# Mathematics - Undergraduate

Chair: Marcia Groszek

Vice Chair: Sergi Elizalde

Professors V. Chernov, P. G. Doyle, A. Gelb, S. Elizalde, M. J. Groszek, P. J. Hanlon, R. C. Orellana, S. D. Pauls, D. Rockmore, T. R. Shemanske, J. Voight, D. I. Wallace, D. L. Webb, D. P. Williams, P. Winkler; Associate Professors C. J. Sutton , J. D. Trout, E. van Erp; Assistant Professor A. Auel, F. Fu, Y. Lee, I. Petkova; Research Instructors S.G. Allen, N. Lafreniere, W.M. Lord, T. Ma, M. Tassy, D. van Wyk, Z. Wang; Visitors E. Assaf, J. Glaubitz, M. Kobayashi, A. Kulkarni, S. Lin, K. Kaveh, J. Petok, C. Schembri, A.J. Song, C. M. Wong, B. Zhou; Adjunct Professor E. Demidenko.

*To view Mathematics Undergraduate courses, click here. *

*To view Mathematics Graduate requirements, click here. *

*To view Mathematics Graduate courses, click here. *

**Introductory Courses**

The four courses MATH 1, MATH 3, MATH 8, and MATH 13 provide a coherent four-term sequence in calculus. MATH 1, MATH 3 and MATH 8 cover the basic calculus of functions of a single variable, as well as vector geometry and calculus of scalar-valued functions of several variables. In addition, the latter two courses are prerequisite for many advanced courses in Mathematics and Computer Science. MATH 13 covers the basic calculus of vector-valued functions of several variables. MATH 11 is a special version of MATH 13 and MATH 9 is a special version of MATH 8, both for first-year students with two terms of advanced placement. Most students planning advanced work in mathematics or the physical sciences will need an additional course in calculus, MATH 23. Students with two terms of advanced placement credit who possibly are interested in a mathematics major or minor should consider MATH 17 as an option in their second term. MATH 17, “An Introduction to Mathematics Beyond Calculus,” is a course designed for students interested in learning about some of the aspects of mathematics not usually encountered in the first years of mathematical studies. Topics change from year to year but may include aspects of combinatorics, algebra, analysis, number theory, geometry, and/or topology. Students planning to take upper-level mathematics courses are encouraged to take MATH 22 or MATH 24 (linear algebra) early in their curriculum. MATH 19, Introduction to Set Theory, is an introductory course that also provides good preparation for upper-level mathematics courses.

A student wishing to devote only two to three terms to the study of mathematics is encouraged to choose among courses MATH 1, MATH 3, MATH 5, and MATH 10. MATH 1 and MATH 3 will introduce the student to the ideas and applications of the differential and integral calculus depending on their background. MATH 5 is a topics and sometimes interdisciplinary course. Recent topics include “Machine Readings: Text Analysis in the Information Age,” “Geometry in Art and Architecture,” “Applications of Calculus to Medicine and Biology,” “Music and Computers,” “The Mathematics of Music and Sound, ”Fundamental Applied Mathematics for the Sciences” and “Geometry and the Imagination.” MATH 10 covers the fundamental concepts of statistics.

**The Mathematics Department offers two distinct majors: (I) The Major in Mathematics and (II) The Major in Mathematical Data Science.**

**I. THE MAJOR IN MATHEMATICS**

The major in mathematics is intended both for students who plan careers in mathematics and related fields, and for those who simply find mathematics interesting and wish to continue its study. The content of the major is quite flexible, and students may select courses largely to reflect their interests. Students who major in mathematics have an opportunity to participate in activities that bring them in close contact with a faculty member—for example, through a small seminar or through an independent research project under the direction of a faculty member. In addition to regular course offerings, a student with specialized interests, not reflected in our current course offerings, can arrange for an independent reading course. Proposals for independent activities should be directed to the Advisor to Majors.

In general, the mathematics major requires the student to pass eight mathematics or computer science courses beyond prerequisites. At least six of the required eight courses must be mathematics, and at least four of these courses must be taken at Dartmouth. In addition, a student must fulfill the College’s requirement for a culminating experience in the major (see below). Additional requirements for honors are described below in a separate section.

