Mathematics

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 decision-making 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 Rose-Hulman 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 non-technical 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 well-rounded 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.  Speak about mathematics in an articulate, sound, and well-organized fashion.
  • SLO 4.3.  Write a formal mathematical thesis or report.

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. A mathematics major must additionally complete an 8-credit hour capstone experience. 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, MA351-MA356, 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.

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 high-performance computing is essential. The Computational Science major provides Rose-Hulman 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.

Computational Science Program Student Learning Objectives:

Graduates with a second major in CPLS will have an ability to:

1.) Develop goals for a computational model such that the results will inform a scientific/engineering decision or provide a desired level of understanding

2.) Choose a computational modelling approach that meets the goals and implement it

3.) Validate a computational model of a complex phenomenon or system and demonstrate that the goals have been met

Requirements for a second major in Computational Science (72 credit hours)

The second major in Computational Science is open to all students. It requires 72 credit hours, including a 52 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 (52 credit hours)

Fundamentals (31 credit hours)

• MA 111, 112, 113 Calculus I, II, III
• MA 221 Matrix Algebra and Differential Equations I
• MA 222 Matrix Algebra and Differential Equations II
• 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 (21 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 and MDS 442 Applied Computational Modeling
• MDS 442 Applied 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 upper-level 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 Discrete-Time 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 MA351-356 and MA450, MA492-494, and MA496-498), all BMTH courses numbered 300 or higher (except BMTH 496-498), 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 well-chosen 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 upper-division 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* Biostatistics
• 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

1. 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.
2. Only one of MA223, MA381, or MA382 can be counted towards any combination of the multiple minors offered by the mathematics department.
3. Mathematics majors or biomathematics majors must have at least 16 credit hours of separation between their major and this minor.
4. No student can take both MA223 and MA382 for credit.
5. Note that MA481, MA483, and EMGT472 have MA381 as a pre-requisite.