Applicant's letter discussing reasons for wishing to pursue
this specific program.
Residence
Six semester hours beyond the master's degree may be transferred
into the program. Two consecutive semesters of full-time study
are required in residence.
Committees and Advising
The student will be advised initially by the departmental
graduate committee. This group will be the student's temporary
advising committee and will assist in the selection of the
student's permanent committee who will supervise the remainder of
the student's program.
Requirements
The program requires a minimum of 48 semester hours distributed
among a subject-matter component, an education component, an
interdisciplinary component, and a teaching internship as
follows:
a. Mathematics Component: Competence in six-sequences is
required:
MATH 541-542 Numerical Analysis 6 cr
MATH 550-551 Probability and Statistics 6 cr
MATH 625-626 Real Analysis 6 cr
MATH 627-628 Complex Analysis 6 cr
MATH 631-632 Abstract Algebra 6 cr
MATH 671-672 Topology 6 cr
These requirements may be met in several ways. Since most
students entering the program will have previously taken several
of these sequences, the departmental graduate committee may
determine that previously taken courses meet the ISU
requirements. Up to four sequences may be satisfied by this
method. If it is the opinion of the advisory group that more work
would be beneficial, they may stipulate that the candidate
satisfy the requirement either by completing the course with a
grade of B or better, or by sitting for and passing a two-hour
exam administered at the semester break.
The two or more sequences not satisfied as outlined above must be
taken as part of the candidate's program along with 12 additional
mathematics credits at the 600 level.
b. Education Component: 3 credits from the College of Education
and 3 credits from the Mathematics Department.
c. Interdisciplinary Component: 6 credits of approved courses
in related academic fields.
d. Thesis: 6 credits.
e. Teaching Internship: 6-9 credits. Each student is required to
serve a teaching internship. By the end of the first year of
residence, a proposal should be submitted to the graduate
committee outlining the details of the internship. From this
proposal a mutually acceptable plan for the internship will be
worked out.
Internships at other colleges are possible. Other possibilities
for internship are work in the computer center, work with
curriculum, work on individual education projects, or
coordination of departmental functions.
Thesis
The thesis consists of an expository or research paper in
mathematics or mathematics education. Six hours' credit is given
for completion of the thesis.
Course Load
Ordinarily, full-time study consists of 9-12 hours of graduate
credits per semester.
Examinations
In addition to demonstrations of competence in the six basic
areas, the student must pass two comprehensive examinations.
Early in a student's program, she/he must pass a written
examination on undergraduate mathematics, administered by the
departmental graduate committee. The second, an oral examination
administered by the candidate's permanent committee, tests
knowledge of the six required areas of competence and the
student's program of study. In the event of unsatisfactory
performance on either or both exams, they may be attempted again
after a semester's wait. A second failure eliminates a candidate
from the program.
Mathematical Topic
Exposition
After successfully completing the Undergraduate Mathematics
Examination and before entering into investigation leading to the
Thesis, the D.A. candidate will complete a Mathematical Topic
Exposition. The purpose of the Exposition is to present the
candidate with the experience of investigating, organizing, and
reporting to the Department on a mathematical subject not
otherwise included in his/her program of study. This is conducted
as follows:
In consultation with the candidate's advisory committee, the
Graduate Committee will select a topic complementary to the
candidate's approved program. Within 11 days of the assignment of
the topic, the candidate will present a 50-minute lecture at a
departmental colloquium and will deliver a written report to the
Graduate Committee. The candidate's oral and written work will be
evaluated according to both their breadth and their depth. Two
failures at the Exposition will eliminate a candidate from the
D.A. Program.
Final Examination
Upon completion of the required course work, the teaching
internship, exams, and the thesis, the candidate must present
him/herself for a final oral examination which is a thesis
defense. The final examination is open to all persons invited by
the major advisor, the department chairperson, or the Dean of the
Graduate School.
