29,340 research outputs found
Diffusion maps for changing data
Graph Laplacians and related nonlinear mappings into low dimensional spaces
have been shown to be powerful tools for organizing high dimensional data. Here
we consider a data set X in which the graph associated with it changes
depending on some set of parameters. We analyze this type of data in terms of
the diffusion distance and the corresponding diffusion map. As the data changes
over the parameter space, the low dimensional embedding changes as well. We
give a way to go between these embeddings, and furthermore, map them all into a
common space, allowing one to track the evolution of X in its intrinsic
geometry. A global diffusion distance is also defined, which gives a measure of
the global behavior of the data over the parameter space. Approximation
theorems in terms of randomly sampled data are presented, as are potential
applications.Comment: 38 pages. 9 figures. To appear in Applied and Computational Harmonic
Analysis. v2: Several minor changes beyond just typos. v3: Minor typo
corrected, added DO
Open mirror symmetry for Pfaffian Calabi-Yau 3-folds
We investigate the open mirror symmetry of certain non-complete intersection
Calabi- Yau 3-folds, so called pfaffian Calabi-Yau. We perform the prediction
of the number of disk invariants of several examples by using the direct
integration method proposed recently and the open mirror symmetry. We treat
several pfaffian Calabi-Yau 3-folds in and branes with two
discrete vacua. Some models have the two special points in its moduli space,
around both of which we can consider different A-model mirror partners. We
compute disc invariants for both cases. This study is the first application of
the open mirror symmetry to the compact non-complete intersections in toric
variety.Comment: 64 pages; v2: typos corrected, minor changes, references added; v3:
published version, minor corrections and improvement
Two-weight norm inequalities for potential type and maximal operators in a metric space
We characterize two-weight norm inequalities for potential type integral
operators in terms of Sawyer-type testing conditions. Our result is stated in a
space of homogeneous type with no additional geometric assumptions, such as
group structure or non-empty annulus property, which appeared in earlier works
on the subject. One of the new ingredients in the proof is the use of a finite
collection of adjacent dyadic systems recently constructed by the author and T.
Hyt\"onen. We further extend the previous Euclidean characterization of
two-weight norm inequalities for fractional maximal functions into spaces of
homogeneous type.Comment: 33 pages, v8 (some typos corrected; clarified the relationship
between the different constants present in the several steps of the proof of
the main result; Lemma 6.18 modified; examples of spaces and operators
included; fixed some technical details; Definition 2.14 and Lemma 2.15
modified; Lemma 6.17 corrected; measures allowed with point masses; some
imprecise arguments clarified
- …