318 research outputs found
On Hamilton decompositions of infinite circulant graphs
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph).
Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}
Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner functions
We demonstrate that the Plancherel transform for Type-I groups provides one
with a natural, unified perspective for the generalized continuous wavelet
transform, on the one hand, and for a class of Wigner functions, on the other.
The wavelet transform of a signal is an -function on an appropriately
chosen group, while the Wigner function is defined on a coadjoint orbit of the
group and serves as an alternative characterization of the signal, which is
often used in practical applications. The Plancherel transform maps
-functions on a group unitarily to fields of Hilbert-Schmidt operators,
indexed by unitary irreducible representations of the group. The wavelet
transform can essentiallly be looked upon as restricted inverse Plancherel
transform, while Wigner functions are modified Fourier transforms of inverse
Plancherel transforms, usually restricted to a subset of the unitary dual of
the group. Some known results both on Wigner functions and wavelet transforms,
appearing in the literature from very different perspectives, are naturally
unified within our approach. Explicit computations on a number of groups
illustrate the theory.Comment: 41 page
Harmonic mappings and conformal minimal immersions of Riemann surfaces into RN
We prove that for any open Riemann surface N, natural number N ≥ 3, non-constant harmonic map h:N→R N−2 and holomorphic 2-form H on N , there exists a weakly complete harmonic map X=(Xj)j=1,…,\scN:N→R\scN with Hopf differential H and (Xj)j=3,…,\scN=h. In particular, there exists a complete conformal minimal immersion Y=(Yj)j=1,…,\scN:N→R\scN such that (Yj)j=3,…,\scN=h . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of CP\scN−1 in general position. (2) There exist complete non-proper embedded minimal surfaces in R\scN, ∀\scN>3.Ministerio de Ciencia y TecnologÃa MTM2007-61775Ministerio de Ciencia y TecnologÃa MTM2007-64504Junta de AndalucÃa P09-FQM-508
Fine shape III: -spaces and -spaces
In this paper we obtain results indicating that fine shape is tractable and
"not too strong" even in the non-locally compact case, and can be used to
better understand infinite-dimensional metrizable spaces and their homology
theories.
We show that every Polish space is fine shape equivalent to the limit of
an inverse sequence of simplicial maps between metric simplicial complexes. A
deeper result is that if is locally finite dimensional, then the simplicial
maps can be chosen to be non-degenerate. They cannot be chosen to be
non-degenerate if is the Taylor compactum.Comment: 25 pages. "Fine shape I" is arXiv:1808.1022
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