Sweeping an oval to a vanishing point

Given a convex region in the plane, and a sweep-line as a tool, what is best way to reduce the region to a single point by a sequence of sweeps? The problem of sweeping points by orthogonal sweeps was first studied in [2]. Here we consider the following \emph{slanted} variant of sweeping recently introduced in [1]: In a single sweep, the sweep-line is placed at a start position somewhere in the plane, then moved continuously according to a sweep vector $\vec v$ (not necessarily orthogonal to the sweep-line) to another parallel end position, and then lifted from the plane. The cost of a sequence of sweeps is the sum of the lengths of the sweep vectors. The (optimal) sweeping cost of a region is the infimum of the costs over all finite sweeping sequences for that region. An optimal sweeping sequence for a region is one with a minimum total cost, if it exists. Another parameter of interest is the number of sweeps. We show that there exist convex regions for which the optimal sweeping cost cannot be attained by two sweeps. This disproves a conjecture of Bousany, Karker, O'Rourke, and Sparaco stating that two sweeps (with vectors along the two adjacent sides of a minimum-perimeter enclosing parallelogram) always suffice [1]. Moreover, we conjecture that for some convex regions, no finite sweeping sequence is optimal. On the other hand, we show that both the 2-sweep algorithm based on minimum-perimeter enclosing rectangle and the 2-sweep algorithm based on minimum-perimeter enclosing parallelogram achieve a $4/\pi \approx 1.27$ approximation in this sweeping model.Comment: 9 pages, 4 figure

BSP-fields: An Exact Representation of Polygonal Objects by Differentiable Scalar Fields Based on Binary Space Partitioning

The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of the function-based modeling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples

Analytic continuations of de Sitter thick domain wall solutions

We perform some analytic continuations of the de Sitter thick domain wall solutions obtained in our previous paper hep-th/0201130 in the system of gravity and a scalar field with an axion-like potential. The obtained new solutions represent anti-de Sitter thick domain walls and cosmology. The anti-de Sitter domain wall solutions are periodic, and correspondingly the cosmological solutions represent cyclic universes. We parameterize the axion-like scalar field potential and determine the parameter regions of each type of solutions.Comment: Additons in section 5, 8 pages, 7 figures, RevTe

New species and records of Stenus (Nestus) of the canaliculatus group, with the erection of a new species group (Insecta: Coleoptera: Staphylinidae: Steninae)

The canaliculatus species group of Stenus (Nestus) is redefined. Four new Palaearctic species of the group are described and illustrated: S. (N.) alopex sp. nov. from the Putorana Highland and Taymyr Peninsula, Russia; S. (N.) canalis sp. nov. from SE Siberia and the Russian Far East; S. (N.) canosus sp. nov. from the Narat Mt Ridge, Chinese Tien Shan; S. (N.) delitor sp. nov. from C & SE Siberia. New distributional data as well as brief analyses of old records for fourteen species described earlier are provided from both Palaearctic and Nearctic material. S. (N.) milleporus Casey, 1884 (= sectilifer Casey, 1884) is revalidated as a species propria. S. (N.) sphaerops Casey, 1884 is redescribed; its aedeagus is figured for the fi rst time; the aedeagus of S. (N.) caseyi Puthz, 1972 as well as aedeagi of eight previously described Palaearctic species are illustrated anew. A key for the identification of all the known Palaearctic species of the group is given. A morphology and ecology based analysis of the main evolutionary trends within the group is provided. A lectotype is designated for S. (N.) melanopus Marsham, 1802; its Siberian and NE European records are supposed to be erroneous; the monotypic melanopus species group is erected

Electronic Structure of Three-Dimensional Superlattices Subject to Tilted Magnetic Fields

Full quantum-mechanical description of electrons moving in 3D structures with unidirectional periodic modulation subject to tilted magnetic fields requires an extensive numerical calculation. To understand magneto-oscillations in such systems it is in many cases sufficient to use the quasi-classical approach, in which the zero-magnetic-field Fermi surface is considered as a magnetic-field-independent rigid body in k-space and periods of oscillations are related to extremal cross-sections of the Fermi surface cut by planes perpendicular to the magnetic-field direction. We point out cases where the quasi-classical treatment fails and propose a simple tight-binding fully-quantum-mechanical model of the superlattice electronic structure.Comment: 8 pages, 7 figures, RevTex, submitted to Phys. Rev.

An Exact Representation of Polygonal Objects by C1-continuous Scalar Fields Based on Binary Space Partitioning

The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of function-based modelling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples