1,273 research outputs found
Arcwise Analytic Stratification, Whitney Fibering Conjecture and Zariski Equisingularity
In this paper we show Whitney's fibering conjecture in the real and complex,
local analytic and global algebraic cases.
For a given germ of complex or real analytic set, we show the existence of a
stratification satisfying a strong (real arc-analytic with respect to all
variables and analytic with respect to the parameter space) trivialization
property along each stratum. We call such a trivialization arc-wise analytic
and we show that it can be constructed under the classical Zariski
algebro-geometric equisingularity assumptions. Using a slightly stronger
version of Zariski equisingularity, we show the existence of Whitney's
stratified fibration, satisfying the conditions (b) of Whitney and (w) of
Verdier. Our construction is based on Puiseux with parameter theorem and a
generalization of Whitney interpolation. For algebraic sets our construction
gives a global stratification.
We also give several applications of arc-wise analytic trivialization, mainly
to the stratification theory and the equisingularity of analytic set and
function germs. In the real algebraic case, for an algebraic family of
projective varieties, we show that Zariski equisingularity implies local
triviality of the weight filtration.Comment: 45 pages, new constructive proof ot the main resul
Standard State Space Models of Unawareness
The impossibility theorem of Dekel, Lipman and Rustichini has been thought to demonstrate
that standard state-space models cannot be used to represent unawareness. We first show that Dekel,
Lipman and Rustichini do not establish this claim. We then distinguish three notions of awareness,
and argue that although one of them may not be adequately modeled using standard state spaces,
there is no reason to think that standard state spaces cannot provide models of the other two notions.
In fact, standard space models of these forms of awareness are attractively simple. They allow us
to prove completeness and decidability results with ease, to carry over standard techniques from
decision theory, and to add propositional quantifiers straightforwardly
Introduction to Cirquent Calculus and Abstract Resource Semantics
This paper introduces a refinement of the sequent calculus approach called
cirquent calculus. While in Gentzen-style proof trees sibling (or cousin, etc.)
sequents are disjoint sequences of formulas, in cirquent calculus they are
permitted to share elements. Explicitly allowing or disallowing shared
resources and thus taking to a more subtle level the resource-awareness
intuitions underlying substructural logics, cirquent calculus offers much
greater flexibility and power than sequent calculus does. A need for
substantially new deductive tools came with the birth of computability logic
(see http://www.cis.upenn.edu/~giorgi/cl.html) - the semantically constructed
formal theory of computational resources, which has stubbornly resisted any
axiomatization attempts within the framework of traditional syntactic
approaches. Cirquent calculus breaks the ice. Removing contraction from the
full collection of its rules yields a sound and complete system for the basic
fragment CL5 of computability logic. Doing the same in sequent calculus, on the
other hand, throws out the baby with the bath water, resulting in the strictly
weaker affine logic. An implied claim of computability logic is that it is CL5
rather than affine logic that adequately materializes the resource philosophy
traditionally associated with the latter. To strengthen this claim, the paper
further introduces an abstract resource semantics and shows the soundness and
completeness of CL5 with respect to it.Comment: To appear in Journal of Logic and Computatio
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