3D Snap Rounding

Let P be a set of n polygons in R^3, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps P to a simplicial complex Q whose vertices have integer coordinates. Every face of P is mapped to a set of faces (or edges or vertices) of Q and the mapping from P to Q can be done through a continuous motion of the faces such that (i) the L_infty Hausdorff distance between a face and its image during the motion is at most 3/2 and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worst case, the size of Q is O(n^{15}) and the time complexity of the algorithm is O(n^{19}) but, under reasonable hypotheses, these complexities decrease to O(n^{5}) and O(n^{6}sqrt{n})

Nature of the vortex-glass order in strongly type-II superconductors

The stability and the critical properties of the three-dimensional vortex-glass order in random type-II superconductors with point disorder is investigated in the unscreened limit based on a lattice {\it XY} model with a uniform field. By performing equilibrium Monte Carlo simulations for the system with periodic boundary conditions, the existence of a stable vortex-glass order is established in the unscreened limit. Estimated critical exponents are compared with those of the gauge-glass model.Comment: Error in the reported value of the exponent eta is correcte

Increasing Copyright Protection for Social Media Users by Expanding Social Media Platforms\u27 Rights

Social media platforms allow users to share their creative works with the world. Users take great advantage of this functionality, as Facebook, Instagram, Flickr, Snapchat, and WhatsApp users alone uploaded 1.8 billion photos per day in 2014. Under the terms of service and terms of use agreements of most U.S. based social media platforms, users retain ownership of this content, since they only grant social media platforms nonexclusive licenses to their content. While nonexclusive licenses protect users vis-à-vis the social media platforms, these licenses preclude social media platforms from bringing copyright infringement claims on behalf of their users against infringers of user content under the Copyright Act of 1976. Since the average cost of litigating a copyright infringement case might be as high as two million dollars, the average social media user cannot protect his or her content against copyright infringers. To remedy this issue, Congress should amend 17 U.S.C. § 501 to allow social media platforms to bring copyright infringement claims against those who infringe their users’ content. Through this amendment, Congress would create a new protection for social media users while ensuring that users retain ownership over the content they create

Intrusive growth of sclerenchyma fibers

Intrusive growth is a type of cell elongation when the rate of its longitudinal growth is higher than that of surrounding cells; therefore, these cells intrude between the neighboring cells penetrating the middle lamella. The review considers the classical example of intrusive growth, e.g., elongation of sclerenchyma fibers when the cells achieve the length of several centimeters. We sum the published results of investigations of plant fiber intrusive growth and present some features of intrusive growth characterized by the authors for flax (Linum usitatissimum L.) and hemp (Cannabis sativa L.) fibers. The following characteristics of intrusive growth are considered: its rate and duration, relationship with the growth rate of surrounding cells, the type of cell elongation, peculiarities of the fiber primary cell wall structure, fibers as multinucleate cells, and also the control of intrusive growth. Genes, which expression is sharply reduced at suppression of intrusive growth, are also considered. Arguments for separation of cell elongation and secondary cell wall formation in phloem fibers and also data indicating diffuse type of cell enlargement during intrusive growth are presented

Snap Rounding of Bézier Curves

We present an extension of snap roundingfrom straight-line segments (see Guibas and Marimont, 1998)to Bézier curves of arbitrary degree, and thus the first method for geometric roundingof curvilinear arrangements.Our algorithm takes a set of intersecting Bézier curvesand directly computes a geometric rounding of their true arrangement, without the need of representing the true arrangement exactly.The algorithm's output is a deformation of the true arrangementthat has all Bézier control points at integer pointsand comes with the same geometric guarantees as instraight-line snap rounding: during rounding, objects do not movefurther than the radius of a pixel, and features of thearrangement may collapse but do not invert

Snap Rounding with Restore: an Algorithm for Producing Robust Geometric Datasets

This paper presents a new algorithm called Snap Rounding with Restore (SRR), which aims to make ge- ometric datasets robust and to increase the quality of geometric approximation and the preservation of topological structure. It is based on the well-known Snap Rounding algorithm, but improves it by eliminat- ing from the snap rounded arrangement the configurations in which the distance between a vertex and a non-incident edge is smaller than half-the-width of a pixel of the rounding grid. Therefore, the goal of SRR is exactly the same as the goal of another algorithm, Iterated Snap Rounding (ISR), and of its evolution, Iterated Snap Rounding with Bounded Drift (ISRBD). However, SRR produces an output with a quality of approximation that is on average better than ISRBD, both under the viewpoint of the distance from the original segments and of the conservation of their topological structure. The paper also reports some cases where ISRBD, notwithstanding the bounded drift, produces strong topological modifications while SRR does not. A statistical analysis on a large collection of input datasets confirms these differences. It follows that the proposed Snap Rounding with Restore algorithm is suitable for applications that require both robustness, a guaranteed geometric approximation and a good topological approximation