Meetings

Click on the theme item for the meeting plan for that theme.Click on the meeting item for references, exercises, and additional reading related to it.

**Theme 1 : Logic & Proofs**- 15 meetings- Meeting 01 : Mon, Jul 30, 11:00 am-11:50 am
- Meeting 02 : Tue, Jul 31, 10:00 am-10:50 am
- Meeting 03 : Wed, Aug 01, 09:00 am-09:50 am
- Meeting 04 : Thu, Aug 02, 12:00 pm-12:50 pm
- Meeting 05 : Mon, Aug 06, 11:00 am-11:50 am
- Meeting 06 : Tue, Aug 07, 10:00 am-10:50 am
- Examples of direct proofs
- Section 1.6 in [KR] Book.
- Meeting 07 : Wed, Aug 08, 09:00 am-09:50 am
- Meeting 08 : Thu, Aug 09, 12:00 pm-12:50 pm
- Meeting 09 : Mon, Aug 13, 09:00 am-09:50 am
- Notes on "Cardinality" - Michael Thomas.
- Infinity and Countability - Luca Trevisan's notes.
- Meeting 10 : Tue, Aug 14, 10:00 am-10:50 am
- Notes on "Cardinality" - Michael Thomas.
- Infinity and Countability - Luca Trevisan's notes.
- Meeting 11 : Tue, Aug 21, 10:00 am-10:50 am
- Notes on "Cardinality" - Michael Thomas.
- Infinity and Countability - Luca Trevisan's notes.
- Meeting 12 : Wed, Aug 22, 09:00 am-09:50 am
- Notes on "Cardinality" - Michael Thomas.
- Infinity and Countability - Luca Trevisan's notes.
- Meeting 13 : Thu, Aug 23, 12:00 pm-12:50 pm
- Meeting 14 : Mon, Aug 27, 11:00 am-11:50 am
- Meeting 15 : Tue, Aug 28, 10:00 am-10:50 am

Admn : Announcements. Overview of the course work and evaluation process.<br> The language of math. Propositions. Operators. References Section 1.1 in [KR] Exercises Exercises in section 1.1 in [KR] Reading <ul> <li><a href="http://en.wikipedia.org/wiki/Pentium_FDIV_bug">Pentium FDIV bug</a></li> <li><a href="http://en.wikipedia.org/wiki/The_Laws_of_Thought">Laws of Thought</a>.</li> <li><a href=" http://www.vordenker.de/ggphilosophy/mcgill_contradiction-excl-middle.pdf">On the law of excluded middle</a></li> </ul> Admn : Announcements. Overview of the course work and evaluation process.

The language of math. Propositions. Operators.References : Section 1.1 in [KR] Exercises : Exercises in section 1.1 in [KR] Reading : Admn : TA introduction, Grouping. <br> Implication, Double Implication, Logical Equivalences. References Section 1.2 of [KR] Exercises Reading Admn : TA introduction, Grouping.

Implication, Double Implication, Logical Equivalences.References : Section 1.2 of [KR] Arguments and Argument forms. Validity of Arguments. References Exercises Reading Arguments and Argument forms. Validity of Arguments.References : None Rules of inference. Examples of Deductions. Applications. References Section 1.5 in [KR] Exercises Exercises for section 1.5 Reading Rules of inference. Examples of Deductions. Applications.References : Section 1.5 in [KR] Exercises : Exercises for section 1.5 Deduction. Quantifiers and Predicate Logic, Examples, Negations. References Exercises Reading Deduction. Quantifiers and Predicate Logic, Examples, Negations.References : None Notion of a Proof. Proof by picture and pitfalls in proofs. Methods of Proofs : Direct Proof and an example. References <ul><li><a href="http://zimmer.csufresno.edu/~larryc/proofs/proofs.direct.html">Examples of direct proofs</a></li> <li>Section 1.6 in [KR] Book.</li> Exercises Reading <ul> <li><a href="http://www.math.uconn.edu/~hurley/math315/proofgoldberger.pdf">Article - What are mathematical proofs and why are they important?</a></li> <li><a href="http://www.math.wustl.edu/~sk/eolss.pdf">Article - History and concept of a mathematical proof</a></li> </ul> Notion of a Proof. Proof by picture and pitfalls in proofs. Methods of Proofs : Direct Proof and an example.References : Reading : One more example of direct proofs. Indirect Proofs with examples. Proof by contradiction. The structure and validity. Identifying the contradiction. Eg: A proof that square root of 2 is irrational. References Section 1.7 in [KR Book] Exercises Show that x^2 - y^2 does not have distinct positive integer solutions.<br> Show the converse of Pythagorean theorem. Reading <ul> <li><a href="http://www.math.uconn.edu/~hurley/math315/proofgoldberger.pdf">Article - What are mathematical proofs and why are they important?</a></li> <li><a href="http://www.math.wustl.edu/~sk/eolss.pdf">Article - History and concept of a mathematical proof</a></li> </ul> One more example of direct proofs. Indirect Proofs with examples. Proof by contradiction. The structure and validity. Identifying the contradiction. Eg: A proof that square root of 2 is irrational.References : Section 1.7 in [KR Book] Exercises : Show that x^2 - y^2 does not have distinct positive integer solutions.

