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22 Oct 2016: 6. Let X, Y be manifolds. A submersion is defined to be a smooth map f : X Y such thatdfx is surjective for all x X. ... zero outside the ball of radius b > a, and such that f(Rn) [0, 1].These functions are very useful for gluings.

3 Nov 2020: 2. Given (fi)iI an arbitrary collection of continuous functions [0, 1] R such thatsupiI |fi(x)| < at each x [0, 1], show that there is an interval [a,b] [0, ... 1] witha < b such that supx[a,b] supiI |fi(x)| <. 3. Let X be a closed subspace of 1. Assume

23 Oct 2020: 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) 6 c log b d.

25 Nov 2021: Prove that A. contains an infinite set B such that no point of B is a convex combination of other points of B.

1 May 2019: By definition, there exists some Zariskiopen affine subscheme Ux = Spec B(x) X containing x such that x1,. ... skd.Fix k such that x Spec Afk. Then B(x)fk A 1(X) is freely generated bysk1,.

23 Jan 2023: Suppose there exists a class [b] B(G,{µ}) such that ν[b] = µ(such a class is necessarily unique if it exists by the above remark). ... When G is quasi-split,this class is just [b]µ. However if G is not quasi-split, there may be no [b] B(G,{µ})such

12 Aug 2008: 5. Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linear functionalθ V such that θ(x) 6= θ(y). ... fn for V such that when x =. m1xiei and y =. n1yjfj we. have ψ(x, y) =r. 1xkyk, where r is the rank of ψ.

4 Aug 2009: Theorem 5.12. If f : [a,b] R is continuous then we can find x1,x2 [a,b]such that. ... If f : [a,b] R is a continuousfunction with f differentiable on (a,b), then we can find a c (a,b) such that.

11 Oct 2017: ii) Let α be such that |K(α) : K| is odd. ... and let β = α4.Find. rationals a, b, c such that β = a bα cα2.

27 Nov 2013: 6. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a. ... 9. (i) Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linearfunctional θ V such that θ(x) 6=

15 Oct 2021: Proof. Since is open, there exists ,x such that the ball B(x) = {y Rn : |y x| < }is contained in. ... Chapter 1 Test functions 3. a) Show that χ C(R). b) Show that there exists a function ψ C0 (Rn) such that.

27 Jan 2010: Use a suitableLagrange resolvent to find a Q(µ3) such that K = Q(ζ3, 3. ... Use (ii) to show that there exists b K such thatdet(σiσjQ1(b)) ̸= 0.

3 Nov 2013: Find a, b, c Q such thaty = a bx cx2. ... 15. Let K and L be subfields of a field M such that M/K is finite.

31 Jan 2024: G = H1 BH2 BH3 BH4 B BHn = {e}. such that each quotient Hi1/Hi is simple. ... Since they have the same cycle type, there is a σ Sn such that (a b c) = σ(1 2 3)σ1.

6 Mar 2008: vertices into sets B andR such that the only edges lie between vertices in different sets.(i) Show that if any finite subgraph of a graph is bipartite, then so is ... Suppose that any such formula is equivalent modulo T to aquantifier-free formula.

27 Nov 2019: Let a, b K such that thepolynomials f = xm a, g = xm b are irreducible. ... Show that G is solvable if and only if there is an i such that.

23 Feb 2009: A graph is bipartite or 2-colourable if we can partition the set of vertices into sets B andR such that the only edges lie between vertices in different sets.(i) ... Suppose that any such formula is equivalent modulo T to aquantifier-free formula.

9 Oct 2018: Show that f isbounded. 11. Show that there is an infinite set S of positive integers such that the sum of any two distinct elements. ... b) a collection of 100 horizontal lines and 100 vertical lines such that the 10,000 points of intersection are.

25 Nov 2023: Let θ : A B be a -homomorphism from a Banach -algebra A to a C-algebraB. ... H) such thatz(T) = T, where z(λ) = λ for all λ K.

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.

20 Oct 2014: such as this. (1) The condition about the number of quadruples a b = c d can be interpreted as. ... Then there exist sets A X and B Y of density α and β such thatη(A,B) αβδ(1 c).

15 Oct 2021: a b = b a, a b = b a. iv) There exist unique, distinct, additive and multiplicative identity elements: there exist0 Φ and 1 Φ with 0 6= 1 such that for ... By the definition of τ,there exists B such that x B W and B has the form.

29 Feb 2016: 8. By a selection function for a set a, we mean a function s: Pa Pa such that s(b) b for allb a, both inclusions being strict unless b is ... First show that there exists β < α such that |f(β) f(γ)| < 1 for allγ > β.].

