This article defines and studies a natural deduction system for partially commutative intuitionistic multiplicative linear logic, that is a combination of intuitionistic commutative linear logic with the Lambek calculus, which is noncommutative, and was first introduced as a sequent calculus by de Groote. In this logic, the hypotheses are endowed with a series-parallel partial order: the parall...

We give some basic results of the theory of proof nets for multiplicative linear logic and multiplicative exponential linear logic with mix. The relation between proof nets and the lambda-calculus is precisely described. 1 Multiplicative Proof Nets 1.1 Multiplicative Linear Logic We assume given a denumerable set of atoms X, Y , . . . The formulas of multiplicative linear logic (MLL) are define...

This paper focuses on the study of existence a mild solution to time and space-fractional stochastic equation perturbed by multiplicative white noise. The required results are obtained analysis techniques, fractional calculus, semigroup theory Leray-Schauder nonlinear alternative.

The paper develops multiplicative compensation for complex-valued semimartingales and studies some of its consequences. It is shown that the stochastic exponential any semimartingale with independent increments becomes a true martingale after when such meaningful. This generalization L\'evy--Khintchin formula fills an existing gap in literature. allows, example, computation Mellin transform sig...

The aim of this work is to give an alternative presentation for the multiplicative fragment of Yetter’s cyclic linear logic. The new presentation is inspired by the calculus of structures, and has the interesting feature of avoiding the cyclic rule. The main point in this work is to show how cyclicity can be substituted by deepness, i.e. the possibility of applying an inference rule at any poin...

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...