Show the converse of Pythagorean theorem.Reading : Exhaustive Proof. Existence proofs. Constructive and non-constructive proof. References Section 1.7 in [KR Book].<br> Example 10 and 11. Exercises Reading <ul> <li><a href="http://www.math.uconn.edu/~hurley/math315/proofgoldberger.pdf">Article - What are mathematical proofs and why are they important?</a></li> <li><a href="http://www.math.wustl.edu/~sk/eolss.pdf">Article - History and concept of a mathematical proof</a></li> </ul> Exhaustive Proof. Existence proofs. Constructive and non-constructive proof.References : Section 1.7 in [KR Book].

Example 10 and 11.Reading : Sets - the membership symbol. Axiom of unrestricted comprehension. Russel's Paradox. Cardinality. Countable and countably infinite sets. References <ul> <li><a href="http://www.ma.utexas.edu/users/mwilliams/cardinality.pdf">Notes on "Cardinality"</a> - Michael Thomas.</li> <li><a href="http://inst.eecs.berkeley.edu/~cs70/sp07/lec/lecture27.pdf">Infinity and Countability</a> - Luca Trevisan's notes.</li> </ul> Exercises Reading <ul> <li>For computability inclined : <a href="http://www.scs.ryerson.ca/~mth110/Handouts/PD/russell.pdf">Russels Paradox and the Halting Problem</a></li> <li><a href="http://people.umass.edu/klement/OriginsPFParadox.pdf">The origins of the proposition function versions of Russel's Paradox.</a></li> <li> <a href="http://www.youtube.com/watch?v=83X1YWDXdDA">Youtube video on Russel's Paradox</a>. </li> </ul> Sets - the membership symbol. Axiom of unrestricted comprehension. Russel's Paradox. Cardinality. Countable and countably infinite sets.References : Reading : Union of countably infinite sets. Countably infinite union of them - Set of integers and set of rationals are countably infinite. References <ul> <li><a href="http://www.ma.utexas.edu/users/mwilliams/cardinality.pdf">Notes on "Cardinality"</a> - Michael Thomas.</li> <li><a href="http://inst.eecs.berkeley.edu/~cs70/sp07/lec/lecture27.pdf">Infinity and Countability</a> - Luca Trevisan's notes.</li> </ul> Exercises Reading Union of countably infinite sets. Countably infinite union of them - Set of integers and set of rationals are countably infinite.References : Cantors Diagonalization Argument. The set of real numbers are uncountable. References <ul> <li><a href="http://www.ma.utexas.edu/users/mwilliams/cardinality.pdf">Notes on "Cardinality"</a> - Michael Thomas.</li> <li><a href="http://inst.eecs.berkeley.edu/~cs70/sp07/lec/lecture27.pdf">Infinity and Countability</a> - Luca Trevisan's notes.</li> </ul> Exercises Reading Cantors Diagonalization Argument. The set of real numbers are uncountable.References : Are there sets of "larger" cardinality? There cannot exist a bijection from a set to its power set. Diagonalization again. Application of the proof technique to show undecidability of halting problem. References Exercises Reading Are there sets of "larger" cardinality? There cannot exist a bijection from a set to its power set. Diagonalization again. Application of the proof technique to show undecidability of halting problem.References : Induction Principle as an axiom. Well-ordering Axiom, Well-ordering thoerem, Well-Ordering of N implies the Principle of Mathematical Induction. References <a href="http://public.csusm.edu/aitken_html/m378/Ch1PeanoAxioms.pdf">Peano's Axioms</a> <br> <a href="http://dwick.org/pages/acwop.pdf">Well-ordering principle and the Axiom of Choice</a>. Exercises Reading Induction Principle as an axiom. Well-ordering Axiom, Well-ordering thoerem, Well-Ordering of N implies the Principle of Mathematical Induction.Structure of a proof by induction. Two examples.<br> All horses are of the same color - pitfalls in inductions proofs. References Section 4.1 in [KR] Book. Exercises Exercises in Page 289-293 Reading Structure of a proof by induction. Two examples.

