Combinatorics and Optimization
This degree option differs from the BA in Mathematics without an option only in the Option Requirements.
Bachelor of Arts  Mathematics; Combinatorics & Optimization Option
College Humanities & Sciences
Catalog Year: 20142015
Degree Specific Credits: 67
Required Cumulative GPA: 2.0
Note: The degree specific credits are much lower for doublemajors and for students completing an additional minor (in another subject): 41 credits for students completing a second major, and 46 credits for students completing a minor.
Mathematics Core Courses
Calculus I
Rule: Take 1 of the following 2 courses.
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M 171  Calculus I
Offered autumn and spring. Prereq., M 122 or 151 or ALEKS placement >= 5. Differential calculus, including limits, continuous functions, Intermediate Value Theorem, tangents, linear approximation, inverse functions, implicit differentiation, extreme values and the Mean Value Theorem. Integral Calculus including antiderivatives, definite integrals, and the Fundamental Theorem of Calculus.

4 Credits 
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M 181  Honors Calculus I
Offered autumn. Prereq., consent of instr. Coreq., Honors Calculus Seminar, a section of M 294. Honors version of M 171.

4 Credits 
Minimum Required Grade: C  4 Total Credits Required 
Calculus II
Rule: Take 1 of the following 2 courses.
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M 172  Calculus II
Offered autumn and spring. Prereq., M 171 or 181. Techniques of Integration. Area computations. Improper integrals. Infinite series and various convergence tests. Power series. Taylor's Formula. Polar coordinates. Parametric curves.

4 Credits 
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M 182  Honors Calculus II
Offered spring. Prereq., M 181 or consent of instr. Coreq., Honors Calculus Seminar, a section of M 294. Honors version of M 172.

4 Credits 
Minimum Required Grade: C  4 Total Credits Required 
Other Mathematics Core Courses
Rule: Take all of the following courses.
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M 210  Intro to Mathematical Software
Offered spring. Prereq., one of M 162, 171, or 181, or consent of instr. Software packages useful for doing and writing mathematics. Introduction to a computer algebra system (such as Maple or Mathematica), a numerical package (such as MATLAB or R), and elementary programming. Writing and communicating mathematics using the mathematical typesetting system LaTeX.

3 Credits 
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M 221  Introduction to Linear Algebra
Offered autumn and spring. Prereq., M 172 or 182. Vectors in the plane and space, systems of linear equations and Gauss–Jordan elimination, matrices, determinants, eigenvalues and eigenvectors, vector spaces, linear transformations. Calculators and/or computers used where appropriate.

4 Credits 
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M 273  Multivariable Calculus
Offered autumn and spring. Prereq., M 172 or 182. Calculus of functions of several variables; differentiation and elementary integration. Vectors in the plane and space.

4 Credits 
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M 300  Undergraduate Mathematics Sem
(R–6) Offered every semester. Prereq., M 171 or 181. Discussion seminar focused on topics and issues of interest to students in the mathematical sciences.

1 Credits 
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M 307  Intro to Abstract Mathematics
Offered autumn and spring. Prereq., M 172 or 182. Designed to prepare students for upper–division proof–based mathematics courses. Topics include proof techniques, logic, sets, relations, functions and axiomatic methods. Students planning to take both M 221 and 307 are encouraged to take M 221 first.

3 Credits 
Minimum Required Grade: C  15 Total Credits Required 
UpperDivision Mathematics Requirement
Rule: Take 23 credits in this category.
Note: (1) Students completing a minor (in another subject) need take only 20 credits.
(2) Students completing a second major need take only 18 credits.
UpperDivision Elective Courses
Rule: Take 7 courses from the following list; at least 3 of them must be at the 400 level.
Note: (1) Students completing a minor (in another subject) or a second major need take only 6 courses (totaling 18 credits or more).
(2) Residency Requirement: At least 4 of the courses in this category must be taken at UMMissoula (only 3 if M 307 is taken at UMMissoula).
(3) Note that STAT 451 does not count toward this requirement.
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M 301  Math Technology for Teachers
Offered autumn. Prereq., M 221. Discrete and continuous mathematical models from a variety of disciplines using appropriate technology.

