Mathematics
Why study mathematics? Many of the new wonders that we take for granted in our modern technological society have mathematical ideas and applications as their basis, though this role is often hidden from view. Complex economic and planning decisions, scientific discoveries that improve our lives, and new technologies and products are often possible only after mathematical or statistical analysis, or a computer visualization, simulation, design and implementation based on mathematics. Therefore, mathematicians, as well as mathematically educated scientists, engineers and economists, make important daily contributions in the understanding and advancement of science, the improvement and discovery of new technology, and decisionmaking and planning in business, industry and government. Students interested in using their mathematical skills in solving real world problems are well prepared, by majoring or minoring in mathematics, for careers such as in the insurance industry, software design, data and systems analysis, scientific computing, combustion research, the animated movie industry, and cryptanalysis to name a few, or a graduate degree in a related technical field. Those students with a very strong interest in mathematics itself can pursue graduate study in mathematics in preparation for careers as university or college mathematics teachers and in the development of new mathematical and statistical concepts and methods as researchers in academia, government and industry.
The curriculum of the program in the Department of Mathematics is designed to provide a broad education in both theoretical and applied mathematics. It also develops the scientific knowledge and the problem solving, computing, and communications skills that are critical to a successful mathematically based career. This preparation is greatly enhanced by taking advantage of the wide variety of science and engineering courses available to students and developing good communications skills, both through technical courses and the strong humanities program. The program offers a solid grounding in the foundational areas of calculus, differential equations, linear algebra, discrete and combinatorial algebra, and probability and statistics. These basic courses are complemented by a varied selection of upper division courses for further elective study in areas such as numerical analysis, operations research, advanced statistics, mathematical modeling, optimization, and other advanced topics in mathematics. Students are encouraged to develop a strong background in an area of science or engineering through election of courses leading to a minor or double major. By appropriate course selection students may complete a double major in mathematics and another field such as computer science, physics, chemistry, applied biology, or economics.
PROGRAM GOALS AND OBJECTIVES
To provide a foundation for further learning as well as contributing to the general education of students, the programs at RoseHulman all have a heavy investment in mathematics and science in the first two years. The freshman and sophomore mathematics curriculum is designed to contribute to this foundation by ensuring that students are familiar with basic mathematical and statistical concepts, and mathematical and statistical reasoning and modeling. Students will also understand the use of mathematics in other disciplines as well as developing an appreciation of mathematics as a discipline in its own right. In addition, students will learn to be competent users of mathematics, especially in problem solving, and be able to effectively communicate mathematically. The curriculum makes strong use of computer methods to develop students’ mathematical understanding and to enhance their ability to use the computer in modeling, computation and problem solving.
For students seeking a major in mathematics, the curriculum prepares them for a mathematically based career after graduation or further graduate study. The major builds upon the goals and objectives of the freshman and sophomore curriculum. In addition to a deeper and broader study of mathematics, majors will further develop their ability to formulate and solve problems from a mathematical perspective, become familiar with the use of mathematics in other fields, and develop competence at the application of mathematics to at least one other field. Graduates will also be able to use technology effectively in mathematics and the application of mathematics. To complement these technical skills graduates will learn the professional skills of effective communication with both technical and nontechnical audiences and the ability to work cooperatively with others.
Mathematics Program Goals and Student Learning Outcomes
Mathematics majors should be able to meet the following Student Learning Outcomes
Goal 1: Students will learn the fundamental principles underlying the major areas of mathematics
 SLO 1.1. Perform calculations and prove statements about objects, morphisms, and structure theorems in vector spaces and abstract algebra.
 SLO 1.2. Write rigorous proofs and state key counterexamples in the areas of sequences, continuity, differentiability, and integrability.
 SLO 1.3. Select, and produce appropriate characterizations for, a model for a random process through functions of random variables (e.g., expectation or probability statements).
Goal 2: Students will have a wellrounded scientific and mathematical background
 SLO 2.1. Develop focused further knowledge of at least one of the fields of mathematical sciences.
 SLO 2.2. Synthesize new and previous knowledge in cooperation with a faculty member.
 SLO 2.3. Demonstrate basic literacy in a number of scientific areas
Goal 3: Students will be able to use technology in a mathematical setting.
 SLO 3.1. Describe and implement computational approaches to solving a mathematical model.
 SLO 3.2. Use mathematical technology to analyze mathematical problems in at least one area.
