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Holly Krieger — Cambridge
https://www.dpmms.cam.ac.uk/~hk439/research.html9 May 2023: Juul, N. Looper, M. Manes, B. Thompson, L. Walton), Research Directions in Number Theory, Association for Women in Mathematics Series, Vol. ... Some consequences of Schanuel's conjecture (with C. Cheng, B. Dietel, M. -
The Cauchy-Schwarz inequality
https://www.dpmms.cam.ac.uk/~wtg10/csineq.html4 Oct 2001: 1. /b. 1. rather than some other a. i. /b. i. , ... i. are equal. This still leaves a minor problem that some of the b. -
Recent papers and preprints
https://www.dpmms.cam.ac.uk/~ik355/pubs.html17 May 2024: pdf]. I. Kontoyiannis. "Some information-theoretic computations related to the distribution of prime numbers." In Festschrift in Honor of Jorma Rissanen, (P. ... Mutual information, synergy, and some curious phenomena for simple channels." 2005 IEEE -
Lent 2024 Number Fields – Example Sheet 3 P.P. ...
https://www.dpmms.cam.ac.uk/~pv270/example3.pdf8 Mar 2024: 2(1. 71). Find an element of OK with norm 2a 3b 5 for some. ... a,b 0. Hence prove that the class group of K is cyclic and find its order. -
Lent 2024 LOGIC AND SET THEORY – EXAMPLES 4 ...
https://www.dpmms.cam.ac.uk/study/II/Logic/2023-2024/sheet4.pdf3 Apr 2024: Show further that if f : X Y is a continuous function between Polish spacesand f is injective on a Borel set B X, then f(B) is Borel. ... 14. Prove that a set A (in some Polish space) is analytic if and only if there exist closedsets An1,.,nk (indexed by -
The existence of root two
https://www.dpmms.cam.ac.uk/~wtg10/roottwo.html20 Sep 2001: We know that some numbers are irrational, but we can still describe them, by means of their decimal expansions. ... 2. in some way, consistent with the principle that 0 < a < b implies that a. -
Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 4 PAR 1. Let ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2023-2024/4.pdf24 Nov 2023: 10. 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. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a,b), and. -
Definitions
https://www.dpmms.cam.ac.uk/~wtg10/definition.html24 Nov 2001: In the end, in order to do this one is forced to come up with some non-canonical construction that satisfies the axiom. ... four times, being periodic with period some number (defined to be 2 pi), satisfying the addition law -
Fermat's little theorem
https://www.dpmms.cam.ac.uk/~wtg10/fermat.html6 Mar 2002: b in B (that is, some non-zero b (mod p)) such that the set of all bx. ... Hence, if we can find some b that does not belong to A, then we have constructed the desired injection. -
Lent 2024 Number Fields – Example Sheet 3 P.P. ...
https://www.dpmms.cam.ac.uk/study/II/NumberFields/2023-2024/example3.pdf23 Feb 2024: 2(1. 71). Find an element of OK with norm 2a 3b 5 for some. ... a,b 0. Hence prove that the class group of K is cyclic and find its order. -
MATHEMATICAL TRIPOS: PART IA Lent 2024 PROBABILITY JRNExample Sheet…
https://www.dpmms.cam.ac.uk/study/IA/Probability/2023-2024/pex1.pdf11 Jan 2024: 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}. -
The fundamental theorem of arithmetic
https://www.dpmms.cam.ac.uk/~wtg10/FTA.html15 Jun 2000: The definition is that it is not divisible by 2. However, another, more positive definition is that a can be written 2b1 for some integer b. ... b. Having had some success with addition, we could try multiplication. What happens if we replace a by ha? -
Riemann SurfacesExample Sheet 1 Michaelmas 2023 Please send any ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2023-2024/RS1M23.pdf12 Oct 2023: 6. By considering the singularity at or otherwise, show that any injective holo-morphic map f C C has the form f(z) = az b, for some a C and b ... 0,1,a} to be conformallyequivalent to C{0,1,b}, where a,b C{0,1}.[Don’t forget the cross ratio from IB - -
Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 1 PAR 1. Let ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2023-2024/1.pdf12 Oct 2023: a, b], and suppose that fn converges pointwise to a continuous function f. ... Show that there is some subinterval [a, b] of [0, 1] with a < b on which (fn) is uniformly. -
Holly Krieger - vita
https://www.dpmms.cam.ac.uk/~hk439/CV.pdf19 Apr 2023: Juul, N. Looper, M. Manes, B. Thompson, L. WaltonTo appear in Research Directions in Number Theory - Women in Numbers 4. ... 23, 5498-5525. 2. 2009 Some Consequences of Schanuel’s Conjecture,with C. Cheng, B. -
The mean value theorem
https://www.dpmms.cam.ac.uk/~wtg10/meanvalue.html13 Dec 2001: Then there is some point c in (a,b) such that f'(c) equals (f(b)-f(a))/(b-a). ... Here, I take f to be real valued and defined on some interval of real numbers. -
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. -
GEOMETRY Keith Carne t.k.carne@dpmms.cam.ac.uk…
https://www.dpmms.cam.ac.uk/~tkc10/Geometry_2009/Overhead_1.pdf28 Jan 2009: Proposition 1.2 Isometries of E2. An orientation preserving isometry of E2 is:(a) The identity.(b) A translation.(c) A rotation about some point c E2. ... a) The identity.(b) A translation.(c) A rotation about some line. (d) A screw rotation, that is a -
STATISTICS Part IBExample Sheet 1 (of 3) Lent 2022 ...
