Abstract:
$\beta$-skeletons, a prominent member of the neighborhood graph family, have interesting geometric properties and various applications ranging from geographic networks to archeology. This paper focuses on developing a new, more general than the present one, definition of $\beta$-skeletons based only on the distance criterion. It allows us to consider them in many different cases, e.g. for weighted graphs or objects other than points. Two types of $\beta$-skeletons are especially well-known: the Gabriel Graph (for $\beta = 1$) and the Relative Neighborhood Graph (for $\beta = 2$). The new definition retains relations between those graphs and the other well-known ones (minimum spanning tree and Delaunay triangulation). We also show several new algorithms finding $\beta$-skeletons.

Abstract:
Semiclassical wave functions in billiards based on the Maslov-Fedoriuk approach are constructed. They are defined on classical constructions called skeletons which are the billiards generalization of the Arnold tori. Skeletons in the rational polygon billiards considered in the phase space can be closed with a definite genus or can be open being a cylinder-like or Moebius-like bands. The skeleton formulation is applied to calculate semiclassical wave functions and the corresponding energy spectra for the integrable and pseudointegrable billiards as well as in the limiting forms in some cases of chaotic ones. The superscars of Bogomolny and Schmit are shown to be simply singular semiclassical solutions of the eigenvalue problem in the billiards well built on the singular skeletons in the billiards with flat boundaries in both the integrable and the pseudointegrable billiards as well as in the chaotic cases of such billiards.

Abstract:
Well known from the sixties, the pressure of e.g. massless phi-four theory may be written as a series of 2PI-diagrams (skeletons) with the lines fully dressed. Varying the self-energy Pi in this expression, it turns into a functional U[Y] having a maximum in function space at Y=Pi. There is also the Feynman-Jensen thermal variational principle V[S], a potentially non-perturbative tool. Here actions S are varied. It is shown, through a few formal but exact steps, that the functional U is covered by V. The corresponding special subset of trial actions is made explicit.

Abstract:
Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model $\mathscr X$ of $X$ plus a suitable horizontal divisor $H$ a skeleton $S(\mathscr X,H)$ in the analytification of $X$. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H. It also generalizes constructions by Tyomkin and Baker--Payne--Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on $S(\mathscr X, H)$. For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a well-known result by Sturmfels--Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.

Abstract:
four steroidal saponins and three sapogenins were identified from aerial parts of brachiaria decumbens. their structures were established by chemical and spectroscopic means (1h and 13c nmr, hmbc, hmqc) as 3b-methoxy-lanost-9(11)-ene (1), diosgenin (2a), yamogenin (2b), 3-o-b-d-glucopyranosyl-24(s)-ethyl-22e -dihydrocholesterol (3a), 3-o-b-d-glucopyranosyl-24(r)-ethyl-22e -dihydrocholesterol (3b), dioscin (4a) and 3-o-{a-l-rhamnopyranosyl-(1？4)-[a-l-rhamnopyranosyl-(1 ？2)]-b-d-glucopyranosyl}-25(s)-spirost-5-en-3 b-ol (4b). all these compounds were isolated for the first time from b. decumbens and compound 4b is described for the first time as far as we know.

Abstract:
Four steroidal saponins and three sapogenins were identified from aerial parts of Brachiaria decumbens. Their structures were established by chemical and spectroscopic means (1H and 13C NMR, HMBC, HMQC) as 3beta-methoxy-lanost-9(11)-ene (1), diosgenin (2a), yamogenin (2b), 3-O-beta-D-glucopyranosyl-24(S)-ethyl-22E -dihydrocholesterol (3a), 3-O-beta-D-glucopyranosyl-24(R)-ethyl-22E -dihydrocholesterol (3b), dioscin (4a) and 3-O-{alpha-L-rhamnopyranosyl-(1->4)-[alpha-L-rhamnopyranosyl-(1 ->2)]-beta-D-glucopyranosyl}-25(S)-spirost-5-en-3 beta-ol (4b). All these compounds were isolated for the first time from B. decumbens and compound 4b is described for the first time as far as we know.

Abstract:
In analogy to the skeletons of a simplicial complex and their Stanley--Reisner ideals we introduce the skeletons of an arbitrary monomial ideal $I\subset S=K[x_1,...,x_n]$. This allows us to compute the depth of $S/I$ in terms of its skeleton ideals. We apply these techniques to show that Stanley's conjecture on Stanley decompositions of $S/I$ holds provided it holds whenever $S/I$ is Cohen--Macaulay. We also discuss a conjecture of Soleyman-Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution.

Abstract:
We survey a collection of closely related methods for generalizing fans of toric varieties, include skeletons, Kato fans, Artin fans, and polyhedral cone complexes, all of which apply in the wider context of logarithmic geometry. Under appropriate assumptions these structures are equivalent, but their different realizations have provided for surprisingly disparate uses. We highlight several current applications and suggest some future possibilities.

Abstract:
构建了α-半乳糖苷酶（Mel1p）的多拷贝质粒pRS426-TEFpr-MEL1，转入酿酒酵母野生型AN120及osw2Δ突变株并进行固定化酶活性测定。实验分别选取蜜二糖、棉子糖（三糖）与水苏糖（四糖）3种低聚寡糖作为Mel1p的底物，检测两种菌株孢子固定化酶对这3种底物的催化活性，结果表明野生型AN120酵母孢子壁的孔径最大只允许介于三糖和四糖之间的底物通过，四糖不能通过；而osw2?驻突变株酵母孢子壁的孔径可以允许四糖通过。通过孢子固定化酶保护性实验，发现孢子固定化酶可以完全抵抗2% SDS的处理，也可部分抵抗10% SDS的处理。通过8 mol/L尿素处理后，酶活检测发现野生型AN120酵母孢子可阻挡尿素对孢子固定化酶的影响，而osw2Δ孢子不能；蛋白免疫印迹分析进一步表明，尿素处理后野生型AN120酵母孢子固定化酶的含量几乎不变，而osw2Δ突变株孢子固定化酶的含量明显降低，说明尿素可进入osw2Δ突变株孢子内部溶解并降低固定化酶的含量，但对AN120酵母孢子影响。本研究也在多种突变株孢子中表达Mel1p，并进行活性比较，发现osw2，osw2Δydl186wΔ及osw2Δydr326cΔ突变株固定化酶均有很高的催化活性。 In this study，the multicopy plasmid pRS426-TEFpr-MEL1 expressing α- galactosidase enzyme（Mel1p） was transformed into the Saccharomyces cerevisiae AN120 wild type and osw2Δ mutant strains. Three different oligosaccharides（melibiose，raffinose and stachyose） were used as substrates to detect the catalytic activity of immobilized enzyme on yeast spores. The results indicated that the pore size of AN120 wild-type spore wall was larger than trisaccharide but smaller than tetrasaccharide；however，the pore size of osw2Δ mutant yeast spore wall was larger than tetrasaccharide. The protective assay of immobilized enzyme suggested that enzyme immobilized on wild type and osw2?驻 mutant spores can completely resist 2% SDS treatment and partially resist 10% SDS treatment. Wild type spore can resist the treatment by 8 mol/L urea，whereas osw2Δ mutant spore was sensitive to urea.Furthermore，western blot analysis showed that the amounts of enzyme in the wild type spores were stable after urea treatment，but the amounts of enzyme onosw2Δ mutant spores decreased significantly. These results demonstrated that urea could enter into osw2Δ mutant spores and dissolve the enzyme，leading to the decrease of the amount of enzyme. Moreover，the immobilized enzymes on osw2Δ，osw2Δydl186Δand osw2Δydr326cΔspores all have high catalytic activity