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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: Let a,b K such that the polynomials f = Xm a, g = Xm b are irreducible. ... Use asuitable Lagrange resolvent to find a Q(ζ3) such that K = Q(ζ3, 3. -
Numbers and Sets (2014–15) Example Sheet 1 of 4 ...
https://www.dpmms.cam.ac.uk/study/IA/Numbers%2BSets/2014-2015/examples-NS-14-1.pdf14 Oct 2014: 8. Let A1, A2, A3,. be sets such that A1 A2. ... c there is an x with a < x < b such that f(x) = c. -
Mich. 2019 ANALYSIS AND TOPOLOGY – EXAMPLES 2 AZ ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2019-2020/sheet2.pdf5 Nov 2019: A = {x M : (xn) in A such that xn x} ,. Define the interior of A (in M) to be the set. ... Deduce that if A and B are disjoint closed subsets of M, then there exist disjoint opensubsets U and V of M such that A U and B V. -
Analysis I Lent term 2020 Example Sheet 4 1. ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2019-2020/AnalysisI_2020_Ex4.pdf3 Mar 2020: Show that f is integrable. What is. 10f(x)dx? 4. Give an example of a continuous function f : [0,) [0,) such that0f(x)dx. ... Show. that for every > 0 and every closed interval I [a,b] of positive length, there isa closed interval J I such that J has -
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 Example Sheet 3 Michaelmas 2006 MJW Corrections ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2006-2007/ex3.pdf10 Nov 2006: Does it follow that every open set must be. closed? Give an example of such a metric space. ... Show that thereexists a norm on C(I) such that the map sending f to y0. -
Mich. 2020 ANALYSIS AND TOPOLOGY – EXAMPLES 3 AZ ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2020-2021/sheet3.pdf12 Nov 2020: 5. Let A Rn be such that every continuous function f : A R is bounded. ... V such that A Uand B V. -
Analysis II Example Sheet 4 Michaelmas 2006 MJW Corrections ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2006-2007/ex4.pdf28 Nov 2006: ii) Suppose that f : R2 R is such that D1f = f/x exists and is bounded near (a,b), and. ... is a function l : U M such that exp(l(A)) = A for all A U. -
COMPLEX ANALYSIS EXAMPLES 2, LENT 2021 Neshan Wickramasekera. Please…
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2020-2021/ca_IB_ex2_2021.pdf12 Feb 2021: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential -
Part IB COMPLEX ANALYSIS (Lent 2009): Example Sheet 2 ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2008-2009/cxanalysis2.pdf23 Feb 2009: Show that for any b C, there exists a sequence of pointszn D(a,r), with zn 6= a, such that zn a and f(zn) b, as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2. -
Analysis I Lent term 2019 Example Sheet 4 1. ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2018-2019/AnalysisI_2019_Ex4.pdf23 Apr 2019: Show that f is integrable. What is. 10f(x)dx? 4. Give an example of a continuous function f : [0,) [0,) such that0f(x)dx. ... Show. that for every > 0 and every closed interval I [a,b] of positive length, there isa closed interval J I such that J has -
ANALYSIS II—EXAMPLES 1 Mich. 2014 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2014-2015/14sheet1.pdf20 Oct 2014: On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)? -
COMPLEX ANALYSIS EXAMPLES 2 G.P. Paternain Lent 2018 Comments ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2017-2018/ca_IB_ex2_2018.pdf5 Feb 2018: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential -
Analysis II Michaelmas 2017 Example Sheet 2 1. Let ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2017-2018/AnalysisII_2017_Ex2.pdf23 Oct 2017: b) If (fn) does not converge uniformly, show that there is a convergent sequence (xm) x [0, 1] such that (fn(xn)) does not converge to f(x). ... a) The space Cb(R) of bounded continuous functions on R.(b) The space C0(R) of continuous functions f : R R -
ANALYSIS 1 EXAMPLES SHEET 4 Lent Term 2015 W. ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2014-2015/analysis1examples4.pdf4 Mar 2015: Prove that f is. integrable. What is 10f(x)dx? 6. Let a < b and let f : [a,b] R be a Riemann integrable function such that f(x) 0for every ... 1, 1], such that f(x) = g(x) h(x) for every x. -
STATISTICAL MODELLING Part IIC / Michaelmas 2023Example Sheet 1 ...
https://www.dpmms.cam.ac.uk/study/II/StatisticalModelling/2023-2024/ExampleSheet1.pdf27 Nov 2023: X̃1, that is, β̃1 = (X̃T1 X̃1). 1X̃T1 Ỹ. (c) Find a condition on X0 and X1 such that β̃0 = β̂0. ... 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 -
Automata & Formal LanguagesMichaelmas Term 2023 Part II of ...