https://www.dpmms.cam.ac.uk/study/IB/Statistics/2021-2022/Qu1.pdf13 Jan 2022: b) For some n 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T , and compute. ... 4. For some n 2 let X1,. ,Xniid Unif(θ, 2θ) for some θ > 0. -
Michaelmas Term 2023 T.A. Fisher Galois Theory: Example Sheet ...
https://www.dpmms.cam.ac.uk/study/II/Galois/2023-2024/galois_theory-23-4.pdf24 Nov 2023: 4. Let K be a field containing a primitive mth root of unity for some m > 1. ... Showthat f and g have the same splitting field if and only if b = cmar for some c Kand r N with gcd(r,m) = 1. -
Lent Term 2024 O. Randal-Williams IB Groups, Rings, and ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2023-2024/Sheet3.pdf15 Jan 2024: 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. -
Solving cubic equations
https://www.dpmms.cam.ac.uk/~wtg10/cubic.html25 Apr 2002: Then, with some embarrassment, we note to ourselves that we can solve the equation when a = 0. ... 1. Is there a similar way to simplify a cubic so that some of the coefficients become zero? -
slide 0 Modular formsand motivesA J Scholl slide 1 ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/banff.pdf29 Jan 2010: slide 12. Lemma j = isomn on HM(;)(") Z[ 12k!].(each component of complement is xedby some with "( ) = 1.)[D]ix(pr0#D [ [prk#D)(") 7!y2 2Hk1M (Uky ;k 1)(")'Hk1M ... E fyg)k;k 1)(")Eisenstein symbol (B)ny = 0, some n > 1[n]y 2 Hk1M ((E 0)k;k 1)(")?yoHk1M -
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.pdf11 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. 2. 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. -
ANALYSIS OF FUNCTIONS EXAMPLE SHEET 4Richard Nickl 〈rn289@cam.ac.uk〉…
https://www.dpmms.cam.ac.uk/study/II/AnalysisofFunctions/2023-2024/example4-RN.pdf7 Mar 2024: space. What if 𝑠 1/2? Exercise 6. a) Suppose 𝑠 = 𝑛2 𝛾 for some 0 < 𝛾 < 1. ... ANALYSIS OF FUNCTIONS EXAMPLE SHEET 4. b) Show that if 𝑠 = 𝑛2 𝑘 𝛾 for some 𝑘 Z>0, 0 < 𝛾 < 1, then. -
Mich. 2007 ANALYSIS II—EXAMPLES 4 PAR 1. At which ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2007-2008/sheet4.pdf21 Nov 2007: 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 exists and is bounded on some open ball around (a, b), and that for -
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 2023 Julian Sahasrabudhe Linear Algebra: Example…
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2023-2024/lin-alg-ex4-2023.pdf24 Nov 2023: 1. The square matrices A and B over the field F are congruent if B = PT AP for some invertible matrixP over F. ... 7. Let S be an n n real symmetric matrix with Sk = I for some k 1. -
GEOMETRY AND GROUPS – Example Sheet 2TKC Michaelmas 2012 ...
https://www.dpmms.cam.ac.uk/~tkc10/GeometryandGroups/Exercise2.pdf23 Oct 2012: J : z 7 az bcz d. for some complex numbers a, b, c, d with ad bc = 1. ... For which choices of a, b, c, d is this map J aninvolution, that is J 2 = I? -
Publications | Department of Pure Mathematics and Mathematical…
https://www.dpmms.cam.ac.uk/publications?page=1872 Jul 2024: Lecture Notes in Mathematics. (1979). 280. SOME MISLEADING ARGUMENTS INVOLVING CONDITIONAL INDEPENDENCE. ... 9,. 171. CRITICALLY PARTITIONABLE GRAPHS. 1. AG THOMASON. – Journal of Combinatorial Theory, Series B. -
Publications | Department of Pure Mathematics and Mathematical…
https://www.dpmms.cam.ac.uk/publications?page=1802 Jul 2024: 1985). 629. Some probabilistic aspects of network flow. FP Kelly. (1985). ... STOCHASTIC-MODELS OF COMPUTER-COMMUNICATION SYSTEMS. FP KELLY. – JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-METHODOLOGICAL. -
Analysis II Michaelmas 2016 Example Sheet 2 1. Let ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2016-2017/16sheet2.pdf27 Oct 2016: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... vn} C such that{B(v1),. ,B(vn)} cover C. (b) Show that there exists some > 0 such that for every v C, B(v) Uα for some α.Deduce that -
GEOMETRY AND GROUPS – Example Sheet 1TKC Michaelmas 2012 ...