Goal 4: Students will develop and utilize effective written and oral communication skills in a mathematical setting
 SLO 4.1. Rigorously write and critique mathematical proofs.
 SLO 4.2. Write and speak about mathematics in an articulate, sound and wellorganized fashion.
Goal 5: Students will develop a broad appreciation for mathematics both as a discipline and as a tool for solving real world problems.
 SLO 5.1. Construct, analyze, and interpret a mathematical model to explain and predict relationships for given deterministic systems and random systems from science or industry.
 SLO 5.2. Create, manipulate, and interpret discrete models and continuous models to illustrate dynamics of a system over time.
 SLO 5.3. Generalize properties of numbers and functions to other algebraic, analytic, topological, or geometric structures.
DEGREE REQUIREMENTS
Major Concentrations: Mathematics majors choose to complete their program in one of four concentrations: Mathematics, Continuous Applied Mathematics, Discrete Applied Mathematics, or Statistics and Operations Research. The Mathematics concentration provides the foundational mathematical depth of a traditional mathematics major and is intended for students planning on graduate study in an area of mathematics. In applied mathematics there are two areas: the Continuous Applied Mathematics concentration and the Discrete Applied Mathematics concentration. Students selecting these concentrations may tailor their programs to interface with another major or to enhance industrial employment or graduate school opportunities. The Statistics and Operations Research concentration is recommended for students pursuing careers in actuarial science, graduate study in statistics, or employment in government or industry in a statistical capacity. It is strongly recommended that students considering graduate education in mathematics include MA 376 Abstract Algebra among their elective mathematics courses. Upon graduation a student may request the Head of the Mathematics Department to issue a letter attesting to the fact that the requirements in the chosen concentration have been completed.
Mathematics Coursework Requirements: All mathematics majors must complete a common core consisting of 39 credit hours of mathematics coursework, which provides breadth across the main areas of mathematics. A mathematics major must also complete an additional 12 credit hours of mathematics coursework specified for the selected major concentration plus an additional 12 credit hours earned in free elective mathematics or biomathematics courses. In addition, a mathematics major must complete 8 credit hours of either a senior thesis or project, meant as a capstone experience to the major. A total of 71 credit hours of mathematics courses is required for the major. None of the credits in the 71 hours above may be taken from the courses MA190, MA351MA356, MA450 or MA223 (unless approved by the department head). These courses (except MA190) may be taken as free electives. Finally, a student taking a degree program in which mathematics is the primary major must also take MA190. A student whose second major is mathematics is not required to take MA 190, but is strongly encouraged to do so.
Common Required Core 
39 hrs. 

MA 111, 112, 113 Calculus I, II, III  15 hrs. 
MA 221 Matrix Algebra and Differential Equations I  4 hrs. 
MA 222 Matrix Algebra and Differential Equations II  4 hrs. 
MA 276 Introduction to Proofs  4 hrs. 
MA 366 Introduction to Real Analysis  4 hrs. 
MA 371 Linear Algebra I  4 hrs. 
MA 381 Introduction to Probability with Applications to Statistics  4 hrs. 
Mathematics Concentration Core 
12 hrs. 


Three courses selected as follows:  
MA 367  Functions of a Complex Variable 
4 hrs. 
MA 376  Abstract Algebra 
4 hrs. 
One of the following 
4 hrs. 

MA 433  Numerical Analysis  
MA 436  Introduction to Partial Differential Equations  
MA 446  Combinatorial Optimization  
MA 481  Introduction to Mathematical Statistics 
Continuous Applied Mathematics Concentration Core  12 hrs.  

Three courses selected per the list below. Students completing the Continuous Applied Mathematics Concentration are strongly urged to complete mathematics coursework in statistics as elective coursework.  
MA 330  Vector Calculus  4 hrs. 
MA 336  Boundary Value Problems  4 hrs. 
MA 433  Numerical Analysis  4 hrs. 
Discrete Applied Mathematics Concentration Core  12 hrs.  

Three courses selected per the list below. Students completing the Discrete Applied Mathematics Concentration are strongly urged to complete mathematics coursework in statistics as elective coursework.  
MA 374  Combinatorics  4 hrs. 
MA 444  Deterministic Models in Operations Research  4 hrs. 
One of the following  4 hrs.  
