In 1958 J. Lambek introduced a calculus L of syntactic types and defined an equivalence relation on types: “x ≡ y means that there exists a sequence x = x1, . . . , xn = y (n ≥ 1), such that xi → xi+1 or xi+1 → xi (1 ≤ i < n)”. He pointed out that x ≡ y if and only if there is join z such that x → z and y → z. This paper gives an effective characterization of this equivalence for the Lambek cal...

We showed that is possible to define a derivative and a integral of a sequence which satisfy similar proprieties of the derivative and integral of the usual differential and integral calculus. We also study convexity of sequences that can be defined as usual by the second derivative. 1 A little introduction on calculus of finite differences We will introduce some definitions and notation from t...

martindale proved that under some conditions every multiplicative isomorphism between two rings is additive. in this paper, we extend this theorem to a larger class of mappings and conclude that every multiplicative $(alpha, beta)-$derivation is additive.

For public key cryptosystems multiplication on elliptic curves can be used instead of exponentiation in finite fields. One attack to such a system is: embedding the elliptic curve group into the multiplicative group of a finite field via weilpairing; calculating the discrete logarithm on the curve by solving the discrete logarithm in the finite field. This attack can be avoided by constructing ...

We present the logic of bunched implications, BI, in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication, and may be viewed as a merging of intuitionistic logic and multiplicative, intuitionistic linear logic. The predi...

We extend the multiplicative fragment of linear logic with a non-commutative connective (called before), which, roughly speaking, corresponds to sequential composition. This lead us to a calculus where the conclusion of a proof is a Partially Ordered MultiSET of formulae. We rstly examine coherence semantics, where we introduce the before connective, and ordered products of formulae. Secondly w...

This paper will review recent advances in the formulation of a discrete version of geometric mechanics, based on the discretization of Hamilton’s variational principle, and progress that has been made in reduction theory for discrete variational mechanics in the form of Discrete Routh Reduction. To place discrete geometric mechanics on a firm mathematical foundation, we propose to develop a dis...

We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations using finite element or discrete exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other things, this allows for localization near topol...

Mimetic discretization methods for the numerical solution of continuum mechanics problems rely on analogs of vector calculus and differential forms identities in their derivation and analysis. Fully mimetic discretizations satisfy discrete analogs of the continuum theory results used to derive energy inequalities. Consequently, continuum arguments carry over and can be used to show that discret...