We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several recent results involving fillability, planarity, and non-separating contact type embeddings. The cobordisms are built from symplectic handles of the form Σ × D and Σ × [−1, 1] × S1, which have symplectic cores and can be attached to contact 3-manifolds along sufficiently large neighborhoods of transverse links and preLagrangian tori. We also sketch a construction of J-holomorphic foliations in these cobordisms and formulate a conjecture regarding maps induced on Embedded Contact Homology with twisted coefficients.

Some new characterizations of the class of positive measures γ on$R$^{$n$}such that$H$_{$p$}^{$l$}∉L_{$p$}(γ) are given where$H$_{$p$}^{$l$}(1<$p$<∞ 0<$l$∞) is the space of Bessel potentials This imbed ding as well as the corresponding trace inequality $$||J_l u||_{L_p (\gamma )} \leqslant C||u||_{L_p } $$ for Bessel potentials$J$_{$l$}=(1-Δ)^{-1/2}is shown to be equivalent to one of the following conditions(a)$J$_{$l$}($J$_{$l$γ})^{$p$}≤$CJ$_{$lγ$}a e(b)$M$_{$l$}($M$_{$l$γ})^{$p’$}≤$CM$_{$lγ$}a e(c)For all compact subsets$E$of$R$^{$n$} $$\int_E {(J_{l\gamma } )^p dx} \leqslant C{\text{ }}cap (E H_p^l )$$ where 1/$p$+1/$p'$=1$M$_{$l$}is the fractional maximal operator and cap ($H$_{$p$}^{$l$}) is the Bessel capacity In particular it is shown that the trace inequality for a positive measure \gg holds if and only if it holds for the measure$(J$_{$l\gg$})^{$p'$}$dx$Similar results are proved for the Riesz potentials$I$_{$l$γ}=|$x$|^{$l-n$}* γ

We present a framework for generating street networks and parcel layouts. Our goal is the generation of high-quality layouts that can be used for urban planning and virtual environments. We propose a solution based on hierarchical domain splitting using two splitting types: streamline-based splitting, which splits a region along one or multiple streamlines of a cross field, and template-based splitting, which warps pre-designed templates to a region and uses the interior geometry of the template as the splitting lines. We combine these two splitting approaches into a hierarchical framework, providing automatic and interactive tools to explore the design space.