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Groups Example Sheet 1Michaelmas 2015 Julia Goedecke Please send ...
https://www.dpmms.cam.ac.uk/study/IA/Groups/2015-2016/GroupsSheet1-2015.pdf21 Oct 2015: 3. Show that the set of functions on R of the form f(x) = ax b, where a and b are realnumbers and a 6= 0, forms a group under composition ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the -
Homotopy Theory, Examples 4 Oscar Randal-Williams Lent 2015 All ...
https://www.dpmms.cam.ac.uk/~or257/teaching/IIIHtyThy2015/Sheet4.pdf25 Feb 2015: that 〈Sq(x) y, [M]〉 = 〈x Sq1(y) v, [M]〉 for all x,y H(M). ... i) Show that Sqi : Hni(M) Hn(M) = F2 is zero for all i > 0. -
PART II AUTOMATA AND FORMAL LANGUAGES MICHAELMAS 2015-16 EXAMPLE ...
https://www.dpmms.cam.ac.uk/study/II/AutomataAndFormalLanguages/2015-2016/AFL-es-3.pdf12 Nov 2015: b) All words w {a,b,c} consisting of some number of a’s (possiblynone), followed by some number of b’s (at least one), followed bysome number of c’s ... c) All words w {0, 1} which contain a 1 somewhere in the last 5positions. -
JPM Michaelmas 2015 Probability and Measure 1 1.1. Let ...
https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2015-2016/ex1.pdf14 Oct 2015: 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 ... Show that. (a) C is uncountable and has Lebesgue measure 0,(b) for all x [0, 1], the limit F(x) = -
ANALYSIS II—EXAMPLES 4 Mich. 2015 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2015-2016/15sheet4.pdf30 Nov 2015: for some K and all t [a,b], x,y BR(x0).We showed in lecture that for each t0 [a,b], there is a unique f C([a,b]; ... tt0F(s,f(s)) ds, t [a,b]. Assuming that F. extends to all of [a,b]Rn as a continuous function, show that this f is in fact the -
HW3.dvi
https://www.dpmms.cam.ac.uk/study/II/AlgebraicGeometry/2014-2015/HW3.pdf16 Feb 2015: Part II Algebraic Geometry. Example Sheet III, 2015(For all questions, assume k is algebraically closed. ... 22 x. 33 k[x0,. ,x3]. Find the dimension of the tangent space at all the. -
ANALYSIS 1 EXAMPLES SHEET 4 Lent Term 2015 W. ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2014-2015/analysis1examples4.pdf4 Mar 2015: In particular,. prove that L(ab) = L(a) L(b) for all positive a and b. ... 2)n cos(θx)dx. Prove that θ2In =. 2n(2n1)In1 4n(n1)In2 for all n 2, and hence that θ2n1In(θ) = n!(Pn(θ) sin θ Qn(θ) cos -
CODES AND CRYPTOGRAPHY – Example Sheet 1TKC Lent 2015 ...
https://www.dpmms.cam.ac.uk/study/II/Coding/2014-2015/CC1.pdf5 Feb 2015: Find an optimal binarycode. Determine whether there are optimal binary codes with either (a) all but one codeword ofthe same length, or (b) each codeword a different length. ... Show that in the case where all of the odds are equal this maximum andthe -
CODES AND CRYPTOGRAPHY – Example Sheet 1TKC Lent 2015 ...
https://www.dpmms.cam.ac.uk/~tkc10/CodesandCryptography/Exercise_1.pdf4 Feb 2015: Find an optimal binarycode. Determine whether there are optimal binary codes with either (a) all but one codeword ofthe same length, or (b) each codeword a different length. ... Show that in the case where all of the odds are equal this maximum andthe -
PART II REPRESENTATION THEORYSHEET 4 Unless otherwise stated, all ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2014-2015/repex4.pdf19 Jan 2015: b) Find all 1-dimensional representations of G.(c) Let ψ : Fp C be a non-trivial 1-dimensional representation of the cyclic group. ... d) Prove that the collection of representations constructed in (b) and (c) gives a com-plete list of all irreducible -
Numbers and Sets (2015–16) Example Sheet 4 of 4 ...
