scholarly journals Quadrature domains and the real quadratic family

2021 ◽  
Vol 25 (6) ◽  
pp. 104-125
Author(s):  
Kirill Lazebnik
Keyword(s):  
Congruence Subgroup ◽  
Mandelbrot Set ◽  
Quadrature Domain ◽  
Symmetric Polynomials ◽  
Schwarz Function ◽  
Content Type ◽  
Quadrature Domains ◽  
Hyperbolic Component ◽  
Simply Connected ◽  
Real Quadratic

We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component of the Mandelbrot set can be conformally mated with a congruence subgroup of P S L ( 2 , Z ) \mathrm {PSL}(2,\mathbb {Z}) , and that this conformal mating is the Schwarz function of a simply connected quadrature domain.

scholarly journals Asymptotic properties of unbounded quadrature domains the plane

2015 ◽  
Vol 26 (2) ◽  
pp. 175-191 ◽  
Author(s):  
LAVI KARP
Keyword(s):  
Compact Support ◽  
Asymptotic Properties ◽  
Quadrature Domain ◽  
Quadrature Domains ◽  
Straight Line ◽  
Simply Connected

We prove that if Ω is a simply connected quadrature domain (QD) of a distribution with compact support and the point of infinity belongs to the boundary, then the boundary has an asymptotic curve that is a straight line, parabola or infinite ray. In other words, such QDs in the plane are perturbations of null QDs.

scholarly journals Unstructured PEEC method with the use of surface impedance boundary condition

2020 ◽  
Vol 39 (5) ◽  
pp. 1017-1030 ◽  
Author(s):  
Gerard Meunier ◽  
Quang-Anh Phan ◽  
Olivier Chadebec ◽  
Jean-Michel Guichon ◽  
Bertrand Bannwarth ◽  
...  
Keyword(s):  
Finite Element ◽  
Surface Impedance ◽  
Computational Effort ◽  
Impedance Boundary Condition ◽  
Surface Mesh ◽  
Impedance Boundary ◽  
Content Type ◽  
Magnetic Effects ◽  
Coupled Circuits ◽  
Simply Connected

Purpose This paper aims to study unstructured-partial element equivalent circuit (PEEC) method for modelling electromagnetic regions with surface impedance condition (SIBC) is proposed. Two coupled circuits representations are used for solving both electric and/or magnetic effects in thin regions discretized by a finite element surface mesh. The formulation is applied in the context of low frequency problems with volumic magnetic media and coils. Non simply connected regions are treated with fundamental branch independent loop matrices coming from the circuit representation. Design/methodology/approach Because of the use of Whitney face elements, two coupled circuits representations are used for solving both electric and/or magnetic effects in thin regions discretized by a finite element surface mesh. The air is not meshed. Findings The new surface impedance formulation enables the modeling of volume conductive regions to efficiently simulate various devices with only a surface mesh. Research limitations/implications The propagation effects are not taken into account in the proposed formulation. Originality/value The formulation is original and is efficient for modeling non simply connected conductive regions with the use of SIBC. The unstructured PEEC SIBC formulation has been validated in presence of volume magnetic nonconductive region and compared with a SIBC FEM approach. The computational effort is considerably reduced in comparison with volume approaches.

scholarly journals Goodwillie Approximations to Higher Categories

2021 ◽  
Vol 272 (1333) ◽  
Author(s):  
Gijs Heuts
Keyword(s):  
Homotopy Theory ◽  
Symmetric Groups ◽  
Homotopy Groups ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Content Type ◽  
Simply Connected ◽  
Derivatives Of ◽  
Connected Spaces

We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of ∞ \infty -categories C \mathcal {C} and classify such Goodwillie towers in terms of the derivatives of the identity functor of C \mathcal {C} . As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras in spectra with Tate diagonals. Our classification of Goodwillie towers simplifies considerably in settings where the Tate cohomology of the symmetric groups vanishes. As an example we apply our methods to rational homotopy theory. Another application identifies the homotopy theory of p p -local spaces with homotopy groups in a certain finite range with the homotopy theory of certain algebras over Ching’s spectral version of the Lie operad. This is a close analogue of Quillen’s results on rational homotopy.

scholarly journals Stress fields around two pores in an elastic body: exact quadrature domain solutions

2015 ◽  
Vol 471 (2180) ◽  
pp. 20150240
Author(s):  
Darren Crowdy
Keyword(s):  
Elastic Body ◽  
Analytical Solutions ◽  
Plane Elasticity ◽  
Stress Fields ◽  
Quadrature Domain ◽  
Mathematical Approach ◽  
Quadrature Domains ◽  
Planar Domains ◽  
Elastic Bodies ◽  
Prime Function

Analytical solutions are given for the stress fields, in both compression and far-field shear, in a two-dimensional elastic body containing two interacting non-circular pores. The two complex potentials governing the solutions are found by using a conformal mapping from a pre-image annulus with those potentials expressed in terms of the Schottky–Klein prime function for the annulus. Solutions for a three-parameter family of elastic bodies with two equal symmetric pores are presented and the compressibility of a special family of pore pairs is studied in detail. The methodology extends to two unequal pores. The importance for boundary value problems of plane elasticity of a special class of planar domains known as quadrature domains is also elucidated. This observation provides the route to generalization of the mathematical approach here to finding analytical solutions for the stress fields in bodies containing any finite number of pores.