All horses are of the same color - pitfalls in inductions proofs.References : Section 4.1 in [KR] Book. Exercises : Exercises in Page 289-293 Creativity in induction. Survivor in a knife fight game. Strong induction. Examples from graph theory. References Example 12, Page 286, [KR] Book. Exercises Page 300-302 [KR] Book. Reading <a href="http://inst.eecs.berkeley.edu/~cs70/sp07/lec/lecture04.pdf">Notes</a> by Luca Trevisan. Creativity in induction. Survivor in a knife fight game. Strong induction. Examples from graph theory.References : Example 12, Page 286, [KR] Book. Exercises : Page 300-302 [KR] Book. Reading : Notes by Luca Trevisan. **Theme 2 : Counting & Combinatorics**- 15 meetings- Meeting 16 : Mon, Sep 03, 11:00 am-11:50 am
- Meeting 17 : Tue, Sep 04, 10:00 am-10:50 am
- Meeting 18 : Wed, Sep 05, 09:00 am-09:50 am
- Meeting 19 : Thu, Sep 06, 12:00 pm-12:50 pm
- Meeting 20 : Wed, Sep 12, 09:00 am-09:50 am
- Meeting 21 : Thu, Sep 13, 12:00 pm-12:50 pm
- Meeting 22 : Mon, Sep 17, 11:00 am-11:50 am
- Meeting 23 : Tue, Sep 18, 10:00 am-10:50 am
- Meeting 24 : Thu, Sep 20, 09:00 am-09:50 am
- Meeting 25 : Mon, Sep 24, 11:00 am-11:50 am
- Meeting 26 : Tue, Sep 25, 10:00 am-10:50 am
- Meeting 27 : Wed, Sep 26, 09:00 am-09:50 am
- Meeting 28 : Thu, Sep 27, 12:00 pm-12:50 pm
- Meeting 29 : Thu, Oct 04, 12:00 pm-12:50 pm
- Meeting 30 : Sat, Oct 06, 02:00 pm-03:00 pm

Basic Counting : Sum rule, generalised sum rule, Product rule, Inclusion-Exclusion Principle. Applications. Pigeon Hole Principle. Simple Applications of PHP. References Section 5.1 and 5.2 of Rosen's Textbook. Exercises Reading Basic Counting : Sum rule, generalised sum rule, Product rule, Inclusion-Exclusion Principle. Applications. Pigeon Hole Principle. Simple Applications of PHP.References : Section 5.1 and 5.2 of Rosen's Textbook. More applications of PHP. References Exercises Reading More applications of PHP.References : None More applications of PHP. Some proofs from the THE Book. Ramsey type theorems. Impossibility of "always length non-increasing" "lossless" "compression" algorithms. Counting arguments. References Example 13 from the textbook.<br> Exercises Reading More applications of PHP. Some proofs from the THE Book. Ramsey type theorems. Impossibility of "always length non-increasing" "lossless" "compression" algorithms. Counting arguments.References : Example 13 from the textbook. Combinatorial identities. Double counting as a proof technique. References [KR] textbook (Around page 365, 7th edition). <br> <a href="http://www.cse.iitm.ac.in/~jayalal/teaching/CS6030/2011/counting.pdf">Proofs that really counts</a>. Exercises Reading Combinatorial identities. Double counting as a proof technique.References : [KR] textbook (Around page 365, 7th edition).