https://www.dpmms.cam.ac.uk/study/IA/Numbers%2BSets/2015-2016/examples-NS-15-4.pdf20 Nov 2015: An 6= holds for all n. Must it bethat. n=1 An 6=? ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T? -
ANALYSIS II—EXAMPLES 2 Mich. 2015 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2015-2016/15sheet2.pdf2 Nov 2015: 1. Quickies: (a) Describe all continuous functions f : [0, 1] Rn satisfying ‖ 10f‖ =. 10‖f‖. (b) Show that two norms ‖ ‖, ‖ ‖′ on a vector space V are Lipschitz equivalent if and only ... denote the vector space of all bounded Riemann -
STATISTICAL MODELLING Part IICExample Sheet 3 (of 4) RDS/Lent ...
https://www.dpmms.cam.ac.uk/study/II/StatisticalModelling/2014-2015/ex3_2015.pdf23 Feb 2015: Wii =1. aiV (µi){g′(µi)}2,. (you need not specify i(σ2), and you may assume /βjσ2 = /σ2βj for all j). ... c) How do the expressions in (a) and (b) simplify when g(µi) is the canonical linkfunction? -
PART II REPRESENTATION THEORYSHEET 3 Unless otherwise stated, all ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2014-2015/repex3.pdf19 Jan 2015: 7 Let ρ : G GL(V ) be a representation of G of dimension d.(a) Compute the dimension of SnV and ΛnV for all n.(b) Let g G and let ... constituent of IndGKχ, then all |G : K| conjugates of χ are constituents of ResGKφ. -
Complex Analysis IB, Lent 2015 Example sheet 2 1 ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2014-2015/CxAnalIB-2015-Sheet2.pdf10 Feb 2015: Show that f is a polynomial, of degree k, if and only if there is a constantM for which |f(z)| < M(1 |z|)k for all z.(ii) Show that ... Show that if there existsk such that |f(z)| |z|k for all z with |z| sufficiently large, then f is a rational function -
Michaelmas Term 2015 SJW Linear Algebra: Example Sheet 4 ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2015-2016/lin_alg-15-4.pdf23 Nov 2015: 3. (i) Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive definite bilinear form on the spaceMatn(R) of all nn real matrices. ... sn forms a basis for Pn.(iii) For all 1 k n, sk spans the orthogonal complement of Pk1 in Pk.(iv) sk is an -
Metric and Topological Spaces Easter 2015 Example Sheet 2 ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2014-2015/MetTop_2015_Ex2.pdf31 Aug 2015: for every x X, there is an open neighborhood U of x such thatf(y) = f(x) for all y U. ... Hint:first find a subsequence (fni ) such that fni (x) converges for all x Q [0, 1].)Deduce that the closure of B in (C[0, 1], d) is compact. -
Metric and Topological Spaces Easter 2015 Example Sheet 1 ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2014-2015/MetTop_2015_Ex1.pdf27 Apr 2015: Show that f is continuous ifand only if f(A) f(A) for all A X. ... Show thatthis is a topology, that all points of R are closed with respect to it, but thatthe topology is not Hausdorff. -
Michaelmas Term 2015 SJW Linear Algebra: Example Sheet 1 ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2015-2016/lin_alg-15-1.pdf15 Oct 2015: a) The set C of continuous functions. (b) The set {f C : |f(t)| 1 for all t [0, 1]}.(c) The set {f C : f(t) 0 as t }.(d) ... i Ui. Give an example where Ui Uj = {0} for all i 6= j, yet U1. -
ANALYSIS II—EXAMPLES 4 Mich. 2014 The questions marked with ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2014-2015/14sheet4revised.pdf27 Feb 2015: continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I). ... b) = inf (γ), wherethe inf is taken over all continuous γ: [0, 1] E with γ(0) = a, γ(1) = b. -
STATISTICAL MODELLING Part IICExample Sheet 1 (of 4) RDS/Lent ...
https://www.dpmms.cam.ac.uk/study/II/StatisticalModelling/2014-2015/ex1_2015.pdf23 Jan 2015: Show that for all v Rn, ‖v‖2 ‖ΠWv‖2 ‖ΠV v‖2. (b) Consider the linear model (1) but where only the first p0 components of β are non-zero. ... Now prove that if A1,. ,Am and B1,. ,Bm are all independent real-valuedrandom variables and A1 B1,. -
Michaelmas Term 2015 T.A. Fisher Number Theory: Example Sheet ...
