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.
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 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 211 Differential Equations 
4 hrs. 
MA 212 Matrix Algebra and Systems of Differential Equations 
4 hrs. 
MA 275 Discrete and Combinatorial Algebra I 
4 hrs. 
MA 366 Functions of a Real Variable 
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 375 
Discrete and Combinatorial Algebra II 
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 coursework in courses numbered 300 or above (MA 351 MA 356, MA 450 excepted).
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 MA 190, 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 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,

The project must involve work done by the student(s) to solve a problem presented by an external sponsor. The written project submission must be signed by the student's project advisor (who must be a member of the mathematics department) and two additional members (who are approved by the project advisor), and must be presented publicly to the department. The additional members of the committee may include representatives of the sponsor.
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 written submission must be signed by the student's thesis advisor (who must be a member of the mathematics department) and two additional faculty members (who are approved by the thesis advisor), and must be presented publicly to the department.

Summary of Requirements 

Mathematics Coursework  core, concentration and electives (MA351MA356, MA450 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 and Social Science (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. 

AB 101 — Essential Biology (or higherlevel AB 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 instead of CSSE 220 but then MA 332 cannot be counted towards the 63 hours of mathematics coursework 


*** 
200 level or above nonmathematics 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. 
24 hrs. 

**** 
CLSK 100 — College and Life Skills MA 200 Career Preparation 
1 hr. 1 hr. 
SUGGESTED SCHEDULE
The schedule below is a suggested schedule only. Scheduling of courses may be altered, subject to 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 basic chemistry course is CHEM 105 unless a student is taking a second major or minor requiring CHEM 111 or credit for CHEM 111 has already been received. If CHEM 111 is taken instead of CHEM 105 then the order of the basic science electives in the freshman and sophomore is the second science course listed. Two science courses are to be taken in the winter quarter of freshman year
MATHEMATICS
Freshman Year 
Fall Term 
Credit

MA PH
CHEM RH HSS CLSK CSSE 
111 111
111 131
100 120 
Calculus I Physics I or General Chemistry I Rhetoric and Composition or Elective College and Life Skills Introduction to Software Development 
5
4 4
1 4 



18 
Winter Term 
Credit

MA PH
PH CHEM AB HSS
RH 
112 112
111 111 101
131 
Calculus II Physics II or Physics I General Chemistry I or Essential Biology (or higher level AB course) Elective or Rhetoric Composition 
5
4
4
4 



17

Spring Term 
Credit

MA PH
PH MA
HSS 
113 113
112 190 
Calculus III Physics III or Physics II Contemporary Mathematics Problems Elective 
5
4
2
4 



15 

Sophomore Year 
Fall Term 
Credit

MA MA AB
PH *CSSE 
211 275 101
113 220 
Differential Equations Disc. & Comb. Algebra I Essential Biology (or higher level AB course) or Physics III ObjectOriented Software Development 
4 4
4 4 



16





Winter Term 
Credit

MA
HSS **MA 
212
200 
Matrix Algebra and Systems of Differential Equations Science Elective Technical Elective Elective Career Preparation 
4
4 4 4 1 



17





Spring Term 
Credit

MA MA HSS 
381 371 
Introduction to Probability Linear Algebra I Technical Elective Elective 
4 4 4 4 



16






Junior Year 
Fall Term 
Credit

MA


Elective Technical Elective Technical Elective 
4 4 4 
HSS 

Elective or 
4 
RH 
330 
Technical and Professional Communication 




16 
Winter Term 
Credit

MA MA
HSS 
366 
Functions of a Real Variable Elective Technical Elective Elective or 
4 4 4
4 
RH 
330 
Technical and Professional Communication 




16 




Spring Term 
Credit

MA MA 

Elective Elective Technical Elective 
4 4 4 
HSS 

Elective or 
4 
RH 
330 
Technical and Professional Communication 




16 

Senior Year 
Fall Term 
Credit

MA
MA
MA
HSS 
491
492
496 
Intro to Math Modeling (2 hours) Senior Project I (2 hours) or Senior Thesis I (4 hours) Free Elective Free Elective HSS 
4
4 4 4 



16





Winter Term 
Credit

MA
MA MA 
493
497 
Senior Project II (2 hours) or Senior Thesis II (2 hours) Elective Free Elective Free Elective Free Elective 
2
4 4 4 4 



18





Spring Term 
Credit

MA
MA MA 
494
498 
Senior Project III (2 hours) or Senior Thesis III (2 hours) Elective Free Elective Free Elective 
2
4 4 4 



