It is shown that the Coxeter-Todd lattice is the unique strongly perfect lattice in dimension 12.

Gabriele Nebe Boris Venkov

Relations between elementary particles masses are given using only known physical constants, without any arbitrary number.

B. Tatischeff I. Brissaud

Let k be an algebraically closed field of characteristic 0. We prove that any division algebra over k(x,y) whose ramification locus lies on a quartic curve is cyclic.

Boris E. Kunyavskii Louis H. Rowen Sergey V. Tikhonov Vyacheslav I. Yanchevskii

Let $b \ge 2$ be an integer. We prove that the $b$-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.

Boris Adamczewski Yann Bugeaud

For k>=3 let A \subset [1,N] be a set not containing a solution to a_1 x_1+...+a_k x_k=a_1 x_{k+1}+...+a_k x_{2k} in distinct integers. We prove that there is an epsilon>0 depending on the coefficients of the equation such that every such A has O(N^{1/2-epsilon}) elements. This answers a question of I. Ruzsa.

Boris Bukh

Recently T. Kieu (arXiv:quant-ph/0110136) claimed a quantum algorithm computing some functions beyond the Church-Turing class. He notes that "it is in fact widely believed that quantum computation cannot offer anything new about computability" and claims the opposite. However, his quantum algorithm does not work, which is the point of my short note. I still believe that quantum computation leads to new complexity but retains the old computability.

Boris Tsirelson

This paper is based on the author's paper "Koszul duality in deformation quantization, I", with some improvements. In particular, an Introduction is added, and the convergence of the spectral sequence in Lemma 2.1 is rigorously proven. Some informal discussion in Section 1.5 is added.

Boris Shoikhet

Nonclassical noises over the plane (such as the black noise of percolation) consist of sigma-fields corresponding to some planar domains. One can treat less regular domains as limits of more regular domains, thus extending the noise and its set of sigma-fields. The greatest extension is investigated in a new general framework.

Boris Tsirelson

The noise-type completion C of a noise-type Boolean algebra B is generally not the same as the closure of B. As shown in Part I (Introduction, Theorem 2), C consists of all complemented elements of the closure. It appears that C is the whole closure if and only if B is classical (as defined in Part II, Sect. 1a), which is the main result of this Part III.

Boris Tsirelson

It is shown that the model of vacuum flavour oscillations is in disagreement with quantum mechanics theorems and postulates. Features of the model are analyzed. It is noted that apart from the number of mixed mass states neutrino oscillations are forbidden by Fock-Krylov theorem. A possible reason of oscillation model inadequacy is discussed.

Boris I. Goryachev