Proofs that really counts.Recurrence relations. Linear Homogeneous Recurrence Relations - Distinct roots case, examples References Exercises Reading Recurrence relations. Linear Homogeneous Recurrence Relations - Distinct roots case, examplesReferences : None Linear Homogeneous Recurrence Relations - higher multiplicity. References Exercises Reading Linear Homogeneous Recurrence Relations - higher multiplicity.References : None General form with multiple roots with higher multiplicities. Linear Non-homogeneous Recurrence Relations, Examples. References Exercises Reading General form with multiple roots with higher multiplicities. Linear Non-homogeneous Recurrence Relations, Examples.References : None More examples of solving Linear non-homogeneous Recurrence Relations. References Exercises Reading More examples of solving Linear non-homogeneous Recurrence Relations.References : None Generating Functions Method I References Exercises Reading Generating Functions Method IReferences : None Generating Functions Method II References Exercises Reading Generating Functions Method IIReferences : None Catalan Numbers using generating functions. <br> The general strategy of solutions. Recap of the theme. The mathematical tools and techniques seen so far. Cardinality of the sets. References Exercises Reading Catalan Numbers using generating functions.

The general strategy of solutions. Recap of the theme. The mathematical tools and techniques seen so far. Cardinality of the sets.References : None Structured sets. Groups, Groupoid, Monoids, Semi-groups. References Class notes. First chapter of any algebra book will be sufficient. A particularly suggested book is "Topics in Algebra" - Herstein. Exercises Reading Structured sets. Groups, Groupoid, Monoids, Semi-groups.References : Class notes. First chapter of any algebra book will be sufficient. A particularly suggested book is "Topics in Algebra" - Herstein. Groups of symmetry in graphs. Subgroups. Lagrange's theorem. References <a href="http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/coset.pdf">Cosets and Lagranges Theorem</a> Exercises Reading Groups of symmetry in graphs. Subgroups. Lagrange's theorem.References : Cosets and Lagranges Theorem Cyclic groups. Generator. Order of an element. Abelian groups. All cyclic groups are abelian. Examples. References Class notes. First chapter of any algebra book will be sufficient. A particularly suggested book is "Topics in Algebra" - Herstein. Exercises Reading Cyclic groups. Generator. Order of an element. Abelian groups. All cyclic groups are abelian. Examples.References : Richer Algebraic Structures. Rings and Fields. Common examples and non-examples. An application of the algebraic structure : ISBN numbers. References Class notes <br> {KR] Book, 7th edition, Chapter 7 Exercises Reading Richer Algebraic Structures. Rings and Fields. Common examples and non-examples. An application of the algebraic structure : ISBN numbers.References : Class notes

{KR] Book, 7th edition, Chapter 7**Theme 3 : Linear Algebraic Structures & Applications**- 16 meetings- Meeting 31 : Sat, Oct 06, 03:00 pm-04:00 pm
- Meeting 32 : Mon, Oct 08, 11:00 am-11:50 am
- Meeting 33 : Tue, Oct 09, 10:00 am-10:50 am
- Meeting 34 : Wed, Oct 10, 09:00 am-09:50 am
- Meeting 35 : Mon, Oct 15, 11:00 am-11:50 am
- Meeting 36 : Tue, Oct 16, 10:00 am-10:50 am
- Meeting 37 : Wed, Oct 17, 09:00 am-09:50 am
- Meeting 38 : Thu, Oct 18, 12:00 pm-12:50 pm
- Meeting 39 : Mon, Oct 22, 11:00 am-11:50 am
- Meeting 40 : Tue, Oct 23, 10:00 am-10:50 am
- Meeting 41 : Thu, Oct 25, 12:00 pm-12:50 pm
- Meeting 42 : Mon, Oct 29, 11:00 am-11:50 am
- Meeting 43 : Tue, Oct 30, 10:00 am-10:50 am
- Meeting 44 : Wed, Oct 31, 09:00 am-09:50 am
- Meeting 45 : Sat, Nov 03, 02:00 pm-02:50 pm
- Meeting 46 : Sat, Nov 03, 03:00 pm-04:20 pm
- Length and Perpendicularity as linear algebraic notions. Norm and Orthogonality. Orthogonal vectors, Orthogonal basis, Orthogonal spaces. (Section 3.1 and 3.4 of [S])
- Row space is orthogonal to Nullspace. (Section 3.1 of [S])
- Are there orthogonal basis for any vector space? Gram-Schmidt Orthogonalization algorithm. (Section 3.4 of [S])
- Symmetric matrices have orthogonal eigen spaces corresp to distinct eigen values (An observation from the proof that symmetric matrices have real eigen values).
- Diagonalization of Symmetric matrices. (Section 5.2, Page 345 in [S], See Example 1 and 2)
- Computing Eigen values?
- Concluding remarks on Linear Algebra Section
- Orthogonality : Problem Set 3.1 in [S], 1-52 .
- Gram-Schmidt Orthogonalization : Problems 1, 6 and 32 in Problem Set 3.4
- (optional) Digaonalizability - Problem Set 5.2, 1-28.

