1,327 research outputs found
Successive Coefficients of Univalent Functions
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135332/1/jlms0448.pd
An Arclength Problem for Close‐to‐Convex Functions
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135569/1/jlms0181.pd
Interface growth in two dimensions: A Loewner-equation approach
The problem of Laplacian growth in two dimensions is considered within the
Loewner-equation framework. Initially the problem of fingered growth recently
discussed by Gubiec and Szymczak [T. Gubiec and P. Szymczak, Phys. Rev. E 77,
041602 (2008)] is revisited and a new exact solution for a three-finger
configuration is reported. Then a general class of growth models for an
interface growing in the upper-half plane is introduced and the corresponding
Loewner equation for the problem is derived. Several examples are given
including interfaces with one or more tips as well as multiple growing
interfaces. A generalization of our interface growth model in terms of
``Loewner domains,'' where the growth rule is specified by a time evolving
measure, is briefly discussed.Comment: To appear in Physical Review
Predictability of band-limited, high-frequency, and mixed processes in the presence of ideal low-pass filters
Pathwise predictability of continuous time processes is studied in
deterministic setting. We discuss uniform prediction in some weak sense with
respect to certain classes of inputs. More precisely, we study possibility of
approximation of convolution integrals over future time by integrals over past
time. We found that all band-limited processes are predictable in this sense,
as well as high-frequency processes with zero energy at low frequencies. It
follows that a process of mixed type still can be predicted if an ideal
low-pass filter exists for this process.Comment: 10 page
Injectivity of sections of convex harmonic mappings and convolution theorems
In the article the authors consider the class of
sense-preserving harmonic functions defined in the unit disk
and normalized so that and , where
and are analytic in the unit disk. In the first part of the article we
present two classes and of
functions from and show that if
and , then the harmonic convolution is a univalent
and close-to-convex harmonic function in the unit disk provided certain
conditions for parameters and are satisfied. In the second
part we study the harmonic sections (partial sums) where , and denote the -th partial sums of
and , respectively. We prove, among others, that if
is a univalent harmonic convex mapping,
then is univalent and close-to-convex in the disk for
, and is also convex in the disk for
and . Moreover, we show that the section of is not convex in the disk but is shown to be convex
in a smaller disk.Comment: 16 pages, 3 figures; To appear in Czechoslovak Mathematical Journa
Hydrodynamic object recognition using pressure sensing
Hydrodynamic sensing is instrumental to fish and some amphibians. It also represents, for underwater vehicles, an alternative way of sensing the fluid environment when visual and acoustic sensing are limited. To assess the effectiveness of hydrodynamic sensing and gain insight into its capabilities and limitations, we investigated the forward and inverse problem of detection and identification, using the hydrodynamic pressure in the neighbourhood, of a stationary obstacle described using a general shape representation. Based on conformal mapping and a general normalization procedure, our obstacle representation accounts for all specific features of progressive perceptual hydrodynamic imaging reported experimentally. Size, location and shape are encoded separately. The shape representation rests upon an asymptotic series which embodies the progressive character of hydrodynamic imaging through pressure sensing. A dynamic filtering method is used to invert noisy nonlinear pressure signals for the shape parameters. The results highlight the dependence of the sensitivity of hydrodynamic sensing not only on the relative distance to the disturbance but also its bearing
Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schroedinger evolution
In non relativistic quantum mechanics time enters as a parameter in the
Schroedinger equation. However, there are various situations where the need
arises to view time as a dynamical variable. In this paper we consider the
dynamical role of time through the construction of a Lyapunov variable - i.e.,
a self-adjoint quantum observable whose expectation value varies monotonically
as time increases. It is shown, in a constructive way, that a certain class of
models admit a Lyapunov variable and that the existence of a Lyapunov variable
implies the existence of a transformation mapping the original quantum
mechanical problem to an equivalent irreversible representation. In addition,
it is proved that in the irreversible representation there exists a natural
time ordering observable splitting the Hilbert space at each t>0 into past and
future subspaces.Comment: Accepted for publication in JMP. Supercedes arXiv:0710.3604.
Discussion expanded to include the case of Hamiltonians with an infinitely
degenerate spectru
Approximate resonance states in the semigroup decomposition of resonance evolution
The semigroup decomposition formalism makes use of the functional model for
class contractive semigroups for the description of the time evolution
of resonances. For a given scattering problem the formalism allows for the
association of a definite Hilbert space state with a scattering resonance. This
state defines a decomposition of matrix elements of the evolution into a term
evolving according to a semigroup law and a background term. We discuss the
case of multiple resonances and give a bound on the size of the background
term. As an example we treat a simple problem of scattering from a square
barrier potential on the half-line.Comment: LaTex 22 pages 3 figure
Fingered growth in channel geometry: A Loewner equation approach
A simple model of Laplacian growth is considered, in which the growth takes
place only at the tips of long, thin fingers. In a recent paper, Carleson and
Makarov used the deterministic Loewner equation to describe the evolution of
such a system. We extend their approach to a channel geometry and show that the
presence of the side walls has a significant influence on the evolution of the
fingers and the dynamics of the screening process, in which longer fingers
suppress the growth of the shorter ones
Growth of fat slits and dispersionless KP hierarchy
A "fat slit" is a compact domain in the upper half plane bounded by a curve
with endpoints on the real axis and a segment of the real axis between them. We
consider conformal maps of the upper half plane to the exterior of a fat slit
parameterized by harmonic moments of the latter and show that they obey an
infinite set of Lax equations for the dispersionless KP hierarchy. Deformation
of a fat slit under changing a particular harmonic moment can be treated as a
growth process similar to the Laplacian growth of domains in the whole plane.
This construction extends the well known link between solutions to the
dispersionless KP hierarchy and conformal maps of slit domains in the upper
half plane and provides a new, large family of solutions.Comment: 26 pages, 6 figures, typos correcte
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