84,362 research outputs found
Properties of the Class of Measure Separable Compact Spaces
We investigate properties of the class of compact spaces on which every
regular Borel measure is separable. This class will be referred to as MS.
We discuss some closure properties of MS, and show that some simply defined
compact spaces, such as compact ordered spaces or compact scattered spaces, are
in MS. Most of the basic theory for regular measures is true just in ZFC. On
the other hand, the existence of a compact ordered scattered space which
carries a non-separable (non-regular) Borel measure is equivalent to the
existence of a real-valued measurable cardinal less or equal to c.
We show that not being in MS is preserved by all forcing extensions which do
not collapse omega_1, while being in MS can be destroyed even by a ccc forcing
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Thin-very tall compact scattered spaces which are hereditarily separable
We strengthen the property of a function considered by Baumgartner and Shelah. This allows us
to consider new types of amalgamations in the forcing used by Rabus, Juh\'asz
and Soukup to construct thin-very tall compact scattered spaces. We
consistently obtain spaces as above where is hereditarily separable
for each . This serves as a counterexample concerning cardinal
functions on compact spaces as well as having some applications in Banach
spaces: the Banach space is an Asplund space of density which
has no Fr\'echet smooth renorming, nor an uncountable biorthogonal system.Comment: accepted to Trans. Amer. Math. Soc
Indestructibility of compact spaces
In this article we investigate which compact spaces remain compact under
countably closed forcing. We prove that, assuming the Continuum Hypothesis, the
natural generalizations to -sequences of the selection principle and
topological game versions of the Rothberger property are not equivalent, even
for compact spaces. We also show that Tall and Usuba's "-Borel
Conjecture" is equiconsistent with the existence of an inaccessible cardinal.Comment: 18 page
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
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