Students are encouraged to take MATH 22/ MATH 24 as soon as feasible, since not only is it a prerequisite to many upper-division courses, but also the level of mathematical sophistication developed in MATH 22/ MATH 24 will be presumed in many upper-division courses for which MATH 22/ MATH 24 is not an explicit prerequisite.

**Mathematics Major Requirements**

Prerequisite Courses: MATH 3; MATH 8; MATH 13; MATH 22 or MATH 24

Requirements: A major in mathematics comprises at least eight courses in addition to the prerequisites, as well as a culminating experience (which may or may not be part of the eight major courses). These eight courses must include:

- (Algebra) MATH 31 or MATH 71;
- (Analysis) At least one of MATH 35, MATH 43, or MATH 63;
- Six additional Mathematics/Computer Science courses numbered 20 or above for Mathematics, and 30 or above for Computer Science.

Caveats:

Also acceptable: MATH 17, MATH 19

Not acceptable: MATH 97, COSC 99

At most two Computer Science courses may be used. The culminating experience requirements are described in a separate section below.

**Choosing Courses for the Major**

While the student interested only in a general exposure to mathematics may choose their major courses subject only to the constraints above, those with more focused interests (pure mathematics, applied mathematics, and mathematics education), will want to consider the course recommendations below.

- (Pure Mathematics) For students interested in pure mathematics, MATH 24 is preferable to MATH 22 as prerequisite.
We recommend that the following courses be included among the eight courses needed for the major:

(Algebra) MATH 71 and MATH 81;

(Analysis) MATH 63, and MATH 43 or MATH 73;

(Topology/Geometry) MATH 54, and at least one of MATH 32, MATH 42 or MATH 72, MATH 74.

Students planning to attend graduate school should take substantially more than the minimum requirements for the major. In particular, such students are strongly urged to take both MATH 43 and MATH 73; moreover, undergraduates with adequate preparation are encouraged to enroll in graduate courses.

- (Applied Mathematics) Applied mathematics now encompasses a wide expanse of mathematical activity in the sciences, ranging across finance, sociology, psychology, biology, physics, computer science, and engineering. Students interested in applied mathematics, especially those considering graduate school in applied mathematics or any of the sciences, are advised to take MATH 23, MATH 20 or MATH 60, MATH 46, and MATH 40.
We recommend choosing additional courses from among the following: MATH 26, MATH 28, MATH 36, MATH 38, MATH 42, MATH 43, MATH 46, MATH 53, MATH 75, MATH 76.

We do not make any specific recommendations concerning the choice of MATH 22 versus MATH 24 as prerequisite and the choices for requirements (1) Algebra and (2) Analysis; these choices depend on the interest of the student.

All students planning to attend graduate school should take substantially more than the minimum requirements for the major. In particular, undergraduates with adequate preparation are encouraged to enroll in graduate courses.

- (Mathematics Education) Certification as a public school Mathematics teacher is available through partnership with the Education Department. Contact the Education Department for details about course requirements.
Students who are considering a career in teaching should pay close attention to the recommendations of the National Council of Teachers of Mathematics (NCTM). The NCTM has endorsed a series of recommendations for a suggested course of study for those people interested in teaching mathematics at the secondary level. In general, their recommendations (www.nctm.org) are for a vigorous course of study. At the moment, these recommendations far exceed the requirements for obtaining a teaching certificate, but indicate the direction in which the NCTM hopes that educators will proceed. Highly qualified teachers in the elementary and secondary schools are of vital national importance, and these guidelines should be carefully considered. Dartmouth courses that closely fit the recommendations of the NCTM are (in addition to the prerequisites): MATH 20 or MATH 60; MATH 23 or MATH 36; MATH 25 or MATH 75; MATH 28, MATH 38 or MATH 68; MATH 31 or MATH 71; MATH 32 or MATH 42 or MATH 72; MATH 35 or MATH 43 or MATH 63; MATH 40

**Culminating Experience**

The Department will accept any of the following in satisfaction of the requirement of a culminating experience:

- Submission of an Honors thesis acceptable for honors or high honors.
- Satisfactory completion of any graduate course in mathematics except MATH 147.
- Satisfactory completion of a one-term independent research project (subject to approval by the advisor to majors).
- Satisfactory completion of an advanced undergraduate course from among: MATH 66, MATH 68, MATH 69, MATH 70, MATH 72, MATH 73, MATH 74, MATH 75, MATH 76, MATH 81, MATH 86, MATH 96.

