50,157 research outputs found
Bounded -calculus for cone differential operators
We prove that parameter-elliptic extensions of cone differential operators
have a bounded -calculus. Applications concern the Laplacian and the
porous medium equation on manifolds with warped conical singularities
Universal Malliavin calculus in Fock and Levy-Ito spaces
We review and extend Lindsay's work on abstract gradient and divergence operators in Fock space over a general complex Hilbert space. Precise expressions for the domains are given, the L2-equivalence of norms is proved and an abstract version of the It^o-Skorohod isometry is established. We then outline a new proof of It^o's chaos expansion of complex Levy-It^o space in terms of multiple Wiener-Levy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform
now identies Levy-It^o space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general Levy process. In particular we establish maximal domains of these operators and obtain
the It^o-Skorohod isometry on its maximal domain
Denotational Semantics of the Simplified Lambda-Mu Calculus and a New Deduction System of Classical Type Theory
Classical (or Boolean) type theory is the type theory that allows the type
inference (the type counterpart of
double-negation elimination), where is any type and is
absurdity type. This paper first presents a denotational semantics for a
simplified version of Parigot's lambda-mu calculus, a premier example of
classical type theory. In this semantics the domain of each type is divided
into infinitely many ranks and contains not only the usual members of the type
at rank 0 but also their negative, conjunctive, and disjunctive shadows in the
higher ranks, which form an infinitely nested Boolean structure. Absurdity type
is identified as the type of truth values. The paper then presents a new
deduction system of classical type theory, a sequent calculus called the
classical type system (CTS), which involves the standard logical operators such
as negation, conjunction, and disjunction and thus reflects the discussed
semantic structure in a more straightforward fashion.Comment: In Proceedings CL&C 2016, arXiv:1606.0582
On First-Order μ-Calculus over Situation Calculus Action Theories
In this paper we study verification of situation calculus action theories against first-order mu-calculus with quantification across situations. Specifically, we consider mu-La and mu-Lp, the two variants of mu-calculus introduced in the literature for verification of data-aware processes. The former requires that quantification ranges over objects in the current active domain, while the latter additionally requires that objects assigned to variables persist across situations. Each of these two logics has a distinct corresponding notion of bisimulation. In spite of the differences we show that the two notions of bisimulation collapse for dynamic systems that are generic, which include all those systems specified through a situation calculus action theory. Then, by exploiting this result, we show that for bounded situation calculus action theories, mu-La and mu-Lp have exactly the same expressive power. Finally, we prove decidability of verification of mu-La properties over bounded action theories, using finite faithful abstractions. Differently from the mu-Lp case, these abstractions must depend on the number of quantified variables in the mu-La formula
Incompleteness of States w.r.t. Traces in Model Checking
Cousot and Cousot introduced and studied a general past/future-time
specification language, called mu*-calculus, featuring a natural time-symmetric
trace-based semantics. The standard state-based semantics of the mu*-calculus
is an abstract interpretation of its trace-based semantics, which turns out to
be incomplete (i.e., trace-incomplete), even for finite systems. As a
consequence, standard state-based model checking of the mu*-calculus is
incomplete w.r.t. trace-based model checking. This paper shows that any
refinement or abstraction of the domain of sets of states induces a
corresponding semantics which is still trace-incomplete for any propositional
fragment of the mu*-calculus. This derives from a number of results, one for
each incomplete logical/temporal connective of the mu*-calculus, that
characterize the structure of models, i.e. transition systems, whose
corresponding state-based semantics of the mu*-calculus is trace-complete
L^2-Theory for non-symmetric Ornstein-Uhlenbeck semigroups on domains
We present some new results on analytic Ornstein-Uhlenbeck semigroups and use
them to extend recent work of Da Prato and Lunardi for Ornstein-Uhlenbeck
semigroups on open domains O to the non-symmetric case. Denoting the generator
of the semigroup by L_O, we obtain sufficient conditions in order that the
domain Dom(\sqrt{-L_O}) be a first order Sobolev space.Comment: 23 pages, revised version, to appear in J. Evol. Eq. The main change
is a correction in Theorem 5.5: the second assertion has been withdrawn due
to a gap in the original proo
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