## Department of Mathematics & Statistics

Distinguished Teaching Professor

Edward S. Thomas Jr., Ph.D.

University of California, Riverside

Professors Emeritae/i

Louis Brickman, Ph.D.

University of PennsylvaniaVincent Cowling, Ph.D.

Rice UniversityEdward D. Davis, Ph.D.

University of ChicagoJoe W. Jenkins, Ph.D.

University of IllinoisMelvin L. Katz, Ph.D.

University of California, BerkeleyViolet H. Larney, Ph.D.

University of WisconsinThomas H. MacGregor, Ph.D.

University of PennsylvaniaGeorge E. Martin, Ph.D.

University of MichiganHajimu Ogawa, Ph.D.

University of California, Berkeley

Professors

Lindsay N. Childs, Ph.D.

Cornell UniversityNathaniel A. Friedman, Ph.D.

Brown UniversityRichard Z. Goldstein, Ph.D.

University of PennsylvaniaBenton N. Jamison, Ph.D.

University of California, BerkeleyBoris Korenblum, Sc.D.

Moscow State UniversityTimothy L. Lance, Ph.D.

Princeton UniversityRichard C. O'Neil, Ph.D.

University of ChicagoR. Michael Range, Ph.D.

University of California, Los AngelesHoward H. Stratton, Ph.D.

University of California, RiversideEdward C. Turner, Ph.D.

University of California, Los AngelesDonald R. Wilken, Ph.D.

Tulane UniversityKehe Zhu, Ph.D.

State University of New York at Buffalo

Associate Professors Emeritae/i

Guy D. Allaud, Ph.D.

University of WisconsinRobert Luippold, M.A.

University of BuffaloRicardo Nirenberg, Ph.D.

New York UniversityErich Nussbaum, Ph.D.

University of VirginiaJohn T. Therrien, M.A.

University at Albany

Associate Professors

Herbert I. Brown, Ph.D.

Rutgers UniversityHara Charalambous, Ph.D.

University of Illinois, Urbana-ChampaignWilliam F. Hammond, Ph.D.

Johns Hopkins UniversityLloyd L. Lininger, Ph.D.

University of IowaSteven Plotnick, Ph.D.

University of MichiganCarlos C. Rodriguez, Ph.D.

Columbia UniversityMalcolm J. Sherman, Ph.D.

University of California, BerkeleyAnupam Srivastav, Ph.D.

University of Illinois, Urbana-ChampaignMark Steinberger, Ph.D.

University of ChicagoMichael I. Stessin, Ph.D.

Moscow State University

Assistant Professors

Boris Goldfarb, Ph.D.

Cornell UniversityMartin Victor Hildebrand, Ph.D.

Harvard UniversityCristian Lenart, Ph.D.

University of CambridgeKarin B. Reinhold-Larsson, Ph.D.

Ohio State UniversityJennifer Taback, Ph.D.

University of Chicago

Adjuncts (estimated): 0

Teaching Assistants (estimated): 30## Careers

The objective of the department is to serve the needs of students aspiring to careers that require mathematical background: physical, biological, social, and management sciences; statistics, actuarial work, computer science, applied mathematics; secondary school teaching; graduate work; college and university teaching; and research in mathematics. In most cases, training beyond the bachelor's degree is desirable and can often be obtained after the graduate has secured employment. The department also welcomes students who wish to study mathematics as part of a traditional liberal arts education.

## Placement and Proficiency Credit

The University awards up to 8 credits and advanced placement in its sequences of calculus courses based on performance on the advanced placement calculus examinations administered by the College Board. Details concerning the decisions on credit and placement are available from the Admissions Office.

## Admission

Students may not declare a major in either mathematics or actuarial and mathematical science until they have completed at least one of A Mat 113, 119, or 214 with a grade of A, B, C, or S. Transfer credits and grades may be used to satisfy the requirement.