https://www.dpmms.cam.ac.uk/~tkc10/GeometryandGroups/Exercise1.pdf13 Oct 2012: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. w Z Z can be written as mu nv for some integers m and n. -
Mathematical Tripos: Part IB DJS/Lent 2015 Statistics: Example Sheet…
https://www.dpmms.cam.ac.uk/study/IB/Statistics/2014-2015/ex1.pdf22 Jan 2015: Xn be independent Poisson random variables, with Xi having meaniθ, for some θ > 0. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ). -
Abstract set theory
https://www.dpmms.cam.ac.uk/~wtg10/settheory.html9 Jun 2003: b. 3. for some pair of positive integers a,b}. (vi) X. ... n. , but for a general proof, we must specify some method that always works. -
Mapping class groupsProblem sheet 2 Lent 2021 1. What ...
https://www.dpmms.cam.ac.uk/study/III/MappingClassGroups/2020-2021/MCGs%20Sheet%202.pdf19 Jan 2021: a) Show that there is a surjection π1Sg G, for some g.(b) Show that G is a subgroup of Mod(Sh), for some h. ... a) Prove that TαTβ(α) = β. (b) Prove the braid relation: TαTβTα = TβTαTβ. -
Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 4 PAR 1. Let ...
https://www.dpmms.cam.ac.uk/~par31/at/4.pdf21 Nov 2023: 10. 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. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a,b), and. -
Publications | Department of Pure Mathematics and Mathematical…
https://www.dpmms.cam.ac.uk/publications?page=125%2C2 Jul 2024: Is equal to. Is not earlier than. Is between. And. Mutual Information, Synergy and Some Curious Phenomena for Simple Channels. ... I Kontoyiannis, B Lucena. – IEEE International Symposium on Information Theory - Proceedings. -
Mich. 2012 ANALYSIS II—EXAMPLES 4 PAR 1. Use the ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2012-2013/12sheet4.pdf26 Nov 2012: a) Suppose that D1f exists and is continuous in some open ball around (a, b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a, b), and. -
Mich. 2013 ANALYSIS II—EXAMPLES 4 PAR 1. (a) Let ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2013-2014/13sheet4.pdf29 Nov 2013: 7. 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. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a,b), and. -
Mapping class groupsProblem sheet 2 Michaelmas 2019 1. What ...
https://www.dpmms.cam.ac.uk/study/III/MappingClassGroups/2019-2020/MCGs%20Sheet%202.pdf12 Nov 2019: 1. 5. Let G be any finite group. (a) Show that there is a surjection π1Sg G, for some g.(b) Show that G is a subgroup of Mod(Sh), for -
Mich. 2013 ANALYSIS II—EXAMPLES 1 PAR 1. Which of ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2013-2014/13sheet1.pdf23 Oct 2013: 4. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and. ... f. Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f is bounded. -
PRINCIPLES OF STATISTICS – EXAMPLES 3/4 Part II, Michaelmas ...
https://www.dpmms.cam.ac.uk/study/II/PrinciplesOfStatistics/2023-2024/examples3-prob.pdf2 Oct 2023: What happensif Θ = [a,b] for some 0 < a < b <? 9. Let X be multivariate normal N(θ,I), where θ Θ = Rp,p 3, and I is the pp identitymatrix. ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.(b) Let θ̂ be the -
DIFFERENTIAL GEOMETRY (PART III) MICHAELMAS 2021 EXAMPLE SHEET 2 ...
https://www.dpmms.cam.ac.uk/files/study/III/DifferentialGeometry/2021-2022/DGSheet2.pdf1 Nov 2021: Show that M M is trivial.(b) Show that for all n we have TSn R = Rn1 over Sn. ... 4.† Let α be a nowhere-zero 1-form on a manifold X, and let β be a p-form.(a) Show that if β = α γ for some (p -
Mich. 2012 ANALYSIS II—EXAMPLES 2 PAR 1. Let f ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2012-2013/12sheet2.pdf29 Oct 2012: a sequence of points in [a, b] with xm x then fn(xn) f (x). ... f. Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f is bounded. -
problemsPDE
https://www.dpmms.cam.ac.uk/files/study/III/AnalysisPDE/2021-2022/Sheet1.pdf14 Oct 2021: Show that ifeb|k|f̂(k) L2(R) for some b > 0 then f is real analytic at each point in R. ... u1. 1We write A B to mean that there exists a universal constant C such that A CB. -
Part III Commutative Algebra Example Sheet II, 2021 Note: ...
https://www.dpmms.cam.ac.uk/study/III/CommutativeAlgebra/2021-2022/HW2.pdf29 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 ... a) Show that x is a zero divisor in A if and only if x p for some p D(A).(b) Let S be a multiplicatively -
NUMBER FIELDS, LENT 2024 PÉTER P. VARJÚ Disclaimer,…
https://www.dpmms.cam.ac.uk/~pv270/NumberFields.pdf8 Mar 2024: a1α1. adαdq. for some a1,. ,ad Z. Proof. We take. q =(disc(α1,. , ... In what follows, we will exhibit asuitable α, but this requires some preparations. -
Mich. 2007 ANALYSIS II—EXAMPLES 1 PAR 1. Which of ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2007-2008/sheet1.pdf12 Oct 2007: 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. -
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.
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