11,815 research outputs found
Dimension zero at all scales
We consider the notion of dimension in four categories: the category of
(unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and
the category of (unbounded) separable metric spaces and (metrically proper)
uniform maps. A unified treatment is given to the large scale dimension and the
small scale dimension. We show that in all categories a space has dimension
zero if and only if it is equivalent to an ultrametric space. Also,
0-dimensional spaces are characterized by means of retractions to subspaces.
There is a universal zero-dimensional space in all categories. In the Lipschitz
Category spaces of dimension zero are characterized by means of extensions of
maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is
coarsely equivalent to a direct sum of cyclic groups. We construct uncountably
many examples of coarsely inequivalent ultrametric spaces.Comment: 17 pages, To appear in Topology and its Application
Hurewicz Theorem for Assouad-Nagata dimension
Given a function of metric spaces, its {\it asymptotic
dimension} \asdim(f) is the supremum of \asdim(A) such that
and \asdim(f(A))=0. Our main result is \begin{Thm} \label{ThmAInAbstract}
\asdim(X)\leq \asdim(f)+\asdim(Y) for any large scale uniform function
. \end{Thm}
\ref{ThmAInAbstract} generalizes a result of Bell and Dranishnikov in which
is Lipschitz and is geodesic. We provide analogs of
\ref{ThmAInAbstract} for Assouad-Nagata dimension and asymptotic
Assouad-Nagata dimension \ANasdim. In case of linearly controlled asymptotic
dimension \Lasdim we provide counterexamples to three questions in a list of
problems of Dranishnikov.
As an application of analogs of \ref{ThmAInAbstract} we prove \begin{Thm}
\label{ThmBInAbstract} If is an exact sequence of
groups and is finitely generated, then \ANasdim (G,d_G)\leq \ANasdim
(K,d_G|K)+\ANasdim (H,d_H) for any word metrics metrics on and
on . \end{Thm}
\ref{ThmBInAbstract} extends a result of Bell and Dranishnikov for asymptotic
dimension
Baryon Self-Energy With QQQ Bethe-Salpeter Dynamics In The Non-Perturbative QCD Regime: n-p Mass Difference
A qqq BSE formalism based on DB{\chi}S of an input 4-fermion Lagrangian of
`current' u,d quarks interacting pairwise via gluon-exchange-propagator in its
{\it non-perturbative} regime, is employed for the calculation of baryon
self-energy via quark-loop integrals. To that end the baryon-qqq vertex
function is derived under Covariant Instantaneity Ansatz (CIA), using Green's
function techniques. This is a 3-body extension of an earlier q{\bar q}
(2-body) result on the exact 3D-4D interconnection for the respective BS wave
functions under 3D kernel support, precalibrated to both q{\bar q} and qqq
spectra plus other observables. The quark loop integrals for the neutron (n) -
proton (p) mass difference receive contributions from : i) the strong SU(2)
effect arising from the d-u mass difference (4 MeV); ii) the e.m. effect of the
respective quark charges. The resultant n-p difference comes dominantly from
d-u effect (+1.71 Mev), which is mildly offset by e.m.effect (-0.44), subject
to gauge corrections. To that end, a general method for QED gauge corrections
to an arbitrary momentum dependent vertex function is outlined, and on on a
proportionate basis from the (two-body) kaon case, the net n-p difference works
out at just above 1 MeV. A critical comparison is given with QCD sum rules
results.Comment: be 27 pages, Latex file, and to be published in IJMPA, Vol 1
3D-4D Interlinkage Of qqq Wave Functions Under 3D Support For Pairwise Bethe-Salpeter Kernels
Using the method of Green's functions within a Bethe-Salpeter framework
characterized by a pairwise qq interaction with a Lorentz-covariant 3D support
to its kernel, the 4D BS wave function for a system of 3 identical relativistic
spinless quarks is reconstructed from the corresponding 3D form which satisfies
a fully connected 3D BSE. This result is a 3-body generalization of a similar
2-body result found earlier under identical conditions of a 3D support to the
corresponding qq-bar BS kernel under Covariant Instaneity (CIA for short). (The
generalization from spinless to fermion quarks is straightforward).
To set the CIA with 3D BS kernel support ansatz in the context of
contemporary approaches to the qqq baryon problem, a model scalar 4D qqq BSE
with pairwise contact interactions to simulate the NJL-Faddeev equations is
worked out fully, and a comparison of both vertex functions shows that the CIA
vertex reduces exactly to the NJL form in the limit of zero spatial range. This
consistency check on the CIA vertex function is part of a fuller accounting for
its mathematical structure whose physical motivation is traceable to the role
of `spectroscopy' as an integral part of the dynamics.Comment: 20 pages, Latex, submitted via the account of K.-C. Yan
- …