A net (
x
α
) in a vector lattice
X
is said to uo-converge to
x
if
|
x
α
−
x
|
∧
u
→
o
0
for every
u
≥ 0. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. ...We prove that uoconvergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in 27, 26. In the second part, we use uo-convergence to study convergence of Cesàro means in Banach lattices. In particular, we establish an intrinsic version of Komlós’ Theorem, which extends the main results of 35, 16, 31 in a uniform way. We also develop a new and unified approach to Banach–Saks properties and Banach–Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in 20, 24, 25.
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Unbounded norm convergence in Banach lattices Deng, Y.; O’Brien, M.; Troitsky, V. G.
Positivity : an international journal devoted to the theory and applications of positivity in analysis,
09/2017, Volume:
21, Issue:
3
Journal Article
Peer reviewed
Open access
A net
(
x
α
)
in a vector lattice
X
is unbounded order convergent to
x
∈
X
if
|
x
α
-
x
|
∧
u
converges to 0 in order for all
u
∈
X
+
. This convergence has been investigated and applied in several ...recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net
(
x
α
)
in a Banach lattice
X
is unbounded norm convergent to
x
if
for all
u
∈
X
+
. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.
ABSTRACT
Baikal-GVD has recently published its first measurement of the diffuse astrophysical neutrino flux, performed using high-energy cascade-like events. We further explore the Baikal-GVD cascade ...data set collected in 2018–2022, with the aim to identify possible associations between the Baikal-GVD neutrinos and known astrophysical sources. We leverage the relatively high angular resolution of the Baikal-GVD neutrino telescope (2–3 deg.), made possible by the use of liquid water as the detection medium, enabling the study of astrophysical point sources even with cascade events. We estimate the telescope’s sensitivity in the cascade channel for high-energy astrophysical sources and refine our analysis prescriptions using Monte-Carlo simulations. We primarily focus on cascades with energies exceeding 100 TeV, which we employ to search for correlation with radio-bright blazars. Although the currently limited neutrino sample size provides no statistically significant effects, our analysis suggests a number of possible associations with both extragalactic and Galactic sources. Specifically, we present an analysis of an observed triplet of neutrino candidate events in the Galactic plane, focusing on its potential connection with certain Galactic sources, and discuss the coincidence of cascades with several bright and flaring blazars.
Results of the search for ∼(1016–1017.5) eV primary cosmic-ray photons with the data of the Moscow State University (MSU) Extensive Air Shower (EAS) array are reported. The full-scale reanalysis of ...the data with modern simulations of the installation does not confirm previous indications of the excess of gamma-ray candidate events. Upper limits on the corresponding gamma-ray flux are presented. The limits are among the most stringent published ones at energies ∼1017 eV.
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In an effort to locate the sites of emission at different frequencies and physical processes causing variability in blazar jets, we have obtained high time-resolution observations of BL Lacertae over ...a wide wavelength range: with the Transiting Exoplanet Survey Satellite (TESS) at 6000-10000 with 2 minute cadence; with the Neil Gehrels Swift satellite at optical, UV, and X-ray bands; with the Nuclear Spectroscopic Telescope Array at hard X-ray bands; with the Fermi Large Area Telescope at γ-ray energies; and with the Whole Earth Blazar Telescope for measurement of the optical flux density and polarization. All light curves are correlated, with similar structure on timescales from hours to days. The shortest timescale of variability at optical frequencies observed with TESS is ∼0.5 hr. The most common timescale is 13 1 hr, comparable with the minimum timescale of X-ray variability, 14.5 hr. The multiwavelength variability properties cannot be explained by a change solely in the Doppler factor of the emitting plasma. The polarization behavior implies that there are both ordered and turbulent components to the magnetic field in the jet. Correlation analysis indicates that the X-ray variations lag behind the γ-ray and optical light curves by up to ∼0.4 day. The timescales of variability, cross-frequency lags, and polarization properties can be explained by turbulent plasma that is energized by a shock in the jet and subsequently loses energy to synchrotron and inverse Compton radiation in a magnetic field of strength ∼3 G.
ABSTRACT
Blazar S5 0716+714 is well-known for its short-term variability, down to intraday time-scales. We here present the 2-min cadence optical light curve obtained by the TESS space telescope in ...2019 December–2020 January and analyse the object fast variability with unprecedented sampling. Supporting observations by the Whole Earth Blazar Telescope Collaboration in B, V, R, and I bands allow us to investigate the spectral variability during the TESS pointing. The spectral analysis is further extended in frequency to the UV and X-ray bands with data from the Neil Gehrels Swift Observatory. We develop a new method to unveil the shortest optical variability time-scales. This is based on progressive de-trending of the TESS light curve by means of cubic spline interpolations through the binned fluxes, with decreasing time bins. The de-trended light curves are then analysed with classical tools for time-series analysis (periodogram, autocorrelation, and structure functions). The results show that below 3 d there are significant characteristic variability time-scales of about 1.7, 0.5, and 0.2 d. Variability on time-scales $\lesssim 0.2$ d is strongly chromatic and must be ascribed to intrinsic energetic processes involving emitting regions, likely jet substructures, with dimension less than about 10−3 pc. In contrast, flux changes on time-scales $\gtrsim 0.5$ d are quasi-achromatic and are probably due to Doppler factor changes of geometric origin.
