where (a) m is nonnegative integer, (b) β= a(0)(1+a(0))/(1−|a(0)|2) and therefore, β >−1/2, (c) h and g are analytic in U , g(0)= 1, and h(0)≠ 0. Univalent logharmonic mappings on the unit disc have been studied extensively. For details see [1, 2, 3, 4, 5, 6, 7, 8]. Suppose that f is a univalent logharmonic mapping defined on the unit disc U . Then, if f(0) = 0, the function F(ζ) = log(f (eζ)) ...

We review two versions of a topology preserving algorithm one of which we had previously [1] found to be more successful in defining smooth manifolds and tight clusters. In the context of outlier detection, however, the other is shown to be more successful. Nevertheless, we show that, by using local kernels for calculation of responsibilities, the first one can also be used in this manner.

This article is an expanded version of lectures given by the first author at the International Workshop on Harmonic and Quasiconformal Mappings (Chennai, August 2010). They are based on several papers written by Tadeusz Iwaniec and the authors, some of which are also joint with Ngin-Tee Koh.

The index theorem for a harmonic mapping of riemannian manifolds is given. Let /: M -> N be a holomorphic mapping of Kaehler manifolds. Then it is shown that the index of / is zero and that a Jacobi field along/is a holomorphic section of the bundle f*T(N) induced by/.

in this paper we obtain coefficient characterization‎, ‎extreme points and‎ ‎distortion bounds for the classes of harmonic $p-$valent functions‎ ‎defined by certain modified operator‎. ‎some of our results improve‎ ‎and generalize previously known results‎.

Tapping-mode has been the most widely used mode of operation in atomic force microscopy. Recent studies have shown that higher harmonics of the cantilever vibrations carries information about material properties such as stiffness, viscoelasticity or capillary forces. A major problem with higher harmonic imaging is the low signal to noise ratio. Here we present a micromachined cantilever that en...

Motivated by the success of the Janowski starlike function, we consider here closely related functions for log-harmonic mappings of the form f z zh z g z defined on the open unit disc U. The functions are in the class of the generalized Janowski starlike log-harmonic mapping, Slh A,B, α , with the functional zh z in the class of the generalized Janowski starlike functions, S∗ A,B, α . By means ...

Accurate molecular imaging via high-order harmonic generation relies on comparing harmonic emission from a laser-irradiated molecule and an adequate reference system. However, an ideal reference atom with the same ionization properties as the molecule is not always available. We show that for suitably designed, very short laser pulses, a one-to-one mapping from high-order harmonic frequencies t...

In this paper, we present an efficient and robust technique for surface remeshing based on harmonic maps. We show how to ensure a one-to-one mapping for the discrete harmonic map and introduce a cubic representation of the geometry based on curved PN triangles. Topological and geometrical limitations of harmonic maps are also put to the fore and discussed. We show that, with the proposed approa...

We investigate macroscopic interference effects in high-order harmonic generation using a Ti:sapphire laser operating at a 100 kHz repetition rate. The structure and behavior of spectral and spatial interference fringes are explained and analytically described by transient phase matching of the long electron trajectory contribution. Time-frequency mapping due to the temporal chirp of the harmon...