Set Partition Complexes

The Hom complexes were introduced by Lovász to study topological obstructions to graph colorings. The vertices of Hom(G,K _n ) are the n-colorings of the graph G, and a graph coloring is a partition of the vertex set into independent sets. Replacing the independence condition with any hereditary condition defines a set partition complex. We show how coloring questions arising from, for example, Ramsey theory can be formulated with set partition complexes. It was conjectured by Babson and Kozlov, and proved by Čukić and Kozlov, that Hom(G,K _n ) is (n−d−2)-connected, where d is the maximal degree of a vertex of G. We generalize this to set partition complexes.     

Epidifferentiability of the map of infima in Hilbert spaces

This article deals with derivatives for set-valued maps that take values in ordered vector spaces, in particular it concerns about the relationship between the epiderivatives of a set-valued map and its associated map of infima. When the image space is a real separable Hilbert space ordered by an orthonormal basis, by using a variational technique based on a decoupling of the ordering cone into half-spaces, we show that both epiderivatives coincide under certain hypothesis of compactness and stability. Furthermore we obtain some computation formulas for these derivatives in terms of associated scalar set-valued maps.     

A major index for matchings and set partitions

We introduce a statistic pmaj(P) for partitions of [n], and show that it is equidistributed with cr₂, the number of 2-crossings, over all partitions of [n] with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inv and maj.     

Fixed point theorems for generalized contractions in ordered metric spaces

The purpose of this paper is to present some fixed point results for self-generalized contractions in ordered metric spaces. Our results generalize and extend some recent results of A.C.M. Ran, M.C. Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443], J.J. Nieto, R. Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], J.J. Nieto, R.L. Pouso, R. Rodríguez-López [J.J. Nieto, R.L. Pouso, R. Rodríguez-López, Fixed point theorem theorems in ordered abstract sets, Proc. Amer. Math. Soc. 135 (2007) 2505-2517], A. Petru?el, I.A. Rus [A. Petru?el, I.A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411-418] and R.P. Agarwal, M.A. El-Gebeily, D. O'Regan [R.P. Agarwal, M.A. El-Gebeily, D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., in press]. As applications, existence and uniqueness results for Fredholm and Volterra type integral equations are given.     

环上的模糊理想幂格及其强截集诱导的格

给出了环上模糊理想幂格的概念,得到相关性质。进而由环上模糊理想强截集诱导出格,讨论了它不是分配格,但为模格。同时,将环上升为主理想环,得到了模糊理想乘积的强截集等于强截集的乘积,且都为环的理想。最后,讨论了Boolean环的模糊子集为模糊理想与其伴随构成交半格的关系。     

Colorful flowers

For a set A let k[A] denote the family of all k-element subsets of A. A function f:k[A]→C is a local coloring if it maps disjoint sets of A into different elements of C. A family F⊆k[A] is called a flower if there exists E∈[A]^k−1 so that |F∩F^′|=E for all F,F^′∈F, F≠F^′. A flower is said to be colorful if f(F)≠f(F^′) for any two F,F^′∈F. In the paper we find the smallest cardinal γ such that there exists a local coloring of k[A] containing no colorful flower of size γ. As a consequence we answer a question raised by Pelant, Holický and Kalenda. We also discuss a few results and conjectures concerning a generalization of this problem.     

Fully ordered groups in partially ordered linear algebras

No abstract. Received September 2, 1999; accepted in final form October 23, 2000.     

A fuzzy-stochastic OWA model for robust multi-criteria decision making

All realistic Multi-Criteria Decision Making (MCDM) problems face various kinds of uncertainty. Since the evaluations of alternatives with respect to the criteria are uncertain they will be assumed to have stochastic nature. To obtain the uncertain optimism degree of the decision maker fuzzy linguistic quantifiers will be used. Then a new approach for fuzzy-stochastic modeling of MCDM problems will be introduced by merging the stochastic and fuzzy approaches into the OWA operator. The results of the new approach, entitled FSOWA, give the expected value and the variance of the combined goodness measure for each alternative. Robust decision depends on the combined goodness measures of alternatives and also on the variations of these measures under uncertainty. In order to combine these two characteristics a composite goodness measure will be defined. The theoretical results will be illustrated in a watershed management problem. By using this measure will give more sensitive decisions to the stakeholders whose optimism degrees are different than that of the decision maker. FSOWA can be used for robust decision making on the competitive alternatives under uncertainty.     

Shadows and intersections: Stability and new proofs

We give a short new proof of a version of the Kruskal-Katona theorem due to Lovász. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lovász's theorem that answers a question of Frankl and Tokushige.     

一种“冻地”鸡血石相似玉石的宝石学及谱学研究

研究对象是一种与“冻地”鸡血石外观高度相似的玉石,该种玉石半透明“地”中含有橙红色矿物。利用X射线粉晶衍射仪、扫描电子显微镜、红外光谱仪、拉曼光谱仪对该玉石的宝石学及谱学特征进行研究。结果表明：该玉石“地”的主要组成矿物为有序度较高的地开石、橙红色矿物为雄黄;地开石晶体为自形假六边形片状,约15～20 μm,厚2～4 μm,粒径均一且形态一致,集合体在三维空间无序排列;部分样品“地”中含有少量黄铁矿、萤石、石英、方解石等矿物。“地”的红外光谱指纹区具有高岭石族矿物的主要特征峰,分别位于430,470,540,698,755,795,913,937,1 002,1 034和1 118 cm-1;官能团区以3 622,3 653和3 706 cm-1处的吸收峰为特征,3 622 cm-1吸收峰由内羟基OH1的面内伸缩振动引起,3 653 cm-1归属于内表面羟基OH2和OH4的同相伸缩振动;从高频峰到低频峰强度依次增大,且内表面羟基OH3伸缩振动引起的吸收峰位于3 706 cm-1,表明“地”为有序地开石;拉曼光谱测试表明“血”为雄黄,具有186,222,235,273,346和355 cm-1的特征拉曼位移,其中186和222 cm-1归属于S-As-S的弯曲振动,346和355 cm-1由As-S的伸缩振动引起;拉曼光谱同样可用于“地”的矿物组成研究,低频区具有133,241,266,336,436,463,747,792和914 cm-1的高岭石族矿物的特征位移,高频区可见三个与红外光谱相似的阶梯状谱峰,3 624 cm-1强度最大,归属于OH1的伸缩振动,次强峰3 646 cm-1由OH2和OH4的同相伸缩振动引起,归属于OH3的伸缩振动峰强度最小且位于3 706 cm-1,高频区拉曼位移特点指示“地”为地开石,且有序度较高,与红外光谱测试结论一致。尽管研究样品的“地”与“冻地”鸡血石的主要组成矿物相同,为地开石,且具有外观细腻、温润等特点,但其“血”并非辰砂而是雄黄,所以不应与鸡血石混淆,其正确的珠宝玉石名称应为“粘土矿物质玉”。