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Electron. Commun. Probab. 0 (2016), no. 0, DOI: 10.1214/ECP.vVOL-PID
https://www.dpmms.cam.ac.uk/~ik355/PAPERS/f-norm.pdf5 Jun 2020: iii) There exists a closed, small set C and an extended-valued non-negative function Vsatisfying V (x0) < for some x0 X, such that Condition (V3) holds. ... Theorem 5.3 of [9] implies that (1.3) will holdif f is subject to this additional bound: For some -
Prof. A. Baker Number Fields Examples Michaelmas Term 2002 ...
https://www.dpmms.cam.ac.uk/study/II/NumberFields/ex01.pdf21 May 2005: for some x,y Z with (x,y) = 1, then there exist a,b Z such that p = a2 b2. ... subgroup of Q/Q2 generated by a,b. Determine whether the field Q(. -
Riemann SurfacesExample Sheet 1 Lent 2023 Please send any ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2022-2023/RS%20Sheet%201.pdf6 Mar 2023: By considering the singularity at or otherwise, show that any injective an-alytic map f C C has the form f(z) = az b, for some a C and b ... 9. Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has -
Riemann SurfacesExample Sheet 1 Lent 2023 Please send any ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2021-2022/RS%20Sheet%201.pdf6 Mar 2023: By considering the singularity at or otherwise, show that any injective an-alytic map f C C has the form f(z) = az b, for some a C and b ... 9. Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has -
RIEMANN SURFACES EXAMPLES 1 Michaelmas 2017 Comments on and/or ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2017-2018/es1.pdf4 Oct 2017: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form f(z) = az b, for some a C and b C. ... conformally equivalent). 10. Define an equivalence relation on C by z w iff z = 2sw for some s Z. -
Level raising and symmetric power functoriality, I Laurent Clozel ...
https://www.dpmms.cam.ac.uk/~jat58/lrspi.pdf20 Feb 2013: For every isomorphism ι : Ql = C, the image of the residual representation rι(π′) contains SL2(Flb),up to conjugation, for some b a. • ... This contradiction shows that rι(π′) must in fact have residual imagecontaining SL2(Flb), some b a, and -
Lent Term 2018 O. Randal-Williams IB Groups, Rings, and ...
https://www.dpmms.cam.ac.uk/~or257/teaching/IBGRM/2018/Sheet3.pdf23 Feb 2018: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R. -
COMPLEX ANALYSIS EXAMPLES 1 Lent 2010 J. M. E. ...
https://www.dpmms.cam.ac.uk/~jmeh1/Teaching/complex10-1.pdf1 Feb 2010: COMPLEX ANALYSIS EXAMPLES 1. Lent 2010 J. M. E. Hyland. These questions are a mix of questions from recent versions of the course together with some specificto my own take on ... 16. Let f be an entire function such that for some a C and r > 0, f takes -
sumset.dvi
https://www.dpmms.cam.ac.uk/~par31/preprints/sumset.pdf25 Jul 2017: By the pigeonhole principle, we must haveca = cb for some a and b with 0 6 a < b 6 k. ... First, what if instead of simply requiring that X be infinite, we seek an Xof cardinality 1, say, or of some larger specified cardinality? -
exsh3.dvi
https://www.dpmms.cam.ac.uk/study/II/AutomataAndFormalLanguages/2019-2020/1920autflno3.pdf15 Nov 2019: a) All words w {0, 1} consisting of either the string 01 repeated some number oftimes (possibly none), or the string 010 repeated some number of times (possiblynone). ... b) All words w {a, b, c} consisting of some number of a’s (possibly none), -
Number Fields IID, Lent 2022: Example Sheet 3 of ...
https://www.dpmms.cam.ac.uk/study/II/NumberFields/2021-2022/ex-sheet3.pdf28 Feb 2022: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder. -
Cubical small-cancellation theory and large-dimensional hyperbolic…
https://www.dpmms.cam.ac.uk/~mcr59/thesis.pdf22 Aug 2023: Let us provide some motivation for both of these endeavours. Cubical small-cancellation as a means for producing examples. ... 2. N = G/K for some K < G,. 3. max{cd(G),2} cd(Γ) cd(G)1. -
PART II AUTOMATA AND FORMAL LANGUAGES MICHAELMAS 2016-17 EXAMPLE ...
