S.-T.Yau College Student Mathematics Contests 2010Analysis and Differential EquationsTeam(Please select 5problems to solve)1.a)Let f (z )be holomorphic in D :|z |<1and |f (z )|≤1(z ∈D ).Prove that|f (0)|−|z |1+|f (0)||z |≤|f (z )|≤|f (0)|+|z |1−|f (0)||z |.(z ∈D )b)For any finite complex value a ,prove that 12π 2π0log |a −e iθ|dθ=max {log |a |,0}.2.Let f ∈C 1(R ),f (x +1)=f (x ),for all x ,then we have ||f ||∞≤ 10|f (t )|dt + 10|f (t )|dt.3.Consider the equation¨x +(1+f (t ))x =0.We assume that ∞|f (t )|dt <∞.Study the Lyapunov stability of the solution (x,˙x )=(0,0).4.Suppose f :[a,b ]→R be a L 1-integrable function.Extend f to be 0outside the interval [a,b ].Letφ(x )=12h x +h x −hf Show thatb a |φ|≤ b a |f |.5.Suppose f ∈L 1[0,2π],ˆf (n )=12π 2π0f (x )e −inx dx ,prove that 1)∞ |n |=0|ˆf(n )|2<∞implies f ∈L 2[0,2π],2)n |n ˆf (n )|<∞implies that f =f 0,a.e.,f 0∈C 1[0,2π],where C 1[0,2π]is the space of functions f over [0,1]such that both f and its derivative f are continuous functions.126.SupposeΩ⊂R3to be a simply connected domain andΩ1⊂Ωwith boundaryΓ.Let u be a harmonic function inΩand M0=(x0,y0,z0)∈Ω1.Calculate the integral:II=−Γu∂∂n(1r)−1r∂u∂ndS,where 1r=1(x−x0)2+(y−x0)2+(z−x0)2and∂∂ndenotes theout normal derivative with respect to boundaryΓof the domainΩ1.(Hint:use the formula∂v∂n dS=∂v∂xdy∧dz+∂v∂ydz∧dx+∂v∂zdx∧dy.)S.-T.Yau College Student Mathematics Contests 2010Applied Math.,Computational Math.,Probability and StatisticsTeam(Please select 5problems to solve)1.Let X 1,···,X n be independent and identically distributed random variables with continuous distribution functions F (x 1),···,F (x n ),re-spectively.Let Y 1<···<Y n be the order statistics of X 1,···,X n .Prove that Z j =F (Y j )has the beta (j,n −j +1)distribution (j =1,···,n ).2.Let X 1,···,X n be i.i.d.random variable with a continuous density f at point 0.LetY n,i =34b n (1−X 2i /b 2n )I (|X i |≤b n ).Show that n i =1(Y n,i −EY n,i )(b n n i =1Y n,i )1/2L −→N (0,3/5),provided b n →0and nb n →∞.3.Let X 1,···,X n be independently and indentically distributed ran-dom variables with X i ∼N (θ,1).Suppose that it is known that |θ|≤τ,where τis given.Showmin a 1,···,a n +1sup |θ|≤τE (n i =1a i X i +a n +1−θ)2=τ2n −1τ2+n −1.Hint:Carefully use the sufficiency principle.4.The rules for “1and 1”foul shooting in basketball are as follows.The shooter gets to try to make a basket from the foul line.If he succeeds,he gets another try.More precisely,he make 0baskets by missing the first time,1basket by making the first shot and xsmissing the second one,or 2baskets by making both shots.Let n be a fixed integer,and suppose a player gets n tries at “1and 1”shooting.Let N 0,N 1,and N 2be the random variables recording the number of times he makes 0,1,or 2baskets,respectively.Note that N 0+N 1+N 2=n .Suppose that shots are independent Bernoulli trails with probability p for making a basket.(a)Write down the likelihood for (N 0,N 1,N 2).12(b)Show that the maximum likelihood estimator of p is ˆp =N 1+2N 2N 0+2N 1+2N 2.(c)Is ˆp an unbiased estimator for p ?Prove or disprove.(Hint:E ˆp is a polynomial in p ,whose order is higher than 1for p ∈(0,1).)(d)Find the asymptotic distribution of ˆp as n tends to ∞.5.When considering finite difference schemes approximating partial differential equations (PDEs),for example,the scheme(1)u n +1j =u n j −λ(u n j −u n j −1)where λ=∆t ∆x ,approximating the PDE (2)u t +u x =0,we are often interested in stability,namely(3)||u n ||≤C ||u 0||,n ∆t ≤T for a constant C =C (T )independent of the time step ∆t and the spa-tial mesh size ∆x .Here ||·||is a given norm,for example the L 2norm orthe L ∞norm,of the numerical solution vector u n =(u n 1,u n 2,···,u n N ).The mesh points are x j =j ∆x ,t n =n ∆t ,and the numerical solutionu n j approximates the exact solution u (x j ,t n )of the PDE (2)with aperiodic boundary condition.(i)Prove that the scheme (1)is stable in the sense of (3)for boththe L 2norm and the L ∞norm under the time step restriction λ≤1.