Let G be a transitive permutation group on a set Ω such that, for ω∈Ω, the stabiliser Gω induces on each of its orbits in Ω\{ω} a primitive permutation group (possibly of degree 1). Let N be the normal closure of Gω in G. Then (Theorem 1) either N factorises as N=GωGδ for some ω, δ∈Ω, or all unfaithful Gω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the Gω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω.

It is shown that known solvability conditions and several new existence results for semilinear elliptic boundary value problems at resonance can be considered from a unified point of view. The proofs apply Leray–Schauder degree arguments.

It is proved that Vaught's conjecture is true for modules over an arbitrary countable Dedekind prime ring. It follows from the structural result that every module with few types over a countable Dedekind prime ring is ω-stable.

Commutative rings form a very special subclass of rings, which shows quite different behaviour from the general case. For example, in a (non-trivial) commutative ring, the absence of zero-divisors is sufficient as well as necessary for the existence of a field of fractions, whereas for general rings, another infinite set of conditions is needed to characterize subrings of skew fields. This suggests the study of a class of rings which includes all commutative rings as well as all integral domains: reversible rings, where a ring is called reversible if ab = 0 implies ba = 0. It turns out that this condition helps to simplify other ring conditions, as we shall see in Section 2, although most of these results are at a somewhat superficial level. We therefore introduce a more technical notion, full reversibility, in Section 3, and show that this is the precise condition for the least matrix ideal to be proper and consist entirely of non-full matrices. Further, we show in Section 4 that a fully reversible ring is embeddable in a skew field if and only if it is an integral domain.

In what follows, all rings are associative, with a unit element 1 which is preserved by ring homomorphisms, inherited by subrings and acts unitally on modules.

I am grateful to V. de O. Ferreira for his comments, in particular the suggestion of using the notion of unit-stable rings.

A special Pisot number is a Pisot number α such that α/(α−1) is also a Pisot number. Lagarias, Porta and Stolarsky found 11 special Pisot numbers. Here it is proved that there are no more. Also, the one known Salem number τ with the property that τ/(τ−1) is a Pisot number is shown to be the only such number.

A flip can be thought of as a diagram X− → X ← X+ of complex threefolds satisfying some conditions. One often thinks of a flip as being in two parts: the first part, X− → X, is the given, while the second, X ← X+, is the unknown. I calculate cohomological properties of the canonical classes, K− = KX− and so on, and in particular properties of the function

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In the case of the toric flips of Danilov [3] and Reid [7], this function can be expressed in terms of lattice points of multiplies of a polyhedron giving sharpness for the more general result.

In [1], Hirano gives a method for constructing families of curves with a large number of singularities. The idea is to consider an abelian covering of ℙ2 ramified along three lines in general position, and to take the pull-back of a curve C intersecting the lines non-generically. Similar constructions are used by Shimada in [10] and Oka in [8]. We apply this method for the case where C is a conic, constructing a family of curves with the following asymptotic behaviour (see [9]):

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The goal of this paper is to calculate the fundamental group for the curves in this family as well as their Alexander polynomial.

We study finite centralizing extensions A⊂H of noetherian Hopf algebras. Our main results provide necessary and sufficient conditions for the fibres of the surjection spec H[Rarr ]spec A to coincide with the X-orbits in spec H, where X denotes the finite group of characters of H that restrict to the counit of A. In particular, all of the fibres are X-orbits if and only if the fibre over the augmentation ideal of A is an X-orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.

The L1 means of various exponential sums with arithmetically interesting coefficients have been investigated in many recent papers. For example, Balog and Perelli proved in [1] that

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for a suitable positive number c. The method of proving the lower bound in [1] is rather flexible and can work well with many multiplicative functions in place of μ(n), the Möbius function, whose Dirichlet series have a suitable expression by the Riemann ζ-function.

In this short note we improve on the above lower bound.

We prove that if A and B are semisimple Banach algebras, then the separating subspace of every Lie isomorphism from A onto B is contained in the centre of B.

In this note we give some sharp estimates for norms of polynomials via the products of norms of their linear terms. Different convex norms on the unit disc are considered.

For a wide class of set ideals (including, for example, all uniform ideals), a criterion of completeness of their symmetry groups is provided in terms of ideal quotients (polars). We apply it to partition ideals, and derive the extended Sierpiński–Erdös duality principle. We demonstrate that if the measure and category ideals I0 and I1 on the real line ℝ are partition (or, equivalently, if just I0 is partition), then not only are they isomorphic via an involution, but they also have complete (and distinct) symmetry groups coinciding, respectively, with the symmetry groups of the polars I⊥0 and I⊥1. The measure and category ideals on ℝ (and in more general spaces) are partition (Oxtoby) ideals assuming Martin's Axiom. In this case their polars are, respectively, the ideals generated by c-Sierpiński and c-Lusin sets. It is well known that the isomorphism of the measure and category ideals is not provable in ZFC. We show that the isomorphism of their symmetry groups is likewise unprovable.

