Combinatorics of the Gauss digitization under translation in 2D

International audienceThe action of a translation on a continuous object before its digitization generates several digital objects. This paper focuses on the combinatorics of the generated digital objects up to integer translations. In the general case, a worst-case upper bound is exhibited and proved to be reached on an example. Another upper bound is also proposed by making a link between the number of the digital objects and the boundary curve, through its self-intersections on the torus. An upper bound, quadratic in digital perimeter, is then derived in the convex case and eventually an asymptotic upper bound, quadratic in the grid resolution, is exhibited in the polygonal case. A few signicant examples finish the paper

Analysis of shape grammars: continuity of rules

The rules in a shape grammar apply in terms of embedding to take advantage of the parts that emerge visually in the appearance of shapes. While the shapes are kept unanalyzed throughout a computation, their descriptions can be defined retrospectively based on how the rules are applied. An important outcome of this is that continuity for rules is not built-in but it is "fabricated" retrospectively to explain a computation as a continuous process. An aspect of continuity analysis that has not been addressed in the literature is how to decide which mapping forms to use to study the continuity of rule applications. This is addressed in this paper in a new approach to continuity analysis, which uses recent results on shape topology and continuous mappings. A characterization is provided that distinguishes the suitable mapping forms from those that are inherently discontinuous or practically inconsequential for continuity analysis. It is also shown that certain inherent properties of shape topologies and continuous mappings provide an effective method of computing topologies algorithmically.Comment: 23 pages, 6 Figures, 6 Tables. Research Report, 2020, MIT. Preprint of Journal Article (2021

Genetic Algorithms for the Imitation of Genomic Styles in Protein Backtranslation

Several technological applications require the translation of a protein into a nucleic acid that codes for it (``backtranslation''). The degeneracy of the genetic code makes this translation ambiguous; moreover, not every translation is equally viable. The common answer to this problem is the imitation of the codon usage of the target species. Here we discuss several other features of coding sequences (``coding statistics'') that are relevant for the ``genomic style'' of different species. A genetic algorithm is then used to obtain backtranslations that mimic these styles, by minimizing the difference in the coding statistics. Possible improvements and applications are discussed.Comment: 17 pages, 13 figures. Submitted to Theor. Comp. Scienc

Optimal polygonal L1 linearization and fast interpolation of nonlinear systems

The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using polygonal (continuous piecewise linear) models under the L1 norm. A thorough error analysis is developed to guide an optimal design of two kinds of polygonal approximations in the asymptotic case of a large budget of evaluation subintervals N. The method allows the user to obtain the level of linearization (N) for a target approximation error and vice versa. It is suitable for, but not limited to, an efficient implementation in modern Graphics Processing Units (GPUs), allowing real-time performance of computationally demanding applications. The quality and efficiency of the technique has been measured in detail on two nonlinear functions that are widely used in many areas of scientific computing and are expensive to evaluate

Method of Riemann surfaces in modelling of cavitating flow

This dissertation is concerned with the applications of the Riemann-Hilbert problem on a hyperelliptic Riemann surface to problems on supercavitating flows of a liquid around objects. For a two-dimensional steady irrotational flow of liquid it is possible to introduce a complex potential w(z) which allows to apply the powerful methods of complex analysis to the solution of fluid mechanics problems. In this work problems on supercavitating flows of a liquid around one or two wedges have been stated. The Tulin single-spiral-vortex model is employed as a cavity closure condition. The flow domain is transformed into an auxiliary domain with known boundaries using the conformal mapping method. After that the problems have been reduced to the solution of Riemann-Hilbert problems on elliptic or hyperelliptic Riemann surfaces. The final step is to solve a system of transcendental equations which is accomplished numerically. The numerical results are presented. To the best of the author’s knowledge no numerical results were available for non-linear problems on supercavitating flows in multiply connected domains before