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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.(.

5 Jun 2020: n1 obtained byminimizing the ideal Shannon code-lengths over the family {Qθ}θΘ,. ... Integrating over distortion-balls effectively changes a continuous alphabet intoa discrete one and log Q(B(xn1,D)) still makes sense as an idealized code-length.

9 Nov 2020: lemma. Since the slope filtration descend to G u(a,b), for a a′ b′ b we have a subquotient over. ... Def (a,b) over s, and Defs(a,b1) to be the fiber of Defa,b1 Def (a,b) over s.

31 Jul 2020: Lemma 13. Let l,a,b pZp and m,c Zp be fixed, subject to the condition that z2 mz c isirreducible over Fp. ... where s is a binaryquadratic form over Fp with a double root over Fp.

9 Nov 2020: separable extension of K. Let G be the Galois group of L over K and let B be the integral. ... Spec(B) Spec(A) has finite fibers. Hint: reduce to the two cases (1) L is separable over.

4 Mar 2020: ii) The group 〈S | E〉 is stable.Proof. Use the fact that dn (cad,cbd) = dn (a,b), and fix for each γ Γ = 〈S | E〉 a word over S1in the ... Define. pn = PxXn ((Xn,x) 'r (B,b)) and p = P(Y,y)µ ((Y,y) 'r (B,b)).

5 Jun 2020: ixs xs. Here B B denotes the algebra generated by B and B and xs xs. ... is the realization over Z n s s obtained by glueing realizations xs and xs.

16 Nov 2020:  : a,b,x Fp. of matrices over the finite field Fp (p prime). ... Which are indecomposable? Remark: Over C irreducibility and indecomposability coincide but this can fail for mod-ular representations.

11 Nov 2020: The key idea is to use theLang–Weil estimate to relate the cardinality of JΣtop to the asymptotic behaviorof the number of points on Xµ(b) over a finite field, as ... Define the group scheme Jb over F by. Jb(R) ={g G(RF L) | g1bσ(g) = b.

3 Mar 2020: Provethat the affine cone over P1P1 in its Segre embedding inside P3 is not smooth. ... Which of the following morphisms separates points resp. tangent vectors?(Assume we’re working over a characteristic 0 field).(a) A1 A2 sending t 7 (t, t2).(b) A1 A2

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.

8 Jun 2020: A. B Cg. f. and a cone over it consists of a completion of the cospan to a commutative square:. ... 16. A. B 1. is the same thing as a cone over the discrete pair (A,B), so a pullback for the above diagram is aproduct AB.

9 Nov 2020: Show that the minimum polynomial of A,over the complex numbers, has real coefficients. ... 1. 10. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

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.

5 Jun 2020: In order to “amplify” the effects of regions of potential dependency in various segmentsof the zmSRp32 gene, we computed the mutual information estimates Îj on the originalstrings over the regular ... Plot (a) shows estimates over the original

28 May 2020: A chain complex (C,d) over Rconsists of:. (i) R-modules Ci for i Z,. ... B ' AC.Theorem 3.23. If (C,d) is a free chain complex over a PID R, then it is isomorphic to a directsum of short injective complexes.

10 Mar 2020: Prove thatif Dfx 6= 0 for every x Σ, then Σ is locally given by the graph of a smooth function over one of the co-ordinateplanes in R3.(b) Let ... Prove that near any point p Σ, the surface Σ is given locallyas the graph of a smooth function over the

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;

5 Jun 2020: We show that, at the expense of a minor increase in routing over-head, the BDAR algorithm gives a substantial improvement in networkperformance, in terms both of network congestion and of ... 1 1µ. )dn12. Applying the. Chernoff bound for u and summing

6 May 2020: ii) For all (a,b) , (a′,b′) R, we have |dA (a,a′) dB (b,b′)| ε. ... We define. Dh(A,B) = infA,BX. DH(A,B),. where the infimum is taken over all compact metric spaces X containing isometric copiesof A and B, and DH is the