14







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 either the winter or spring 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.
 Free Elective: Any course.
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 55 credit hour core and a 16 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
I. Computational Science Core (55 credit hours)
Fundamentals (35 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 (1) Introduction to Software Development, or any of BE 100, CE 110, CHE 110 (plus 2 cr.), ME 123
 CSSE 132 or ECE130 Introduction to Computer Systems, Introduction to Logic Design
 MA 332 (2) Introduction to Computational Science, or any of CE 310 (plus 2 cr.) CHE 310, ME 323(plus 2 Credits)
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 upperlevel computational science course (see list below)
List of Advanced Electives
 BE 510 Biomedical Signal and Image Processing
 CHE 521 Advanced Chemical Engineering Computation
 CSSE 304 Programming Language Concepts
 ECE 480/PH 437 Introduction to Image Processing
 ECE 483 DSP System Design
 ECE 582/PH 537 Advanced Image Processing
 EMGT 526 Technology Forecasting
 EMGT 534/MA 534 Management Science
 MA 323 Geometric Modeling
 MA 348 Continuous Optimization
 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
 MA 490 Mathematical Biology
 ME 422 Finite Element Methods for Engineering Applications
 ME 427 Introduction to Computational Fluid Dynamics
 ME 522 Advanced Finite Elements Analysis
 Other courses may be counted as advanced electives upon approval by the Chair of Computational Science.
II. Area of Concentration (16 credit hours): Each student must complete 16 credit hours in 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.
Computational Methods
 MA 371 or MA 373 Linear Algebra I, Applied Linear Algebra for Engineers
 MA 433 Numerical Analysis
 Eight credit hours chosen from CSSE 304, CSSE/MA 473, MA 348, 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/PH 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 Specroscopy
 OE 570 Nanomedicine
 PH 540 Computer Physics
Computational Biomedics
 BE 482/MA 482 Bioengineering Statistics
 BE 535/OE 535 Biomedical Optics
 BE 541/ECE 597 Medical Imaging
 MA 49001 Mathematical Biology 1
 MA 49002 Mathematical Biology 2
AREA MINOR IN MATHEMATICS
A student, not pursuing a major or second major in mathematics may obtain an area minor in mathematics by taking 10 or more mathematics courses as follows:
 Six courses in foundational mathematics
Calculus, Differential Equations and Matrix Algebra: MA 111, MA 112, MA 113, MA 211, MA 212
Basic Probability and Statistics or Basic Statistics: one of MA 223, MA 381, or MA382
 Sixteen additional credit hours of “upper division” courses:
Courses selected from MA 275, all MA courses numbered 300 or higher (except MA351356 and MA450), or other MA courses approved by the area minor advisor for mathematics.
Approval and Math Minor Form
All area minors must be approved by the area 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, MA211 and one of MA223, MA381, or MA382 can be counted towards both a statistics minor and a mathematics minor.
 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 Basic Probability and Statistics.
 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.
AREA MINOR IN COMPUTATIONAL SCIENCE
Any student may obtain an area minor in Computational Science by taking the following courses:
 Five courses in foundational mathematics: MA111, MA112, MA113, MA211, MA212
 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
 MA348  Continuous Optimization
 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 area 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.
AREA MINOR IN STATISTICS
A student, not pursuing a major or second major in mathematics may obtain an area minor in statistics by taking ten or more mathematics courses (40 credit hours) including the following:
 Globally required mathematics courses
MA 111 Calculus I
MA 112 Calculus II
MA 113 Calculus III
MA 212 Differential Equations
 Required Introductory Statistics/Probability Courses:
MA 381 Introduction to Probability with Applications to Statistics,
One of MA223 Engineering Statistics I or MA382 Introduction to Statistics with Probability. If MA 381 is taken before MA223/MA382 the student will be strongly recommended to take MA382.
 Required Second Statistics Course
One of MA383 Engineering Statistics II or MA482 Bioengineering Statistics
 Electives
3 courses (12 credits) selected from the following list, at least two of which must be starred. Statistics courses not on this list may count towards the minor if approved by statistics area minor advisor.
 MA 385* Quality Methods
 MA 386* Statistical Programming
 MA 387* Statistical Methods in Six Sigma
 MA 371 Linear Algebra or MA373 Applied Linear Algebra for Engineers
 MA 445 Stochastic Models in Operations Research
 MA 481* Mathematical Statistics
 MA 485* Regression and Time Series Analysis
 MA 487* Design of Experiments
 MA 480* Topics in Probability and Statistics
All area minors in Statistics must be approved by the statistics area minor advisor and the student's advisor. The department has a form for the planning and approval of a statistics minor.
Notes and Limitations on Requirements
 Almost all students are required to take the four globally required mathematics courses plus one probability or statistics course as a requirements for their major, therefore only five "extra courses” are required for most students
 Only MA111, MA112, MA113, MA212 and one of M223, MA381, or MA382 can be counted towards both a statistics minor and a mathematics minor.
 No student can take both MA 371 and MA 373 for credit.
 No student can take both MA223 and MA382 for credit.