**II. THE MAJOR IN MATHEMATICAL DATA SCIENCE**

Statistics has become a ubiquitous tool not only in traditional areas in the natural and social sciences, but in emerging cross-disciplinary fields in data science. The major combines a solid theoretical foundation with application to one or more fields of study.

Prerequisites: MATH 3, MATH 8, MATH 13; MATH 22 or 24; COSC 1 or other programming experience.

Four course are required: MATH 20, MATH 40, MATH 50, and MATH 70.

Students must also satisfy a computational requirement for the major, selecting at least one of either COSC 35 or COSC 74. Students taking both of these courses may count additional selections as partially satisfying the next requirement.

Further students must select three courses from the following list. At the discretion of the Adviser to Majors, other courses can be substituted.

MATH 76, MATH 86, MATH 96, MATH 116, MATH 120, MATH 126;

QSS 41, QSS 45;

QBS 149.

Culminating Experience: Majors in Mathematical Data Science must complete a data intensive research project to satisfy the culminating experience. Students may complete this by writing a thesis, completing an independent research project, or completing a course with a significant statistical project. The culminating experience must be approved in advance by the Adviser to Majors.

**Minors in Mathematics **

The following minors are available to all students who are not majoring in mathematics and who do not have a modified major with the Mathematics Department. For each minor, the prerequisites and required courses are listed below. Approval of a minor can be obtained through the Department’s Advisor to Mathematics Majors.

I. Mathematics

*Prerequisites:*MATH 3, MATH 8, MATH 13, MATH 22

*Required Courses (4 courses):*MATH 31 or MATH 71; MATH 35 or MATH 43 or MATH 63; plus two other Mathematics courses numbered 20 or above. MATH 17 is also acceptable.

II. Applied Mathematics for Physical and Engineering Sciences

*Prerequisites:*MATH 3, MATH 8, MATH 13 or MATH 22, COSC 1

*Required Courses (4 courses):*MATH 23, MATH 46, MATH 40* or MATH 60, MATH 43 or MATH 53 or MATH 76.

III. Applied Mathematics for Biological and Social Sciences

*Prerequisites:*MATH 3, MATH 8, MATH 13, MATH 22

*Required Courses (5 courses): *MATH 20, MATH 23, MATH 27, MATH 28 or MATH 36, MATH 40* or MATH 53 or MATH 76.

IV. Mathematical Biology

*Prerequisites:*MATH 3, MATH 8, MATH 13, and either MATH 22 or MATH 23; MATH 10 or BIOL 29

*Required Courses (4 courses): *Two courses chosen from among MATH 26, MATH 27, MATH 36, MATH 40*; and two courses chosen from among BIOL 21/BIOL 51, BIOL 39, BIOL 47, BIOL 59, COSC 75, MATH 37, MATH 47 and ENGS 41.

V. Mathematical Logic

*Prerequisites:*MATH 3, MATH 8, MATH 24 (MATH 22 by permission)

*Required Courses (5 courses): *MATH 19; MATH 29 or COSC 39; MATH 39 or MATH 69; MATH 63 (not MATH 35); one additional course chosen from among MATH 31, MATH 71, MATH 54, PHIL 32, MATH 29 if COSC 39 is taken as a required course.

VI. Mathematical Physics

This minor is sponsored by the faculty in Mathematics and Physics. It may be combined with majors in either of the two departments, or any other department. Students majoring in both physics and mathematics cannot take the minor.

*Prerequisites:*PHYS 13, PHYS 14, PHYS 19 (or PHYS 15 and PHYS 16), and PHYS 40, MATH 3, MATH 8, MATH 13, and MATH 22 or MATH 24.

*Requirements:* A total of four additional courses are required. These must include MATH 23 and MATH 46. Mathematics majors must choose two elective physics courses from the following list; physics majors must choose two elective mathematics courses; students majoring in a department other than mathematics or physics must choose one math and one physics course.

PHYS 30, PHYS 31, PHYS 41, PHYS 43, PHYS 44, PHYS 47, PHYS 66, PHYS 50, PHYS 72, PHYS 75, PHYS 77, PHYS 90. [NOTE: PHYS 50 requires PHYS 40]

MATH 31 or MATH 71, MATH 42, MATH 43, MATH 53, MATH 54, MATH 63, MATH 66, MATH 73, MATH 76.

