Search results for "Mathematica"

Hadamard-type theorems for hypersurfaces in hyperbolic spaces

2006

Abstract We prove that a bounded, complete hypersurface in hyperbolic space with normal curvatures greater than −1 is diffeomorphic to a sphere. The completeness condition is relaxed when the normal curvatures are bounded away from −1. The diffeomorphism is constructed via the Gauss map of some parallel hypersurface. We also give bounds for the total curvature of this parallel hypersurface.

Stability conditions and related filtrations for $(G,h)$-constellations

2017

Given an infinite reductive algebraic group $G$, we consider $G$-equivariant coherent sheaves with prescribed multiplicities, called $(G,h)$-constellations, for which two stability notions arise. The first one is analogous to the $\theta$-stability defined for quiver representations by King and for $G$-constellations by Craw and Ishii, but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for $(G,h)$-constellations, and depends on some finite subset $D$ of the isomorphy classes of irreducible representations of $G$. We show that these two stability notions do not coincide, answering negatively a question raise…

2-Potenzen der ordnung von Einheitengruppen unimodularer definiter ?-Gitter

1990

Holomorphic Maps and Pencils of Circles

2008

(2008). Holomorphic Maps and Pencils of Circles. The American Mathematical Monthly: Vol. 115, No. 8, pp. 690-700.

The Composition Operation on Spaces of Holomorphic Mappings

2020

AbstractWe discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.

Lower Semi-frames, Frames, and Metric Operators

2020

AbstractThis paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

Rigidity of commutators and elementary operators on Calkin algebras

1998

LetA=(A 1,...,A n ),B=(B 1,...,B n )eL(l p ) n be arbitraryn-tuples of bounded linear operators on (l p ), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators e a,b on the Calkin algebraC(l p )≡L(l p )/K(l p ); $$\varepsilon _{\alpha ,b} (s) = \sum\limits_{i = 1}^n {a_i sb_i } $$ , where quotient elements are denoted bys=S+K(l p ) forSeL(l p ). It is shown among other results that the kernel Ker(e a,b ) is a non-separable subspace ofC(l p ) whenever e a,b fails to be one-one, while the quotient $$C(\ell ^p )/\overline {\operatorname{Im} \left( {\varepsilon _{\alpha ,b} } \right)} $$ is non-separable whenever e a,b fails to be onto. These re…

On Serrin’s overdetermined problem in space forms

2018

We consider Serrin’s overdetermined problem for the equation $$\Delta v + nK v = -\,1$$ in space forms, where K is the curvature of the space, and we prove a symmetry result by using a P-function approach. Our approach generalizes the one introduced by Weinberger to space forms and, as in the Euclidean case, it provides a short proof of the symmetry result which does not make use of the method of moving planes.

Interpolation properties of Besov spaces defined on metric spaces

2010

Let X = (X, d, μ)be a doubling metric measure space. For 0 < α < 1, 1 ≤p, q < ∞, we define semi-norms When q = ∞ the usual change from integral to supremum is made in the definition. The Besov space Bp, qα (X) is the set of those functions f in Llocp(X) for which the semi-norm ‖f ‖ is finite. We will show that if a doubling metric measure space (X, d, μ) supports a (1, p)-Poincare inequality, then the Besov space Bp, qα (X) coincides with the real interpolation space (Lp (X), KS1, p(X))α, q, where KS1, p(X) is the Sobolev space defined by Korevaar and Schoen [15]. This results in (sharp) imbedding theorems. We further show that our definition of a Besov space is equivalent with the definiti…

Dorronsoro's theorem in Heisenberg groups

2020

A theorem of Dorronsoro from the 1980s quantifies the fact that real-valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last variable. As an application, we deduce new proofs for certain vertical vs. horizontal Poincare inequalities for real-valued functions on the Heisenberg group, originally due to Austin-Naor-Tessera and Lafforgue-Naor.