Local Geometry of the Fatou Set

1993 ◽  
pp. 109-122
Author(s):  
Lennart Carleson ◽  
Theodore W. Gamelin
Keyword(s):  
Local Geometry ◽  
Fatou Set

scholarly journals The geometry of some natural conjugacies inℂndynamics

2004 ◽  
Vol 2004 (67) ◽  
pp. 3685-3693
Author(s):  
John W. Robertson
Keyword(s):  
Projective Space ◽  
Topological Conjugacy ◽  
Local Geometry ◽  
Green Functions ◽  
Fatou Set ◽  
Absolute Value ◽  
Dimensional Projective Space ◽  
Complex Valued

We show that under some simple conditions a topological conjugacyhbetween two holomorphic self-mapsf1andf2of complexn-dimensional projective spaceℙnlifts canonically to a topological conjugacyHbetween the two corresponding polynomial self-maps ofℂn+1, and this conjugacy relates the two Green functions off1andf2. These conjugacies are interesting because their geometry is not inherited entirely from the geometry of the conjugacy onℙn. Part of the geometry of such a conjugacy is given (locally) by a complex-valued function whose absolute value is determined by the Green functions for the two maps, but whose argument seems to appear out of thin air. We work out the local geometry of such conjugacies over the Fatou set and over Fatou varieties of the original map.

Field equations in the neighbourhood of a particle in a conformal theory of gravitation. II

1969 ◽  
Vol 313 (1512) ◽  
pp. 71-82 ◽  
Keyword(s):  
Local Geometry ◽  
Field Equations ◽  
Schwarzschild Solution ◽  
Conformal Theory ◽  
Conformally Invariant ◽  
Coordinate Singularity ◽  
Previous Solution ◽  
Theory Of Gravitation ◽  
Spherically Symmetric Metric ◽  
Radial Variable

The field equations in the neighbourhood of a particle for a spherically symmetric metric in the conformal theory of gravitation put forward by Hoyle & Narlikar are examined. As the theory is conformally invariant, one can use different but physically equivalent conformal frames to study the equations. Previously these equations were studied in a conformal frame which, though suitable far away from the isolated particle, turns out not to be suitable in the neighbourhood of the particle. In the present paper a solution in a conformal frame is obtained that is suitable for considering regions near the particle. The solution thus obtained differs from the previous one in several respects. For example, it has no coordinate singularity for any non-zero value of the radial variable, unlike the previous solution or the Schwarzschild solution. It is also shown with the use of this solution that in this theory distant matter has an effect on local geometry.

scholarly journals Geological and groundwater flow model of a submarine groundwater discharge site at Hanko (Finland), northern Baltic Sea

2021 ◽  
Author(s):  
Samrit Luoma ◽  
Juha Majaniemi ◽  
Arto Pullinen ◽  
Juha Mursu ◽  
Joonas J. Virtasalo
Keyword(s):  
Hydraulic Conductivity ◽  
Groundwater Flow ◽  
Baltic Sea ◽  
Submarine Groundwater Discharge ◽  
Flow Model ◽  
Groundwater Discharge ◽  
Coarse Grained ◽  
Local Geometry ◽  
Groundwater Flow Model ◽  
Submarine Groundwater

AbstractThree-dimensional geological and groundwater flow models of a submarine groundwater discharge (SGD) site at Hanko (Finland), in the northern Baltic Sea, have been developed to provide a geological framework and a tool for the estimation of SGD rates into the coastal sea. The dataset used consists of gravimetric, ground-penetrating radar and shallow seismic surveys, drill logs, groundwater level monitoring data, field observations, and a LiDAR digital elevation model. The geological model is constrained by the local geometry of late Pleistocene and Holocene deposits, including till, glacial coarse-grained and fine-grained sediments, post-glacial mud, and coarse-grained littoral and aeolian deposits. The coarse-grained aquifer sediments form a shallow shore platform that extends approximately 100–250 m offshore, where the unit slopes steeply seawards and becomes covered by glacial and post-glacial muds. Groundwater flow preferentially takes place in channel-fill outwash coarse-grained sediments and sand and gravel interbeds that provide conduits of higher hydraulic conductivity, and have led to the formation of pockmarks on the seafloor in areas of thin or absent mud cover. The groundwater flow model estimated the average SGD rate per square meter of the seafloor at 0.22 cm day−1 in autumn 2017. The average SGD rate increased to 0.28 cm day−1 as a response to an approximately 30% increase in recharge in spring 2020. Sensitivity analysis shows that recharge has a larger influence on SGD rate compared with aquifer hydraulic conductivity and the seafloor conductance. An increase in recharge in this region will cause more SGD into the Baltic Sea.