MA 376  Abstract Algebra  
MA 475  Topics in Discrete Mathematics  
MA 476  Algebraic Codes  
MA 477  Graph Theory 
Statistics and Operations Research Concentration Core 
12 hrs. 


Three courses selected per the list below. Students completing the Statistics and Operations Research Concentration are strongly urged to complete mathematics coursework in applied mathematics as elective coursework.  
MA 382  Introduction to Statistics with Probability  4 hrs. 
MA 444  Deterministic Models in Operations Research 
4 hrs. 
One of the Following  4 hrs.  
MA 445  Stochastic Models in Operations Research  
MA 446  Combinatorial Optimization  
MA 481  Introduction to Mathematical Statistics  
MA 485  Applied Regression Analysis and Introduction to Time Series  
MA 487  Design of Experiments  
It is strongly suggested that the student take as many of the above courses as possible. 
Free Mathematics Electives—12 hrs.
Additional mathematics and biomathematics coursework in courses numbered 300 or above (MA 351 MA 356, MA 450, BMTH496498 not allowed).
MA 190 – Contemporary Mathematical Problems (2 hrs.) A student taking a degree program in which mathematics is the primary major must also take MA 190. A student whose second major is mathematics is not required to take MA190, but is strongly encouraged to do so.
Senior Project or Thesis (8 hrs.) A student must complete either a Senior Project, equivalent to the 8 credit hours of MA 491 – 494, or a Senior Thesis, equivalent to the 8 credit hours of MA 496 – 498. The project and thesis are each important capstone experiences for the mathematics major, representing sustained efforts to solve a complex problem from industry, mathematics modeling or application, or mathematical research.
Senior Project Option: Students seeking to do a senior project must complete a written project involving effort equivalent to the 8 credit hours of MA491 – 494. Specifically,
 MA 493 and MA 494 must be taken in separate terms.
 The requirement of MA 491492 may be fulfilled through some project experience (such as an internship) and another 300level or above mathematics course (4 hours), as approved by the project advisor. The course substitution procedure must be used.
 The project must involve work done by the student(s) to solve a problem presented by an external sponsor or a problem with a substantial mathematical modeling, application and/or computational content. The written project submission must be approved by the advisor and/or sponsor and must be presented publicly to the department.
Senior Thesis Option: Students seeking to do a senior thesis must complete a written thesis involving effort equivalent to the 8 credit hours of MA496 – MA 498. Specifically,
 MA 497 and MA 498 must be taken in separate terms.
 The requirement of MA 496 may be fulfilled through some undergraduate research experience and an additional 300level or above mathematics course (4 hours), as approved by the thesis advisor. The course substitution procedure must be used.
 The thesis must involve creative work done by the student and a significant portion of this work must have been done by the student individually (not as part of a team).
 The approved written thesis must be submitted to the department for archiving and must be publicly presented to the department.
Summary of Requirements  

Mathematics Coursework  core, concentration and electives (MA351MA356, MA450, BMTH496498 not allowed)  63 hrs.  
Mathematics Senior Project/Thesis  8 hrs.  
MA 190  Contemporary Mathematical Problems (primary major only) 
2 hrs.  
Physical and Life Sciences*  24 hrs.  
Computer Science**  8 hrs.  
Humanities, Social Sciences, and the Arts (standard requirement, one course must be RH330)  36 hrs.  
Technical Electives***  24 hrs.  
Free Electives  28 hrs.  
Miscellaneous****  2 hr.  
Total hours required for graduation 
195 hrs. 

*  PH 111, 112, and 113 — Physics I, II, and III  12 hrs. 
BIO 101 — Essential Biology (or higherlevel BIO course)  4 hrs.  
CHEM 111 — General Chemistry I  4 hrs.  
4 additional credit hours in Physical or Life Sciences  4 hrs.  
**  CSSE 120 — Introduction to Software Development  4 hrs. 
CSSE 220 — ObjectOriented Software Development  4 hrs.  
MA 332  Introduction to Computational Science  may be taken instead of CSSE 220 but then MA 332 cannot be counted towards the 63 hours of mathematics coursework  
***  200 level or above coursework, approved by the major advisor, in areas of science, engineering, or economics in which 12 credit hours constitute a coherent set of three courses representing a specific area of technical depth and 12 credit hours represent technical breadth. Coursework in mathematics and biomathematics is not allowed.  24 hrs. 
