We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders. Supported by NSA...

We investigate several fragments of multiplicative linear logic, in a natural deduction setting and with the aim of a better understanding of the par connective. We study, first, a pre-tensorial calculus, which is strengthened then in the standard tensorial fragment. The addition of a further pre-tensorial connective yields (a natural deduction version of) Full Intuitionistic Linear Logic. A fu...

The paper presents a way to transform pregroup grammars into contextfree grammars using functional composition. The same technique can also be used for the proof-nets of multiplicative cyclic linear logic and for Lambek calculus allowing empty premises.

In this paper, a novel variational model with strict convexity for removing multiplicative noise from images is proposed and studied. Firstly, by applying maximum likelihood estimation method and the Bayesian formulation, the variational model is derived. Then, we use an alternating minimization algorithm to find out the minimizer of the objective function, and prove the existence of the minimi...

In this article, a class of second order differential equations on [0, 1], driven by a γ-Hölder continuous function for any value of γ ∈ (0, 1) and with multiplicative noise, is considered. We first show how to solve this equation in a pathwise manner, thanks to Young integration techniques. We then study the differentiability of the solution with respect to the driving process and consider the...

We examine ‘weak-distributivity’ as a rewriting rule ; WD defined on multiplicative proof-structures (so, in particular, on multiplicative proof-nets: MLL). This rewriting does not preserve the type of proofsnets, but does nevertheless preserve their correctness. The specific contribution of this paper, is to give a direct proof of completeness for ; WD : starting from a set of simple generator...

We consider a classical (propositional) version, CBI, of O’Hearn and Pym’s logic of bunched implications (BI) from a modeland prooftheoretic perspective. We present a class of classical models of BI which extend the usual BI-models, based on partial commutative monoids, with an algebraic notion of “resource negation”. This class of models gives rise to natural definitions of multiplicative fals...

The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: In contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than any other formalisms supporting analytical proofs. However, deep applicability of the inference rules causes greater nondeterminism th...

We present two rewriting systems that define labelled explicit substitution λ -calculi. Our work is motivated by the close correspondence between Lévy’s labelled λ -calculus and paths in proof-nets, which played an important role in the understanding of the Geometry of Interaction. The structure of the labels in Lévy’s labelled λ -calculus relates to the multiplicative information of paths; the...