82 research outputs found
Subexponential estimations in Shirshov's height theorem (in English)
In 1993 E. I. Zelmanov asked the following question in Dniester Notebook:
"Suppose that F_{2, m} is a 2-generated associative ring with the identity
x^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponential
growth?" We show that the nilpotency degree of l-generated associative algebra
with the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l
(nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov by
this result. It is the consequence of one fact, which is based on combinatorics
of words. Let l, n and d>n be positive integers. Then all the words over
alphabet of cardinality l which length is greater than Psi(n,d,l) are either
n-divided or contain d-th power of subword, where a word W is n-divided, if it
can be represented in the following form W=W_0 W_1...W_n such that W_1 >'
W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshov
proved that the set of non n-divided words over alphabet of cardinality l has
bounded height h over the set Y consisting of all the words of degree <n.
Original Shirshov's estimation was just recursive, in 1982 double exponent was
obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation.
We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Our
proof uses Latyshev idea of Dilworth theorem application.Comment: 21 pages, Russian version of the article is located at the link
arXiv:1101.4909; Sbornik: Mathematics, 203:4 (2012), 534 -- 55
Bose-Einstein condensation of quasiparticles in graphene
The collective properties of different quasiparticles in various graphene
based structures in high magnetic field have been studied. We predict
Bose-Einstein condensation (BEC) and superfluidity of 2D spatially indirect
magnetoexcitons in two-layer graphene. The superfluid density and the
temperature of the Kosterlitz-Thouless phase transition are shown to be
increasing functions of the excitonic density but decreasing functions of
magnetic field and the interlayer separation. The instability of the ground
state of the interacting 2D indirect magnetoexcitons in a slab of superlattice
with alternating electron and hole graphene layers (GLs) is established. The
stable system of indirect 2D magnetobiexcitons, consisting of pair of indirect
excitons with opposite dipole moments, is considered in graphene superlattice.
The superfluid density and the temperature of the Kosterlitz-Thouless phase
transition for magnetobiexcitons in graphene superlattice are obtained.
Besides, the BEC of excitonic polaritons in GL embedded in a semiconductor
microcavity in high magnetic field is predicted. While superfluid phase in this
magnetoexciton polariton system is absent due to vanishing of
magnetoexciton-magnetoexciton interaction in a single layer in the limit of
high magnetic field, the critical temperature of BEC formation is calculated.
The essential property of magnetoexcitonic systems based on graphene (in
contrast, e.g., to a quantum well) is stronger influence of magnetic field and
weaker influence of disorder. Observation of the BEC and superfluidity of 2D
quasiparticles in graphene in high magnetic field would be interesting
confirmation of the phenomena we have described.Comment: 13 pages, 5 figure
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