Vector spaces. Three example sets which are vector spaces : Arrows in plane from origin, Cartesian Product of fields, Space of polynomials. References Chapter 2 in [GS] Exercises Reading Vector spaces. Three example sets which are vector spaces : Arrows in plane from origin, Cartesian Product of fields, Space of polynomials.References : Chapter 2 in [GS] Vector spaces of matrices. Notion of linear combination. Solution of linear equations. Practical Examples. References Chapter 1 in [S] Exercises Reading Problem Set 2.2 in [S] Vector spaces of matrices. Notion of linear combination. Solution of linear equations. Practical Examples.References : Chapter 1 in [S] Reading : Problem Set 2.2 in [S] Solutions of systems Linear Equations. Matrix form. Review of Elimination procedure. Vector multiplication- dot product. Matrix multiplication by a vector. Systems of linear equations as a vector equation. References Chapter 1 in [GS] Exercises Problem Set 1.1, 1.2, 1.3, 1.4 in [S] Reading Solutions of systems Linear Equations. Matrix form. Review of Elimination procedure. Vector multiplication- dot product. Matrix multiplication by a vector. Systems of linear equations as a vector equation.References : Chapter 1 in [GS] Exercises : Problem Set 1.1, 1.2, 1.3, 1.4 in [S] Gaussian Elimination procedure in terms of matrices. LU decomposition of matrices. References Chapter 1 in [S] Exercises Problem Set 1.5 of [S] Reading Gaussian Elimination procedure in terms of matrices. LU decomposition of matrices.References : Chapter 1 in [S] Exercises : Problem Set 1.5 of [S] Under permutations. PA = LU. References Exercises Problem Set 1.5 of [S] Reading Under permutations. PA = LU.References : None Exercises : Problem Set 1.5 of [S] (Solving Ax = b) vs (Checking if b is a linear combination of columns of A) - equivalence. Span of a set S, Examples and non-Examples. Properties of the Span. References Exercises Problem Set 2.3, 11-18 in [S] Reading (Solving Ax = b) vs (Checking if b is a linear combination of columns of A) - equivalence. Span of a set S, Examples and non-Examples. Properties of the Span.References : None Exercises : Problem Set 2.3, 11-18 in [S] Span(S) forms a vector subspace. Linear dependence and independence. Examples. Definition of basis. References Exercises Problem Set 2.3, 1-10 in [S] Reading Span(S) forms a vector subspace. Linear dependence and independence. Examples. Definition of basis.References : None Exercises : Problem Set 2.3, 1-10 in [S] Any two bases must be of the same size. Dimension of a vector space. References Exercises Problem Set 2.3, 19-37 Reading Any two bases must be of the same size. Dimension of a vector space.References : None Exercises : Problem Set 2.3, 19-37 Row space, column space of matrices. Rank of a matrix. Column Rank and Row Rank. References Exercises Problem Set 2.4 in [S] 1-41 Reading Row space, column space of matrices. Rank of a matrix. Column Rank and Row Rank.References : None Exercises : Problem Set 2.4 in [S] 1-41 Row Rank = Column Rank. References <a href="http://www.mtts.org.in/userapps/download-expo.php?fileid=64">Notes by Prof. Kumaresan</a>. Exercises Problem Set 2.4 in [S] 1-41 Reading Row Rank = Column Rank.References : Notes by Prof. Kumaresan. Exercises : Problem Set 2.4 in [S] 1-41 Computing the matrix rank. Row-reduced form. <br> Linear transformations. Viewing matrices as linear transformations. <br> Null-spaces of matrices. References Exercises Problem Set 2.6, 1-30, Review Exercises 2.1 to 2.3 of [S] Reading Computing the matrix rank. Row-reduced form.

