please see below for a message from Dr. Markaryan regarding his math refresher course:
My name is Tigran Markaryan and I will be your instructor for the Math Review/Refresher sessions
on August 21, August 28 and September 11. You have already received communication from the Department and Dr. Slawski about the importance and objectives. I would like to welcome you to this series of sessions and provide some additional information on the
format and contents of these sessions. Please feel free to send me your questions. Please make sure to send your emails to me and not to "reply to all".
I understand that attending three full days may be challenging for some of you, especially in the current situation. I encourage you to attend for the portions you can even if you are unable to attend
the sessions in their entirety. If all you can do is to attend a few hours, I still recommend you do so.
We will start the sessions at 9 a.m. and finish around 4:00 p.m. At the end, I will leave some time for Q&A and will also stay longer if there are questions.
LOCATION: Johnson Center, Meeting Room F
BREAKFAST & LUNCH: The breakfast is provided by the Department and is scheduled to be brought in close to 8:30-9.00 a.m. You are welcome (and encouraged!) to arrive a bit early to get settled and
enjoy the breakfast. I will bring some snacks and sweets (chips, crackers, chocolate, cookies) for your enjoyment after the lunch break.
CONTENT: These sessions are going to cover Algebra, Basics of Set Theory, Univariate and Multivariable Calculus, Optimization and Matrix Algebra. Solid foundation in Algebra and Set Theory is necessary
not only for Calculus or Matrix Algebra, but for most advanced mathematical, statistical and quantitative disciplines. The first session on
August 21 is, therefore, going to be on Algebra and Set Theory. Detailed outlines of these sessions are provided below.
FORMAT: As you can easily imagine, 7 hours is a long time for any meeting. I would like you to approach these sessions as an opportunity to ask questions, clarify concepts, share your comments and
observations and otherwise be actively engaged. It will help me to adjust the focus effectively and drill down on topics and issues that are more important. Please make these sessions maximally beneficial for yourselves. Try to get to the why's and how's as
those are going to be my focus.
ATTENDANCE: It is up to you to attend any or all of the 3 sessions; however. I strongly recommend that you attend to the degree possible. If you are unable to attend a whole session, attend a part
of it (that is just fine). I understand that some of you may have to skip some portions of the sessions. Feel free to check with me about the detailed agenda prior to the sessions. Even if you attended some of the sessions in the past years, you can still
benefit from these sessions. I have had many students who attended 2-3 times.
PRETEST: I am attaching a PDF file that contains some problems in Algebra, Set Theory, Calculus and Matrix Algebra. Please try to solve the problems in the pretest. That would help identify any potential
soft areas. The only purpose of the pretest is to help yourselves assess strengths and softer areas. The date on the test is 2019 as that is when I updated the file.
DETAILED AGENDA: Here is the preliminary agenda for the three sessions. I would like to preserve flexibility and spend more or less time on various topics depending on the actual demand in the audience.
Please treat this as directional only. In the past years when we covered Calculus, there was a large interest in foundational concepts such as open and closed sets, limit points, metrics, distance, inverse functions and composite functions, etc. I am happy
to see this level of interest. Given the time constraints we have, I decided to create a short document devoted to this topics and send to the students prior to the August 28 session and then address questions during the session.
August 21, 2021
** The method of mathematical induction
** Different methods of factorization and simplifying fractions (common factor, grouping, completing square, etc.)
** Arithmetic and geometric progressions
* Basics of set theory
** Cardinality of sets
** Set Algebra (Union, Intersection, complement, etc.) and their properties. Translations to events in Probability.
** Cartesian products
** The Inclusion-exclusion principle
August 28, 2021
* Handout on "Topology of Euclidian Spaces" will be sent to the students at least a week prior to this session. I will answer questions in the beginning of the session and also do a quick review.
* Univariate Calculus
** Limits of sequences and functions
** Continuity and uniform continuity; properties of continuous functions
** Derivatives, properties of derivatives, Taylor's formula
** Indefinite integral
** Definite integral; definition; properties
** Evaluation of definite integrals (Fundamental Thm. of Calculus, Integration By Parts, Integration by Substitution)
** Improper Integrals [which are very common in Probability and Statistics]
** [If time permits] Infinite series (convergence, term-by-term differentiation and integration)
* Multivariable Calculus
** Regions in R^n and Functions: R^n->R^m (Linear mappings R^n->R^m are the subject of Matrix Algebra)
** Polar coordinates
** Limits and Continuity
** Partial derivatives and higher order derivatives
** Integration (Fubini's Theorem, Substitution, Examples from Probability)
Sep 11, 2021
* Matrix Algebra
** Definitions; matrices as linear transformations R^n->R^m
** Algebra of matrices (addition, multiplication, linear forms, quadratic forms, differentiation of linear & quadratic forms)
** Definite and semi-definite matrices
** Matrix inverses, determinants and their properties; linear independence; rank of a matrix
** Partitioned matrices
** Special matrices (symmetric, orthogonal, idempotent, permutation, diagonal, triangular)
** Column reduction
** Eigenvalues and eigenvectors
** Formulation of OLS in matrix form and derivation of OLS estimators
* Optimization (If time permits)
** Types and examples of optimization problems
** Linear, convex and general nonlinear problems
** Methods for finding extrema of functions
*** Grid search and enumeration
*** Exact methods (Fermat's theorem, Lagrange multipliers)
*** The Newton-Raphson method (also in R^2)
HANDOUTS: I will bring copies of handouts with me. I do not send electronic copies (except for Handout on "Topology of Euclidian Spaces"). You are welcome to make an arrangement with me or with one
of the students to get a copy for yourself in case you miss a session.
ADDITIONAL TOPICS: Please let me know if there are topics you would like to see covered if they are not listed above. To the extent possible, I will try to address them or provide pointers to useful
COVID-19: We will be following all the requirements for the safe return to campus.
I look forward to seeing you soon!