9,437 research outputs found
The weight and Lindel\"of property in spaces and topological groups
We show that if is a dense subspace of a Tychonoff space , then
, where is the Nagami number of . In
particular, if is a Lindel\"of -space, then .
Better upper bounds for the weight of topological groups are given. For
example, if a topological group contains a dense subgroup such that
is a Lindel\"of -space, then . Further,
if a Lindel\"of -space generates a dense subgroup of a topological
group , then .
Several facts about subspaces of Hausdorff separable spaces are established.
It is well known that the weight of a separable Hausdorff space can be as
big as , where . We prove on the
one hand that if a regular Lindel\"of -space is a subspace of a
separable Hausdorff space, then , and the same conclusion
holds for a Lindel\"of -space . On the other hand, we present an example
of a countably compact topological group which is homeomorphic to a
subspace of a separable Hausdorff space and satisfies , i.e. has the maximal possible weight.Comment: 15 pages, submitte
Four-manifold systoles and surjectivity of period map
P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space
of Riemann surfaces of genus s, of the least conformal length of a
nonseparating loop, is logarithmic in s. We present an application of
(polynomially) dense Euclidean packings, to estimates for an analogous
2-dimensional conformal systolic invariant of a 4-manifold X with indefinite
intersection form. The estimate turns out to be polynomial, rather than
logarithmic, in \chi(X), if the conjectured surjectivity of the period map is
correct. Such surjectivity is targeted by the current work in gauge theory. The
surjectivity allows one to insert suitable lattices with metric properties
prescribed in advance, into the second de Rham cohomology group of X, as its
integer lattice. The idea is to adapt the well-known Lorentzian construction of
the Leech lattice, by replacing the Leech lattice by the Conway-Thompson
unimodular lattices which define asymptotically dense packings. The final step
can be described, in terms of the successive minima \lambda_i, as deforming a
\lambda_2-bound into a \lambda_1-bound.Comment: 16 page
Superluminal motion and Lorentzian symmetry breaking and repairing in two-metric theories
The new results by OPERA collaboration claim the discovery of superluminal
neutrinos. Superluminal particles have to break Lorentzian symmetry or
causality principle. The method discussed gives us the possibility to
reintroduce Lorentzian symmetry without breaking of causality.Comment: 5 pages. Extra references are added in v
A quantitative obstruction to collapsing surfaces
We provide a quantitative obstruction to collapsing surfaces of genus at
least 2 under a lower curvature bound and an upper diameter bound. Keywords:
curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distanceComment: 4 pages. Published in Open Mathematic
On uniqueness of quantum measurement theory
The paper discuss the structure of quantum mechanics and uniqueness of its
postulates.
The Born rule for quantum probabilities is fixed by requirement of
nonexistence of quantum telepathy.
Von Neumann projection postulate describes the transformation of quantum
state under the condition of no-interaction measurement. Projection postulate
could be considered as transition to conditional probability under the
condition of a certain result of quantum measurement.Comment: 8 page
Torus cannot collapse to a segment
In earlier work, we analyzed the impossibility of codimension-one collapse
for surfaces of negative Euler characteristic under the condition of a lower
bound for the Gaussian curvature. Here we show that, under similar conditions,
the torus cannot collapse to a segment. Unlike the torus, the Klein bottle can
collapse to a segment; we show that in such a situation, the loops in a short
basis for homology must stay a uniform distance apart.Comment: 8 pages, Journal of Geometry 111, Article number: 13 (2020
Systolic inequalities and Massey products in simply-connected manifolds
We show that the existence of a nontrivial Massey product in the cohomology
ring H^*(X) imposes global constraints upon the Riemannian geometry of a
manifold X. Namely, we exhibit a suitable systolic inequality, associated to
such a product. This generalizes an inequality proved in collaboration with Y.
Rudyak, in the case when X has unit Betti numbers, and realizes the next step
in M. Gromov's program for obtaining geometric inequalities associated with
nontrivial Massey products. The inequality is a volume lower bound, and depends
on the metric via a suitable isoperimetric quotient. The proof relies upon W.
Banaszczyk's upper bound for the successive minima of a pair of dual lattices.
Such an upper bound is applied to the integral lattices in homology and
cohomology of X. The possibility of applying such upper bounds to obtain volume
lower bounds was first exploited in joint work with V. Bangert. The latter work
deduced systolic inequalities from nontrivial cup-product relations, whose role
here is played by Massey products.Comment: 14 pages, to appear in Israel J. Mat
Physics and technology system of units for electrodynamics
The contemporary practice is to favor the use of the SI units for electric
circuits and the Gaussian CGS system for electromagnetic field. A modification
of the Gaussian system of units (the Physics and Technology System) is
suggested. In the Physics and Technology System the units of measurement for
electrical circuits coincide with SI units, and the equations for the
electromagnetic field are almost the same form as in the Gaussian system. The
XXIV CGMP (2011) Resolution "On the possible future revision of the
International System of Units, the SI" provides a chance to initiate gradual
introduction of the Physics and Technology System as a new modification of the
SI.Comment: 12 pages. Misprints in table at page 10 are correcte
Search for photon bubble oscillations in V0332+53
We report results of our search for fast oscillations in lightcurve of one of
the brightest accretion powered pulsars on the sky V0332+53 with the help of
data of the PCA spectrometer of the RXTE observatory. In course of this search
we have carefully explored complications appearing if one uses only sub-bands
of the total bandpass of the PCA spectrometer. We show that lightcurves
collected in the soft sub-band of the PCA spectrometer contains an additional
instrumental noise, lightcurves of harder sub-bands lack some fraction of the
anticipated Poisson noise. We show that this noise is caused by a cross-talk of
energy bands, which lasts up to ~200usec. One hypothesis is that these effects
are caused by temporarily drop of the PCA detector gain after any occurred
event due to slowly moving ions in the detector volume. In order to avoid this
effect we searched for fast oscillations in flux of V0332+53 only in the total
bandpass of the PCA spectrometer 2-60 keV. We have not detected any
quasi-periodic oscillations in lightcurve of the source with an upper limit at
the level of 0.5% in the Fourier frequency range 200-1500 Hz.Comment: 7 pages, 11 figures, accepted for publication in MNRA
Lattices of homomorphisms and pro-Lie groups
Early this century K. H. Hofmann and S. A. Morris introduced the class of
pro-Lie groups which consists of projective limits of finite-dimensional Lie
groups and proved that it contains all compact groups, all locally compact
abelian groups, and all connected locally compact groups and is closed under
the formation of products and closed subgroups. They defined a topological
group to be almost connected if the quotient group of by the connected
component of its identity is compact.
We show here that all almost connected pro-Lie groups as well as their
continuous homomorphic images are -factorizable and
\textit{-cellular}, i.e.~every family of -sets contains a
countable subfamily whose union is dense in the union of the whole family. We
also prove a general result which implies as a special case that if a
topological group contains a compact invariant subgroup such that the
quotient group is an almost connected pro-Lie group, then is
-factorizable and -cellular.
Applying the aforementioned result we show that the sequential closure and
the closure of an arbitrary -set in an almost connected
pro-Lie group coincide.Comment: 22 page
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