Linear transformations. Viewing matrices as linear transformations.

Null-spaces of matrices.References : None Exercises : Problem Set 2.6, 1-30, Review Exercises 2.1 to 2.3 of [S] Range of Linear Transformations. Rank-nullity theorem. References Exercises Problem Set 2.6, 1-30, Review Exercises 2.1 to 2.3 of [S] Reading Range of Linear Transformations. Rank-nullity theorem.References : None Exercises : Problem Set 2.6, 1-30, Review Exercises 2.1 to 2.3 of [S] Viewing Linear Transformations as matrices. Determinant of a matrix. References Exercises Problem Set 2.6, 1-30, Review Exercises 2.1 to 2.3 of [S] <br> Problem Set 4.2, Problem 1-6, 19 in [S] Reading Viewing Linear Transformations as matrices. Determinant of a matrix.References : None Exercises : Problem Set 2.6, 1-30, Review Exercises 2.1 to 2.3 of [S]

Problem Set 4.2, Problem 1-6, 19 in [S]Eigen values of a matrix. Eigen spaces, Examples and significance. References Exercises Problem Set 5.1, 1-30 in [S] Reading Eigen values of a matrix. Eigen spaces, Examples and significance.References : None Exercises : Problem Set 5.1, 1-30 in [S] Dimensions of the eigen spaces. Eigen decomposition of R^n space. Application : Image compression (when the image is nice). Symmetric matrices have real eigen values. References <a href="http://www.quandt.com/papers/basicmatrixtheorems.pdf">For the theorem about symmatric matrices</a><br> For the application to image compression, see section 6.4 of [S] on SVDs - what was presented in class is a variant of that. Exercises Problem Set 5.1, 1-30 Reading Dimensions of the eigen spaces. Eigen decomposition of R^n space. Application : Image compression (when the image is nice). Symmetric matrices have real eigen values.References : For the theorem about symmatric matrices

For the application to image compression, see section 6.4 of [S] on SVDs - what was presented in class is a variant of that.Exercises : Problem Set 5.1, 1-30 Orthogonality. Gram-Schmidt Orthogonalization process. Diagonalization of Symmetric matrices. Computing Eigen values? Concluding Linear Algebra Section. References <ul> <li>Length and Perpendicularity as linear algebraic notions. Norm and Orthogonality. Orthogonal vectors, Orthogonal basis, Orthogonal spaces. (Section 3.1 and 3.4 of [S])</li> <li>Row space is orthogonal to Nullspace. (Section 3.1 of [S]) </li> <li>Are there orthogonal basis for any vector space? Gram-Schmidt Orthogonalization algorithm. (Section 3.4 of [S]) </li> <li>Symmetric matrices have orthogonal eigen spaces corresp to distinct eigen values (An observation from the <a href="http://www.quandt.com/papers/basicmatrixtheorems.pdf">proof</a> that symmetric matrices have real eigen values). </li> <li>Diagonalization of Symmetric matrices. (Section 5.2, Page 345 in [S], See Example 1 and 2)</li> <li>Computing Eigen values?</li> <li>Concluding remarks on Linear Algebra Section</li> </ul> Exercises <ul> <li>Orthogonality : Problem Set 3.1 in [S], 1-52</li>. <li>Gram-Schmidt Orthogonalization : Problems 1, 6 and 32 in Problem Set 3.4</li> <li>(optional) Digaonalizability - Problem Set 5.2, 1-28.</li> </ul> Reading Section 5.2, Section 6.3 Orthogonality. Gram-Schmidt Orthogonalization process. Diagonalization of Symmetric matrices. Computing Eigen values? Concluding Linear Algebra Section.References : Exercises : Reading : Section 5.2, Section 6.3 **Theme 4 : Basic Probability Theory & Applications**- 8 meetings- Meeting 47 : Mon, Nov 05, 11:00 am-11:50 am
- Meeting 48 : Mon, Nov 05, 03:00 pm-03:50 pm
- Meeting 49 : Wed, Nov 07, 09:00 am-09:50 am
- Meeting 50 : Thu, Nov 08, 12:00 pm-12:50 pm
- Meeting 51 : Sat, Nov 10, 12:00 pm-01:00 pm
- Meeting 52 : Sat, Nov 10, 04:15 pm-05:15 pm
- Meeting 53 : Wed, Nov 14, 09:00 am-09:50 am
- Meeting 54 : Thu, Nov 15, 12:00 pm-01:25 pm

