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12 Mar 2008: If g is another such function then f(z) = g(z) exp h(z)for some entire function h. ... If g were another such function then g/f would be an entire function with no zeros and thereforeequal to exp h for some entire function h.

30 Nov 2008: Find the Taylor series of g(I H) about I. 9. Let f : R2 R and (a, b) R2.(a) Suppose that D1f exists and is continuous in some open ball ... around (a, b), and that D2f exists at (a, b).Show that f is differentiable at (a, b).(b) Suppose instead that D1f

12 Aug 2008: vary in difficulty but cover some instructive points. 1. For what values of a and b does the system of simultaneous linear equations. ... What are the eigenvalues and eigenvectors?Show that if an endomorphism β of V commutes with α then β = p(α) for

27 Feb 2008: C = {M M(A) : ResAB(M) P(B)}.Since any P P(A) is a direct summand of An for some n > 0, the restrictionResAB(P) is a finitely generated projective ... It follows thatwe can find P,Q P(H) such that [P] [Q] = m[1] inside G0(H) for some m > 0depending only

5 Nov 2008: 1) History. [If you decide to include some history, keep it down to acouple of sentences. ... This is a mathematics exam.]. (2) Preliminary definitions. [You may decide to give some illustrativeexamples.].

2 Feb 2008: A′,B′) = (m11A N m12B N,m21A N m22B N). for some (M,N) G4(K) := GL2(K) GL4(K). ... Q = ξ2for some ξ F and F[x1,x2,x3,x4] a linear form.

14 Oct 2008: Show thatf(x) = x for some x S1. 3. Which of the letters A,B,. ... 1,1) π1(S1,f(1)) when. (a) f(eiθ) = ei(θπ/2). (b) f(eiθ) = einθ for some n Z.

19 Mar 2008: only if U = Y V for some V open in (X, dX ).(ii Show that A is closed in (Y, dY ) if and only if A = Y B for some B ... a contraction? Deduce that the differential equation. dy. dx= 2xy , with y = 1 when x = 0 ,. has a unique solution in some interval

4 Feb 2008: Show that f and g have the same splitting field if andonly if b = cmar for some c K and r N with gcd(r, m) = 1. ... Deduce that every finite abelian group is the Galois group of some Galois extensionK/Q.

8 Mar 2008: In some sense a topological group is an object were the the statement‘the neighbourhood of every point looks the same’ is (a) meaningful and (b)true. ... Theorem B. (Bochner’s theorem.) A continuous function f : G C ispositive definite if and only

23 Feb 2008: fundamental group of some space. (5) Let L be the “infinite ladder of circles” given by the subset ofR2 consisting of the union of the circle of radius 1/2 aroundeach ... in C which maps to 0 in C′. (1) Show that x maps to 0 in Dand hence that x is

27 Aug 2008: L3;! A B! A (!R)! AFig. 1. (Multiplicative) Intuitionistic Linear Logicpackets, using some predened set of inference rules. ... Some thought that this represented a mismatch betweenthe semantics and syntax of linear logic.

17 Oct 2008: Show that if fn f uniformly and (xm) isa sequence of points in [a, b] with xm x then fn(xn) f (x). ... Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f is bounded.

4 Feb 2008: 6) Let f : B B be a continuous map from the open disc B = {(x, y) R2 : x2 y2 < 1} into it-self. ... 10) Suppose f : R R has the intermediate value property that if a, b R, f (a) < c < f (b),then f (y) = c for some y between a and b.

3 Nov 2008: 3. Let a,b,c be three positive integers. Are the following true or false? ... b) Now let F = Zp be the field of integers mod p, for some prime p, and suppose d dividesp 1.

21 Nov 2008: f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C. ... N. m,n=0. am,nf(z)mg(z)n 0, for some am,n C. Hence show that any two meromorphic functions f,g on a complextorus C/Λ are

17 Nov 2008: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Given any prime p. Find an example of a representation of some finite group oversome field of characteristic p, which is not completely reducible.

14 Oct 2008: Hence x1 = x2 = = xk1, so theresult is true for n = k 1.(b) Clearly, some positive integers are interesting: for example, 6 is the smallest perfectnumber, 1729 is the smallest number ... Suppose some positive integers were not interesting: then, by the

27 Aug 2008: L3;! A B! A (!R)! AFig. 1. (Multiplicative) Intuitionistic Linear Logicpackets, using some predened set of inference rules. ... Some thought that this represented a mismatch betweenthe semantics and syntax of linear logic.

