We establish new sampling representations for linear integral transforms associated with arbitrary general Birkhoff regular boundary value problems. The new approach is developed in connection with the analytical properties of Green’s function, and does not require the root functions to be a basis or complete. Unlike most of the known sampling expansions associated with eigenvalue problems, the results obtained are, generally speaking, of Hermite interpolation type.

For a class of convex curves in ℝd we prove that the corresponding maximal operator and Hilbert transform are of weak type Llog L. The point of interest here is that this class admits curves which are infinitely flat at the origin. We also prove an analogous weak type result for a class of nonconvex hypersurfaces.

A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.

Let G be a finite group and let δ(G) be the number of prime order subgroups of G. We determine the groups G with the property δ(G)≥∣G∣/2−1, extending earlier work of C. T. C. Wall, and we use our classification to obtain new results on the generation of near-rings by units of prime order.

We study an important class of Finsler metrics, namely, Randers metrics. We classify Randers metrics of scalar flag curvature whose S-curvatures are isotropic. This class of Randers metrics contains all projectively flat Randers metrics with isotropic S-curvature and Randers metrics of constant flag curvature.

In this note, we provide an example of a Banach space X for which that is not isomorphic to any Hilbert space, where denotes the infimum of all von Neumann–Jordan constants for equivalent norms of X.

This paper continues the investigation of semigroup constructions motivated by applications in data mining. We give a complete description of the error-correcting capabilities of a large family of clusterers based on Rees matrix semigroups well known in semigroup theory. This result strengthens and complements previous formulas recently obtained in the literature. Examples show that our theorems do not generalize to other classes of semigroups.

In basic homological algebra, projective, injective and flat modules play an important and fundamental role. In this paper, we discuss some properties of Gorenstein projective, injective and flat modules and study some connections between Gorenstein injective and Gorenstein flat modules. We also investigate some connections between Gorenstein projective, injective and flat modules under change of rings.

For a class of ‘linear’ sudoku solutions, we construct mutually orthogonal families of maximal size for all square orders, and we show that all such solutions must lie in the same orbit of a symmetry group preserving sudoku solutions.

We investigate an inverse spectral problem for the singular rank-one perturbations of a Hill operator. We give a necessary and sufficient condition for a real sequence to be the spectrum of a singular rank-one perturbation of the Hill operator.