The characterisation of the bulk energy density of the relaxation in W1, P(Ω; ℝd) of a functional

is obtained for p > q – q/N, where u ∈ W1, P(Ω; ℝd), and f is a continuous function on the set of d × N matrices verifying

for some constant C > 0 and 1 ≦ q < + ∞. Typical examples may be found in cavitation and related theories. Standard techniques cannot be used due to the gap between the exponent q of the growth condition and the exponent p of the integrability of the macroscopic strain ∇u. A recently introduced global method for relaxation and fine Sobolev trace and extension theorems are applied.

Sharp weak type (1,1) and Lp estimates in dimension one are obtained for uncentred maximal functions associated with Borel measures which do not necessarily satisfy a doubling condition. In higher dimensions, uncentred maximal functions fail to satisfy such estimates. Analogous results for centred maximal functions are given in all dimensions.

We first prove existence and uniqueness of non-negative solutions of the equation

in in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the form

where g = ga satisfies

After uniqueness is proved, the asymptotic behaviour of solutions of

is studied. In particular, we show that

The case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

We extend Zygmund's Theorem characterising the Bloch functions via a generalised Libera transform and so we answer an open problem formulated by N. Danikas, S. Ruscheweyh and A. Siskakis. Furthermore, we show some differences between the holomorphic Zygmund class and the class of holomorphic functions whose derivatives are of logarithmic growth on the unit disk.

This paper consists of two main parts. The first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results. The second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum.

An infinite system of reaction–diffusion equations that represents a particular case of the discrete coagulation–fragmentation model with diffusion is studied. The reaction part of the model describes the rate of clusters break-up into smaller particles. Diffusion constants are assumed to be different in each equation and concentration-dependent fragmentation coefficients are considered. Existence of solutions is studied under fairly general assumptions on fragmentation coefficients and initial data. Uniqueness in the class of mass-preserving solutions is proved. Convergence of solutions to spatially homogeneous equilibrium state is obtained.

The maximum and anti-maximum principles are extended to the case of eigenvalue Sturm–Liouville problems

with boundary conditions of Dirichlet type (if possible) on a bounded interval [a, b]. The function r is assumed to be continuous and > 0 on ]a, b[, but the function 1/r is not necessarily integrable on [a, b]. The conditions on the functions p, m and h depend on the integrability or nonintegrability of 1/r on [a, c] and/or [c, b], for some c ∈ ]a, b[. The weight function m is not necessarily of constant sign.

We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.

Results by Simader, Brézis and Cycon of the genre ‘locally essentially self-adjoint implies globally essentially self-adjoint’ are generalised to Schrödinger operators that are not necessarily bounded from below.

We prove that a class of weighted semilinear reaction diffusion equations on RN generates gradient-like semiflows on the Banach space of bounded uniformly continuous functions on RN. If N = 1 we show convergence to a single equilibrium. The key for getting the result is to show the exponential decay of the stationary solutions, which is obtained by means of a decay estimate of the kernel of the underlying semigroup.

Problems associated with m-ary trees have been studied by computer scientists and combinatorialists. It is well known that a simple generalization of the Catalan numbers counts the number of m-ary trees on n nodes. In this paper we consider τm, n, the number of m-ary search trees on n keys, a quantity that arises in studying the space of m-ary search trees under the uniform probability model. We prove an exact formula for τm, n, both by analytic and by combinatorial means. We use uniform local approximations for sums of i.i.d. random variables to study the asymptotic development of τm, n for fixed m as n→∞.

The Turán Number T(n, k, r) is the smallest possible number of edges in a k-graph of ordern such that every set of r vertices contains an edge. The limit

formula here

exists, but there is no pair (k, r) with r>k[ges ]3 for which this function could be determined as yet. We give a constructive proof of the upper bound

formula here

for every k and r with r[ges ]k[ges ]2. In the case k=6, r=11 we improve this result, refuting thereby a conjecture of Turán.

In this paper the method of inner and outer sums [5], together with the computational power of computer symbolic manipulation, are used to extend to high order the asymptotic expansions in an appropriate limit of some infinite series arising in low Reynolds-number fluid mechanics. The enhanced applicability of the expansions is demonstrated, and the method is extended to treat alternating series.

The forms under discussion are integral positive definite quadratic forms in three variables. Such a form g is called regular if g represents every integer represented by the genus of g. This can be recast in elementary terms: g is regular if the solvability of g≡a (mod n) for every n implies the solvability of g = a.

The motivation for this work arises from the study of the processes involved in the manufacturing of a class of composite materials, in particular, those that are obtained by injecting a resin through a porous preform. A one-dimensional model that describes the non-isothermal filtration of an incompressible fluid is presented, and it also includes the possibility of curing, i.e. the polymerization of the penetrating resin. It comes out a fully coupled system consisting of the heat diffusion equation, Darcy's law and an equation related to the kinetics of the chemical reaction. The system is regarded as a free boundary problem for the heat equation with non-constant discontinuous coefficients. Its weak formulation is studied and the local existence of solutions is proved.

The maximal zero-free intervals for chromatic polynomials of graphs are precisely (−∞, 0), (0, 1), (1, 32/27]. We also investigate the distribution of zeros of chromatic polynomials in various classes of graphs closed under minors. For example, the zeros of chromatic polynomials of graphs of tree-width at most k consist of 0, 1 and a dense subset of the interval (32/27, k].

A graph G is called an H-type graph for some graph H if there is a mapping from V(G) to V(H) preserving edges. In this paper, we shall prove that: (1) every triangle-free graph G of order n with χ(G)[les ]3 and δ(G)>n/3 is of Fd-type for some d[ges ]1, where Fd is a certain d-regular triangle-free Hamiltonian Cayley graph of order 3d−1, (2) every triangle-free graph G of order n with χ(G)[ges ]4 and δ(G)>n/3 contains the Mycielski graph (see Figure 2) as a subgraph.

Rabinowitz' global bifurcation theorem shows that for a large class of nonlinear eigenvalue problems a continuum (i.e., a closed, connected set) of solutions bifurcates from the trivial solution at each eigenvalue (or characteristic value) of odd multiplicity of the linearized problem (linearized at the trivial solution). Each continuum must either be unbounded, or must meet some other eigenvalue. This paper considers a class of such nonlinear eigenvalue problems having simple eigenvalues and a “weak” nonlinear term. A result regarding the location of the continua is obtained which shows, in particular, that in this case the bifurcating continua must be unbounded. Also, under further differentiability conditions it is shown that the continua are smooth, 1-dimensional curves and that there are no non-trivial solutions of the equation other than those lying on the bifurcating continua.

In this paper a generalized model for the growth of avascular tumours is presented. The formulation leads naturally to the incorporation of free boundaries which define the outer tumour surface explicitly and various inner surfaces implicitly. A combination of numerical simulations, asymptotic analysis and perturbation techniques is used to study the model and yields results, which agree well with experimentally-observed phenomena.