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29 Jan 2010: Therefore there are constants a, b, c k such that if v, v′ Spand w IndGB δ1/2, then. ... π′′ = I(1)0, then we can choose w such that HomGw (πw.

29 May 2010: Let Γ Aut(X′) be a finite subgroup such that f factorsas X′ X′/Γ g X, where g is a radicial alteration. ... Then one knows that there exists a finite extension K′/K such that ρ(IK′) isunipotent.

23 Mar 2007: Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic ... Chung, Graham, Frankl and Shearer have proved that, in the case where B is a block of length t, we can do

12 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

16 Jul 2014: Mark Gross Mirror symmetry and the SYZ conjecture. Definition. A tropical affine manifold with singularities is a real (C 0) manifoldB along with an open subset B0 B such that := B ... For each integral point v of a two-face, we can choose anopen

9 Jun 2003: To give a simple example: if I take two subsets A and B, I can define their union to be the set of all n such that n belongs either to ... a. 1. ,b. 1. ] is any closed interval such that a.

23 May 2001: then construct a new sequence v in such a way that the first two terms of v mean that it is not a multiple of v. ... If the vector space V is infinite-dimensional, then this means that V contains a subset B such that every vector v in V is a linear

5 Nov 2003: AxB, the Cartesian product of A and B, is defined to be the set of all ordered pairs (x,y) such that x is an element of A and y is ... possible such that r is non-negative) with the same highest common factor.

10 Mar 2017: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... 6. Show that every graph G has a partition of its vertex-set as X Y such that the numberof edges from X to Y is at least 12e(G).

18 Feb 2015: ri 1 such thati ri = r. For example, in the quaternionic case H = B the plectic reflex Galois group isisomorphic to (St n ΓtF ) (Srt n Γ. ... The cup productof two such classes is an element g1 g2 H2(Y, 2).

29 Jan 2010: W2 E = G. Q(1). and GrWj E = 0 for j > 0 or j < 2, such that. ... The reader who is sceptical of the general finiteness of groups such as IIIwill be reassured that we do not require equality in (b).).

23 May 2024: Tho15, Lemma 2.8]):. Lemma 2.11. There exists a non-zero N0 OF such that for all primes p and for all b B(Op) wehave N0 κb V (Op). ... There exists b′ B(Op) such that ordp (b tb′) = 1,where t is a uniformiser of Op.

29 Jan 2021: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

22 Oct 2006: iii) For any two subsets A, B V (G) the number of pairs (x, y) A B such thatxy E(G) differs from p|A||B| by at most c2n2. ... number of pairs (x, y) A B such.

23 Nov 2016: Mich. 2016 NUMBERS AND SETS—EXAMPLES 4 PAR. 1. The symmetric difference A4B of two sets A and B is defined to be the set of elements x such that xis ... 11. Let S be a collection of subsets of N such that for all A, B S we have A B or B A.

14 Oct 2016: c) Let x be a non-zero rational number. Show that there is a rational number y such thatxy = 1. ... d) Show that if a and b are natural numbers such that a is a factor of b then a 6 b.

7 Sep 2017: On the other hand, as the signature andEuler characteristic agree modulo 2, and the latter is multiplicative, it is true thatσ(E) σ(B) σ(F) mod 2 for all such ... More recently, Hambleton,Korzeniewski, and Ranicki [5, Theorem A] proved that σ(E) σ(B)

13 Oct 2006: b) Suppose that f is bounded. Prove that there exists M and n0 such that |fn(x)| < M for all. ... a,b] such that fn(xn) does not converge to f(x). 7. Which of the following functions f are (a) uniformly continuous, (b) bounded on [0,)?

11 Jan 2024: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

17 Feb 2010: B(µp,νp) spanned by tensors pφ̂p such that. φ̂p belongs to Bs(µp,νp) for at least one p, and this is the unique invariant subspace of B(µ,ν) ... Then C is finite.Proof. Choose a prime p 7 such that GZp K.