AFPP vs FPP

The main theme of this paper is that almost fixed point properties of discrete structures and fixed point properties of (topological) spaces are interdeducible via a suitable category which contains both graphs and spaces as objects. To carry out the program, we have to consider (almost) fixed points of multifunctions, and for this we need a preliminary discussion of power structures for graphs and simplicial complexes. Specific applications developed are: a digital convexity (discrete) version of Kakutani's fixed point theorem for convex-valued multifunctions; and fixed point properties of dendrites in terms of those of finite discrete trees.     

Topological graph dimension

In the invited chapter Discrete Spatial Models of the book Handbook of Spatial Logics, we have introduced the concept of dimension for graphs, which is inspired by Evako’s idea of dimension of graphs [A.V. Evako, R. Kopperman, Y.V. Mukhin, Dimensional properties of graphs and digital spaces, J. Math. Imaging Vision 6 (1996) 109-119]. Our definition is analogous to that of (small inductive) dimension in topology. Besides the expected properties of isomorphism-invariance and monotonicity with respect to subgraph inclusion, it has the following distinctive features:

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Local aspect. That is, dimension at a vertex is basic, and the dimension of a graph is obtained as the sup over its vertices.

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Dimension of a strong product G×H is dim(G)+dim(H) (for non-empty graphs G,H).

In this paper we present a short account of the basic theory, with several new applications and results.     

Mixed Valence Chemistry of Pyrimidine Bridged Binuclear Complex of Pentacyanoferrate and Pentaammineruthenium

The pyrimidine bridged binuclear complex (CN)₅FepymRu(NH₃)₅- (I) was prepared in aqueous solution by mixing cquimolar of Fe(CN)₅OH₂^3? and Ru(NH₃)₅pym²⁺. Its mixed valence state molecule (CN)₅FepymRu(NH₃)₅(II) was obtained upon oxidation of I by one equivalent of peroxydisulfate ion. Both binuclear complexes and corresponding Fe(II) and Ru(II) mononuclear complexes displayed a metal-to-ligand charge transfer absorption in 400–450 nm region. Rate constants of formation and dissociation of I and II were measured, and the values of k_f (?10³M^?1s^?1) and k_d (?10^?3-10^?4 s^?1) were consistent with kinetic results expected for the substitution of Fe(CN)₅OH₂^3? with di- and trivalent ligands. Cyclic voltammetry of I exhibited two one-electron steps of oxidation corresponding to [III, L, II] + e → [II, L, II] and [III, L, III] + e → [III, L, II], respectively. The mixed valence binuclear complex II showed an intervalence band at 955 nm with a molar extinction coefficient 5.80 × 10² M^?1cm^?1 and a half-width 5100 cm^?l. The properties of the IT band conform to Hush's theory. Spectroscopic, electrochemical and kinetic results of II suggest that the mixed valence complex features a trapped - valence formulation with localized oxidation states of Fe(II) and Ru(III).     

Decision risk analysis for an interval TOPSIS method

TOPSIS is a multi-attribute decision making (MADM) technique for ranking and selection of a number of externally determined alternatives through distance measures. When the collected data for each criterion is interval and the risk attitude for a decision maker is unknown, we present a new TOPSIS method for normalizing the collected data and ranking the alternatives. The results show that the decision maker with different risk attitude ranks the different alternatives.     

Fuzzy portfolio model with different investor risk attitudes

We propose a fuzzy portfolio model designed for efficient portfolio selection with respect to uncertain or vague returns. Although many researchers have studied the fuzzy portfolio model, no researcher has yet attempted a behavioral analysis of the investor in the fuzzy portfolio model. To address this problem, we examined investor risk attitudes—risk-averse, risk-neutral, or risk-seeking behaviors—to discover an efficient method for fuzzy portfolio selection. In this study, we relied on the advantages of possibilistic mean–standard deviation models that we believed would fit the risk attitudes of investors. Thus, we developed a fuzzy portfolio model that focuses on different investor risk attitudes so that fuzzy portfolio selection for investors who possess different risk attitudes can be achieved more easily. Finally, we presented a numerical example of a portfolio selection problem to illustrate ways to address problems presented by a variety of investor risk attitudes.     

Electromagnetic plane-wave scattering from a sectorial groove in a perfectly conducting plane

Scattering characteristics of plane waves by a sectorial groove in a perfectly conducting plane are investigated. Both the transverse magnetic (TM) and transverse electric (TE) polarizations of the incident wave are considered. Judicious use of the region-matching technique provides a rigorous series solution to the problem. The analyzed region is separated into two sub-regions by choosing a semi-circular auxiliary boundary. Thefield in each sub-region is expanded as a summationof proper wave functions with unknown coefficients. Enforcing the matching of conditions on the auxiliary boundary and of boundary condition on the circular-arc surface of the groove leads to a linear set of equations and the unknown coefficients are then determined. Numerical results demonstrate the influence of central angles of the sectorial groove on echo width, far-field pattern and near-field distribution. The presented geometry is easily applicable to the design and fabrication of a grating structure for optical switches and tunable filters.      

Diffraction patterns for a transmission volume hologram under Bragg mismatch

We have presented simulations and experiments for 2-dimensional diffraction distributions of a transmission volume hologram. We sample the input pattern with a few elementary points and calculate the diffraction of the corresponding elementary grating. The simulations can describe well the diffraction pattern obtained in the experiment and can be used to predict the tolerance for angular deviation of the reading beam. The tolerance for angular deviation depends on its direction. It is affected by the Bragg degeneracy. The simulation results provide a clear picture of the spatial tolerance for reading a transmission volume hologram.     

Synthesis of analogs of 5(4)-aminoimidazole-4(5)-carboxamide and purines

4-Chloroimidazo[4,5-d]-1,2,3-triazine was obtained by oxidative chlorinatlon from imidazo[4,5-d]-1,2,3-triazine-4-thione, and its reaction with nucleophilic agents and cleavage of the triazine ring to give 5-diazoimidazole-4-carbonitrile were studied. The newly synthesized diazoimidazole was subjected to C- and N-diazo coupling. It was established that 3-methylimidazo[4,5-d]-1,2,3-triazine-4-imine, formed by reaction of the diazo compound with methylamine, is capable of recyclization to 4-methylaminoimidazo[4,5-d]-1,2,3-triazine.See [1] for communication IV.Translated from Khimiya Getereotsiklicheskikh Soedinenii, No. 4, pp. 556–559, April, 1976.