I am honored to have this opportunity to give the final talk at this wonderful meeting. I was originally asked to talk about “Lattice gauge theory: past, present, and future.” Well, I really don’t know much about the future, and if I did, I would be out trying to develop the ideas. As for the present, well you have for the last few days been hearing all about it, and there is too much for me to...

In this paper, a self-contained electromagnetic theory is derived on a regular lattice. The discretized form of integral and diierential calculus, which is called`discrete calculus' here, is used to describe this theory. It is shown that discrete forms of Gauss' theorem, Stokes' theorem, Green's theorem, and Huygens' principle can be derived. Moreover, many electromagnetic theorems can also be ...

I discuss a study performed by the SCRI lattice gauge theory group which compared light hadron spectroscopy using the Wilson and Clover fermionic actions. We showed that a Clover coefficient chosen using tadpole-improved tree-level perturbation theory effectively eliminates the O(a) discretization errors present in the Wilson action. We found that discretization errors in light spectroscopy for...

The rare kaon decays K → πνν̄ are strongly suppressed in the standard model and widely regarded as processes in which new phenomena, not predicted by the standard model, may be observed. Recognizing such new phenomena requires precise standard model prediction for the braching ratio of K → πνν̄ with controlled uncertainty for both short-distance and long-distance contributions. In this work we de...

For the first time, we compute non–perturbatively, i.e. without lattice perturbation theory, the renormalization constants of two–fermion operators in the quenched approximation at β = 6.0, 6.2 and 6.4 using the Wilson and the tree–level improved SW–Clover actions. We apply these renormalization constants to fully non–perturbatively estimate quark masses in the MS scheme from lattice simulation...

This work focuses on lattice-theoretical foundations of rough set theory. It consists of the following sections: 1: Introduction 2: Basic Notions and Notation, 3: Orders and Lattices, 4: Distributive, Boolean, and Stone Lattices, 5: Closure Systems and Topologies, 6: Fixpoints and Closure Operators on Ordered Sets, 7: Galois Connections and Their Fixpoints, 8: Information Systems, 9: Rough Set ...

In this tutorial, I will introduce probability theory from the perspective of lattice theory where the fundamental notion is based on a partial ordering of logical statements ordered by implication. This construction forms a Boolean lattice, which is related to the fact that the corresponding logical operations form a Boolean algebra. I generalize the concept of logical implication to degrees o...

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also include a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations. 2010 MSC: 14M25, 14G50,...

The tools of lattice theory are applied to readdress the static state feedback linearization problem for discrete-time nonlinear control systems. Unlike the earlier results that are based on differential geometry, the new tools are also applicable for nonsmooth systems. In case of analytic systems, close connections are established between the new results and those based on differential one-for...