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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

5 Jun 2020: Part (ii) follows from (i) on multiplying each side of (19) by αeαT , andintegrating over T R. ... Dκv = κ[κRκv v] δκ. δ κv. κ. δ κb0v. This gives (31) on choosing the worst-case over κ 1:.

22 Oct 2020: The model category we work inwill be the category of simplicial algebras over a ring. ... Furthermore, given a sequence:. Fi> E. p> B. where E is a Serre fibration and i is the inclusion of the fibre F over the base pointB, then there is a

5 Jun 2020: where the infimum is taken over all jointly distributed random variables (X,Y ) such that X hasdistribution P and Eρ(X,Y ) D. ... H(W‖P1Q) λ1EW(ρ) λ2EQ[log M(Y )] ΛP1,Q(λ). Taking the supremum over all λ R2 and recalling (23) this becomes.

5 Jun 2020: codebook; equivalently, thiscan be thought of as the rate loss of the straightforward schemewhich uses a mismatched codebook without entropy coding.b) The rate loss due to quantizer mismatch, over the ... bits/symbol (5). where the expectation is taken

5 Jun 2020: b) The rate loss due to quantizer mis-match, over the optimal rate-distortion function: We will derive a universal upper bound for this rate. ... where the expectation is taken over both the message Xn1 and the random codebook Cn.

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

5 Jun 2020: support; or (b) Qhas tails heavy enough so that, for some ρ,β > 0 and N0 1, we have, Q(x) ρx. ... As can be seen there, conditions (a) and (b) areintroduced purely for technical reasons, and can probably be significantly relaxed; see Section 7 for

5 Jun 2020: and the infimum is taken over all joint distributions onsuch that the -marginal of is and. ... where the infimum is over all stationary probabilitymeasures on such that the -marginal of isand.

3 Jun 2020: The problem is to minimize the sparsity over all cuts. This is NP-hard. ... over all cuts with nonzero demand, where dS is the cut semimetric (c.f.

5 Jun 2020: 2011 Royal Statistical Society 1369–7412/12/74133. J. R. Statist. Soc. B (2012)74, Part 1, pp. ... b) non-conditionally conjugate samplers, where the conditional expectations P G.x/ for theclass of linear basis functions cannot be computed in closed

5 Jun 2020: Theorem 8: Averaging uniformly over all binning com-pressors f: X {1, 2,. ... 1d ). (141). α(d) = sup{|P(B A) P(B)P( A)|} (142)where the sup is over A F0 and B Fd.

5 Jun 2020: Clearly, the semigroup {P̂ t } = {P̂ tα} is, in general, nonprobabilistic.The definitions of ψ -irreducibility and aperiodicity carry over to nonprobabilisticsemigroups immediately. ... With R̂θ replacing Rθ , thedefinitions of small functions,

5 Jun 2020: some reference measure, uniformly over all initial conditions x in thesublevel set CW (r) of W. ... Assume more-over that X = Rn; that condition (DV3+) holds with V : X [1,) continuous; W unbounded;and the kernels {ICW (r)P.

5 Jun 2020: Over the past years the practical requirement for e cient data compression al. ... scientists over the past several decades there are by now a number of very e cient.