We prove that if q is an integer greater than one, r and s are any positive rationals, then

formula here

is irrational and is not a Liouville number.

Introduite par Hall [4] en 1979 et étudiée de manière systématique par Hall et Tenenbaum ([6, 7]), la fonction

formula here

est une mesure quadratique de la proximité des diviseurs.

On a trivialement τ(n)[les ]T(n, α)[les ]T(n, 1)=τ(n)2 pour tout α, où, ici et dans la suite, τ(n) désigne le nombre de diviseurs d'un entier générique n: Le statut de l'inégalité de gauche pour presque tout entier n est une question difficile qui a inspiré une conjecture d'Erdős (cf. [1]) assez connue. Convenons de désigner par pp (presque partout) une relation valable sur un ensemble d'entiers de densité unité et posons

formula here

(Cette fonction est mentionnée dans [4], mais sa définition fait l'objet d'une coquille.)

We compute the (uni)versal deformation ring of an odd Galois representation ρ: Gal (M/Q) → Gl2(Fp) with an upper triangular image, where M is the maximal abelian pro-p-extension of F∞ unramified outside a finite set of places S, F∞ being a free pro-p-extension of a subextension F of the field K fixed by Ker ρ. We establish a link between the latter (uni)versal deformation ring and the (uni)versal deformation ring of ρ: Gal (KS/Q) → Gl2(Fp), where KS is the maximal pro-p-extension of K unramified outside S. We then give some examples.

The Mullineux map is an involutory bijection on the set of p-regular partitions of any given integer n, where a partition is called p-regular if no part of it is repeated p or more times. Many combinatorial properties of the Mullineux map make it reasonable to view this map as a p-analogue of the transposition map T on the set of all partitions.

Based on the work of Kleshchev [7], it has been shown [2, 5, 14] that the Mullineux map M has the following property.

If λ is a p-regular partition of n and Dλ is the p-modular representation of the symmetric group Sn labelled by λ (see [6]) then

formula here

where sgn is the sign representation of Sn. Thus, from a representation theoretic point of view as well, the Mullineux map is a p-analogue of the transposition map T. The Mullineux map plays a vital role not only in the representation theory of symmetric groups but also in other contexts. The definition of M as given in [9] is quite complicated and to study various questions involving this map it is desirable to have other descriptions. One alternative inductive description of M using the concept of good boxes in a p-regular partition was given by Kleshchev [7] and this was used by Walker to prove a result contained in Theorem 4·5 below; this work was motivated by the investigation of Schur modules. In this paper we study a third description of M based on the operator J on the set of p-regular partitions defined in [13].

Let G be a simple algebraic group over an algebraically closed field K of characteristic p. If Σ is the root system of G and Uα is the root subgroup of G corresponding to a long root α∈Σ, then 〈Uα, U−α〉 is an image of SL2 and any G-conjugate of this subgroup is called a fundamental subgroup of G. In [LS1], the closed connected semisimple subgroups of G generated by long root elements were determined. Of course, long root elements are unipotent elements of fundamental subgroups. In this paper we consider subgroups of G which are generated by semisimple elements lying in fundamental subgroups.

Our first three results (Theorems 1–3) concern semisimple connected subgroups of G which contain a maximal torus of a fundamental subgroup; we call such a torus a fundamental torus. Notice that for classical groups, the elements in fundamental tori have fixed spaces of small codimension in the natural module; indeed, for the groups SL(V) and Sp(V), the elements of fundamental tori are precisely those semisimple elements with fixed space of codimension 2, while for SO(V), the codimension is 4.

As consequences of these results, we obtain information on subgroups (finite or infinite) of classical groups which are generated by conjugates of a single element of a fundamental torus of order at least 5 (see Theorems 4, 5 and Corollary 6); for example, Theorem 5 determines those finite irreducible subgroups containing such an element which are quasisimple and of Lie type in characteristic p.

We now state our results in detail. Recall from [LS1] that a subsystem subgroup of G is a connected semisimple subgroup which is invariant under a maximal torus of G.

Let F be a free group. We explain the classification of finitely presented subgroups of F×F in geometric terms. The classification emerges as a special case of results concerning the structure of 2-complexes which are built out of squares and have the property that the link of each vertex has no reduced circuits whose length is odd or less than four. We obtain these results using tower arguments and elements of the theory of non-positively curved spaces.

It is a well known fact that every group G has a presentation of the form G=F/R, where F is a free group and R the kernel of the natural epimorphism from F onto G. Driven by the desire to obtain a similar presentation of the group of automorphisms Aut (G), we consider the subgroup Stab (R) ⊆ Aut (F) of those automorphisms of F that stabilize R and ask whether the natural homomorphism Stab (R) → Aut (G) is onto; if it is, we can try to determine its kernel.

Both parts of this task are usually quite hard. The former part received considerable attention in the past, whereas the more difficult part (determining the kernel) seemed unapproachable. Here we approach this problem for a class of one-relator groups with a special kind of small cancellation condition. Then, we address a somewhat easier case of 2-generator (not necessarily one-relator) groups and determine the kernel of the above-mentioned homomorphism for a rather general class of those groups.

We obtain a result compatible with Bloch's conjecture for Mumford's fake plane. In the process some new surfaces associated with Mumford's surface are found.

Let n≥5. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball in (n+2)-space.) Furthermore, the projection is not the projection of any unknotted sphere in (n+2)-space. There are two inequivalent embeddings of an n-manifold in (n+2)-space such that the singular point set of the projection of each embedding consists of double points and that the number of components of the singular point set of each embedding is one.

If G is a compact Lie group and M a Mackey functor, then Lewis, May and McClure [4] define an ordinary cohomology theory H*G(−; M) on G-spaces, graded by representations. In this article, we compute the ℤ/p-rank of the algebra of integer-degree stable operations [Ascr ]M, in the case where G=ℤ/p and M is constant at ℤ/p. We also examine the relationship between [Ascr ]M and the ordinary mod-p Steenrod algebra [Ascr ]p.

The main result implies that while [Ascr ]M is quite large, its image in [Ascr ]p consists of only the identity and the Bockstein. This is in sharp contrast to the case with M constant at ℤ/p for q≠p; there [Ascr ]M≅[Ascr ]q.

We introduce an analogue of Calderón's first commutator along a parabola and establish its L2 boundedness under essentially sharp hypotheses.

We present a class of approximately finite dimensional C*-algebras whose local multiplier algebras are simple with real rank zero and stable rank one.

Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with g∘f=h∘g. We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g−1(J(h))=J(f). Here J(·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h, there are only countably many entire functions g such that g∘f=h∘g.