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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.

24 Nov 2023: 45) Show that there is a w such that Ww = {w} and a w such that |Ww| = |w|. ... Show that the following sets arecomputably enumerable:. (a) {w ; there are three distinct words v such that f(w, v)};(b) {w ; there is a word v of even length such

9 Nov 2020: Now suppose b G(L) is such that νb X(T) is dominant, hence defined over Zp. ... Itsuffices to prove that also descends to R(a,b). The coordinates ti induce a Frobenius-equivariant map RG R(a,b), such that the compositeR(a,b) RG

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).

8 Mar 2024: αd OK such that. OK = α1Z. αdZ.Such a collection of elements α1,. , ... 14 PÉTER P. VARJÚ. of this form. So no such π1 may exist.

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).).

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.

25 Aug 2021: Moreover, forthe given b there exists µ1 X(T) such that |Jb(F)Σtop(Xµ1 (b))| = 1. ... It is a reductivegroup over F such that. Jb(R) = {g G(F̆ F R) | g1bσ(g) = b}.

11 Nov 2020: We now let µ X(T) and let b G(L) such that [b] B(G,µ). ... Reversing the above process we obtain Z′ Σtop(Xw′(b′))) such that StabU′(J′) = StabZ′(J′).

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.

13 Dec 2017: Show that E|B0 'E|B1 as follows. (a) For each b B show there is an open neighborhood Vb of b such that E|VbI istrivial. ... b) Choose a finite cover V = V1,. ,Vm of B such that E|VbI is trivial.

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)

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.

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

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).

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.

17 Apr 2017: If A, B, U are subspaces of a vector space V such that U = (A U) (B U)then (A U) (B U) = (AB) U. ... 19. (2.4.11) The main inductive step is the following: let n = a b, and V = WnV be anobject of Crk such that gr.

25 Jun 2015: Foreach open set A Ψ(U )cs and each M A, we shall show that there is a smalleropen neighbourhood M B A Ψ(U )cs such that. ... Then we may choose smooth functions ϕi : M [0, 1] such that supp(ϕi) int(Vi) and p(ν K) (. ϕi)1(1) and choose B =. B(A, ϕ1

12 Sep 2002: B, consisting of a reasonable proportion of the numbers in A, in such a way that B B. ... larger than A. Will we always be able to pick out a reasonably large set B in such a way.

22 Oct 2020: We write w ̃σw′ if there exists τ ̆ such that w σ τw′σ(τ)1. ... scheme over Fq, i.e. a scheme over Fq such that the absolute Frobenius is an isomorphism.

24 Nov 2003: Deducethat there exists a pair of irrational numbers a, b such that ab is rational. ... 3. 2 100. 3. 2 is irrational. 19. Construct a function f : R R that takes every value on every interval - in otherwords, such that for every a < b and every

8 Jan 2020: iii) Let w = a1b2a2b1a F. Using the method of Stallings folding, orotherwise, produce an explicit homomorphism f : F Q to a finite groupQ such that f(b) / f(〈w〉). ... b) For a commutative ring R with unity, define. T(R) =. {(a b0 d. ) SL2(R). }. (i)

22 Oct 2020: The maps h and h′ then induce a map k : A′ B such that kj′′0 = f0 andkj′′1 = f2. ... By a correspondence between h and k we mean amap H : A I B such that Hj0 = k,Hj1 = sg,i0H = h and i1H = gσ.

19 Jan 2021: for every x X there exists some B B such that x B; and. ... Leta,b G. Suppose that X has non-empty disjoint subsets U and V such that:.

23 Mar 2009: d : R{0} N. such that for all a R and b 6= 0, b R we can write. ... e f2 Q. [2. ]. Pick g, h Z such that |e g|, |f h| 12 and set.

11 Nov 2004: 2. Let x1,. , xn be real numbers such that. ni=1 xi = 0 and. ... Prove that there are constants a and b such thatP1(x) = P2(ax b) for every x.7.

19 Dec 2022: Of course, Corollary 2 is only useful if we can find such an algebra homomorphism. ... Since our proof does not rely on the details of the construction of ks, it can readily accommodatemodifications such as bulk deformations and B-fields, which we

29 Sep 2003: 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

22 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

19 Jan 2021: Show that φis isotopic to the identity. 8. Let α,β be a pair of simple closed curves on a surface S, such thati(α,β) = 1. ... Let α,β be astandard pair of simple closed curves on S such that i(α,β) = 1, andlet γ be the boundary curve.

23 Jan 2019: But for such a fibre bundle we have. B. f(ch4kd(ξH) σ4kd) = 0,. ... Suppose that the fibration is in addition equipped with a map τ : E BOand a section s : B E, such that τ s : B BO is the constant map to thebasepoint.

5 Feb 2017: 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

25 Oct 2023: Then there is a point y / E such. that B(y,r) E 6= whenever r > 0. ... Since Uj1 isopen we can find a δj1 > 0 with δj1 δj/4 such that B(xj1,δj1) Uj1.

19 Jan 2021: i) Prove that fm is a group homomorphism. (ii) Let a,b Z be such that ab 1 modulo m. ... Deduce thatthere exists r > 0 such that. {n such that p | 2n 1} = rZ.

5 Feb 2019: 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.

19 Oct 2004: Suppose p is anattracting fixed point for f. Show that there is a neighbourhood U of p such that the forward orbitof every point in U converges to p. ... 11. The Haar measure of a compact metrisable topological group G is the unique probabilitymeasure µ

21 Apr 2013: Still assume that R Zp. There is a unique γp 1 pZp such thatγmp = δ. ... a(n)qn/m =nZ. b(n)tn, b(n) o. Let H =rj=0 AjT. j o[T] such that the image of f in DR(Γ,o,k) is

5 Dec 2019: compact. 5. Are the invertible operators dense in B(l2)? 6. Construct a self-adjoint operator T on a (non-zero) Hilbert space H such that T hasno eigenvalues. ... 7. Construct a compact operator T on a (non-zero) Hilbert space H such that T has

14 Feb 2017: Λ > 0 there is a constant C thatdepends only on Λ such that if n 5 and if M is a stable minimal hypersurface of B with 0 Mand Hn(M) ... Such an estimate cannot hold for stablehypersurfaces when n 7, nor can it hold without stability even when n = 2).

18 Feb 2021: criterion. Let F : xCO Ñ Sets be a continuous functor such that Fpkq is a singleton.For A, B, C P xCO and maps α : A Ñ B and β : B ... It will be enough to find a P kerpGLnpB Iq Ñ GLnpkqq such thataρpΓqa1 GLnpBq, so we can assume that B I A.Now

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,)?

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.