Search results for "Remainder"

Chiral condensates from tau decay: a critical reappraisal

2006

The saturation of QCD chiral sum rules is reanalyzed in view of the new and complete analysis of the ALEPH experimental data on the difference between vector and axial-vector correlators (V-A). Ordinary finite energy sum rules (FESR) exhibit poor saturation up to energies below the tau-lepton mass. A remarkable improvement is achieved by introducing pinched, as well as minimizing polynomial integral kernels. Both methods are used to determine the dimension d=6 and d=8 vacuum condensates in the Operator Product Expansion, with the results: {O}_{6}=-(0.00226 \pm 0.00055) GeV^6, and O_8=-(0.0053 \pm 0.0033) GeV^8 from pinched FESR, and compatible values from the minimizing polynomial FESR. Som…

Finite energy chiral sum rules in QCD

2003

The saturation of QCD chiral sum rules of the Weinberg-type is analyzed using ALEPH and OPAL experimental data on the difference between vector and axial-vector correlators (V-A). The sum rules exhibit poor saturation up to current energies below the tau-lepton mass. A remarkable improvement is achieved by introducing integral kernels that vanish at the upper limit of integration. The method is used to determine the value of the finite remainder of the (V-A) correlator, and its first derivative, at zero momentum: $\bar{\Pi}(0) = - 4 \bar{L}_{10} = 0.0257 \pm 0.0003 ,$ and $\bar{\Pi}^{\prime}(0) = 0.065 \pm 0.007 {GeV}^{-2}$. The dimension $d=6$ and $d=8$ vacuum condensates in the Operator P…

2003

In this article we apply the S(M, g)–calculus of L. Hormander and, in particular, results concerning the spectral invariance of the algebra of operators of order zero in ℒ(L2(ℝn)) to study generators of Feller semigroups. The core of the article is the proof of the invertibility of λ Id + P for a strongly elliptic operator P in Ψ(M, g) and suitable weight functions M and metrics g. The proof depends highly on precise estimates of the remainder term in asymptotic expansions of the product symbol in Weyl and Kohn–Nirenberg quantization. Due to the Hille–Yosida–Ray theorem and a theorem of Courrege, the result concerning the invertibility of λ Id + P is applicable to obtain sufficient conditio…

Predicting human performance in interactive tasks by using dynamic models

2017

The selection of an appropriate sequence of activities is an essential task to keep student motivation and foster engagement. Usually, decisions in this respect are made by taking into account the difficulty of the activities, in relation to the student's level of competence. In this paper, we present a dynamic model that aims to predict the average performance of a group of students at solving a given series of maths problems. The system takes into account both student- and task-related features. This model was built and validated by using the data gathered in an experimental session that involved 64 participants solving a sequence of 26 arithmetic problems. The data collected from the fir…