We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on p and on the symmetric part of a gradient of u, namely, it is represented by a stress tensor T (Du, p):= v(p, |D|2)D which satisfies r-growth condition with r \in (1, 2]. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for example in the paper Caffarelli, Peral (1998)., Václav Mácha., and Obsahuje seznam literatury
As said by Mareš and Mesiar, necessity of aggregation of complex real inputs appears almost in any field dealing with observed (measured) real quantities (see the citation below). For aggregation of probability distributions Sklar designed his copulas as early as in 1959. But surprisingly, since that time only a very few literature have appeared dealing with possibility to aggregate several different pairwise dependencies into one multivariate copula. In the present paper this problem is tackled using the well known Iterative Proportional Fitting Procedure. The proposed solution is not an exact mathematical solution of a marginal problem but just its approximation applicable in many practical situations like Monte Carlo sampling. This is why the authors deal not only with the consistent case, when the iterative procedure converges, but also with the inconsistent non-converging case. In the latter situation, the IPF procedure tends to cycle (when combining three pairwise dependencies the procedure creates three convergent subsequences), and thus the authors propose some heuristics yielding a "solution'' of the problem even for inconsistent pairwise dependence relations.
Perez's approximations of probability distributions by dependence structure simplification were introduced in 1970s, much earlier than graphical Markov models. In this paper we will recall these Perez's models, formalize the notion of a compatible system of elementary simplifications and show the necessary and sufficient conditions a system must fulfill to be compatible. For this we will utilize the apparatus of compositional models.
Most satellite laser stations have used extermal terrestrial targets at distances from several hundred metres to kilometres for the determination of the calibration constant. The disadvantage of this method is the incomplete overlap of the transmitter and receiver angular fields and the necessity to monitor accurately the optical distance of the remote target. At Potsdam station, one year ago we introduced a very simple calibration link based on dual diffuse scattering, which can be easily attached in front of any laser radar. No optical components other than diffuse reflectors are used. The inherent attenuation is about 10^13, so that a 1000 : 1 additional filter is sufficient to reach the single photoelectron level. The method has been used for routing operations during the MERIT campaign and for investigating some error sources as well.
A simple yet powerful procedure for an echo attenuation in signals is introduced. The presented method involves no external reference signal. It is based on comb FIR filtering. To the advantages of the described method belong the simplicity and performance which are beneficial in real time implementations. For illustration, a simulation of the procedure is included. The efficiency of the presented method is demonstrated by a real time implementation on a digital signal processor.
Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_\varphi $ of Wiener measure and for a partition $ 0=t_0< t_1< \cdots < t_n <t_{n+1}= T$ of $[0, T]$, let $X_n\: C[0,T]\to \mathbb R^{n+1}$ and $X_{n+1} \: C [0, T]\to \mathbb R^{n+2}$ be given by $X_n(x) = ( x(t_0), x(t_1), \cdots , x(t_n))$ and $X_{n+1} (x) = ( x(t_0), x(t_1), \cdots , x(t_{n+1}))$, respectively. \endgraf In this paper, using a simple formula for the conditional $w_\varphi $-integral of functions on $C[0, T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_\varphi $-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions of the form $F_m(x) = \int _0^T (x(t))^m d t$ for $x\in C[0, T]$ and for any positive integer $m$. Moreover, with the conditioning $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions in a Banach algebra $\mathcal S_{w_\varphi }$ which is an analogue of the Cameron and Storvick's Banach algebra $\mathcal S$. Finally, we derive the conditional analytic Feynman $w_\varphi $-integrals of the functions in $\mathcal S_{w_\varphi }$.
We show that a transformation method relating planar first-order differential systems to second order equations is an effective tool for finding non-liouvillian first integrals. We obtain explicit first integrals for a subclass of Kukles systems, including fourth and fifth order systems, and for generalized Liénard-type systems.
Measurement of leaf area is commonly used in many horticultural research experiments, but it is generally destructive, requiring leaves to be removed for measurement. Determining the individual leaf area (LA) of bedding plants like pot marigold (Calendula officinalis L.), dahlia (Dahlia pinnata), sweet William (Dianthus barbatus L.), geranium (Pelargonium × hortorum), petunia (Petunia × hybrida), and pansy (Viola wittrockiana) involves measurements of leaf parameters such as length (L) and width (W) or some combinations of these parameters. Two experiments were carried out during spring 2010 (on two pot marigold, four dahlia, three sweet William, four geranium, three petunia, and three pansy cultivars) and summer 2010 (on one cultivar per species) under greenhouse conditions to test whether a model could be developed to estimate LA of bedding plants across cultivars. Regression analysis of LA versus L and W revealed several models that could be used for estimating the area of individual bedding plants leaves. A linear model having LW as the independent variable provided the most accurate estimate (highest R2, smallest mean square error, and the smallest predicted residual error sum of squares) of LA in all bedding plants. Validation of the model having LW of leaves measured in the summer 2010 experiment coming from other cultivars of bedding plants showed that the correlation between calculated and measured bedding plants leaf areas was very high. Therefore, these allometric models could be considered simple and useful tools in many experimental comparisons without the use of any expensive instruments. and F. Giuffrida ... [et al.].
Let $T$ be a locally compact Hausdorff space and let $C_0(T)$ be the Banach space of all complex valued continuous functions vanishing at infinity in $T$, provided with the supremum norm. Let $X$ be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of $X$-valued $\sigma $-additive Baire measures on $T$ is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map $u\: C_0(T) \rightarrow X$ when $c_0 \lnot \subset X$ are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].