****  RHIT 100 — Foundations of RoseHulman Success MA 200  Career Preparation (primary major only) 
1 hr. 1 hr. 
SUGGESTED SCHEDULE
The schedule (Course Sequence) on the right is a suggested schedule only. Scheduling of courses may be altered, subject to the approval of the advisor, in order to take advantage of advanced placement or to accommodate a second major, area minor or other special program. However, note that some courses are offered only at certain times during the year, and all prerequisites must be met. In the schedule an MA elective is either a concentration elective or free math elective, as described above, and a science elective is a physical or life science elective as defined on this page.
Alternate Science Schedule: The recommended science schedule of six science courses starts with PH 111. If CHEM 111 is required in the fall quarter because of a double major or minor, then the alternate science sequence may be completed by taking the second science course in each place where a choice is given. Two science courses are to be taken in the winter quarter of freshman year.
COMPUTATIONAL SCIENCE MAJOR (CPLS) (Second Major Only)
Computational methods are widely employed in science and engineering for simulation, experimentation, analysis, and design. In many areas the use of highperformance computing is essential. The Computational Science major provides RoseHulman students with the opportunity to add to their primary major a second major that increases their knowledge and skill in applied scientific and engineering computation.
Requirements for a second major in Computational Science (71 credit hours)
The second major in Computational Science is open to all students. It requires 71 credit hours, including a 51 credit hour core and a 20 credit hour specialization. The courses used to satisfy the requirements in the Advanced Core may not be counted toward any other major or minor. All Computational Science programs of study are subject to approval by the Chair of the Computational Science Steering Committee.
Computational Science Core (51 credit hours)
Fundamentals (31 credit hours)
 MA 111, 112, 113 Calculus I, II, III
 MA 211 Differential Equations
 MA 212 Matrix Algebra and Systems of Differential Equations
 CSSE 120 Introduction to Software Development, or any of BE 100, CE 111*, CHE 110*, ENGD 120, ME 123
 MA 332 Introduction to Computational Science, or any of CHE 310, ME 323*, ME 327
Advanced (20 credit hours; these courses may not be counted toward any other major or minor)
 CSSE/MA 335 Introduction to Parallel Computing
 MA 336 Boundary Value Problems
 MA 342 Computational Modeling
 MA 435 or ME 422 Finite Difference Methods, Finite Element Methods for Engineering Applications
Any course from the list of Approved Computational Science Electives (or another upperlevel course if approved by the Chair of the Computational Science Steering Committee):
 BE 340 Biomedical Signal Processing
 BE 510 Biomedical Signal and Image Processing
 BMTH 312 Bioinformatics
 BMTH 413 Computational Biology
 CHE 310 Numerical Methods for Chemical Engineers
 CE 310 Computer Applications in Civil Engineering
 CSSE 304 Programming Language Concepts
 ECE 480/OE 437 Introduction to Image Processing
 ECE 483 DSP System Design
 EMGT 534/MA 534 Management Science
 MA 323 Geometric Modeling
 MA 384 Data Mining
 MA 433 Numerical Analysis
 MA 434 Topics in Numerical Analysis
 MA 435 Finite Difference Methods
 MA 439 Mathematical Methods of Image Processing
 MA 444 Deterministic Models in Operations Research
 MA 446 Combinatorial Optimization
 ME 422 Finite Element Methods for Engineering Applications
 ME 427 Introduction to Computational Fluid Dynamics
 ME 430 Mechatronic Systems
 ME 522 Advanced Finite Elements Analysis
 ME 536 Computational Intelligence in Control Engineering
 PH 540 Computer Physics
Area of Concentration (20 credit hours): Each student must complete 20 credit hours of advanced work reflecting an Area of Concentration within Computational Science. Courses used to satisfy the core requirements may not be used to satisfy the area of concentration requirements. The 20 credit hours shall consist of at least 16 credit hours within a single Area of Concentration, as specified below, and an additional 4 credit hours from any of the Areas of Concentration, or from the list of Approved Computational Science Electives. Exceptions may be made on occasion (e.g. when an appropriate special topics course has been taken).