## The Mathematics Major

Students majoring in mathematics elect either the general program or the teacher education program. Under either program, a student may choose to complete the requirements for either the B.A. or B.S. degree. Under any of the four program-degree combinations, a student may apply for admission to the honors program.

Students considering a major in mathematics are encouraged to visit the department office (ES-110) for advice. Information is also available at the web site http://math.albany.edu.

## Degree Requirements for the Major in Mathematics

General Program B.A.:A minimum of 36 credits from the Department of Mathematics and Statistics in courses numbered above 110, including A Mat 214, 220, and a 3-credit course numbered above 300 in each of these four areas: algebra, analysis, geometry/topology, and probability/statistics.

General Program B.S.:A minimum of 36 credits from the Department of Mathematics and Statistics in courses numbered above 110, including A Mat 214, 220, and any two of the following four options: (1) A Mat 326 and 327, (2) either (a) both A Mat 314 and 315 or (b) any two of 312, 412, 413, or 414, (3) any two of A Mat 342, 441, or 442, (4) either (a) A Mat 368 and 369, (b) A Mat 367 and 464, or (c) A Mat 467 and 468. With departmental approval, other 400-level or 500-level courses may be substituted for the courses listed above. In addition, each student must complete: 6 credits in computer science from A Csi 101N, 201N, 203, 204, 205, 310; and a minor in atmospheric science, biology, business, chemistry, computer science, economics, electronics, geology, or physics.

Note:The Statistics minor is not open to students with a major in mathematics.

General ProgramStudents, with suitable advisement, can design programs that will best meet their particular interests and career goals. Note, however, that those who plan to do graduate work in any mathematical field-pure or applied-should obtain as strong an undergraduate background as possible in the basic areas of mathematics: algebra, analysis, and geometry/topology. In particular, they should make every effort to include A Mat 413 and 414 (Advanced Calculus) in their programs.

To guide students in their planning, a number of options, some of a general nature and others to meet specific career objectives, are presented here.

1. Liberal Arts (B.A.)

Some professional careers and many jobs require a mathematical background characterized more by breadth than by concentration in any particular area of the mathematical sciences. The purpose of the B.A. program is to assure that the student acquires a broad view of mathematics and statistics. Each B.A. major is required to complete a 3-credit course numbered above 300 in each of these areas: algebra, analysis, geometry/topology, and probability/statistics. The following lists those courses that can be taken to fulfill that requirement:Algebra: A Mat 326, 326Z, 327, 327Z, 424

Analysis: A Mat 311, 312, 312Z, 314, 409, 412, 412Z, 413, 413Z, 414

Geometry/Topology: A Mat 331, 331Z, 342, 342Z, 432, 432Z, 441, 442

Probability/Statistics: A Mat 367, 367Z, 368, 369, 464, 465, 465Z, 467, 468

Students are urged to explore in greater depth, preferably at the 400 level. Since students will have different goals, it is impossible to provide useful sample programs. Students are encouraged to devise their own plans in consultation with their advisers. However, if a student is to graduate on time, the calculus sequence and linear algebra should be completed during the freshmen and sophomore years.

2. Graduate School Preparation

The department offers excellent opportunities for students who plan to go on to graduate work in mathematics and statistics as well as other areas such as computer science, the natural sciences, and the social and behavioral sciences.Students whose goal is to obtain a graduate degree in mathematics should include in their programs as many of the following core courses as possible in each of the designated areas:

Algebra: A Mat 326, 327, 424

Analysis: A Mat 413, 414

Geometry/Topology: A Mat 342

Probability/Statistics: A Mat 467, 468

Those hoping to do graduate work should also consider entering the honors program.

3. Applied Mathematics

Although it is common to classify mathematics as either "pure" or "applied," the division is often arbitrary. Some extremely abstract mathematics in recent years has turned out to be useful in areas outside mathematics. Students preparing for a career in applied mathematics would be well advised to acquire as strong a background as possible in the pure mathematical areas of analysis, algebra, and geometry/topology. On the other hand, students concentrating in pure mathematics should have some understanding of how to apply mathematical methods to other disciplines.Listed here are the mathematical subjects that are more commonly applied to problems in other fields along with the corresponding courses in which methodology or applications are treated.

