Smoothness equivalence properties of univariate subdivision schemes and their projection analogues

We study the following modification of a linear subdivision scheme S: let M be a surface embedded in Euclidean space, and P a smooth projection mapping onto M. Then the P-projection analogue of S is defined as T := P ◦ S. As it turns out, the smoothness of the scheme T is always at least as high as the smoothness of the underlying scheme S or the smoothness of P minus 1, whichever is lower. To prove this we use the method of proximity as introduced by Wallner et al. (Constr Approx 24(3):289–318, 2006; Comput Aided Geom Design 22(7):593–622, 2005). While smoothness equivalence results are already available for interpolatory schemes S, this is the first result that confirms smoothness equivalence properties of arbitrary order for general non-interpolatory schemes.     

Approximation order of interpolatory nonlinear subdivision schemes

Linear interpolatory subdivision schemes of C^r smoothness have approximation order at least r+1. The present paper extends this result to nonlinear univariate schemes which are in proximity with linear schemes in a certain specific sense. The results apply to nonlinear subdivision schemes in Lie groups and in surfaces which are obtained from linear subdivision schemes. We indicate how to extend the results to the multivariate case.     

Ruled Laguerre minimal surfaces

A Laguerre minimal surface is an immersed surface in ${\mathbb{R}^3}$ being an extremal of the functional ${\int (H^2/K-1)dA}$ . In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces ${\mathbf{R}(\varphi,\lambda) = ( A\varphi,\, B\varphi,\, C\varphi + D\cos 2\varphi\, ) + \lambda\left(\sin \varphi,\, \cos \varphi,\, 0\,\right)}$ , where ${A,B,C,D\in \mathbb{R}}$ are fixed. To achieve invariance under Laguerre transformations, we also derive all Laguerre minimal surfaces that are enveloped by a family of cones. The methodology is based on the isotropic model of Laguerre geometry. In this model a Laguerre minimal surface enveloped by a family of cones corresponds to a graph of a biharmonic function carrying a family of isotropic circles. We classify such functions by showing that the top view of the family of circles is a pencil.     

Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames

In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions \(\{\psi _{\lambda }\}_{\lambda \in \Lambda }\subset L^2(\mathbb {R}^d)\) that constitutes a semi-discrete frame, we ask whether any real-valued function \(f \in L^2(\mathbb {R}^d)\) can be uniquely recovered from its unsigned convolutions \({\{|f *\psi _\lambda |\}_{\lambda \in \Lambda }}\). We find that under some mild assumptions on the semi-discrete frame and if f has exponential decay at \(\infty \), it suffices to know \(|f *\psi _\lambda |\) on suitably fine lattices to uniquely determine f (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of \(L^2(\mathbb {R}^d)\), \(d=1,2\), we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.     

Continuous Shearlet Tight Frames

Based on the shearlet transform we present a general construction of continuous tight frames for L ²(ℝ²) from any sufficiently smooth function with anisotropic moments. This includes for example compactly supported systems, piecewise polynomial systems, or both. From our earlier results in Grohs (Technical report, KAUST, 2009) it follows that these systems enjoy the same desirable approximation properties for directional data as the previous bandlimited and very specific constructions due to Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719–2754, 2009). We also show that the representation formulas we derive are in a sense optimal for the shearlet transform.     

Refinable functions for dilation families

We consider a family of d × d matrices W _e indexed by e?∈?E where (E, μ) is a probability space and some natural conditions for the family (W _e ) _e?∈?E are satisfied. The aim of this paper is to develop a theory of continuous, compactly supported functions $\varphi: {{\mathbb R}}^d \to {\mathbb{C}}$ which satisfy a refinement equation of the form $$ \varphi (x) = \int_E \sum\limits_{\alpha \in {{\mathbb Z}}^d} a_e(\alpha)\varphi\left(W_e x - \alpha\right) d\mu(e) $$ for a family of filters $a_e : {{\mathbb Z}}^d \to {\mathbb{C}}$ also indexed by e?∈?E. One of the main results is an explicit construction of such functions for any reasonable family (W _e ) _e?∈?E . We apply these facts to construct scaling functions for a number of affine systems with composite dilation, most notably for shearlet systems.     

Dynamics of optical and electrooptical switching in photo-thermal bistable CdS crystals

We investigate the dynamical properties of photo-thermal Self Electrooptic Effect Devices (SEEDs) fabricated of thin single crystal platelets in a wide region of excitation intensities at room temperature. Detailed characteristics of the switching processes and dynamics are precented for optical and for the first time for electro-optical types of bistable operation using an improved contact geometry. The role of the substrate of the SEED is also taken into account. It is shown that in the case of rectangular excitation pulses of light or of voltage the switching process can be divided into two regions, namely the crystal reaction time and the switching time itself. Both depend sensitively on the applied optical or electrical pulse height and also on the initial preheating. Critical slowing down is observed in the optical and the electrical case.Experimental results are discussed in terms of the thermal reaction of the crystal and of the substrate. The frequency dependencies of the switching processes are given. A quantitative theoretical analysis based on the heat conduction equation is done. Simple analytical formulas are deduced and discussed together with the experimental data.     

Self oscillations of a bistable element in two coupled hybrid ring resonators

A CdS crystal showing thermally induced optical bistability is incorporated into two coupled hybrid ring resonators with different delay times. Both delay times are much longer than the relaxation time of the nonlinearity. The resulting self-oscillations are investigated both experimentally and theoretically. We find two different types of oscillation modes. If the crystal is on the lower branch of the bistability during the longer delay time, step like oscillations similar to the case of a single resonator occur. If the crystal is in the lower state onlh for the shorter delay time (also the shorter of the two delay times is much longer than the relaxation time) we find more complicated modes with plateaus and stairs because the long resonator acts as a memory for the system state before the switching process. We find complex mode locking structures exhibiting Farey-tree like transitions between different oscillation modes as well as mode coexistence. Based on an adiabatic theory we compute the regions of extstence of the different oscillation modes and compare them with experimental results.     

<Emphasis Type="Italic">ε</Emphasis>-subgradient algorithms for locally lipschitz functions on Riemannian manifolds

This paper presents a descent direction method for finding extrema of locally Lipschitz functions defined on Riemannian manifolds. To this end we define a set-valued mapping \(x\rightarrow \partial _{\varepsilon } f(x)\) named ε-subdifferential which is an approximation for the Clarke subdifferential and which generalizes the Goldstein- ε-subdifferential to the Riemannian setting. Using this notion we construct a steepest descent method where the descent directions are computed by a computable approximation of the ε-subdifferential. We establish the global convergence of our algorithm to a stationary point. Numerical experiments illustrate our results.