# Department of Mathematical Sciences

#### Leonid Kalachev, Chair

Mathematics is studied both as a tool and for its own sake. Its usefulness in the sciences - physical, biological, social, behavioral, and environmental - and in decision-making processes is so established as to make it an indispensable part of many curricula.

Mathematics is chosen as a major area of study by individuals who find it challenging, fascinating, and beautiful. It is also appreciated by many who seek primarily to use mathematics as a tool.

A career in mathematics, except for teaching at the secondary level, generally requires a graduate degree as preparation. Careers include teaching, research, and the application of mathematics to diverse problems in institutions of higher learning, business, industry, and government.

The Bachelor of Arts, Master of Arts, and Doctor of Philosophy degrees are offered as well as a Bachelor of Science in Mathematical Sciences–Computer Science.

**High School Preparation: **For studying mathematics at the university level, it is recommended that the high school course work consist of four years of college-preparatory mathematics, including geometry, trigonometry, and college algebra or precalculus. A course in calculus or statistics is helpful, but not necessary. It is unusual to complete an undergraduate degree in mathematics in four years without the necessary background to take Calculus I (M 171) during the freshman year (preferably during the first semester at the university).

### Special Degree Requirements

Refer to graduation requirements listed previously in the catalog. See index.

#### Mathematics Requirements for B.A. Degree with a Major in Mathematical Sciences

To obtain a B.A. degree with a major in Mathematical Sciences, the required courses are M 171 or 181, 172 or 182 (MATH 152, 153), M 210 (except for students in the Mathematics Education option), M 221 (MATH 221), M 273 (MATH 251) (except for students in the Mathematics Education option), M 300 (MATH 300) (except for students in the Mathematics Education option), M 307 (MATH 305) and six additional courses from the following list (at least three of the six must be numbered 400 or above): M 301, 311, 325, 326, 361, 362, 381, 412, 414, 429, 431, 432, 439, 440, 445, 472, 473, 485 and STAT 341, 421, 422 (MATH 301, 311, 325, 326, 341, 351, 381, 382, 406, 412, 414, 421, 422, 431, 441, 442, 451, 452, 471, 475, 485). Four of the seven required 3- or 4-credit upper-division mathematical sciences courses must be taken from UM-Missoula. All mathematical sciences courses counted toward the major must be passed with a grade of C– or better and a 2.00 grade point average is required for these courses. In addition, if a special option is desired, the minimum requirements listed below for that option must be met. Additional courses should be chosen in consultation with a mathematics advisor.

### Requirements for the Special Options

#### Applied Mathematics Option

M 311, 412, 414 (MATH 311, 412, 414) and one of M 440 or 472 (MATH 452 or 471). (M 381 and M485 (MATH 485) are recommended.)

#### Combinatorics and Optimization Option

M 361, 362, 485 (MATH 381, 382, 485); and one course chosen from STAT 341 (MATH 341), M 414, 440 (MATH 414, 471), or CSCI 332 (CS 332).

#### Mathematics Education Option

M 301, 326, 429, 431, 439 (MATH 301, 326, 406, 421, 431), and STAT 341 (MATH 341); either M 273 (MATH 251) or one additional course chosen from the above list for the six-course requirement; and the completion of licensure requirements for teaching in secondary schools to include EDU 497 (C&I 430).

#### Pure Mathematics Option

Four courses chosen from M 381, M431, 432, 472, 473 (MATH 421, 422, 451, 452).

#### Statistics Option

STAT 341, 421, 422 (MATH 341, 441, 442). (Additional mathematics and statistics courses chosen with advisor.)

#### Major Requirements in Courses Outside Mathematics

- Except for students in the Mathematics Education option and for students presenting a second major within the University, students must either complete a two-semester language sequence as specified under "Group III: Modern and Classical Languages" in the General Education section of the Catalog, or take one course chosen from CSCI 100, 135, 136, 250 (CS 101, 131, 132, 177).
- All mathematics majors, except those selecting the mathematics education option, must complete 18 credits in at most three sciences selected from astronomy, biology, chemistry, computer science, economics, forestry, geosciences, and physics. Students selecting the mathematics education option must complete 12 credits in at most two sciences selected from astronomy, biology, chemistry, computer science, geosciences, and physics. (Note that ‘biology’ includes all courses with prefixes BIOL, MICB, BIOB, BIOE, BIOH, BIOM, and BIOO.) An alternative to the science requirement is for the student to present a minor or second major within the University, or for the student with a mathematics education option to complete an additional teaching minor or major.
- The upper–division writing requirement for Mathematical Sciences majors consists of: M 429 (MATH 406), or any other approved General Education upper–division Writing course, or a senior thesis (M 499 (MATH 499)).

