# Attachment 2

## Attachment 2: Examinations

Information on the Ph.D. qualifying exam can be found in Sections (3.6) and (3.7).

**1. The M.S. Basic Exam**

Information on the Basic Exam: The Basic Exam is given to M.S. students prior to beginning their program in order to assess their background in introductory real analysis and linear algebra.

(a) MS Basic Exam Format: Each exam consists of a 2-hour written exam, a take-home exam, and an oral discussion on the take-home part.

(b) MS Basic Exam Schedule: The exam will be given on the Friday before the start of the Fall semester. The take-home part should be turned in by noon of the Tuesday of the first week of the semester. The oral discussion is optional and at the discretion of the graders in case more information is needed. The oral discussion will be scheduled on the Thursday of the first week of the semester.

(c) Content of the MS Basic Exam

(i) Topics to be covered in Advanced Calculus: Elementary properties of Open/closed/compact/connected sets in R^n. Numerical sequences and series. Limits, Cauchy sequences, convergence. Continuity. Continuity and compactness/connectedness. Uniform continuity. Sequences and series of functions; uniform convergence. Calculus of real-valued functions: Differentiation, mean value theorems, Taylor's theorem. Definition and existence of the Riemann integral. Fundamental Theorem of Calculus. Integration and differentiation of series/sequences of functions.

(ii) Topics to be covered in Linear Algebra: Vector spaces, linear independence, basis, dimension, linear transformation, and matrix representations, rank, range space, null space, eigenvalues and eigenvectors, diagonalizations, canonical forms, inner product spaces, othogonal basis, symmetric and hermitian matrices and properties.

(d) Grading of the MS Basic Exam: The examination committees will send the graduate program committee their course recommendations within a 7 day period after the written exam is conducted. These may include advanced calculus Math 451, or real analysis Math 551, and/or linear algebra Math 343, Math 441, Math 543. The recommendation will be based on the student’s background, and performance on the exam.

Textbooks:

__Advanced Calculus__

Elementary Analysis: The Theory of Calculus, by Kenneth Ross (used for Math 451)

Principles of Mathematical Analysis, Rudin (a standard advanced calculus text)

__Linear Algebra__

Elementary Linear Algebra, Kolman (used for undergraduate linear algebra)

Introduction to Linear Algebra, Strang (used for applied linear algebra)

Linear Algebra, Hoffmann & Kunze (used)

**2. The M.S. Advanced Exam/Ph.D. Entrance Exam**

Two times each year, in the late spring and early fall, the Department administers four subject area exams over a one-week period. The four subjects are real analysis, algebra, topology and differential equations. Each exam is a three hour written exam. These subject area exams together constitute the M.S. Advanced Exam and the Ph.D. Entrance Exam. Each subject area exam taken by the student is graded, in terms of decreasing performance, as either Ph.D. level pass, M.S. level pass, or fail. For a student in the M.S. program, the M.S. Advanced Exam is passed by achieving at least an M.S. level pass in two subject areas. For a student in the Ph.D. program, the Ph.D. Entrance Exam is passed by achieving a Ph.D. level pass in two subject areas.

**3. Policies and time limits for the M.S. Advanced Exam/Ph.D. Entrance Exam**

(a) A student may attempt any set of subject area exams each time they are administered (i.e. one subject area exam or up to four).

(b) A student may attempt a given subject area exam no more than three times. This is a cumulative overall limit that applies while the student is in either degree program. Thus, for instance, if a student attempts a subject area exam twice while in the M.S. program, they may attempt that subject area exam at most once more if they subsequently enter the Ph.D. program.

(c) Passing the M.S. Advanced exam is a graduation requirement in Option A and Option C of the M.S. program. Students have up to three academic years in which to pass from when they first enroll in the M.S. program. Specifically, a student first enrolling in the M.S. program in Year N must pass the M.S. Advanced exam by the end of the Spring semester in Year N+3.

(d) Students entering the Ph.D. program must pass the Ph.D. Entrance Exam within two full academic years from when they first enroll. Specifically, a student first enrolling in the Ph.D. program in Year N must pass the Ph.D. Entrance Exam by the end of the Spring semester in Year N+2.

