Solutions for the Hyperbolic Dirac Equation on R1, m

In this article the hyperbolic unit ball in R ^m will be identified with the manifold of rays in the future null cone in R ^m+1. By means of the induced Clifford algebra structure there, one can introduce a definition of Dirac operators on sections of homogeneous line bundles. An infinite class of solutions for the resulting hyperbolic Dirac-equation will be constructed, in case of an odd dimension. In order to obtain these solutions a geometrical picture will be used, because each ray in the future cone will be identified with a point on a surface Σ in the future cone. The original Dirac-equation can then be rewritten in terms of the coordinates on this surface, and the resulting equation will be solved by means of Frobenius' method.     

Minimal Gröbner bases and the predictable leading monomial property

We focus on Gröbner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the “predictable leading monomial (PLM) property” that is shared by minimal Gröbner bases of modules in F[x]^q, no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property (= row reducedness), a terminology introduced by Forney in the 70’s. Because of the presence of zero divisors, minimal Gröbner bases over a finite ring of the type Z_p^r (where p is a prime integer and r is an integer >1) do not necessarily have the PLM property. In this paper we show how to derive, from an ordered minimal Gröbner basis, a so-called “minimal Gröbner p-basis” that does have a PLM property. We demonstrate that minimal Gröbner p-bases lend themselves particularly well to derive minimal realization parametrizations over Z_p^r. Applications are in coding and sequences over Z_p^r.     

Analytic aspects of Delannoy numbers

The Delannoy numbers $d(n,k)$ count the number of lattice paths from $(0,0)$ to $(n?k,k)$ using steps $(1,0),(0,1)$ and $(1,1)$. We show that the zeros of all Delannoy polynomials $d_n\left(x\right)={\sum }_{k=0}^{n}d(n,k)x^k$ are in the open interval $(?3?2\sqrt{2},?3+2\sqrt{2})$ and are dense in the corresponding closed interval. We also show that the Delannoy numbers $d(n,k)$ are asymptotically normal (by central and local limit theorems).     

Objection and Counter-Objection Equilibrium In Differential Game Under Uncertainty

For a linear-quadratic differential game under uncertainty, sufficient conditiors for the existence of Slater's, Pareto's and Geoffrion's equilibrium of objection and counter-objection are given under matrix Riccati differential equations form     

On two generalizations of the Alon–Tarsi polynomial method

In a seminal paper (Alon and Tarsi, 1992 [6]), Alon and Tarsi have introduced an algebraic technique for proving upper bounds on the choice number of graphs (and thus, in particular, upper bounds on their chromatic number). The upper bound on the choice number of G obtained via their method, was later coined the Alon–Tarsi number of G and was denoted by AT(G) (see e.g. Jensen and Toft (1995) [20]). They have provided a combinatorial interpretation of this parameter in terms of the eulerian subdigraphs of an appropriate orientation of G. Their characterization can be restated as follows. Let D be an orientation of G. Assign a weight ωD(H) to every subdigraph H of D: if H⊆D is eulerian, then ωD(H)=(−1)^e(H), otherwise ωD(H)=0. Alon and Tarsi proved that AT(G)?k if and only if there exists an orientation D of G in which the out-degree of every vertex is strictly less than k, and moreover _∑H⊆DωD(H)≠0. Shortly afterwards (Alon, 1993 [3]), for the special case of line graphs of d-regular d-edge-colorable graphs, Alon gave another interpretation of AT(G), this time in terms of the signed d-colorings of the line graph. In this paper we generalize both results. The first characterization is generalized by showing that there is an infinite family of weight functions (which includes the one considered by Alon and Tarsi), each of which can be used to characterize AT(G). The second characterization is generalized to all graphs (in fact the result is even more general—in particular it applies to hypergraphs). We then use the second generalization to prove that χ(G)=ch(G)=AT(G) holds for certain families of graphs G. Some of these results generalize certain known choosability results.     

Simple approach to approximate predictions of the vapor–liquid equilibrium curve near the critical point and its application to Lennard-Jones fluids

We propose a simple analytical form of the vapor–liquid equilibrium curve near the critical point for Lennard-Jones fluids. Coexistence densities curves and vapor pressure have been determined using the Van der Waals and Dieterici equation of state. In described method the Bernoulli differential equations, critical exponent theory and some type of Maxwell?s criterion have been used. Presented approach has not yet been used to determine analytical form of phase curves as done in this Letter. Lennard-Jones fluids have been considered for analysis. Comparison with experimental data is done. The accuracy of the method is described.     

Magnetic Field Effects on Diffusion In 2-Dimensional Electron Fluid

Molecular Dynamics calculations have been performed on a 2-dimensional (2D) electron fluid interacting with a l/ r potential, and subjected to a perpendicular, uniform and constant magnetic field, for various values of the plasma parameter o . A new and more accurate algorithm, that takes into account the effects of the magnetic field exactly, has been used in the simulation. The diffusion coefficient has been calculated and the results show a monotonic decrease as a function of the magnetic field strength. A simple theoretical model based on a generalized Langevin equation predicts a Lorentzian dependence. The model results are in reasonable agreement with the Molecular Dynamics results, especially for higher values of o.