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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.

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.

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