22,335 research outputs found
On side lengths of corners in positive density subsets of the Euclidean space
We generalize a result by Cook, Magyar, and Pramanik [3] on three-term
arithmetic progressions in subsets of to corners in subsets of
. More precisely, if , ,
and is large enough, we show that an arbitrary measurable set
of positive upper Banach density
contains corners , , such that the -norm of
the side attains all sufficiently large real values. Even though we closely
follow the basic steps from [3], the proof diverges at the part relying on
harmonic analysis. We need to apply a higher-dimensional variant of a
multilinear estimate from [5], which we establish using the techniques from [5]
and [6].Comment: 17 pages; v2: several computations expanded, references added and
update
Area law and vacuum reordering in harmonic networks
We review a number of ideas related to area law scaling of the geometric
entropy from the point of view of condensed matter, quantum field theory and
quantum information. An explicit computation in arbitrary dimensions of the
geometric entropy of the ground state of a discretized scalar free field theory
shows the expected area law result. In this case, area law scaling is a
manifestation of a deeper reordering of the vacuum produced by majorization
relations. Furthermore, the explicit control on all the eigenvalues of the
reduced density matrix allows for a verification of entropy loss along the
renormalization group trajectory driven by the mass term. A further result of
our computation shows that single-copy entanglement also obeys area law
scaling, majorization relations and decreases along renormalization group
flows.Comment: 15 pages, 6 figures; typos correcte
Forecasting inflation with consumer survey data â application of multi-group confirmatory factor analysis to elimination of the general sentiment factor
This paper (1) examines the properties of survey based householdsâ inflation expectations and investigates their forecasting performance. With application of the individual data from the State of the Householdsâ Survey (50 quarters between 1997Q4 and 2010Q1) it was shown that inflation expectations were affected by the consumer sentiment. Multi-Group Confirmatory Factor Analysis (MGCFA) was employed to verify whether a set of proxies provides a reliable basis for measurement of two latent phenomena â consumer sentiment and inflation expectations. Following the steps proposed by Davidov (2008) and Steenkamp and Baumgartner (1998), it appeared that it was possible to specify and estimate a MGCFA model with partial measurement invariance. Thus it was possible to eliminate the influence of consumer sentiment on inflation expectations and at the same time to obtain individually corrected answers concerning the inflation expectations. Additionally, it was shown that the linear relation between consumer sentiment and inflation expectations was stable over time. As a by-product of analysis, it was possible to show that respondents during the financial crisis were much less consistent in their answers to the questions of the consumer questionnaire. In the next step of the analysis, data on inflation expectations were applied to modelling and forecasting inflation. It was shown that with respect to standard ARIMA processes, inclusion of the information on the inflation expectations significantly improved the in-sample and out-of-sample forecasting performance of the time-series models. Especially out-of-sample performance was significantly better as the average absolute error in forecasts of headline and core inflation was reduced by half. It was also shown that models with inflation expectations based on the CFA method (after elimination of the consumer sentiment factor) provided better in-sample forecasts of inflation. Nevertheless, it was not confirmed for the out-of-sample forecasts. (1) Project financed by the National Bank of Poland. Polish title of the project: "Prognozowanie inflacji na podstawie danych koniunktury gospodarstw domowych. Zastosowanie konfirmacyjnej analizy czynnikowej dla wielu grup do oczyszczenia prognoz inflacji z czynnika ogĂłlnego nastroju gospodarczego."Inflation expectations, Inflation forecasts, Confirmatory Factor Analysis
The Bounded L2 Curvature Conjecture
This is the main paper in a sequence in which we give a complete proof of the
bounded curvature conjecture. More precisely we show that the time of
existence of a classical solution to the Einstein-vacuum equations depends only
on the -norm of the curvature and a lower bound on the volume radius of
the corresponding initial data set. We note that though the result is not
optimal with respect to the standard scaling of the Einstein equations, it is
nevertheless critical with respect to its causal geometry. Indeed, bounds
on the curvature is the minimum requirement necessary to obtain lower bounds on
the radius of injectivity of causal boundaries. We note also that, while the
first nontrivial improvements for well posedness for quasilinear hyperbolic
systems in spacetime dimensions greater than 1+1 (based on Strichartz
estimates) were obtained in [Ba-Ch1] [Ba-Ch2] [Ta1] [Ta2] [Kl-R1] and optimized
in [Kl-R2] [Sm-Ta], the result we present here is the first in which the full
structure of the quasilinear hyperbolic system, not just its principal part,
plays a crucial role. To achieve our goals we recast the Einstein vacuum
equations as a quasilinear -valued Yang-Mills theory and introduce a
Coulomb type gauge condition in which the equations exhibit a specific new type
of \textit{null structure} compatible with the quasilinear, covariant nature of
the equations. To prove the conjecture we formulate and establish bilinear and
trilinear estimates on rough backgrounds which allow us to make use of that
crucial structure. These require a careful construction and control of
parametrices including error bounds which is carried out in [Sz1]-[Sz4],
as well as a proof of sharp Strichartz estimates for the wave equation on a
rough background which is carried out in \cite{Sz5}.Comment: updated version taking into account the remarks of the refere
On the Use of Underspecified Data-Type Semantics for Type Safety in Low-Level Code
In recent projects on operating-system verification, C and C++ data types are
often formalized using a semantics that does not fully specify the precise byte
encoding of objects. It is well-known that such an underspecified data-type
semantics can be used to detect certain kinds of type errors. In general,
however, underspecified data-type semantics are unsound: they assign
well-defined meaning to programs that have undefined behavior according to the
C and C++ language standards.
A precise characterization of the type-correctness properties that can be
enforced with underspecified data-type semantics is still missing. In this
paper, we identify strengths and weaknesses of underspecified data-type
semantics for ensuring type safety of low-level systems code. We prove
sufficient conditions to detect certain classes of type errors and, finally,
identify a trade-off between the complexity of underspecified data-type
semantics and their type-checking capabilities.Comment: In Proceedings SSV 2012, arXiv:1211.587
Interaction of vortices in viscous planar flows
We consider the inviscid limit for the two-dimensional incompressible
Navier-Stokes equation in the particular case where the initial flow is a
finite collection of point vortices. We suppose that the initial positions and
the circulations of the vortices do not depend on the viscosity parameter \nu,
and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex
system is well-posed on the interval [0,T]. Under these assumptions, we prove
that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a
superposition of Lamb-Oseen vortices whose centers evolve according to a
viscous regularization of the point vortex system. Convergence holds uniformly
in time, in a strong topology which allows to give an accurate description of
the asymptotic profile of each individual vortex. In particular, we compute to
leading order the deformations of the vortices due to mutual interactions. This
allows to estimate the self-interactions, which play an important role in the
convergence proof.Comment: 39 pages, 1 figur
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