Applied algebra: A Mat 326, 372

Applied analysis: A Mat 311, 314, 315, 409, 412, 416

Numerical Methods: A Mat 313, 401

Probability/Statistics: A Mat 367, 368, 369, 464, 465

4. StatisticsStatistics is a widely applied branch of mathematics and the demand for statisticians is high. Preparation for a career or for advanced study in statistics should include one of the following two combinations of courses: (1) probability (A Mat 367 or 367Z, 464) and statistics (A Mat 368 or 368Z, 369 or 369Z, 465 or 465Z), or (2) probability (A Mat 367 or 367Z, 464) and statistics (A Mat 467, 468). Sequence (2) is recommended as the more advanced and thorough treatment. A Mat 424 (advanced linear algebra) is highly recommended. Also useful are A Mat 401, 409, 413 or 413Z, and 414. Because computing is a close adjunct to statistics, students are strongly advised to include A Csi 201N, 205, and 310 as a minimal introduction.

## Teacher Education Program

Students interested in a career in secondary school teaching must apply for and be admitted to the Teacher Education Program, which is administered by the School of Education, before they can be officially enrolled in the teacher education program with a major in mathematics. The admission requirements and the certification requirements are described in this bulletin under the School of Education. Note that students in the teacher education program must complete both the education professional requirements described under the School of Education and those courses within the major and related fields that are listed under "Degree Requirements for the Major in Mathematics."

- In addition to the education professional requirements, specific course requirements for the mathematics teacher education program are: 6 credits in the same one of these sciences-atmospheric science, biology, chemistry, geology, physics; and in mathematics: A Mat 326 or 326Z, 327 or 327Z, 331 or 331Z, 367 or 367Z, 368 or 368Z, 342 or 342Z, and 452 or 452Z; in addition, either A Mat 363 or 363Z or 367 or 367Z (A Mat 312 or 312Z is recommended for those wishing to teach calculus).

- Typical schedule of mathematics courses:

Year Fall Spring Fresh. 112 or 118 113 or 119 Soph. 214 & 220 312 & 331 Junior 326 & 367 327 & 368 Senior 342 & 452 Student Teaching Note particularly that at least two courses should be taken each semester beginning in the sophomore year.

## Honors Program

The honors program is designed for the talented and committed student of mathematics. Successful completion of the program is excellent preparation for graduate work in mathematics.

Students entering the University with strong mathematical backgrounds should consider taking Honors Calculus, A Mat 118 and 119, in place of the standard Calculus, A Mat 112 and 113.

A student may be admitted formally to the honors program at any time after the sophomore year, and then will be formally advised by the Director of the Honors Program. However, any student who is interested in the program should see the Director of the Honors Program as early as possible for informal advisement.

To be admitted, the applicant must have an academic average in all University courses of at least 3.30, and an academic average in all mathematics courses of at least 3.40. Specific course requirements are: A Mat 413 or 413Z, 414, 424, and 9 additional credits from among A Mat 327 or 327Z, 416, 420, 425, 432 or 432Z, 441, 442, 464, 467, 468, 510A, 513A, 520A, 520B, 540A, 557A, 557B, and independent study (maximum of 3 credits).

To be recommended for graduation with honors, the candidate must write an acceptable honors thesis and also maintain an academic average of at least 3.30 in all University courses and at least 3.40 in all mathematics courses numbered 400 or above.

## The Actuarial Major

Mathematics has useful and interesting applications in actuarial science. The department offers courses that prepare students for several examinations given by the Society of Actuaries. The following University at Albany courses correspond to the indicated actuarial society examinations.

