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15 Oct 2020: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... Can such a group beinnite? 8. Let G be a group of even order.

5 Jun 2020: According to Occam’s razor, the simplest such explanation isthe one that should be favored. ... Given a string x, suppose wefocus on a special sub-class of programs p that produce x, namely, programs thatfirst describe a set B of strings such that x B,

5 Jun 2020: such thatlim supnn1 log Qn(B(Xn1 ,D)) R < a.s. ... The question then becomes how to find such a Q when P is unknown.

11 Feb 2020: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

22 Oct 2020: Let B(G,F) be the (extended) Bruhat–Tits building of G(F). For any x B(G,F), there is asmooth affine group scheme Gx over OF such that ... a choice of basis b for Λ such that the embeddings of buildings (3.3.1) and (3.3.2) induceembeddings of the

20 Nov 2020: payout Y is called attainable if there exists a scalar a andportfolio b Rd such that Y = a b>S1. ... For the general case, show that there exists aportfolio θ Rd such that Y = θ S1 Z where Z and S1 are independent.] Verify thatY π(Y ) is concave.

5 Jun 2020: 15, 18]. This means that there is a σ-finite measure ψ on(X,B) such that, for any A B with ψ(A) > 0, and any x X,. ... There is a function V : X [1,], a small set C X,and constants δ > 0, b < , such that:.

20 Nov 2020: S aSbb | T, T bTaa | S | ǫ. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF (giving a justification for the transformation). ... b) Show that the language L := {anbncn | n 1} is not a CFL.

5 Jun 2020: π(f) :=. π(dx)f(x) <. (ii) There exists a closed, small set C B such that,. ... x X and every B B satisfying ψ(B) > 0, there is a constantcB < such that,.

12 Nov 2020: 5. Let A Rn be such that every continuous function f : A R is bounded. ... V such that A Uand B V.

5 Jun 2020: But Wn, the “waiting time” until the firstD-close match for Xn1 , is approximately equal to the reciprocalof the probability of finding such a match. ... Given such a code, we candefine a (sub-)probability distribution on Ân by.

10 Oct 2020: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

28 Oct 2020: if any), and why? (b) By replacing A in the construction of B with a suitably chosen set, construct a setC N such that neither C nor N C are ... and such that Wn is recursive foreach n I but. nI Wn is not r.e.

5 Jun 2020: This means thatthere is aσ-finite measureψ on (X,B) such that,for any A B satisfying ψ(A) > 0 and any initialconditionx,. ... There exists an integern0 1, a constanta0 > 0, and a probability measure̺ on(X,B), such that,.

3 Mar 2020: Show that f is integrable. What is. 10f(x)dx? 4. Give an example of a continuous function f : [0,) [0,) such that0f(x)dx. ... Show. that for every > 0 and every closed interval I [a,b] of positive length, there isa closed interval J I such that J has

5 Jun 2020: This isformalized by requiring that{Xn} has a unique stationarydistribution π such that. ... Recall [13]that an irreducible, aperiodic chain{Xn} with values in thecountable alphabetA is geometrically ergodicif there existsa functionV : A [1, ), positive

5 Jun 2020: what happens when we look for longer and longer such initial patterns how. ... ii If then there exists a standard Brownian motion fBt t g such that.

5 Jun 2020: probability:. Doeblin Condition (DC). There exists an integer r 1 and a real numberβ > 0 such that,. ... This gives the first boot-strap sample. Then the whole process is repeated to generate a total of B such blocksL1, L2,. ,

5 Jun 2020: probability:. Doeblin Condition (DC). There exists an integer r 1 and a real number β > 0 such that,. ... This gives the first bootstrap sample. Then the whole process is repeated to generate a totalof B such blocks L1, L2,. ,

5 Jun 2020: Indeed, Comets [23] provides such anargument when and are both i.i.d. ... We assume the existence of aref-erence letter, i.e., an such that.

6 May 2020: d. (p, Im c|[b′,b]. ) D0. Therefore there exist s [a,a′] and t [b′,b] such that d (p,c(s)) ,d (p,c(t)) D0. ... By definition, form,n 1, there exists bm,n B such that amRnbm,n.

28 Jan 2020: Regarding T as a map from C into C by identifyingR2 with C in the usual way, show that there exist unique complex numbers A, B such thatfor every z C, ... Can such a map everbe a Möbius transformation? 9. Let U C be open and let f = uiv : U C.

2 Dec 2020: identity). (G3) For each a G there exists a b G such that a b = e. ... ii) If a G then e a = a. (iii) If a,b,b′ G are such that a b = e and a b′ = e, then b = b′.

30 May 2020: Abigon for α,β is an embedded (closed) disc D S such that D(αβ) = D = ab where a αand b β are arcs. ... Therefore, after iterating, we eventually findψ Zd 6 Mod(S) such that φαi ψ αi.

6 May 2020: By definition, form,n 1, there exists bm,n B such that amRnbm,n. ... R such that dX (ai,b) =dA (ai,αi) ε2 dB (βi,b).

