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  2. Tony Scholl's home page

    https://www.dpmms.cam.ac.uk/~ajs1005/
    11 Feb 2023: In: The Arithmetic and Geometry of Algebraic Cycles ed. B. B.
  3. Commutativity of Multiplication

    https://www.dpmms.cam.ac.uk/~wtg10/commutative.html
    25 Nov 2002: But the map (a,b)-->(b,a) is easily seen to be a bijection between AxB and BxA, from which it follows that BxA has cardinality mn. ... If m=1 then the result is obvious, since the map (1,x)-->x is a bijection between AxB and B, B has cardinality n, and
  4. Mich. 2012 ANALYSIS II—EXAMPLES 4 PAR 1. Use the ...

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2012-2013/12sheet4.pdf
    26 Nov 2012: D2 det |I as a bilinear map. 9. Let f : R2 R and (a, b) R2. ... D2f exists at (a, b). Show that f is differentiable at (a, b).
  5. Suggested accommodation within walking distance to CMS Arundel House…

    https://www.dpmms.cam.ac.uk/~jat58/mordell/accommodation.pdf
    16 Jun 2022: Tel: 01223 367701. Website https://www.arundelhousehotels.co.uk/. Email: info@arundelhousehotel.co.uk. Benson House (B&B). 24 Huntingdon Road, Cambridge.
  6. DIFFERENTIAL GEOMETRY (PART III) MICHAELMAS 2021 EXAMPLE SHEET 1 ...

    https://www.dpmms.cam.ac.uk/files/study/III/DifferentialGeometry/2021-2022/DGSheet1.pdf
    13 Oct 2021: A and B are smoothly equivalent then they have the same smooth functions, in. ... b) Using this diffeomorphism to identify RP1 with S1, show that M is isomorphic to the tauto-.
  7. Mirror symmetry and the SYZ conjecture

    https://www.dpmms.cam.ac.uk/~mg475/current.pdf
    16 Jul 2014: symplectic structure as before. Torus bundle X (B) B Torus bundle X̌ (B) B. ... Kähler form 2īK. complex coordinates x̌i K/yi. Torus bundle X (B) B Torus bundle X̌ (B) B.
  8. Lent Term 2024 O. Randal-Williams II Algebraic Topology // ...

    https://www.dpmms.cam.ac.uk/study/II/AlgebraicTopology/2023-2024/Sheet4.pdf
    16 Jan 2024: Show that the linear map A : R2 R2 peserves theequivalence relation (a,b) (a′,b′) (aa′,bb′) Z2, and so induces a continuous map fAfrom the torus T to itself.
  9. Solving cubic equations

    https://www.dpmms.cam.ac.uk/~wtg10/cubic.html
    25 Apr 2002: Next, we perhaps note that if b=a. 2. then we have the equation x. ... a-3t)y. 2. (b-2at3t. 2. y) c-btat. 2. t. 3. This gives us a'=a-3t and b'=b-2at3t.
  10. Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 4 PAR 1. Let ...

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2023-2024/4.pdf
    24 Nov 2023: b) Define an equivalence relation on X Y by (x,y) (x′,y′) iff x = x′. ... D2f exists at (a,b). Show that f is differentiable at (a,b).
  11. SS (ss2871@maths.cam.ac.uk) Michaelmas 2023 Probability and Measure 1 …

    https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2023-2024/ex1.pdf
    11 Oct 2023: 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 ... 1.8. Let B be a Borel subset of the interval [0, 1].
  12. Michaelmas Term 2023-24 Numbers and Sets: Examples Sheet 4 ...

    https://www.dpmms.cam.ac.uk/study/IA/Numbers%2BSets/2023-2024/numset4_2023.pdf
    16 Nov 2023: 3. The symmetric difference of two sets A and B is A4B = (AB) (B A). ... 8. Define a relation R on N by setting aRb if a divides b or b divides a.
  13. Mathematical Tripos: Part IB DJS/Lent 2015 Statistics: Example Sheet…

    https://www.dpmms.cam.ac.uk/study/IB/Statistics/2014-2015/ex1.pdf
    22 Jan 2015: likelihood estimate of θ is (2a b)/(b 2c). 4. (a) Let X1,. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).
  14. ANALYSIS II EXAMPLES 2 G.P. Paternain Mich. 2003 The ...

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/es2-03.pdf
    21 May 2005: 6. If A and B are subsets of Rn, we write A B for the set {a b : a A,b B}. ... Show that if Aand B are both closed and one of them is bounded, then A B is closed.
  15. ANALYSIS II EXAMPLES 2 G.P. Paternain Mich. 2002 The ...

