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Discrete models of the Dirac-K��hler equation and the Dirac equation in the Hestenes form are discussed. A discrete version of the plane wave solutions to a discrete analogue of the Hestenes equation is established., 20 pages. arXiv admin note: text overlap with arXiv:1411.7673

A discrete version of the plane wave solution to some discrete Dirac type equations in the spacetime algebra is established. The conditions under which a discrete analogue of the plane wave solution satisfies the discrete Hestenes equation are briefly discussed., 12 pages. arXiv admin note: text overlap with arXiv:1609.04594

A relationship between the discrete Dirac-K��hler equation and discrete analogues of some Dirac type equations in the geometric spacetime algebra is discussed. We show that a solution of the discrete Dirac-K��hler equation can be represented as the sum and difference of solutions of discrete Dirac type equations with the corresponding sign of the mass term., 13 pages, corrected typos

A discrete analogue of the Dirac equation in the Hestenes form is constructed by introduction the Clifford product on the space of discrete forms. We discuss the relation between the discrete Dirac-K��hler equation and a discrete Hestenes equation., 9 pages

We discuss a discrete analogue of the Dirac-K��hler equation in which chiral properties of the continual counterpart are captured. We pay special attention to a discrete Hodge star operator. To build one a combinatorial construction of double complex is used. We describe discrete exterior calculus operations on a double comlex and obtain the discrete Dirac-K��hler equation using these tools. Self-dual and anti-self-dual discrete inhomogeneous forms are presented. The chiral invariance of the massless discrete Dirac-K��hler equation is shown. Moreover, in the massive case we prove that a discrete Dirac-K��hler operator flips the chirality., 18 pages. arXiv admin note: substantial text overlap with arXiv:1307.1220

We construct a new discrete analog of the Dirac-K��hler equation in which some key geometric aspects of the continuum counterpart are captured. We describe a discrete Dirac-K��hler equation in the intrinsic notation as a set of difference equations and prove several statements about its decomposition into difference equations of Duffin type. We study an analog of gauge transformations for the massless discrete Dirac-K��hler equations., 19 pages; added references; to appear in Rept. Math. Phys

We prove essential self-adjointness for a semibounded from below discrete magnetic Schr��dinger operator in a space that represents a combinatorial model of the two-dimensional Euclidean space. The Dezin discretization scheme is used for constructing a discrete model., 9 pages

We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preserves the geometric structure of the original continual object. This means that, speaking of a discrete model, we do not mean just the direct replacement of differential operators by difference ones but also a discrete analog of the Riemannian structure. We consider this structure on the appropriate combinatorial analog of differential forms. Self-adjointness and boundness for a discrete Laplacian are proved. We define the Green function for this operator and also derive an explicit formula of the one., 12 pages

An intrinsically defined gauge-invariant discrete model of the Yang-Mills equations on a combinatorial analog of $\Bbb{R}^4$ is constructed. We develop several algebraic structures on the matrix-valued cochains (discrete forms) that are analogs of objects in differential geometry. We define a combinatorial Hodge star operator based on the use of a double complex construction. Difference self-dual and anti-self-dual equations will be given. In the last section we discus the question of generalizing our constructions to the case of a 4-dimensional combinatorial sphere, Comment: LaTeX, 17 pages

In the case of a gauge-invariant discrete model of Yang-Mills theory difference self-dual and anti-self-dual equations are constructed.

We construct some intrinsically defined discrete model of the magnetic Laplacian. The existence and uniqueness of solutions of the Dirichlet problem for the difference Poisson type equation are proved. We study in detail properties of the discrete model including the limiting process in the two-dimensional Euclidean case., 13 pages

Using methods of differential geometry, a discrete analog of the Yang-Mills equations in Minkowski space is constructed. The gauge transformation law in a discrete formulation is given and gauge invariance of discrete Yang-Mills equations is studied. Difference self-dual and anti-self-dual equations with respect to the Lorentz metric are presented., Comment: 13 pages