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Location of the critical points of certain polynomials

Somjate Chaiya, Aimo Hinkkanen (2013)

Annales UMCS, Mathematica

Let D¯ denote the unit disk {z : |z| < 1} in the complex plane C. In this paper, we study a family of polynomials P with only one zero lying outside D¯. We establish criteria for P to satisfy implying that each of P and P' has exactly one critical point outside D¯.

Loewner chains and quasiconformal extension of holomorphic mappings

Hidetaka Hamada, Gabriela Kohr (2003)

Annales Polonici Mathematici

Let f(z,t) be a Loewner chain on the Euclidean unit ball B in ℂⁿ. Assume that f(z) = f(z,0) is quasiconformal. We give a sufficient condition for f to extend to a quasiconformal homeomorphism of 2 n onto itself.

Logarithmic capacity is not subadditive – a fine topology approach

Pavel Pyrih (1992)

Commentationes Mathematicae Universitatis Carolinae

In Landkof’s monograph [8, p. 213] it is asserted that logarithmic capacity is strongly subadditive, and therefore that it is a Choquet capacity. An example demonstrating that logarithmic capacity is not even subadditive can be found e.gi̇n [6, Example 7.20], see also [3, p. 803]. In this paper we will show this fact with the help of the fine topology in potential theory.

Logarithmic derivative of the Euler Γ function in Clifford analysis.

Guy Laville, Louis Randriamihamison (2005)

Revista Matemática Iberoamericana

The logarithmic derivative of the Γ-function, namely the ψ-function, has numerous applications. We define analogous functions in a four dimensional space. We cut lattices and obtain Clifford-valued functions. These functions are holomorphic cliffordian and have similar properties as the ψ-function. These new functions show links between well-known constants: the Eurler gamma constant and some generalisations, ζR(2), ζR(3). We get also the Riemann zeta function and the Epstein zeta functions.

Long time asymptotics of the Camassa–Holm equation on the half-line

Anne Boutet de Monvel, Dmitry Shepelsky (2009)

Annales de l’institut Fourier

We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u t - u t x x + 2 u x + 3 u u x = 2 u x u x x + u u x x x on the half-line x 0 . The paper continues our study of IBV problems for the CH equation, the key tool of which is the formulation and analysis of associated Riemann–Hilbert factorization problems. We specify the regions in the quarter space-time plane x &gt; 0 , t &gt; 0 having qualitatively different asymptotic pictures, and give the main terms of the asymptotics in terms of spectral data...

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