Sketches from the life of hypercomplex numbers

In the article, the main ideas of the induction construction arXiv:1204.0194, arXiv:1110.4737, arXiv:1202.0941, arXiv:1208.4466 are considered in the form of sketches. The article establishes a link between Clifford algebras and alternative-elastic algebras at the level of connectors.Comment: MiKTeX v2.9, 16 page

On the classification of metric hypercomplex group alternative-elastic algebras for n=8

In this article, the clarification to Note 4 (arXiv:1202.0941) for n=8 is considered. In this connection, answers to the following questions are given. 1. How to classify the metric hypercomplex orthogonal group alternative-elastic algebras for n=8? 2. How to associate the metric hypercomplex orthogonal group alternative-elastic algebra to the symmetric controlling spinor for n=8? 3. How technically to construct the symmetric controlling spinor for n=8? 4. What class does the octonion belong to, and how to describe its?Comment: MiKTeX v2.9, 13 pages, 6 tables, 1 listin

Using of Phenomenological Piecewise Continuous Map for Modeling of Neurons Behaviour

A piecewise continuous map for modeling bursting and spiking behaviour of isolated neuron is proposed. The map was created from phenomenological viewpoint. The map demonstrates oscillations, which are qualitatively similar to oscillations generating by Rose-Hindmarsh model. The synchronization in small ensembles of the maps is investigated. It is considered the different number of elements in the ensemble and different connectivity topologies.Comment: 4 pages RevTeX, 7 figures PostScript, revised versio

Two-fluid dynamics of one-dimensional quantum liquids in the absence of Galilean invariance

Luttinger liquid theory of one-dimensional quantum systems ignores exponentially weak backscattering of particles. This endows Luttinger liquids with superfluid properties. The corresponding two-fluid hydrodynamic description available at present applies only to Galilean-invariant systems, whereas most experimental realizations of one-dimensional quantum liquids lack Galilean invariance. Here we develop the two-fluid theory of such quantum liquids. In the low-frequency limit the theory reduces to single-fluid hydrodynamics. However, the absence of Galilean invariance brings about three new transport coefficients. We obtain expressions for these coefficients in terms of the backscattering rate

Equilibration of a spinless Luttinger liquid

We study how a Luttinger liquid of spinless particles in one dimension approaches thermal equilibrium. Full equilibration requires processes of backscattering of excitations which occur at energies of order of the bandwidth. Such processes are not accounted for by the Luttinger liquid theory. We treat the high-energy excitations as mobile impurities and derive an expression for the equilibration rate in terms of their spectrum. Our results apply at any interaction strength

Second sound in systems of one-dimensional fermions

We study sound in Galilean invariant systems of one-dimensional fermions. At low temperatures, we find a broad range of frequencies in which in addition to the waves of density there is a second sound corresponding to ballistic propagation of heat in the system. The damping of the second sound mode is weak, provided the frequency is large compared to a relaxation rate that is exponentially small at low temperatures. At lower frequencies the second sound mode is damped, and the propagation of heat is diffusive

Equilibration of Luttinger liquid and conductance of quantum wires

Luttinger liquid theory describes one-dimensional electron systems in terms of non-interacting bosonic excitations. In this approximation thermal excitations are decoupled from the current flowing through a quantum wire, and the conductance is quantized. We show that relaxation processes not captured by the Luttinger liquid theory lead to equilibration of the excitations with the current and give rise to a temperature-dependent correction to the conductance. In long wires, the magnitude of the correction is expressed in terms of the velocities of bosonic excitations. In shorter wires it is controlled by the relaxation rate

Anisotropic re-entrant spin-glass features in a metallic kagome lattice, Tb3Ru4Al12

We report the results of ac and dc magnetic susceptibility and isothermal magnetization measurements (T= 2-300 K) on the single crystals of a metallic kagome lattice, Tb3Ru4Al12, reported recently to undergo reentrant magnetism with the onset of long range antiferromagnetic order below (TN=) 22K. The magnetization data obtained on the crystal with the c-axis orientation along magnetic-field reveal spin-glass-like characteristics near 17 K (below TN). However, for the orientation along basal plane, such glassy anomalies are not observable above 2 K. In this respect, this compound behaves like an anisotropic reentrant spin-glass. Possible implications of this finding to the field of geometrically frustrated magnetism is considered.Comment: 9 pages, 4 figure

Equilibration of a one-dimensional Wigner crystal

Equilibration of a one-dimensional system of interacting electrons requires processes that change the numbers of left- and right-moving particles. At low temperatures such processes are strongly suppressed, resulting in slow relaxation towards equilibrium. We study this phenomenon in the case of spinless electrons with strong long-range repulsion, when the electrons form a one-dimensional Wigner crystal. We find the relaxation rate by accounting for the Umklapp scattering of phonons in the crystal. For the integrable model of particles with inverse-square repulsion, the relaxation rate vanishes

On the spinor formalism for even n

Spinor formalism is the formalism induced by solutions of the Clifford equation (the connecting operators). For the space-time manifold (n = 4), these operators, connecting the tangent and spinor bundle, are operators that are represented by the Dirac matrices in the special basis. Reduced connecting operators are represented by the Pauli matrices. In order to uniquely prolong the Killing equation from the tangent bundle onto the spinor bundle over the space-time manifold, it is necessary to pass to the complexification of the manifold and the corresponding bundles, and then to pass to the real representation. Returning the reverse motion, one can already obtain two copies of the spinor bundle. Their set (the pair-spinor) allows to construct the Lie operator analogues for the spinors (and the pair-spinors). Similar procedure is feasible for any even n. For n=6, the specified formalism is closely connected with the Bogolyubov-Valatin transformations. For n mod 8=0, being based on the Bott periodicity, the reduced connecting operators generate the structural constants of an hypercomplex algebra (without division for n> 8) with the alternative-elastic, flexible (Jordan), and "norm" identities. For n = 8, such the algebra is the octonion algebra. In addition, in the article the various options of the prolonging of the connection to the spinor bundle with even-dimensional base are considered, and the corresponding curvature spinors are constructed.Comment: MiKTeX v2.7, 268 pages, 6 tables, 4 figures. Rus edition: On the spinor formalism for the base space of even dimension. VINITI-298-B-11, Jun 2011. 138pp. Paper deponed on Jun 16, 2011 at VINITI (Moscow), ref. No 298-B 11. Version 2: Correct pp.27,47-48,54-59,94-95(old 94),101(old 100),117-119(old 116-118). Version 3: Add Russian editio