An advanced undergraduate or graduate level physics or mathematics course may be substituted, with permission from the physics or mathematics department undergraduate advisor. No course may count towards both the major and minor.

VII. Mathematical Finance

Mathematical Finance is an interdisciplinary minor that will provide students with the opportunity to see how mathematics, economics and computer science can be used to study theoretical and applied problems arising in economics, finance and risk management. The minor requires students to take 5 courses beyond the prerequisites. To allow for maximum flexibility in scheduling, students are encouraged to complete (either MATH 60 or both MATH 20 and MATH 40*), MATH 23 and COSC 1 by the end of their sophomore year as these courses are requirements for MATH 86, which serves as the capstone for the minor.

*Prerequisites:* MATH 3, MATH 8, MATH 13, MATH 22 or MATH 24, COSC 1, ECON 21.

*Required Courses (5 courses): *MATH 60 or both Math 20 and Math 40*, MATH 23, MATH 86, ECON 26, ECON 36

VIII. Complex Systems

Complex Systems is an interdisciplinary field that integrates ideas and techniques from mathematics and the sciences to study emergent phenomena, generally characterized by an evolutionary nature in which the “whole is more than the sum of its parts.” Examples include the collective of species-species interactions that give rise to an ecosystem, the aggregate of buyer-seller interactions that create economies or markets, the neuron-neuron signalings that create the brain and mind, or individual social relationships that result in a coherent society, all of which display properties of adaptation and selection and multiscale structure. The study of complex systems is highly interdisciplinary, at its best, using insights into the etiology of one phenomenon to inform another, a kind of analogical reasoning made possible through the use of common mathematical and computational tools.

The minor requires students to take 5 courses beyond the prerequisites and includes the accomplishment of an integrative independent project, advised by a faculty member in mathematics, as evidence of the ability to integrate these ideas into a coherent whole.

*Prerequisites:* MATH 3, MATH 8, MATH 13, MATH 22 or MATH 24; One of COSC 1, COSC 3, COSC 8, COSC 10 or ENGS 20; BIOL 11 or PHYS 30/ENGS 30.

*Required Courses:*

(a) Two of MATH 27, MATH 36, MATH 53, MATH 76, at least one of which must be either MATH 36 or MATH 76.

(b) MATH 87 (Note that students will need to find an advisor for their MATH 87 project, which must be integrative in nature);

(c) One course from among the following: BIOL 15, BIOL 16: ECON 29, ECON 49, ECON 76; SOCY 16, SOCY 27; EARS 15, EARS 67; PSYC 40/COSC 79, PSYC 46; PHYS 30, PHYS 43; COSC 58, COSC 75, COSC 79, COSC 81; CHEM 41, CHEM 75; ENGS 30, ENGS 35, ENGS 114. Students may substitute for the additional course either in biology, chemistry, economics, sociology, environmental sciences, computer science, physics, or psychological and brain sciences with the approval of the advisor to majors.

IX. Minor in Statistics

It is difficult to overstate the importance of statistical training in the twenty-first century, especially in today's (and the future) job market. Students completing this minor will have the skills to perform advanced statistical analyses and with the accompanying accreditation will be able to compete successfully for employment and professional school admission.

*Prerequisites:*MATH 3, MATH 8, MATH 13; MATH 22; COSC 1 or ENGS 20 or Permission of the Instructor.

*Required Courses (4 courses): *MATH 20 (or MATH 60), MATH 40*, MATH 70 and one other major level course in Mathematics. At the discretion of the Advisor to Majors, students may substitute MATH 30, BIOL 59, COSC 74, or ENGS 107 for MATH 70.

**NOTE: Prior to Fall 2014, Math 40 was numbered Math 50 and can be used accordingly. *

*The Honors Program in Mathematics *

A student who satisfies the requirements of the College for admission to the Honors Program and is interested in doing independent work is strongly encouraged to participate in the departmental Honors Program. Students who successfully complete the Honors Program will have their degrees conferred with ‘Honors’ or ‘High Honors’ in mathematics; high honors is awarded only if the student submits a written thesis. Interested students should read this section of the ORC carefully and consult the Advisor to Majors. This program can be especially important to those who contemplate graduate work in mathematics or a related field.

