Embedding surfaces into S3S^3 with maximum symmetry

We restrict our discussion to the orientable category. For $g > 1$, let $OE_g$ be the maximum order of a finite group $G$ acting on the closed surface $\Sigma_g$ of genus $g$ which extends over $(S^3, \Sigma_g)$, where the maximum is taken over all possible embeddings $\Sigma_g\hookrightarrow S^3$. We will determine $OE_g$ for each $g$, indeed the action realizing $OE_g$. In particular, with 23 exceptions, $OE_g$ is $4(g+1)$ if $g\ne k^2$ or $4(\sqrt{g}+1)^2$ if $g=k^2$, and moreover $OE_g$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and $481$.Comment: 42 pages, 37 figures, 6 tables of figure

Graphs in the 3--sphere with maximum symmetry

We consider the orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Gamma)$, where $\Gamma$ is a connected graph of genus $g>1$, embedded in $S^3$. For each $g$ we give the maximum order $m_g$ of such $G$ acting on $(S^3, \Gamma)$ for all such $\Gamma\subset S^3$. Indeed we will classify all graphs $\Gamma\subset S^3$ which realize these $m_g$ in different levels: as abstract graphs and as spatial graphs, as well as their group actions. Such maximum orders without the condition "orientation-preserving" are also addressed.Comment: 34 pages, to appear in Discrete Comput. Geo

Bordered surfaces in the 3-sphere with maximum symmetry

We consider finite group actions on the 3-sphere which leave invariant an embedded, compact, bounded surface of algebraic genus g > 1 (orientable or non-orientable), and determine for each g the maximum order of such an action. For example, the maximal possibility 12(g-1) is obtained for the finitely many values g = 2, 3, 4, 5, 9, 11, 25, 97, 121 and 241. For each g > 1, we classify the topological types of the surfaces and their embeddings into the 3-sphere

Characterization and Quantification of Livestock Odorants using Sorbent Tube Sampling and Thermal Desorption coupled with Multidimensional Gas Chromatography–Mass Spectrometry–Olfactometry

Characterization and quantification of livestock odorants is one of the most challenging analytical tasks because odor-causing gases are very reactive, polar and often present at very low concentrations in a complex matrix of less important or irrelevant gases. The objectives of this research is to develop a novel analytical method for characterization of the livestock odorants including their odor character, odor intensity, and hedonic tone and further quantitative analysis of the key odorants responsible for livestock odor emissions. Sorbent tubes packed with Tenax TA were employed for sampling. The automated one-step thermal desorption coupled with multidimensional gas chromatography-mass spectrometry-olfactometry system was developed for simultaneous chemical and odor analysis. Fifteen odorants identified from different livestock species operations are quantified. In addition, odor character, odor intensity and hedonic tone associated with each of the target compounds are also analyzed. The method developed in this research is being used on a multistate, multispecies project focused on quantifying odor and chemical analysis of odor

Embedding compact surfaces into the 3-dimensional Euclidean space with maximum symmetry

We give the maximum orders of finite group actions on Euclidean 3-space which leave invariant an embedded compact bordered surface (orientable or non-orientable), in terms of the algebraic genus of the surface. We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces