We consider a new way of looking at multisets, involving linear rather than multiplicative notation and operations which mirror those on natural numbers. The motivation is to create a framework for mathematics which more closely connects with scientific usage in which repetition and relative concepts of equality play an important role. The tropical calculus is introduced as an important tool. a...

We present a proofs-as-programs correspondence between linear logic and process calculi that permits non-deterministic behaviours. Processes are translated into wireless proof nets, i.e. proof nets of multiplicative linear logic without cut wires. Typing a term using them consists in typing some of its possible determinisations in standard sequent calculus. Generalized cut-elimination steps in ...

This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classical logic in that its sequent-calculus formulation has intrinsic multiple conclusions. From the intrin...

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by J....

The Langevin equation with a multiplicative Lévy white noise is solved. The noise amplitude and the drift coefficient have a power-law form. A validity of ordinary rules of the calculus for the Stratonovich interpretation is discussed. The solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the Stratonovich interpretation. The problem of escapin...

there are several kinds of hyperrings, for example, krasnerhyperrings, multiplicative hyperring, general hyperrings and$h_v$-rings. in a multiplicative hyperring, the multiplication isa hyperoperation, while the addition is a binary operation. in this paper, the notion of derivation on multiplicative hyperrings is introduced and some related properties are investigated. {bf keywords:} multiplic...

System NEL is a conservative extension of multiplicative exponential linear logic (MELL) by a self-dual non-commutative connective called seq which lives between the par and the times. In this paper, I will show that system NEL is undecidable by encoding two counter machines into NEL. Although the encoding is quite simple, the proof of the faithfulness is a little intricate because there is no ...

System NEL is a conservative extension of multiplicative exponential linear logic (extended by the rules mix and nullary mix) by a self-dual noncommutative connective called seq which has an intermediate position between the connectives par and times. In this paper, I will show that system NEL is undecidable by encoding two counter machines into NEL. Although the encoding is simple, the proof o...

A many-valued modal logic is introduced that combines the standard (crisp) Kripke frame semantics of the modal logic K with connectives interpreted locally as abelian group operations over the real numbers. A labelled tableau system and a sequent calculus admitting cut elimination are then defined for this logic and used to establish completeness of an axiomatic extension of the multiplicative ...

Let V be a symplectic vector space and LG be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in V . We give a presentation for the (small) quantum cohomology ring QH∗(LG) and show that its multiplicative structure is determined by the ring of Q̃-polynomials. We formulate a ‘quantum Schubert calculus’ which includes quantum Pieri and Giambelli formulas, as well as algor...