MAT 000 - Tutorial

Syllabus for MAT 000 – Tutorial

This is a compulsory course for BS Mathematics students in their 1^st semester.

Credits (Lec:Tut:Lab) = 0:3:0 (3 hours of discussion weekly)

Prerequisites: None

Overview: This course is open only to undergraduates majoring in Mathematics and is a compulsory course during their 1^st semester at SNU. Students will be introduced in a tutorial setting to issues regarding the nature and uses of Mathematics. The intent is to ease the transition from high school to university education, as well as to initiate the student into a more holistic view of Mathematics.

Detailed Syllabus: This course will take up issues such as the concepts of axioms and proof, the role of counter-examples, problem solving techniques, geometric intuition, the process of abstraction, etc. Some time will also be set aside for discussion of topics being studied in other courses.

References:

1. What is Mathematics? by Richard Courant and Herbert Robbins. 2^nd edition, Oxford University Press, 2007
2. How to Solve It by G. Polya. 2^nd edition, Prentice Hall India, 2007
3. The Princeton Companion to Mathematics by T. Gowers, J. Barrow-Green and I. Leader (editors). Princeton University Press, 2008.
4. Mathematical Vistas by Peter Hilton, Derek Holton and Jean Pedersen. Springer International Edition, 2010.

MAT 240 - Algebra I

Syllabus for MAT 240 – Algebra I

This is a compulsory course for BS Mathematics students in their 3^rd semester.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Overview: Learning traditional Abstract Algebra in a contemporary style. The course will cover the standard algebraic structures of groups, rings and fields up to the Fundamental Theorem of Algebra.

Detailed Syllabus:

Module I: Groups

1. Definition and examples, abelian and non-abelian groups, finite and infinite groups
2. Subgroups: characterisations, subgroup generated by a subset, commutator subgroup, center
3. Cyclic Groups: Properties, classification of subgroups
4. Permutation Groups: definition and notation, examples, properties, Symmetric group on n letters (S_n), Alternating group (A_n) on n letters
5. Cosets and Lagrange's theorem
6. External Direct Product: Definition and examples, properties, criteria for external direct product to be cyclic, finitely generated abelian groups

Module II: Morphisms

7. Normal subgroups, factor groups, internal direct products
8. Group homomorphism: Definition and examples, properties
9. Isomorphism, First Isomorphism Theorem, automorphism, properties, examples

Module III: Rings

10. Introduction to Rings: Definition, examples, properties
11. Subrings
12. Ideals, factor rings, prime ideals and maximal ideals
13. Polynomial Rings: Notation and terminology, division algorithm

Module IV: Extension Fields

10. Integral Domain, definitions and examples, Fields, Characteristic
11. Examples of Fields, algebraic and transcendental elements, degree of a field extension
12. Finite Fields: examples, Fundamental Theorem of Algebra

Main Reference:

• Contemporary Abstract Algebra by Joseph A. Gallian, 4^th edition. Narosa, 1999.

Other References:

• Topics in Algebra by I.N. Herstein, 2^nd Edition. Wiley India, 2006.
• Algebra by Michael Artin, 2^nd Edition. Prentice Hall India, 2011.
• A First Course in Abstract Algebra by John B. Fraleigh, 7^th Edition. Pearson, 2003.
• Undergraduate Algebra by Serge Lang, 2^nd Edition. Springer India, 2009.

MAT 210 Programming

Syllabus for MAT 210 – Programming

This is a compulsory course for BS Mathematics students in their 3^rd semester.

From XKCD. Thanks to Aman Agarwal for the reference!

Credits (Lec:Tut:Lab)= 1:0:1 (One lecture hour and three lab hours weekly)

Prerequisites: None

Overview: This course provides an introduction to formal programming languages via the medium of Python 3.0. The programming activities will be centered around mathematical models involving differential equations, algebraic systems, iterative processes, linear transformations, random processes etc. The course begins with Python language constructs and moves to an in-depth exploration of the SCIPY and NUMPY packages that hold the key to the desired mathematical simulations.

Detailed Syllabus:

1. Basics of the PYTHON programming language:

• Input and output statements, formatting output, copy and assignment, arithmetic operations, string operations, lists and tuples, control statements
• User defined functions, call by reference, variable number of arguments
• One dimensional arrays, two dimensional arrays, random number generation
• Classes, static data, private data, inheritance, scope of variables, nested functions

2. The NUMPY and SCIPY packages:

• Numpy numerical types, data type objects, character codes, dtype constructors.
• Mathematical libraries, plotting 2D and 3D functions, ODE integrators, charts and histograms, image processing functions.
• File I/O, loading data from CSV files
• Using SCIPY/NUMPY to solve models involving difference equations, differential equations, finding limit at a point, approximation using Taylor series, interpolation, definite integrals.