(ii)Since the numerical solution u n is in a finite dimensional space,Student A argues that the stability (3),once proved for a spe-cific norm ||·||a ,would also automatically hold for any other norm ||·||b .His argument is based on the equivalency of all norms in a finite dimensional space,namely for any two norms ||·||a and ||·||b on a finite dimensional space W ,there exists a constant δ>0such thatδ||u ||b ≤||u ||a ≤1δ||u ||b .Do you agree with his argument?If yes,please give a detailed proof of the following theorem:If a scheme is stable,namely (3)holds for one particular norm (e.g.the L 2norm),then it is also stable for any other norm.If not,please explain the mistake made by Student A.6.We have the following 3PDEs(4)u t +Au x =0,(5)u t +Bu x =0,3 (6)u t+Cu x=0,C=A+B.Here u is a vector of size m and A and B are m×m real matrices. We assume m≥2and both A and B are diagonalizable with only real eigenvalues.We also assume periodic initial condition for these PDEs.(i)Prove that(4)and(5)are both well-posed in the L2-norm.Recall that a PDE is well-posed if its solution satisfies||u(·,t)||≤C(T)||u(·,0)||,0≤t≤Tfor a constant C(T)which depends only on T.(ii)Is(6)guaranteed to be well-posed as well?If yes,give a proof;if not,give a counter example.(iii)Suppose we have afinite difference schemeu n+1=A h u nfor approximating(4)and another schemeu n+1=B h u nfor approximating(5).Suppose both schemes are stable in theL2-norm,namely(3)holds for both schemes.If we now formthe splitting schemeu n+1=B h A h u nwhich is a consistent scheme for solving(6),is this scheme guar-anteed to be L2stable as well?If yes,give a proof;if not,givea counter example.S.-T.Yau College Student Mathematics Contests2010Geometry and TopologyTeam(Please select5problems to solve)1.Let S n⊂R n+1be the unit sphere,and R n⊂R n+1the equator n-plane through the center of S n.Let N be the north pole of S n.Define a mappingπ:S n\{N}→R n called the stereographic projection that takes A∈S n\{N}into the intersection A ∈R n of the equator n-plane R n with the line which passes through A and N.Prove that the stereographic projection is a conformal change,and derive the standard metric of S n by the stereographic projection.2.Let M be a(connected)Riemannian manifold of dimension2.Let f be a smooth non-constant function on M such that f is bounded from above and∆f≥0everywhere on M.Show that there does not exist any point p∈M such that f(p)=sup{f(x):x∈M}.3.Let M be a compact smooth manifold of dimension d.Prove that there exists some n∈Z+such that M can be regularly embedded in the Euclidean space R n.4.Show that any C∞function f on a compact smooth manifold M (without boundary)must have at least two critical points.When M is the2-torus,show that f must have more than two critical points.5.Construct a space X with H0(X)=Z,H1(X)=Z2×Z3,H2(X)= Z,and all other homology groups of X vanishing.6.(a).Define the degree deg f of a C∞map f:S2−→S2and prove that deg f as you present it is well-defined and independent of any choices you need to make in your definition.(b).Prove in detail that for each integer k(possibly negative),there is a C∞map f:S2−→S2of degree k.1S.-T.Yau College Student Mathematics Contests 2010Algebra,Number Theory andCombinatoricsTeam(Please select 5problems to solve)1.For a real number r ,let [r ]denote the maximal integer less or equal than r .Let a and b be two positive irrational numbers such that 1a +1b = 1.Show that the two sequences of integers [ax ],[bx ]for x =1,2,3,···contain all natural numbers without repetition.2.Let n ≥2be an integer and consider the Fermat equationX n +Y n =Z n ,X,Y,Z ∈C [t ].Find all nontrivial solution (X,Y,Z )of the above equation in the sense that X,Y,Z have no common zero and are not all constant.3.Let p ≥7be an odd prime number.(a)Evaluate the rational number cos(π/7)·cos(2π/7)·cos(3π/7).(b)Show that (p −1)/2n =1cos(nπ/p )is a rational number and deter-mine its value.4.For a positive integer a ,consider the polynomialf a =x 6+3ax 4+3x 3+3ax 2+1.Show that it is irreducible.Let F be the splitting field of f a .Show that its Galois group is solvable.5.Prove that a group of order 150is not simple.6.Let V ∼=C 2be the standard representation of SL 2(C ).(a)Show that the n -th symmetric power V n =Sym n V is irre-ducible.(b)Which V n appear in the decomposition of the tensor productV 2⊗V 3into irreducible representations?1。
