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GEOMETRY AND GROUPS – Example Sheet 4TKC Michaelmas 2006 ...
https://www.dpmms.cam.ac.uk/study/II/Geometry%2BGroups/2006-2007/Exercise4.pdf24 Nov 2006: Show that, for such a map,. dimH f (M ) 61α. ... Prove that the limit set for such a group is totallydisconnected. -
ANALYSIS II (Michaelmas 2009): EXAMPLES 4 The questions are ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2009-2010/anII-09-4.pdf8 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = f/y exists at (a,b). ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that D1f = f/x exists and is bounded near (a,b), -
ANALYSIS II—EXAMPLES 3 Mich. 2015 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2015-2016/15sheet3.pdf13 Nov 2015: every polynomial p of degree n there is x0 [0, 1/n]such that |p(x)| C|p(x0)| for every x [0, 1].(b) If (X,d) is a metric ... That is to say, showthat there is no metric d on the set of continuous functions f : [a,b] R such that pointwise convergenceof -
PART II REPRESENTATION THEORYSHEET 4 Unless otherwise stated, all ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2016-2017/repex4.pdf10 Jan 2017: 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 -
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: Does it follow that thereexists c (a,b) such that f(b) f(a) = f′(c)(ba)? ... containing (x0,y0) and V containing F(x0,y0) = (x0, 0) such that F maps U bijectively to. -
ANALYSIS II (Michaelmas 2010): EXAMPLES 3 The questions are ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2010-2011/anII_ex_2010_3.pdf19 Nov 2010: 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 at (a,b). ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that f/x exists and is bounded near (a,b), and that fora -
CODES AND CRYPTOGRAPHY – Example Sheet 4TKC Lent 2015 ...
https://www.dpmms.cam.ac.uk/~tkc10/CodesandCryptography/Exercise_4.pdf7 Mar 2015: n) is the output from a LFSR of length d. (c) The period of (xn)n>0 is the least integer r > 1 such that xnr = xn for all sufficiently largen. ... a) Show that N ϕ(N) < 3N. (b) Let k = (de 1)/ϕ(N). -
CT1.dvi
https://www.dpmms.cam.ac.uk/study/III/CategoryTheory/2018-2018/CT1.pdf10 Oct 2018: Ch. > A. g. f. Dk. > B. with g epic, there exists a (necessarily unique) t: D A such that ft = k and tg = h. ... in [C, Set]), there exists i I such that the restriction of αto Gi is still epimorphic. -
ANALYSIS II EXAMPLES 3 Michaelmas 2004 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2004-2005/an04-3.pdf21 May 2005: 2 ANALYSIS II EXAMPLES 3. 7. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in a neighbourhood of(a,b), and D2f = f/y exists ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that D1f = f/x exists and is -
Ordinals and Ordinal Hierarchies
https://www.dpmms.cam.ac.uk/~wtg10/ordinals.html10 Sep 2002: Then either [a] is minimal or there is a non-empty subset B consisting of all [b] such that b is isomorphic to an initial segment I(t) of a. ... Pick [b] such that t is minimized (which can be done as a is a well-ordered set of reals). -
ANALYSIS II EXAMPLES 3 Michaelmas 2005 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2005-2006/an05-3.pdf22 Nov 2005: 2 ANALYSIS II EXAMPLES 3. 7. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in a neighbourhood of(a,b), and D2f = f/y exists ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that D1f = f/x exists and is -
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: b) The model is to be reparametrised in such a way that it can be specialised to (i)two parallel lines for the two groups, (ii) two lines with the same ... Z21 Z2np,. where the Zi are independent and Zi N(λi,σ2) for some λi such that. -
Tensor products
https://www.dpmms.cam.ac.uk/~wtg10/tensors3.html14 May 2003: Proof. Applying the hypothesis about b to the bilinear map g:UxV-->U@V, we obtain a linear map i:X-->U@V such that g=ib. ... Similarly, applying the universal property of g to the bilinear map b, we obtain a linear map j:U@V-->X such that b=jg. -
The Bloch-Kato conjecture on special values of L-functions Jef ...