5 Jun 2020: According to Occam’s razor, the simplest such explanation isthe one that should be favored. ... Given a string x, suppose wefocus on a special sub-class of programs p that produce x, namely, programs thatfirst describe a set B of strings such that x B,

16 Oct 2019: 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) 6 c log b d.

17 Oct 2016: ii) Let α be such that |K(α) : K| is odd. ... and let β = α4.Find. rationals a, b, c such that β = a bα cα2.

4 Oct 2022: b) Now let X and Y be jointly Gaussian random vectors such that Cov(X) is positivedefinite. ... Let Qbe the equivalent probability measure such that dQ. dP eX.

5 Jun 2020: Theorem Almost sure invariance principle Let X be a nitevalued stationary process such. ... fB t t g such that for any. R t B t O t as.

5 Jun 2020: N = M2 pY11 pY22 pYπ(n)π(n) , (6). where M 1 is the largest integer such that M2 divides N, and theYi are now binary. ... Actually Chebyshev came pretty close to proving the PNT. Forexample, using Theorem 1 in a slightly refined form, he was ableto

4 Oct 2022: n such that Ax = 0.• There exists a y Rm with (AT y)i 0 for all i = 1,. , ... Make this assumption for the rest of the problem.(b) Why must there exist an equivalent measure Q such that 1.

7 Aug 2015: B/A. such that for any Banach B-module M there is a natural isomorphism. ... Then thereis a unique A-linear map. ψ : DerK(A) DerK(B). such that ψ(u) ϕ = ϕu for each u DerK(A).

8 Jan 2008: v) If ‖a b‖1 = ‖a‖1 ‖b‖1, does it follow that we can find α, β F,not both zero, such that βaj = αbj for all 1 j n? ... Give reasons. (vi) If ‖ab‖ = ‖a‖ ‖b‖, does it follow that we can find α, β F,not both zero, such that

27 Oct 2019: 4) (a b) (cd) = (a c) (bd),. and such that each has a two-sided unit (1 and 1 respectively). ... x0,x1) sα0 (x0)holds for all pairs (α0,α1) of such α.

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.

31 Jan 2013: 16. Find X R2 such that (a) any two points x,y X can be joined by a continuouspath γ : [0, 1] X and (b) for x 6= y the length ... of any such path is infinite.

21 May 2005: for some x,y Z with (x,y) = 1, then there exist a,b Z such that p = a2 b2. ... αn be algebraicintegers in K such that the discriminant D(α1,. ,αn) is a square-free non-zerointeger.

16 Nov 2004: Corollary 39. If f and g are differentiable functions (a,b) R withf ′(t) = g′(t) for all a < t < b then we can find a constant c such thatf(t) = ... with γ(a1) = a2 and γ(b1) = b2 such that r1(t) =r2(γ(t)).

12 Mar 2013: Show that I T hasa square root, i.e., there exists S B(X) such that S2 = I T. ... invertible operator T : X X such that Txn = yn for all n N.

4 Feb 2008: Let P be the set of all p Q such that 2 < p2 < 3. ... c) If A and B are closed subsets of R such that 1 A, 1 B and [1, 1] A B, thenA B 6=.

22 Oct 2001: 3. =. (π. p4. )3. 6= 1. Proof. Let B {1, 4, 7} such that{p1p2B 6 1 (mod 9)p3p4B 6 1 (mod 9). ... Take a prime π Z[ω] such that N(π) B (mod 9).

15 Oct 2021: b) Let f L2(U). For each γ > µ, denote by uγ the unique solution of (†). i) Show that for any γ > µ there exists wγ H10 (U) such that:. ... b) Show that there exists a constant C > 0, depending only on n, such that:.

9 Jan 2015: The measure ν can be written explicitly as follows. There is an integer b 1 such that ν(f) =S1fF dµ, where F(z) = 1 2. ... Let b denote the smallest value of n such that F.

19 May 2003: matrix P and a non singularn n matrix Q such that B = QAP. ... Two n nmatrices A and B can represent the same endomorphism of V (with respectto appropriately chosen bases) if and only if there exists a non singular nnmatrix P such that

22 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

14 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

29 Nov 2023: 10. )〉. So. R2 =〈(. 10. )〉〈(. 01. )〉is the only such decomposition in this case. ... Ws such that W = sj=1Wj. Thus. V =ri=1. Ui sj=1.

15 Oct 2020: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... Can such a group beinnite? 8. Let G be a group of even order.

15 Oct 2021: Let a, b K such that thepolynomials f = xm a, g = xm b are irreducible. ... Show that G is solvable if and only if there is an i such that.

5 Jun 2020: with ψ( ) := R(x0, ): For all x X and all A B such thatR(x0,A) > 0, we have,. ... constants δ > 0, b < , such that,.

1 Feb 2022: b) Find a condition on s such that x 2 Hs(Rn). ... 2u u = f. b) Show further that there exists C > 0 such that kukHs4.