(e) Ph.D. students that do not pass the Ph.D. Entrance Exam within the allowed period will be dropped from the Ph.D. program. In this event they may be eligible for an M.S. degree based on their course work and exam outcomes, or continue work toward an M.S. degree.

(f) The awarding of a teaching assistantship carries with it the expectation of higher standards. In this regard, a GTA in the Ph.D. program is expected to require no more than two attempts to receive a Ph.D. pass on any given subject area exam attempted. If they fail to achieve this standard, GTA support may be withdrawn. Also note the expectations in (2.5.6) that apply to students continuing with GTA support from the M.S. to the Ph.D. program.

(h) Part-time students are expected to comply with the deadlines above. Extensions of the time in which to pass can be considered by the Graduate Committee based on the particular circumstances involved, including the overall number of credits taken by the student.

**4. Exam Schedule of M.S. Advanced Exam/Ph.D. Entrance Exam**

(a) Second week of classes in Fall.

(b) Two weeks before the final exam week of the Spring term.

(c) Students must notify the graduate director of their intention in writing to take the exams a month before the exams are given, specifying the subject area(s) to be taken.

(d) The one month advanced notification stated above may be waived by the graduate director.

**5. Examining Committees**

(a) Each subject area exam has an examining committee consisting of two designated faculty members and the graduate director as a member ex officio.

(b) Members will serve on the committee for a two year period, with renewal possible.

(c) Members (including replacements for absent members) of each committee will be appointed by the Graduate Director in consultation with the Graduate Committee.

(d) Each examining committee acts independently from each other examining committee in preparing and grading its exam.

(e) Grades are assigned to exam papers based on the judgment of the examiners and not on particular percentages.

(f) Instructors who regularly teach the related subject should be expected to serve on an examining committee in the field if asked.

**6. Content of Exams**

(a) Lists of topics for each subject area exam are to be made up by appropriate faculty (committees) and posted online. The lists are effective for a 5-year period, and will be reviewed and revised (if necessary) every 5 years. Topics are to be standard.

(b) Faculty teaching the related courses are expected to cover a reasonable amount of the material listed in (6a). However, the exams are topic oriented, not course oriented and students should not assume that a given course sequence will cover exam content in its entirety.

(c) Samples of previous exams will be made available to students.

(d) Students are allowed to take the exams without taking the related courses at WVU.

**7. Notification**

(a) A memo from the graduate director announcing the time, place, and the examiners is to be circulated to all faculty and graduate students at the beginning of the Fall semester and at the end of March in the Spring semester.

(b) Each examining committee is to report to the graduate director the outcome of each exam for each student within 5 (five) working days upon the completion of the exams.

(c) Written notification from the graduate director concerning the decision of the examination committee is to be sent to the student within 10 (Ten) working days upon the completion of all the exams.

**8. Appeals and Exceptions**

Students may speak with examiners, as available and agreeable, concerning their performance on a given subject area exam, and may review their answers. The exams themselves remain in the custody of the department and are not returned to the students. Since the grade on an exam is a collective decision of the examiners, students will normally not see the scores assigned by individual examiners. A student may appeal their grade to the Graduate Committee. However if the examiners concur in the assigned grade, the Graduate Committee will usually defer to the examination committee’s judgment unless there is strong evidence that the grade given is inconsistent with usual practice or is based on a mistake or a misreading of the solutions.

Exceptions to the policies in (3) above may be considered by the Graduate Committee on rare occasions when warranted. Personal circumstances, including medical problems, accidents, hardships, etc. may play a role. However the general outlook of the Graduate Committee is that the allowed three attempts and the time period provided in which to pass the respective exams already build in substantial opportunity to overcome most individual difficulties. In considering exceptions, the Graduate Committee will review not just the reasons for the exception, but also the student’s academic and exam performance and their potential for success. Normally, to consider an extension of time to pass the Ph.D. Entrance Exam, the student will be expected to have obtained at least a Ph.D. pass in one subject area and an M.S. pass in another; to have taken advantage of prior opportunities to attempt the exams; to have a strong academic record; and to have made reasonable progress in identifying a research area and thesis supervisor.