A Study of Strain and Strain-Rate Dependent Properties of a Superplastic Zn-Al Alloy Under Biaxial Stresses

1982 ◽  
Vol 104 (1) ◽  
pp. 41-46
Author(s):  
T. C. Hsu ◽  
I. M. Bidhendi
Keyword(s):  
Strain Rate ◽  
Stress Ratio ◽  
Local Stress ◽  
Biaxial Stress ◽  
Al Alloy ◽  
Local Geometry ◽  
Forming Process ◽  
Rate Dependent ◽  
Strain And Strain Rate ◽  
Liquid Behavior

A superplastic Zn-Al alloy in sheet form is formed into a bulge over a circular hole by pneumatic pressure. The geometry, the stress, the strain, and the strain-rate are determined at various points covering the whole specimen and at various stages of the forming process. The complicated shape, and its complicated changes, are represented by introducing an index for the local geometry, called “prolateness,” which is also related to the local stress ratio in a simple way. The biaxial stress is analyzed into a strain-proportional and a strain-rate-proportional component, which represent, respectively, the quasi-solid and the quasi-liquid behavior of the superplastic material.

Local geometry for two Mn2+ paramagnetic centers in α-LiIO3 crystal

1999 ◽  
Vol 271 (1-4) ◽  
pp. 252-255 ◽  
Author(s):  
Wen-Chen Zheng ◽  
Shao-Yi Wu
Keyword(s):  
Local Geometry ◽  
Paramagnetic Centers

On uniformly perfect boundary of stable domains in iteration of meromorphic functions II

2002 ◽  
Vol 132 (3) ◽  
pp. 531-544 ◽  
Author(s):  
ZHENG JIAN-HUA
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Deficient Value ◽  
Uniformly Perfect ◽  
Simply Connected ◽  
Stable Domains ◽  
Uniform Perfectness

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

scholarly journals On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions

2010 ◽  
Vol 661 ◽  
pp. 446-481 ◽  
Author(s):  
YUE YANG ◽  
D. I. PULLIN
Keyword(s):  
Incompressible Flows ◽  
Initial Conditions ◽  
Scalar Fields ◽  
Local Geometry ◽  
Optimization Approach ◽  
Vortex Reconnection ◽  
Barotropic Flow ◽  
Lagrangian Surfaces ◽  
Definition Of ◽  
Surface Fields

For a strictly inviscid barotropic flow with conservative body forces, the Helmholtz vorticity theorem shows that material or Lagrangian surfaces which are vortex surfaces at time t = 0 remain so for t > 0. In this study, a systematic methodology is developed for constructing smooth scalar fields φ(x, y, z, t = 0) for Taylor–Green and Kida–Pelz velocity fields, which, at t = 0, satisfy ω·∇φ = 0. We refer to such fields as vortex-surface fields. Then, for some constant C, iso-surfaces φ = C define vortex surfaces. It is shown that, given the vorticity, our definition of a vortex-surface field admits non-uniqueness, and this is presently resolved numerically using an optimization approach. Additionally, relations between vortex-surface fields and the classical Clebsch representation are discussed for flows with zero helicity. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Both uniqueness and the distinction separating the evolution of vortex-surface fields and Lagrangian fields are discussed. By tracking φ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes so that neighbouring portions approach each other, with subsequent rolling up of structures near the interface, which reveals more information on dynamics than the iso-surfaces of vorticity magnitude. The non-local geometry in the evolution is quantified by two differential geometry properties. Rolled-up local shapes are found in the Lagrangian structures that were initially vortex surfaces close to the time of vortex reconnection. It is hypothesized that this is related to the formation of the very high vorticity regions.

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

scholarly journals Study of Influence of Electrode Geometry on Impedance Spectroscopy

2010 ◽  
Author(s):  
Riaz Ahmed ◽  
Kenneth Reifsnider
Keyword(s):  
Impedance Spectroscopy ◽  
High Frequency ◽  
Low Frequency ◽  
Effective Area ◽  
Ac Impedance ◽  
Local Geometry ◽  
Electrode Geometry ◽  
Electrode Area ◽  
Impedance Response ◽  
Working Electrode

Electrochemical Impedance Spectroscopy (EIS) is a powerful and proven tool for analyzing AC impedance response. A conventional three electrode EIS method was used to perform the investigation in the present study. Saturated potassium chloride solution was used as the electrolyte and three different material rods were used as working electrodes. Different configurations of electrode area were exposed to the electrolyte as an active area to investigate electrode geometry effects. Counter to working electrode distance was also altered while keeping the working electrode effective area constant to explore the AC response dependence on the variation of ion travel distance. Some controlled experiments were done to validate the experimental setup and to provide a control condition for comparison with experimental results. A frequency range of 100 mHz to 1 MHz was used for all experiments. In our analysis, we have found a noteworthy influence of electrode geometry on AC impedance response. For all electrodes, impedance decreases with the increase of effective area of the electrolyte. High frequency impedance is not as dependent on geometry as low frequency response. The observed phase shift angle drops in the high frequency region with increased working electrode area, whereas at low frequency the reverse is true. Resistance and capacitive reactance both decrease with an increase of area, but resistance response is more pronounce than reactance. For lower frequencies, small changes in working area produce very distinctive EIS variations. Electrode material as well as geometry was systematically varied in the present study. From these and other studies, we hope to develop a fundamental foundation for understanding specific changes in local geometry in fuel cell (and other) electrodes as a method of designing local morphology for specific performance.

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