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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.

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.

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.

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.

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.].

20 Feb 2009: Show that every such transformationis an isometry for the hyperbolic metric. ... Every such quaternion q can be written asq0 q1i q2j q3k where.

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.

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.

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).

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.

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.

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

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=.

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.

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,

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.

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.

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

22 Oct 2020: Let B(G,F) be the (extended) Bruhat–Tits building of G(F). For any x B(G,F), there is asmooth affine group scheme Gx over OF such that ... a choice of basis b for Λ such that the embeddings of buildings (3.3.1) and (3.3.2) induceembeddings of the

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

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.

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.

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 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

22 Nov 2012: Show that, for such a map,. dimH f (M ) 61α. ... Prove that the limit set for such a group is totallydisconnected.

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.

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

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).

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.

1 Feb 2010: Find a conjugate harmonic functionv(x, y), that is, a function v such that u and v together satisfy the Cauchy-Riemann equations. ... Show that there is an analyticfunction H : C C such that h(z) = exp H(z) for all z.

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.

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)).

21 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.

28 Oct 2022: 1. Let T B(H) with |||T ||| < 1. Show that there is R B(H) such that R2 = 1 T. ... 11. Let T B(B) compact and λ 6 (σp(T) {0}). Show that T λ is bounded below.

9 Feb 2015: That is, for everya < b and every t there should exist x (a,b) such that f(x) = t. ... 14. Let f : R R be a function that has the intermediate value property: that is, iff(a) < c < f(b) then there exists x (a,b) such that f(x) =

12 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

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

18 Sep 2007: a) = g(b), then we can find a c (a, b)such that g′(c) = 0. ... Then there is a unique solution to the differential equationg′(t) = f (t) [t (a, b)] such that g(u) = c.

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.

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,.

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.

6 Feb 2023: a) Construct a continuous surjection p : T2 T2 from the torus to itself such that for every x T2, p1(x)consists of exactly two points, and such that p is ... Such a map is calleda covering map.]. (b) Show that T2 also admits a covering map p̂ : T2 K to

20 Feb 2011: Using partial fractions evaluate the integral. Hence deduce that f (a) = f (b). ... 5. Let f be an entire function such that for some a C and r > 0, f takes no values in the discD(a, r).

8 Aug 2021: b) If yr = 0 whenever ar = 0 and there exists an i such that ai = 0,then there are an infinity of solutions with xs = a. ... m n matrix such thatAx Bx = Cx. If A is an m n matrices and B is an m p matrix, then AB = Cwhere C is the n p matrix such

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).

5 Oct 2022: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

5 Oct 2004: pn log p B′n.(vi)There exists a constant B such that π(n) Bn(log n)1. ... ii) Find A and B such that g(s) = Aζ(s)B2sζ(s) for all <s > 1.

15 Oct 2021: Incomponents we write B. j,γ,δ= ((bklj,γ,δ)) and c. γ,δ = (c. ... j, c such that the corresponding. series for u converges near x = 0.

24 Jul 2012: Theorem 137 (Chebychev) There exist constants A and B with 0 < A B such that. ... if and only if there exist constants A′ and B′ with 0 < A B such thatA′n β(n) B′n.

13 Nov 2011: Explain geometrically why this is thecase. 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 ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that f/x exists and is