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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.

In a former paper we proposed a model for the quantization of gravity by working in a bundle $E$ where we realized the Hamilton constraint as the Wheeler-DeWitt equation. However, the corresponding operator only acts in the fibers and not in the base space. Therefore, we now discard the Wheeler-DeWitt equation and express the Hamilton constraint differently, either with the help of the Hamilton equations or by employing a geometric evolution equation. There are two possible modifications possible which both are equivalent to the Hamilton constraint and which lead to two new models. In the first model we obtain a hyperbolic operator that acts in the fibers as well as in the base space and we can construct a symplectic vector space and a Weyl system. In the second model the resulting equation is a wave equation in $\so \times (0,\infty)$ valid in points $(x,t,\xi)$ in $E$ and we look for solutions for each fixed $\xi$. This set of equations contains as a special case the equation of a quantized cosmological Friedman universe without matter but with a cosmological constant, when we look for solutions which only depend on $t$. Moreover, in case $\so$ is compact we prove a spectral resolution of the equation.

This research provides insights for the separation of cryptocurrencies from other assets. Using dimensionality reduction techniques, we show that most of the variation among cryptocurrencies, stocks, exchange rates, commodities, bonds, and real estate indexes can be explained by the tail, memory and moment factors of their log-returns. By applying various classification methods, cryptocurrencies are categorized as a separate asset class, mainly due to the tail factor. The main result is the complete separation of cryptocurrencies from the other asset types, using the Maximum Variance Components Split method. Additionally, we show that cryptocurrencies tend to exhibit similar characteristics over time and become more distinguished from other asset classes (synchronic evolution).