We take a second look at two basic topics in the study of weighted convolution algebras L1(ω) on ℝ+. An early result showed that one could replace the weight $\omega$ with a very well-behaved weight without changing the space L1(ω) as long as L1(ω) was an algebraffi We prove the analogous result for measure algebras when M(ω) is an algebraffi This allows us to preserve not only the norm topology but also the relative weak* topology on L1(ω). A homomorphism between weighted convolution algebras is said to be standard if it preserves generators of dense principal ideals. The original proofs of standardness and its variants are all based on finding the generator of a particular strongly continuous convolution semigroup. In this paper we give much simpler direct proofs of these results. We also improve the statement and proof of the theorem, giving useful properties equivalent to the standardness of a homomorphism.

This paper is mainly concerned with the critical extinction and blow-up exponents for the homogeneous Dirichlet boundary-value problem of the fast diffusive polytropic filtration equation with reaction sources.

In this paper we prove that, for a type-II1 factor N with a Cartan maximal abelian subalgebra, the Hochschild cohomology groups Hn(N,N)=0 for all n≥1. This generalizes the result of Sinclair and Smith, who proved this for all N having a separable predual.

We deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid and the elliptic hyperboloid in ℝn implies that for the cone in ℝn+1. We also prove a new restriction estimate for any surface in ℝ3 locally isometric to the plane and of finite type.

Using the framework of overpartitions, we give a combinatorial interpretation and proof of the q-Bailey identity. We then deduce from this identity a couple of facts about overpartitions. We show that the method of proof of the q-Bailey identity also applies to the (first) q-Gauss identity.

We extend the group theoretic notions of transfer and stable elements to graded centres of triangulated categories. When applied to the centre Z*(Db(B) of the derived bounded category of a block algebra B we show that the block cohomology H*(B) is isomorphic to a quotient of a certain subalgebra of stable elements of Z*(Db(B)) by some nilpotent ideal, and that a quotient of Z*(Db(B)) by some nilpotent ideal is Noetherian over H*(B).

Direct and converse theorems are established for the q-Bernstein polynomials introduced by G. M. Phillips. The direct approximation theorems are given for the second-order Ditzian–Totik modulus of smoothness. The converse results are theorems of Berens–Lorentz type.

We study the boundedness properties of Rudin–Forelli-type operators associated to tubular domains over symmetric cones. As an application, we give a characterization of the topological dual space of the weighted Bergman space .

In this paper, a theory is developed of generalized oscillatory integrals (OIs) whose phase functions and amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need for a general framework for partial differential operators with non-smooth coefficients and distribution dataffi The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wavefront sets.

We present a new method for solving the minimization problem in ferromagnetism. Our method is based on replacing the non-local non-convex total energy of magnetization by a new local non-convex energy of divergence-free fields. Such a general method works in all dimensions. However, for the two-dimensional case, since the divergence-free fields are equivalent to the rotated gradients, this new energy can be written as an integral functional of gradients and hence the minimization problem can be solved by some recent non-convex minimization procedures in the calculus of variations. We focus on the two-dimensional case in this paper and leave the three-dimensional situation to future work. Special emphasis is placed on the analysis of the existence/non-existence depending on the applied field and the physical domain.

In this paper we obtain the existence of a radial solution for some elliptic non-local problem with constraints. The problem arises from some mean field equation which models, among other things, a system of self-gravitating particles when one looks for its stationary solutions. We include the cases of Maxwell—Boltzmann, Fermi—Dirac and polytropic statistics.

In this article, we study the existence of travelling waves for a class of epidemic models structured in space and with respect to the age of infection. We obtain a necessary and sufficient condition for the existence of travelling waves for such a class of problems. As a consequence of our main result, we also derive the existence of travelling waves of a class of functional partial derivative equations.

We construct a duality theory for (a, k)-regularized resolvents, extending some of the known theorems for dual semigroups. We present several classes of spaces, which in the semigroup case correspond to the Favard class and the sun-dual space. By duality arguments spectral inclusion theorems for regularized resolvents are also obtained.

We prove that the triangle inequalities in general are not equivalent to lower semi-continuity of surface energy. In 1990, Ambrosio and Braides proved that the two conditions are equivalent when the number of regions, s, is three. They gave an example in the plane showing that the triangle inequalities do not imply lower semi-continuity when s ≥ 6. The cases when s = 4 and s = 5 have remained open. In this paper, we resolve these open questions and show that, in ℝm, the triangle inequalities are sufficient for lower semi-continuity if and only if s = 3.

We establish some regularity criteria in terms of the velocity field with weight for the Navier—Stokes equations in ℝ3. It is proved that if the weak solution satisfies

or

then the weak solution actually is regular up to time T.

Let X be a finite spectrum. We prove that R(X(p)), the endomorphism ring of the p-localization of X, is a semi-perfect ring. This implies, among other things, a strong form of unique factorization for finite p-local spectra. The main step in the proof is that the Jacobson radical of R(X(p)) is idempotent-lifting, which is proved by a combination of geometric properties of finite spectra and algebraic properties of the p-localization.

This paper concerns the homogenization problem for fully nonlinear first-order equations of Hamilton—Jacobi type with a finite number of scales. Under some coercivity and periodicity assumptions we provide an estimate of the rate of convergence. Finally, some examples arising from optimal control and deterministic differential games are discussed.

Let a, c ≥ 0 and let B be a compact set of scalars. We show that if X is a Banach space such that the canonical projection π from X*** onto X* satisfies the inequality

and 1 ≤ λ < max |B| + c, then every λ-commuting bounded compact approximation of the identity of X is shrinking. This generalizes a theorem by Godefroy and Saphar from 1988. As an application, we show that under the conditions described above both X and X* have the metric compact approximation property (MCAP). Relying on the Willis construction, we show that the commuting MCAP does not imply the approximation property.

We consider a singular Sturm—Liouville expression with the indefinite weight sgn x. There is a self-adjoint operator in some Krein space associated naturally with this expression. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgn x)(−d2/dx2 + q) with non-real spectrum accumulating to a real point. The results obtained are applied to several classes of Sturm—Liouville operators.

We investigate real convex-transitive Banach spaces X, which admit a one-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. For example, if uSX is a big point such that there is a bicontractive linear projection P : X → [u] and X* is weak*-locally uniformly rotund, then X is a Hilbert space. The results obtained here are motivated by the well-known Banach—Mazur rotation problem, as well as a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.