Properties of the Conformal Radius of a Domain

We consider the function , where is the conformal radius of a simply connected domain D at a point . We study relations between the values of the function at various points of the domain D. In Sec. 1, we establish exact inequalities relating the values of the function in an arbitrary simply connected domain with the position of the conformal center and with the maximal conformal radius of the domain D. The same values are related to the values of at another two points of the domain D. In Sec. 2, similar results are established for convex domains. This work supplements some recent results of Emel'yanov and Kovalev. Bibliography: 9 titles.     

START in a Five-Dimensional Conformal Domain

In this paper we give a brief review of the pseudo-Riemannian geometry of the five-dimensional homogeneous space for the conformal group O(4, 2). Its topology is described and its relation to the conformally compactified Minkowski space is discussed. Its metric and geodesics are calculated using a generalized half-space representation. Compactification via Lie-sphere geometry is outlined. Possible applications to Jaime Keller’s START theory may follow by using its predecessor - the 5-optics of Yu. B. Rumer. The point of view of Rumer is given extensively in the last section of the paper.     

Inner Radius of Univalence for a Strongly Starlike Domain

 The inner radius of univalence of a domain D with Poincaré density ρ _D is the possible largest number σ such that the condition ∥ S _f ∥ _D  = sup _{w∈ D} ρ _D (w) ⁻²∥ S _f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S _f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α.     

Inner Radius of Univalence for a Strongly Starlike Domain

 The inner radius of univalence of a domain D with Poincaré density ρ _D is the possible largest number σ such that the condition ∥ S _f ∥ _D  = sup _{w∈ D} ρ _D (w) ⁻²∥ S _f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S _f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α. The author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 11740088. A part of this work was carried out during his visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS. Received November 26, 2001; in revised form September 24, 2002 Published online May 9, 2003     

The Inner Radius of Univalency of the Rhom busDom ain

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Estimates of Conformal Radius and Distortion Theorems for Univalent Functions

A simple proof of the recent result by E. G. Emel'yanov concerning the maximum of the conformal radius r(D,1) for a family of simply connected domains with a fixed value r(D,0) is given. A similar problem is solved for a family of convex domains. Exact estimates for functionals of the form are obtained for families of functions inverse to elements of the classes S and S_m, where S={f:f is regular and univalent in the disk {z:|z| < 1} and f(0)=f'(0)-1=0} and S_M= for . Bibliography: 7 titles.     

Radiation and Scattering by Complex Conformal Antennas on a Circular Cylinder

A technique for characterizing and designing complex conformal antennas flush-mounted on a singly-curved surface is presented. This approach is based on the hybrid finite element–boundary integral (FE–BI) method. A related method was proposed in the past utilizing cylindrical-shell finite element and roof-top rectangular basis functions for the boundary integral. Although that method proved very powerful for analyzing cylindrical–rectangular patch arrays flush-mounted to a circular cylinder, the requirement for uniform meshing in the aperture ultimately limited its usefulness. In this present formulation, tetrahedral elements are used to expand the volumetric electric fields while similar basis functions are used for the boundary integral. The curvature of the aperture is explicitly included via the use of the circular cylinder dyadic Green's function. After presentation of the formulation and validation using several well-understood examples, an example is presented that illustrates the capabilities of this method for modeling complex conformal antennas heretofore not examined by rigorous methods due to inherent limitations of the various published methods.     

Energy Minimization and Flux Domain Structure in the Intermediate State of a Type-I Superconductor

The intermediate state of a type-I superconductor involves a fine-scale mixture of normal and superconducting domains. We take the viewpoint, due to Landau, that the realizable domain patterns are (local) minima of a nonconvex variational problem. We examine the scaling law of the minimum energy and the qualitative properties of domain patterns achieving that law. Our analysis is restricted to the simplest possible case: a superconducting plate in a transverse magnetic field. Our methods include explicit geometric constructions leading to upper bounds and ansatz-free inequalities leading to lower bounds. The problem is unexpectedly rich when the applied field is near-zero or near-critical. In these regimes there are two small parameters, and the ground state patterns depend on the relation between them.     

Approximating the Conformal Maps of Elongated Quadrilaterals by Domain Decomposition