scholarly journals Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block

2018 ◽  
Vol 21 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Donna M. Testerman ◽  
Alexandre E. Zalesski
Keyword(s):  
Algebraic Groups ◽  
Jordan Block ◽  
Linear Algebraic Group ◽  
Irreducible Representations ◽  
Closed Field ◽  
Jordan Normal Form ◽  
Unipotent Element ◽  
Content Type ◽  
Simply Connected ◽  
Graphic Content

AbstractLetGbe a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed fieldFof characteristic{p\geq 0}, and let{u\in G}be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation ofG. Then the Jordan normal form of{\phi(u)}contains at most one non-trivial block if and only ifGis of type{G_{2}},uis a regular unipotent element and{\dim\phi\leq 7}. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I. D. Suprunenko [21].

scholarly journals Computational Geometry of Period-3 Hyperbolic Components in the Mandelbrot Set

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2519
Author(s):  
Young-Hee Geum ◽  
Young-Ik Kim
Keyword(s):  
Computational Geometry ◽  
Euclidean Plane ◽  
Theoretical Background ◽  
Mandelbrot Set ◽  
Boundary Line ◽  
Computational Results ◽  
Boundary Equation ◽  
Geometrical Properties ◽  
Cubic Equation ◽  
Hyperbolic Component

A parametric theoretical boundary equation of a period-3 hyperbolic component in the Mandelbrot set is established from a perspective of Euclidean plane geometry. We not only calculate the interior area, perimeter and curvature of the boundary line but also derive some relevant geometrical properties. The budding point of the period-3k component, which is born on the boundary of the period-3 component, and its relevant period-3k points are theoretically obtained by means of Cardano’s formula for the cubic equation. In addition, computational results are presented in tables and figures to support the theoretical background of this paper.

Describing quadratic Cremer point polynomials by parabolic perturbations

1998 ◽  
Vol 18 (3) ◽  
pp. 739-758 ◽  
Author(s):  
DAN ERIK KRARUP SØRENSEN
Keyword(s):  
Infinite Order ◽  
Julia Set ◽  
Main Idea ◽  
Mandelbrot Set ◽  
Basic Tool ◽  
Quadratic Polynomials ◽  
Hyperbolic Component ◽  
External Rays ◽  
External Ray ◽  
Single Comb

We describe two infinite-order parabolic perturbation procedures yielding quadratic polynomials having a Cremer fixed point. The main idea is to obtain the polynomial as the limit of repeated parabolic perturbations. The basic tool at each step is to control the behaviour of certain external rays.Polynomials of the Cremer type correspond to parameters at the boundary of a hyperbolic component of the Mandelbrot set. In this paper we concentrate on the main cardioid component. We investigate the differences between two-sided (i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove the existence of polynomials having an explicitly given external ray accumulating both at the Cremer point and at its non-periodic preimage. We think of the Julia set as containing a ‘topologist's double comb’.In the one-sided case we prove a weaker result: the existence of polynomials having an explicitly given external ray accumulating at the Cremer point, but having in the impression of the ray both the Cremer point and its other preimage. We think of the Julia set as containing a ‘topologist's single comb’.By tuning, similar results hold on the boundary of any hyperbolic component of the Mandelbrot set.

Accumulation theorems for quadratic polynomials

1996 ◽  
Vol 16 (3) ◽  
pp. 555-590 ◽  
Author(s):  
Dan Erik Krarup Sørensen
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Quadratic Polynomials ◽  
Hyperbolic Component ◽  
Non Local ◽  
External Ray ◽  
The One ◽  
Image Position ◽  
Symmetrical Points

AbstractWe consider the one-parameter family of quadratic polynomials:i.e. monic centered quadratic polynomials with an indifferent fixed point αtand prefixed point −αt. LetAt, be any one of the sets {0, ±αt}, {±αt}, {0, αt}, or {0, −αt}. Then we prove that for quadratic Julia sets corresponding to aGδ-dense subset ofthere is an explicitly given external ray accumulating onAt. In the caseAt= {±αt} the theorem is known as theDouady accumulation theorem.Corollaries are the non-local connectivity of these Julia sets and the fact that all such Julia sets contain a Cremer point. Existence of non-locally connected quadratic Julia sets of Hausdorff dimension two is derived by using a recent result of Shishikura. By tuning, the results hold on the boundary of any hyperbolic component of the Mandelbrot set.Finally, we concentrate on quadratic Cremer point polynomials. Here we prove that any ray accumulating on two symmetrical points of the Julia set must accumulate the origin. As a consequence, the denseGδsets arising from the first two possible choices ofAtare the same. We also prove that, if two distinct rays accumulate both to two distinct points, then the rays must accumulate on a common continuum joining the two points. This supports the conjecture that αtand –αtmay be joined by an arc in the Julia set.

scholarly journals Linear combinations of harmonic measures and quadrature domains of signed measures with small supports

1999 ◽  
Vol 42 (3) ◽  
pp. 433-444 ◽  
Author(s):  
Makoto Sakai
Keyword(s):  
Harmonic Functions ◽  
Positive Measure ◽  
Signed Measure ◽  
Quadrature Domain ◽  
Arbitrary Domain ◽  
Linear Combinations ◽  
Total Measure ◽  
Quadrature Domains ◽  
Harmonic Measures ◽  
Small Support

In this paper we discuss the shape of the quadrature domain of a signed measure for harmonic functions. It is known that the quadrature domain of a positive measure with small support is like a ball if the total measure is large enough. We show that, on the contrary, if the measure is not positive then the quadrature domain can be close to an arbitrary domain. This follows from a lemma concerning linear combinations of harmonic measures.

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