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4 Dec 2014: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... 7. Let S be an nn real symmetric matrix with Sk = I for some k 1.

24 Jan 2014: Xn be independent Poisson random variables, with Xi having meaniθ, for some θ > 0. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).

18 Sep 2014: components. Consider a map. : 1()! Z=pZ. for some prime p that maps all the boundary. ... If G F and H G then H is. measure-equivalent to some free group.

11 May 2014: AB| = |A| |B| 1. This looks fair enough, and in some sense it is, but we need to be clear aboutthe conventions we are using. ... All this is well and good, but it can cause some confusionwhen we start thinking about expressions like: |AB|.

16 Jul 2014: that the transition functions ψi ψ1j are in factaffine linear transformations v 7 Tv v0 for some T GL(Λ) andv0 V. ... This conjecture remains unproven other than some verystraightforward cases (abelian varieties, K3 surfaces).However, it has led to a

18 Jul 2014: Two. models Φ and Φ′ are R-equivalent if Φ′ = [A,B]Φ for some A,B GL5(R). ... If Φ′ = [A,B]Φ for some A,B GL5(K) then A,B K GL5(OK).In particular.

17 Mar 2014: There is some overlap between these examples andthe examples in my lecture notes. ... 6. This example arose from some psychiatry data provided by Professor I.Goodyerof Cambridge University.

10 Feb 2014: 3. An element r of a ring R is nilpotent if rn = 0 for some n. ... 6. Let A and B be elements of the power-set ring P(S), for some set S.

16 Apr 2014: Since B() is a local ring, it follows that. B() = (̂̃Ai,mi,k)() for some pair (i,k) which is uniquely determined by , and then = ri,k. ... GB ιχv,1 ιχv,n, for some smooth. characters χv,i : Fv Q.

24 Feb 2014: domain. Could there be some other Euclidean function φ making Z[3] into a Euclidean. ... Does it follow that the ring R is either a field or a polynomial ringF[X] for some field F?

18 Sep 2014: The first claim to be made is that, for some δ 0 depending on S, for everybase b, b S has length at least δ. ... 3. The toral case. Some base b is a carrier and is carried infinitely often,and has length that does not converge to 0.

15 Apr 2014: 11) σ((a,b,c)T ) = λσρ(ξσ)(a,b,c)T. for some λσ K. Now ρ(ξσ) G, whereas c4 and c6 are homogeneous polyno-mials of degrees 14 ... First we compute some covariant columns. Let x1 = (a,b,c,d,e)T. If γ.

15 Jan 2014: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... b) Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible.

24 Jan 2014: 5. Let f : B B be a continuous function from the open disc B = {(x,y) R2 :x2 y2 < 1} to itself. ... 11. Suppose f : R R has the intermediate value property that if a,b R andf(a) < c < f(b), then f(y) = c for some y between a and b.

20 Oct 2014: 4. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and. ... a) If Dn is the set of discontinuities of fn and D is the set of discontinuities of f, show that D n=1j=nDj.(b) Suppose that for some finite

13 Oct 2014: 5.1. Observations from obtained data. — We record here some data about Σ5from our experiments. ... 19)Some of this is implementation-dependent; the discriminant of an auxiliary conic has a factor.

18 Sep 2014: 1 Some basics of topological and geometric grouptheory. 1.1 Presentations and complexes. ... All the γn arecontained in some compact ball B of radius R, and so d(γn(s),γn(t)) < 2R|st|for all s,t.

7 Feb 2014: a) g(z) = z for some z S1.(b) g(z) = z for some z S1.

14 Oct 2014: 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. ... How large can theirproduct be? 6. Prove that A (B C) = (AB) (AC).

24 Oct 2014: 2. Find the convergents to the fraction 5744. Prove that if x and y are integers such that 57x 44y = 1, then x = 1744k and y = 57k 22 for some k ... Z. 3. Let a, b, c N. Must the numbers (a, b)(c, d) and (ac, bd) be equal?

4 Mar 2014: 71). Find an element of OK with norm 2a 3b 5 for some. ... Hint: Let x1,. ,xm generate OL as an OK -module. If pOL = OLthen we can write xi =. aijxj for some aij p.] Deduce that if a and b are ideals.

21 Nov 2014: Eo = {x X : B(x) E for some > 0}. ... and that. E = {x X : xn x for some sequence (xn) in E}.

26 Nov 2014: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... 7. Let S be an nn real symmetric matrix with Sk = I for some k 1.

21 Jan 2014: average)of the orbit of the origin under G, show that G fixes some point of Rn. ... Show that γ′′(s) = κ(s)n(s) for some κ(s) : [a,b] R.

26 Nov 2014: Deduce that every finite abelian groupis the Galois group of some Galois extension Q L. ... 2) Let K be a field containing an n-th primitive root of unity for some n > 1.Let a,b K such that the polynomials f(t) = tn a and

17 Feb 2014: iv) Identify the quotient space H2/〈A,B〉. [If you know some algebraictopology, prove directly that the quotient space has fundamental group afree group of rank 2.] If G′ is the ... Deduce that traces of all words in A and B and theirinverses (i.e.

19 Dec 2014: by some real r ;(b) 20 < ω. ... Let a = min{| A |: A is an infinite maximalalmost disjoint family of subsets of ω}, b = min{| B |: B is unbounded in (ωω,)},and d = min{| D |: D is dominating in (

19 Dec 2014: P(I) : Y X} for some Y P(I) ; otherwise F isnon-principal. ... Prove thatthe following are equivalent:. 5. (a) h is one-to-one on some club C of δ ;(b) h is strictly increasing on some club D of δ ;(c)

19 Dec 2014: Note that g M and P(P 2) hascardinality β for some β < κ (by strong inaccessibility). ... Say that A and B arepartially isomorphic, denoted A 'p B if some non-empty family F Part(A, B) of thepartial isomorphisms from A to B is a

20 Oct 2014: of writing some number in Zp as a sum of two elements a,b of A, taking their order intoaccount, and that is clearly the number of ways of choosing a ... This is a problem for. some applications, but often the sets A and B that we are interested in have a