28 Nov 2008: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... λn.(i) Show that if Sk = I for some k 1 then S2 = I.(ii) Find the eigenvalues of the endomorphism of Matn(R) given by X 7 SXT S.

16 Jul 2008: Ψ′ =(. 0 λxnbλxnb 0 A. AT Ψ. )for some 0 6= λ k, b R and A Mat2,n2(R). ... So (4.3) becomes. (4.4) A(xng h)T = (λxn b)(f2f1. )for some linear form b R.

13 Aug 2008: a,b,c), (a′,b,c′). for some b. Corollary 2.3 Let (a0,c0), (a1,b1),. , ... As we explain in Section 4 this leads us to hope that wecan in some sense borrow an exponential from the relationalmodel.

20 Jan 2008: 7) Let R be a commutative ring with unit. For a,b R we say that a divides b (notation a|b) ifb = ac for some c R. ... If R is a domain, then a b (a|b b|a). We call an elementa RR irreducible if for any factorization a = bc one of b,c is a unit

21 Aug 2008: iii) there is a generic object G 2 E. P. for some distinguished object P 2 B,. ... i) P =? =) (P! Q) = U. Suppose there are some a; b 2 U for which ab is undened and.

21 Aug 2008: iii) there is a generic object G 2 E. P. for some distinguished object P 2 B,. ... i) P =? =) (P! Q) = U. Suppose there are some a; b 2 U for which ab is undened and.

20 Feb 2008: 1.6. Some special cases. We next show that the rank of αi attains the upperbound given in Theorem B in some special cases. ... Hence there exists h H suchthat xh := h1xh = ya for some a > 1.

18 Aug 2008: A B) (C D)are not equal. (There are some similar issues for the triangle diagrams.). ... V andW for a conjunction and disjunction according to some bracketings.In these circumstances we say thatiA;B, i>, iC;D and i?

21 Aug 2008: ForestForest. of A of BFigure 1: The forest of questions of A ) B.At any stage of a play, not every move is necessarilyavailable to the players; however some moves may ... Suppose at some point afterinteracting with the environment, the name a has been in

19 Aug 2008: def. (〈a〉, 〈〉. ), S. (ι2(b). )=def. (〈〉, 〈b〉. )for a A and b B. ... We introduce some abbreviations for future reference. For x P(A B),we write (x.1, x.2) for S](x), so that we have x.1 PA and x.2

8 Jan 2008: However, the work we have already done enables us to produce some niceinfinite dimensional Banach spaces. ... For historical reasons Baire’s theorem is associated with some rather un-helpful nomenclature.

7 May 2008: x =V for some n, and of course this cannot happen in ZF. ). ... If w is a molecular term of rank n then, in full generality, it is w′g forsome constant g and some boolean combination w′ of B(zi) for various i.

22 Jan 2008: It is best to think of what is presented here as another reworking of some basicmaterial. ... To have identities for this composition, weneed to contract some silent actions3 and these actions are insignificant.

29 Sep 2008: TµAB? µAB- T (A B). µAB? There may appear to be some redundancy in the definition, and if our commu-tativity is symmetric in the sense of section 3.6, that ... We record some basic properties. Proposition 13. The following equalities hold in T-Alg.(i)

1 Feb 2008: Now fix some w D, choose r > 0 such that D(w,r) D. ... Remark. A function f : U C is said to be analytic if for every a U, f canbe represented by a convergent power series on some B(a,r) U.

27 Aug 2008: Some thought that this represented a mismatch between the semantics andsyntax of linear logic. ... theory of some basic intuitionistic logic via the notion of a cartesianclosed category.

27 Aug 2008: Some thought that this represented a mismatch between the semantics andsyntax of linear logic. ... theory of some basic intuitionistic logic via the notion of a cartesianclosed category.

13 May 2008: N.b. some texts use the term alphabet instead of vocabulary, letter insteadof word, and word instead of sentence.). ... Can also have n-tuples: A B. N. Any binary relation R on the members two sets A and B is some subset oftheir Cartesian product:.

22 Aug 2008: a model of T (of some unspe ied kind); we write [[ for the interpretation fun tion. ... when [[s is a value in [[P implies s vfor some v with [[s = [[v 2 [[P.

22 Aug 2008: a model of T (of some unspe ied kind); we write [[ for the interpretation fun tion. ... when [[s is a value in [[P implies s vfor some v with [[s = [[v 2 [[P.

5 Feb 2008: 71.2 Do some ordinals here. 9. 2 Some model theory 112.1 Products. ... Sacks is right—omitting types is hard!First some definitions. DEFINITION 20. 1.