We consider positive linear operators of probabilistic type L1f acting on real functions f defined on the positive semi-axis. We deal with the problem of uniform convergence of L1f to f, both in the usual sup-norm and in a uniform Lp type of norm. In both cases, we obtain direct and converse inequalities in terms of a suitable weighted first modulus of smoothness of f. These results are applied to the Baskakov operator and to a gamma operator connected with real Laplace transforms, Poisson mixtures and Weyl fractional derivatives of Laplace transforms.

The classical notion of a two-dimensional develpable surface in Euclidean three-space is extended to the case of arbitrary dimension and codimension. A collection of characteristic properties is presented. The theorems are stated with the minimal possible integer smoothness. The main tool of the investigation is Cartan's moving frame method.

Suppose that G is a π-separable group. Let N be a normal π1-subgroup of G and let H be a Hall π-subgroup of G. In this paper, we prove that there is a canonical basis of the complex space of the class functions of G which vanish of G-conjugates ofHN. When N = 1 and π is the complement of a prime p, these bases are the projective indecomposable characters and set of irreduciblt Brauer charcters of G.

An n-hedral tiling of ℝd is a tiling with each tile congruent to one of the n distinct sets. In this paper, we use the iterated function systems (IFS) to generate n-hedral tilings of ℝd. Each tile in the tiling is similar to the attractor of the IFS. These tiles are fractals and their boundaries have the Hausdorff dimension less than d. Our results generalize a result of Bandt.

In this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.

Let Tβ be the β-transformation on [0, 1). When β is an integer Tβ is ergodic with respect to Lebesgue measure and almost all orbits {} are uniformly distributed. Here we consider the non-integer case, determine when Tα, Tβ have the same invariant measure and when (appropriately normalised) orbits are uniformly distributed.