7,872 research outputs found
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
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