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10 Jan 2011: b) Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible. ... Determine all finite groups which have a faithful 2-dimensional representation over R.

3 Nov 2013: 17. Let L/K be an extension, and x,y L transcendental over K. ... Show that x is algebraicover K(y) iff y is algebraic over K(x).

21 Nov 2016: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

24 Nov 2010: Suppose that the Galois group ofP over Q doesn’t contain an n-cycle. ... 4.11. Compute the Galois group of X5 2 over Q. 4.12.

2 Feb 2008: We factor A and B over K as. A = (x1 α1x2 β1x3 γ1x4)(x1 α3x2 β3x3 γ3x4)B = (x1 α2x2 β2x3 γ2x4)(x1 α4x2 β4x3 γ4x4). ... Let S = (λ1A µ1B,λ2A µ2B) with λi,µi L′. Then κ L, whereas if Aand B are not dened over L then Gal(L′/L)

15 Oct 2021: a a1 = 1. vi) The multiplication operation is distributive over addition: the following identity holdsfor all a,b Φ:. ... Definition B.3. Suppose X is a vector space over Φ, where Φ is either R or C.

29 Jan 2024: But what about the engine that reveals mathematics to us over the ages? ... In this connection consider the language over the alphabet {a,b} whereno block of k (k fixed in advance) consecutive characters contains three evenlyspaced ‘a’s or three

12 Sep 2013: We note that in the case H = Zp an immediate consequence of Theorem B isthe well-known algebraic independence of the logarithmic series log(1 T) over theIwasawa algebra OK[[T ]], ... note that because these groups are defined over OL we can findL-uniform

12 Apr 2005: Summing over all vertices of U we obtain the result. The next step in the chain of reasoning is the following innocent-looking statement. ... the following idea. If f is a function of three variables x, y and z that range over sets X,.

23 Nov 2015: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

3 Jun 2007: c) Zω1 Zω2 = {nω1 mω2 : n, m Z} for some ω1, ω2 C which are linearly independent over R. ... Let this be ω2. Suppose that ω1, ω2 were not linearly independent over R.

22 Oct 2020: We can alsoconsider the corresponding objects over L. Then for x B(G,L), we have Gx(OL) = Gx(OL)ker κ̃Gwhere. ... If G isan unramified group (i.e. G is quasi-split and split over an unramified extension of F), then everyparahoric subgroup is connected;

22 Oct 2020: let T̆ be its centralizer. By Steinberg’s theorem, G is quasi split over F̆. ... Let Z be a finite type affinescheme over OF and Z its generic fiber.

1 Nov 2013: . 2. Let A and B be n n matrices over a field F. ... Show that the minimum polynomial of A,over the complex numbers, has real coefficients.

6 Aug 2021: αn! The spaces Ck() and C() are vector spaces over C, where addition and scalarmultiplication are defined pointwise. ... 1.1). Exercise(). Show that with the definitions (1.1) the space Ck() is a vectorspace over C, and that Cl() is a vector subspace of

30 Oct 2009: 5. Let A and B be n n matrices over a field F. ... 13. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

25 Oct 2007: 5. Let A and B be n n matrices over a field F. ... 13. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

27 Oct 2010: 5. Let A and B be n n matrices over a field F. ... 13. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

31 Oct 2008: 5. Let A and B be n n matrices over a field F. ... 13. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

23 Jan 2023: over OF. For the rest of 2.3, we fix a geometric conjugacy class of cocharacters{µ} of G and assume the existence of [b]µ B(G,{µ}). ... schemes. Let G be a reductive group over F and G̃ a Bruhat–Tits stabilizer schemecorresponding to x B(G,F).

1 May 2019: In section 3, we prove Theorem B. In section 4, we show that the sections ÙD(X) over a smooth affinoid X are of. ... If N is a D(X)[f1]-module, finitely generated over A[f1]by m1,. ,mr such that all the roots of the associated b-functions b1,.

28 Nov 2016: Let G be a split reductive group over Z, and let T Gbe a split maximal torus, B a Borel subgroup containing T. ... We write W for the rigid space over L which represents the functor.

19 Nov 2012: ii) Find the Galois group of X4 4X 2 over Q and over Q(i). ... polynomial of α over Q(µ3). Determine the possible groups for Gal(F/Q(µ3)).

18 May 2005: More-over, each point of the action spectrum arises as a singularity of Υ for an appropriatechoice of ϕ. ... PATERNAIN. Integrating Pestov’s identity over SM against the Liouville measure dµ, we get.

14 Nov 2013: Prove it?). 7. Which of the following symmetric matrices are congruent to the identity matrix (a) over C, (b) over Rand (c) over Q? ... 4 44 5. ). 1. 8. Find the rank and signature of the following quadratic forms over R.

24 Nov 2011: ii) Find the Galois group of X4 4X 2 over Q and over Q(i). ... polynomial of α over Q(µ3). Determine the possible groups for Gal(F/Q(µ3)).