3 Credits 
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M 311  Ordinary Diff Equations/System
Offered autumn. Prereq., M 273. Ordinary differential equations. Systems of linear differential equations from a matrix viewpoint. Series solutions. Existence and uniqueness for initial value problems. Numerical methods. Stability and selected topics. M 317 computer lab recommended.

3 Credits 
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M 325  Discrete Mathematics
Offered spring. Prereq., M 171 and 225 or 307. Continuation of 225 and topics from graph theory, Boolean algebras, automata theory, coding theory, computability and formal languages.

3 Credits 
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M 326  Number Theory
Offered spring. Prereq., M 225 or 307. Congruences, Diophantine equations, properties of primes, quadratic residues, continued fractions, algebraic numbers.

3 Credits 
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M 361  Discrete Optimization
Offered spring. Prereq., one of M 162, 172 or 182 (221 or 225 recommended). Intended for non–mathematics majors as well as mathematics majors. Introduction to discrete optimization and modeling techniques with applications. Topics from combinatorics and graph theory, including enumeration, graph algorithms, matching problems and networks.

3 Credits 
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M 362  Linear Optimization
Offered autumn. Prereq., one of M 162, 172 or 182 (221 recommended). Coreq., M 363 recommended. Intended for non–mathematics majors as well as majors. Introduction to linear programming and modeling techniques with applications. Topics include the simplex method, duality, sensitivity analysis and network models.

3 Credits 
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M 381  Advanced Calculus I
Offered autumn . Prereq., M 307. Rigorous development of singlevariable calculus with formal proof. Functions, sequences, limits, continuity, differentiation, and integration.

3 Credits 
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M 412  Partial Differential Equations
Offered spring. Prereq., M 311. Fourier series, Sturm–Liouville and boundary value problems. Partial differential equations: Cauchy problems and the method of characteristics, separation of variables and Laplace transform methods. Numerical methods and selected topics. M 418 computer lab recommended.

3 Credits 
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M 414  Deterministic Models
Offered autumn or spring. Prereq., M 274 or 311 or consent of instr. Linear and nonlinear difference and differential equations: stability, phase–plane analysis, oscillatory behavior, limit cycles, and chaos. Eigenvalues and eigenfunctions. Emphasis on models in biology.

3 Credits 
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M 429  History of Mathematics
Offered spring. Prereq., M307. Historical study of the development of mathematics from the Egyptian and Babylonian eras to the 20th century.

3 Credits 
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M 431  Abstract Algebra I
Offered autumn. Prereq., M 221 and 307 or consent of instr. An introduction to modern ideas of algebra through the study of groups, rings, and fields.

4 Credits 
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M 432  Abstract Algebra II
Offered spring. Prereq., M 431. Continues the investigation of groups, rings, and fields begun in M 431. Further topics include vector spaces and field extensions.

4 Credits 
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M 439  Euclidean & NonEuclidean Geo
Offered autumn. Prereq., M 307; M 231 recommended. Euclidean geometry from a rigorous, axiomatic viewpoint and Non–Euclidean geometries chosen from Lobachevskian, projective, finite and Riemannian.

3 Credits 
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M 440  Numerical Analysis
Offered intermittently. Prereq., M 307, 311, one computer language. Error analysis; approximation and interpolation, numerical solution of linear and nonlinear equations, numerical integration of ordinary and partial differential equations.

4 Credits 
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M 445  Stat/Math/Comp Modeling
Offered autumn oddnumbered years. Prereq., consent of instr. An interdisciplinary course on the integration of statistical and dynamical models with applications to biological problems. Linear and nonlinear models, estimation, systems of ordinary differential equations, numerical integration, bootstrapping, MCMC methods. Intended both for students in mathematics and the natural sciences.

4 Credits 
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M 472  Intro to Complex Analysis
Offered spring. Prereq., M 273, M 307. Analytic functions, complex integration, singularities and application to contour integration, harmonic functions, spaces of analytic functions.