https://www.dpmms.cam.ac.uk/study/II/NumberTheory/2015-2016/number_theory-15-4.pdf26 Nov 2015: 4. Find all bases for which 39 is an Euler pseudoprime. ... More generally, showthat if p and 2p1 are both prime numbers, then N = p(2p1) is a pseudoprimefor precisely half of all bases. -
ON FAMILIES OF 9-CONGRUENT ELLIPTIC CURVES TOM FISHER Abstract. ...
https://www.dpmms.cam.ac.uk/~taf1000/papers/congr9.pdf2 Jun 2015: Let Kn be the function field of X(n) over K. Let B K3 /(K3 )3 be thesubgroup generated by all rational functions on X(3) with zeros and poles only ... All computer calculations in supportof this work were carried out using Magma [2]. -
What is the probability that a random integral quadraticform ...
https://www.dpmms.cam.ac.uk/~taf1000/papers/isotropic.pdf15 Sep 2015: Q(x1,x2,. ,xn) =. 1ijn. cijxixj, (1). where all coefficients cij lie in Z. ... Thus we have established items (a)–(c), for all n, in the case that D = U(a,b) is a constantdistribution supported on a box [a,b]. -
INTRODUCTION TO PLECTIC COHOMOLOGY J. NEKOVÁŘ AND A. J. ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/introplec.pdf18 Feb 2015: In the best of all possible worlds one would expect the following picture. ... for all infinite primes v of F and product decompositions. G(R) =v|. -
Holmes.dvi
https://www.dpmms.cam.ac.uk/~twk10/Holmes.pdf22 Sep 2015: 3. for all a and b. 41. Exercise 2.3.1. (Only part (ii).). ... for some constant B. 106. If h(t) = C exp at for all t,. -
ÙD-MODULES ON RIGID ANALYTIC SPACES IKONSTANTIN ARDAKOV AND SIMON ...
https://www.dpmms.cam.ac.uk/~sjw47/DCapOne.pdf7 Aug 2015: bx,b′x′] := bb′[x,x′] b′(x′ b)x b(x b′)x′. for all b,b′ B and x,x′ L. ... bx,c(b′x′)] = c[bx,b′x′] (1 σ)(bx)(c)b′x′. for all b,b′,c B and x,x′ L. -
LINEAR ALGEBRA SIMON WADSLEY Contents 1. Vector spaces 21.1. ...
https://www.dpmms.cam.ac.uk/~sjw47/LinearAlgebra.pdf20 May 2015: a) for all u1,u2 U, u1 u2 U;(b) for all λ F and u U, λu U;(c) 0 U. ... fn〉. Then B = P1AP. Proof. This is a special case of the change of basis formula for all linear mapsbetween f.d. -
LINEAR ALGEBRA SIMON WADSLEY Contents 1. Vector spaces 21.1. ...
https://www.dpmms.cam.ac.uk/~sjw47/LinearAlgebraM15.pdf2 Dec 2015: x X is linear.(3) For all x [a,b], δx : C([a,b], R) R; f 7 f(x) is linear.(4) I : C([a,b], R) C([a,b], ... Therefore for all v ker α, φ(v) = θα(v) = θ(0) = 0. -
Level-raising and symmetric power functoriality, III Laurent Clozel…
https://www.dpmms.cam.ac.uk/~jat58/lrspiii.pdf10 Dec 2015: We also thank all the refereesfor corrections and criticisms which, we hope, have substantially improved the exposition. ... Proof. All the representations we construct will have non-zero B-invariant vectors. -
Automorphy of some residually dihedral Galois representations Jack A. …
https://www.dpmms.cam.ac.uk/~jat58/version_over_F.pdf3 Apr 2015: Choosean element x R such that ρ(x) = 0. Then x = x, hence yx = yx for all y R. ... then the action of Uv commutes with all of the inclusions (n m 1):. -
Department of Pure Mathematics and Mathematical StatisticsUniversity…
https://www.dpmms.cam.ac.uk/~tkc10/CodesandCryptography/CodesandCryptography.pdf10 Mar 2015: We will need to consider all of these. SENDER. m 7 c(m). ... We thenwish to decode this. Because of the errors introduced by the channel, the word we receive may not bea code word at all and certainly not the code word b
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