Problems of Harmonic Analysis related to finite type hypersurfaces in R^3, and Newton polyhedra

This article, which grew out of my lecture at the conference "Analysis and Applications: A Conference in Honor of Elias M. Stein" in May 2011, is intended to give an overview on a collection of results which have been obtained jointly with I.I. Ikromov, and in parts also with M. Kempe, and at the same time to give a kind of guided tour through the rather comprehensive proofs of the major results that I shall address. All of our work is highly influenced by the pioneering ideas developed by E.M. Stein.Comment: 44 pages, a picture

Uncertainty-Aware Principal Component Analysis

We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertainty-aware PCA. In addition to the accuracy and performance gained by our approach over sampling-based strategies, our formulation allows us to perform sensitivity analysis with regard to the uncertainty in the data. For this, we propose factor traces as a novel visualization that enables to better understand the influence of uncertainty on the chosen principal components. We provide multiple examples of our technique using real-world datasets. As a special case, we show how to propagate multivariate normal distributions through PCA in closed form. Furthermore, we discuss extensions and limitations of our approach

Detecting Similarity of Rational Plane Curves

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.Comment: 22 page

Analyze Large Multidimensional Datasets Using Algebraic Topology

This paper presents an efficient algorithm to extract knowledge from high-dimensionality, high- complexity datasets using algebraic topology, namely simplicial complexes. Based on concept of isomorphism of relations, our method turn a relational table into a geometric object (a simplicial complex is a polyhedron). So, conceptually association rule searching is turned into a geometric traversal problem. By leveraging on the core concepts behind Simplicial Complex, we use a new technique (in computer science) that improves the performance over existing methods and uses far less memory. It was designed and developed with a strong emphasis on scalability, reliability, and extensibility. This paper also investigate the possibility of Hadoop integration and the challenges that come with the framework

Robust Registration of Calcium Images by Learned Contrast Synthesis

Multi-modal image registration is a challenging task that is vital to fuse complementary signals for subsequent analyses. Despite much research into cost functions addressing this challenge, there exist cases in which these are ineffective. In this work, we show that (1) this is true for the registration of in-vivo Drosophila brain volumes visualizing genetically encoded calcium indicators to an nc82 atlas and (2) that machine learning based contrast synthesis can yield improvements. More specifically, the number of subjects for which the registration outright failed was greatly reduced (from 40% to 15%) by using a synthesized image

Deep Big Simple Neural Nets Excel on Handwritten Digit Recognition

Good old on-line back-propagation for plain multi-layer perceptrons yields a very low 0.35% error rate on the famous MNIST handwritten digits benchmark. All we need to achieve this best result so far are many hidden layers, many neurons per layer, numerous deformed training images, and graphics cards to greatly speed up learning.Comment: 14 pages, 2 figures, 4 listing

Regularity results for shortest billiard trajectories in convex bodies in Rn\mathbb{R}^n

We derive properties of closed billiard trajectories in convex bodies in $\mathbb{R}^n$. Building on techniques introduced by K. and D. Bezdek we establish two regularity results for length minimizing closed billiard trajectories: one for billiard trajectories in general convex bodies, the other for billiard trajectories in the special case of acute convex polytopes. Moreover, we attach particular importance to various examples, also including examples which show the sharpness of the first regularity result. Finally, we show how our results can be used in order to calculate (analytically and by computer) length minimizing closed regular billiard trajectories in convex polytopes.Comment: 34 pages, 9 figures, 1 table, chapter 6 adde

Shearing of loose granular materials: A statistical mesoscopic model

A two-dimensional lattice model for the formation and evolution of shear bands in granular media is proposed. Each lattice site is assigned a random variable which reflects the local density. At every time step, the strain is localized along a single shear-band which is a spanning path on the lattice chosen through an extremum condition. The dynamics consists of randomly changing the `density' of the sites only along the shear band, and then repeating the procedure of locating the extremal path and changing it. Starting from an initially uncorrelated density field, it is found that this dynamics leads to a slow compaction along with a non-trivial patterning of the system, with high density regions forming which shelter long-lived low-density valleys. Further, as a result of these large density fluctuations, the shear band which was initially equally likely to be found anywhere on the lattice, gets progressively trapped for longer and longer periods of time. This state is however meta-stable, and the system continues to evolve slowly in a manner reminiscent of glassy dynamics. Several quantities have been studied numerically which support this picture and elucidate the unusual system-size effects at play.Comment: 11 pages, 15 figures revtex, submitted to PRE, See also: cond-mat/020921

Currents and Superpotentials in classical gauge invariant theories I. Local results with applications to Perfect Fluids and General Relativity

E. Noether's general analysis of conservation laws has to be completed in a Lagrangian theory with local gauge invariance. Bulk charges are replaced by fluxes of superpotentials. Gauge invariant bulk charges may subsist when distinguished one-dimensional subgroups are present. As a first illustration we propose a new {\it Affine action} that reduces to General Relativity upon gauge fixing the dilatation (Weyl 1918 like) part of the connection and elimination of auxiliary fields. It allows a comparison of most gravity superpotentials and we discuss their selection by the choice of boundary conditions. A second and independent application is a geometrical reinterpretation of the convection of vorticity in barotropic nonviscous fluids. We identify the one-dimensional subgroups responsible for the bulk charges and thus propose an impulsive forcing for creating or destroying selectively helicity. This is an example of a new and general Forcing Rule.Comment: 64 pages, LaTeX. Version 2 has two more references and one misprint corrected. Accepted in Classical and Quantum Gravit