Course Exam 100: A Mat 112 or 118, 113 or 119, 214, 220, 204

Course Exam 110: A Mat 467, 468, 469

Course Exam 120: A Mat 465

Course Exam 130: A Mat 372, 374

Course Exam 135: A Mat 401. A small amount of material on linear systems can be obtained by self study.

Course Exam 140: A Mat 301

Note:The Society of Actuaries will modify its examinations in the year 2000. Corresponding changes in courses and syllabi are in the process of being implemented by the department.Students who wish to graduate in four years should try to adhere to the following schedule for required mathematics courses.

Year Fall Spring Fresh. 112 or 118 113 or 119 Soph. 214 & 220 204 & 301 Junior 367 & 401 368 & 372 Senior 465 & 467 374 & 468 & 469

## Degree Requirements for the Major in Actuarial and Mathematical Sciences

General Program B.S.A combined major and minor sequence consisting of 61 credits as follows:42 credits in mathematics: A Mat 112 (or 118), 113 (or 119), 214, 220, 301, 367 (or 367Z), 368 (or 368Z), 372 or 372Z, 374, 401, 465 or 465Z, 467, and 468

7 credits: A Csi 201N and 203

3 credits: B Acc 211

9 credits: as advised, in courses from the School of Business or having prefix A Csi or A Eco. Acceptable courses include A Eco 110M and 111M, B Fin 300 and 301, and A Csi 310. Students who wish to substitute other courses must secure the permission of their adviser.

## Combined B.A./M.A. and B.S./M.A. Programs

The combined B.A./M.A. and B.S./M.A. programs in mathematics provide an opportunity for students of recognized academic ability and educational maturity to fulfill integrated requirements of undergraduate and master's degree programs from the beginning of their junior year. A carefully designed program can permit a student to earn the B.A or B.S. and the M.A. degrees within nine semesters.

The combined programs require a minimum of 138 credits, of which at least 30 must be graduate credits. In qualifying for the B.A. or B.S., students must meet all University and college requirements, including the requirements of the undergraduate major described previously, the minimum 90- or 60- credit liberal arts and sciences requirement, general education requirements, and residence requirements. In qualifying for the M.A., students must meet all University and college requirements as outlined in the Graduate Bulletin, including completion of a minimum of 30 graduate credits and any other conditions such as a research seminar, thesis, comprehensive examination, professional experience, and residence requirements. Up to 12 graduate credits may be applied simultaneously to both the B.A. and M.A. programs or to both the B.S. and M.A. programs.

Students are considered as undergraduates until completion of 120 graduation credits and satisfactory completion of all B.A. or B.S. requirements. Upon meeting B.A. or B.S. requirements, students are automatically considered as graduate students.

Students may apply to the graduate committee of the department for admission to either combined program in mathematics at the beginning of their junior year or after the successful completion of 56 credits, but no later than the accumulation of 100 credits. A cumulative grade point average of 3.2 or higher and three supportive letters of recommendation from faculty are required for consideration.

## Combined Mathematics and Master of Business Administration Program:

In this program a student is able to obtain a B.S. degree in mathematics and a M.B.A. degree in a total of five years by taking a coordinated program in mathematics and business administration during the senior year. Application should be made during the second semester of the junior year to the director of the M.B.A. program, School of Business.

## Related Program: Interdisciplinary Major in Computer Science and Applied Mathematics:

This major prepares a student to handle mathematically oriented computer applications in engineering and business. Details of the program are listed under Computer Science.

## Courses

A Mat 100 Precalculus Mathematics (3)

This course provides a background in those topics that are needed for success in calculus. Topics include graphing techniques, systems of equations, functions, logarithms, and trigonometry. May not be taken for credit by students with credit in any calculus course. Student with credit for the former A Mat 103 (College Algebra) may not take A Mat 100 for credit). Prerequisite(s): three years of high school mathematics or permission of department. May not be offered during 1999-2000.

