Combinatorial data of a free arrangement and the Terao conjecture

We present a combinatorial structure of generators of $D(\mathcal{A}).$ This structure permits us to detect the relationship between the combinatorial determined property and the singularity of vector field. Consequently, by using only combinatorial data, we have a basis of the module in free case and that yields a proof for the Terao's conjecture. We also verify the example of Ziegler and give a sufficient condition on combinatorial determined property of generators

Relations between various boundaries of relatively hyperbolic groups

Suppose a group $G$ is relatively hyperbolic with respect to a collection \PP of its subgroups and also acts properly, cocompactly on a \CAT(0) (or $\delta$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative boundary \partial(G,\PP). The \CAT(0) structure provides a different boundary at infinity $\partial X$. In this article, we examine the connection between these two spaces at infinity. In particular, we show that \partial (G,\PP) is $G$--equivariantly homeomorphic to the space obtained from $\partial X$ by identifying the peripheral limit points of the same type.Comment: 22 page

Energy analysis of hydraulic fracturing

In this paper, numerical simulations of circular boreholes under internal hydraulic pressure are carried out to investigate the energy transferred to the surrounding rock and the breakdown pressure. The simulations are conducted by using a micromechanical continuum damage model proposed by Golshani et al. (2006). The simulation results suggest that the borehole breakdown pressure and the energy transferred to the surrounding rock are dependent on the mechanical properties of the rock and borehole size. Although the energy transferred to the surrounding rock increases with increasing borehole size, the borehole breakdown pressure decreases

A second-order continuity domain-decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems

This paper presents a second-order continuity non-overlapping domain decomposition (DD) technique for numerically solving second-order elliptic problems in two-dimensional space. The proposed DD technique uses integrated Chebyshev polynomials to represent the solution in subdomains. The constants of integration are utilized to impose continuity of the second-order normal derivative of the solution at the interior points of subdomain interfaces. To also achieve a C2 (C squared) function at the intersection of interfaces, two additional unknowns are introduced at each intersection point. Numerical results show that the present DD method yields a higher level of accuracy than conventional DD techniques based on differentiated Chebyshev polynomials