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Example Sheet 1 Analysis of FunctionsClaude Warnick Lent 2021 ...
https://www.dpmms.cam.ac.uk/study/II/AnalysisofFunctions/2020-2021/Examples%201.pdf3 Feb 2021: 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. 3. b) () Suppose {Bj : j 2 J} is an arbitrary collection of balls in Rn such that each -
BT08 Part II Representation Theory Sheet 4 Unless otherwise ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2007-2008/repex4.pdf28 Feb 2008: In this and the following questions, youare not being asked to find such an isomorphism explicitly.)Decompose V 23 , Λ. ... b) Explicitly construct such a representation ρ : SU (2) SO(3) by showing that SU (2)acts on the vector space of matrices of the -
Example sheet 4, Galois Theory, 2007. 1. (i) Let ...
https://www.dpmms.cam.ac.uk/study/II/Galois/2007-2008/ex4.pdf4 Feb 2008: Let a, b K such that thepolynomials f = xm a, g = xm b are irreducible. ... Show that G is solvable if and only if there is an i such that. -
Lent Term 2005 C.J.B. Brookes IB Groups, Rings and ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2004-2005/bex3.pdf21 May 2005: 2. Let R be an integral domain. The highest common factor of non-zero elements a and b in R is an elementd in R such that d divides both a ... be identified with the set Fq in such a way that GL2(Fq) acts on as the group ofMöbius transformations z 7 azb -
PART II REPRESENTATION THEORYSHEET 4 Unless otherwise stated, all ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2013-2014/repex4.pdf15 Jan 2014: In this and the followingquestions, you are not being asked to find such an isomorphism explicitly.). ... Here O(n) denotesthe subgroup of GLn(R) preserving the standard (positive definite) symmetric bilinear form.(b) Explicitly construct such a -
exsh2.dvi
https://www.dpmms.cam.ac.uk/study/II/AutomataAndFormalLanguages/2021-2022/2122autflno2.pdf27 Oct 2021: 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. -
ALGEBRAIC GEOMETRY, SHEET III: LENT 2020 Throughout this sheet, ...
https://www.dpmms.cam.ac.uk/study/II/AlgebraicGeometry/2019-2020/AGExample3.pdf3 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] -
Example Sheet 3, Geometry 2005 pmhw@dpmms.cam.ac.uk (1) Show the ...
https://www.dpmms.cam.ac.uk/study/IB/Geometry/2004-2005/Geom05.3.pdf21 May 2005: Prove that there exists such a reparametrization Γ̃ = Γ ffor which ‖dΓ̃/dt‖ is constant, namely 1/l. ... 10) For a surface of revolution S, corresponding to a curve η : (a,b) R3 given byη(u) = (f(u),0,g(u)), where η is parametrized in such a -
Michaelmas Term 2020 Linear Algebra: Example Sheet 2 of ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2020-2021/example-sheet-2.pdf19 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. -
Linear Analysis Example Sheet 5 1. Define normed vector ...
https://www.dpmms.cam.ac.uk/study/II/LinearAnalysis/2004-2005/examplesheet5.pdf21 May 2005: Thenthere is a non-empty subinterval [a,b], with a < b, such that {fi} is uniformly bounded. ... λn, such that. T(v) =. i. λie(vi,v)vi,. where {λi} = σ(T). -
Michaelmas Term 2015 T.A. Fisher Number Theory: Example Sheet ...
https://www.dpmms.cam.ac.uk/study/II/NumberTheory/2015-2016/number_theory-15-1.pdf15 Oct 2015: 1. Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d. -
STATISTICAL MODELLING Part IIC / Michaelmas 2019Example Sheet 1 ...
https://www.dpmms.cam.ac.uk/study/II/StatisticalModelling/2019-2020/Qu1.pdf11 Oct 2019: b) The model is to be reparametrised in such a way that it can be specialised to (i) two parallellines for the two groups, (ii) two lines with the same intercept, ... Z21 Z2np,where the Zi are independent and Zi N(λi,σ2) for some λi such that. -
Example Sheet 3 Analysis of FunctionsClaude Warnick Lent 2021 ...
