THOUGH the details of mathematical science, \>y their very nature, elude the comprehension of the layman, and therefore fail to arouse his interest, yet the mathematician may profitably indicate certain general points of view from which he surveys his science, especially if these points of view determine his attitude to kindred subjects. I propose therefore on the present occasion to explain my...

In this brief review we describe the coming of age of Magnus expansions as an asymptotic and numerical tool in the investigation of linear differential equations in a Lie-group and homogeneous-space setting. Special attention is afforded to the many connections between modern theory of geometric numerical integration and other parts of mathematics: from abstract algebra to differential geometry...

In this paper we survey various set-theoretic approaches that have been proposed over the last thirty years as foundational frameworks for the use of nonstandard methods in mathematics. Introduction. Since the early developments of calculus, infinitely small and infinitely large numbers have been the object of constant interest and great controversy in the history of mathematics. In fact, while...

Phase space can be constructed for N equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy S(BG) identical with -k Sigma(i) p(i) ln p(i) to be extensive, i.e., S(BG)(N) proportional, variant N for N --> infinity. In particular, if they are independent, S(BG) is s...

Our world has been shaped by the outcome of the scientific revolution and its offspring, the industrial revolution. The essence of this scientific revolution, shaped by its pioneers – Newton, Gallileo, Descartes, Kepler, Huygens, Boyle, Leibnitz – was in the rigorous understanding of the physical world, the laws underlying matter, energy and their interaction. The word “rigorous” is a hint to t...

With each coassociative coalgebra, we associate an oriented graph. The coproduct ∆, obeying the coassociativity equation (∆ ⊗ id)∆ = (id ⊗ ∆)∆ is then viewed as a physical propagator which can convey information. We notice that such a coproduct is non local. To recover locality we have to break the coassociativity of the coproduct and restore it, in some sense by introducing another coproduct ∆̃...

The fundamental result of stochastic calculus is Itô’s formula (2.1) firstly established by Itô [5] for a standard Brownian motion and then later extended to continuous semimartingales by Kunita and Watanabe [7]. [For simplicity in this article we will not consider semimartingales with jumps.] The function F appearing in Itô’s formula is C2 in the space variable, and the correction to the class...

Abstract In this survey, we review the classical Hamilton–Jacobi theory from a geometric point of view in different backgrounds. We propose equation for structures attending to one particular characterization: whether they fulfill Jacobi and Leibniz identities simultaneously, or if at least satisfy them. This property reviews work present authors such way that it is presented according hierarch...