The cubic chessboard

We present a survey of recent results, scattered in a series of papers that appeared during past five years, whose common denominator is the use of cubic relations in various algebraic structures. Cubic (or ternary) relations can represent different symmetries with respect to the permutation group S_3, or its cyclic subgroup Z_3. Also ordinary or ternary algebras can be divided in different classes with respect to their symmetry properties. We pay special attention to the non-associative ternary algebra of 3-forms (or ``cubic matrices''), and Z_3-graded matrix algebras. We also discuss the Z_3-graded generalization of Grassmann algebras and their realization in generalized exterior differential forms. A new type of gauge theory based on this differential calculus is presented. Finally, a ternary generalization of Clifford algebras is introduced, and an analog of Dirac's equation is discussed, which can be diagonalized only after taking the cube of the Z_3-graded generalization of Dirac's operator. A possibility of using these ideas for the description of quark fields is suggested and discussed in the last Section.Comment: 23 pages, dedicated to A. Trautman on the occasion of his 64th birthda

Stochastic Description of Agglomeration and Growth Processes in Glasses

We show how growth by agglomeration can be described by means of algebraic or differential equations which determine the evolution of probabilities of various local configurations. The minimal fluctuation condition is used to define vitrification. Our methods have been successfully used for the description of glass formation.Comment: 9 pages, 1 figure, LaTeX 2e, uses ws-ijmpb.cls ; submitted to International Journal of Modern Physics

Z_3-graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d^3=0, but d^2\not=0. The first and second order differentials generate an associative algebra; we shall suppose that there are no binary relations between first order differentials, while the ternary products will satisfy the cyclic relations based on the representation of cyclic group Z_3 by cubic roots of unity. We shall attribute grade 1 to the first order differentials and grade 2 to the second order differentials; under the associative multiplication law the grades add up modulo 3. We show how the notion of covariant derivation can be generalized with a 1-form A, and we give the expression in local coordinates of the curvature 3-form. Finally, the introduction of notions of a scalar product and integration of the Z_3-graded exterior forms enables us to define variational principle and to derive the differential equations satisfied by the curvature 3-form. The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensor F_{ik} and its covariant derivatives D_i F_{km}.Comment: 13 pages, no figure

On the glass transition temperature in covalent glasses

We give a simple demonstration of the formula relating the glass transition temperature, $T_g$, to the molar concentration $x$ of a modifier in two types of glasses: binary glasses, whose composition can be denoted by $X_nY_m+xM_pY_q$, with ^$X$ an element of III-rd or IV-th group (e.g. B, or Si, Ge), while $M_pY_q$ is an alkali oxide or chalcogenide; next, the network glasses of the type $A_xB_{1-x}$, e.g. $Ge_xSe_{1-x}$, $Si_xTe_{1-x}$, etc. After comparison, this formula gives an exact expression of the parameter $\beta$ of the modified Gibbs-Di Marzio equation.Comment: 15 pages, LateX; ([email protected]), ([email protected]

Probabilistic Description of Traffic Breakdowns

We analyze the characteristic features of traffic breakdown. To describe this phenomenon we apply to the probabilistic model regarding the jam emergence as the formation of a large car cluster on highway. In these terms the breakdown occurs through the formation of a certain critical nucleus in the metastable vehicle flow, which enables us to confine ourselves to one cluster model. We assume that, first, the growth of the car cluster is governed by attachment of cars to the cluster whose rate is mainly determined by the mean headway distance between the car in the vehicle flow and, may be, also by the headway distance in the cluster. Second, the cluster dissolution is determined by the car escape from the cluster whose rate depends on the cluster size directly. The latter is justified using the available experimental data for the correlation properties of the synchronized mode. We write the appropriate master equation converted then into the Fokker-Plank equation for the cluster distribution function and analyze the formation of the critical car cluster due to the climb over a certain potential barrier. The further cluster growth irreversibly gives rise to the jam formation. Numerical estimates of the obtained characteristics and the experimental data of the traffic breakdown are compared. In particular, we draw a conclusion that the characteristic intrinsic time scale of the breakdown phenomenon should be about one minute and explain the case why the traffic volume interval inside which traffic breakdown is observed is sufficiently wide.Comment: RevTeX 4, 14 pages, 10 figure

Cellular automata approach to three-phase traffic theory

The cellular automata (CA) approach to traffic modeling is extended to allow for spatially homogeneous steady state solutions that cover a two dimensional region in the flow-density plane. Hence these models fulfill a basic postulate of a three-phase traffic theory proposed by Kerner. This is achieved by a synchronization distance, within which a vehicle always tries to adjust its speed to the one of the vehicle in front. In the CA models presented, the modelling of the free and safe speeds, the slow-to-start rules as well as some contributions to noise are based on the ideas of the Nagel-Schreckenberg type modelling. It is shown that the proposed CA models can be very transparent and still reproduce the two main types of congested patterns (the general pattern and the synchronized flow pattern) as well as their dependence on the flows near an on-ramp, in qualitative agreement with the recently developed continuum version of the three-phase traffic theory [B. S. Kerner and S. L. Klenov. 2002. J. Phys. A: Math. Gen. 35, L31]. These features are qualitatively different than in previously considered CA traffic models. The probability of the breakdown phenomenon (i.e., of the phase transition from free flow to synchronized flow) as function of the flow rate to the on-ramp and of the flow rate on the road upstream of the on-ramp is investigated. The capacity drops at the on-ramp which occur due to the formation of different congested patterns are calculated.Comment: 55 pages, 24 figure

Algebras with ternary law of composition and their realization by cubic matrices

We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions we construct a ternary algebra which is used as a building block for a Lie algebra. We construct ternary algebras of cubic matrices and find four different totally associative ternary multiplications of second kind of cubic matrices. It is proved that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. We describe a ternary analog of Lie algebra of cubic matrices of second order which is based on a notion of j-commutator and find all commutation relations of generators of this algebra.Comment: 17 pages, 1 figure, to appear in "Journal of Generalized Lie Theory and Applications