Computational Methods
 MA 371 or MA 373 Linear Algebra I, Applied Linear Algebra for Engineers
 MA 433 Numerical Analysis
 Eight credit hours chosen from BMTH 413, CSSE 304, CSSE/MA 473, MA 384, MA 386, MA 434, MA 435, MA439, MA 444, MA 446, MA 485, ME 422
Computational Mechanics
 MA 435 or ME 422 Finite Difference Methods, Finite Element Methods for Engineering Applications
 ME 401 Foundations of Fluid Mechanics
 ME 427 Introduction to Computational Fluid Dynamics
 ME 522 Advanced Finite Element Analysis
Computational Signals and Image Processing
 ECE 380 DiscreteTime Signals and Systems
 ECE 480/OE 437 Introduction to Image Processing
 ECE 483 DSP System Design
 MA 439 Mathematical Methods of Image Processing
Computational Physics and Chemistry
 CHEM 361 Physical Chemistry I*
 CHEM 362 Physical Chemistry II*
 CHEM 363 Quantum Chemistry & Molecular Spectroscopy
 OE 450 Nanomedicine
 PH 540 Computer Physics
*For CHE students, CHEM 361 and CHEM 362 may be substituted by CHE 303, CHE 304 and CHEM 360
Computational Biomedics
 BE 482/MA 482 Bioengineering Statistics
 BE 535/OE 535 Biomedical Optics
 BE 541/ECE 584 Medical Imaging Systems
 BMTH 310 Mathematical Biology
 BMTH 413 Computational Biology
DATA SCIENCE MAJOR (SECOND MAJOR ONLY)
Data Science is open to all students as a second major; this means that the student will have some other discipline as their primary major. Students whose primary major is in Computer Science, Software Engineering or Mathematics will find the Data Science program the easiest since there is considerable overlap between those programs and the Data Science requirements Students from other disciplines are also encouraged to participate, but will have to take more courses. All students are encouraged to take the individual courses in the program, regardless of whether they wish to fulfill the second major requirements. Learn more about Data Science requirements.
MINOR IN MATHEMATICS
Any student not pursuing a major or second major in either mathematics or in biomathematics may obtain a minor in mathematics by taking 10 or more mathematics courses as follows:
 Six courses in foundational mathematics
 Calculus, Matrix Algebra and Differential Equations, Introduction to Proofs : MA 111, MA 112, MA 113, MA 221, and either MA 222 or MA276.
 Introductory Statistics or Probability: either MA 223 or MA 381
 Sixteen additional credit hours of “upper division” courses:
 Courses selected from MA 222, MA 223, MA 276, all MA courses numbered 300 or higher (except MA351356 and MA450, MA492494, and MA496498), all BMTH courses numbered 300 or higher (except BMTH 496498), or other MA courses approved by the minor advisor for mathematics. Computer Science majors cannot use either MA 473 or MA 474 to satisfy both their computer science major requirements and the requirements of the mathematics minor.
Approval and Math Minor Form
All minors must be approved by the minor advisor and the student’s advisor. The department has a form for the planning and approval of a mathematics minor.
Notes and Limitations on Requirements:
 Almost all students are required to take six foundational courses as a requirement for their major; therefore only four "extra courses" are required for most students.
 Only MA111, MA112, MA113, MA221, MA222 and one of MA223, MA381, or MA382 can be counted towards any combination of the multiple minors offered by the mathematics department.
 No student can take both MA 371 and MA 373 for credit.
 No student can take both MA223 and MA382 for credit
 Except as noted above, if MA 381 is being counted towards the four additional courses then, MA 223 may be taken and counted towards the Introductory Statistics and Probability.
 Science and engineering, especially the most recent "high tech" developments, have sophisticated mathematical and statistical concepts and methodologies as their foundation. Thus a wellchosen set of courses for a mathematics minor (or a second major in mathematics) will greatly enhance a student's analytical and computational skills. Students thinking of going on to graduate school should especially give consideration to this option.
