CS6030 - Mathematical Concepts for Computer Science

Meetings  

• Theme 1 : Logic & Proofs - 15 meetings

  • Meeting 01 : Mon, Jul 30, 11:00 am-11:50 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	:	Pentium FDIV bug Laws of Thought. On the law of excluded middle

  • Meeting 02 : Tue, Jul 31, 10:00 am-10:50 am

  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]

  • Meeting 03 : Wed, Aug 01, 09:00 am-09:50 am

  Arguments and Argument forms. Validity of Arguments.

  References
  Exercises
  Reading

  Arguments and Argument forms. Validity of Arguments.
  

  References

  :

  None

  • Meeting 04 : Thu, Aug 02, 12:00 pm-12:50 pm

  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

  • Meeting 05 : Mon, Aug 06, 11:00 am-11:50 am

  Deduction. Quantifiers and Predicate Logic, Examples, Negations.

  References
  Exercises
  Reading

  Deduction. Quantifiers and Predicate Logic, Examples, Negations.
  

  References

  :

  None

  • Meeting 06 : Tue, Aug 07, 10:00 am-10:50 am

  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	:	Examples of direct proofs Section 1.6 in [KR] Book.
  Reading	:	Article - What are mathematical proofs and why are they important? Article - History and concept of a mathematical proof

  • Meeting 07 : Wed, Aug 08, 09:00 am-09:50 am

  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	:	Article - What are mathematical proofs and why are they important? Article - History and concept of a mathematical proof

  • Meeting 08 : Thu, Aug 09, 12:00 pm-12:50 pm

  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	:	Article - What are mathematical proofs and why are they important? Article - History and concept of a mathematical proof

  • Meeting 09 : Mon, Aug 13, 09:00 am-09:50 am

  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	:	Notes on "Cardinality" - Michael Thomas. Infinity and Countability - Luca Trevisan's notes.
  Reading	:	For computability inclined : Russels Paradox and the Halting Problem The origins of the proposition function versions of Russel's Paradox. Youtube video on Russel's Paradox.

  • Meeting 10 : Tue, Aug 14, 10:00 am-10:50 am

  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

  :

  Notes on "Cardinality" - Michael Thomas. Infinity and Countability - Luca Trevisan's notes.

  • Meeting 11 : Tue, Aug 21, 10:00 am-10:50 am

  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

  :

  Notes on "Cardinality" - Michael Thomas. Infinity and Countability - Luca Trevisan's notes.

  • Meeting 12 : Wed, Aug 22, 09:00 am-09:50 am

  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	<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

  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

  :

  Notes on "Cardinality" - Michael Thomas. Infinity and Countability - Luca Trevisan's notes.

  • Meeting 13 : Thu, Aug 23, 12:00 pm-12:50 pm

  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.
  

  References

  :

  Peano's Axioms Well-ordering principle and the Axiom of Choice.

  • Meeting 14 : Mon, Aug 27, 11:00 am-11:50 am

  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

  • Meeting 15 : Tue, Aug 28, 10:00 am-10:50 am

  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

  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.

  • Meeting 17 : Tue, Sep 04, 10:00 am-10:50 am

  More applications of PHP.

  References
  Exercises
  Reading

  More applications of PHP.
  

  References

  :

  None

  • Meeting 18 : Wed, Sep 05, 09:00 am-09:50 am

  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.

  • Meeting 19 : Thu, Sep 06, 12:00 pm-12:50 pm

  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.

  • Meeting 20 : Wed, Sep 12, 09:00 am-09:50 am

  Recurrence relations. Linear Homogeneous Recurrence Relations - Distinct roots case, examples

  References
  Exercises
  Reading

  Recurrence relations. Linear Homogeneous Recurrence Relations - Distinct roots case, examples
  

  References

  :

  None

  • Meeting 21 : Thu, Sep 13, 12:00 pm-12:50 pm

  Linear Homogeneous Recurrence Relations - higher multiplicity.

  References
  Exercises
  Reading

  Linear Homogeneous Recurrence Relations - higher multiplicity.
  

  References

  :

  None

  • Meeting 22 : Mon, Sep 17, 11:00 am-11:50 am

  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

  • Meeting 23 : Tue, Sep 18, 10:00 am-10:50 am

  More examples of solving Linear non-homogeneous Recurrence Relations.

  References
  Exercises
  Reading

  More examples of solving Linear non-homogeneous Recurrence Relations.
  