Master of
Science in
Mathematics
The Master of Science degree program is designed to provide a
broad and in-depth background and prepare the student for further
study at the doctoral level or for industrial or academic
careers.
Admission
For full admission to the M.S. degree program in mathematics, the
applicant must have completed all requirements for a bachelor's
degree in mathematics at an accredited institution. The applicant
should have a grade point average of at least 3.0 over the last
two years of undergraduate work and have taken the Graduate
Record Examination. The student should have completed course work
in modern algebra, differential equations, advanced calculus, and
introductory analysis. Applicants not fully meeting these
requirements may be allowed to make up deficiencies at ISU.
Requirements
Two 600-level sequences are required. The department routinely
offers the following sequences:
MATH 625-626 Real Analysis 6 cr
MATH 627-628 Complex Analysis 6 cr
MATH 631-632 Abstract Algebra 6 cr
MATH 641-642 Numerical Analysis 6 cr
MATH 662-663 Differential Equations 6 cr
MATH 671-672 Topology 6 cr
Of the remaining 18 credits at least 12 must be taken in graduate
mathematics and at most 6 may, subject to departmental approval,
be chosen from graduate courses in other disciplines. The student
must complete a written examination in one of the four required
sequences, and must pass a final oral examination which is
intended to verify satisfactory understanding of the major field.
No thesis is required; the emphasis is on course work. No
language study is required. Each student's program must be
approved by the departmental graduate committee.
Master of Natural Science in Mathematics
The degree of Master of Natural Science with a major in
mathematics is designed specifically for people who hold a
standard secondary school teaching certificate for the teaching
of mathematics. The objective of the program is to enhance the
mathematical training of secondary teachers and to equip such
teachers with a broad and modern background in mathematics.
Admission
For full admission to the M.N.S. program in mathematics the
applicant must hold a bachelor's degree and a standard secondary
school teaching certificate. The applicant must have a GPA of at
least 2.75 for the last two years of undergraduate work and must
have taken the Graduate Record Examination (GRE).
Prerequisites
Applicants should have completed undergraduate work in analytic
geometry and calculus, a first course in linear algebra and
modern algebra, and at least one other mathematics course at the
upper-division level.
Requirements
for the Master of Natural Science in Mathematics degree must meet
the following criteria:
- Possession of a standard secondary teaching certificate or
the equivalent.
- Completion of a program of study approved by the graduate
committee of the Mathematics Department and the Dean of the
Graduate School.
- A minimum of 30 credits beyond the bachelor's degree in
courses numbered 300 or above. At least 22 credits must be in
residence.
- Satisfactory performance on final written and oral
examinations.
Required coursework will depend upon the student's background in
mathematics. No thesis or foreign language is required.
Mathematics Graduate Courses
g326 Elementary Analysis 3 credits. Rigorous calculus on real
line. Completeness, compactness and connectedness. Continuity,
images of compact and connected sets. Series, uniform
convergence. Differentiability, inverse functions, chain rule.
Integration, fundamental theorem, improper integrals. PREREQ:
MATH 223 and MATH 287.
g327 Vector Analysis 3 credits. Calculus of vector functions of
several variables. Derivative matrix. Chain rule. Inverse
function theorem. Multiple integration. Change of variables.
Integrals over curves and surfaces. Green's, Stokes' and
divergence theorems. Applications to physics. PREREQ: MATH 223.
g330 Linear Algebra 3 credits. Fields, vector spaces, linear
transformations and matrices, triangular and Jordan forms,
eigenvalues, dual spaces and tensor products, bilinear forms,
inner product spaces. PREREQ: MATH 222 and MATH 230.
g331-g332 Modern Algebra 3 credits each. Rings, fields, groups,
algebras, and selected topics in abstract algebra. PREREQ: MATH
287 and MATH 330.
g343 Modern Geometry 3 credits. Projective, Euclidean, and
non-Euclidean geometries from an axiomatic point of view. PREREQ:
MATH 230 or permission of the instructor.