Main References:

1. John Zelle, Python Programming: An Introduction to Computer Science. Franklin, Beedle & Associates Inc., Second Edition, 2010.
2. Ivan Idris, Numpy 1.5 Beginner’s Guide. Packt Publishing, 2011.
3. Hans Petter Langtangen, A Primer on Scientific Programming on Python. Springer, Second Edition, 2011.

Other References:

1. Hans Petter Langtangen, Python Scripting for Computational Science. Springer, 2010.
2. David M. Beazley, Python Essential Reference, 3^rd Edition. Pearson, 2009.
   
   

BS Mathematics - Calculus I - Course Instructions

• The main text for the course is Essential Calculus: Early Transcendentals by James Stewart, 1^st edition, Cengage, 2011. Please immediately issue a copy from the Library. You will be required to bring this book to each class.
• There will be three lectures and two tutorials every week.
• There will be fortnightly assignments. Part A of each assignment will be discussed in tutorials, and students will be allotted problems for presentation. Part B is meant for written submission. The problems will mainly be assigned from the exercises in Stewart.
• It is expected that about 10 assignments will be distributed during the course. You will then be allowed to drop your worst two assignment marks.
• No late submissions will be accepted. The only leeway for taking care of special circumstances is provided by the above-mentioned dropping of your worst two assignment marks.
• Attendance will be taken in each class. Be aware that SNU requires a minimum of 70% attendance separately in lectures and tutorials. No further waiver is given beyond this 30%, even for illness.

Content:

1. Functions & Graphs, Limits, Continuity, Derivatives, L’Hôpital’s Rule

2.  Higher derivatives, Maxima/ Minima, Curve sketching

3. Area & Integration, Fundamental Theorem of Calculus, Techniques of integration, Improper integrals

4. Applications to Area, Volume, Arc-Length

5. Parametric Equations and Polar Coordinates

6. Several variables: Level curves, limits & continuity, partial derivatives, tangent planes, chain rule, directional derivative, gradient, Maxima/Minima, Lagrange multipliers

Further References:

1.       The Calculus Lifesaver by A Banner, Princeton, 2007.

2.       Calculus and Analytic Geometry by G B Thomas and R L Finney, 9^th edition, Pearson.

3.       Basic Multivariable Calculus by J E Marsden, A J Tromba and A Weinstein, 1^st edition, Springer (India), 2011.

4.       Calculus by Ken Binmore and Joan Davies, 1^st edition, Cambridge, 2010.

Assessment:

Final Exam	50%
Midterm Exam	25%
Assignments	15%
Class Performance	10%

Penrose Tiling video

As a follow up to the impromptu description of Penrose tiles by Prof. Fozia Qazi in our Thursday class, here is an animation of their properties:

Part 1 - Symmetries



Part 2 - Scaling



The creators of these videos, Maurizio Paolini and Alessandro Musesti, teach Mathematics at the Department of Mathematics and Physics Niccolò Tartaglia at the Catholic University of the Sacred Heartk in Brescia, Italy. The website for this project is frecceaquiloni.dmf.unicatt.it/

Precalculus - Notes on Cardinality 1

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Cardinality_1

Precalculus Notes - Logic 2

Logic 2

Precalculus Notes - Logic I

Logic 1

Precalculus Notes 3 - Images & Preimages of Sets

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Functions 3

Precalculus - Assignment 1

Submit by: September 12

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1. Let $f:A\to B$ and $g:B\to A$ such that $g\circ f =\mathrm{id}_A$. State whether the following conclusions are valid. For valid conclusions give a proof. For invalid conclusions give a counterexample.
   
   1. $f$ is a bijection.
   2. $f$ is onto.
   3. $f$ is one-one.
   4. $g$ is a bijection.
   5. $g$ is onto.
   6. $g$ is one-one.
2. Let $A$ and $B$ be sets with 3 and 4 elements respectively. Can there be a bijection between these sets?
   
3. Exhibit a bijection from the set of even natural numbers $2\mathbb N$ to the set of all natural numbers $\mathbb N$.
   
4. Exhibit a bijection from $\mathbb N$ to the set of integers $\mathbb Z$.
   
5. Are the following functions one-one or onto?
   
   1. $f:[0,1]\to[a,b]$, $f(x)=bx+(1-x)a$.
   2. $f:\mathbb R\to\mathbb R$, $f(x)=x^2+x+1$.
   3. $f:\mathbb R\to\mathbb R$, $f(x)=x+|x|$.
6. Let $f:\mathbb N\to A$ and $g:\mathbb N\to B$ be surjective. Show there is a surjective map $h:\mathbb N\to A\cup B$.
   
7. Let $f:\mathbb R^2\to\mathbb R$ be defined by $f(x,y)=xy$. What are $f^{-1}(r)$ for $r\in\mathbb R$ and $f^{-1}([a,b])$? Draw pictures of these inverse images.
   
8. Let $f:X\to Y$. Show that
   
   1. $f$ is onto iff $f(f^{-1}(B))=B$ for every $B\subset Y$.
   2. $f$ is one-one iff $f^{-1}(f(A))=A$ for every $A\subset X$.