https://www.dpmms.cam.ac.uk/study/II/AutomataAndFormalLanguages/2023-2024/M23_AFL_ES2.corrected.pdf23 Oct 2023: Show that:. (a) There is a nondeterministic automaton N with n 1 states such that L(N) = Ln and(b) if D is a deterministic automaton with fewer than 2n states, ... Prove that there is a proper excerpt e of v such that xey L. -
COMPLEX ANALYSIS EXAMPLES 2, LENT 2022 Neshan Wickramasekera. Please…
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2021-2022/ca_IB_ex2_2022.pdf22 Feb 2022: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential -
Mich. 2021 ANALYSIS AND TOPOLOGY – EXAMPLES 3 AZ ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2021-2022/sheet3.pdf21 Nov 2021: and B, there exist disjoint open sets U and V such that A Uand B V. ... Deducethat B′ is compact. 15. Show that there exist topological spaces X and Y with continuous bijections f : X Yand g : Y X such that X and Y are not homeomorphic. -
ANALYSIS II—EXAMPLES 1 Michaelmas 2018 Please email comments,…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2018-2019/18sheet1.pdf12 Oct 2018: x). On the other hand, show that if fn does not convergeuniformly to f then we can find a convergent sequence xn x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b) -
ANALYSIS 1 EXAMPLES SHEET 4 Lent Term 2014 W. ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2013-2014/analysis1examples4.pdf5 Mar 2014: Prove that f is. integrable. What is 10f(x)dx? 6. Let a < b and let f : [a,b] R be a Riemann integrable function such that f(x) 0for every ... 1, 1], such that f(x) = g(x) h(x) for every x. -
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: 11 fn 6 0. Is it possible to find such a sequence with |fn(x)| 1 for all n and for all x? ... On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a, b] such that fn(xn) 6 f -
ANALYSIS II EXAMPLES 4 G.P. Paternain Mich. 2003 The ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/es4-03.pdf21 May 2005: Tripos IB 96401(b), modified] (i) Let (X,d) be a nonempty complete metric space, and let f :X X be a continuous map such that, for any two points ... xn} is a finite set of points inX such that the balls B(xi,) cover X, consider the 2n 1 points of HX -
COMPLEX ANALYSIS EXAMPLES 1, LENT 2020 Neshan Wickramasekera. Please…
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2019-2020/2ca_IB_ex1_2020.pdf28 Jan 2020: Regarding T as a map from C into C by identifyingR2 with C in the usual way, show that there exist unique complex numbers A, B such thatfor every z C, ... Can such a map everbe a Möbius transformation? 9. Let U C be open and let f = uiv : U C. -
ANALYSIS 1 EXAMPLES SHEET 3 Lent Term 2015 W. ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2014-2015/analysis1examples3.pdf19 Feb 2015: Deducethe more general result that if f ′(a) 6= f ′(b) and z lies between f ′(a) and f ′(b), then thereexists c (a,b) such that f ′(c) = z. ... f(a) < 0 and f(b) > 0 then there exists c (a,b)such that f(c) = 0. -
Complex Analysis IB, 2008 Example sheet 2 1 (i) ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2007-2008/ex-sheet2.pdf10 Mar 2008: Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2. -
problemsPDE
https://www.dpmms.cam.ac.uk/files/study/III/AnalysisPDE/2021-2022/Sheet1.pdf14 Oct 2021: y2= 0 (3). Construct a sequence of real analytic functions (uk)k=1 with uk : R2 R solving (4) such. ... u1. 1We write A B to mean that there exists a universal constant C such that A CB. -
ANALYSIS II EXAMPLES 4 G.P. Paternain Mich. 2002 The ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/es4-02.pdf21 May 2005: Why would such a thesis be unlikely to be consideredworth a Ph.D? ... xn} is a finite set of points inX such that the balls B(xi, ) cover X, consider the 2n 1 points of HX which are the nonemptysubsets of X0.]. -
ANALYSIS 1 EXAMPLES SHEET 3 Lent Term 2014 W. ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2013-2014/analysis1examples3.pdf5 Mar 2014: Deducethe more general result that if f ′(a) 6= f ′(b) and z lies between f ′(a) and f ′(b), then thereexists c (a,b) such that f ′(c) = z. ... f(a) < 0 and f(b) > 0 then there exists c (a,b)such that f(c) = 0. -
ANALYSIS II EXAMPLES 1 G.P. Paternain Mich. 2003 The ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/es1-03.pdf21 May 2005: 1. 2 ANALYSIS II EXAMPLES 1. (ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a,b] such that fn(xn) does ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0, -
ANALYSIS II EXAMPLES 1 G.P. Paternain Mich. 2002 The ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/es1-02.pdf21 May 2005: 1. 2 ANALYSIS II EXAMPLES 1. (ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a,b] such that fn(xn) does ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0, -
Complex Analysis IB: 2015-16 – Sheet 1 1. Let ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2015-2016/CxAnalIB-2016-Sheet1.pdf18 Jan 2016: Show that T can be written Tz = Az Bz̄for unique A,B C. ... Can such a map be therestriction to the strip of a Möbius map? -