4 Credits 
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M 473  Introduction to Real Analysis
Offered autumn oddnumbered years. Prereq., M 273, M 307. Theory of metric spaces and point set topology, RiemannStieltjes integral, sequences and series of functions. StoneWeierstrass theorem, theorem of ArzelaAscoli, introduction to Lebesgue integration.

4 Credits 
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M 485  Graph Theory
Offered autumn. Prereq., M 325, or M 307 and M 361, or consent of instr. Theory and applications of graphs. Topics chosen from trees, matchings, connectivity, coloring, planarity, Ramsey theory, random graphs, combinatorial designs and matroid theory.

3 Credits 
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STAT 341  Intro to Probability and Stat
Offered autumn and spring. Prereq., one of M 162, 172 or 182. Probability, probability models and simulation, random variables, density functions, special distributions, and a brief survey of estimation and hypothesis testing. Computer use integrated throughout.

3 Credits 
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STAT 421  Probability Theory
Offered autumn. Prereq., M 273 and STAT 341 or consent of instr. An introduction to probability, random variables and their probability distributions, estimation and hypothesis testing. This course is the foundation on which more advanced statistics courses build.

3 Credits 
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STAT 422  Mathematical Statistics
Offered spring. Prereq., STAT 421. Continuation of 421.

3 Credits 
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STAT 452  Statistical Methods II
Offered spring. Prereq., STAT 451. Continuation of STAT 451. May not be counted toward a major in mathematics. Multiple regression, experimental design, analysis of variance, other statistical models.

3 Credits 
Minimum Required Grade: C  21 or more Total Credits Required 
UpperDivision Elective Computer Labs
Rule: Computer labs from the following list are optional; if taken, they count toward the total number of credits required for the UpperDivision Mathematics Requirement.
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M 317  ODE Computer Lab
Offered autumn. Coreq., M 311 or consent of instr. Intended primarily for student in M 311.

1 Credits 
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M 363  Linear Optimization Lab
Offered autumn. Coreq., M 362. Introduction to linear optimization software.

1 Credits 
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M 418  PDE Computer Lab
Offered spring. Coreq., M 412 or consent of instr. Intended primarily for students in M 412.

1 Credits 
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STAT 457  Computer Data Analysis I
Offered autumn. Coreq., STAT 451 or consent of instr. An introduction to software for doing statistical analyses. Intended primarily for students in STAT 451.

1 Credits 
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STAT 458  Computer Data Analysis II
Offered spring. Coreq., STAT 452 or consent of instr. Continuation of STAT 457. Intended primarily for students in STAT 452.

1 Credits 
Minimum Required Grade: C  05 Total Credits Required 
Science Requirement
Rule: Take 18 credits in at most 3 sciences selected from astronomy, biology, chemistry, computer science, economics, forestry, geosciences, and physics.
Note: (1) Students completing a minor (in another subject) or a second major are exempt from this requirement.
(2) Note that 'biology' includes courses with prefixes BIOL, MICB, BIOB, BIOE, BIOH, BIOM, and BIOO.
UpperDivision Writing Requirement
Rule: Take 1 of the following 2 courses, or any other approved General Education upperdivision writing course.
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M 429  History of Mathematics
Offered spring. Prereq., M307. Historical study of the development of mathematics from the Egyptian and Babylonian eras to the 20th century.

3 Credits 
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M 499  Senior Thesis
(R–12) Offered autumn and spring. Prereq., consent of instr. Senior thesis for mathematics majors and/or Watkins Scholars.

1 To 12 Credits 
Minimum Required Grade: C  3 Total Credits Required 
GPA Requirement
Note: (1) A cumulative GPA of 2.0 is required for all courses used to fulfill major requirements.
(2) In addition, a cumulative GPA of 2.0 is required for all mathematical sciences courses used to fulfill major requirements. (Mathematical sciences courses are those with a prefix of M or STAT.)
Foreign Language/Computer Science Requirement
Rule: Either complete the General Education Requirement "Group III: Modern and Classical Language" (not the symbolic systems exception), or take one course from the following list.
Note: Students completing a second major are exempt from this requirement.
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CSCI 100  Intro to Programming
Offered autumn and spring. Elementary programming techniques using the Visual BASIC programming language. A wide range of primarily nonmathematical programs will be written by the student and run on a computer. (Two hours independent lab per week.) Credit not allowed for both CSCI 100 and CSCI 110.