We prove that Yuzvinskii's entropy addition formula cannot be extended for the action of certain finitely generated nonamenable groups.

The context of this note is as follows. One considers a connected reductive group G and a Frobenius endomorphism F[ratio ]G→G defining G over a finite field of order q. One denotes by GF the associated (finite) group of fixed points.

Let [lscr ] be a prime not dividing q. We are interested in the [lscr ]-blocks of the finite group GF. Such a block is called unipotent if there is a unipotent character (see, for instance, [6, Definition 12.1]) among its representations in characteristic zero. Roughly speaking, it is believed that the study of arbitrary blocks of GF might be reduced to unipotent blocks (see [2, Théorème 2.3], [5, Remark 3.6]). In view of certain conjectures about blocks (see, for instance, [9]), it would be interesting to further reduce the study of unipotent blocks to the study of principal blocks (blocks containing the trivial character). Our Theorem 7 is a step in that direction: we show that the local structure of any unipotent block of GF is very close to that of a principal block of a group of related type (notion of ‘control of fusion’, see [13, §49]).

For a very general class of ordinary quasi-differential expressions M with matrix-valued coefficients, the maximal operator TM and the minimal operator T0M are defined as operators of a subspace of Lp into Lq for arbitrary p, q∈[1, ∞], where the weak topologies induced by the dualities 〈Lp, Lp′〉 and 〈Lq, Lq′〉 with 1/p+1/p′=1 and 1/q+1/q′=1 are used instead of the norm topologies. It is shown that these operators and their adjoints possess the usual properties which are known for scalar differential expressions and the two cases p, q∈[1, ∞) and p, q∈(1, ∞].

We describe completely the possible Gorenstein Artin quotients of the coordinate ring of a set of points in projective space. In addition, we give a sufficient condition for an ideal of a Cohen–Macaulay graded k-algebra A to be isomorphic to the canonical module of A. This leads to a description of the canonical module in the case when A is the coordinate ring of points.

One of the basic facts of group theory is that each finite group contains a Sylow p-subgroup for each prime p which divides the order of the group. In this note we show that each vertex-transitive self- complementary graph has an analogous property. As a consequence of this fact, we obtain that each prime divisor p of the order of a vertex-transitive self-complementary graph satisfies the congruence pm ≡ 1(mod 4), where pm is the highest power of p which divides the order of the graph.

For a finite group G and a subset S of G with 1∉S and S = S−1, the Cayley graph Cay(G, S) is the graph with vertex set G such that {x, y} is an edge if and only if yx−1∈S. The group G is called a CI-group if, for all subsets S and T of G[setmn ]{1}, Cay(G, S) ≅ Cay(G, T) if and only if Sσ = T for some σ∈Aut(G). In this paper, for each prime p ≡ 1 (mod 4), a symmetric graph Γ(p) is constructed from PSL(2, p) such that Aut Γ(p) = ℤ2 × PSL(2, p); it is then shown that A5 is not a CI-group, and that all finite CI-groups are soluble.

En 1914, Littlewood a montré, contre l'opinion courante à l'époque, qu'il existe des valeurs de x pour lesquelles le nombre de nombres premiers inférieurs à x, π(x), dépasse le logarithme intégral de x, li x. Plus précisément, il a établi que, pour un K>0 convenable, il existe une infinité de valeurs de x pour lesquelles

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et aussi [1] une infinité de valeurs de x pour lesquelles

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En 1937, Beurling a instauré une nouvelle manière de considérer les problèmes sur les nombres premiers, en introduisant les ‘nombres premiers généralisés’ [2]. L'idée est de partir d'une fonction croissante P(x) (x[ges ]0), nulle sur [0, 1], qui joue le rôle de π(x). On lui associe la fonction dzeta et la fonction croissante N(x) (qui joue le rôle de la partie entière de x) selon la formule

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l'hypothèse est toujours que les intégrales ci-dessus existent lorque σ>1 (s = σ+it). Le but de Beurling est, partant de propriétés convenables de la fonction N(x), d'obtenir pour P(x) le ‘théorème des nombres premiers’, P(x) ∼ li x (x → ∞). Nous allons, au contraire, partir d'une hypothèse simple sur P(x) (par exemple, P(x) < li x) et en tirer des conséquences pour la fonction ζ(x), en laissant de côté la fonction N(x). Bien entendu, nous verrons en passant que l'hypothèse P(x) < li x est incompatible avec le fait que ζ(s) soit la fonction dzeta de Riemann.

Il sera commode d'associer à la fonction ζ(s) la fonction

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elle aussi définie pour σ > 1. Ainsi

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Nous nous servirons seulement des deux faits suivants.

We give a structure theorem for pointed Hopf algebras of dimension 2n, having coradical kC2, where k is an algebraically closed field of characteristic zero.