2,849 research outputs found
Bounded variation and the strength of Helly's selection theorem
We analyze the strength of Helly's selection theorem HST, which is the most
important compactness theorem on the space of functions of bounded variation.
For this we utilize a new representation of this space intermediate between
and the Sobolev space W1,1, compatible with the, so called, weak*
topology. We obtain that HST is instance-wise equivalent to the
Bolzano-Weierstra\ss\ principle over RCA0. With this HST is equivalent to ACA0
over RCA0. A similar classification is obtained in the Weihrauch lattice
Non-principal ultrafilters, program extraction and higher order reverse mathematics
We investigate the strength of the existence of a non-principal ultrafilter
over fragments of higher order arithmetic.
Let U be the statement that a non-principal ultrafilter exists and let
ACA_0^{\omega} be the higher order extension of ACA_0. We show that
ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that
ACA_0^{\omega}+\U is conservative over PA.
Moreover, we provide a program extraction method and show that from a proof
of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U
a realizing term in G\"odel's system T can be extracted. This means that one
can extract a term t, such that A(f,t(f))
An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts
Even a cursory inspection of the Hodge plot associated with Calabi-Yau
threefolds that are hypersurfaces in toric varieties reveals striking
structures. These patterns correspond to webs of elliptic-K3 fibrations whose
mirror images are also elliptic-K3 fibrations. Such manifolds arise from
reflexive polytopes that can be cut into two parts along slices corresponding
to the K3 fibers. Any two half-polytopes over a given slice can be combined
into a reflexive polytope. This fact, together with a remarkable relation on
the additivity of Hodge numbers, explains much of the structure of the observed
patterns.Comment: 30 pages, 15 colour figure
On the strength of weak compactness
We study the logical and computational strength of weak compactness in the
separable Hilbert space \ell_2.
Let weak-BW be the statement the every bounded sequence in \ell_2 has a weak
cluster point. It is known that weak-BW is equivalent to ACA_0 over RCA_0 and
thus that it is equivalent to (nested uses of) the usual Bolzano-Weierstra{\ss}
principle BW. We show that weak-BW is instance-wise equivalent to the
\Pi^0_2-CA. This means that for each \Pi^0_2 sentence A(n) there is a sequence
(x_i) in \ell_2, such that one can define the comprehension functions for A(n)
recursively in a cluster point of (x_i). As consequence we obtain that the
Turing degrees d > 0" are exactly those degrees that contain a weak cluster
point of any computable, bounded sequence in \ell_2. Since a cluster point of
any sequence in the unit interval [0,1] can be computed in a degree low over
0', this show also that instances of weak-BW are strictly stronger than
instances of BW.
We also comment on the strength of weak-BW in the context of abstract Hilbert
spaces in the sense of Kohlenbach and show that his construction of a solution
for the functional interpretation of weak compactness is optimal
Excitation Enhancement of a Quantum Dot Coupled to a Plasmonic Antenna
Plasmonic antennas are key elements to control the luminescence of quantum
emitters. However, the antenna's influence is often hidden by quenching losses.
Here, the luminescence of a quantum dot coupled to a gold dimer antenna is
investigated. Detailed analysis of the multiply excited states quantifies the
antenna's influence on the excitation intensity and the luminescence quantum
yield separately
Patterns in Calabi-Yau Distributions
We explore the distribution of topological numbers in Calabi–Yau manifolds, using the Kreuzer–Skarke dataset of hypersurfaces in toric varieties as a testing ground. While the Hodge numbers are well-known to exhibit mirror symmetry, patterns in frequencies of combination thereof exhibit striking new patterns. We find pseudo-Voigt and Planckian distributions with high confidence and exact fit for many substructures. The patterns indicate typicality within the landscape of Calabi–Yau manifolds of various dimension
HERA-B Framework for Online Calibration and Alignment
This paper describes the architecture and implementation of the HERA-B
framework for online calibration and alignment. At HERA-B the performance of
all trigger levels, including the online reconstruction, strongly depends on
using the appropriate calibration and alignment constants, which might change
during data taking. A system to monitor, recompute and distribute those
constants to online processes has been integrated in the data acquisition and
trigger systems.Comment: Submitted to NIM A. 4 figures, 15 page
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