Admission: Admission to the Honors Program requires a general College average of B, and a B average in the Mathematics Department at the time of admission and at the time of graduation. Moreover, a B+ average is required in the work of the Honors Program. The B average in the Department is computed as follows: Courses prerequisite to the major and undergraduate research courses (MATH 97) are not counted, but all other courses titled (or cross listed with) mathematics which the student has taken are counted, whether or not these courses form part of the student’s formal major. In the case of a modified major, this average may include courses outside the Mathematics Department. The B+ average required in the work of the Honors Program is defined to be a grade of B+ given by the faculty advisor on the research project. Questions about this requirement should be directed to the Advisor to Majors.

Requirements: Under the supervision of a faculty member, the student must complete an independent research project or thesis beyond what is required as part of a course. Often the subject of the project or thesis will be motivated by concepts or the content of an advanced seminar or course in which the student has participated, and, typically, the project or thesis will be completed over a period of three terms. The student should consult with his/her prospective faculty advisor and submit to the Advisor to Mathematics Majors a brief written proposal of the project that has the written approval of the faculty advisor. The Advisor to Majors will then review the student’s proposal and the courses that have been selected for the Honors major. Approval of the proposal and course selection constitutes formal admission into the Honors Program. This procedure should be completed by the beginning of fall term of the student’s senior year. The student may then register for (at most two terms of) MATH 97, Undergraduate Research.

In the first week of the student’s final term in residence, the student must register with his/her faculty advisor for ‘Honors Thesis/Project Supervision.’ This is not an official College course; rather, it represents a declaration of intent to the Department that the student wishes to be considered for honors at the time of graduation. Forms for this purpose are available from the Advisor to Majors. No student who has failed to file this intent form with the Advisor to Majors will be considered for honors in the major.

After the thesis is completed and submitted to the faculty advisor, the student will give a short presentation of their results. The advisor can then offer a recommendation for honors or high honors on behalf of the student; this recommendation must be ratified by a vote of the Department faculty.

*Modified Majors *

* *

*Modified Major with Mathematics as the primary Department*

Prerequisite: Same as mathematics major plus some additional prerequisites from modifying major (subject to approval of Advisor to Majors).

Requirements: An algebra and an analysis course that satisfy the requirements of the mathematics major, together with four additional courses that normally count towards the major in mathematics, including one course that satisfies the culminating experience requirement (choice subject to approval of Advisor to Majors). Subject to the approval of the Advisor to Majors, the algebra course can be replaced by one of the following courses: MATH 28, MATH 38, MATH 54, MATH 69.

Four additional courses from the secondary department selected with the approval of the Advisor to Majors and the secondary department. In particular, these ten non-prerequisite courses must form a coherent unit that renders the modified major academically more valuable than an abbreviated major together with a minor in the secondary department.

*Mathematics Modified with Biology *

Prerequisite: MATH 3, MATH 8, MATH 13, MATH 22. Students may replace MATH 22 with MATH 24 if they prefer.

Requirements: All students pursuing this modified major must complete an algebra and an analysis course which would satisfy the requirements for the Mathematics major, together with four additional courses that normally count towards the major in mathematics. One of these courses must fulfill the culminating experience requirement and two of these courses must be from the following list: MATH 20, MATH 27, MATH 36, MATH 46, MATH 40*, MATH 53 and MATH 76.

All students pursuing the modified major must take one course from among BIOL 12 (Cell Structure and Function), BIOL 13 (Gene Expression and Inheritance), BIOL 14 (Physiology), BIOL 15 (Genetic Variation and Evolution) or BIOL 16 (Ecology) and three other biology courses from the list below. One additional course from among BIOL 12 - 16 may be used as one of the three additional courses. These should be chosen in consultation with the departments. Some possible areas of focus include:

Genomics: BIOL 47 (From Data to Analysis),

Biostatistics & Experimental Design: BIOL 22 (Methods in Ecology), BIOL 29 (Biostatistics), BIOL 59 (Advanced Biostatistics).

Ecology: BIOL 21/51 (Population Ecology), BIOL 27 (Animal Behavior), BIOL 58 (Advanced Community Ecology), BIOL 60.01 (Evolutionary Ecology).