https://www.dpmms.cam.ac.uk/study/II/AnalysisofFunctions/2020-2021/Examples%203.pdf24 Feb 2021: there exists 2 C such that:. u[] =. Z. R(x)dx, for all 2 D(R). () ... Rm(x, y)dy. b) Show that there exists a sequence of smooth functions fn 2 C1c (Rn) such that Tfn! -
Example Sheet 3 Analysis of FunctionsClaude Warnick Lent 2020 ...
https://www.dpmms.cam.ac.uk/study/II/AnalysisofFunctions/2019-2020/ExamplesAoF3.pdf4 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. -
TOPICS IN ANALYSIS (Lent 2020): Example Sheet 3 Comments, ...
https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2019-2020/topics-sheet3.pdf26 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. -
EXAMPLE SHEET 3 (LECTURES 13–18) GALOIS THEORY MICHAELMAS 2009 ...
https://www.dpmms.cam.ac.uk/study/II/Galois/2009-2010/ex3.pdf28 Mar 2010: 4. Let F/K be a finite extension. Show that there is a unique intermediate fieldK L F such that L/K is separable and F/L is purely inseparable. ... Let a,b K such that the polynomials P(X) = Xn a and Q(X) = Xn b are irreducible.Show that P and Q have the -
Representation Theory Sheet 4 4.1 Question Show that any ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2005-2006/sheet4.pdf13 Mar 2006: a) LetG be a compact group. Show that there is a group homomorphismρ : G On(R) if and only if G has ann dimensional representation overR.(b) Explicitly construct such a ... subset ofH given by matrices such thatb = 0, and letφ : Fp C be a non-trivial -
Lent Term 2021 T.A. Fisher Groups Rings and Modules: ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2020-2021/grm-21-3.pdf26 Feb 2021: 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. -
sheet4.dvi
https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2007-2008/sheet4.pdf10 Mar 2008: Prove that the set S of points x [0, 1] such that x Aj for infinitely many j is dense.Must S be open? ... b) If f : (0, 1) R is continuous and bounded, prove that there exists a sequence of polynomialsQn such that Qn are uniformly bounded and Qn f -
Logic and Set Theory Examples 1 PTJ Lent 2016 ...
https://www.dpmms.cam.ac.uk/study/II/Logic/2015-2016/LSqns1.pdf8 Feb 2016: p], and deducethat if f,g : P P are order-preserving maps such that f(x) g(x) for all x P , then µ(f) µ(g).[Consider the restriction of ... The largest such c is usually denoted ab; this defines a binary operationonL, which is called Heyting implication -
Topics in Analysis: Example Sheet 2 Michaelmas 2009-10 N. ...
https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2009-2010/09-10sheet2.pdf30 Oct 2009: n. LetβJn (x) = (xx0)(xx1)(xx2). (xxn). Prove that for each x [1, 1], there exists ζ (1, 1)such that. ... Thus, for any fixed δ (0, 1/2), In,δ/In 0 asn. (b) Choose numbers a1, b1 such that 0 < a1 < a < b < b1 < 1, and set, for x R -
Logic and Set Theory Examples 1 PTJ Lent 2012 ...
https://www.dpmms.cam.ac.uk/study/II/Logic/2011-2012/LSqns1.pdf31 Jan 2012: ii) Find an example of a finite lattice L which admits at least one homomorphism L 2,but does not have enough such homomorphisms to separate its elements. ... The largest such c is usually denoted a b; this defines a binaryoperation on L, which is called -
Michaelmas Term 2020 Linear Algebra: Example Sheet 4 of ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2020-2021/example-sheet-4.pdf9 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. -
LSqns1.dvi
https://www.dpmms.cam.ac.uk/study/II/Logic/2012-2013/LSqns1.pdf30 Oct 2012: ii) Find an example of a finite lattice L which admits at least one homomorphism L 2,but does not have enough such homomorphisms to separate its elements. ... b. (The largest such c is usually denoted a b; this defines a binaryoperation on L, which is -
exsh4.dvi
https://www.dpmms.cam.ac.uk/study/II/AutomataAndFormalLanguages/2019-2020/1920autflno4.pdf27 Nov 2019: 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. -
JPM Michaelmas 2016 Probability and Measure 1 1.1. Let ...