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/es2-02.pdf
    21 May 2005: 6. If A and B are subsets of Rn, we write A B for the set {a b : a A, b B}. ... Show that if Aand B are both closed and one of them is bounded, then A B is closed.
  16. Mich. 2007 ANALYSIS II—EXAMPLES 4 PAR 1. At which ...

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2007-2008/sheet4.pdf
    21 Nov 2007: Show that f = gR f gLand hence, or otherwise, show that f is differentiable at B. ... Show that f is differentiable at (a, b). (b) Suppose instead that D1f exists and is bounded on some open ball around (a, b), and that for fixed x.
  17. ANALYSIS II EXAMPLES 2 G.P. Paternain Mich. 2003 The ...

    https://www.dpmms.cam.ac.uk/~gpp24/es2-03.pdf
    29 Sep 2003: 6. If A and B are subsets of Rn, we write A B for the set {a b : a A,b B}. ... Show that if Aand B are both closed and one of them is bounded, then A B is closed.
  18. TECHNIQUES IN COMBINATORICS: EXAMPLES SHEET 1 Michaelmas Term 2014 ...

    https://www.dpmms.cam.ac.uk/study/III/2014-15/TechniquesinCombinatorics/techniques2014examples1.pdf
    20 Oct 2014: 14. Let A and B be subsets of Zp of size αp and βp respectively. ... Prove that there existu,v Zp such that. |A (uB v)| αβp αβp, where uB v denotes the set{ub v : b B}.
  19. Hypergraph Regularity and the multidimensional Szemerédi Theorem. W. …

    https://www.dpmms.cam.ac.uk/~wtg10/hypersimple4.pdf
    12 Apr 2005: If B B, then fB(ω) is definedto be the value of f(φ) for any φ with φ(A) = ω(B). ... 3.1 as follows. Let K1 be the chain of all subsets of all sets B B.
  20. Relaxing stratification Thomas ForsterDPMMS, CMS Wilberforce…

    https://www.dpmms.cam.ac.uk/~tef10/relaxing.pdf
    29 Mar 2011: Our pointof departure is formula 6 which says there is an external isomorphism between〈A,A〉 and 〈B,B〉.(. ... We have now established that if there is an external isomorphism between〈A,A〉 and 〈B,B〉 then there is precisely one.
  21. Example sheet 1 Example sheet 1 Problem 1. Show ...

    https://www.dpmms.cam.ac.uk/study/IB/Optimization/2022-2023/example1_easter23.pdf
    17 Jul 2023: a) minimize cT x subject to Ax = b. (b) minimize cT x subject to Ax b. ... d) minimize cT x subject to Ax b, x 0. In each case.
  22. Dear Mr. Körner, Sehr geehrter Herr Michael Basler, I ...

    https://www.dpmms.cam.ac.uk/~twk10/German.html
    20 May 2002: 15. p. 243, lines 6 and 9: replace "(., f((r - 1)b/N /- |B_r| (b/N)2))" by "(., f((r - 1)b/N) /- |B_r| (b/N)2)". ... the brackets are misplaced). 16. p. 247, line 6: delete "<= B b3 N(-2)" (I am not quite sure here.).
  23. 18 Jan 2005: Lïiì Õ Þòð;ñ;à Õ Ð5 Ðîbª5àÏTÐbÑÑEÑEÐàÒ Â5ËE»b«»b¿ª¥bÔwb¿ ¥ÂµwEÁmY«=ª¥«'á;Sû!YE»lª=5bi=T¥«=µ¥ ðSÔw}Á'TþwEÁÁEíYE=¥i¥w bªEÁmª¥@¥Âµ(ðÔs«=ÓªEüÖwE¿ª@ªÁm¥w¥Âµ¥w¥«=
  24. E. Breuillard Michaelmas 2017 Probability and Measure 1 1.1. ...

    https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2017-2018/ex1bis.pdf
    12 Oct 2017: 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 ... 1.8. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.
  25. Optimisation Michael TehranchiExample sheet 1 - Easter 2017 1. ...

    https://www.dpmms.cam.ac.uk/study/IB/Optimization/2016-2017/example1.pdf
    1 May 2017: 5. Consider the following problems:. (a) minimise c>x subject to Ax = b;. ... Let P() be the linear programming problem when the right-hand side b = (2, 5, 6)> is.
  26. TOPICS IN SET THEORY: Example Sheet 1 1 Department ...

    https://www.dpmms.cam.ac.uk/study/III/2014-15/TopicsinSetTheory/TST%20Example%20Sheet%201%202014.pdf
    19 Dec 2014: Let a = min{| A |: A is an infinite maximalalmost disjoint family of subsets of ω}, b = min{| B |: B is unbounded in (ωω,)},and d = min{| D |: D is dominating in ( ... i) Prove that 1 min{a, b, d} max{a, b, d} 20.
  27. A note on trilinear forms for reducible representations and ...

