18,806 research outputs found
Partial b_{v}(s) and b_{v}({\theta}) metric spaces and related fixed point theorems
In this paper, we introduced two new generalized metric spaces called partial
b_{v}(s) and b_{v}({\theta}) metric spaces which extend b_{v}(s) metric space,
b-metric space, rectangular metric space, v-generalized metric space, partial
metric space, partial b-metric space, partial rectangular b-metric space and so
on. We proved some famous theorems such as Banach, Kannan and Reich fixed point
theorems in these spaces. Also, we give definition of partial v-generalized
metric space and show that these fixed point theorems are valid in this space.
We also give numerical examples to support our definitions. Our results
generalize several corresponding results in literature.Comment: 15 page
Probabilistic embeddings of the Fr\'echet distance
The Fr\'echet distance is a popular distance measure for curves which
naturally lends itself to fundamental computational tasks, such as clustering,
nearest-neighbor searching, and spherical range searching in the corresponding
metric space. However, its inherent complexity poses considerable computational
challenges in practice. To address this problem we study distortion of the
probabilistic embedding that results from projecting the curves to a randomly
chosen line. Such an embedding could be used in combination with, e.g.
locality-sensitive hashing. We show that in the worst case and under reasonable
assumptions, the discrete Fr\'echet distance between two polygonal curves of
complexity in , where , degrades
by a factor linear in with constant probability. We show upper and lower
bounds on the distortion. We also evaluate our findings empirically on a
benchmark data set. The preliminary experimental results stand in stark
contrast with our lower bounds. They indicate that highly distorted projections
happen very rarely in practice, and only for strongly conditioned input curves.
Keywords: Fr\'echet distance, metric embeddings, random projectionsComment: 27 pages, 11 figure
Non-Abelian Duality for Open Strings
We examine non-abelian duality transformations in the open string case. After
gauging the isometries of the target space and developing the general
formalism, we study in details the duals oftarget spaces with SO(N) isometries
which, for the SO(2) case, reduces to the known abelian T-duals. We apply the
formalism to electrically and magnetically charged 4D black hole solutions and,
as in the abelian case, dual coordinates satisfy Dirichlet conditions.Comment: 18 pages, Latex. Some formulas are added. Final version to appear in
Nucl. Phys.
What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics
The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which the former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions
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