8 Jun 2020: A representation ofthis functor, if it exists, is called a product of A and B: it consists of an object AB together withmorphisms AB π1 A and AB π2 B such ... iI of objects of (A G) indexed by a set I, such that every B ob (A G) has amorphism Bi B in

19 Oct 2020: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

28 May 2020: F :X I Y (where I = [0, 1]) such that F(, 0) = f0 and F(, 1) = f1. ... Let f : Sn Sn and q Sn be such that f1 (f(q)) = {q1,.

30 Nov 2020: f. There is a chart (φ,U) about p withφ(p) = 0 such that. ... of isolated zeroes, there is an analytic functiong with g(0) 0 such that.

5 Jun 2020: Q(B(Xn1 ;D)) that such a match indeed occurs. In the case when no distortion is allowed, Q(B(Xn1 ;D)). ... the supremum is taken over all integers r 1 and all pairs of events A and B such that B 2 (Y 1rk),.

3 Jun 2020: iii) f is a bilipschitz embedding if there exist a,b > 0 such that. ... A {1,. ,n} with |A| = t such that A(s) = {B A, |B| = s}is monochromatic.Definition 2.17 (Trees).

26 Feb 2020: n}, let the non-negative numbers A(n)k andthe ‘nodes’ x. (n)k [a,b] be given such that for each polynomial P , the error. ... such that for every f C([1, 1]), there exists a unique series.

5 Jun 2020: Theorem Almost sure invariance principle Let X be a nitevalued stationary process such. ... fB t t g such that for any. R t B t O t as.

9 Nov 2020: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 13. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

10 Jun 2020: jJ Gj such that pj f = qj, and we have f(Z) L. ... Let (Gj)jJ be a surjective inverse system such that G = limjJ Gj.

6 Jun 2020: I A family of transition kernels {Pθ,θ Θ} such that, for all θ Θ,the target distribution π? ... Θ and a set A such that P(A) = 1 and.

10 Oct 2020: b) If A is finite show that we can find such a B with B A. ... a) Can you define a ordered pair such that ρ(〈x,y〉) = max(ρ(x),ρ(y)) 1?(b) Can you define a ordered pair such that ρ(〈x,y〉) = max(

3 Mar 2020: Con-struct a morphism of smooth curves X P1 such that the preimage of consists ofexactly n points, with ramification indices given by a1,. ... Conclude thatf,g,f g,f λg must be perfect squares. (b) Prove the following: If f,g are polynomials in k[t]

5 Jun 2020: b) There exist (distinct) points fx0;x1; : : : ;xkg in I such that, for every permutation ofthe indices f0;1; : : : ;kg with not equal to the identity, we have. ... Then by Lemma 1 there is a (unique) < 0 such that (a) holds, and letting W be dened as in

24 Feb 2020: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elementsa and b in R is an element d in R such that d divides both ... a and b, and if c dividesboth a and b then c divides d.

12 Feb 2020: b) We say that a measurable f : Rn! C is in weak-Lp(Rn), written f 2 Lp,w(Rn) ifthere exists a constant C such that. ... Deduce. N[. i=1. Bi. 6 3n. kX. j=1. Bij. b) () Suppose {Bj : j 2 J} is an arbitrary collection of balls in Rn such that each ballhas

5 Jun 2020: Since suffixes of the form “100 0” are the most commonnon-zero suffixes produced by the CTW, we note that in 22 outof the 28 neurons the percentage of such suffixes ... Using the empirical ISI distribution is problematicsince such estimates are

29 Sep 2020: Choose then a and b such that. 1. a. def=. 1. ... 29. Proof of Lemma 3.1.1. x K , rx > 0 such that B(x,rx).

4 Mar 2020: a) Show that S is a dense subset of Hs(Rn). b) Find a condition on s such that x 2 Hs(Rn). ... 2u u = f. b) Show further that there exists C > 0 such that kukHs4.

5 Jun 2020: Thenthere exists "0 > 0 such that for each z 2 B("0) there exists an eigenvaluez 2 C for Qz satisfying jzj =. Qz , and is real for real 2 (0;"0). ... i) For any Æ > 0, there exists b1 = b1(Æ) < 1 such that.

4 Feb 2020: b) Construct a continuous surjection p : T2 K such that for every x K, p1(x) consists of exactly two points.(c) Draw inside the identification square for the Klein bottle ... Suppose that N N is a positive integer such that2E/F < N for every possible

5 Jun 2020: b) If f(X1) = 0 with probability one for a sequence ofdistortion values Dn (0,Dmax) such that Dn 0, thenP is the uniform distribution and ρ is a permutation ... 2. Proof of Theorem 1, (b): Let Dn, n 1, be a sequenceof of distortion values in (0,Dmax)

5 Jun 2020: such that, for all 0 i 6= j k, with j 6= 0, we haverj(xj) > rj(xi):(b) There exist (distinct) points fx0;x1; : : : ;xkg in I such. ... Then by Lemma 1 there is a (unique) < 0 such that(a) holds, and letting W be dened as in (b) we also have.

5 Jun 2020: what happens when we look for longer and longer such initial patterns { how. ... nding such a match (see Theorem 4.1, Chapter 4):. log Wn(D) log Q(B(Xn1 ;D)):.

5 Jun 2020: with ψ( ) := R(x0, ): For all x X and all A B such thatR(x0,A) > 0, we have,. ... constants δ > 0, b < , such that,.