Narrow Results

In this paper, we prove a Ros–Rosenberg theorem in the setting of Special Weingarten surfaces. We show that a compact, connected, embedded, Special Weingarten surface in ${ℝ}_{+}^3$ with planar convex boundary is a topological disk under mild suitable assumptions.

We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and Lárusson’s elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov’s condition ${\mathrm{\text{Ell}}}_1$, and the equivariant localization principle is also given.

We study the propagation of a linearly polarized laser beam in the external transverse magnetic field taking into consideration the vacuum polarization by arbitrary spin particles. Induced ellipticity of the beam is evaluated using the effective Lagrangian. With the help of the PVLAS experimental data, we obtain bounds on masses of charged higher spin particles contributed to ellipticity.

In this paper, we considered four observationally constrained, one-parameter dark energy models to see their plausible impacts on the tidal ellipticity and related statistics of cosmic voids. Our main question is to see if different models of dark energy could be distinguished through ellipticity of cosmic voids. In this way, we employed an analytic method on the base of local tidal tensor. In this method, the center of a void is located at local minimum of the linear density field. At first, we obtained the effect of the one-parameter DE models on the distribution function of the cosmic voids and simultaneously compared the results with the standard cosmology ($\mathrm{\Lambda }$CDM). We found that the distribution function of these models is different with respect to that of $\mathrm{\Lambda }$CDM. In addition, we obtained the mean ellipticity of cosmic voids, $〈\epsilon 〉$, against redshift and found that in all models the $〈\epsilon 〉$ as well as ${\epsilon }_{max}$ has a decaying trend. Besides, we considered the mean ellipticity evolution versus the smoothing scale of voids ($R_L$). We found that the differences between dark energy models are significantly discernible via the ellipticity evolution versus the void scale ($R_L$). We saw that the studied models result in different ellipticity evolutions (versus $R_L$) up to $5\%$ smaller with respect to that of the $\mathrm{\Lambda }$CDM.

In this paper, the optical properties of the novel windmill-shaped chiral metamaterial (CMM) with low ellipticity have been studied numerically in THz band. The dispersionless optical activity can be achieved in a wide frequency range (i.e. from 1.7 THz to 2.7 THz). The dependence of CMM’s optical properties on the structural parameters is studied systematically and the frequency range of low ellipticity can be controlled dynamically through adjusting the width of the metal. The excitation mechanism of optical activity based on the current distribution is also analyzed in detail.

In 1993, it was shown by Geymonat, Müller and Triantafyllidis that, in the setting of linearized elasticity, a Γ-convergence result holds for highly oscillating sequences of elastic energies whose functional coercivity constant in ℝ^N is zero while the corresponding coercivity constant on the torus remains positive. We illustrate the range of applicability of that result by finding sufficient conditions for such a situation to occur. We thereby justify the degenerate laminate construction given by Gutiérrez in 1999. We also demonstrate that the predicted loss of strict strong ellipticity resulting from the construction by Gutiérrez is unique within a "laminate-like" class of microstructures. It will only occur for the specific micro-geometry investigated there. Our results thus confer both a rigorous, and a canonical character to those of Gutiérrez.

We introduce an individual-based model for fiber elements having the ability to cross-link or unlink each other and to align with each other at the cross links. We first formally derive a kinetic model for the fiber and cross-links distribution functions. We then consider the fast linking/unlinking regime in which the model can be reduced to the fiber distribution function only and investigate its diffusion limit. The resulting macroscopic model consists of a system of nonlinear diffusion equations for the fiber density and mean orientation. In the case of a homogeneous fiber density, we show that the model is elliptic.

This paper considers the problem of constructing shape measures; we start by giving a short overview of areas of practical application of such measures.

Shapes can be characterised in terms of a set of geometric properties, some of which are Boolean in nature. We give two general principles for turning Boolean properties into continuous measures of that property. (For example, we may turn the Boolean property conves into a convexity value in the range [0–1].) We demonstrate how these principles can be applied to construct various shape measures, including ones for convexity, circularity, ellipticity, triangularity, rectilinearity, rectangularity and symmetry in two dimensions, and 2.5D-ness, stability, and imperforateness in three dimensions. Some of these measures are new; others are well known and we show how they fit into this general framework.

We also show how such measures for a single shape can be generalised to multiple shapes, and consider as examples measures for containment, resemblance, congruence, and similarity.

Different stent structures lead to different deformations of blood vessels, such as different cross-sectional shapes, which further influence the blood flow patterns. In this paper, six non-commercial stents with different link structures called I-, C-, S-, U-, N- and W-types were considered. Their influences on arteries with five different curvatures (i.e., 0${}^{\circ }$, 15${}^{\circ }$, 30${}^{\circ }$, 45${}^{\circ }$ and 60${}^{\circ }$) were studied using finite element method. Four indices including the maximum plastic strain of stents, the rate of expansion, the maximum von Mises stress and the ellipticity of arteries, were compared for all cases. The results showed that the S-type or U-type stents, with larger plastic strain and lower von Mises stress on the arteries, provided the optimal outcome. As the link structures became complex, the arterial expansion increased monotonically, while the ellipticity of arteries showed a decreasing tendency in the vessel models. The present study could be useful for the commercial design and clinic selection of a stent with different link structures for different curved arteries.

In this paper, we give a simple proof of the ellipticity of Koiter's model for a shell, based on the three-dimensional Korn inequality applied to Kirchhoff–Love displacement fields.

In this paper, we establish sufficient conditions for transversality of two hypersurfaces of a given pseudo-Riemannian manifold. These conditions are the ellipticity of some operators obtained by means of Newton transformations.

In this paper, the mechanical behavior of incompressible transversely isotropic materials is modeled using a strain energy density in the framework of Ball’s theory. Based on this profound theory and with respect to physical and mathematical aspects of deformation invariants, a new polyconvex constitutive model is proposed for the mechanical behavior of these materials. From the physical viewpoint, it is assumed that the proposed model is additively decomposed into three parts nominally representing the energy contributions from the matrix, fiber and fiber–matrix interaction where each of the parts should be presented in terms of the invariants consistent with the physics of the deformation. From the mathematical viewpoint, the proposed model satisfies the fundamental postulates on the form of strain energy density, specially polyconvexity and coercivity constraints. Indeed, polyconvexity ensures ellipticity condition, which in turn provides material stability and in combination with coercivity condition, guarantees the existence of the global minimizer of the total energy. In order to evaluate the performance of the proposed strain energy density function, some test data of incompressible transverse materials with pure homogeneous deformations are used. It is shown that there is a good agreement between the test data and the obtained results from the proposed model. At the end, the performance of the proposed model in the prediction of the material behavior is evaluated rather than other models for two representative problems.

We establish a calculus of boundary value problems (BVPs) on a manifold $N$ with boundary and edge, based on Boutet de Monvel’s theory of BVPs in the case of a smooth boundary and on the edge calculus, where in the present case the model cone has a base which is a compact manifold with boundary. The corresponding calculus with boundary and edge is a unification of both structures and controls different operator-valued symbolic structures, in order to obtain ellipticity and parametrices.