    https://www.dpmms.cam.ac.uk/~ajs1005/preprints/trilinear.pdf
    29 Jan 2010: 1/2 it is the subspace spanned by the functionφ: bk 7 δ(b) (for b B, k K), and the linear form in (1.2) can be nor-malised so ... hence b = c since (,) is symmetric. Likewise b = a = c hence a = b =c = 0.
  28. Topics in Analysis, Examples Sheet 1, 2013 1. For ...

    https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2012-2013/examples1.pdf
    15 Feb 2013: 2. Suppose that A and B are disjoint, closed, bounded subsets of R2. ... Show that. there is a δ > 0 so that |a b| > δ for all a A and b B.
  29. Mathematical Tripos: Part IB DJS/Lent 2016 Statistics: Example Sheet…

    https://www.dpmms.cam.ac.uk/study/IB/Statistics/2015-2016/ex-S1B-16-1.pdf
    20 Jan 2017: likelihood estimate of θ is (2a b)/(b 2c). 4. (a) Let X1,. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).
  30. Part IID DIFFERENTIAL GEOMETRY (Lent 2011): Example Sheet 2 ...

    https://www.dpmms.cam.ac.uk/study/II/DifferentialGeometry/2010-2011/dg2e2.pdf
    21 Feb 2011: Show that if there exists a vector e such that e,n(s),b(s) are linearly dependent forall s I, then α is a plane curve. ... ϕ(s,v) = α(s) r(n(s) cos v b(s) sin v. ),. where n = n(s) and b = b(s) are the normal and binormal vectors of α.
  31. Hypersurfaces and the Weil conjectures A J Scholl Abstract ...

    https://www.dpmms.cam.ac.uk/~ajs1005/preprints/weil-preprint1.pdf
    29 May 2010: γ = γa,b,n : nCa,b nCa,b1. the alternating sum of the Gysin maps. ... Assume:. (a) RH(d 1) holds;(b) YK and ZK satisfy the weight-monodromy conjecture.
  32. Part III Commutative Algebra Example Sheet II, 2021 Note: ...

    https://www.dpmms.cam.ac.uk/study/III/CommutativeAlgebra/2021-2022/HW2.pdf
    29 Oct 2021: finitely generated A-module, then Supp(B A M) =. (f)1(Supp(M)). [Note: You may find useful the identity on tensor products (M A B) B C = M A C ... given an A-module M,an A-algebra B and a B-algebra C.].
  33. snmeiwseis-ga.dvi

    https://www.dpmms.cam.ac.uk/~md384/snmeiwseis-ga.pdf
    5 Apr 2007: If V is a normed vector space, then B is bounded iff B B(t) forsome t > 0, where B(t) denotes the open unit ball around the origin of radius ... Tα(B(ǫ)) Tαx0 B(n). Let n0 = supα |Tαx0|. We have then.
  34. Cubulating spaces and groups; example sheet 1 Macarena Arenas ...

    https://www.dpmms.cam.ac.uk/~mcr59/index_files/examplesheet1.pdf
    29 Jan 2024: b b. bb. b. b. b. b. b b. b. b. ... b b. b. b. 6. Let Γ and L, respectively, be the graph and the link pictured below.
  35. Scrapbook on Set Theory with a Universal Set Thomas ...

    https://www.dpmms.cam.ac.uk/~tef10/NFnotesredux.pdf
    14 Jul 2024: The last rule in the proof of A B C can only be a -R. ... B. So then theboolean ring B′ corresponding to B believes the ring formula φ′ corresponding.
  36. THE IMPORTANCE OF MATHEMATICS W. T. Gowers It is ...

    https://www.dpmms.cam.ac.uk/~wtg10/importance.pdf
    12 Sep 2002: B, consisting of a reasonable proportion of the numbers in A, in such a way that B B. ... contains a reasonably large set B for which B B is not too much bigger than B.
  37. Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 4 PAR 1. Let ...

    https://www.dpmms.cam.ac.uk/~par31/at/4.pdf
    21 Nov 2023: b) Define an equivalence relation on X Y by (x,y) (x′,y′) iff x = x′. ... D2f exists at (a,b). Show that f is differentiable at (a,b).
  38. Remarks on special values of L-functions Anthony J. Scholl* ...

    https://www.dpmms.cam.ac.uk/~ajs1005/preprints/remarks.pdf
    29 Jan 2010: denotes the set of rational primes. Hypothesis (b) includes a weak form of. ... Moreover we have. ords=0 L(M′, s) = 0. Now by hypothesis (b).
  39. Mich. 2008 ANALYSIS II—EXAMPLES 4 PAR 1. Show that ...

