16,938,407 research outputs found
Isospectral deformations of the Dirac operator
We give more details about an integrable system in which the Dirac operator
D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a
Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) =
d(t) + b(t) + d(t)^* defines a new exterior derivative d(t) and a new Dirac
operator C(t) = d(t) + d(t)^* and Laplacian M(t) = d(t) d(t)^* + d(t)* d(t) and
so a new distance on G or a new metric on M.Comment: 32 pages, 8 figure
Ramond-Ramond Cohomology and O(D,D) T-duality
In the name of supersymmetric double field theory, superstring effective
actions can be reformulated into simple forms. They feature a pair of vielbeins
corresponding to the same spacetime metric, and hence enjoy double local
Lorentz symmetries. In a manifestly covariant manner --with regard to O(D,D)
T-duality, diffeomorphism, B-field gauge symmetry and the pair of local Lorentz
symmetries-- we incorporate R-R potentials into double field theory. We take
them as a single object which is in a bi-fundamental spinorial representation
of the double Lorentz groups. We identify cohomological structure relevant to
the field strength. A priori, the R-R sector as well as all the fermions are
O(D,D) singlet. Yet, gauge fixing the two vielbeins equal to each other
modifies the O(D,D) transformation rule to call for a compensating local
Lorentz rotation, such that the R-R potential may turn into an O(D,D) spinor
and T-duality can flip the chirality exchanging type IIA and IIB
supergravities.Comment: 1+37 pages, no figure; Structure reorganized, References added, To
appear in JHEP. cf. Gong Show of Strings 2012
(http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Thursday/Gongshow/Lee.pdf
Number of Common Sites Visited by N Random Walkers
We compute analytically the mean number of common sites, W_N(t), visited by N
independent random walkers each of length t and all starting at the origin at
t=0 in d dimensions. We show that in the (N-d) plane, there are three distinct
regimes for the asymptotic large t growth of W_N(t). These three regimes are
separated by two critical lines d=2 and d=d_c(N)=2N/(N-1) in the (N-d) plane.
For d<2, W_N(t)\sim t^{d/2} for large t (the N dependence is only in the
prefactor). For 2<d<d_c(N), W_N(t)\sim t^{\nu} where the exponent \nu=
N-d(N-1)/2 varies with N and d. For d>d_c(N), W_N(t) approaches a constant as
t\to \infty. Exactly at the critical dimensions there are logaritmic
corrections: for d=2, we get W_N(t)\sim t/[\ln t]^N, while for d=d_c(N),
W_N(t)\sim \ln t for large t. Our analytical predictions are verified in
numerical simulations.Comment: 5 pages, 3 .eps figures include
D-branes and T-duality
We show how the --duality between --branes is realized (i) on
--brane solutions of IIA/IIB supergravity and (ii) on the
--brane actions ( that act as source terms for the
--brane solutions. We point out that the presence of a cosmological constant
in the IIA theory leads, by the requirement of gauge invariance, to a
topological mass term for the worldvolume gauge field in the 2--brane case.Comment: 14 pages, Late
Lifetime asymptotics of iterated Brownian motion in R^{n}
Let be the first exit time of iterated Brownian motion from a
domain D \subset \RR{R}^{n} started at and let be its distribution. In this paper we establish the exact asymptotics of
over bounded domains as an improvement of the results
in \cite{deblassie, nane2}, for \begin{eqnarray}
\lim_{t\to\infty} t^{-1/2}\exp({3/2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3})
P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber \end{eqnarray} where
. Here
is the first eigenvalue of the Dirichlet Laplacian
in , and is the eigenfunction corresponding to .
We also study lifetime asymptotics of Brownian-time Brownian motion (BTBM),
, where and are independent
one-dimensional Brownian motions
On Keller's conjecture in dimension seven
A cube tiling of is a family of pairwise disjoint cubes
such that . Two cubes , are called a
twin pair if for some and
for every . In , Keller conjectured that in
every cube tiling of there is a twin pair. Keller's conjecture
is true for dimensions and false for all dimensions . For
the conjecture is still open. Let , , and
let be the set of all th coordinates of vectors
such that and . It is
known that if for some and every or for some and , then
Keller's conjecture is true for . In the present paper we show that it is
also true for if for some and . Thus, if there is a counterexample to Keller's conjecture in dimension
seven, then for some and .Comment: 37 pages, 7 figures. arXiv admin note: substantial text overlap with
arXiv:1304.163
- …