A rigidity theorem for a space-like graph¶of higher codimension

We prove a rigidity theorem for a space-like graph with parallel mean curvature of arbitrary dimension and codimension in pseudo-Euclidean space via properties of its harmonic Gauss map. We also give an estimate of the squared norm of the second fundamental form in terms of the mean curvature and the image diameter under the Gauss map for space-like submanifolds with parallel mean curvature in pseudo-Euclidean space. The estimate also implies the former theorem. Received: 10 December 1999     

A Kruskal–Katona type theorem for graphs

A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal–Katona theorem. A bound on non-consecutive clique numbers is also proven.     

A Lindeberg–Feller theorem for stable laws

We prove a stable version of the Lindeberg–Feller theorem and apply this result to an approximation of stable processes that are represented by stochastic integrals.     

A Radon–Nikodym type theorem for forms

The Lebesgue decomposition theorem and the Radon–Nikodym theorem are the cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. Hassi, Sebestyén, and de Snoo recently proved a Lebesgue type decomposition theorem for nonnegative sesquilinear forms defined on complex linear spaces. The main purpose of this paper is to formulate and prove also a Radon–Nikodym type result in this setting. As an application, we present a Lebesgue type decomposition theorem and solve a special case of the infimum problem for densely defined (not necessarily bounded) positive operators.     

A local Borg–Marchenko theorem for complex potentials

The Borg–Marchenko theorem states that the Weyl–Titchmarsh m-function of the differential expression −d²/dx²+q with a real-valued potential q determines this potential uniquely. We investigate the validity of the Borg–Marchenko theorem (and its local version) for complex-valued potentials.     

A Grothendieck–Lefschetz theorem for equivariant Picard groups

We prove a Grothendieck–Lefschetz theorem for equivariant Picard groups of non-singular varieties with finite group actions.     

A Rudin–Carleson theorem for planar vector fields

This paper generalizes the Rudin–Carleson theorem for homogeneous solutions of locally solvable real analytic vector fields.     

A Fathi–Siconolfi theorem for codimension one transport

Some features of the Monge–Kantorovich transport problem can be extended to currents of all dimensions; we show that the “Fathi–Siconolfi” theorem is one of them.     

A Wiener–Wintner theorem for the Hilbert transform

We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows (X,μ,T _t ) and f∈L ^p (X,μ), there is a set X _f ⊂X of probability one, so that for all x∈X _f ,

A Cartier–Gabriel–Kostant structure theorem for Hopf algebroids

In this paper we give an extension of the Cartier–Gabriel–Kostant structure theorem to Hopf algebroids.     

A Borel–Weil–Bott type theorem for group completions

We prove a character formula of cohomologies of line bundles on the wonderful completion of an adjoint semisimple algebraic group in the sense of De Concini and Procesi (in our case, we need the general construction of De Concini and Springer).     

A Birch–Goldbach theorem

We prove an analogue of a theorem of Birch with prime variables.     

A theorem of Ambrose for Bakry–Emery Ricci tensor

In this short note, we prove a theorem of Ambrose (or Myers) for the Bakry–Emery Ricci tensor with the potential function at most linear growth. We also prove a complete manifold $(M, g, f)$ with the Bakry–Emery Ricci tensor bounded from below by a uniform positive constant and the potential function at most quadratic growth is compact.     

A Serre–Swan theorem for bundles of bounded geometry

The Serre–Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. In this context it states that the category of Hilbert bundles of bounded geometry is equivalent to the category of operator ?-modules over the operator ?-algebra of continuously differentiable functions which vanish at infinity. Operator ?-modules are generalizations of Hilbert C^?$C^{?}$-modules where the category of C^?$C^{?}$-algebras has been replaced by a more flexible category of involutive algebras of bounded operators: The operator ?-algebras. Operator ?-modules play an important role in the study of the unbounded Kasparov product.