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2008-2009/08sheet4.pdf
    30 Nov 2008: around (a, b), and that D2f exists at (a, b).Show that f is differentiable at (a, b).(b) Suppose instead that D1f exists and is bounded on some open ball ... around (a, b), and that for fixed xthe function y 7 f (x, y) is continuous.
  40. 23 May 2024: b B(OF ) whose discriminant is strongly (resp. weakly) divisible by I 2. ... 2. Let b B(Op) be an element with discriminant weakly divisible by p2.
  41. VERMA MODULES FOR IWASAWA ALGEBRAS ARE FAITHFUL KONSTANTIN ARDAKOV ...

    https://www.dpmms.cam.ac.uk/~sjw47/VermaIwasawa.pdf
    12 Sep 2013: That is b x = θ(b)x for b B+ and x Kθ. ... We can compute that for b B+,log b pn1b+ acts by log θ(b) pR.
  42. NUMBER FIELDS, LENT 2024 PÉTER P. VARJÚ Disclaimer,…

    https://www.dpmms.cam.ac.uk/~pv270/NumberFields.pdf
    8 Mar 2024: Indeed, the minimalpolynomial of a/b with gcd(a,b) = 1 is xa/b. ... OK =. {a b 1 m. 1/2. 2: a,b Z. }if m 1 mod 4.
  43. Computer Science Tripos 2002 Paper 1 Question 8 Thomas ...

    https://www.dpmms.cam.ac.uk/~tef10/cam_only/2002p1q8.pdf
    4 Feb 2021: BBB = {x : (B B)(x B)}. (a) Let B P() and C P(). ... B C) (BB. B) (CC. C). (b) This part seems to have caused problems for some.
  44. BEILINSON’S THEOREM ON MODULAR CURVES. Norbert Schappacher and…

    https://www.dpmms.cam.ac.uk/~ajs1005/preprints/RSS.pdf
    17 Feb 2010: φ̂(bng) = |b|fφ̂(g). for all b BQ, n NẐ, g Gf. ... Now if Im(z) > 0 then. η̄ ω =(b>0. Wη. ((b 00 1.
  45. Ramsey Theory I.B. Leader Michaelmas 2000 1 Monochromatic Systems ...

    https://www.dpmms.cam.ac.uk/~par31/notes/ramsey.pdf
    8 Dec 2005: say with active coordinate set I, such. that for all a [m]n and all b, b′ L{L+}, we have c(a, b) = c(a, b′) = c′(a),say. ... ii) A B U if and only if A U or B U;.
  46. 30 Aug 2022: I was asleep at the wheel. Should read(iv) If a a′ and b b′ then a b a′ b′ mod n.Page 124 Proof of Lemma 4.3.10 (ii), ... which simplifies to. b(3 b) (b 1)2c > 1. Page 254 Replace first part of Exercise 9.2.3 as followsExercise 9.2.3 (i) Show that
  47. Analysis II Michaelmas 2016 Example Sheet 2 1. Let ...

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2016-2017/16sheet2.pdf
    27 Oct 2016: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... a) f(x) = sin x2 (b) f(x) = inf{|xn2| |n Z} (c) (sin x3)/(x 1).
  48. Computer Science Tripos 2018 Paper 2 Question 9 A ...

    https://www.dpmms.cam.ac.uk/~tef10/cam_only/2018p2q9.pdf
    26 Apr 2021: This means we have proved. GCD(b Fn4 a Fn3, b Fn2 a Fn1) = GCD(a b, b). ... But of course GCD(a b, b) = GCD(a, b) so we get.
  49. Lectures.dvi

    https://www.dpmms.cam.ac.uk/~sjw47/RepThLecturesMich2012.pdf
    28 Nov 2012: Forexample if a,b C and x,y,z V , then. (x,ay bz) =1. ... Then 〈Pa,Pb〉 = (a,b) for a,b V. So for each g G.
  50. More on Church-Oswald Models for Set Theory Thomas Forster ...

    https://www.dpmms.cam.ac.uk/~tef10/COmodels.pdf
    8 Apr 2021: NF0 is NF2 plus an axiom saying B(x) exists for all x. ... E D I T B E L O W H E R E.
  51. Lent Term 2024 O. Randal-Williams II Algebraic Topology // ...

    https://www.dpmms.cam.ac.uk/~or257/teaching/IIAlgTop2024/Sheet4.pdf
    11 Jan 2024: Show that the linear map A : R2 R2 peserves theequivalence relation (a,b) (a′,b′) (aa′,bb′) Z2, and so induces a continuous map fAfrom the torus T to itself.

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