22,335 research outputs found

    On side lengths of corners in positive density subsets of the Euclidean space

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    We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of Rd\mathbb{R}^d to corners in subsets of Rd×Rd\mathbb{R}^d\times\mathbb{R}^d. More precisely, if 1<p<∞1<p<\infty, p≠2p\neq 2, and dd is large enough, we show that an arbitrary measurable set A⊆Rd×RdA\subseteq\mathbb{R}^d\times\mathbb{R}^d of positive upper Banach density contains corners (x,y)(x,y), (x+s,y)(x+s,y), (x,y+s)(x,y+s) such that the ℓp\ell^p-norm of the side ss 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

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    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 – application of multi-group confirmatory factor analysis to elimination of the general sentiment factor

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    This paper (1) examines the properties of survey based households’ inflation expectations and investigates their forecasting performance. With application of the individual data from the State of the Households’ 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 – 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 ogólnego nastroju gospodarczego."Inflation expectations, Inflation forecasts, Confirmatory Factor Analysis

    The Bounded L2 Curvature Conjecture

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    This is the main paper in a sequence in which we give a complete proof of the bounded L2L^2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L2L^2-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, L2L^2 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 so(3,1)so(3,1)-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 L2L^2 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

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    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

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    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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