Search

The variability and interdependence of local porosity and local mean geodesic tortuosity, which is a measure for the sinuosity of shortest transportation paths, is investigated at the example of the microstructure in sack paper. By means of statistical image analysis, these two morphological characteristics are computed for several cutouts of 3D image data obtained by X-ray microcomputed tomography. Considering cutouts of different sizes allows us to study the influence of the sample size on the local variability of the considered characteristics. Moreover, the interdependence between local porosity and local mean geodesic tortuosity is quantified by modeling their joint distribution parametrically using Archimedean copulas. It turns out that the family of Gumbel copulas is an appropriate model type, which is formally validated by a goodness of fit test. Besides mean geodesic tortuosity, we consider further related morphological characteristics, describing the sinuosity of those shortest transportation paths, whose minimum diameter exceeds a predefined threshold. Moreover, we show that the copula approach investigated in this paper can also be used to quantify the negative correlation between local porosity and these modified versions of local mean geodesic tortuosity. Our results elucidate the impact of local porosity on various kinds of morphological characteristics, which are not experimentally accessible and which are important for local air permeance – a key property of sack paper.

Let a(n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of \(SL_2({\mathbb R})\). This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of \(a(n)a(n+r)\) for a fixed positive integer r.

Place a square sheet of paper with side length 1 into the first quadrant of a coordinate system in such a way that the lower-left corner coincides with the origin. Look at the set of points that ar...

In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.

An error has been detected (and also corrected) in Theorem 2.8 of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity, (2005)9: 207–223; DOI 10.1007/s11117-003-2778-5). Accordingly, some of the results of the paper have been modified. Also, a notion of C*-matricially, Riesz normed spaces has been introduced.

We study the problem of characterizing the effective (homogenized) properties of materials whose diffusive properties are modeled with random fields. Focusing on elliptic PDEs with stationary and ergodic random coefficient functions, we provide a gentle introduction to the mathematical theory of homogenization of random media. We also present numerical examples to elucidate the theoretical concepts and results.

On a recent paper Benchohra and Mostefai [2] presented some existence results for an integral equation of fractional order with multiple time delay in Banach spaces. In contrast to the classical case, when assumptions are expressed in terms of the strong topology, they considered another case, namely with the weak topology. It has some consequences for the proof. We present here some comments and corrections.

We have considered the process of inscribing triangles between parallel lines in the Euclidean plane using an iterative process. It is a familiar result that the limiting triangle is equilateral. In this Note, we demonstrate that the same result holds when the geometry is hyperbolic and the lines are asymptotically parallel. We further extend the process to the sphere, where it is found that the limiting triangle is isosceles.

The authors of [KMM] consider a system of two functional inequalities for a function $$f : {\mathbb{R}} \rightarrow {\mathbb{R}}$$ , and they show that, if certain arithmetical conditions and inequalities for certain parameters are fulfilled, f has to be a polynomial provided that f is continuous at some point x0. This result is derived here under the weaker condition that for some x0 the limit $${\rm lim}_{x \rightarrow x_0} f(x)$$ exists. Moreover, another system of inequalities is given leading to the same result on the nature of f. The methods used also give natural explanations for the fact that Bernoulli numbers play an important role in this context.

Many solid materials exhibit stress–induced phase transformations. Such phenomena can be modelled with the aid of the nonlinear elasticity theory with appropriate choices of the strain–energy function. It is known that if a two–phase deformation (with gradient F ) in a finite elastic body is a local energy minimizer, then given any point p of the surface of discontinuity, the piecewise–homogeneous deformation corresponding to the two values F ± ( p ) of F ( p ) is a global energy minimizer. Thus, instability of the latter state would imply instability of the former state. In this paper we investigate the stability properties of such piecewise–homogeneous deformations. More precisely, we are concerned with two joined half–spaces that correspond to two different phases of the same material. We first show how such a two–phase deformation can be constructed. Then the stability of the piecewise–homogeneous deformation is investigated with the aid of two test criteria. One is a kinetic stability criterion based on a quasi–static approach and on the growth/decay behaviour of the interface in the undeformed configuration when it is perturbed; the other, referred to as the energy criterion, is used to determine whether the deformation is a minimizer of the total energy with respect to perturbations of the interface in both the current and undeformed configurations. We clarify the differences between the two criteria, and provide a compact formula which can be used to establish the stability/instability of any two–phase piecewise–homogeneous deformations.

We considered in Example 3.1 of the paper [1] an S-structure on R2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.

Kolmogorov-Arnold-Moser (or KAM) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and KAM theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov's pioneering work.

Let S be a surface obtained from a plane polygon by identifying infinitely many pairs of segments along its boundary. A condition is given under which the complex structure in the interior of the polygon extends uniquely across the quotient of its boundary to make S into a closed Riemann surface. When this condition holds, a modulus of continuity is obtained for a uniformizing map on S., 36 pages, 11 figures

This note concerns the paper "On the controllability of a coupled system of two Korteweg–de Vries equations" by Micu et al. [2]. They study a nonlinear coupled system of two Korteweg–de Vries equations and prove that the system is controllable by using four boundary controls. Here, we prove that in some cases it is possible to get the controllablity of the system by using only two controls. This can be done depending on both the spatial domain and the control time.