We discuss the isoperimetric problem in planes with density. In particular, we examine planes with generalized curvature zero. We solve the isoperimetric problem on the plane with density ex, as well as on the plane with density rp for p<0. The Appendix provides a proof by Robert Bryant that the Gauss plane has a unique closed geodesic.

In this paper we give a nonexistence theorem for real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) with anti-commuting shape operator.

In this paper, we study the discreteness criteria for nonelementary subgroups of U(1,n;ℂ) acting on complex hyperbolic space. Several discreteness criteria are obtained. As applications, we obtain a classification of nonelementary subgroups of U(1,n;ℂ) and show that any dense subgroup of SU(1,n;ℂ) contains a dense subgroup generated by at most n elements when n≥2. We also obtain a necessary and sufficient condition for the normalizer of a discrete and nonelementary subgroup in SU(1,n;ℂ) to be discrete.

Sharp bounds for the deviation of a real-valued function f defined on a compact interval [a,b] to the chord generated by its end points (a,f(a)) and (b,f(b)) under various assumptions for f and f′, including absolute continuity, convexity, bounded variation, and monotonicity, are given. Some applications for weighted means and f-divergence measures in information theory are also provided.

In this paper we introduce a class of left shift semigroups that are differentiable. With the help of perturbation theory for differentiable semigroups we show that solutions of an integrodifferential equation can be infinitely differentiable if the convolution kernel is sufficiently smooth and regular.

We construct families of irreducible representations for a class of quantum groups Uq(fm(K,H). First, we realize these quantum groups as hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for Uq(fm(K,H)). Second, we study the relationship between Uq(fm(K,H)) and Uq(fm(K)). As a result, any finite-dimensional weight representation of Uq(fm(K,H)) is proved to be completely reducible. Finally, we study the Whittaker model for the center of Uq(fm(K,H)), and a classification of all irreducible Whittaker representations of Uq(fm(K,H)) is obtained.

Let N be a complete Riemannian manifold isometrically immersed into a Hadamard manifold M. We show that the immersion cannot be bounded if the mean curvature of the immersed manifold is small compared with the curvature of M and the Laplacian of the distance function on N grows at most linearly. The latter condition is satisfied if the Ricci curvature of N does not approach too fast. The main tool in the proof is a modification of Yau’s maximum principle.

Much research has been done on the geometry of Teichmüller space and Hamilton sequences of extremal Beltrami differentials. This paper discusses some problems concerning infinitesimal Teichmüller geodesic discs and Hamilton sequences of extremal Beltrami differentials in the tangent space of an infinite-dimensional Teichmüller space.

A nine-dimensional exponential Lie group G and a linear form ℓ on the Lie algebra of G are presented such that for all Pukanszky polarizations 𝔭 at ℓ the canonically associated unitary representation ρ=ρ(ℓ,𝔭) of G has the property that ρ(ℒ1(G)) does not contain any nonzero operator given by a compactly supported kernel function. This example shows that one of Leptin’s results is wrong, and it cannot be repaired.

In a recent paper by Wang and Ding, it was stated that any ring which is generalized supplemented as a left module over itself is semiperfect. The purpose of this note is to show that Wang and Ding’s claim is not true and that the class of generalized supplemented rings lies properly between the classes of semilocal and semiperfect rings. Moreover, we propose a corrected version of the theorem by introducing a wider notion of ‘local’ for submodules.

We study sums involving multiplicative functions that take values over a nonhomogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas for quantities such as the number of integers in a Beatty sequence that are representable as a sum of two squares up to a given magnitude.

Near exact Banach frames are introduced and studied, and examples demonstrating the existence of near exact Banach frames are given. Also, a sufficient condition for a Banach frame to be near exact is obtained. Further, we consider block perturbation of retro Banach frames, and prove that a block perturbation of a retro Banach frame is also a retro Banach frame. Finally, it is proved that if E and F are both Banach spaces having Banach frames, then the product space E×F has an exact Banach frame.

Let R be a left coherent ring, S a right coherent ring and RU a generalized tilting module, with S=End(RU) satisfying the condition that each finitely presented left R-module X with ExtRi(X,U)=0 for any i≥1 is U-torsionless. If M is a finitely presented left R-module such that ExtRi(M,U)=0 for any i≥0 with (where n is a nonnegative integer), then and ExtSi(ExtRn(M,U),U)=0 for any i≥0 with . A duality is thus induced between the category of finitely presented holonomic left R-modules and the category of finitely presented holonomic right S-modules.