This is the third in a series of papers in which we develop a unified method for treating the discrete logarithm problem (DLP) in various contexts. In [HR1], we described a formalism using global duality for a unified approach to the DLP for the multiplicative group and for elliptic curves over finite fields. The main tool to be employed is what we call signature calculus. In [HR2], we used sig...

The language of modern mechanics is calculus on manifolds, and exterior calculus is an important part of that. It consists of objects like differential forms, general tensors and vector fields on manifolds, and operators that act on these. While the smooth exterior calculus has a long history going back to Cartan, Lie, Grassmann, Hodge, de Rham and many others, the need for a discrete calculus ...

linTAP is a tableau prover for the multiplicative and exponential fragment M?LL of Girards linear logic. It proves the validity of a given formula by constructing an analytic tableau and ensures the linear validity using prefix unification. We present the tableau calculus used by linTAP, an algorithm for prefix unification in linear logic, the linTAP implementation, and some experimental result...

The unification and extension of continuous calculus, discrete calculus, q-calculus, and indeed arbitrary real-number calculus to time-scale calculus, where a time scale is simply any nonempty closed set of real numbers, were first accomplished by Hilger in [4]. Since then, time-scale calculus has made steady inroads in explaining the interconnections that exist among the various calculuses, an...

Context dependent rules are an obstacle to cut elimination. A generalised sequent style formulation using deep inferences is helpful and may indeed be essential. First this is shown for a substructural, multiplicative, pure propositional calculus. Then we consider the extra problems induced by non-logical axioms and extend the results to additive connectives and quantifiers.

We introduce proof nets and sequent calculus for the multiplicative fragment of non-commutative logic, which is an extension of both linear logic and cyclic linear logic. The two main technical novelties are a third switching position for the non-commutative disjunction, and the structure of order variety.

Interaction nets are graph rewriting systems which are a generalisation of proof nets for classical linear logic. The linear chemical abstract machine (CHAM) is a term rewriting system which corresponds to classical linear logic, via the Curry-Howard isomorphism. We can obtain a textual calculus for interaction nets which is surprisingly similar to linear CHAM based on the multiplicative fragme...

A general methodology for the solution of partial differential equations is described in which the discretization of the calculus is exact and all approximation occurs as an interpolation problem on the material constitutive equations. The fact that the calculus is exact gives these methods the ability to capture the physics of PDE systems well. The construction of both node and cell based meth...

A discrete total variation calculus is presented in this paper. Using this calculus, we prove that the solution of the system of difference equations in Lee s discrete mechanics preserves a symplectic structure in the space–time sense. Numerical experiments are also reported. 2005 Elsevier Inc. All rights reserved.

Continuous-time models play a central role in the modern theoretical finance literature, while discrete-time models are often used in the empirical finance literature. The continuous-time models are diffusions governed by stochastic differential equations. Most of the discrete-time models are autoregressive conditionally heteroscedastic (ARCH) models and stochastic volatility (SV) models. The d...