Schreier rewriting beyond the classical setting Dedicated to our teacher Alfred Lvovich Shmelkin on his 70th birthday

Using actions of free monoids and free associative algebras, we establish some Schreier-type formulas involving ranks of actions and ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish a generalization of the Schreier formula in the case of subgroups of infinite index. We also study and apply large modules over free associative and free group algebras.     

Analysis of ordered categorical data from repeated measurements assuming a quantitative latent variable

Summary The purpose of the present paper is to propose an analytical method for ordered categorical responses obtained from a repeated measurement/longitudinal experiment. The ordered categorical scale is assumed to be a manifestation of a latent quantitative variable. A linear model is assumed for location parameters of the underlying distributions. Weighted least square method is applied to parameter estimation and subsequent analysis. Two data sets are analyzed to show several aspects of analysis by the proposed model and to discuss comparative characteristics of analysis compared with earlier analysis. A mention is made for a computer software program for the proposed model.     

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.     

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.     

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.     

Axiomatizability and completeness of some classes of partially ordered polygons

We study partially ordered monoids over which a class of free (over sets and over posets), projective, and (strongly, weakly) flat partially ordered polygons is axiomatizable, complete, or model complete. Similar issues for polygons were dealt with in papers by V. Gould and A. Stepanova. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-2810.2008.1). Translated from Algebra i Logika, Vol. 48, No. 1, pp. 90–121, January–February, 2009.     

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”。     

自组装有序纳米银线表面增强拉曼光谱检测牛奶中三聚氰胺

制备具有表面增强拉曼散射(SERS)增强效果的基底是获得灵敏SERS检测的基础.本研究以多元醇法合成、流体流动法组装制备的有序纳米银线作为三聚氰胺的SERS增强基底,实现了样品中痕量三聚氰胺的快速灵敏检测.通过理论计算及实验考察得到了三聚氰胺的拉曼特征峰,优化了三聚氰胺在有序纳米银线基底上的SERS检测条件.在pH=8、水为溶剂、溶剂挥发时间为14 min的最佳条件下,三聚氰胺特征峰强度与浓度在0.05 ～ 1.00 mg/L范围内呈现良好的线性关系,其线性相关系数R=0.997,检测限为0.05 mg/L.在牛奶中添加不同浓度的三聚氰胺,其回收率为89.7％～ 109.2％,相对标准偏差低于6.8％.本方法对三聚氰胺检测具有很好的灵敏度和稳定性,为其它小分子化合物的SERS检测提供技术支持.