A Mat 101 Algebra and Calculus I (3)

An integrated approach to precalculus and calculus. Elements of algebra and analytic geometry necessary to study calculus of one variable. Functions, limits, continuity, differentiation of algebraic functions, applications of differentiation. May not be taken for credit by students with credit for A Mat 100, 106, 112 or 118. Prerequisite(s): three years of high school mathematics or permission of the department.

A Mat 102N Mathematics by Visualization (3)

General Education: NS

This is a nontraditional course introducing contemporary mathematics primarily by visualization rather than algebra. This will enable the student to learn to see the way mathematicians see. Thus the student will be able to experience creative visualization in mathematics. The content of the course will include fractals; chaos; 4-dimensional geometry; Platonic solids; color maps; Escher tesselations. and impossible figures. A Mat 102F is the writing intensive version of A Mat 102N; only one of these may be taken for credit. Prerequisite(s): three years of high school mathematics or permission of instructor.

A Mat 102F Mathematics by Visualization (3)

General Education: NS & WI

The course is writing intensive and each student will keep a journal (notebook). A Mat 102F is the writing intensive version of A Mat 102N; only one of these may be taken for credit. Prerequisite(s): three years of high school mathematics or permission of instructor.

A Mat 105 Finite Mathematics (3)

An introduction to topics of interest to students of the social sciences; sets and logic, partitions and counting, probability, vectors and matrices, theory of games. Prerequisite(s): three years of high school mathematics.

A Mat 106 Survey of Calculus (3)

An intuitive approach to differentiation and integration of algebraic and transcendental functions, intended only for students who plan to take no more calculus. Does not yield credit toward the major or minor in mathematics. May not be taken for credit by students with credit for A Mat 111, 112 or 118. Prerequisite(s): A Mat 100 or satisfactory performance on the mathematics placement exam.

A Mat 108 Elementary Statistics (3)

Frequency distributions, measures of central tendency and dispersion, probability and sampling, estimation, testing of hypotheses, linear regression and correlation. Prerequisite(s): three years of high school mathematics. Only one of A Mat 108 and B Msi 220 may be taken for credit.

A Mat 109 Applied Matrix Algebra (3)

Matrix algebra as applied to solving systems of linear equations. Markov chains, linear programming. Emphasizes calculations and applications rather than theory. Prerequisite(s): three years of high school mathematics.

A Mat 110 Introduction to Maple (2)

A hands-on introduction to the computer algebra system Maple. Basic commands are introduced by way of examples from the areas of algebra, calculus, number theory, graphics, business mathematics, and numerical analysis. Intended for transfer students having no background in Maple. Does not yield credit toward a major in mathematics. Prerequisite(s): A Mat 101 or a semester of calculus.

A Mat 111 Algebra and Calculus II (4)

The second semester of an integrated approach to precalculus and calculus; serves as a prerequisite to A Mat 113. Applications of differentiation, the definite integral, antiderivatives, logarithms, trigonometry, exponential functions. Only one of A Mat 111, 112 & 118 may be taken for credit. Prerequisite(s): A Mat 101.

A Mat 112 Calculus I (4)

Calculus of one variable. Limits, continuity, differentiation of algebraic functions, applications of differentiation, antiderivatives, the definite integral, transcendental functions. Prerequisite(s): A Mat 100 or satisfactory performance on the mathematics placement exam.

A Mat 113 Calculus II (4)

Techniques of integration, applications of the definite integral, conics, polar coordinates, improper integrals, infinite series. Prerequisite(s): A Mat 111 or 112.

A Mat 118 Honors Calculus I (4)

Honors version of first semester calculus. Same topics as A Mat 112, but topics are covered in greater depth. This course is for students with more than average ability and more than average interest in mathematics. Presidential Scholars with a strong interest in mathematics or the physical sciences should consider taking A Mat 118 instead of A Mat 112. A Mat 118 substitutes for A Mat 112 toward the prerequisite in any course. Only one of A Mat 112 & 118 may be taken for credit. Prerequisite(s): three years of secondary school mathematics and permission of the instructor. Offered fall semester only.