952,107 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
Space of spaces as a metric space
In spacetime physics, we frequently need to consider a set of all spaces
(`universes') as a whole. In particular, the concept of `closeness' between
spaces is essential. However, there has been no established mathematical theory
so far which deals with a space of spaces in a suitable manner for spacetime
physics.
Based on the scheme of the spectral representation of geometry, we construct
a space of all compact Riemannian manifolds equipped with the spectral measure
of closeness. We show that this space of all spaces can be regarded as a metric
space. We also show other desirable properties of this space, such as the
partition of unity, locally-compactness and the second countability. These
facts show that this space of all spaces can be a basic arena for spacetime
physics.Comment: To appear in Communications in Mathematical Physics. 20 page
Strongly non embeddable metric spaces
Enflo constructed a countable metric space that may not be uniformly embedded
into any metric space of positive generalized roundness. Dranishnikov, Gong,
Lafforgue and Yu modified Enflo's example to construct a locally finite metric
space that may not be coarsely embedded into any Hilbert space. In this paper
we meld these two examples into one simpler construction. The outcome is a
locally finite metric space which is strongly non
embeddable in the sense that it may not be embedded uniformly or coarsely into
any metric space of non zero generalized roundness. Moreover, we show that both
types of embedding may be obstructed by a common recursive principle. It
follows from our construction that any metric space which is Lipschitz
universal for all locally finite metric spaces may not be embedded uniformly or
coarsely into any metric space of non zero generalized roundness. Our
construction is then adapted to show that the group
admits a Cayley graph which
may not be coarsely embedded into any metric space of non zero generalized
roundness. Finally, for each and each locally finite metric space
, we prove the existence of a Lipschitz injection .Comment: 10 page
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