Interaction of [Mo6Cl14]2− with H2Se: Selective Preparation of [Mo6SeCl13]3−

Reaction of [Mo₆(μ₃-Cl)₈Cl₆]²⁻ with H₂Se, generated in situ from ZnSe and 4 M HCl under hydrothermal conditions lead to the substitution of one or two bridging chlorides, depending on the reagents ratio. With the Mo₆/ZnSe 1:3 molar ratio [Mo₆(μ₃-SeCl₇)Cl₆]³⁻ forms selectively in high yield. Further substitution is more hindered, and even at 1:20 cluster-to-selenide molar ratio a mixture of [Mo₆(μ₃-SeCl₇)Cl₆]³⁻ and [Mo₆(μ₃-Se₂Cl₆)Cl₆]⁴⁻ is formed. The products were characterized by X-ray, Raman spectra and electrospray ionization mass spectrometry.     

Note on the natural system of units

We propose to substitute Newton’s constant G _N for another constant G ₂, as if the gravitational force would fall off with the 1/r law, instead of the 1/r ²; so we describe a system of natural units with G ₂, c and ℏ. We adjust the value of G ₂ so that the fundamental length L = L _Pl is still the Planck’s length and so G _N = L × G ₂. We argue for this system as (1) it would express longitude, time and mass without square roots; (2) G ₂ is in principle disentangled from gravitation, as in (2 + 1) dimensions there is no field outside the sources. So G ₂ would be truly universal; (3) modern physics is not necessarily tied up to (3 + 1)-dim. scenarios and (4) extended objects with p = 2 (membranes) play an important role both in M-theory and in F-theory, which distinguishes three (2, 1) dimensions. As an alternative we consider also the clash between gravitation and quantum theory; the suggestion is that non-commutative geometry [x _i , x _j ] = Λ² θ _ij would cure some infinities and improve black hole evaporation. Then the new length Λ shall determine, among other things, the gravitational constant G _N.     

Potential modularity for elliptic curves and some applications

In this paper we prove the simultaneous potential modularity for a finite number of elliptic curves defined over a totally real field. As an application we prove the meromorphic continuation of some L-functions associated to elliptic curves and Tate conjecture for a product of 2 or 4 elliptic curves defined over a totally real field.     

On Higher Riesz Transforms for Gaussian Measures

Given an n × n matrix B, positive definite and symmetric, let L_B be the differential operator in Rⁿ given by A class of higher Riesz transforms associated with L_B is defined by means of higher gradients of order k. It is shown that these transformations are bounded in the space with respect to the measure that makes L_B selfadjoint. The constants obtained are independent of the dimension n and depend only on k,p, and the number of different eigenvalues of the matrix B. The proof of the results uses an extension of the Littlewood-Paley-Stein theory of square functions to the vector-valued case and inequalities previously proved by one of the authors in the context of the Riesz transforms of order one.     

Language-theoretic complexity of disjunctive sequences

A sequence over an alphabet ∑ is called disjunctive if it contains all possible finite strings over ∑ as its substrings. Disjunctive sequences have been recently studied in various contexts. They abound in both category and measure senses. In this paper we measure the complexity of a sequence x by the complexity of the language P(x) consisting of all prefixes of x. The languages P(x) associated to disjunctive sequences can be arbitrarily complex. We show that for some disjunctive numbers x the language P(x) is context-sensitive, but no language P(x) associated to a disjunctive number can be context-free. We also show that computing a disjunctive number x by rationals corresponding to an infinite subset of P(x) does not decrease the complexity of the procedure, i.e. if x is disjunctive, then P(x) does not have an infinite context-free subset. This result reinforces, in a way, Chaitin's thesis (1969) according to which perfect sets, i.e. sets for which there is no way to compute infinitely many of its members essentially better (simpler/quicker) than computing the whole set, do exist. Finally we prove the existence of the following language-theoretic complexity gap: There is no x ε ∑^w such that P(x) is context-free but not regular. If S(x), the set of all finite substrings of a sequence x ε ∑^w, is slender, then the set of all prefixes of x is regular, that is P(x) is regular if and only if S(x) is slender.     

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x)$u(\mathbf{x})$ and |∇u|²${|\mathrm{\nabla }u|}^2$, where u(x)$u(\mathbf{x})$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R^N+1${\mathbb{R}}^{N+1}$.     

On the conjecture of Birch and Swinnerton-Dyer

In this paper we complete Rubin's partial verification of the conjecture for a large class of elliptic curves with complex multiplication by .

     

Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields

We show that, for all characteristic p global fields k and natural numbers n coprime to the order of the non-p-part of the Picard group Pic⁰(k) of k, there exists an abelian extension L/k whose local degree at every prime of k is equal to n. This answers in the affirmative in this context a question recently posed by Kisilevsky and Sonn. As a consequence, we show that, for all n and k as above, the n-torsion subgroup Br_n(k) of the Brauer group Br(k) of k is algebraic, answering a question of Aldjaeff and Sonn in this context.     

The Rubin-Stark conjecture for a special class of function field extensions

We prove a strong form of the Brumer-Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum k_p∞?k_p·k_∞ of the maximal pro-p abelian extension k_p/k and the maximal constant field extension k_∞/k of k, which happens to sit inside the maximal abelian extension k^ab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321-367).