2,757 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
Sub-wavelength imaging at infrared frequencies using an array of metallic nanorods
We demonstrate that an array of metallic nanorods enables sub-wavelength
(near-field) imaging at infrared frequencies. Using an homogenization approach,
it is theoretically proved that under certain conditions the incoming radiation
can be transmitted by the array of nanorods over a significant distance with
fairly low attenuation. The propagation mechanism does not involve a resonance
of material parameters and thus the resolution is not strongly affected by
material losses and has wide bandwidth. The sub-wavelength imaging with
resolution by silver rods at 30 THz is demonstrated numerically
using full-wave electromagnetic simulator.Comment: 12 pages, 16 figures, submitted to PR
Enhanced Efficiency of Light-Trapping Nanoantenna Arrays for Thin Film Solar Cells
We suggest a novel concept of efficient light-trapping structures for
thin-film solar cells based on arrays of planar nanoantennas operating far from
plasmonic resonances. The operation principle of our structures relies on the
excitation of chessboard-like collective modes of the nanoantenna arrays with
the field localized between the neighboring metal elements. We demonstrated
theoretically substantial enhancement of solar-cell short-circuit current by
the designed light-trapping structure in the whole spectrum range of the
solar-cell operation compared to conventional structures employing
anti-reflecting coating. Our approach provides a general background for a
design of different types of efficient broadband light-trapping structures for
thin-film solar-cell technologically compatible with large-area thin-film
fabrication techniques
Schnabl's L_0 Operator in the Continuous Basis
Following Schnabl's analytic solution to string field theory, we calculate
the operators for a scalar field in the
continuous basis. We find an explicit and simple expression for them
that further simplifies for their sum, which is block diagonal in this basis.
We generalize this result for the bosonized ghost sector, verify their
commutation relation and relate our expressions to wedge state representations.Comment: 1+16 pages. JHEP style. Typos correcte
Spontaneous radiation of a finite-size dipole emitter in hyperbolic media
We study the radiative decay rate and Purcell effect for a finite-size dipole
emitter placed in a homogeneous uniaxial medium. We demonstrate that the
radiative rate is strongly enhanced when the signs of the longitudinal and
transverse dielectric constants of the medium are opposite, and the
isofrequency contour has a hyperbolic shape. We reveal that the Purcell
enhancement factor remains finite even in the absence of losses, and it depends
on the emitter size.Comment: 6 pages, 3 figure
Normalization anomalies in level truncation calculations
We test oscillator level truncation regularization in string field theory by
calculating descent relations among vertices, or equivalently, the overlap of
wedge states. We repeat the calculation using bosonic, as well as fermionic
ghosts, where in the bosonic case we do the calculation both in the discrete
and in the continuous basis. We also calculate analogous expressions in field
level truncation. Each calculation gives a different result. We point out to
the source of these differences and in the bosonic ghost case we pinpoint the
origin of the difference between the discrete and continuous basis
calculations. The conclusion is that level truncation regularization cannot be
trusted in calculations involving normalization of singular states, such as
wedge states, rank-one squeezed state projectors and string vertices.Comment: 1+20 pages, 6 figures. v2: Ref. added, typos correcte
Witten's Vertex Made Simple
The infinite matrices in Witten's vertex are easy to diagonalize. It just
requires some SL(2,R) lore plus a Watson-Sommerfeld transformation. We
calculate the eigenvalues of all Neumann matrices for all scale dimensions s,
both for matter and ghosts, including fractional s which we use to regulate the
difficult s=0 limit. We find that s=1 eigenfunctions just acquire a p term, and
x gets replaced by the midpoint position.Comment: 24 pages, 2 figures, RevTeX style, typos correcte
Temperature evolution of magnetic structure of HoFeO by single crystal neutron diffraction
We have investigated the temperature evolution of the magnetic structures of
HoFeO by single crystal neutron diffraction. The three different magnetic
structures found as a function of temperature for \hfo\ are described by the
magnetic groups Pbn, Pbn and Pbn and are stable in the
temperature ranges 600-55~K, 55-37~K and 35~K respectively. In
all three the fundamental coupling between the Fe sub-lattices remains the same
and only their orientation and the degree of canting away from the ideal axial
direction varies. The magnetic polarisation of the Ho sub-lattices in these two
higher temperature regions, in which the major components of the Fe moment lie
along and , is very small. The canting of the moments from the axial
directions is attributed to the antisymmetric interactions allowed by the
crystal symmetry. They include contributions from single ion anisotropy as well
as the Dzyaloshinski antisymmetric exchange. In the low temperature phase two
further structural transitions are apparent in which the spontaneous
magnetisation changes sign with respect to the underlying antiferromagnetic
configuration. In this temperature range the antisymmetric exchange energy
varies rapidly as the the Ho sub-lattices begin to order. So long as the
ordered Ho moments are small the antisymmetric exchange is due only to Fe-Fe
interactions, but as the degree of Ho order increases the Fe-Ho interactions
take over whilst at the lowest temperatures, when the Ho moments approach
saturation the Ho-Ho interactions dominate. The reversals of the spontaneous
magnetisation found in this study suggest that in \hfo\ the sums of the Fe-Fe
and Ho-Ho antisymmetric interactions have the same sign as one another, but
that of the Ho-Fe terms is opposite
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