ANALYSIS II—EXAMPLES 2 Mich. 2018 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2018-2019/18sheet2.pdf29 Oct 2018: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f -
MATHEMATICAL TRIPOS: PART IA Lent 2022 PROBABILITY Dominic YeoExample …
https://www.dpmms.cam.ac.uk/study/IA/Probability/2021-2022/pex22_sheet2.pdf3 Feb 2022: 1. Let A,B F be two events such that P(B) = 0 or 1. ... Ie, at least one day d such that at least k people were born on day d.). -
ANALYSIS II—EXAMPLES 1 Mich. 2015 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2015-2016/15sheet1.pdf16 Oct 2015: On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)? -
Complex Analysis IB – 2014 – Sheet 1 1. ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2013-2014/CxAnalIB-2014-Sheet1.pdf14 Jan 2014: Show that T can be written Tz = Az Bz̄for unique A,B C. ... Can such a map be therestriction to the strip of a Möbius map? -
Mich. 2019 ANALYSIS AND TOPOLOGY – EXAMPLES 4 AZ ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2019-2020/sheet4.pdf29 Nov 2019: V such that A Uand B V. ... I) Mn such that g(A)2 = A for all A Dr(I). -
Complex Analysis IB, 2015 Example sheet 1 1 Let ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2014-2015/CxAnalIB-2015-Sheet1.pdf10 Feb 2015: a). n=1. nenz on {z. 0 < r Re(z)}; (b) n=1. ... Can such a map ever be a Möbius trans-formation? 9 Calculateγ z sin z dz when γ is the straight line joining 0 to i. -
COMPLEX ANALYSIS EXAMPLES 1, LENT 2022 Neshan Wickramasekera. Please…
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2021-2022/ca_IB_ex1_2022.pdf6 Feb 2022: Regarding T as a map from C into C by identifying R2with C in the usual way, show that there exist unique complex numbers A, B such that for everyz C, ... that there need notexist holomorphic f : U C such that u = Re (f) on all of U. -
Complex Analysis IB: 2015-16 – Sheet 2 Let B(a; ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2015-2016/CxAnalIB-2016-Sheet2.pdf1 Feb 2016: Show that for any b C, there exists a sequence of points zn B(a; r)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2. -
CxAnalIB-2013-Sheet2.dvi
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2012-2013/CxAnalIB-2013-Sheet2.pdf22 Feb 2013: Show that for any b C, there exists a sequence of points zn Ba(r) withzn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2. -
Mich. 2021 ANALYSIS AND TOPOLOGY – EXAMPLES 2 AZ ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2021-2022/sheet2.pdf28 Oct 2021: A = {x M : (xn) in A such that xn x} and A = {x M : r > 0 Dr(x) A}. ... 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 -
Part III Commutative Algebra Example Sheet I, 2021 Note: ...
https://www.dpmms.cam.ac.uk/study/III/CommutativeAlgebra/2021-2022/HW1.pdf14 Oct 2021: Show thatS1M = 0 if and only if there exists an s S such that sM = 0. ... ii) There exists nonzero elements e1,e2 A such that e1e2 = 0, e21 = e1, e22 = e2 and e1 e2 = 1.iii) A is isomorphic to A1 A2 for some rings -
COMPLEX ANALYSIS EXAMPLES 1, LENT 2021 Neshan Wickramasekera. Please…
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2020-2021/ca_IB_ex1_2021.pdf19 Feb 2021: Regarding T as a map from C into C by identifyingR2 with C in the usual way, show that there exist unique complex numbers A, B such thatfor every z C, ... thatthere need not exist holomorphic f : U C such that u = Re (f) on all of U. -
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: Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 1 PAR. 1. Let (zn) be a sequence in R2 such that zn a and zn b. ... On the other hand, show that if fn does not converge uniformly to f thenwe can find a convergent sequence xm x in [a, b] such that fn(xn) 6 -
Complex Analysis IB – 2014 – Sheet 2 Recall ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2013-2014/CxAnalIB-2014-Sheet2.pdf3 Feb 2014: Show that for any b C, there exists a sequence of points zn Ba(r) withzn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2. -
ANALYSIS II—EXAMPLES 3 Mich. 2018 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2018-2019/18sheet3.pdf13 Nov 2018: Supposethat there is a constant K such that ‖F(t,x) F(t,y)‖ K‖xy‖ for all t [0, 1] and all x,y B. ... Letx1,x2 Rn. By the Picard–Lindelöf theorem, we know that there is (0, 1] and differentiable functionsf1,f2 : [0,] B such that dfjdt = F(t,fj -
Part IB COMPLEX ANALYSIS (Lent 2009): Example Sheet 1 ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2008-2009/cxanalysis1.pdf21 Jan 2009: a)1z. (0 < |z| < ); (b) z1 z2. (1 < |z| < ). 18. For each C {0} and w C, such that exp(w) = , find a continuously differentiablecurve γ : [0, 1] C {0} joining ... 1 and , such that. -
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 2 Mich. 2015 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2015-2016/15sheet2.pdf2 Nov 2015: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f
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