Why study probability. Motivation from Communication Theory, Algorithm Analysis. <br> Structure in the uncertain - isnt there a contradiction?. The definition based on the limits. Shortcomings. Axioms of Modern Probability Theory. References [R] : Chapter 2, Sections 2.1, 2.2, 2.3, 2.4 Exercises All the basic ones listed in Page 54-66 in [R] Reading Why study probability. Motivation from Communication Theory, Algorithm Analysis.

Structure in the uncertain - isnt there a contradiction?. The definition based on the limits. Shortcomings. Axioms of Modern Probability Theory.References : [R] : Chapter 2, Sections 2.1, 2.2, 2.3, 2.4 Exercises : All the basic ones listed in Page 54-66 in [R] Independence and Conditional Probability, Multiplication rule. Examples. References Chapter 2 & 3 : Section 2.4, 3.1 and 3.2 Exercises Work out the examples from those sections. We did a few in class. But only a few !. Reading Independence and Conditional Probability, Multiplication rule. Examples.References : Chapter 2 & 3 : Section 2.4, 3.1 and 3.2 Exercises : Work out the examples from those sections. We did a few in class. But only a few !. Baye's Theorem, Example. Generalization to multiple events. Examples. References Chapter 3 in [R]: Section 3.3, 3.4 Example 3h. Exercises Work out the examples in [R] section 3.3 and 3.4 and exercises at the end of chapter 3. Reading Baye's Theorem, Example. Generalization to multiple events. Examples.References : Chapter 3 in [R]: Section 3.3, 3.4 Example 3h. Exercises : Work out the examples in [R] section 3.3 and 3.4 and exercises at the end of chapter 3. Random Variables and Expectation. References Chapter 4 in [R], Section 4.2, 4.3 and 4.4 Exercises Work out the examples from the textbook and exercises at the end of chapter 4. Reading Random Variables and Expectation.References : Chapter 4 in [R], Section 4.2, 4.3 and 4.4 Exercises : Work out the examples from the textbook and exercises at the end of chapter 4. Functions of random variables. Linearity of Expectation. Variance. Markov's inequality. <br> Four Random Variables - Bernoulli, Binomial, Poisson and Geometric. Parameters, Expected Value and Variance in each case. References Chapter 4 in [R] - Section 4.5 and 4.6.<br> <a href="http://www2.informatik.hu-berlin.de/alcox/lehre/lvss11/rapm/linearity_expectation.pdf">Notes</a> (section 2.1)<br> Sections 4.7, 4.8 and 4.9.1 in [R]. Exercises Work out the examples from the textbook [R] and try exercises at the end of chapter 4 in [R]. Reading <a href="http://www2.informatik.hu-berlin.de/alcox/lehre/lvss11/rapm/linearity_expectation.pdf">Notes</a> (section 2.2 and 2.3) Functions of random variables. Linearity of Expectation. Variance. Markov's inequality.

Four Random Variables - Bernoulli, Binomial, Poisson and Geometric. Parameters, Expected Value and Variance in each case.An application of Markov's Inequality for a Bernoulli Random Variable case. <br> Concentration of Measure. More Tail Inequalities : Chebychev Bounds - proof and applications. Better bounds for the Bernoulli case. References <a href="http://www2.informatik.hu-berlin.de/alcox/lehre/lvss11/rapm/concentration_measure.pdf">Notes from the web</a> (we did only section 3.1 and 3.2 in detail and mentioned some in section 3.3).<br> <a href="http://www.statlect.com/subon/inequa1.htm"> Probabilistic Inequalities</a>. Exercises Reading <a href="http://www2.informatik.hu-berlin.de/alcox/lehre/lvss11/rapm/concentration_measure.pdf">Notes from the web</a>.<br> <a href="http://www.statlect.com/subon/inequa1.htm"> Probabilistic Inequalities</a>.<br> (Extra Stuff) - <a href="http://mark.reid.name/iem/behold-jensens-inequality.html">Behold! Jensen's Inequality</a> - A demonstration by Mark Reid. An application of Markov's Inequality for a Bernoulli Random Variable case.

