesian representation.Monotone functions.Linear functions.Power, exponential, logarithm.Trigonometric functions.Mathematical induction. 2) Sequences. Limits. Operations with limits.Indeterminate forms.Napier's number e.3) Continuous functions.Limits of functions.Definition of continuous function: examples and properties.Weierstrass theorem.Zeros of continuous functions. 4) Differentiation. Geometrical meaning of the derivative.Differentiation rules.Higher order derivatives.5) The fundamental theorems of differential calculus.Rolle, Lagrange, Cauchy theorems and applications.Points of growth, of degression, maximum and minimum of afunction.Convex functions.Taylor's formula. 6) Theory of Riemann integration. Integral of a continuous function.Definite Integrals: geometric interpretation. The fundamental theorem of calculus. Fundamental formula of calculus. Indefinite integrals. Integration sum decomposition.Integration by parts. Integration by substitution.
This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Conner and Aaron are working on their homework together to find the distance between two numbers, $a$ and $b$, on a number line. Conner count the units...
Students use number lines to solve addition and subtraction problems involving positive and negative fractions and decimals. They then verify that the same rules they found for integers apply to fractions and decimals as well. Finally, they solve some real-world problems.Key ConceptsThe first four lessons of this unit focused on adding and subtracting integers. Using only integers made it easier for students to create models and visualize the addition and subtraction process. In this lesson, those concepts are extended to positive and negative fractions and decimals. Students will see that the number line model and rules work for these numbers as well.Note that rational number will be formally defined in Lesson 15.Goals and Learning ObjectivesExtend models and rules for adding and subtracting integers to positive and negative fractions and decimals.Solve real-world problems involving addition and subtraction of positive and negative fractions and decimals.
In CK-12 Middle School Math Concepts Grade 8, the learning content is divided into concepts. Each concept is complete and whole providing focused learning on an indicated objective. Theme-based concepts provide students with experiences that integrate the content of each concept. Students are given opportunities to practice the skills of each concept through real-world situations, examples, guided practice and explore more practice. There are also video links provided to give students an audio/visual way of connecting with the content.
An introductory course in problem solving and computer programming using the programming language Java. The course focuses on the fundamental concepts of problem solving and the techniques associated with the development of algorithms and their implementation as computer programs. This course or its equivalent is required for all additional courses in CMPSCI. Three hours of lecture/recitation per week. About 6 programming problems are assigned. In addition there are assigned homework problems, a midterm exam and a final. No computer science prerequisite, although high school algebra and basic math skills (e.g. R1) are assumed. Use of computer is required. 3 credits.
Lecture, lab. The architecture and machine-level operations of modern computers at the logic, component, and system levels: topics include binary arithmetic, logic gates, Boolean algebra, arithmetic-logic unit, control unit, system bus, memory, addressing modes, interrupts, input-output, floating point arithmetic, and virtual memory. Simple assembly language is used to explore how common computational tasks are accomplished by a computer. Three hours of lecture and one or two lab sessions per week. Laboratory exercises, homework exercises, two midterm exams, a final exam, and occasional quizzes are required. Prerequisite: CMPSCI 187 or equivalent. 4 credits.
In the CMPSCI 391F seminar "HTML for Poets" we cover HTML in the first half, and Javascript during the second half. HTML, Javascript, and Java, form an extremely powerful trio of platform independent languages that allow practically anyone to propagateideas, create business and retrieve massive amounts of information from the World Wide Web through the creation of Web sites and the use of Web browsers. Throughout the seminar HTML and Javascript are taught as tools to implement elements of design to realize web sites whose content needs to be communicated in a professional and appealing way. Possible political and philosophical consequences are also discussed. Students are required to use HTML and Javascript constructs learned in the seminar to create/improve a Web page of their own. Students are also required to review a professional Web site each week and analyze its content, form, appeal and tools used in its construction. A final project is also required. A large number of working examplesare posted on the Web for student perusal. TAs for the Seminar must know and be able to explain the following material: Intro: Overview of the World Wide Web, Browsers and Servers. Basic HTML: Distributing Information with HTML, building blocks of HTML, creating an HTML document, linking HTML documents, graphics and Images. Advanced HTML: Client-side Imagemaps, Tables and Math, Frames, Embedded objects, Sound, MIDI, Animation, Forms and CGI scripts, Dynamic documents. Javascript: Javascript as an event driven, object-based, scripting language. Javascript versus Java. Events and event handlers. Programming basics: Variables, identifiers, types, scope of variables. Expressions, operators: arithmetic, string, logical, assignment, comparison. Statements: variable declaration, comments, conditional statements, loops, object manipulation, Functions. Objects: Objects, classes, inheritance. Objects built in Javascript: navigator, location, buttons, reset, select, submit, text and more. Arrays:built in Arrays, creating Arrays and filling them with data. Methods, properties and functions: math methods, string methods, date, window methods, blur, focus, select and more. Knowledge of Unix and mail filters is also required. 3 credits.
This course explores the basic problems in the translation of programming languages focusing on theory and common implementation techniques for compiling traditional block structured programming languages to produce assembly or object code for typical machines. The course involves a substantial laboratory project in which the student constructs a working compiler for a considerable subset of a realistic programming language. The lectures are augmented by an optional laboratory section that covers details of the programming language used to build the compiler, the operating system, the source language, and various tools. Use of computer required. Text: Crafting a Compiler in C, by Fischer and LeBlanc. Prerequisite: CMPSCI 250, 377. CMPSCI 491A(410) offers an honors section (CMPSCI H0x), the requirements for which are completed by fulfilling the additional assignments and lectures for CMPSCI 610 (one extra hour of discussion section per week, plus some additional homework problems and the implementation of more features in the project). 3 credits.
Artificial neural networks, also called connectionist systems, are networks of relatively simple processing elements that mimic some of the properties of biological neurons. By studying these abstract nervous systems, researchers hope to improve our understanding of how real nervous systems function as well as to advance the field of artificial intelligence. This course provides an in-depth treatment of the central artificial neural network methods, with an emphasis on their computational capabilities. Prerequisites: basic calculus, linear algebra, basic computer skills. 3 credits. 2ff7e9595c
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