Countable compact Hausdorff spaces need not be metrizable in ZF

We show that the existence of a countable, first countable, zero-dimensional, compact Hausdorff space which is not second countable, hence not metrizable, is consistent with ZF.

     

A note on circle patterns on surfaces

In this paper we give two different proofs of Bobenko and Springborn’s theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface up to isometry (or similarity). This work is partially supported by NSF Grant #0625935.     

Fragments of Martin's Maximum in generic extensions

We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly (ω₁ + 1)‐game‐closed forcings. PFA can be destroyed by a strongly (ω₁ + 1)‐game‐closed forcing but not by an ω₂‐closed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hausdorff properties of topological algebras

Let P be a property of topological spaces. Let [P] be the class of all varieties having the property that any topological algebra in has underlying space satisfying property P. We show that if P is preserved by finite products, and if is preserved by ultraproducts, then [P] is a class of varieties that is definable by a Maltsev condition.?The property that all T ₀ topological algebras in are j-step Hausdor. (Hj) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize the Maltsev condition associated to by showing that this topological implication holds in every (2j + 1)-permutable variety, but not in every (2j + 2)-permutable variety.?Finally, we show that the topological implication holds in every k-permutable, congruence modular variety. Received March 1, 2000; accepted in final form October 18, 2001.     

Diffuse interface tracking of immiscible fluids: Improving phase continuity through free energy density selection

Diffuse interface (DI) tracking methods frequently adopt the double-well energy density function to describe the free energy variation across an interface, leading to phase interpenetration and spontaneous drop shrinkage when applied to immiscible two-phase systems. While the observed continuity losses can be limited by constraints placed on the interfacial width and mobility parameter, the associated increase in computational cost and mesh requirements has limited DI methods to 2D planar and axi-symmetric flow.     

The topology of a semisimple Lie group is essentially unique

We study locally compact group topologies on simple and semisimple Lie groups. We show that the Lie group topology on such a group S is very rigid: every “abstract” isomorphism between S and a locally compact and σ-compact group Γ is automatically a homeomorphism, provided that S is absolutely simple. If S is complex, then noncontinuous field automorphisms of the complex numbers have to be considered, but that is all. We obtain similar results for semisimple groups.     

Continuity of spectrum and approximate point spectrum on operator matrices

Let X and Y be given Banach spaces. For A∈B(X), B∈B(Y) and C∈B(Y,X), let MC be the operator defined on X⊕Y by . In this paper we give conditions for continuity of τ at MC through continuity of τ at A and B, where τ can be equal to the spectrum or approximate point spectrum.     

Hölder Continuity of Roots of Complex and p-adic Polynomials

Let f be a polynomial with coefficients in an algebraically closed, valued field. We show a refinement of the principle of continuity of roots, namely, that each root α of f is locally Hölder continuous of order 1/μ as a function of the coefficients of f, where μ is the root multiplicity of α. This is derived as a consequence of a principle that could be called continuity of factors, namely, that if f = gh is a factorisation with (g, h) = 1, then the coefficients of g and h are locally Lipschitz continuous as functions of the coefficients of f. The proofs are elementary and of an algebraic nature.     

The Boolean prime ideal theorem and products of cofinite topologies

We show:

1. The Boolean Prime Ideal theorem is equivalent to each one of the statements:
   1. “For every family of compact spaces, for every family of basic closed sets of the product with the fip there is a family of subbasic closed sets () with the fip such that for every ”.
   2. “For every compact Loeb space (the family of all non empty closed subsets of has a choice function) and for every set X the product is compact”.
2. (: the axiom of choice restricted to families of finite sets) implies “every well ordered product of cofinite topologies is compact” and “every well ordered basic open cover of a product of cofinite topologies has a finite subcover”.
3. (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”.
4. (: every filter of extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space having a base of size and for every set X of size the product is compact”.

     

Perpetual Bermudan Continuity Corrections and a Multi-Dimensional Wiener–Hopf Type Result

Abstract

In this article, Wiener–Hopf type results by Feller are generalized to higher dimensions, in order to derive bounds on the continuity corrections at the exercise boundary for certain perpetual Bermudan options on multiple assets. We assume that the vector of logarithmic price processes of the underlyings is a Lévy process under some risk-neutral measure. In addition, we have to impose the condition that the payoff functions of these perpetual Bermudans have been chosen in such a way that the corresponding optimal exercise regions of the Bermudan options are, up to translation, a half-space. (This is, of course, a fairly restrictive assumption for higher dimensions, but none for dimension one.)