Molecular & Cellular Biology: BIOL 38 (Experimental Genetic Analysis), BIOL 40 (Biochemistry), BIOL 45 (Molecular Biology), BIOL 66 (Molecular Basis of Cancer, BIOL 69 (Cell Signaling), BIOL 71 (Advanced Topics in Cell Biology).

In every case, the collection of courses must be approved. Majors should demonstrate a coherent intellectual rationale*.*

*Modified Major in Mathematics with Philosophy*

Prerequisite: MATH 3, MATH 8, MATH 13, MATH 24 (MATH 22 with permission); PHIL 1, PHIL 6.

Requirements: To complete the major, it is necessary to complete successfully at least six mathematics courses and four philosophy courses (as described below) in addition to the prerequisites, including a culminating experience. MATH 69 satisfies the culminating experience requirement. Modified majors may participate in the honors program and write an honors thesis. The required courses are: MATH 19, MATH 31 or MATH 71, MATH 35 or MATH 63, MATH 29 (COSC 39 may be substituted), MATH 39 or MATH 69, and an additional mathematics course numbered 20 or above, excluding MATH 97 (MATH 17 is also acceptable). Four philosophy courses chosen from among PHIL 26, PHIL 27, PHIL 29, PHIL 32, PHIL 33, PHIL 34.

* *

*Modified Major for Complex Systems
*

* *

Complex Systems is an interdisciplinary field that integrates ideas and techniques from mathematics and the sciences to study emergent phenomena, generally characterized by an evolutionary nature in which the “whole is more than the sum of its parts.” Examples include the collective of species-species interactions that give rise to an ecosystem, the aggregate of buyer-seller interactions that create economies or markets, the neuron-neuron signalings that create the brain and mind, or individual social relationships that result in a coherent society, all of which display properties of adaptation and selection and multiscale structure. The study of complex systems is highly interdisciplinary, at its best, using insights into the etiology of one phenomenon to inform another, a kind of analogical reasoning made possible through the use of common mathematical and computational tools.

The major requires students to take 10 courses beyond the prerequisites, 6 in mathematics and 4 in other departments, and includes the accomplishment of an integrative independent project, advised by a faculty member in mathematics, as evidence of the ability to integrate these ideas into a coherent whole. This independent project satisfies the culminating experience requirement. Modified majors may participate in the mathematics department honors program; with the approval of the advisor to majors, the independent project may comprise part of an honors thesis project.

In every case, the collection of courses should be approved. Majors should demonstrate a coherent intellectual rationale.

Prerequisites: MATH 3, MATH 8, MATH 13, MATH 22 or MATH 24; One of COSC 1, COSC 3, COSC 8, COSC 10, or ENGS 20; BIOL 11 or PHYS 30/ENGS 30.

Required Courses:

(a) A course in differential equations (MATH 23 or MATH 46);

(b) A course in probability or statistics (MATH 20, MATH 40*, or MATH 60);

(c) Two courses from among MATH 27, MATH 36, MATH 53, MATH 76, at least one of which must be either MATH 36 or MATH 76.

(d) MATH 87 (Note that students will need to find an advisor for their MATH 87 project, which must be integrative in nature);

(e) One course from among MATH 31, MATH 35, MATH 43, MATH 63, and MATH 71.

(f) Four courses chosen from BIOL 15, BIOL 16; ECON 29, ECON 49, ECON 76; SOCY 16, SOCY 27; EARS 15, EARS 67; PSYC 40/COSC 79, PSYC 46; PHYS 30, PHYS 43; COSC 58, COSC 75, COSC 81; CHEM 41, CHEM 75; ENGS 30, ENGS 35, ENGS 114. With the approval of the advisor to majors, students may replace up to two of these courses with other appropriate courses in biology, chemistry, economics, sociology, environmental sciences, computer science, physics, or psychology, or, with a compelling rationale, another department.

*Courses *

Course Numbering System: For most courses numbered 20 or above, the last digit in the course number indicates the field of mathematics as follows: probability and statistics, 0; algebra, 1; geometry, 2; analysis, 3; topology, 4; number theory, 5; applications, 6; combinatorics, 8; logic and foundations, 9.

Course Prerequisites: In all cases in which a prerequisite to a course is listed, the honors or advanced placement equivalent of that course may be substituted. For example, wherever MATH 13 appears as a prerequisite, MATH 14 will serve. MATH 11 and MATH 12 also serve in place of MATH 13 as a prerequisite.

* *