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.

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

formula here

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.

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

formula here

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

formula here

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

formula here

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

formula here

elle aussi définie pour σ > 1. Ainsi

formula here

Nous nous servirons seulement des deux faits suivants.

Continuing work of Duret, we treat the relation between isomorphism and elementary equivalence of function fields over algebraically closed fields. For function fields of curves, these are ‘usually’ the same, but in characteristic zero, for elliptic curves with complex multiplication, a weak variant of elementary equivalence of their function fields corresponds to isomorphism of the endomorphism rings of the curves, not to isomorphism of the curves themselves.

The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By definition, these zeta functions are rational functions in one variable, and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non-degenerate with respect to its Newton polyhedron, we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; we give a similar result concerning the global zeta function. Moreover, for any f we show that poles of maximal possible order are always of the form −1/N with N a positive integer.

We prove that if n>66 and (n, 30) = 1, then there exist uncountably many infinite simple (2, 3, n)- groups, that is, groups generated by a pair of elements x, y, say, where the orders of x, y and xy are 2, 3 and n, respectively. This extends previous results of Schupp and the authors.

These results are used to prove the existence of subgroups of the modular group with special arithmetic properties.

In the representation theory of finite groups, it is useful to know which ordinary irreducible representations remain irreducible modulo a prime p. For the symmetric groups [Sfr ]n, this amounts to determining which Specht modules are irreducible over a field of characteristic p. Throughout this note we work in characteristic 2, and in this case we classify the irreducible Specht modules, thereby verifying the conjecture in [3, p. 97].

Recall that a partition is 2-regular if all of its non-zero parts are distinct; otherwise the partition is 2-singular. The irreducible Specht modules Sλ with λ a 2-regular partition were classified in [2]. Let λ′ denote the partition conjugate to λ. If Sλ is irreducible, then Sλ′ is irreducible, since Sλ′ is isomorphic to the dual of Sλ tensored with the sign representation. It turns out that if neither λ nor λ′ is 2-regular, then Sλ is irreducible only if λ = (2, 2).

In order to state our theorem, we let l(k), for an integer k, be the least non-negative integer such that k<2l(k).

We solve the functional equations

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for suitable functions f, g and h subject to x + y + z = 0. These equations essentially characterise the Weierstrass [pscr ]-function and its degenerations.

Every ε-isometry between real normed spaces of the same finite dimension which maps the origin to the origin may be uniformly approximated to within 2ε by a linear isometry. Under a smoothness hypothesis, necessary and sufficient conditions are obtained for the same conclusion to hold for a given ε-isometry between infinite-dimensional Banach spaces.

Let P be an n-dimensional polytope admitting a finite reflection group G as its symmetry group. Consider the set [Hscr ]P(k) of all continuous functions on Rn satisfying the mean value property with respect to the k-skeleton P(k) of P, as well as the set [Hscr ]G of all G-harmonic functions. Then a necessary and sufficient condition for the equality [Hscr ]P(k)=[Hscr ]G is given in terms of a distinguished invariant basis, called the canonical invariant basis, of G.

We shall construct a map Ω0(G/H)→ΠαS2nα−1× ΠβΩS2nβ−1 which is a rational equivalence with applications to phantom maps, spaces of the same n-type and genus sets. This answers in the affirmative a question of C. McGibbon.

Kunihiko Kodaira, who died on 26 July 1997, was the outstanding Japanese mathematician of the post-war period, his fame established by the award of the Fields Medal at the Amsterdam Congress in 1954.

He was born on 16 March 1915, the son of an agricultural scientist who at one time was Vice Minister of Agriculture in the Japanese Government and had also played an active role in agricultural developments in South America. Kodaira studied at Tokyo University, taking degrees in both mathematics and physics. From 1944 to 1951 he was an associate professor of physics at the University. His PhD thesis was published in the Annals of Mathematics [18], and it immediately attracted international attention. Essentially this filled a significant lacuna in the basic theorem of W. V. D. Hodge on harmonic integrals. Kodaira had worked on this for many years but, because of the war, his research was carried out in isolation from the international community and did not become known until much later.

Hermann Weyl, who had been a keen supporter of Hodge's work, realised the importance of Kodaira's thesis, and arranged for him to come to the Institute for Advanced Study in Princeton in 1949. This was the start of Kodaira's 18-year residence in the United States, a fruitful period which saw the full blossoming of his research, much of it in collaboration with Donald Spencer. Kodaira spent many years at Princeton, divided between the Institute and the University, but the years 1961–67 were more unsettled, seeing him successively at Harvard, Johns Hopkins and finally Stanford. In 1967 he returned to a professorship at the University of Tokyo, where he remained until the normal retiring age. From 1975 to 1985 he worked at Gakushuin University, where retirement restrictions did not apply.