13 Nov 2019: Show that the minimum polynomial of A,over the complex numbers, has real coefficients. ... 11. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

28 Feb 2008: BT08. Part II Representation Theory Sheet 4. Unless otherwise stated, all vector spaces are finite dimensional over C. ... Q.8 (a) Let G be a compact group. Show that there is a continuous group homomorphismρ : G O(n) if and only if G has an

24 Oct 2022: 1. 10. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. ... 14. Let E be a vector space over C of dimension n 1.

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. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

21 May 2005: In whichof these cases is U a vector space over R?(a) x1 > 0.(b) either x1 = 0 or x2 = 0.(c) x1 x2 = 0.(d) x1 x2 = 1.(e) x1 ... 8. For what values of a and b does the system of simultaneous linear equations.

26 Feb 2019: a,b 0. Hence prove that the class group of K is cyclic and find its order. ... a) Show that f is irreducible over Q and compute its discriminant.(b) Show that 3OK = P3 where P = (α 1) is a prime ideal in OK with residue.

24 Oct 2022: Prof. Dr. B. Löwe. Example Sheet #2. (15) Work over the alphabet Σ = {0, 1} and for n 1 consider. ... 17) Let L and M be languages over an alphabet Σ and consider the set equation X = LX M.

13 Mar 2006: on SU2 = {(. a bb̄ ā. )| aā bb̄ = 1}, and normalize it so that the integral over the group is one. ... 4.8 Question. Let G = {(. a b0 1. )| a F p , b Fp}, with multiplication given by matrix multiplication.

10 May 2006: d. dt. t=0. log Jut dν. where ν runs over all φ-invariant Borel probability measures and hν (φ) is the measuretheoretic entropy of φ with respect to ν. ... Then. (12). N. X(f) Θ =. N. f LX Θ. Integrating the Pestov identity over SM against the

15 Oct 2021: PL(f,P). Af(x)dx := inf. PU(f,P). where the supremum and infimum are taken over all partitions of [a,b]. ... af(t)dt = F(b) F(a). This result is the simplest of a family of results which relate integrals over a regionA Rn with quantities evaluated on the

15 Oct 2021: Itasserts that a function which is continuous on [a,b] and differentiable on (a,b) mustsomewhere have a slope equal to the ‘average slope’ over the interval, i.e. ... However, our control over theremainder term becomes too weak for us to sum the

8 Nov 2018: amk50@cam.ac.uk - 1 - November 2018. 10. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. ... Deduce that if two n n real matrices P and Q are similar whenregarded as matrices over C, then they are similar as matrices

2 Dec 2013: 7. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis. ... 1. 11. For A an n m and B an m n matrix over the field F , let τA(B) denote trAB.

12 Sep 2013: ii) The Pontrjagin–Thom construction provides a map. αg : B Diff (W2ng , D. ... iv) This means that the fibration sequence. R+(Wg ) R+(Wg )//Diff (Wg, D ) B Diff (Wg, D )is pulled back from a fibration over the -constructionB Diff (Wg, D ). , which

24 Nov 2011: Suppose thatψ|U is non-singular. Show that ψ is non-singular. 2. Which of the following symmetric matrices are congruent to the identity matrix (a) over C, (b) over Rand ... 4 44 5. ). 3. Find the rank and signature of the following quadratic forms

15 Oct 2021: f. l. (y). [Hint: Take the supremum over x and infimum over y in the previous inequality]. ... where 2 R. b) For t 2 R, show that:. ||f tg||2L.

28 Oct 2011: 4. Let A and B be n n matrices over a field F. ... . . Which of the matrices are diagonalizable over C? Which over R?

22 Sep 2014: It is characterised up to homotopyequivalence over BO(2n) by these properties. ... Secondly, let hAut(θ) denote the grouplike topological monoid of self-homotopy. equivalences of B over BO(2n), i.e.

13 Apr 2022: the map{. maps f : B M(W),up to homotopy. }. {smooth W-bundles over B,. ... 5. Stabilising and scanning. Outline. For manifolds of even dimension d = 2n, a powerful technique forunderstanding the cohomology of M(W) has arisen over the last 10years.

12 Aug 2008: 1. Which of the following symmetric matrices are congruent to the identity matrix (a) over Q, (b) over Rand (c) over C? (. ... Try to get away with the minimum calculation.]. 2. Find the rank and signature of the following quadratic forms over R.

18 Jan 2018: PART II REPRESENTATION THEORYSHEET 4. Unless otherwise stated, all vector spaces are finite-dimensional over C. ...  : a,b,x Fp. of matrices over the finite field Fp (p prime).

9 Nov 2020: The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

19 May 2007: This is an expectation over pairs (x, x′) of terms that are real and non-negative. ... 2. Now let us take the expectation over y of both sides.

5 Jun 2020: the supremum is taken over all integers r 1 and all pairs of events A and B such that B 2 (Y 1rk),. ... and our choice of N, P̂n. (D) is given by the supremum of [DP̂n()] over.