#### Requirements for a B.S. Degree with a Combined Major in Mathematical Sciences–Computer Science

The purpose for the combined program is to provide a thorough background in both allied disciplines and to inculcate a deeper understanding of their goals and methods. A student must complete 60 credits in the two disciplines: 30 of these credits in mathematical sciences courses and 30 of these credits in computer science courses. A minimum grade of “C–” and a 2.0 grade point average is required in all courses which follow.

**The mathematical sciences requirements are:** M 171 (or 181), 172 (or 182), 221, 273, 307 (or 225) (MATH 152,153, 221, 251, 305 (or 225)), and twelve credits of mathematical sciences electives selected from the following list: M 311, 325, 326, 361, 362, 381, 412, 414, 429, 431, 432, 439, 440, 445, 472, 473, 485 and STAT 341, 421, 422, 451, 452 (MATH 311, 325, 326, 341, 351, 381, 382, 406, 412, 414, 421, 422, 431, 441, 442, 444, 445, 451, 452, 471, 475, 485).

**The computer science requirements are:** CSCI 106, 135-136 or 137, 205, 232, 332, 361 (CS 121, 131–132 or 133, 242, 241, 332, 281) and nine credits of CSCI (CS) electives selected from courses numbered 300 and above. A total of at most three of the nine credits of CSCI (CS) electives may be in CSCI 398 or 498 (CS 398 or 498).

The combined nine additional credits of computer science electives and twelve additional credits of mathematical sciences electives must include at least three 3– or 4–credit courses numbered 400 or above, with at least one chosen from each department (not including M 429 (MATH 406) and STAT 451, 452 (MATH 444 and 445)).

**Other requirements are:** One of the sequences BIOB 160N, 170N, 171N (BIOL 110N, 108N, 109N); CHMY 141N, 143N (CHEM 161N, 162N); or PHSX 215N-218N (PHYS 211N–214N). In addition, WRIT 222 (FOR 220), and either COMX 111A or COMX 242 (COMM 111A or COMM 242).

Each student plans a program in consultation with a computer science and a mathematical sciences advisor. Students planning to attend graduate school in computer science or the mathematical sciences should consult with their respective advisors.

The upper–division writing requirement is one of the following: CSCI 315E (CS 415E), M 429 (MATH 406), any other approved General Education upper–division writing course, or a senior thesis (CSCI 499 (CS 499) or M 499 (MATH 499)).

**Suggested Curricula:**

Applied Math–Scientific Programming: M 311, 412, 414 (MATH 311, 412, 414), and one course chosen from STAT 341 (MATH 341), M 381, M 473, 472, 440 (MATH 451, 452, 471). Three courses chosen from CSCI 460, 441, 477, 444 (CS 344, 446, 477, 486).

Combinatorics and Optimization–Artificial Intelligence: M 361, 362 (MATH 381, 382), and two courses chosen from M 325, 414, 485 (MATH 325, 414, 485) and STAT 341 (MATH 341); and CSCI 460, 446, and 447 (CS 344, 455, and 457).

Statistics–Machine Learning: STAT 341, 421 (MATH 341, 441), and two courses chosen from M 325, 362, 485 (MATH 325, 382, 485) and STAT 422 (MATH 442). Three courses chosen from CSCI 340, 446, 447, 451, and 444 (CS 365, 455, 457, 458, and 486).

Algebra–Analysis: M 381, M 431 (MATH 421), and two courses chosen from M 326, 432, 473, 472 (MATH 326, 422, 451, 452); CSCI 460, 426 (CS 344, 441), and one other course.

### Suggested Course of Study

First Year |
A |
S |

M 171-172 or 181-182 (MATH 152-153) Calculus I, II or Honors Calculus I, II |
4 |
4 |

M 210 Introduction to Mathematical Software |
- |
3 |

WRIT 101 (ENEX 101) Composition and other General Education Courses (including two sciences courses) |
12 |
9 |

| 16 |
16 |

Second Year |
A |
S |

M 221 (MATH 221) Introduction to Linear Algebra |
4 |
– |

M 273 (MATH 251) Multivariable Calculus |
4 |
– |

M 307 (MATH 305) Introduction to Abstract Mathematics |
– |
3 |

General Education courses, additional science courses and electives |
9 |
13 |

| 17 |
16 |

#### Requirements for a Minor

To earn a minor in mathematics the student must earn 23 credits in M, MATH, or STAT courses listed in a UM-Missoula Catalog (or in transfer courses equivalent to such courses). M courses must be numbered 115 or higher (excluding M 118), and MATH courses must be numbered 111 or higher. Courses must include: (a) one of M 162 or 172 or 182 (MATH 150 or 153), and (b) at least three 3– or 4– credit courses at the 300 level or above. M 172 or 182 (MATH 153) (Calculus II) is recommended since it is a prerequisite for many upper–division mathematics courses. All courses counted toward the minor must be passed with a grade of C– or better and a 2.00 grade point average is required for these courses. A handout with detailed advice for math minors, including suggested curricula, is available on the math department’s home page.

