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We define locally trivial quantum vector bundles (QVB) and QVB associated to locally trivial quantum principal fibre bundles. There exists a differential structure on the associated vector bundle coming from the differential structure on…
For a more general notion of Cartan connection we define characteristic classes, we investigate their relation to usual characteristic classes.
We will give a short introduction to discrete or lattice soliton equations, with the particular example of the Korteweg-de Vries as illustration. We will discuss briefly how B\"acklund transformations lead to equations that can be…
Pure gauge lattice QCD at arbitrary D is considered. Exact integration over link variables in an arbitrary D-volume leads naturally to an appearance of a set of surfaces filling the volume and gives an exact expression for functional of…
The discretization of Cartan's exterior calculus of differential forms has been fruitful in a variety of theoretical and practical endeavors: from computational electromagnetics to the development of Finite-Element Exterior Calculus, the…
The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The…
In this paper we introduce the Integration Problem for principal connections. Just as a principal connection on a principal bundle $\phi:Q\rightarrow M$ may be used to split $TQ$ into horizontal and vertical subbundles, a discrete…
We describe the basic lattice structures of attractors and repellers in dynamical systems. The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and…
I discuss some aspects of a lattice approach to canonical quantum gravity in a connection formulation, discuss how it differs from the continuum construction, and compare the spectra of geometric operators - encoding information about…
We shall define the relative $\dbar$-complex and study the curvature properties of the associated vector bundles. As an application, we shall prove that Yamaguchi's theory on subharmonicity of the Green operator can be seen as a curvature…
Quark number susceptibility on the lattice, obtained by merely adding a $\mu N$ term with $\mu$ as the chemical potential and $N$ as the conserved quark number, has a quadratic divergence in the cut-off $a$. We show that such a divergence…
We study the diffusion phenomena on the negatively curved surface made up of congruent heptagons. Unlike the usual two-dimensional plane, this structure makes the boundary increase exponentially with the distance from the center, and hence…
We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…
We study reversible deterministic dynamics of classical charged particles on a lattice with hard-core interaction. It is rigorously shown that the system exhibits three types of transport phenomena, ranging from ballistic, through diffusive…
The calculus of finite differences is a solid foundation for the development of operations such as the derivative and the integral for infinite sequences. Here we showed a way to extend it for finite sequences. We could then define…
We investigate path integral formalism for continuum theory. It is shown that the path integral for the soft modes can be represented in the form of a lattice theory. Kinetic term of this lattice theory has a standard form and potential…
Continuous limits of discrete systems with long-range interactions are considered. The map of discrete models into continuous medium models is defined. A wide class of long-range interactions that give the fractional equations in the…
Formally capturing the transition from a continuous model to a discrete model is investigated using model based refinement techniques. A very simple model for stopping (eg. of a train) is developed in both the continuous and discrete…
A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential…
We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of curvature in discrete spaces. An appealing feature of this discrete version seems to be that it is fairly straightforward to compute this…