1381170045_768__232_lecture_4.4

1381170045_768_232_lecture_4.4

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Unformatted text preview: u lty of science d epa r tm ent of m athema tic s Week Date 1 Sept 7 Sections Part/ References Topic/Sections MATH 232 S ECTION #4.4 Notes/Speaker from FS2009 The numbers r1 , r2 , . . . , rn are called the roots of the polynomial p(λ). I.1, I.2, I.3 Combinatorial Structures FS: Part A.1, A.2 Comtet74 Handout #1 (self study) Symbolic methods Combinatorial 3 parameters FS A.III (self-study) Combinatorial 14 I.4, Unlabelled structures A2numberI.5, I.6 might appear more than once on this list. 3 21 II.1, II.2, II.3 4 28 II.4, II.5, II.6 5 Oct 5 III.1, III.2 6 12 IV.1, IV.2 For example Labelled structures I Labelled structures II Asst #1 Due arameters x − 6x2P+ 9x − 4 = (x − 1)(x − 1)(x − 4) Multivariable GFs so the roots of this polynomial Analysis 1 and 4. 7 19 IV.3, IV.4 Complex are Analytic Methods 8 26 IV.5 V.1 FS: Part B: IV, V, VI Appendix B4 Stanley 99: Ch. 6 Handout #1 (self-study) 9 The Nov 2 4 is called a root Singularity Analysis Asymptotic methods Asst #2 Due 9 VI.1 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 Quasi-Powers and Gaussian limit laws Sophie 14 Dec 10 10 The root 1 is called a 11 Random Structures Sophie and Limit a The multiplicity ofLaws root is the number of times it appears in the factorizaFS: Part C Combinatorial 23 IX.3 Mariolys (rotating instances of discrete tion. 12 presentations) Presentations Asst #3 Due If you add up the multiplicities of all the roots, you should get the degree of the polynomial Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 C EDRIC C HAUVE , FALL 2013 12 MATH 895-4 Fall 2010 Course Schedule f a cu lty of science d epa r tm ent of m athema tic s Week 5 Date Sections from FS2009 Part/ References Topic/Sections Finding Integer Roots 1 Sept 7 I.1, I.2, I.3 2 14 I.4, I.5, I.6 Combinatorial Structures FS: Part A.1, A.2 Comtet74 Handout #1 (self study) Combinatorial Parameters Analytic Methods Complex Analysis Stanley 99: Ch. 6 Handout #1 (self-study) −1 n Asymptotic methods Notes/Speaker Symbolic methods Combinatorial parameters FS A.III (self-study) MATH 232 S ECTION #4.4 Unlabelled structures We can always find the roots structures quadratic polynomial with the quadratic 3 21 II.1, II.2, II.3 Labelled of a I 4 28 II.4, II.5, II.6 Labelled polynomials, it can be difficult. formula, but for higher degreestructures II 5 Oct 5 III.1, III.2 Asst #1 Due Sometimes integer roots can be found by trial and error. 6 12 IV.1, IV.2 Multivariable GFs 7 19 IV.3, IV.4 FS: Part B: To do this, we makeB4IV, V, VI ofSingularityfollowing fact: use the Analysis 8 26 Appendix 9 Nov 2 IV.5 V.1 n Asst #2 Due Sophie If10p(9 ) = λ + pn−1 λ λ VI.1 + · · · + p1 λ + p0 is a polynomial, and the coefficients 12 A.3/ C Introduction to Prob. Mariolys p0 , p1 , . . . , pn−1 are all integers, then p(λ) does not need to have integer roots, 18 IX.1 Limit Laws and Comb Marni 11 but if it does have any, they must be divisors of p0 . Random Structures 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 30 IX.5 and Limit Laws FS: Part C (rotating 0 presentations) Combinatorial Mariolys instances of 12 By a “divisor” of p , we mean an discrete integer n (which can be positive or negative), 25 IX.4 Continuous Limit Laws Marni so that p0 /n is also an integer. 13 Quasi-Powers and Gaussian limit laws 14 Dec Presentations So, for10instance, the divisors of 6 are Sophie Asst #3 Due Example. Let’s see if p(λ) = λ3 − 5λ2 + 13λ − 21 has any integer roots. Dr. Marni MISHNA, Department of...
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This note was uploaded on 12/08/2013 for the course MATH 232 taught by Professor Russel during the Fall '10 term at Simon Fraser.

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