https://www.dpmms.cam.ac.uk/study/II/AutomataAndFormalLanguages/2016-2017/AFLex3.pdf7 Nov 2016: a) All words w {0, 1} consisting of either the string 01 repeated some number oftimes (possibly none), or the string 010 repeated some number of times (possiblynone). ... b) All words w {a,b,c} consisting of some number of a’s (possibly none), followed -
COMPLEX ANALYSIS EXAMPLES 2 Lent 2011 J. M. E. ...
https://www.dpmms.cam.ac.uk/~jmeh1/Teaching/complex11-2.pdf20 Feb 2011: 5. Let f be an entire function such that for some a C and r > 0, f takes no values in the discD(a, r). ... Show that |f (z)| as |z| ,and deduce that |f (z)| takes a minimum at some a C. -
Handout: The Cauchy-Kovalevskaya Theorem Part III: Analysis of…
https://www.dpmms.cam.ac.uk/~cmw50/resources/PartIIIPDE/PDECauchyKov.pdf15 Oct 2021: Bj,γ,δ. zγxδ, j = 1,. ,n1,. andc(z,x′) =. γ,δ. cγ,δzγxδ,. where these power series converge if |z| |x′| < s for some small s > 0. ... cγ,δzγxδ,. 4. with these power series converging for |x| |x′| < s for some (possibly smaller) s. -
MAT 449: Representation theory These lecture notes are in ...
https://www.dpmms.cam.ac.uk/~jcsl5/notes.pdf24 Oct 2023: We say X is a G-set. Some highlights of group actions include the following:. • ... The proof will be postponed to next lecture. We first look at some staggering consequences. -
RN (nickl@maths.cam.ac.uk) Michaelmas 2022 Probability and Measure 1…
https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2022-2023/ex1.pdf5 Oct 2022: 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 ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0. -
IB Statistics Michael TehranchiExample sheet 1 of 3 Lent ...
https://www.dpmms.cam.ac.uk/study/IB/Statistics/2018-2019/Example1.pdf28 Feb 2019: b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T, and. ... 4. For some n 2 let X1,. ,Xniid U[θ, 2θ], for some θ > 0. -
MATHEMATICAL TRIPOS Part III Before Thursday 11th June, 2007 ...
https://www.dpmms.cam.ac.uk/~tkc10/complex_2007/MockExam2007.pdf3 Jun 2007: The questions carry equal weight. This is a Mock examination, intended to give you some idea of the sortof questions you will face in the proper Part III examination. ... The function f : DR C is analytic on some disc DR = {z C : |z| < R} withR > 1 and -
Mich. 2021 ANALYSIS AND TOPOLOGY – EXAMPLES 1 AZ ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2021-2022/sheet1.pdf7 Oct 2021: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded. -
Lent Term 2017 O. Randal-Williams IB Groups, Rings, and ...
https://www.dpmms.cam.ac.uk/~or257/teaching/IBGRM/2017/Sheet3.pdf19 Jan 2017: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R. -
Mich. 2021 ANALYSIS AND TOPOLOGY – EXAMPLES 2 AZ ...
https://www.dpmms.cam.ac.uk/~az10000/2021-mich-partib-aandt-sheet2.pdf26 Oct 2021: Some more questions. 13. (a) A topological space is said to be second countable if there is a countable family Bof open sets such that every open set is a union ... of some members of B. -
RIEMANN SURFACES AND DISCRETE GROUPS TKC Lent 2007 1. ...
https://www.dpmms.cam.ac.uk/~tkc10/complex_2007/Exercise_2007_1.pdf3 Jun 2007: u(z) = b log |z| Re a(z)for some b R and some analytic function a : A C. ... a) Show that Z(T) is a subgroup of Möb.(b) Find which groups (up to isomorphism) can arise as Z(T) for some Möbius transformation T. -
ANALYSIS II (Michaelmas 2011): EXAMPLES 3 The questions are ...
