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ANALYSIS I EXAMPLES 2 G.P. Paternain Lent 2010 Comments ...
https://www.dpmms.cam.ac.uk/~gpp24/aI2.pdf29 Jan 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. -
ANALYSIS I EXAMPLES 2 G.P. Paternain Lent 2010 Comments ...
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. -
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 -
ALGEBRAIC TOPOLOGY LENT 2010 EXAMPLE SHEET 1 1. Let ...
www.dpmms.cam.ac.uk/study/II/AlgebraicTopology/2009-2010/Examples1.pdf19 Jan 2010: Show thatf(x) = x for some x S1. 3. Which of the letters A,B,. ... 1,1) π1(S1,f(1)) when. (a) f(eiθ) = ei(θπ/2). (b) f(eiθ) = einθ for some n Z. -
STATISTICS Part IB Example Sheet 1 (of 3) RJS/Lent ...
www.dpmms.cam.ac.uk/study/IB/Statistics/2009-2010/Example1.pdf24 Jan 2010: 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. -
EXAMPLE SHEET 3 (LECTURES 13–18) GALOIS THEORY MICHAELMAS 2009 ...
www.dpmms.cam.ac.uk/study/II/Galois/2009-2010/ex3.pdf28 Mar 2010: Deduce that every finiteabelian group is the Galois group of some Galois extension K/Q. ... only if b = cnar for some c Kand r N with gcd(r, n) = 1. -
Part II Riemann Surfaces, Michaelmas 2010 pmhw@dpmms.cam.ac.uk…
www.dpmms.cam.ac.uk/study/II/Riemann/2010-2011/RS10.2.pdf5 Nov 2010: 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 if andonly if τ2 = (aτ1 b)/(cτ1 d), for some matrix. -
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 -
Part II Riemann Surfaces, Michaelmas 2010 pmhw@dpmms.cam.ac.uk…
www.dpmms.cam.ac.uk/study/II/Riemann/2010-2011/RS10.1.pdf13 Oct 2010: 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.Show that the quotient space R = C/ has the natural structure of -
GEOMETRY AND GROUPS – Example Sheet 2TKC Lent 2010 ...
www.dpmms.cam.ac.uk/study/II/Geometry%2BGroups/2009-2010/Exercise2.pdf4 Feb 2010: 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? -
Numbers and Sets (2010–11) Example Sheet 1 of 4 ...
www.dpmms.cam.ac.uk/study/IA/Numbers%2BSets/2010-2011/examples-NS-10-1.pdf19 Oct 2010: about. (ii) m n a b (n m) [(a = 1) (b = 1) (ab 6= n)] ,. 5. The sum of some (not necessarily distinct) natural numbers is 100. ... 10. Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5}? -
GEOMETRY AND GROUPS – Example Sheet 1TKC Lent 2010 ...
www.dpmms.cam.ac.uk/study/II/Geometry%2BGroups/2009-2010/Exercise1.pdf4 Feb 2010: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. v Z Z can be written as mu nv for some integers m and n. -
ANALYSIS II (Michaelmas 2010): EXAMPLES 3 The questions are ...
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). ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, show that there is an open ball B(I; -
Example Sheet 1. Lectures 1–6, Galois Theory Michaelmas 2010 ...
www.dpmms.cam.ac.uk/study/II/Galois/2010-2011/2010_Galois_Ex1.pdf20 Oct 2010: Show that if the characteristic of K is not2, then F = K(x) = {a bx | a, b K} for some x F with x2 K. ... ii) Show that if F/K is a finite extension of infinite fields for which there exist onlyfinitely many intermediate subfields K K′ F, then F = K(x) -
10-11sheet4.dvi
www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2010-2011/10-11sheet4.pdf29 Nov 2010: limz. znf (z) = c,. prove that f (z0) = 0 for some z0 C. ... a) g(z) = z for some z S1. (b) g(z) = z for some z S1. -
Example Sheet 3. Lectures 13–18, Galois Theory Michaelmas 2010 ...
www.dpmms.cam.ac.uk/study/II/Galois/2010-2011/2010_Galois_Ex3.pdf9 Nov 2010: 3.16. Let K be a field containing a primitive n-th root of unity for some n > 1. ... Showthat P and Q have the same splitting field if and only if b = cnar for some c K and r Nwith gcd(r, n) = 1. -
10-11sheet1.dvi
www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2010-2011/10-11sheet1.pdf19 Oct 2010: Hint: argue by contradiction.). (6) Let f : B B be a continuous map from the open disc B = {(x, y) R2 : x2 y2 < 1} into it-self. ... 1. (11) Suppose f : R R has the intermediate value property that if a, b R, f (a) < c < f (b),then f (y) = c for some y -
Lent Term 2010 R. Camina IB Groups, Rings and ...
www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2009-2010/ex10-3.pdf23 Feb 2010: Could there be some. other Euclidean function φ making Z[3] into a Euclidean domain? ... of elements a and b exists and can be writtenas d = ra sb for some r,s R. [ -
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′ -
Michaelmas Term 2010 T.A. Fisher Linear Algebra: Example Sheet ...
www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2010-2011/lin_alg-10-4.pdf26 Nov 2010: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... 7. Let S be an n n real symmetric matrix with Sk = I for some k 1. -
ANALYSIS II (Michaelmas 2010): EXAMPLES 4 The questions are ...
