In this paper we introduce the spaces of Hankel convolutors. We characterize the dual spaces of certain Hankel transformable function spaces as spaces of Hankel convolutors. Here the Hankel convolution and the Hankel transformation play an important role.

In this paper, we show that a commutative Noetherian ring which satisfies the radical formula must be of dimension at most one. From this we give a characterization of commutative Noetherian rings that satisfy the radical formula.

We show in a direct and elementary way that the spherical building at infinity of every rank 3 affine building which satisfies Tits' Moufang condition, is itself a Moufang building. This result is also true for higher rank affine buildings by Tits' classification [4].

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

Suppose that [Cscr]={cij[ratio ]i, j[ges ]1} is a collection of i.i.d. nonnegative continuous random variables and suppose T is a rooted, directed tree on vertices labelled 1,2,[ctdot ],n. Then the ‘cost’ of T is defined to be c(T)=[sum ](i,j)∈Tcij, where (i, j) denotes the directed edge from i to j in the tree T. Let Tn denote the ‘optimal’ tree, i.e. c(Tn)=min{c(T)[ratio ]T is a directed, rooted tree in with n vertices}. We establish general conditions on the asymptotic behaviour of the moments of the order statistics of the variables c11, c12, [ctdot ], cin which guarantee the existence of sequences {an}, {bn}, and {dn} such that b−1n(c(Tn)−an)→N(0, 1) in distribution, d−1nc(Tn)→1 in probability, and d−1nE(c(Tn))→1 as n→∞, and we explicitly determine these sequences. The proofs of the main results rely upon the properties of general random mappings of the set {1, 2, [ctdot ], n} into itself. Our results complement and extend those obtained by McDiarmid [9] for optimal branchings in a complete directed graph.

Two-sided bounds are obtained for the maximal eigenvalue of a positive matrix by iterating computations of row sums. The result provides an algorithm for approximating the maximal eigenvalue of a nonnegative matrix.

For a graph G=(V, E) on n vertices, where 3 divides n, a triangle factor is a subgraph of G, consisting of n/3 vertex disjoint triangles (complete graphs on three vertices). We discuss the problem of determining the minimal probability p=p(n), for which a random graph G∈[Gscr](n, p) contains almost surely a triangle factor. This problem (in a more general setting) has been studied by Alon and Yuster and by Ruciński, their approach implies p=O((log n/n)1/2). Our main result is that p=O(n)−3/5) already suffices. The proof is based on a multiple use of the Janson inequality. Our approach can be extended to improve known results about the threshold for the existence of an H-factor in [Gscr](n, p) for various graphs H.

Let φ: ℋ → be a bilinear form on vector Hardy space. Introduce the symbol φ of Φ by (φ (Z1, Z2), a ⊗ b) = Φ (K21 ⊗ a, K22 ⊗ b ), where Kw is the reproducing kernel for w ∈ D. We show that Φ extends to a bounded bilinear form on provided that the gradient defines a Carleson measure in the bidisc D2. We obtain a sufficient condition for Φ to extend to a Hilbert space. For vectorial bilinear Hankel forms we obtain an analogue of Nehari's Theorem.

B. Rouxel [7] and S. J. Li and C. S. Houh [6] have generalised the notion of an -submanifold (Chen submanifold) to an k-submanifold. In [1] we have studied the relation between their definitions for the Euclidean case.

Let Q be a stochastic matrix and I be the identity matrix. We show by a direct combinatorial approach that the coefficients of the characteristic polynomial of the matrix I−Q are log-concave. We use this fact to prove a new bound for the second-largest eigenvalue of Q.

If G is the group of holomorphic automorphisms of a bounded symmetric domain, then G has a distinguished class of irreducible unitary representations called the holomorphic discrete series of G. These representations have been studied by Harish-Chandra in [7]. On the Lie algebra level, the Harish-Chandra modules corresponding to the holomorphic discrete series representations are highest weight modules. Even for G as above, it turns out that not all the unitary highest weight modules belong to the holomorphic discrete series but there exists a condition on the highest weight which characterizes the holomorphic discrete series among the unitary highest weight representations. They can be defined as those unitary highest weight representations with square integrable matrix coefficients.