Bibasic sequences in Banach lattices Taylor, M.A.; Troitsky, V.G.
Journal of functional analysis,
06/2020, Volume:
278, Issue:
10
Journal Article
Peer reviewed
Open access
Given a Schauder basic sequence (xk) in a Banach lattice, we say that (xk) is bibasic if the expansion of every vector in xk converges not only in norm, but also in order. We prove that, in this ...definition, order convergence may be replaced with uniform convergence, with order boundedness of the partial sums, or with norm boundedness of finite suprema of the partial sums.
The results in this paper extend and unify those from the pioneering paper Order Schauder bases in Banach lattices by A. Gumenchuk, O. Karlova, and M. Popov. In particular, we are able to characterize bibasic sequences in terms of the bibasis inequality, a result they obtained under certain additional assumptions.
After establishing the aforementioned characterizations of bibasic sequences, we embark on a deeper study of their properties. We show, for example, that they are independent of ambient space, stable under small perturbations, and preserved under sequentially uniformly continuous norm isomorphic embeddings. After this we consider several special kinds of bibasic sequences, including permutable sequences, i.e., sequences for which every permutation is bibasic, and absolute sequences, i.e., sequences where expansions remain convergent after we replace every term with its modulus. We provide several equivalent characterizations of absolute sequences, showing how they relate to bibases and to further modifications of the basis inequality.
We further consider bibasic sequences with unique order expansions. We show that this property does generally depend on ambient space, but not for the inclusion of c0 into ℓ∞. We also show that small perturbations of bibases with unique order expansions have unique order expansions, but this is not true if “bibases” is replaced with “bibasic sequences”.
In the final section, we consider uo-bibasic sequences, which are obtained by replacing order convergence with uo-convergence in the definition of a bibasic sequence. We show that such sequences are very common.
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Unbounded norm topology beyond normed lattices Kandić, M.; Li, H.; Troitsky, V. G.
Positivity : an international journal devoted to the theory and applications of positivity in analysis,
07/2018, Volume:
22, Issue:
3
Journal Article
Peer reviewed
Open access
In this paper, we generalize the concept of unbounded norm (un) convergence: let
X
be a normed lattice and
Y
a vector lattice such that
X
is an order dense ideal in
Y
; we say that a net
(
y
α
)
...un-converges to
y
in
Y
with respect to
X
if
|
|
|
y
α
-
y
|
∧
x
|
|
→
0
for every
x
∈
X
+
. We extend several known results about un-convergence and un-topology to this new setting. We consider the special case when
Y
is the universal completion of
X
. If
Y
=
L
0
(
μ
)
, the space of all
μ
-measurable functions, and
X
is an order continuous Banach function space in
Y
, then the un-convergence on
Y
agrees with the convergence in measure. If
X
is atomic and order complete and
Y
=
R
A
then the un-convergence on
Y
agrees with the coordinate-wise convergence.
Unbounded norm topology in Banach lattices Kandić, M.; Marabeh, M.A.A.; Troitsky, V.G.
Journal of mathematical analysis and applications,
07/2017, Volume:
451, Issue:
1
Journal Article
Peer reviewed
Open access
A net (xα) in a Banach lattice X is said to un-converge to a vector x if ‖|xα−x|∧u‖→0 for every u∈X+. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. ...We show that un-topology agrees with the norm topology iff X has a strong unit. Un-topology is metrizable iff X has a quasi-interior point. Suppose that X is order continuous, then un-topology is locally convex iff X is atomic. An order continuous Banach lattice X is a KB-space iff its closed unit ball BX is un-complete. For a Banach lattice X, BX is un-compact iff X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.
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Convergence is a fundamental topic in analysis that is most commonly modeled using topology. However, there are many natural convergences that are not given by any topology; e.g., convergence almost ...everywhere of a sequence of measurable functions and order convergence of nets in vector lattices. The theory of convergence structures provides a framework for studying more general modes of convergence. It also has one particularly striking feature: it is formalized using the language of filters. This paper develops a general theory of convergence in terms of nets. We show that it is equivalent to the filter-based theory and present some translations between the two areas. In particular, we provide a characterization of pretopological convergence structures in terms of nets. We also use our results to unify certain topics in vector lattices with general convergence theory.
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