Intuitively, it is tempting to view a cube in R as a discrete lattice with infinitesimal edges. A similar infinitesimal picture suggests itself with other fractal spaces such as the Sierpinski gasket or the Sierpinski sponge (see [13], [5].) The purpose of this paper is to show certain classes of metric spaces characterized by volume growth properties of balls ([4],[2]) can viewed as graphs wit...

There is an ancient philosphical-religious divide over questions of infinity. This divide extends over mathematics and we see it in many forms. A perfect example is offered by the twin approaches to analysis: limits and infinitesimals. With Robinson’s discovery of nonstandard analysis, both approaches have achieved legitimacy. Today, there is a new approach, non-nonstandard analysis, which, in ...

This paper describes a type constructor for nonstandard extensions of arbitrary types, generalizing the previous formalization of nonstandard analysis in Isabelle by Fleuriot and Paulson [1]. In addition to numeric types, nonstandard extensions of function spaces and set types also prove useful—they provide an elegant way to define internal functions and internal sets. The paper also describes ...

Suppose B is an ultraproduct of finite dimensional C∗-algebras. We consider mapping and injectability properties for separable C∗algebras into B. In the case of approximately finite C∗-algebras, we obtain a classification of these mappings up to inner conjugacy. Using a Theorem of Voiculescu, we show that for nuclear C ∗-algebras injectability into an ultraproduct of finite dimensional C ∗-alge...

where φ, ψ ∈ L, H = −~ 2m ∆+V (~x) is essentially self-adjoint, H̄ is the closure ofH , and φ, ψ, V each carries at most a finite number of singularities and discontinuities. In flavor of physics literature, we will formulate the Feynman path integral with improper Riemann integrals. In hope that with further research we can formulate a rigorous polygonal path integral, we will also provide a No...