Search results for "Geometry"

A Characterization of Bispecial Sturmian Words

2012

A finite Sturmian word w over the alphabet {a,b} is left special (resp. right special) if aw and bw (resp. wa and wb) are both Sturmian words. A bispecial Sturmian word is a Sturmian word that is both left and right special. We show as a main result that bispecial Sturmian words are exactly the maximal internal factors of Christoffel words, that are words coding the digital approximations of segments in the Euclidean plane. This result is an extension of the known relation between central words and primitive Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the set of Sturmian wo…

Construction of 3D Triangles on Dupin Cyclides

2011

This paper considers the conversion of the parametric Bézier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles (meridian arcs, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm …

Comparing the relative volume with a revolution manifold as a model

1993

Given a pair (P, M), whereM is ann-dimensional connected compact Riemannian manifold andP is a connected compact hypersurface ofM, the relative volume of (P, M) is the quotient volume(P)/volume(M). In this paper we give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature ofM and the mean curvature ofP, with respect to that of a model pair\(\left( {\mathcal{P},\mathcal{M}} \right)\) where ℳ is a revolution manifold and\(\mathcal{P}\) a “parallel” of ℳ.

Convex bodies and convexity on Grassmann cones

1962

Optimal Locations and Inner Products

1997

Abstract In a normed space X , we consider objective functions which depend on the distances between a variable point and the points of certain finite sets A . A point where such a function attains its minimum on X is generically called an optimal location. In this paper we obtain characterizations of inner product spaces with properties connecting optimal locations and the convex hull of A or barycenters of points of A with well chosen weights. We thus generalize several classical results about characterization of inner product spaces.

Lifting paths on quotient spaces

2009

Abstract Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X / G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn–Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g ∈ G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the firs…

Spaces of typen on partially ordered sets

1989

This paper contains a generalized approach to incidence geometry on partially ordered sets. A difference to the usual geometrical concepts is that points may have different size. Our main result states that a large class of spaces allows lattice theoretic characterizations. Especially, a generalized version of the Veblen-Young axiom of projective geometry has a lattice theoretic equivalent, called then-generation property (which is a generalization of the ‘Verbindungssatz’). Modularity and distributivity of a lattice of subspaces are reflected in the underlying space. Finally we give specializations and examples.

Blocking sets and partial spreads in finite projective spaces

1980

A t-blocking set in the finite projective space PG(d, q) with d≥t+1 is a set $$\mathfrak{B}$$ of points such that any (d−t)-dimensional subspace is incident with a point of $$\mathfrak{B}$$ and no t-dimensional subspace is contained in $$\mathfrak{B}$$ . It is shown that | $$\mathfrak{B}$$ |≥q t +...+1+q t−1√q and the examples of minimal cardinality are characterized. Using this result it is possible to prove upper and lower bounds for the cardinality of partial t-spreads in PG(d, q). Finally, examples of blocking sets and maximal partial spreads are given.

Sigma-fragmentability and the property SLD in C(K) spaces

2009

Abstract We characterize two topological properties in Banach spaces of type C ( K ) , namely, being σ-fragmented by the norm metric and having a countable cover by sets of small local norm-diameter (briefly, the property norm-SLD). We apply our results to deduce that C p ( K ) is σ-fragmented by the norm metric when K belongs to a certain class of Rosenthal compacta as well as to characterize the property norm-SLD in C p ( K ) in case K is scattered.

Almost Tight Bound for the Union of Fat Tetrahedra in Three Dimensions

2007

For any AND-OR formula of size N, there exists a bounded-error N1/2+o(1)-time quantum algorithm, based on a discrete-time quantum walk, that evaluates this formula on a black-box input. Balanced, or "approximately balanced," formulas can be evaluated in O(radicN) queries, which is optimal. It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.