236 research outputs found
Semicrossed products of the disk algebra and the Jacobson radical
We consider semicrossed products of the disk algebra with respect to
endomorphisms defined by finite Blaschke products. We characterize the Jacobson
radical of these operator algebras. Furthermore, in the case the finite
Blaschke product is elliptic, we show that the semicrossed product contains no
nonzero quasinilpotent elements. However, if the finite Blaschke product is
hyperbolic or parabolic with zero hyperbolic step, the Jacobson radical is
nonzero and a proper subset of the set of quasinilpotent elements.Comment: 12 page
Quantitative estimates of discrete harmonic measures
A theorem of Bourgain states that the harmonic measure for a domain in
is supported on a set of Hausdorff dimension strictly less than
\cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the
distribution of the first entrance point of a random walk into a subset of , . By refining the argument, we prove that for all \b>0 there
exists \rho (d,\b)N(d,\b), any , and any | \{y\in\Z^d\colon \nu_{A,x}(y)
\geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where denotes the
probability that is the first entrance point of the simple random walk
starting at into . Furthermore, must converge to as \b \to
\infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne
Divided Differences & Restriction Operator on Paley-Wiener Spaces for Carleson Sequences
For a sequence of complex numbers we consider the restriction
operator defined on Paley-Wiener spaces
(). Lyubarskii and Seip gave necessary and sufficient conditions on
for to be an isomorphism between and a
certain weighted space. The Carleson condition appears to be necessary.
We extend their result to Carleson sequences (finite unions of disjoint
Carleson sequences). More precisely, we give necessary and sufficient
conditions for to be an isomorphism between and
an appropriate sequence space involving divided differences
Interpolation and harmonic majorants in big Hardy-Orlicz spaces
Free interpolation in Hardy spaces is caracterized by the well-known Carleson
condition. The result extends to Hardy-Orlicz spaces contained in the scale of
classical Hardy spaces , . For the Smirnov and the Nevanlinna
classes, interpolating sequences have been characterized in a recent paper in
terms of the existence of harmonic majorants (quasi-bounded in the case of the
Smirnov class). Since the Smirnov class can be regarded as the union over all
Hardy-Orlicz spaces associated with a so-called strongly convex function, it is
natural to ask how the condition changes from the Carleson condition in
classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of
this paper is to narrow down this gap from the Smirnov class to ``big''
Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences
for a class of Hardy-Orlicz spaces that carry an algebraic structure and that
are strictly bigger than . It turns out that the
interpolating sequences are again characterized by the existence of
quasi-bounded majorants, but now the weights of the majorants have to be in
suitable Orlicz spaces. The existence of harmonic majorants in such Orlicz
spaces will also be discussed in the general situation. We finish the paper
with an example of a separated Blaschke sequence that is interpolating for
certain Hardy-Orlicz spaces without being interpolating for slightly smaller
ones.Comment: 19 pages, 2 figure
Critical assessment and ramifications of a purported marine trophic cascade
When identifying potential trophic cascades, it is important to clearly establish the trophic linkages between predators and prey with respect to temporal abundance, demographics, distribution, and diet. In the northwest Atlantic Ocean, the depletion of large coastal sharks was thought to trigger a trophic cascade whereby predation release resulted in increased cownose ray abundance, which then caused increased predation on and subsequent collapse of commercial bivalve stocks. These claims were used to justify the development of a predator-control fishery for cownose rays, the âSave the Bay, Eat a Rayâ fishery, to reduce predation on commercial bivalves. A reexamination of data suggests declines in large coastal sharks did not coincide with purported rapid increases in cownose ray abundance. Likewise, the increase in cownose ray abundance did not coincide with declines in commercial bivalves. The lack of temporal correlations coupled with published diet data suggest the purported trophic cascade is lacking the empirical linkages required of a trophic cascade. Furthermore, the life history parameters of cownose rays suggest they have low reproductive potential and their populations are incapable of rapid increases. Hypothesized trophic cascades should be closely scrutinized as spurious conclusions may negatively influence conservation and management decision
Partial regularity and t-analytic sets for Banach function algebras
In this note we introduce the notion of t-analytic sets. Using this concept, we construct a class of closed prime ideals in Banach function algebras and discuss some problems related to Allingâs conjecture in H infinity. A description of all closed t-analytic sets for the disk-algebra is given. Moreover, we show that some of the assertions in [8] concerning the O-analyticity and S-regularity of certain Banach function algebras are not correct. We also determine the largest set on which a Douglas algebra is pointwise regular
Are Devaney hairs fast escaping?
Beginning with Devaney, several authors have studied transcendental entire
functions for which every point in the escaping set can be connected to
infinity by a curve in the escaping set. Such curves are often called Devaney
hairs. We show that, in many cases, every point in such a curve, apart from
possibly a finite endpoint of the curve, belongs to the fast escaping set. We
also give an example of a Devaney hair which lies in a logarithmic tract of a
transcendental entire function and contains no fast escaping points.Comment: 22 pages, 1 figur
Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis
We study the behaviour of the Standard map critical function in a
neighbourhood of a fixed resonance, that is the scaling law at the fixed
resonance. We prove that for the fundamental resonance the scaling law is
linear. We show numerical evidence that for the other resonances , , and and relatively prime, the scaling law follows a
power--law with exponent .Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit
Lyapunov exponents, bifurcation currents and laminations in bifurcation loci
Bifurcation loci in the moduli space of degree rational maps are shaped
by the hypersurfaces defined by the existence of a cycle of period and
multiplier 0 or . Using potential-theoretic arguments, we
establish two equidistribution properties for these hypersurfaces with respect
to the bifurcation current. To this purpose we first establish approximation
formulas for the Lyapunov function. In degree , this allows us to build
holomorphic motions and show that the bifurcation locus has a lamination
structure in the regions where an attracting basin of fixed period exists
Dynamics with Infinitely Many Derivatives: The Initial Value Problem
Differential equations of infinite order are an increasingly important class
of equations in theoretical physics. Such equations are ubiquitous in string
field theory and have recently attracted considerable interest also from
cosmologists. Though these equations have been studied in the classical
mathematical literature, it appears that the physics community is largely
unaware of the relevant formalism. Of particular importance is the fate of the
initial value problem. Under what circumstances do infinite order differential
equations possess a well-defined initial value problem and how many initial
data are required? In this paper we study the initial value problem for
infinite order differential equations in the mathematical framework of the
formal operator calculus, with analytic initial data. This formalism allows us
to handle simultaneously a wide array of different nonlocal equations within a
single framework and also admits a transparent physical interpretation. We show
that differential equations of infinite order do not generically admit
infinitely many initial data. Rather, each pole of the propagator contributes
two initial data to the final solution. Though it is possible to find
differential equations of infinite order which admit well-defined initial value
problem with only two initial data, neither the dynamical equations of p-adic
string theory nor string field theory seem to belong to this class. However,
both theories can be rendered ghost-free by suitable definition of the action
of the formal pseudo-differential operator. This prescription restricts the
theory to frequencies within some contour in the complex plane and hence may be
thought of as a sort of ultra-violet cut-off.Comment: 40 pages, no figures. Added comments concerning fractional operators
and the implications of restricting the contour of integration. Typos
correcte
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