On the life and work of Cyrus Derman

This article provides a brief biographical synopsis of the life of Cyrus Derman and a comprehensive summary of his research. Professor Cyrus Derman was known among his friends as Cy.     

Properties of Ahlfors constant in Ahlfors covering surface theory

This paper is a subsequent work of [Invent. Math., 2013, 191: 197-253]. The second fundamental theorem in Ahlfors covering surface theory is that, for each set E_q of q (≥3) distinct points in the extended complex plane \overline{ℂ}; there is a minimal positive constant H₀ (E_q) (called Ahlfors constant with respect to E_q), such that the inequality(q-2)A(\sum )-4\pi ♯(f^{-1}(E_q)\cap U)\le H_0(E_q)L(\partial \sum )holds for any simply-connected surface{\displaystyle \sum \mathrm{=}（f,\bar{U}）} ; where A(\sum) is the area of\sum; L(\partial \sum) is the perimeter of\sum; and # denotes the cardinality. It is difficult to compute H₀ (E_q) explicitly for general set E_q; and only a few properties of H₀(E_q) are known. The goals of this paper are to prove the continuity and differentiability of H₀ (E_q); to estimate H₀ (E_q); and to discuss the minimum of H₀ (E_q) for fiixed q.     

On the absence of the Ahlfors and Sullivan theorems for Kleinian groups in higher dimensions

Khabarovsk and L'vov. Translated fromSibirskii Matematicheskii Zhurnal, Vol. 32, No. 2, pp. 61–73, March–April, 1991.     

Some generalizations of Ahlfors Lemma

One of the most influential versions of the classical Schwarz–Pick Lemma is probably that of Ahlfors. Pulling back a conformal semimetric on a Riemann surface under any holomorphic map from the open unit disk equipped with a Poincaré metric, the curvature of which is assumed to bound from above the curvature of the Riemann surface, he successfully showed that a conformal semimetric to be compared with the Poincaré metric is obtained. In the present paper, we give a comparison theorem between two conformal semimetrics of variable curvature in the same spirit. Our main theorem is a local one by its nature, but global results can be derived therefrom.     

A new proof of the Ahlfors five islands theorem

We deduce the Ahlfors five islands theorem from a corresponding result of Nevanlinna concerning perfectly branched values, a rescaling lemma for non-normal families and an existence theorem for quasiconformal mappings. We also give a proof of Nevanlinna’s result based on the rescaling lemma and a version of Schwarz’s lemma.     

The life and work of Sophie Germain

We discuss the life of Sophie Germain, her struggle to educate herself and to win acceptance within the French mathematical community, and her contributions to number theory and to the theory of elasticity.

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Received: 23 January 2003, Accepted: 25 March 2003, Corespondence to: Gareth Jones     

The life and work of Sophie Germain

We discuss the life of Sophie Germain, her struggle to educate herself and to win acceptance within the French mathematical community, and her contributions to number theory and to the theory of elasticity.     

Exotic baker and wandering domains for Ahlfors islands maps

Let X be a compact Riemann surface of genus at most 1, i.e., the Riemann sphere or a torus, and let W ⊊ X be an arbitrary domain. We construct a variety of examples of holomorphic functions g: W → X that satisfy Epstein’s Ahlfors islands property and that have “pathological” dynamical behaviour. In particular, we show that the accumulation set of any curve tending to the boundary of W can be realized as the ω-limit set of a Baker domain of such a function. We furthermore construct Ahlfors islands maps

Traces of Sobolev functions on the Ahlfors sets of Carnot groups

We prove the converse of the trace theorem for the functions of the Sobolev spaces W _p ^l on a Carnot group on the regular closed subsets called Ahlfors d-sets (the direct trace theorem was obtained in one of our previous publications). The theorem generalizes Johnsson and Wallin’s results for Sobolev functions on the Euclidean space. As a consequence we give a theorem on the boundary values of Sobolev functions on a domain with smooth boundary in a two-step Carnot group. We consider an example of application of the theorems to solvability of the boundary value problem for one partial differential equation.     

Giuseppe Tallini: His life and work

It is known that euclidean or hyperbolic spaces are characterized among certain metric spaces by the property of linearity of the equidistant locus of pairs of points. In this paper, this linearity requirement is replaced by the requirement of convexity of the set of points which are metrically pythagorean orthogonal to a given segment at a given point. As a result a new characterization of real inner product spaces among complete, convex, externally convex metric spaces is obtained.     

The life and work of André Cholesky

Cholesky’s method for solving a system of linear equations with a symmetric positive definite matrix is well known. In this paper, I will give an account of the life of Cholesky, analyze an unknown and unpublished paper of him where he explains his method, and review his other scientific works. I dedicated this work to John (Jack) Todd with esteem and respect at the occasion of his 95th anniversary.     

Most honourable remembrance: the life and work of thomas bayes

The Mathematical Intelligencer -     

Packing dimension and Ahlfors regularity of porous sets in metric spaces

Let X be a metric measure space with an s-regular measure μ. We prove that if A ì X{A\subset X} is r{\varrho} -porous, then dim_p(A) ￡ s-cr^s{{\rm {dim}_p}(A)\le s-c\varrho^s} where dim_p is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ì X{N\subset X} with μ(N) = 0 such that dim_p(A) ￡ dim_p(X)-c(log\tfrac1r)^-1r^t{{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t} for all r{\varrho} -porous sets A ì X\ N{A \subset X{\setminus} N} . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that A ì F{A\subset F} .