Hi I need full solutions to Algebra and Combinatorics and Differential Equations please
School of Mathematics Level I Semester 2 Week 11 Supplementary Assessment Instructions • Answer all questions for modules for which you are registered and which appear in this booklet. • You have a maximum period of time to complete and upload your solutions of 12+ 12q hours, where q is the number of full questions that you are required to complete. Any solutions submitted after your personal deadline will not be marked. • This assessment is not intended to need the full time period you have available; the time period indicates only the timescale within which you need to complete this assessment. The available time period automatically takes into account an adjustment for those stu- dents with Reasonable Adjustment Plans (RAPs) or individual personal circumstances. • Solutions must be neatly handwritten; typed solutions will not be marked. • Make sure that you read and follow the instructions for each module, in particular the page limit for your answers. Solutions will only be marked up to the specified page limit. • Solutions to each question should be uploaded to Canvas individually as a single PDF document and may be submitted at any point within the time period available. Solutions may be re-uploaded, however incorrectly submitted solutions, i.e. uploaded to the wrong question, may not be marked. It is your responsibility to ensure that your solutions have successfully uploaded to Canvas by your personal deadline. • Whilst you may use your notes and other materials available to you, submitted answers should be your own. The School of Mathematics has enhanced processes in place to identify any instances of plagiarism or collusion. There is no reduction in the threshold for plagiarism associated with this assessment. Appropriate University penalties will be applied to any individual found to be involved in any way with an act of plagiarism or collusion. • When your answers to each question are graded the marker will take into account a num- ber of factors to determine your awarded grade including: whether your answers are fully correct to all aspects of the question; the overall level of mathematical precision and rigor demonstrated in your arguments; the appropriateness of your mathematical explanation; and, your overall presentation of the mathematical material. The index uses hyperlinks to aid navigation and on each page there is a “Back to index” link. Page 2 Turn over Index 1 LI Algebra & Combinatorics 2 3 LI Differential Equations 4 LI Statistics 2AC 06 25665 Level I LI Algebra & Combinatorics 2 Full marks may be obtained with complete answers to BOTH the following questions. Each answer must be no more than THREE sides of A4, any work in excess of this will not be marked. 1. You may apply results from the lecture notes, but you should clearly state which results you use and when you apply them. (a) Let I = 〈X2−X〉 be the principal ideal of Q[X ] generated by X2−X and let J = 〈X−1〉 be the principal ideal of Q[X ] generated by X−1. Define θ : Q[X ]→Q[X ]/I by θ( f (X)) = [X f (X)]I . (i) Show that θ is a homomorphism. (ii) Show that X−1 ∈ kerθ . (iii) Show that kerθ = J. (iv) Show that there is a subring S of Q[X ]/I such that Q[X ]/J ∼= S. (b) Let m(X) = X2 +X + 2̄ ∈ Z5[X ]. Let f (X) = X4 + 2X3− 5X2−X + 19 ∈ Z[X ] and let f̄ (X) ∈ Z5[X ] be the polynomial obtained from f (X) by reduction modulo 5. (i) Show that m(X) is irreducible in Z5[X ]. (ii) Show that f̄ (X) = m(X)2. (iii) By using the factorisation of f̄ (X) in (b)(ii), or otherwise, show that f (X) is irreducible in Z[X ]. LI Algebra & Combinatorics 2 Turn over Page 1 Back to the index 2. (a) For a graph G = (V,E) a subset S⊆V is called a vertex cover, if every edge of G has at least one endpoint in S. Let M be a maximum matching in G and let S = {v ∈V : vu ∈M for some u ∈V}. Show that S is a vertex cover of G. (b) Assume that the graph G = (V,E) has maximum degree ∆. Show that G contains an independent set of size at least n/(∆+1). (c) Let G = (V,E) be a graph with |V | = n ≥ 4. Suppose that α(G) ≤ √n. Show that |E| ≥ n/4. (d) Give an example of a bipartite graph on which the greedy algorithm uses 3 colours on a particular ordering of its vertices. End of LI Algebra & Combinatorics 2 Page 2 Back to the index 2DE 06 25670 Level I LI Differential Equations Full marks may be obtained with a complete answer to the following question. Your answer must be no more than SIX sides of A4, any work in excess of this will not be marked. 1. (a) For the dynamical system ẋ = (x−2)(x+1)x, sketch the direction fields and the qualitative trajectories in the domain (−3,4)× (0,10). (b) Consider the dynamical system ẋ = x2 +2x− c−2, x ∈ R, where c ∈ R, constant, is a control parameter of the system. (a) Determine the number and location of the equilibrium points of the above system for all values of the control parameter c. (b) Using linear stability analysis determine the stability of the equilibrium points you have found in part (i). If linear stability analysis fails, use a graphical argument to determine stability. (c) State the location and nature of any bifurcations of the system. (d) Sketch the bifurcation diagram of the system. End of LI Differential Equations Page 3 Back to the index 2S 06 25671 Level I LI Statistics Full marks may be obtained with a complete answer to the following question. Your answer must be no more than SIX sides of A4, any work in excess of this will not be marked. 1. (a) Suppose X1,X2, . . . ,Xn are independent and identically distributed (i.i.d.) with probability density function f (x|θ) = { (θ +1)xθ for 0 < x="">< 1="" 0="" otherwise.="" (i)="" find="" e(x1).="" (ii)="" from="" the="" above="" or="" otherwise,="" find="" a="" method="" of="" moments="" estimator="" (mme)="" for="" θ="" .="" (b)="" suppose="" x1,x2,="" .="" .="" .="" ,xn="" are="" independent="" and="" identically="" distributed="" (i.i.d.)="" with="" probability="" density="" function="" f="" (x|θ)="{" e−(x−θ)="" for="" x=""> θ 0 otherwise. (i) Write down the likelihood function of θ given the data X1, . . . ,Xn. (ii) Obtain the maximum likelihood estimator (MLE) of θ . (c) Suppose X1,X2, . . . ,Xn are independent and identically distributed (i.i.d.) Poisson (λ ) random variables with probability mass function p(x|λ ) = { e−λ λ x x! for x = 0,1,2, . . . 0 otherwise. (i) Find P(X1 = 0). (ii) State E(X1). (No derivation is required). (iii) Obtain a method of moments estimator (MME) for λ . (iv) From the above or otherwise, obtain a method of moments estimator (MME) for P(X1 = 0). Question 1 continued overleaf. LI Statistics Turn over Page 4 Back to the index (d) Suppose X1,X2, . . . ,Xn are independent and identically distributed (i.i.d.) with probability density function f (x|θ) = θeθ x−θ−1 for x > e 0 otherwise. Here θ > 1. (i) Write down the likelihood function of θ given the data X1, . . . ,Xn. (ii) Derive the Cramer-Rao lower bound (CRLB) for the variance of an unbiased estima- tor of θ . (e) Two sets of data have been collected on the number of hours spent watching sports on television by some randomly selected males and females during a week: Males 7 13 33 Females 27 24 16 15 Assume that the number of hours spent by the males watching sports, denoted by Xi, i = 1,2,3, are independent and identically distributed normal random variables with mean µ1 and variance σ2. Also assume that the number of hours spent by the females, Yj, j = 1,2,3,4, are independent and identically distributed normal random variables with mean µ2 and variance σ2. Further, assume that the Xi’s and the Yj ’s are independent. Note that 3 ∑ i=1 (Xi− X̄)2 = 370.7, 4 ∑ j=1 (Yj− Ȳ )2 = 105. Test for H0 : µ1 = µ2 against H1 : µ1 6= µ2 at 5% level of significance. End of LI Statistics Page 5 Back to the index