We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to A4, the alternating group of degree 4 and order 12. There are two such fields with Galois group A4 × C2 (see Theorem 14) and at most one with Galois group SL2(F3) (see Theorem 18); if the Generalized Riemann Hypothesis is true, the...

We analyze further the Magnus-Derek game, a two-player game played on a round table with n positions. The players jointly control the movement of a token. One player, Magnus, aims to maximize the number of positions visited while minimizing the number of rounds. The other player, Derek, attempts to minimize the number of visited positions. We present a new strategy for Magnus that succeeds in v...

Under Martin’s axiom, collapsing of the continuum by Sacks forcing S is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if d = c proving that under this hypothesis there are no small uncountable maximal antichains in S. We also construct a partition of 2 into c perfect sets which is a maximal antichain in S and show that s-sets ...

We show that if the lower central series of the fundamental group of a closed oriented 3-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion 2-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental groups of closed oriented 3-manifolds.

We show that the number of all maximal α-gapped repeats and palindromes of a word of length n is at most 3(π/6 + 5/2)αn and 7(π/6 + 1/2)αn − 5n− 1, respectively.

We consider a real semisimple Lie group G with finite center and K a maximal compact subgroup of G. We prove an Lp −Lq version of Hardy’s theorem for the spherical Fourier transform on G. More precisely, let a, b be positive real numbers, 1 ≤ p, q ≤ ∞, and f a K-bi-invariant measurable function on G such that h−1 a f ∈ Lp(G) and eb‖λ‖ (f )∈ Lq(a∗ +) (ha is the heat kernel on G). We establish th...

It is shown that Martin’s Axiom for σ-centred partial orders implies that every maximal orthogonal family in R is of size 20 . For x, y ∈ R define the inner product

In the present paper I show that one can recover much of the inertia structure of (quasi) divisors of a function field K|k over an algebraically closed base field k from the maximal pro-` abelian-by-central Galois theory of K. The results play a central role in the birational anabelian geometry and related questions.

Let k be a field of positive characteristic. We construct, for each dominant coweight λ of the standard maximal torus in Sln, a closed subvariety D(λ) of the multigraded Hilbert scheme of an affine space over k, such that the k-valued points of D(λ) can be interpreted as lattices in k((z))n endowed with infinitesimal structure. Moreover, for any λ we construct a universal homeomorphism from D(λ...