Topics in mathematics that every educated person needs to know to process, evaluate, and understand the numerical and graphical information in our society. Applications of mathematics in problem solving, finance, probability, statistics, geometry, population growth. Note: This course does not cover the algebra and pre-calculus skills needed for calculus.
Students will learn how to read, understand, devise and communicate proofs of mathematical statements. A number of proof techniques (contrapositive, contradiction, and especially induction) will be emphasized. Topics to be discussed include set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), basic logic (truth tables, negation, quantifiers). Other topics will be included as time allows. Math 300 is designed to help students make the transition from calculus courses to the more theoretical junior-senior level mathematics courses.
a brief course in mathematical statistics solutions manual.zip
The goal of this course is to help students learn the language of rigorous mathematics.Students will learn how to read, understand, devise and communicate proofs of mathematical statements. A number of proof techniques (contrapositive, contradiction, and especially induction) will be emphasized. Topics to be discussed include set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), basic logic (truth tables, negation, quantifiers). Other topics will be included as time allows. Math 300 is designed to help students make the transition from calculus courses to the more theoretical junior-senior level mathematics courses.
While the mathematicians of the pre-internet age often spread their mathematical ideas within the community via written letters prior to publication, modern mathematical correspondence and exposition is rapidly facilitated by a variety of digital tools. Of great importance to the publishing process in mathematical sciences is the LaTeX markup language, used to typeset virtually all modern mathematical publications, even at the pre-print stage. In this course we will develop facility with LaTeX, and develop a variety of writing practices important to participation in the mathematical community. There will be regular written assignments completed in LaTeX, as well as collaborative writing assignments, owing to the importance of collaborative writing in mathematical research. Writing topics may include proofs, assignment creation, pre-professional writing (resumes/cover letters, research and teaching statements), expository writing for a general audience, recreational mathematics, and the history of mathematics. Short writing assignments on such topics will be assigned in response to assigned readings from a variety of accessible/provided sources. Towards the end of the semester groups will complete a research paper of an expository nature and craft a seminar style presentation. This course meets the junior year writing requirement.
Students will develop skills in writing, oral presentation, and teamwork. The first part of the course will focus on pre-professional skills, such as writing a resume, cover letter, and graduate school essay and preparing for interviews. Subsequent topics will include presenting mathematics to a general audience, the role of mathematics in society, mathematics education, and clear communication of mathematical content. The end of the term will be dedicated to the research process in mathematics and will include grant writing, research paper, and professional presentation.
In the year 2000 the Clay Institute listed seven (then) unsolved problems across all areas of mathematics considered the most challenging and important for the new millenium.So far only one of the problems, the Poincare Conjecture, has been solved by Perleman (who refused to collect the one million dollar prize stating that mathematics should never be done for money).In this course we shall focus on the, as of yet unsolved, Birch and Swinnerton-Dyer conjecture. This conjecture pertains to the behavior of integer/rational solutions to certain equations. The area of mathematics is number theory and in particular the study of Diophantine equations. We shall go far back in the history of such problems (from Egyptian, Babylonian and Greek mathematics through the Middle Ages into our millenium) and try to grasp some understanding of the issues in easy to handle cases (Pythagorean numbers, canon ball stacking etc). The course is structured around writing assignments on these topics which will be peer reviewed and/or graded by the instructor and the course TA. During the last quarter of the semester there will be group project presentations.All writing has to be done in the word processing system LaTex, which is the only word processing system capable of producing a professional layout. Templates and some basic tutorials will be provided.We shall NOT spend time on resume and job application writing, since there is ample opportunity to receive expert help from the career center (a representative of which will give a presentation in class).
This course is about how to write and use computer code to explore and solve problems in pure and applied mathematics. The first part of the course will be an introduction to programming in Python. The remainder of the course (and its goal) is to help students develop the skills to translate mathematical problems and solution techniques into algorithms and code. Students will work together on group projects with a variety applications throughout the curriculum.
This is a rigorous introduction to some topics in mathematics that underlie areas in computer science and computer engineering, including: graphs and trees, spanning trees, colorings and matchings, the pigeonhole principle, induction and recursion, generating functions, and (if time permits) combinatorial geometry. The course integrates mathematical theories with applications to concrete problems from other disciplines using discrete modeling techniques. Small student groups will be formed to investigate a modeling problem independently, and each group will report its findings to the class in a final presentation. Satisfies the Integrative Experience for BS-Math and BA-Math majors.
This course can be used to satisfy the UMass Integrative Experience requirement. The main goal of the class is to learn how to translate real-world situations into mathematical terms and use the model to predict, optimize and generally understand the original situation. The course material will concentrate on topics related to social sciences, such as voting and electoral systems. We will use a variety of mathematical techniques and objects, including networks.
This is an introduction to the history of mathematics from ancient civilizations to present day. Students will study major mathematical discoveries in their cultural, historical, and scientific contexts. This course explores how the study of mathematics evolved through time, and the ways of thinking of mathematicians of different eras - their breakthroughs and failures. Students will have an opportunity to integrate their knowledge of mathematical theories with material covered in General Education courses. Forms of evaluation will include a group presentation, class discussions, and a final paper. Satisfies the Integrative Experience requirement for BA-MATH and BS-MATH majors.
This course is an introduction to mathematical analysis. A rigorous treatment of the topics covered in calculus will be presented with a particular emphasis on proofs. Topics include: properties of real numbers, sequences and series, continuity, Riemann integral, differentiability, sequences of functions and uniform convergence.
Time-permitting, we will briefly examine numerical methods for partial differential equations and relevant implementation thereof, e.g., in Matlab, as well as some select examples of nonlinear partial differential equations and the traveling or standing wave solutions possible therein.
This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing financial instruments, or "derivatives." The central topic will be options, culminating in the Black-Scholes formula. The goal is to understand how the models derive from basic principles of economics, and to provide the necessary mathematical tools for their analysis.
A practical overview of computational methods used in science, statistics, industry, and machine learning. Topics will include: an introduction to python programming and software for scientific computing such as NumPy and LAPACK, numerical linear algebra, optimization and root-finding, approximation of functions by splines and trigonometric polynomials, and the Fast Fourier Transform. Applications may include regression problems in statistics, audio and image processing, and the calculation of properties of molecules. Homework will be assigned frequently. Each assignment will involve both mathematical theory and python programming. There will be no exams. Instead, each student will pursue an open-ended project related to a topic discussed in class.
This proof-based course covers the fundamentals of linear optimization and polytopes and the relationship between them. The course will give a rigorous treatment of the algorithms used in linear optimization. The topics covered in linear optimization are graphical methods to find optimal solutions in two and three dimensions, the simplex algorithm, duality and Farkas' lemma, variation of cost functions, an introduction to integer programming and Chvatal-Gomory cuts. The topics covered simultaneously in polytopes are two- and three-dimensional polytopes, f-vectors, equivalence of the vertex and hyperplane descriptions of polytopes, the Hirsch conjecture, the secondary polytope, and an introduction to counting lattice points of polytopes. 2ff7e9595c
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