Concentration of Measure. More Tail Inequalities : Chebychev Bounds - proof and applications. Better bounds for the Bernoulli case.References : Notes from the web (we did only section 3.1 and 3.2 in detail and mentioned some in section 3.3).

Probabilistic Inequalities.Reading : Notes from the web.

Probabilistic Inequalities.

(Extra Stuff) - Behold! Jensen's Inequality - A demonstration by Mark Reid.Stochastic Processes, Markov property. Markov Chains. Time invariance. States and Transition Probability Matrix. Example Markov Chains - Random Walks on Graphs. References <a href="http://www2.informatik.hu-berlin.de/alcox/lehre/lvss11/rapm/markov_chains.pdf">Notes</a> (section 4.1 and a part of 4.2). Exercises Reading Stochastic Processes, Markov property. Markov Chains. Time invariance. States and Transition Probability Matrix. Example Markov Chains - Random Walks on Graphs.References : Notes (section 4.1 and a part of 4.2). Stationary Distribution for Markov Chains. Examples.<br> An important CS application : Google Page Rank Algorithm - Discussion on "when should we consider a page to be important?". A first attempt. Link farm attack. An innovative answer and the connection to Eigen vectors - <b>world's costliest eigen vector</b> - a connection to stationary distribution and hence to the random walk Markov Chain on the web. A few pitfalls and fixes on the idea. Discussion on "How does one compute the pagerank vector?"</b>. References <a href="http://www.ams.org/samplings/feature-column/fcarc-pagerank">How Google Finds Your Needle in the Web's Haystack</a> - David Austin (we followed this notes very closely). Exercises Reading <a href="http://infolab.stanford.edu/~backrub/google.html">The Anatomy of a Large-Scale Hypertextual Web Search Engine</a> - the technical paper by Sergey Brin and Lawrence Page. <br> <a href="http://blog.kleinproject.org/?p=280">How Google works: Markov chains and eigenvalues</a> - Christiane Rousseau. Stationary Distribution for Markov Chains. Examples.

An important CS application : Google Page Rank Algorithm - Discussion on "when should we consider a page to be important?". A first attempt. Link farm attack. An innovative answer and the connection to Eigen vectors -**world's costliest eigen vector**- a connection to stationary distribution and hence to the random walk Markov Chain on the web. A few pitfalls and fixes on the idea. Discussion on "How does one compute the pagerank vector?".References : How Google Finds Your Needle in the Web's Haystack - David Austin (we followed this notes very closely). Reading : The Anatomy of a Large-Scale Hypertextual Web Search Engine - the technical paper by Sergey Brin and Lawrence Page.

How Google works: Markov chains and eigenvalues - Christiane Rousseau.**Evaluation Meetings**- 7 meetings- Meeting 55 : Thu, Aug 16, 12:00 pm-12:50 pm
- Meeting 56 : Thu, Aug 30, 12:00 pm-12:50 pm
- Meeting 57 : Tue, Sep 11, 08:00 am-08:50 am
- Meeting 58 : Wed, Oct 03, 09:00 am-09:50 am
- Meeting 59 : Thu, Oct 11, 08:00 am-08:50 am
- Meeting 60 : Tue, Nov 06, 10:00 am-10:50 am
- Meeting 61 : Fri, Nov 23, 01:00 pm-04:00 pm

Short Exam 1 (5%) References Exercises Reading Short Exam 1 (5%)References : None Short Exam 2 (5%) References Exercises Reading Short Exam 2 (5%)References : None Quiz I (20%) References Exercises Reading Quiz I (20%)References : None Short Exam 3 (5%) References Exercises Reading Short Exam 3 (5%)References : None Quiz - II (20%) References Exercises Reading Quiz - II (20%)References : None Short Exam 4 (5%) References Exercises Reading Short Exam 4 (5%)References : None End-semester Examination (40%) References Exercises Reading End-semester Examination (40%)References : None