540,507 research outputs found
Polyharmonic functions of infinite order on annular regions
Polyharmonic functions f of infinite order and type {\tau} on annular regions
are systematically studied. The first main result states that the
Fourier-Laplace coefficients f_{k,l}(r) of a polyharmonic function f of
infinite order and type 0 can be extended to analytic functions on the complex
plane cut along the negative semiaxis. The second main result gives a
constructive procedure via Fourier-Laplace series for the analytic extension of
a polyharmonic function on annular region A(r_{0},r_{1}) of infinite order and
type less than 1/2r_{1} to the kernel of the harmonicity hull of the annular
region. The methods of proof depend on an extensive investigation of Taylor
series with respect to linear differential operators with constant
coefficients.Comment: 32 page
Local Complexity of Delone Sets and Crystallinity
This paper characterizes when a Delone set X is an ideal crystal in terms of
restrictions on the number of its local patches of a given size or on the
hetereogeneity of their distribution. Let N(T) count the number of
translation-inequivalent patches of radius T in X and let M(T) be the minimum
radius such that every closed ball of radius M(T) contains the center of a
patch of every one of these kinds. We show that for each of these functions
there is a `gap in the spectrum' of possible growth rates between being bounded
and having linear growth, and that having linear growth is equivalent to X
being an ideal crystal. Explicitly, for N(T), if R is the covering radius of X
then either N(T) is bounded or N(T) >= T/2R for all T>0. The constant 1/2R in
this bound is best possible in all dimensions. For M(T), either M(T) is bounded
or M(T) >= T/3 for all T>0. Examples show that the constant 1/3 in this bound
cannot be replaced by any number exceeding 1/2. We also show that every
aperiodic Delone set X has M(T) >= c(n)T for all T>0, for a certain constant
c(n) which depends on the dimension n of X and is greater than 1/3 when n > 1.Comment: 26 pages. Uses latexsym and amsfonts package
Some results on two-sided LIL behavior
Let {X,X_n;n\geq 1} be a sequence of i.i.d. mean-zero random variables, and
let S_n=\sum_{i=1}^nX_i,n\geq 1. We establish necessary and sufficient
conditions for having with probability 1, 0<lim sup_{n\to
\infty}|S_n|/\sqrtnh(n)<\infty, where h is from a suitable subclass of the
positive, nondecreasing slowly varying functions. Specializing our result to
h(n)=(\log \log n)^p, where p>1 and to h(n)=(\log n)^r, r>0, we obtain
analogues of the Hartman-Wintner LIL in the infinite variance case. Our proof
is based on a general result dealing with LIL behavior of the normalized sums
{S_n/c_n;n\ge 1}, where c_n is a sufficiently regular normalizing sequence.Comment: Published at http://dx.doi.org/10.1214/009117905000000198 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Exact Results for the Asymmetric Simple Exclusion Process with a Blockage
We present new results for the current as a function of transmission rate in
the one dimensional totally asymmetric simple exclusion process (TASEP) with a
blockage that lowers the jump rate at one site from one to r < 1. Exact finite
volume results serve to bound the allowed values for the current in the
infinite system. This proves the existence of a gap in allowed density
corresponding to a nonequilibrium ``phase transition'' in the infinite system.
A series expansion in r, derived from the finite systems, is proven to be
asymptotic for all sufficiently large systems. Pade approximants based on this
series, which make specific assumptions about the nature of the singularity at
r = 1, match numerical data for the ``infinite'' system to a part in 10^4.Comment: 18 pages, LaTeX (including figures in LaTeX picture mode
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