https://www.dpmms.cam.ac.uk/~jcsl5/BK-studygroup.pdf10 Mar 2021: There is a distinguished uniformizer t B. dR. such that F kBdR = tkBdR and GQ acts on t through the -adic cyclotomic character. ... Such a functor is known to exist on the subcategory of smooth quasi-projective varieties [Fon92, 6.7]. -
Metric and Topological Spaces Easter 2016 Example Sheet 2 ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2015-2016/MetTop_2016_Ex2.pdf16 May 2016: p. k, where k is the smallest value ofn such that xn 6= yn. ... 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. -
ANALYSIS II (Michaelmas 2009): EXAMPLES 1 The questions are ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2009-2010/anII-09-1.pdf14 Oct 2009: On the other hand, show that if fn does not converge uniformly to f , then we can find aconvergent sequence xm x in [a, b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges -
Part 1B Metric and Topological Spaces 2018 pmhw@dpmms.cam.ac.uk…
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2017-2018/MTS1.18.pdf22 Mar 2018: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y. -
1 Metric & Topological spaces, Sheet 1: 2007 1. ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2006-2007/2007sheet1.pdf8 May 2007: b) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... b) (x, y, z) (x′, y′, z′) x = x′, y = y′, z = 1 z′. -
Part 1B Metric and Topological Spaces 2017 pmhw@dpmms.cam.ac.uk…
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2016-2017/MTS1.17.pdf24 Apr 2017: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y. -
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: p. k, where k is the smallest value ofn such that xn 6= yn. ... 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. -
1 Metric & Topological Spaces, sheet 2: 2007 1. ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2006-2007/2007sheet2.pdf14 May 2007: Is there always an orientable such surface, i.e. one in which“turning clockwise” makes sense globally on the surface? ... b) What is the smallest genus (number of holes) of such a surface with boundary thetrefoil knot, which is the knot which lies on -
ANALYSIS II EXAMPLES 4 Michaelmas 2005 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2005-2006/an05-4.pdf22 Nov 2005: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 9. [Tripos IB 96401(b), modified] (i) Suppose that (X,d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x,y X, the -
The fundamental theorem of arithmetic
https://www.dpmms.cam.ac.uk/~wtg10/FTA.html15 Jun 2000: We must convert a into a new number a' in such a way that p divides a'b and does not divide a'. ... Then we can find H and T such that HaTb=1, and we know that we will now find that p divides b if we continue the argument. -
ANALYSIS II EXAMPLES 1 Michaelmas 2004 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2004-2005/an04-1.pdf21 May 2005: a,b] such that fn(xn) does not converge to f(x). [Hint: Bolzano–Weierstrass.]. ... every point of [0,1] (here we interpret f ′(0) as the ‘one-sided derivative’ limh0+((f(h) f(0))/h),and similarly for f′(1)), but such that f ′ is -
ANALYSIS II EXAMPLES 1 Michaelmas 2005 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2005-2006/an05-1.pdf18 Oct 2005: ii) Give an example of a sequence of uniformly continuous functions fn on R, such that fn con-. ... a,b] such that fn(xn) does not converge to f(x). 12. (i) Suppose f is defined and differentiable on a (bounded or unbounded) interval E R, andthat its -
ANALYSIS II EXAMPLES 4 Michaelmas 2004 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2004-2005/an04-4.pdf21 May 2005: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 10. [Tripos IB 96401(b), modified] (i) Suppose that (X,d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x,y X, the -
ANALYSIS II (Michaelmas 2010): EXAMPLES 1 The questions are ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2010-2011/anII_ex_2010_1.pdf15 Oct 2010: On the other hand, show that if fn does not converge uniformly to f, then we can find aconvergent sequence xm x in [a, b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges -
Part 1B Metric and Topological Spaces 2012 pmhw@dpmms.cam.ac.uk…
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2011-2012/MTS12.1.pdf23 Jun 2012: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y. -
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: b) Find a condition on 𝑠 such that 𝛿𝑥 𝐻𝑠(R𝑛). c) Show that 𝐻𝑡 (R𝑛) is continuously embedded in 𝐻𝑠(R𝑛) for 𝑠 < 𝑡. ... Δ2𝑢 𝑢 = 𝑓. b) Show further that there exists 𝐶 > 0 such that ‖𝑢‖𝐻 𝑠4 -
ANALYSIS II (Michaelmas 2011): EXAMPLES 1 The questions are ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2011-2012/anII_ex_2011_s1.pdf18 Oct 2011: On the other hand, show that if fn does not converge uniformly to f, then we can find aconvergent sequence xm x in [a,b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges -
1 Metric & Topological Spaces, sheet 2: (2006) 1. ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2005-2006/2006sheet2.pdf4 May 2006: Prove that f has a real root. 17. Prove there is no continuous function f : [0, 1] R such that x Q f (x) 6 Q (whereQ denotes the rational numbers). ... 18. A continuous surjective map p : E B is a covering if every point b B has an openneighbourhood V -
TECHNIQUES IN COMBINATORICS: EXAMPLES SHEET 1 Michaelmas Term 2014 ...
