Abelian groups, module theory, and topology: proceedings in by Dikran Dikranjan, Luigi Salce

By Dikran Dikranjan, Luigi Salce

Includes a stimulating choice of papers on abelian teams, commutative and noncommutative jewelry and their modules, and topological teams. Investigates at the moment renowned themes reminiscent of Butler teams and nearly thoroughly decomposable teams.

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With effort, the exponent h(a) can be sharpened— and made more explicit. • Indeed, for a = b or even a ≈ b we now have many other, rapidly convergent options. 45 Proof: With a view to induction, assume that for some constants (n-independent) d(a), g(a) and for n ∈ [1, N − 1] we have qn < dng . 1) mean that An > f (a)/n for an n-independent f . Then we have a bound for the next qN : f qN > d(N − 1)g + d (N − 2)g . N Since for g < 1, 0 < x ≤ 1/2 we have (1 − x)g > 1 − gx − gx2, and the constants d, g can be arranged so that qN > dN g and the induction goes through.

11: The precise domain of convergence for R1(a, b) is D0 = {(a, b) ∈ C×C : (a/b ∈ C ) or (a2 = b2, b ∈ I)}. In particular, for a/b ∈ C we have divergence. Moreover, provably, R1(a, b) converges to an analytic function of both a or b on the domain D2 := {(a, b) ∈ C × C : |a/b| = 1} ⊂ D0. 6 because neither H nor K intersects C . 11 is very subtle. 2) with all three fractions converging. 11 we have sufficient analyticity c to apply Berndt’s technique. 12 follows: Equivalently, a/b belongs to the closed exterior of ∂H, which in polar-coordinates is given by the cardioid-knot r2 + (2 cos φ − 4)r + 1 = 0 drawn in the complex plane (r = |a/b|).

Define • H := {z ∈ C : √ 2 z 1+z < 1}, 2z • K := {z ∈ C : 1+z 2 < 1}. 4: If a/b ∈ K then both R1(a, b) and R1(b, a) converge. 5: H ⊂ K (properly). 6: If a/b ∈ H then R1(a, b) & R1(b, a) converge, and the arithmetic mean √ (a + b)/2 dominates the geometric mean ab in modulus. 2) · · · . • We performed “scatter diagram” analysis to find computationally where the AGM relation holds in the parameter space. The results were quite spectacular! And lead to the Theorems above. 11: The precise domain of convergence for R1(a, b) is D0 = {(a, b) ∈ C×C : (a/b ∈ C ) or (a2 = b2, b ∈ I)}.

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