g352 General Statistics 3 credits. Reviews some essential
material from a first course in applied statistics and proceeds
to additional statistical techniques; estimation, testing
hypotheses, regression and correlation, analysis of variance, and
non-parametric statistics. Oriented toward the behavioral,
social, and managerial sciences. PREREQ: MATH 250, 252 or
equivalent.
g355 Operations Research I 3 credits. Deterministic problems in
operations research oriented towards business. Includes linear
programming, transportation problems, network analysis, PERT,
dynamic programming, and elementary game theory. PREREQ: MATH
230, 250 or permission of the instructor.
g356 Operations Research II 3 credits. Probabilistic models
oriented towards business are treated. Selections from stochastic
processes, Markov chains, queuing theory, inventory theory,
reliability, decision analysis, and simulation. PREREQ: MATH 355.
g360 Differential Equations 3 credits. Theory and applications of
ordinary differential equations. PREREQ: MATH 222 and MATH 230 or
permission of instructor.
g421 Advanced Engineering Mathematics I 3 credits. Cross-listed
as ENGR g421. Analysis of complex linear and non-linear
engineering systems using advanced techniques, including Laplace
transforms, Fourier series and classical partial differential
equations. PREREQ: MATH 360, ENGR 264.
g422 Advanced Engineering Mathematics II 3 credits. Cross-listed
as ENGR g422. Analysis of complex linear and non-linear
engineering systems using advanced techniques, including
probability and statistics, advanced numerical methods and
variational calculus. PREREQ: ENGR 421 or MATH 421.
g423-g424 Introduction to Real Analysis 3 credits each. The real
number system, limits, sequences, series, and convergence; metric
spaces; completeness; and selected topics on measure and
integration theory. PREREQ: MATH 287, MATH 326, MATH 330, and
MATH 360.
g435 Elementary Number Theory 3 credits. Diophantine equations,
prime number theorems, residue systems, theorems of Fermat and
Wilson, and continued fractions. PREREQ: MATH 331.
g441 Introduction to Numerical Analysis 3 credits. Designed to
offer students in any applied science a reasonably broad
introduction to standard numerical techniques for solving
problems dealing with non-linear equations, systems of linear
equations, differential equations, as well as techniques of
interpolation, numerical integration, and differentiation.
PREREQ: MATH 326 and MATH 360 or permission of instructor.
g442 Introduction to Numerical Analysis 3 credits. Extension of
MATH 441 for students who wish to pursue more advanced techniques
with emphasis on analysis. Typical topics covered include
numerical methods applied to partial differential equations,
integral equations, and in-depth treatment of topics covered in
MATH 441. PREREQ: MATH 441.
g450-g451 Probability and Statistics 3 credits each. MATH 450
includes discrete and continuous random variables, central limit
theorem and some special distributions. Other topics may include
Markov chains, branching processes, and random walks. MATH 451
includes interval and point estimation with emphasis on
sufficient statistics, testing hypotheses, including uniformly
most powerful tests, sequential probability ratio tests, Chi
square tests, analysis of variance, regression analysis, tests
for independence, and non-parametric methods. Applications to the
physical social and biological sciences will be stressed. PREREQ
for MATH 450: MATH 223.
g462 Introduction to Complex Variables 3 credits. Introduction to
the study of functions of a complex variable including analytic
functions, power series, integral theorems, and applications.
PREREQ: MATH 360 and either MATH 326 or MATH 421.
g465 Partial Differential Equations 3 credits. Equations of the
first and second orders, methods of solution, Laplace's Equation,
heat equation, and wave equation. Emphasis on applications to
problems in the physical sciences and engineering. PREREQ: MATH
360 and either MATH 326 or MATH 421.
g473 Introduction to Topology 3 credits. Metric spaces;
convergence; notions of continuity; connected, separable and
compact spaces. PREREQ: Permission of the instructor.
g481 Special Problems 1-3 credits. Reading and conference in an
area not usually covered by a regular offering. Individual work
under the supervision and guidance of a professor whose specialty
includes the chosen area. Open to seniors and graduate students
in good standing and with the consent of the instructor. May be
repeated until 6 credits are earned.
g491 Mathematics Seminar 1-3 credits. Advanced reading and
discussion on selected topics in mathematics. May be taken for
credit more than once. PREREQ: Senior standing or equivalent.