https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2016-2017/ex1.pdf27 Oct 2016: 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. -
E. Breuillard Michaelmas 2019 Probability and Measure 3 Exercises ...
https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2019-2020/ex3nosol.pdf29 Nov 2019: Use ψ to build, foreach interval I = [a,b] and > 0 a C-function ψI, on R such that. ... 1[a,b] ψI, 1[a,b+]. Use this to construct for each compact set K Rd and each open set U K a C-function ψK,Uon Rd such that. -
Logic and Set Theory Examples 2 PTJ Lent 2016 ...
https://www.dpmms.cam.ac.uk/study/II/Logic/2015-2016/LSqns2.pdf8 Feb 2016: iv) For each of the following classes of structures, either write down a universal axiomatization ofthe class or show using (ii) that no such axiomatization exists. ... For those which are not universal, eitherwrite down an inductive axiomatization or -
Michaelmas Term 2014 T.A. Fisher Number Theory: Example Sheet ...
https://www.dpmms.cam.ac.uk/study/II/NumberTheory/2014-2015/number_theory-14-1.pdf17 Oct 2014: 1. Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d. -
10-11sheet2.dvi
https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2010-2011/10-11sheet2.pdf3 Nov 2010: Recall that the existence of such a mini-mizer was proved in lecture.). ... b) Choose numbers a1, b1 such that 0 < a1 < a < b < b1 < 1, and set, for x R and n = 1, 2, 3,. ,. -
Example Sheet 3, Geometry 2006 pmhw@dpmms.cam.ac.uk (1) Show the ...
https://www.dpmms.cam.ac.uk/study/IB/Geometry/2005-2006/Geom06.3.pdf7 Feb 2006: Prove that there exists such a reparametrization Γ̃ = Γffor which ‖dΓ̃/dt‖ is constant, namely l. ... 10) For a surface of revolution S, corresponding to a curve η : (a,b) R3 given byη(u) = (f(u),0,g(u)), where η is parametrized in such a way -
Topics in Analysis: Example Sheet 2 Michaelmas 2011-12 N. ...
https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2011-2012/11-12sheet2.pdf5 Nov 2011: Recall that the existence of such a mini-mizer was proved in lecture.). ... Thus, for any fixed δ (0, 1/2), In,δ/In 0 asn. (b) Choose numbers a1, b1 such that 0 < a1 < a < b < b1 < 1, and set, for x R -
Topics in Analysis: Example Sheet 3 Lent 2007-08 N. ...
https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2007-2008/sheet3.pdf28 Feb 2008: n}, let the non-negative numbers A(n)k and the“nodes” x(n)k [a, b] be given such that for each polynomial P , the error. ... such that for every f C([0, 1]), there exists a unique series. -
Michaelmas Term 2019 Linear Algebra: Example Sheet 2 of ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2019-2020/lin_alg-19-2.pdf31 Oct 2019: 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. -
LSqns4.dvi
https://www.dpmms.cam.ac.uk/study/II/Logic/2011-2012/LSqns4.pdf29 Feb 2012: 8. By a selection function for a set a, we mean a function s : Pa Pa such that s(b) bfor all b a, both inclusions being strict unless b is ... First show that there exists β < α such that|f (β) f (γ)| < 1 for all γ > β.]. -
Lent Term 2011 R. Camina IB Groups, Rings and ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2010-2011/ex11-3.pdf24 Feb 2011: 9. Let R be an integral domain. The highest common factor (hcf) of non-zero elements a and b in R is anelement d in R such that d divides both ... ii) Show that can be identified with the set Fq {} in such a way that GL2(Fq) acts on as thegroup of -
Example Sheet 1 Analysis of FunctionsClaude Warnick Lent 2020 ...