https://www.dpmms.cam.ac.uk/~ajs1005/analysis_II/anII_ex_2011_s3.pdf13 Nov 2011: Explain geometrically why this is thecase. 5. (i) Suppose that f : R2 R is such that f/x is continuous in some open ball around (a,b),and f/y exists ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, -
Michaelmas Term 2003 J. M. E. Hyland Linear Algebra: ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2ndla03.pdf21 May 2005: vary in difficulty but cover some instructive points. 1. For what values of a and b does the system of simultaneous linear equations. ... only if it is nilpotent, that is, αk = 0 for some natural number k. -
Riemann Surfaces, Examples 2 Michaelmas 2020 Please send any ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2020-2021/RS%20Sheet%202.pdf5 Oct 2020: 4. If the map f of Question 3 is a conformal equivalence, show that F(z) = λz µfor some λ C. ... 5. Show that complex tori C/⟨1,τ1⟩ and C/⟨1,τ2⟩ are analytically isomorphic ifand only if τ2 = (aτ1 b)/(cτ1 d), for some matrix ( a bc d ) SL2 -
The Eisenstein quotient
https://www.dpmms.cam.ac.uk/~jcsl5/mazur/8.TheEisensteinquotient.pdf18 Jan 2021: Proof. Let f : J0(N) A be the quotient map. There exists a mapg : A J0(N) such that fg = [n] for some n > 0. ... and for a given f, it’s possible that some Vf,p are of the form (trivial) (cyclotomic) while others aren’t. -
Number Fields IID, Lent 2021: Example Sheet 3 of ...
https://www.dpmms.cam.ac.uk/study/II/NumberFields/2020-2021/ex-sheet3.pdf15 Jan 2021: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder. -
STATISTICS Part IB Example Sheet 1 (of 3) RJS/Lent ...
https://www.dpmms.cam.ac.uk/study/IB/Statistics/2017-2018/Example1.pdf12 Feb 2018: Xn be independent Poisson random variables, with Xi having parameteriθ for some θ > 0. ... b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T, and. -
Compressions and Probably IntersectingFamilies Paul A. Russell ∗†…
https://www.dpmms.cam.ac.uk/~par31/preprints/probint.pdf16 Aug 2011: larger family Aand then randomly discard some of the sets in A to form a subfamily B. ... Then A = vf (W )Xfor some W U , X X W2 YW and W X B. -
Proofs for some results inTopics in Analysis T. W. ...
https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2023-2024/Caesar.pdf25 Oct 2023: Proofs for some results inTopics in Analysis. T. W. Körner. October 25, 2023. ... Now P(z) = 0 if and only if z = ajfor some 1 j m. -
RN (nickl@maths.cam.ac.uk) Michaelmas 2021 Probability and Measure 1…
https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2021-2022/ex1.pdf7 Oct 2021: 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 ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0. -
Transitive Sets and Cyclic Quadrilaterals Imre Leader∗† Paul A. ...
https://www.dpmms.cam.ac.uk/~par31/preprints/cycquad.pdf25 Dec 2010: Then the collection ofpairs (α, β) that can occur in quadrilaterals of the form A(w), B(w), C(w), wfor some w 6= 0 is exactly the zero set of ... It follows that the parameters (α, β) occur for some some (non-trivial)irreducible representation of G′ -
ANALYSIS I EXAMPLES 2 G.P. Paternain Lent 2010 Comments ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2009-2010/aI2.pdf2 Feb 2010: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b. -
Bipartite graphs of approximate rank 1. W. T. Gowers ...
