On-line covering a box by cubes.
On-Line Covering a Cube by a Sequence of Cubes.
On-Line Covering the Unit Cube by Cubes.
On-line Covering the Unit Square with Squares
The unit square can be on-line covered with any sequence of squares whose total area is not smaller than 4.
On-line packing regular boxes in the unit cube
We describe a class of boxes such that every sequence of boxes from this class of total volume smaller than or equal to 1 can be on-line packed in the unit cube.
On-line packing sequences of segments, cubes and boxes.
On-line Packing Squares into n Unit Squares
If n ≥ 3, then any sequence of squares of side lengths not greater than 1 whose total area does not exceed ¼(n+1) can be on-line packed into n unit squares.
On-line -adic covering by the method of the -th segment and its application to on-line covering by cubes.
Only ellipsoids have caustics.
Opérades cellulaires et espaces de lacets itérés
L’espace des configurations de points distincts de admet une filtration naturelle qui est induite par les inclusions des dans . Nous caractérisons le type d’homotopie de cette filtration par les propriétés combinatoires d’une structure cellulaire sous-jacente, étroitement liée à la théorie des -opérades de May. Cela donne une approche unifiée des différents modèles combinatoires d’espaces de lacets itérés et redémontre les théorèmes d’approximation de Milgram, Smith et Kashiwabara.
Opérades différentielles graduées sur les simplexes et les permutoèdres
On définit plusieurs opérades différentielles graduées, dont certaines en relation avec des familles de polytopes : les simplexes et les permutoèdres. On obtient également une présentation de l’opérade liée aux associaèdres introduite dans un article antérieur.
Operations between sets in geometry
An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in -dimensional Euclidean space . It is proved that if , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, covariant, and associative if and only if it is addition for some . It is also demonstrated that if ,...
Opérations sur les cartes et métamorphoses de la catégorie des G-ensembles.
We call metamorphosis of a given category an autoequivalence functor up to within natural equivalence. We show that, given a group G, the group of metamorphoses of the category of G-sets (as well as the corresponding group for ?sufficiently big? subcategories) may be naturally identified to the group of outer automorphism of G. We get by this way a natural description of a group of known operations on tessellations of a surface: the identity operation, the Poincaré duality, and four others which...
Oprava k článku „Poznámka k otázce řešitelnosti jisté soustavy nerovností kladnými čísly‟
Optimal bounds for the colored Tverberg problem
We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.
Optimal control in linear systems without the condition of regular controllability
Optimal isometries for a pair of compact convex subsets of ℝⁿ
In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for metrics for all p ≥ 2 and the symmetric difference metric.
Optimal packings for filled rings of circles
General circle packings are arrangements of circles on a given surface such that no two circles overlap except at tangent points. In this paper, we examine the optimal arrangement of circles centered on concentric annuli, in what we term rings. Our motivation for this is two-fold: first, certain industrial applications of circle packing naturally allow for filled rings of circles; second, any packing of circles within a circle admits a ring structure if one allows for irregular spacing of circles...
Optimal Spherical Packing of Circles and Hilbert's 14th Problem
Optimal substructures in optimal and approximate circle packings.