597 Professional Education Development Topics. Variable credit.
May be repeated. A course for practicing professionals aimed at
the development and improvement of skills. May not be applied to
graduate degrees. May be graded S/U.
625-626 Real Analysis 3 credits each. Continuity, convergence,
measurable sets and functions, the Lebesgue integral, measure
spaces, integration, normed linear spaces. Hilbert and Banach
spaces; extension and representation theorems. PREREQ: Permission
of the instructor.
627-628 Complex Analysis 3 credits each. Classical theorems of
Cauchy, Goursat, Mittag-Leffler, Weierstrass, Riemann, and Picard
involving analytic functions, representation theorems, conformal
mappings, entire and meromorphic functions, analytic
continuation, and other topics. PREREQ: Permission of the
instructor.
631-632 Abstract Algebra 3 credits each. Categories, groups,
rings and ideals, polynomials, and fields through Galois Theory,
modules, lattices, advanced linear and multilinear algebra.
PREREQ: MATH 332 and 330 or permission of the instructor.
633 Matrix Theory 3 credits. Modern aspects of matrix theory.
Perron-Frobenius-Wielandt theory of nonnegative matrices,
M-matrices, theory of doubly stochastic matrices, inertia
theorems, canonical forms, elementary divisor theory. PREREQ:
MATH 330 or permission of the instructor.
641-642 Numerical Analysis 3 credits each. Topics selected from
approximation theory, optimization, numerical linear algebra,
differential and integral equations, spline analysis, computer
algorithms, and other areas of current research in numerical
analysis. PREREQ: MATH 423 and MATH 441.
650 Thesis (D.A.) 1-6 credits.
652 Stochastic Processes 3 credits. Poisson processes, renewal
processes, branching processes, continuous and discrete time
Markov chains and queuing theory. Applications of the theory and
methods of model building are stressed. PREREQ: MATH 325 and 450.
655-656 Combinatorics 3 credits each. Theory and applications of:
choice and enumeration techniques, generating functions,
partitions, designs and configurations, graph theory including
digraphs, algebraic graph theory and extremal problems. PREREQ:
Permission of the instructor.
662-663 Differential Equations 3 credits each. Topics selected
from the theory of existence, uniqueness, extension, stability
and behavior of solutions of differential equations. Numerical
techniques, transform theory, expansions of solutions, and
related areas may be studied. PREREQ: MATH 360.
667 Introduction to Functional Analysis 3 credits. Metric spaces
and their completion, convergence, Banach and Hilbert spaces,
linear operators and related topics. PREREQ: MATH 423 or 625 or
permission of the instructor.
668 Topics in Functional Analysis 3 credits. Major results of
functional analysis, such as the Hahn-Banach, uniform
boundedness, open mapping, and fixed point theorems and their
applications to other areas of mathematics. PREREQ: MATH 667.
671-672 Topology 3 credits each. Fundamental theorems of
point-set topology. Metric spaces, compact spaces, topological
spaces, and applications. PREREQ: Permission of the instructor.
691 Seminar 1-3 credits. Advanced readings, problems, and
discussion on selected topics in mathematics. May be taken for
credit more than once on distinct topics.
699 Special Topics in Mathematics 1-3 credits. Each offering
will deal with a topic selected from such fields of mathematics
as algebra, analysis, geometry, number theory, topology, applied
analysis, probability, and mathematical logic. May be taken for
credit more then once. PREREQ: Permission of the instructor.
700 Supervised Teaching Internship. Credit variable up to 9
credits.