3 Credits 
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CSCI 135  Fund of Computer Science I
Offered autumn and spring. Prereq., computer programming experience in a language such as BASIC, Pascal, C, etc.; coreq., M 095 or consent of instr. CSCI 104 highly recommended as prereq. or coreq. Fundamental computer science concepts using the high level structured programming language, Java.

3 Credits 
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CSCI 136  Fund of Computer Science II
Offered autumn and spring. Prereq., CSCI 135; coreq., M 115 or M 151 or consent of instr. Continuation of CSCI 135. Survey of computer science topics including recursion, algorithms, basic data structures, operating systems, artificial intelligence, graphics, user interfaces, and social and ethical implications of computing.

3 Credits 
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CSCI 250  Computer Mdlng/Science Majors
Offered autumn. Prereq., basic computer and spreadsheet literacy; coreq., M 162 or 171. An introduction to computer modeling in the sciences using spreadsheets and a programming language. Integrates principles of math, computer science and science. A student can take at most one of CSCI 172, CSCI 250, CRT 280, and CRT 281 for credit.

3 Credits 
Minimum Required Grade: C  3 Total Credits Required 
Requirements for the Combinatorics & Optimization Option
Combinatorics & Optimization Option: Core Courses
Rule: Take all of the following courses.
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M 361  Discrete Optimization
Offered spring. Prereq., one of M 162, 172 or 182 (221 or 225 recommended). Intended for non–mathematics majors as well as mathematics majors. Introduction to discrete optimization and modeling techniques with applications. Topics from combinatorics and graph theory, including enumeration, graph algorithms, matching problems and networks.

3 Credits 
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M 362  Linear Optimization
Offered autumn. Prereq., one of M 162, 172 or 182 (221 recommended). Coreq., M 363 recommended. Intended for non–mathematics majors as well as majors. Introduction to linear programming and modeling techniques with applications. Topics include the simplex method, duality, sensitivity analysis and network models.

3 Credits 
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M 485  Graph Theory
Offered autumn. Prereq., M 325, or M 307 and M 361, or consent of instr. Theory and applications of graphs. Topics chosen from trees, matchings, connectivity, coloring, planarity, Ramsey theory, random graphs, combinatorial designs and matroid theory.

3 Credits 
Minimum Required Grade: C  9 Total Credits Required 
Combinatorics & Optimization Option: Elective Courses
Rule: Take 1 of the following 2 courses.
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CSCI 332  Design/Analysis of Algorithms
Offered spring. Prereq., CSCI 232 and M 225 or consent of instr. Algorithm design, analysis, and correctness. Commonly used algorithms including searching and sorting, string search, dynamic programming, branch and bound, graph algorithms, and parallel algorithms. Introduction to NPcomplete problems.

3 Credits 
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M 414  Deterministic Models
Offered autumn or spring. Prereq., M 274 or 311 or consent of instr. Linear and nonlinear difference and differential equations: stability, phase–plane analysis, oscillatory behavior, limit cycles, and chaos. Eigenvalues and eigenfunctions. Emphasis on models in biology.

3 Credits 
Show Description 
M 440  Numerical Analysis
Offered intermittently. Prereq., M 307, 311, one computer language. Error analysis; approximation and interpolation, numerical solution of linear and nonlinear equations, numerical integration of ordinary and partial differential equations.

4 Credits 
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STAT 341  Intro to Probability and Stat
Offered autumn and spring. Prereq., one of M 162, 172 or 182. Probability, probability models and simulation, random variables, density functions, special distributions, and a brief survey of estimation and hypothesis testing. Computer use integrated throughout.

3 Credits 
Minimum Required Grade: C  34 Total Credits Required 