闭模糊拟阵模糊圈的极值问题

本文主要方法是通过基本序列、导出拟阵序列和模糊集分解定理,将模糊圈的研究转化为对圈子集套和数组的研究。在闭模糊拟阵中,我们得出三个结论:以同一集合为支撑集的模糊圈的最大模糊圈总是存在;以同一子集串为圈子集套的模糊圈的最大模糊圈不一定存在。但是,找到了存在最大模糊圈的充要条件;以同一集合为支撑集的模糊圈的最小模糊圈,以同一子集串为圈子集套的模糊圈的最小模糊圈都是不存在的。但它们的最小模糊势是存在的,而且找出了计算最小模糊势的公式。我们构造了两个算法:一是构造支撑集最大模糊圈算法。通过这个算法可构造出支撑集最大模糊圈,同时计算出其最大模糊势;二是判断和构造圈子集套最大模糊圈算法。通过这个算法首先判断最大模糊圈是否存在,如果存在就可以找出圈子集套最大模糊圈同时计算出最大模糊势。     

On the sizes of large subgraphs of the binomial random graph

We consider the binomial random graph $G(n,p)$, where p is a constant, and answer the following two questions.First, given $e(k)=p\left(\begin{array}{c}k\\ 2\end{array}\right)+O(k)$, what is the maximum k such that a.a.s. the binomial random graph $G(n,p)$ has an induced subgraph with k vertices and $e(k)$ edges? We prove that this maximum is not concentrated in any finite set (in contrast to the case of a small $e(k)$). Moreover, for every constant $C>0$, with probability bounded away from 0, the size of the concentration set is bigger than $C\sqrt{n/\ln ?n}$, and, for every ${\omega }_n\to \infty$, a.a.s. it is smaller than ${\omega }_n\sqrt{n/\ln ?n}$.Second, given $k>\epsilon n$, what is the maximum μ such that a.a.s. the set of sizes of k-vertex subgraphs of $G(n,p)$ contains a full interval of length μ? The answer is $\mu =\mathrm{\Theta }\left(\sqrt{(n?k)n\ln ?\left(\begin{array}{c}n\\ k\end{array}\right)}\right)$.     

On the inducibility of oriented graphs on four vertices

We consider the problem of determining the inducibility (maximum possible asymptotic density of induced copies) of oriented graphs on four vertices. We provide exact values for more than half of the graphs, and very close lower and upper bounds for all the remaining ones. It occurs that, for some graphs, the structure of extremal constructions maximizing density of its induced copies is very sophisticated and complex.     

Large induced subgraphs with three repeated degrees

For any positive integer k, let $C(k)$ denote the least integer such that any n-vertex graph has an induced subgraph with at least $n?C(k)$ vertices, in which at least $\min ?\{k,n?C(k)\}$ vertices are of the same degree. Caro, Shapira and Yuster initially studied this parameter and showed that $\mathrm{\Omega }(k\log ?k)\le C(k)\le {(8k)}^k$. For the first nontrivial case, the authors proved that $3\le C(3)\le 6$, and the exact value was left as an open problem. In this paper, we first show that $3\le C(3)\le 4$, improving the former result as well as a recent result of Kogan. For special families of graphs, we prove that $C(3)=3$ for $K_5$-free graphs, and $C(3)=1$ for large $C_{2s+1}$-free graphs. In addition, extending a result of Erd?s, Fajtlowicz and Staton, we assert that every $K_r$-free graph is an induced subgraph of a $K_r$-free graph in which no degree occurs more than three times.     

Search for evidence of life in space: Analysis of enantiomeric organic molecules by N,N-dimethylformamide dimethylacetal derivative dependant Gas Chromatography–Mass Spectrometry

Within the context of the future space missions to Mars (MSL 2011 and Exomars 2016), which aim at searching for traces of life at the surface, the detection and quantitation of enantiomeric organic molecules is of major importance. In this work, we have developed and optimized a method to derivatize and analyze chiral organic molecules suitable for space experiments, using N,N-dimethylformamide dimethylacetal (DMF-DMA) as the derivatization agent. The temperature, duration of the derivatization reaction, and chromatographic separation parameters have been optimized to meet instrument design constraints imposed upon space experiment devices. This work demonstrates that, in addition to its intrinsic qualities, such as production of light-weight derivatives and a great resistance to drastic operating conditions, DMF-DMA facilitates simple and fast derivatization of organic compounds (three minutes at 140 °C in a single-step) that is suitable for an in situ analysis in space. By using DMF-DMA as the derivatization agent, we have successfully identified 19 of the 20 proteinic amino acids and been able to enantiomerically separate ten of the potential 19 (glycine being non-chiral). Additionally, we have minimized the percentage of racemized amino acid compounds produced by optimizing the conditions of the derivatization reaction itself. Quantitative linearity studies and the determination of the limit of detection show that the proposed method is also suitable for the quantitative determination of both enantiomeric forms of most of the tested amino acids, as limits of detection obtained are lower than the ppb level of organic molecules already detected in Martian meteorites.