www.dpmms.cam.ac.uk/study/IB/AnalysisII/2010-2011/anII_ex_2010_4.pdf29 Nov 2010: Find this solution to some reasonable accuracy using a pocket calculator or thecalculator on your computer (remember to work in radians!), and justify the claimed accuracy ofyour approximation. ... norm. Show, however, that T is a contraction under the -
ON MODULAR UNITS A. J. Scholl Introduction. In [6], ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/u.pdf29 Jan 2010: It isthen clear that for some non-zero invariant v Vq the mapping Φq,1 is given by. ... may be represented by some k GẐ, and then. σ : ZQDf Nf k 7 ZQDẐNf k. -
Extensions of motives and higher Chow groups A. J. ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/chow.pdf29 Jan 2010: The second is to. construct some extensions of motives related to elements of higher Chow groups and to. ... as in 1 is probably known to some experts (see for example the penultimate paragraph of. -
SOME BOUNDS ON THE COEFFICIENTSOF COVERING CURVES T.A. FISHER ...
https://www.dpmms.cam.ac.uk/~taf1000/papers/ncovbds.pdf29 Jul 2010: For this weneed lower bounds on some of the ||vi||. The hypotheses in parts (a), (b)and (c) of Theorem 1.2 are used to give successively better lower bounds,and ... F(x)| max(|a|, |b|, |c|) ||x||3 < 1. Since F(x) = f(u,v,w) for some u,v,w Z it follows -
Transitive Sets in Euclidean Ramsey Theory Imre Leader∗† Paul ...
https://www.dpmms.cam.ac.uk/~par31/preprints/ert.pdf22 Nov 2010: It follows that A and B lie on some regular polygon Π (withcentre O). ... In the case of agroup G which acts as the symmetry group of some transitive set of order |G|,Conjecture C may be deduced easily from Conjecture B. -
Integral elements in K-theory and products ofmodular curves A ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/k1.pdf29 Jan 2010: Proof. We can assume that c = [Z′/k] for some integral closed subscheme Z′/k. ... 2.1.1. We recall some basic facts about modular curves (see [4] or [9]). -
BEILINSON’S THEOREM ON MODULAR CURVES. Norbert Schappacher and…
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/RSS.pdf17 Feb 2010: ii) For some K, some ω V Kπ , and some u, v O(MK) Z Q,MK (C). ... For some positive integer N, the correspondence NXp(Tp) annihilatesPic0(MK), and since p 1 > 2. -
LOCAL SOLUBILITY AND HEIGHT BOUNDS FOR COVERINGS OF ELLIPTIC ...
https://www.dpmms.cam.ac.uk/~taf1000/papers/htbounds.pdf12 Oct 2010: for some λ,α,β,γ,δ Z[n11,n12,. ,n33,a,b,c,. ,m]. Moreover if N is diagonal. ... for some a,b OK then substitute x xay bz and go to Step (ii).(iv) Return α. -
THE HESSIAN OF A GENUS ONE CURVE TOM FISHER ...
https://www.dpmms.cam.ac.uk/~taf1000/papers/g1hessians.pdf25 Nov 2010: 1 σan τcn and g τn g1 σbn τdn for some a,b,c,d Z/nZ. ... for some A GLm. Putting g = [A,B] wehave φ′′ = gφ. -
Symmetric sets and graph models of set and multiset ...
https://www.dpmms.cam.ac.uk/~tef10/dangthesis.pdf23 Dec 2010: Byuniqueness, we can call x = t(R). Lemma 20. Any x V is t(R) for some R B. ... B,b] to an SPG [Q,q]. 28. By Lemma 23, there is a bijection θ : Dom Q ι5“H for some H M. -
PII: 0003-4843(79)90006-8
https://www.dpmms.cam.ac.uk/~jmeh1/Research/Pub71-80/filter79.pdf17 Dec 2010: But by (b), this means that some finite intersectioc oi’ elements of CD is empty, a contradiction. ... But then the W,, defined has the further property that if W, $ W,, for some B’, then W,.$ W,,,. -
PII: 0003-4843(79)90006-8
https://www.dpmms.cam.ac.uk/~jmeh1/Research/Oldpapers/filter79.pdf17 Dec 2010: But by (b), this means that some finite intersectioc oi’ elements of CD is empty, a contradiction. ... But then the W,, defined has the further property that if W, $ W,, for some B’, then W,.$ W,,,. -
An introduction to Kato's Euler systemsA. J. Scholl to ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/euler.pdf29 Jan 2010: 1 Review of modular forms and elliptic curvesWe review some well-known facts about the moduli of elliptic curves. ... Therefore S = YH := Y (N)=H for some subgroupH GL2(Z=N), E = Euniv f! -
Height Bounds for n-Coverings Graham Sills∗ Trinity College,…
https://www.dpmms.cam.ac.uk/~taf1000/sills-thesis.pdf19 Oct 2010: We give an algorithm which hasbeen implemented using MAGMA and also give some examples. ... α2 = g(s, t),. 31. for some α Q and binary quartic g. -
2009 lectures on Set Theory Thomas Forster April 3, ...
https://www.dpmms.cam.ac.uk/~tef10/cam_only/partiii2009.pdf15 Nov 2010: 91.2 Do some ordinals here. 111.3 Some recursive function theory. 11. ... Sacks is right—omitting types is hard!First some definitions. DEFINITION 22. 1.
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