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/.

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...

A distributed modeling approach has been developed to describe the dynamic behavior of ring resonators. The model includes the effect of large amplitudes around primary resonance frequencies, material and electrostatic nonlinearities. Through a combination of geometric and material nonlinearities, closed-form expression for third-order nonlinearity in mechanical stiffness of bulk-mode ring reso...

Recently, novel application areas in digital geometry processing, such as simulation, dynamics, and medical surgery simulations, have necessitated the representation of not only the surface data but also the interior volume data of a given 3D object. In this paper, we present an efficient framework for the shape approximations of spherical solid objects based on trivariate B-splines. To do this...

We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space W 1,2 and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.