1,360,178 research outputs found
Polynomial identities and noncommutative versal torsors
To any cleft Hopf Galois object, i.e., any algebra H[t] obtained from a Hopf
algebra H by twisting its multiplication with a two-cocycle t, we attach two
"universal algebras" A(H,t) and U(H,t). The algebra A(H,t) is obtained by
twisting the multiplication of H with the most general two-cocycle u formally
cohomologous to t. The cocycle u takes values in the field of rational
functions on H. By construction, A(H,t) is a cleft H-Galois extension of a
"big" commutative algebra B(H,t). Any "form" of H[t] can be obtained from
A(H,t) by a specialization of B(H,t) and vice versa. If the algebra H[t] is
simple, then A(H,t) is an Azumaya algebra with center B(H,t). The algebra
U(H,t) is constructed using a general theory of polynomial identities that we
set up for arbitrary comodule algebras; it is the universal comodule algebra in
which all comodule algebra identities of H[t] are satisfied. We construct an
embedding of U(H,t) into A(H,t); this embedding maps the center Z(H,t) of
U(H,t) into B(H,t) when the algebra H[t] is simple. In this case, under an
additional assumption, A(H,t) is isomorphic to B(H,t) \otimes_{Z(H,t)} U(H,t),
thus turning A(H,t) into a central localization of U(H,t). We work out these
constructions in full detail for the four-dimensional Sweedler algebra.Comment: 39 page
Local times of multifractional Brownian sheets
Denote by a function in
with values in . Let
be an
-multifractional Brownian sheet (mfBs) with Hurst functional .
Under some regularity conditions on the function , we prove the
existence, joint continuity and the H\"{o}lder regularity of the local times of
. We also determine the Hausdorff dimensions of the level sets
of . Our results extend the corresponding results for
fractional Brownian sheets and multifractional Brownian motion to
multifractional Brownian sheets.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ126 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Average mixing of continuous quantum walks
If is a graph with adjacency matrix , then we define to be the
operator . The Schur (or entrywise) product is a
doubly stochastic matrix and, because of work related to quantum computing, we
are concerned the \textsl{average mixing matrix}. This can be defined as the
limit of C^{-1} \int_0^C H(t)\circ H(-t)\dt as . We establish
some of the basic properties of this matrix, showing that it is positive
semidefinite and that its entries are always rational. We find that for paths
and cycles this matrix takes on a surprisingly simple form, thus for the path
it is a linear combination of , (the all-ones matrix), and a permutation
matrix.Comment: 20 pages, minor fixes, added section on discrete walks; fixed typo
Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments,
This paper continues the investigation of Du and Lou (J. European Math Soc,
to appear), where the long-time behavior of positive solutions to a nonlinear
diffusion equation of the form for over a varying
interval was examined. Here and are free
boundaries evolving according to , , and . We answer several intriguing
questions left open in the paper of Du and Lou.First we prove the conjectured
convergence result in the paper of Du and Lou for the general case that is
and . Second, for bistable and combustion types of , we
determine the asymptotic propagation speed of and in the
transition case. More presicely, we show that when the transition case happens,
for bistable type of there exists a uniquely determined such that
, and for
combustion type of , there exists a uniquely determined such that
. Our
approach is based on the zero number arguments of Matano and Angenent, and on
the construction of delicate upper and lower solutions
Uniform susceptibility of classical antiferromagnets in one and two dimensions in a magnetic field
We simulated the field-dependent magnetization m(H,T) and the uniform
susceptibility \chi(H,T) of classical Heisenberg antiferromagnets in the chain
and square-lattice geometry using Monte Carlo methods. The results confirm the
singular behavior of \chi(H,T) at small T,H: \lim_{T \to 0}\lim_{H \to 0}
\chi(H,T)=1/(2J_0)(1-1/D) and \lim_{H \to 0}\lim_{T \to 0} \chi(H,T)=1/(2J_0),
where D=3 is the number of spin components, J_0=zJ, and z is the number of
nearest neighbors. A good agreement is achieved in a wide range of temperatures
T and magnetic fields H with the first-order 1/D expansion results [D. A.
Garanin, J. Stat. Phys. 83, 907 (1996)]Comment: 4 PR pages, 4 figures, submitted to PR
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