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We present a new penalty term approximating the CiarletNe\v{c}as condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For nonsimple materials including a suitable higher order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the CiarletNe\v{c}as condition. Moreover, the penalization can be chosen in such a way that all low energy deformations, selfinterpenetration is completely avoided even for sufficiently small finite values of the penalization parameter. We also present numerical experiments in 2d illustrating our theoretical results.
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The Go Blog looks forward to version 2 of the Go language. "A major difference between Go 1 and Go 2 is who is going to influence the design and how decisions are made. Go 1 was a small team effort with modest outside influence; Go 2 will be much more communitydriven. After almost 10 years of exposure, we have learned a lot about the language and libraries that we didn’t know in the beginning, and that was only possible through feedback from the Go community."
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We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs $H_1$ and $H_2$. By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to $(H_1,H_2)$free graphs for all but a finite number of pairs $(H_1,H_2)$, but without explicitly giving the number of open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomialtime solvable on graph classes of bounded cliquewidth. By combining previously known results for Graph Isomorphism with known results for boundedness of cliquewidth, we reduce the number of open cases to 14. By proving a number of new results we then further reduce this number to seven. By exploiting the strong relationship between Graph Isomorphism and cliquewidth, we also prove that the class of $(\mbox{gem},P_1+2P_2)$free graphs has unbounded cliquewidth. This reduces the number of open cases for boundedness of cliquewidth for $(H_1,H_2)$free grap
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To every topologically transitive Cantor dynamical system $(X, \varphi)$ we associate a group $T(\varphi)$ acting faithfully by homeomorphism on the real line. It is defined as the group of homeomorphisms of the suspension flow of $(X, \varphi)$ which preserve every leaf and acts by dyadic piecewise linear homeomorphisms in the flow direction. We show that if $(X, \varphi)$ is minimal, the group $T(\varphi)$ is simple, and if $(X, \varphi)$ is a subshift the group $T(\varphi)$ is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple leftorderable groups. We show that if the system $(X, \varphi)$ is minimal, every action of the group $T(\varphi)$ on the circle has a fixed point, providing examples of so called "orderable monsters". We additionally have the following: for every subshift $(X, \varphi)$ the group $T(\varphi)$ does not have nontrivial subgroups with Kazhdan's property (T); for every
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Let $G$ be a nontrivial connected, edgecolored graph. An edgecut $S$ of $G$ is called a rainbow cut if no two edges in $S$ are colored with a same color. An edgecoloring of $G$ is a rainbow disconnection coloring if for every two distinct vertices $s$ and $t$ of $G$, there exists a rainbow cut $S$ in $G$ such that $s$ and $t$ belong to different components of $G\setminus S$. For a connected graph $G$, the {\it rainbow disconnection number} of $G$, denoted by $rd(G)$, is defined as the smallest number of colors such that $G$ has a rainbow disconnection coloring by using this number of colors. In this paper, we show that for a connected graph $G$, computing $rd(G)$ is NPhard. In particular, it is already NPcomplete to decide if $rd(G)=3$ for a connected cubic graph. Moreover, we prove that for a given edgecolored (with an unbounded number of colors) connected graph $G$ it is NPcomplete to decide whether $G$ is rainbow disconnected.
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We give a concrete presentation for the general linear group defined over a ring which is a finitely generated free $\mathbb{Z}$module or the integral Clifford group $\Gamma_n(\mathbb{Z})$ of invertible elements in the Clifford algebra with integral coefficients. We then use this presentation to prove that the elementary linear group over $\Gamma_n(\mathbb{Z})$ has a nontrivial decomposition as a free product with amalgamated subgroup the elementary linear group over $\Gamma_{n1}(\mathbb{Z})$. This allows to obtain applications to the unit group $\mathcal{U}(\mathbb{Z} G)$ of an integral group ring $\mathbb{Z} G$ of a finite group $G$. In particular, we prove that $\mathcal{U} (\mathbb{Z} G)$ is hereditary (FA), i.e. every subgroup of finite index has property (FA), or is commensurable with a nontrivial amalgamated product. In the case $\mathcal{U}(\mathbb{Z} G)$ is not hereditary (FA), we investigate subgroups of finite index in $\mathcal{U}(\mathbb{Z} G)$ that have a nontrivial
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We compute the Dolbeault cohomology of certain domains contained in Cousin groups defined by lattices which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of OeljeklausToma manifolds and in particular the fact that the Hodge decomposition holds for their cohomology.
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In this paper we construct families of homology spheres which bound 4manifolds with intersection forms isomorphic to $E_8$. We show that these families have arbitrary large correction terms. This result says that among homology spheres, the difference of the maximal rank of minimal sublattice of definite filling and the maximal rank of even definite filling is arbitrarily large.
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