https://www.dpmms.cam.ac.uk/~wtg10/approxrankone3.pdf19 May 2007: The next lemma gives us some information under weaker hypotheses than those of. ... with some probability p, whereas a graph with the weaker intersections property looks like. -
Analysis I: Example Sheet 1 AFB, Michaelmas 2004 Please ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2004ex1-4.pdf21 May 2005: Is ittrue that for some c in [a,b] we have. 1. ... i) Suppose that λ > 0. Show that there is some interval [c,d] in [a,b] with the property that. -
Analysis of Partial Differential Equations Example sheet II (Chapter…
https://www.dpmms.cam.ac.uk/~md384/example_sheet_PDE_2-v3.pdf11 Nov 2016: u(0,x) =1. 1 x2. 1. 5. (Cauchy-Kovalevskaya Theorem for systems of ODEs) Suppose b > 0 and F : u0 (b,b)d Rd bereal analytic in a neighbourhood of ... u′(t) = F(t,u(t)), u(0) = u0 Rd,. on (a,a) for some a > 0 with u((a,a)) u0 (b,b)d. -
MATHEMATICAL TRIPOS: PART IA Lent 2017 PROBABILITY JRNExample Sheet…
https://www.dpmms.cam.ac.uk/study/IA/Probability/2016-2017/pex1.pdf14 Feb 2017: 6. Let (An : n N) be a sequence of events in some probability space (,F,P). ... Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}. -
Hypergraph Regularity and the multidimensional Szemerédi Theorem. W. …
https://www.dpmms.cam.ac.uk/~wtg10/hypersimple4.pdf12 Apr 2005: intersection of A with (x,y)A has expected size cδ2N2 for some absolute constant c > 0,lives inside the grid [N,N]2, and has the property that B = ... It is straightforwardto show that B has density at least η > 0 for some η that depends on δ only. -
Modular Forms of Weight one Jef Laga Contents 1. ...
https://www.dpmms.cam.ac.uk/~jcsl5/partIIIessay.pdf15 Feb 2021: f(z) =n0. ane2πinz,. where an = O(nα) for some constant α > 0. ... ξ((a)) = sgn(aτ ) (a 1 mod m). for some embedding τ : K R. -
Mich. 2020 ANALYSIS AND TOPOLOGY – EXAMPLES 1 AZ ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2020-2021/sheet1.pdf19 Oct 2020: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded. -
Part IB - Groups, Rings, and Modules
https://www.dpmms.cam.ac.uk/~or257/teaching/notes/GRM.pdf31 Jan 2024: If g2H = g′2H, then we know g. ′2 = g2 h for some h H. ... So we have to show p - ||. This is just some basic counting: we have. || -
Riemann Surfaces, Examples 1 Michaelmas 2020 Please send any ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2020-2021/RS%20Sheet%201.pdf5 Oct 2020: az b, for some a C and b C. ... conformally equivalent). 9. Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has the natural structure of -
MATHEMATICAL TRIPOS, PART II, 2020/2021REPRESENTATION THEORY EXAMPLE…
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2020-2021/IIRT1.pdf14 Oct 2020: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible. -
Homotopy Theory, Examples 3 Oscar Randal-Williams Lent 2015 1. ...
https://www.dpmms.cam.ac.uk/~or257/teaching/IIIHtyThy2015/Sheet3.pdf11 Feb 2015: Fi //. Ep //. p. B. {b0} // B B. and show using the naturality of the Serre spectral sequence that the compositions. ... for some monic polynomial p(x) with coefficients in H(B; Z). Homotopy Theory Example sheet 3. -
Number Fields IID, Lent 2020: Example Sheet 3 of ...
https://www.dpmms.cam.ac.uk/study/II/NumberFields/2019-2020/ex-sheet3.pdf9 Mar 2020: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder. -
Relative Entropy and Exponential Deviation Boundsfor General Markov…
https://www.dpmms.cam.ac.uk/~ik355/PAPERS/KLM-C.pdf5 Jun 2020: Such asymptotic results may be of limited use if we wantto have some sort of guarantee that, after a certain number of. ... the sense that|F (x)| CV (x) for all x A, for some constantC). -
RN Michaelmas 2020 Probability and Measure 1 1.1. Let ...
https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2020-2021/ex1.pdf10 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 ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0. -
Adaptive and non-linear MCMC algorithms
https://www.dpmms.cam.ac.uk/~ik355/AWMCMC/AWMCMC_talks/EM.pdf6 Jun 2020: Two examples. Adaptive SRWM. Adaptive SRWM. I All MCMC algorithms depend on some design parameters. ... PV (x) λV (x) b1C(x) V (x) =(π(x). supX π. )τ(1β)for some λ (0, 1), b < , a set C, τ (0, 1]. -
Lent 2016 ANALYSIS I – EXAMPLES 2 AZ 1. ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2015-2016/sheet2.pdf9 Feb 2016: 5. Let f be a function on a set A and let B A. ... some c (a,b) the vectors(f(b),g(b). )(f(a),g(a). )and. (f ′(c),g′(c). )in R2.
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