https://www.dpmms.cam.ac.uk/study/II/AnalysisofFunctions/2019-2020/ExamplesAoF1.pdf12 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 -
Lent Term 2010 R. Camina IB Groups, Rings and ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2009-2010/ex10-3.pdf23 Feb 2010: 6. Let R be an integral domain. The highest common factor of non-zero elements a and b in R is an elementd in R such that d divides both a ... identified with the set Fq {} in such a way that GL2(Fq) acts on asthe group of Möbius transformations z 7 azb -
LSqns4.dvi
https://www.dpmms.cam.ac.uk/study/II/Logic/2012-2013/LSqns4.pdf30 Oct 2012: 8. By a selection function for a set a, we mean a function s : Pa Pa such that s(b) bfor all b a, both inclusions being strict unless b is ... First show that there exists β < α such that|f (β) f (γ)| < 1 for all γ > β.]. -
Example Sheet 3, Geometry 2007 pmhw@dpmms.cam.ac.uk (1) Show the ...
https://www.dpmms.cam.ac.uk/study/IB/Geometry/2006-2007/Geom3.07.pdf2 Mar 2007: Prove that there exists such a reparametrization Γ̃ = Γffor which ‖dΓ̃/dt‖ is constant, namely l. ... 9) For a surface of revolution S, corresponding to a curve η : (a,b) R3 given byη(u) = (f(u),0,g(u)), where η is parametrized in such a way -
Lent Term 2020 T.A. Fisher Groups Rings and Modules: ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2019-2020/grm-20-3.pdf24 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. -
PART II REPRESENTATION THEORYSHEET 4 Unless otherwise stated, all ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2017-2018/IIRT4.pdf18 Jan 2018: to V4 V3; in this and the followingquestions, you are not being asked to find such an isomorphism explicitly). ... Here O(n) denotesthe subgroup of GLn(R) preserving the standard (positive definite) symmetric bilinear form.(b) Explicitly construct such a -
TOPICS IN ANALYSIS (Lent 2014): Example Sheet 3. Comments, ...
https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2013-2014/sheet3.pdf10 Mar 2014: 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. -
PART II AUTOMATA AND FORMAL LANGUAGES MICHAELMAS 2016-17 EXAMPLE ...
https://www.dpmms.cam.ac.uk/study/II/AutomataAndFormalLanguages/2016-2017/AFLex4.pdf28 Nov 2016: 2) Convert the following CFG to CNF:. S aSbb | T, T bTaa | S |. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF:. ... b) Show that the language L := {anbncn | n 1} is not a CFL. -
Part II Galois theory (2014–2015) Example Sheet…
https://www.dpmms.cam.ac.uk/study/II/Galois/2014-2015/galois-2014-exa-4.pdf2 Dec 2014: If ϕ AutK(L) where L = K(t), show that there exist a, b, c, d K withad 6= bc such that ϕ(t) = (at b)/(ct d), and conversely that ... a) such that f splits over L = Q(. a)( 3α). -
IB GROUPS, RINGS AND MODULES Lent Term 2023Example Sheet ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2022-2023/Example%20sheet%203.pdf24 Jan 2023: 6) Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements aand b in R is an element d in R such that d divides both ... a and b, and if c divides both aand b then c divides d. -
exsh2.dvi
https://www.dpmms.cam.ac.uk/study/II/AutomataAndFormalLanguages/2019-2020/1920autflno2.pdf1 Nov 2019: 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 ... Construct two explicit codes m, m′ such that Tm is recursive, and Tm′ is not r.e.Hint: Use the results of -
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 ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε. -
Michaelmas Term 2018 Linear Algebra: Example Sheet 2 of ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2018-2019/lin_alg-18-2.pdf24 Oct 2018: 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. -
IB GEOMETRY LENT 2012 EXAMPLE SHEET 1 1. Suppose ...
https://www.dpmms.cam.ac.uk/study/IB/Geometry/2011-2012/Geometry1.pdf8 Feb 2012: 14. Exhibit a subset X of R2 such that (a) any two points x, y X can be joined bya continuous path γ : [0, 1] X and (b) for x 6= ... A regular polyhedron is one in which all two-dimensional faces are congruent regularpolygons, and such that for each
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