https://www.dpmms.cam.ac.uk/study/III/2014-15/TechniquesinCombinatorics/techniques2014examples1.pdf20 Oct 2014: metric relation R defined on X such that for any k elements x1,. ... Prove that there existu,v Zp such that. |A (uB v)| αβp αβp, where uB v denotes the set{ub v : b B}. -
ANALYSIS II EXAMPLES 2 Michaelmas 2004 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2004-2005/an04-2.pdf21 May 2005: n=1 = (x1,x2,x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E,F) be the space of bounded linear operators equipped with the operator -
ON MODULAR UNITS A. J. Scholl Introduction. In [6], ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/u.pdf29 Jan 2010: The irreducibility of Sp(χ) for such a prime pin conjunction with a) implies that Φq,χ = 0. ... Since GẐ acts transitively on |MK |,. there exists a positive constant cn Q such that for every x |M|,. -
ANALYSIS II EXAMPLES 2 Michaelmas 2005 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2005-2006/an05-2.pdf4 Nov 2005: n=1 = (x1,x2,x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E,F) be the space of bounded linear operators equipped with the usual -
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: Does there exist c (a,b)such tat f(b) f(a) = f′(c)(ba)?(v) Let f: Rn Rn be a C1 map with ‖Df(x)I‖ 1/2 for ... continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I). -
MTS-2011-Sheet1.dvi
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2010-2011/MTS-2011-Sheet1.pdf6 May 2011: c) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... Prove the following:. (a) G is homogeneous: given any x, y G there is a homeomorphism φ : G G such thatφ(x) = y. -
MTS-2010-Sheet1.dvi
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2009-2010/MTS-2010-Sheet1.pdf29 Apr 2010: c) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... Prove the following:. (a) G is homogeneous: given any x, y G there is a homeomorphism φ : G G such thatφ(x) = y. -
1 Metric & Topological spaces, Sheet 1: 2006 1. ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2005-2006/2006sheet1.pdf4 May 2006: b) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... a) Prove (R, TS ) is Hausdorff. (b) Prove (R, TS ) is not metrisable. -
TECHNIQUES IN COMBINATORICS: EXAMPLES SHEET 2 Michaelmas Term 2014 ...
https://www.dpmms.cam.ac.uk/study/III/2014-15/TechniquesinCombinatorics/techniques2014examples2.pdf20 Oct 2014: Prove that there are. subsets A X and B G such that |η(A,B) δαβ| θ, where θ > 0 depends on γ andδ only. ... x1,. ,xk such that no xi xj with i,j distinct is a quadratic residue mod p. -
Applications of Mirror Symmetry
https://www.dpmms.cam.ac.uk/~mg475/riverside.pdf16 Jul 2014: singularities is a real manifold Bwith an open subset B0 B, with := B B0 of codimension 2, such that B0 has the structure of an integral affine manifold.We also allow ... X Spec k[[t]]. such that the special fibre satisfies. X0 =. σPmax. Pσ. and the -
The philosophy of mathematics
https://www.dpmms.cam.ac.uk/~wtg10/philosophy.html20 Sep 2004: And yet there doesn't seem to be such a fact of the matter. ... Similarly, if B={x:Q(x)} then the statement "A cap B = emptyset" means nothing more than that there is no x such that P(x) and Q(x). -
Solving cubic equations
https://www.dpmms.cam.ac.uk/~wtg10/cubic.html25 Apr 2002: 3. dy. 2. eyf in y and finally (the most important part) cleverly choose u and v in such a way that d. ... d. 2. equal to 3e, or it may be that, although such a choice exists, the dependence of u and v on a, b and c is so complicated that we -
Michaelmas Term 2023 Number Theory: Example Sheet 1 of ...
https://www.dpmms.cam.ac.uk/study/II/NumberTheory/2023-2024/number_theory-23-1.pdf27 Oct 2023: 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) c log b d. -
The fundamental theorem of arithmetic not always true.
https://www.dpmms.cam.ac.uk/~wtg10/meta.fta.html11 May 2001: 1/2. , so two such numbers must multiply to something of modulus at least 5. ... Analogues of Euclid's algorithm work in other contexts, such as the ring of all polynomials with rational coefficients. -
TOPICS IN SET THEORY: Example Sheet 3 1 Department ...
https://www.dpmms.cam.ac.uk/study/III/2014-15/TopicsinSetTheory/TST%20Example%20Sheet%203%202014.pdf19 Dec 2014: if B = {min(Xα) : α < κ} is stationary.(iii) If {Xα : α < κ} is pairwise disjoint, then there exists a [κ]κ such that. ... Prove that there exists a family F of2κ stationary subsets of S such that if A 6= B F , then A B and B A arestationary in -
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.pdf19 Dec 2014: i) Prove that 20 = 1 implies the following assertion SD :there exist sets A and B such that R2 = AB and Al and Bm are countablefor every horizontal line l and ... exist x1 and x2 such that x1 6 f(x2) and x2 6 f(x1). -
Commutativity of Multiplication
https://www.dpmms.cam.ac.uk/~wtg10/commutative.html25 Nov 2002: We also know that CxB is a subset of AxB, and that the set-theoretic difference AxB-CxB consists of all ordered pairs (m,x) such that x is in B. ... me that if m> n then there is a unique positive integer t such that nt=m. -
Continuity and open sets II
https://www.dpmms.cam.ac.uk/~wtg10/easyanalysis2.html7 Aug 2002: A formal statement of the result to be proved. Let X and Y be metric spaces and let f be a function from X to Y such that f. ... in U (for the reason I have just given) and such that [f(y) is in U] implies [d(f(x),f(y)) < a]. -
ANALYSIS I EXAMPLES 4 C. M. Warnick Lent 2024 ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2023-2024/Analysis%20Sheet%204.pdf24 Jan 2024: Does 10f exist? 3. Give an example of a continuous function f : [0,) [0,), such that0f. ... Must f vanishidentically?[We say g vanishes near a and b if there exists > 0 such that g(x) = 0 for x 6(a ,b ).]. 1. 2 ANALYSIS I EXAMPLES 4.
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