Shearlets and Optimally Sparse Approximations

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data", Birkh\"auser-Springe

PoN-S : a systematic approach for applying the Physics of Notation (PoN)

Visual Modeling Languages (VMLs) are important instruments of communication between modelers and stakeholders. Thus, it is important to provide guidelines for designing VMLs. The most widespread approach for analyzing and designing concrete syntaxes for VMLs is the so-called Physics of Notation (PoN). PoN has been successfully applied in the analysis of several VMLs. However, despite its popularity, the application of PoN principles for designing VMLs has been limited. This paper presents a systematic approach for applying PoN in the design of the concrete syntax of VMLs. We propose here a design process establishing activities to be performed, their connection to PoN principles, as well as criteria for grouping PoN principles that guide this process. Moreover, we present a case study in which a visual notation for representing Ontology Pattern Languages is designed

Edge Elimination in TSP Instances

The Traveling Salesman Problem is one of the best studied NP-hard problems in combinatorial optimization. Powerful methods have been developed over the last 60 years to find optimum solutions to large TSP instances. The largest TSP instance so far that has been solved optimally has 85,900 vertices. Its solution required more than 136 years of total CPU time using the branch-and-cut based Concorde TSP code [1]. In this paper we present graph theoretic results that allow to prove that some edges of a TSP instance cannot occur in any optimum TSP tour. Based on these results we propose a combinatorial algorithm to identify such edges. The runtime of the main part of our algorithm is $O(n^2 \log n)$ for an n-vertex TSP instance. By combining our approach with the Concorde TSP solver we are able to solve a large TSPLIB instance more than 11 times faster than Concorde alone

Twisted Conjugacy Classes in Lattices in Semisimple Lie Groups

Given a group automorphism $\phi:\Gamma\to \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-conjugacy classes. One says that $\Gamma$ has the $R_\infty$-property if there are infinitely many $\phi$-conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this paper we show that any irreducible lattice in a connected semi simple Lie group having finite centre and rank at least 2 has the $R_\infty$-property.Comment: 6 page

Unification of Radio Galaxies and Their Accretion/Jet Properties

We investigate the relation between black hole mass, M_bh, and jet power, Q_jet, for a sample of BL Lacs and radio quasars. We find that BL Lacs are separated from radio quasars by the FR I/II dividing line in M_bh-Q_jet plane, which strongly supports the unification scheme of FR I/BL Lac and FR II/radio quasar. The Eddington ratio distribution of BL Lacs and radio quasars exhibits a bimodal nature with a rough division at L_bol/L_Edd~0.01, which imply that they may have different accretion modes. We calculate the jet power extracted from advection dominated accretion flow (ADAF), and find that it require dimensionless angular momentum of black hole j~0.9-0.99 to reproduce the dividing line between FR I/II or BL Lac/radio quasar if dimensionless accretion rate mdot=0.01 is adopted, which is required by above bimodal distribution of Eddington ratios. Our results suggest that black holes in radio galaxies are rapidly spinning.Comment: To appear JAA in Jun

Quantifying Bar Strength: Morphology Meets Methodology

A set of objective bar-classification methods have been applied to the Ohio State Bright Spiral Galaxy Survey (Eskridge et al. 2002). Bivariate comparisons between methods show that all methods agree in a statistical sense. Thus the distribution of bar strengths in a sample of galaxies can be robustly determined. There are very substantial outliers in all bivariate comparisons. Examination of the outliers reveals that the scatter in the bivariate comparisons correlates with galaxy morphology. Thus multiple measures of bar strength provide a means of studying the range of physical properties of galaxy bars in an objective statistical sense.Comment: LaTeX with Kluwer style file, 5 pages with 3 embedded figures. edited by Block, D.L., Freeman, K.C., Puerari, I., & Groess,