**Mathematics Education Minor:** For a teaching minor endorsement in the field of mathematics, a student must complete M 171-172, 221, 301, 307, 326, 439 (MATH 152-153, 221, 301, 305, 326, 431), and STAT 341 (MATH 341). Students also must complete (EDU 497 (C&I 430), gain admission to Teacher Education Programs and meet the requirements for teaching licensure (see the Department of Curriculum and Instruction section of this catalog). All courses counted toward the minor must be passed with a letter grade of C– or better.

### Courses (click Course Descriptions link below to navigate to course description information in CyberBear)

Unless the student has prior written approval of the Mathematical Sciences Department, credit is not allowed for any mathematics course that is a prerequisite for a mathematics course for which credit has already been earned. Students receiving transfer or Advanced Placement credit for STAT 216 (MATH 241) may take M 115 (MATH 117) for credit.

See the Missoula College section for Introductory Algebra, M 090 (MAT 005), and Intermediate Algebra, M 095 (MAT 100).

### Below is a complete listing of courses taught by the Mathematical Sciences Department

R- before the course description indicates the course may be repeated for credit to the maximum indicated after the R. Credits beyond this maximum do not count toward a degree.

104, 105, 115, 118, 121, 122, 135, 136, 151, 162, 171, 172, 181, 182, 191, 210, 221, 225, 231, 273, 274, 291, 292, 294, 300, 301, 307, 311, 317, 325, 326, 361, 362, 363, 381, 391, 392, 394, 398, 412, 414, 418, 429, 431, 432, 439, 440, 445, 472, 473, 485, 490, 491, 492, 494, 498, 499, 500, 501, 504, 506, 510, 511, 512, 514, 520, 521, 522, 524, 526, 530, 531, 532, 550, 551, 555, 564, 570, 572, 573, 574, 578, 581, 582, 584, 593, 595, 596, 597, 598, 599, 600, 602, 605, 606, 609, 610, 620, 630, 650, 670, 680, 690, 691, 694, 699

216, 341, 421, 422, 451, 452, 457, 458, 540, 541, 542, 543, 544, 545, 547, 549, 640

### Faculty

#### Professors

Jonathan Graham, Ph.D., North Carolina State University, 1995

James J. Hirstein, Ed.D., University of Georgia, 1976

Leonid Kalachev, Ph.D., Moscow State University, 1987 (Chair)

P. Mark Kayll, Ph.D., Rutgers University, 1994

Jennifer McNulty, Ph.D., University of North Carolina at Chapel Hill, 1993

D. George McRae, Ph.D., University of Washington, 1967

David A. Patterson, Ph.D., University of Iowa, 1984

Bharath Sriraman, Ph.D., Northern Illinois University, 2002

Emily Stone, Ph.D., Cornell University, 1989

Karel M. Stroethoff, Ph.D., Michigan State University, 1987

Thomas Tonev, Ph.D., Moscow State University, 1973

Nikolaus Vonessen, Ph.D., Massachusetts Institute of Technology, 1988

#### Associate Professors

John Bardsley, Ph.D., Montana State University, 2002

Eric Chesebro, Ph.D., University of Texas at Austin, 2006

Jennifer Halfpap, Ph.D., University of Wisconsin, 2005

Solomon Harrar, Ph.D., Bowling Green State University, 2004

Kelly McKinnie, Ph.D., University of Texas at Austin, 2006

Greg St. George, Ph.D., The University of Montana, 1989

Brian Steele, Ph.D., The University of Montana, 1995

Ke Wu, Ph.D., University of Minnesota, 2008

#### Assistant Professors

Cory Palmer, Ph.D., Central European University, 2008

Matt Roscoe, Ph.D., University of Montana, 2011

#### Lecturers

Lauren Fern, M.S., Northern Illinois University, 1994

Cindy Leary, M.A., The University of Montana, 2006

Regina Souza, Ph.D., Massachusetts Institute of Technology, 1990

Bonnie Spence, M.A., University of Tulsa, 1991

#### Emeritus Professors

William R. Ballard, Ph.D., University of Chicago, 1957

Richard W. Billstein, Ed.D., The University of Montana, 1972

Charles A. Bryan, Ph.D., University of Arizona, 1963

William R. Derrick, Ph.D., Indiana University, 1966

Rudy A. Gideon, Ph.D., University of Wisconsin, 1970

Stanley I. Grossman, Ph.D., Brown University, 1969

Gloria C. Hewitt, Ph.D., University of Washington, 1962

Don O. Loftsgaarden, Ph.D., Montana State University, 1964

Johnny W. Lott, Ph.D., Georgia State University, 1973

Robert W. McKelvey, Ph.D., University of Wisconsin 1954

William M. Myers, Jr., Ph.D., Ohio State University, 1952

Howard E. Reinhardt, Ph.D., University of Michigan, 1959

George F. Votruba, Ph.D., University of Michigan, 1964

I. Keith Yale, Ph.D., University of California, Berkeley, 1966