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...

We show that standard mixed finite element methods for second order elliptic equations can be modified by imposing additional continuity conditions for the flux, which reduces the dimension of the space. This reduced space still gives a stable method with an optimal order of convergence. We recall our postprocessing method and the a posteriori error estimator based on this. AMS subject classifi...

This paper provides a new proof of the consistency of a formal system similar to the one presented by Chuaqui and Suppes in 2, 9]. First, a simpler, yet in some respects stronger, system, called Elementary Re-cursive Nonstandard Analysis (ERNA) will be provided. Indeed, it will be shown that ERNA proves the main axioms of the Chuaqui and Sup-pes system. Then a nitary consistency proof of ERNA w...

This paper provides a new proof of the consistency of the formal system presented by Chuaqui and Suppes in [2, 9). First, a simpler, yet stronger, system, called Elementary Recursive Nonstandard Analysis, ERNA, will be provided. Indeed, it will be shown that ERNA proves all of the theorems of the Chuaqui and Suppes system. Then a finitary consistency proof of ERNA dl be given; in particular, we...

Anisotropic diffusion filters with a diffusion tensor are successfully used in many image processing and computer vision applications, ranging from image denoising over compression to optic flow computation. However, finding adequate numerical schemes is difficult: Implementations may suffer from dissipative artifacts, poor approximation of rotation invariance, and they may lack provable stabil...

We present a short and self-contained proof of Jin’s theorem about the piecewise syndeticity of difference sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultrafilters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following:...

We consider the basic SIR epidemiological model with the Michaelis-Menten formulation of the contact rate. From the study of Michaelis-Menten basic enzymatic reaction, we design two types of Nonstandard Finite Difference (NSFD) schemes for the SIR model: Exact-related schemes based on the Lambert W function and schemes obtained by using Mickens’s rules of more complex denominator functions for ...

We construct a nonstandard finite difference (NSFD) scheme for an SIRS mathematical model of respiratory virus transmission. This discretization is in full compliance with the NSFD methodology as formulated by R. E. Mickens. By use of an exact conservation law satisfied by the SIRS differential equations, we are able to determine the corresponding denominator function for the discrete first-ord...