Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory

This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.Comment: 65 pages, 24 figures. arXiv admin note: text overlap with arXiv:math/040542

PIMs and invariant parts for shape recognition

Journal ArticleWe present completely new very powerful solutions t o two fundamental problems central to computer vision. 1. Given data sets representing C objects to be stored in a database, and given a new data set for an object, determine the object in the database that is most like the object measured. We solve this problem through use of PIMs ("Polynomial Interpolated Measures"), which, is a new representation integrating implicit polynomial curves and surfaces, explicit polynomials, and discrete data sets which may be sparse. The method provides high accuracy at low computational cost. 2. Given noisy 20 data along a curve (or 30 data along a surface), decompose the data into patches such that new data taken along affine transformation-s or Eucladean transformations of the curve (or surface) can be decomposed into corresponding patches. Then recognition of complex or partially occluded objects can be done in terms of invariantly determined patches. We briefly outline a low computational cost image-database indexing-system based on this representation for objects having complex shape-geometry

On Jordan's measurements

The Jordan measure, the Jordan curve theorem, as well as the other generic references to Camille Jordan's (1838-1922) achievements highlight that the latter can hardly be reduced to the "great algebraist" whose masterpiece, the Trait\'e des substitutions et des equations alg\'ebriques, unfolded the group-theoretical content of \'Evariste Galois's work. The present paper appeals to the database of the reviews of the Jahrbuch \"uber die Fortschritte der Mathematik (1868-1942) for providing an overview of Jordan's works. On the one hand, we shall especially investigate the collective dimensions in which Jordan himself inscribed his works (1860-1922). On the other hand, we shall address the issue of the collectives in which Jordan's works have circulated (1860-1940). Moreover, the time-period during which Jordan has been publishing his works, i.e., 1860-1922, provides an opportunity to investigate some collective organizations of knowledge that pre-existed the development of object-oriented disciplines such as group theory (Jordan-H\"older theorem), linear algebra (Jordan's canonical form), topology (Jordan's curve), integral theory (Jordan's measure), etc. At the time when Jordan was defending his thesis in 1860, it was common to appeal to transversal organizations of knowledge, such as what the latter designated as the "theory of order." When Jordan died in 1922, it was however more and more common to point to object-oriented disciplines as identifying both a corpus of specialized knowledge and the institutionalized practices of transmissions of a group of professional specialists

Persistent topology for natural data analysis - A survey

Natural data offer a hard challenge to data analysis. One set of tools is being developed by several teams to face this difficult task: Persistent topology. After a brief introduction to this theory, some applications to the analysis and classification of cells, lesions, music pieces, gait, oil and gas reservoirs, cyclones, galaxies, bones, brain connections, languages, handwritten and gestured letters are shown