Curves with rational chord-length parametrization

It has been recently proved that rational quadratic circles in standard Bezier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limacon of Pascal. The extension to 3D rational curves is also tackled

On multi-degree splines

Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to B\'ezier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines

A characterization of virtually embedded subsurfaces in 3-manifolds

The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this case, the subsurface is virtually a leaf of a taut foliation. Besides other consequences, examples are exhibited that non-geometric 3-manifolds with no Seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces.Comment: 28 pages. Errors of previous Proposition 3.1 and Formula 7.2 correcte

Linear groups in Galois fields. A case study of tacit circulation of explicit knowledge

This preprint is the extended version of a paper that will be published in the proceedings of the Oberwolfach conference "Explicit vs tacit knowledge in mathematics" (January 2012). It presents a case study on some algebraic researches at the turn of the twentieth century that involved mainly French and American authors. By investigating the collective dimensions of these works, this paper sheds light on the tension between the tacit and the explicit in the ways some groups of texts hold together, thereby constituting some shared algebraic cultures. Although prominent algebraists such as Dickson made extensive references to papers published in France, and despite the roles played by algebra and arithmetic in the development of the American mathematical community, our knowledge of the circulations of knowledge between France and the United States at the beginning of the 20th century is still very limited. It is my aim to tackle such issues through the case study of a specific collective approach to finite group theory at the turn of the 20th century. This specific approach can be understood as a shared algebraic culture based on the long run circulation of some specific procedures of decompositions of the analytic forms of substitutions. In this context, the general linear group was introduced as the maximal group in which an elementary abelian group (i.e., the multiplicative group of a Galois field) is a normal subgroup

Life is an Adventure! An agent-based reconciliation of narrative and scientific worldviews\ud

The scientific worldview is based on laws, which are supposed to be certain, objective, and independent of time and context. The narrative worldview found in literature, myth and religion, is based on stories, which relate the events experienced by a subject in a particular context with an uncertain outcome. This paper argues that the concept of “agent”, supported by the theories of evolution, cybernetics and complex adaptive systems, allows us to reconcile scientific and narrative perspectives. An agent follows a course of action through its environment with the aim of maximizing its fitness. Navigation along that course combines the strategies of regulation, exploitation and exploration, but needs to cope with often-unforeseen diversions. These can be positive (affordances, opportunities), negative (disturbances, dangers) or neutral (surprises). The resulting sequence of encounters and actions can be conceptualized as an adventure. Thus, the agent appears to play the role of the hero in a tale of challenge and mystery that is very similar to the "monomyth", the basic storyline that underlies all myths and fairy tales according to Campbell [1949]. This narrative dynamics is driven forward in particular by the alternation between prospect (the ability to foresee diversions) and mystery (the possibility of achieving an as yet absent prospect), two aspects of the environment that are particularly attractive to agents. This dynamics generalizes the scientific notion of a deterministic trajectory by introducing a variable “horizon of knowability”: the agent is never fully certain of its further course, but can anticipate depending on its degree of prospect

A KdV-like advection-dispersion equation with some remarkable properties

We discuss a new non-linear PDE, u_t + (2 u_xx/u) u_x = epsilon u_xxx, invariant under scaling of dependent variable and referred to here as SIdV. It is one of the simplest such translation and space-time reflection-symmetric first order advection-dispersion equations. This PDE (with dispersion coefficient unity) was discovered in a genetic programming search for equations sharing the KdV solitary wave solution. It provides a bridge between non-linear advection, diffusion and dispersion. Special cases include the mKdV and linear dispersive equations. We identify two conservation laws, though initial investigations indicate that SIdV does not follow from a polynomial Lagrangian of the KdV sort. Nevertheless, it possesses solitary and periodic travelling waves. Moreover, numerical simulations reveal recurrence properties usually associated with integrable systems. KdV and SIdV are the simplest in an infinite dimensional family of equations sharing the KdV solitary wave. SIdV and its generalizations may serve as a testing ground for numerical and analytical techniques and be a rich source for further explorations.Comment: 15 pages, 4 figures, corrected sign typo in KdV Lagrangian above equation 3