  References

  :

  None

  • Meeting 24 : Thu, Sep 20, 09:00 am-09:50 am

  Generating Functions Method I

  References
  Exercises
  Reading

  Generating Functions Method I
  

  References

  :

  None

  • Meeting 25 : Mon, Sep 24, 11:00 am-11:50 am

  Generating Functions Method II

  References
  Exercises
  Reading

  Generating Functions Method II
  

  References

  :

  None

  • Meeting 26 : Tue, Sep 25, 10:00 am-10:50 am

  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

  • Meeting 27 : Wed, Sep 26, 09:00 am-09:50 am

  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.

  • Meeting 28 : Thu, Sep 27, 12:00 pm-12:50 pm

  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

  • Meeting 29 : Thu, Oct 04, 12:00 pm-12:50 pm

  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

  :

  Class notes. First chapter of any algebra book will be sufficient. A particularly suggested book is "Topics in Algebra" - Herstein.

  • Meeting 30 : Sat, Oct 06, 02:00 pm-03:00 pm

  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

  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]

  • Meeting 32 : Mon, Oct 08, 11:00 am-11:50 am

  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]

  • Meeting 33 : Tue, Oct 09, 10:00 am-10:50 am

  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]

  • Meeting 34 : Wed, Oct 10, 09:00 am-09:50 am

  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]

  • Meeting 35 : Mon, Oct 15, 11:00 am-11:50 am

  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]

  • Meeting 36 : Tue, Oct 16, 10:00 am-10:50 am

  (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]

  • Meeting 37 : Wed, Oct 17, 09:00 am-09:50 am

  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]

  • Meeting 38 : Thu, Oct 18, 12:00 pm-12:50 pm

  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

  • Meeting 39 : Mon, Oct 22, 11:00 am-11:50 am

  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

  • Meeting 40 : Tue, Oct 23, 10:00 am-10:50 am

  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

  • Meeting 41 : Thu, Oct 25, 12:00 pm-12:50 pm

  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]

  • Meeting 42 : Mon, Oct 29, 11:00 am-11:50 am

  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]

  • Meeting 43 : Tue, Oct 30, 10:00 am-10:50 am

  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]

  • Meeting 44 : Wed, Oct 31, 09:00 am-09:50 am

  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]

  • Meeting 45 : Sat, Nov 03, 02:00 pm-02:50 pm

  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

  • Meeting 46 : Sat, Nov 03, 03:00 pm-04:20 pm

  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	:	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
  Exercises	:	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.
  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

  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]

  • Meeting 48 : Mon, Nov 05, 03:00 pm-03:50 pm

  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 !.

  • Meeting 49 : Wed, Nov 07, 09:00 am-09:50 am

  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.

  • Meeting 50 : Thu, Nov 08, 12:00 pm-12:50 pm

  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.

  • Meeting 51 : Sat, Nov 10, 12:00 pm-01:00 pm

  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.
  

  References	:	Chapter 4 in [R] - Section 4.5 and 4.6. Notes (section 2.1) 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	:	Notes (section 2.2 and 2.3)

  • Meeting 52 : Sat, Nov 10, 04:15 pm-05:15 pm

  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.

  • Meeting 53 : Wed, Nov 14, 09:00 am-09:50 am

  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).

  • Meeting 54 : Thu, Nov 15, 12:00 pm-01:25 pm

  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

  Short Exam 1 (5%)

  References
  Exercises
  Reading

  Short Exam 1 (5%)
  

  References

  :

  None

  • Meeting 56 : Thu, Aug 30, 12:00 pm-12:50 pm

  Short Exam 2 (5%)

  References
  Exercises
  Reading

  Short Exam 2 (5%)
  

  References

  :

  None

  • Meeting 57 : Tue, Sep 11, 08:00 am-08:50 am

  Quiz I (20%)

  References
  Exercises
  Reading

  Quiz I (20%)
  

  References

  :

  None

  • Meeting 58 : Wed, Oct 03, 09:00 am-09:50 am

  Short Exam 3 (5%)

  References
  Exercises
  Reading

  Short Exam 3 (5%)
  

  References

  :

  None

  • Meeting 59 : Thu, Oct 11, 08:00 am-08:50 am

  Quiz - II (20%)

  References
  Exercises
  Reading

  Quiz - II (20%)
  

  References

  :

  None

  • Meeting 60 : Tue, Nov 06, 10:00 am-10:50 am

  Short Exam 4 (5%)

  References
  Exercises
  Reading

  Short Exam 4 (5%)
  

  References

  :

  None

  • Meeting 61 : Fri, Nov 23, 01:00 pm-04:00 pm

  End-semester Examination (40%)

  References
  Exercises
  Reading

  End-semester Examination (40%)
  

  References

  :

  None