Let Q:={ Ω;z ₁ ,z ₂ ,z ₃ ,z ₄ } be a quadrilateral consisting of a Jordan domain Ω and four points z ₁ , z ₂ , z ₃ , z ₄ , in counterclockwise order on \partial Ω and let m(Q) be the conformal module of Q . Then Q is conformally equivalent to the rectangular quadrilateral {R _m(Q) ;0,1,1+\mathop \it im (Q),\mathop \it im (Q)}, where R _m(Q) := {(ξ,η): 0<ξ<1, 0 <η<m(Q)}, in the sense that there exists a unique conformal map f: Ω \rightarrow R _m(Q) that takes the four points z ₁ , z ₂ , z ₃ , z ₄ , respectively, onto the four vertices 0 , 1 , 1+\mathop \it im (Q) , \mathop \it im (Q) of R _m(Q) . In this paper we consider the use of a domain decomposition method (DDM) for computing approximations to the conformal map f , in cases where the quadrilateral Q is ``long.' The method has been studied already but, mainly, in connection with the computation of m(Q) . Here we consider certain recent results of Laugesen \cite{La}, for the DDM approximation of the conformal map f: Ω \rightarrow R _m(Q) associated with a special class of quadrilaterals (viz., quadrilaterals whose two opposite boundary segments (z ₂ , z ₃ ) and (z ₄ , z ₁ ) are parallel straight lines), and seek to extend these results to more general quadrilaterals. By making use of the available DDM theory for conformal modules, we show that the corresponding theory for f can, indeed, be extended to a much wider class of quadrilaterals than those considered by Laugesen. June 1, 2000. Date accepted: September 6, 2000.     

Minimization of the makespan in a two-machine problem under given resource constraints

In the paper the classical two-machine flow-shop problem was generalized to the case when job processing times may be reduced linearly by the application of a limited, continuously divisible resource, e.g. financial outlay, energy, fuel, catalyzer etc. It is proved that the decision form of this problem is NP-complete even for the fixed job processing times on one of the machines and identical job reduction rates on another. Some polynomially solvable cases of the problem are identified. Four simple and modified approximate algorithms are presented together with their worst case and experimental analysis. Also, a fast exact algorithm of the branch and bound type based on the shown elimination properties of the problem considered is presented. Some computational results and generalizations (e.g. bicriterial approach) are included as well.     

Cone Adaptation Strategies for a Finite and Exact Cutting Plane Algorithm for Concave Minimization

In this paper we are concerned with pure cutting plane algorithms for concave minimization. One of the most common types of cutting planes for performing the cutting operation in such algorithm is the concavity cut. However, it is still unknown whether the finite convergence of a cutting plane algorithm can be enforced by the concavity cut itself or not. Furthermore, computational experiments have shown that concavity cuts tend to become shallower with increasing iteration. To overcome these problems we recently proposed a procedure, called cone adaptation, which deepens concavity cuts in such a way that the resulting cuts have at least a certain depth with 0, where is independent of the respective iteration, which enforces the finite convergence of the cutting plane algorithm. However, a crucial element of our proof that these cuts have a depth of at least was that we had to confine ourselves to -global optimal solutions, where is a prescribed strictly positive constant. In this paper we examine possible ways to ensure the finite convergence of a pure cutting plane algorithm for the case where = 0.     

Optimal Interpolation of Scattered Data on a Circular Domain with Boundary Conditions

Optimal interpolation problems of scattered data on a circular domain with two different types of boundary value conditions are studied in this paper. Closed-form optimal solutions, a new type of spline functions defined by partial differential operators, are obtained. This type of new splines is a generalization of the well-known $L_g$-splines and thin-plate splines. The standard reproducing kernel structure of the optimal solutions is demonstrated. The new idea and technique developed in this paper are finally generalized to solve the same interpolation problems involving a more general class of partial differential operators on a general region.     

Minimization of the root of a quadratic functional under an affine equality constraint

We present a way of solving the problem of minimizing the root of quadratic functional subject to an affine constraint. We give an explicit formula for computing the solutions of such a problem. This is of interest for solving significant problems of financial economics as well as some classes of feasibility and optimization problems which frequently occur in tomography and other fields.     

Distortion of the Hyperbolic Robin Capacity under a Conformal Mapping and Extremal Configurations

This paper is connected with recent results of Duren and Pfaltzgraff (J. Anal. Math., 78, 205218 (1999)). We consider the problem on the distortion of the hyperbolic Robin capacity _h(A,) of the boundary set A under a conformal mapping of a domain U into the unit disk U. It is shown that, for sets consisting of a finite number of boundary arcs or complete boundary components, the inequality

圆外平面弹性问题的边界积分公式

将边界上的应力函数及其法向导数展开为罗朗级数,与复应力函数的罗朗级数的表达式对比,可以确定罗朗级数的各系数,再利用傅利叶级数和卷积的几个公式进行计算,得到应力函数边界积分公式．通过边界的应力函数及其法向导数的积分,直接得到圆外应力函数值,并给出几个算例,表明结果用于求解单位圆外平面弹性问题十分方便．     

一族Bloch函数的单叶半径与覆盖半径

研究了Bloch函数族B中的一个子族Bg,给出了Bg中函数的单叶半径.作为应用建立了Bg中函数的覆盖定理,从而刻画了Bg中函数的有关性质.     

复合荷载下波纹圆板的非线性分析*

本文按照各向同性和正交各向异性圆板的大挠度理论,研究了具有光滑中心的波纹圆板在均布和中心集中荷载联合作用下的非线性弯曲问题.应用修正迭代法,我们得到了夹紧固定和滑动固定两种边界条件下十分精确的解析解.