Thomassen [6] conjectured that if I is a set of k−1 arcs in a k-strong tournament T, then T−I has a Hamiltonian cycle. This conjecture was proved by Fraisse and Thomassen [3]. We prove the following stronger result. Let T=(V, A) be a k-strong tournament on n vertices and let X1, X2, [ctdot ], Xl be a partition of the vertex set V of T such that [mid ]X1[mid ][les ][mid ]X2[mid ][les ][ctdot ][les ][mid ]Xl[mid ]. If k[ges ][sum ]l−1i=1[lfloor][mid ]Xi[mid ]/2[rfloor]+[mid ]Xl[mid ], then T−∪li=1{xy∈A[ratio ]x, y∈Xi} has a Hamiltonian cycle. The bound on k is sharp.

Let A and B be regular semisimple commutative Banach algebras; that is to say, regular Banach function algebras. A linear map T denned from A into B is said to be separating or disjointness preserving if f.g = 0 implies Tf.Tg = 0, for all f, g ∈ A In this paper we prove that if A satisfies Ditkin's condition then a separating bijection is automatically continuous and its inverse is separating. If also B satisfies Ditkin's condition, then it induces a homeomorphism between the structure spaces of A and B.

Let D1 and D2 be two dice with k and l integer faces, respectively, where k and l are two positive integers. The game Gn consists of tossing each die n times and summing the resulting faces. The die with the higher total wins the game. We examine the question of which die wins game Gn more often, for large values of n. We also give an example of a set of three dice which is non-transitive in game Gn for infinitely many values of n.

We show that there are almost surely only finitely many times at which there are at least four ‘tied’ favourite edges for a simple random walk. This (partially) answers a question of Erdős and Révész.

In this paper we show that the list chromatic index of the complete graph Kn is at most n. This proves the list-chromatic conjecture for complete graphs of odd order. We also prove the asymptotic result that for a simple graph with maximum degree d the list chromatic index exceeds d by at most [Oscr](d2/3√log d).

In this note we show that every complemented Montel subspace F of a Fréchet space E of Moscatelli type is isomorphic to ω or is finite–dimensional; the last case always occurs when E has a continuous norm. To do this, we first study the topology induced by E on its Montel subspaces, extending a result on Fr6chet-Montel spaces of Moscatelli type in [4].

We recall that the Fréchet spaces of Moscatelli type were introduced and studied by J. Bonet and S. Dierolf in [4]; the general idea behind the construction of such spaces was due to V. B. Moscatelli [7].

We consider both stationary and time-dependent heat equations for a non-convex body or a collection of disjoint conducting bodies with Stefan-Boltzmann radiation conditions on the surface. The main novelty of the resulting problem is the non-locality of the boundary condition due to self-illuminating radiation on the surface. Moreover, the problem is nonlinear and in the general case also non-coercive. We show that the non-local boundary value problem admits a maximum principle. Hence, we can prove the existence of a weak solution assuming the existence of upper and lower solutions. This result is then applied to prove existence under some hypotheses that guarantee the existence of sub- and supersolutions. Some special cases where the problem is coercive are also discussed. Finally, the analysis is extended to cases with nonlinear material properties.

The Ginzburg–Landau model for superconductivity is examined in the one-dimensional case. First, putting the Ginzburg–Landau parameter κ formally equal to infinity, the existence of a minimizer of this reduced Ginzburg–Landau energy is proved. Then asymptotic behaviour for large κ of minimizers of the full Ginzburg–Landau energy is analysed and different convergence results are obtained, according to the exterior magnetic field. Numerical computations illustrate the various behaviours.