MINOR IN COMPUTATIONAL SCIENCE
Any student may obtain a minor in Computational Science by taking the following courses:
 Five courses in foundational mathematics: MA111, MA112, MA113, MA221, MA222
 Basic computing course: CSSE 120 or departmental equivalent of at least 4 credit hours
 Introductory Computational Science courses:
 MA332 Introduction to Computational Science
 MA342 Computational Modeling
 Four credit hours of applied Computational Science course from list A
 Four credit hours of additional Computational Science course from list B
List A: Applied Computational Science courses
 MA323 – Geometric Modeling
 MA439 – Mathematical Methods of Image Processing
 MA444 – Deterministic Models in Operations Research
 CSSE351 – Computer Graphics
 CSSE451  Advanced Computer Graphics
 CSSE413 – Artificial Intelligence
 CSSE453 – Topics in Artificial Intelligence
 CSSE461 – Computer Vision
 CSSE463  Image Recognition
 CE522  Advanced Finite Element Analysis
 ME422 – Finite Elements for Engineering Applications
 ME427  Introduction to Computational Fluid Dynamics
 ME511  Numerical Methods for Dynamic Systems Analysis
 ME522  Advanced Finite Elements Analysis
 4XX – Introduction to MEMS:Fabrication and Applications
 5XX – Advanced Topics in MEMS
 CHE521 – Advanced Chemical Engineering Computation
 BE510 – Biomedical Signal and Image Processing
 EMGT526  Technology Forecasting
 MA534/EMGT534  Management. Science
 ECE420  Nonlinear Control Systems
 ECE480//PH437 – Introduction to Image Processing
 ECE582/PH537 – Advanced Image Processing
 ECE483  DSP System Design
List B: Additional Computational Science courses
 MA/CSSE335  Introduction to Parallel Computing
 MA433  Numerical Analysis
 MA434 – Topics in Numerical Analysis
 MA446  Combinatorial Optimization
 CSSE304  Programming Language Concepts
 CSSE371  Software Requirements and Specification
Electives not on list A or B may be substituted with other courses with the approval of the area minor advisor.
The minor must be approved by the minor advisor for Computational Science and the student's advisor. The department has a form for the planning and approval of a minor.
Notes and limitations on requirements
 Almost all students are required to take the five foundational courses as a requirement for their major
 Most majors should be able to apply the basic computing requirement and/or one of the elective courses towards their major.
 Math majors or double majors are not allowed to count MA332 and MA342 for both the minor and the major.
 A student may not apply the four upperdivision courses toward both this minor and a math or statistics minor.
MINOR IN STATISTICS
Any student may obtain a minor in statistics by taking ten or more mathematics courses (24 credit hours) including the following:
4 credit hours – Foundational Statistics Course:
One of the following:
 MA 223 Engineering Statistics I
 MA 382 Introduction to Statistics with Probability
If MA 381 is taken before MA223/MA382, it will be strongly recommended the student take MA382 instead of MA223.
20 credit hours – Additional Coursework:
Five courses selected from the following list, at least two of which must be starred (*). Courses not on this list may count towards the minor if approved by the statistics minor advisor.
 MA 381 Introduction to Probability with Applications to Statistics
 MA 383 Engineering Statistics II
 MA 386 Statistical Programming
 MA 481 Mathematical Statistics
 MA 482* Bioengineering Statistics
 MA 483* Bayesian Data Analysis
 MA 485* Applied Linear Regression
 MA 487* Design of Experiments
 MA 480 Topics in Probability and Statistics
 EMGT472 Reliability Engineering
All minors must be approved by the minor advisor and the student's advisor. The department has a form for the planning and approval of a minor.
Notes and Limitations on Requirements
 Almost all students are required to take either MA223 or MA381 as a requirement for their major; therefore, only five “extra courses” are required for most students.
 Only one of MA223, MA381, or MA382 can be counted towards any combination of the multiple minors offered by the mathematics department.
 Mathematics majors or biomathematics majors must have at least 16 credit hours of separation between their major and this minor.
 No student can take both MA223 and MA382 for credit.
 Note that MA481, MA483, and EMGT472 have MA381 as a prerequisite.
Plan of Study
Total credits required: 195
Notes:
*MA 332  Introduction to Computational Science  may be taken instead of CSSE 220 but then MA 332 cannot be counted towards the 63 hours of mathematics coursework
**MA 200  Career Preparation  may be taken in the winter quarter of the sophomore year
Notes and Definitions
 The suggested four year plan is a guideline.
 Close consultation with the advisor on electives is required, especially for electives after the freshman year, or if a double major or minor is planned.
The following definitions of electives are specific to the Mathematics Department.
 Math Elective: A course either required by the concentration or a true math elective.
 Science Elective: Any Physical or Life Sciences elective (not Computer Science) at any level.
 Technical Elective: Nonmathematics courses numbered 200 or above in Engineering, Science or Economics; coursework in mathematics and biomathematics is not allowed.
 Free Elective: Any course.