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If (zn) is an infinite sequence of points in ...
https://www.dpmms.cam.ac.uk/~tkc10/ComplexDE/Weierstrass.pdf12 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. -
Mich. 2008 ANALYSIS II—EXAMPLES 4 PAR 1. Show that ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2008-2009/08sheet4.pdf30 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 -
Michaelmas Term 2003 J. M. E. Hyland Linear Algebra: ...
https://www.dpmms.cam.ac.uk/~jmeh1/Teaching/2ndla03.pdf12 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 -
ON THE CARTAN MAP FOR CROSSED PRODUCTS ANDHOPF-GALOIS EXTENSIONS ...
https://www.dpmms.cam.ac.uk/~sjw47/HopfGalois.pdf27 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 -
Thoughts on the Essay Question T. W. Körner November ...
https://www.dpmms.cam.ac.uk/~twk10/Excess.pdf5 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.]. -
Some improvements to 4-descent on an elliptic curve Tom ...
https://www.dpmms.cam.ac.uk/~taf1000/papers/fourdesc.pdf2 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. -
ALGEBRAIC TOPOLOGY MICHELMAS 2008 EXAMPLE SHEET 1 1. Let ...
https://www.dpmms.cam.ac.uk/study/II/AlgebraicTopology/2008-2009/Examples1.pdf14 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. -
ANALYSIS II EXAMPLES 4 Michaelmas 2005 J. M. E. ...
https://www.dpmms.cam.ac.uk/~jmeh1/Teaching/an05-4.pdf19 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 -
Example sheet 4, Galois Theory, 2007. 1. (i) Let ...
https://www.dpmms.cam.ac.uk/study/II/Galois/2007-2008/ex4.pdf4 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. -
Topological GroupsPart III, Spring 2008 T. W. Körner March ...
https://www.dpmms.cam.ac.uk/~twk10/Topg.pdf8 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 -
ALGEBRAIC TOPOLOGY (PART II)EXAMPLE SHEET 2 CAUCHER BIRKAR (1) ...
https://www.dpmms.cam.ac.uk/study/II/AlgebraicTopology/2007-2008/top-exa2.pdf23 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 -
A Term Calculus for Intuitionistic Linear LogicNick Benton1, Gavin ...
https://www.dpmms.cam.ac.uk/~jmeh1/Research/Oldpapers/bbdphtlca93.pdf27 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. -
Mich. 2008 ANALYSIS II—EXAMPLES 1 PAR 1. Which of ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2008-2009/08sheet1.pdf17 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. -
Topics in Analysis: Example Sheet 1 Lent 2007-08 N. ...
https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2007-2008/sheet1.pdf4 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. -
Mich. 2008 NUMBERS AND SETS — EXAMPLES 2 PTJ ...
https://www.dpmms.cam.ac.uk/study/IA/Numbers%2BSets/2008-2009/N+S2.pdf3 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. -
Part IID RIEMANN SURFACES (2008–2009)Example Sheet 3…
https://www.dpmms.cam.ac.uk/study/II/Riemann/2008-2009/rs-example3.pdf21 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 -
PART II REPRESENTATION THEORYSHEET 1 Unless otherwise stated, all ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2008-2009/repex1.pdf17 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. -
Mich. 2008 NUMBERS AND SETS — EXAMPLES 1 PTJ ...
https://www.dpmms.cam.ac.uk/study/IA/Numbers%2BSets/2008-2009/N+S1.pdf14 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 -
Michaelmas Term 2008 T.A. Fisher Linear Algebra: Example Sheet ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2008-2009/lin_alg-08-4.pdf28 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. -
A Term Calculus for Intuitionistic Linear LogicNick Benton1, Gavin ...
https://www.dpmms.cam.ac.uk/~jmeh1/Research/Pub91-00/bbdphtlca93.pdf27 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.
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