On ideals and congruences in bcc-algebras

We introduce a new concept of ideals in BCC-algebras and describe connections between such ideals and congruences.     

Non-traceability of large connected claw-free graphs

Let G be a connected claw-free graph on n vertices. Let ς₃(G) be the minimum degree sum among triples of independent vertices in G. It is proved that if ς₃(G) ≥ n − 3 then G is traceable or else G is one of graphs G_n each of which comprises three disjoint nontrivial complete graphs joined together by three additional edges which induce a triangle K₃. Moreover, it is shown that for any integer k ≥ 4 there exists a positive integer ν(k) such that if ς₃(G) ≥ n − k, n > ν(k) and G is non-traceable, then G is a factor of a graph G_n. Consequently, the problem HAMILTONIAN PATH restricted to claw-free graphs G = (V, E) (which is known to be NP-complete) has linear time complexity O(|E|) provided that ς₃(G) ≥ . This contrasts sharply with known results on NP-completeness among dense graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 75–86, 1998     

A proof of uniqueness of the Gurariĭ space

We present a short and elementary proof of isometric uniqueness of the Gurari? space.     

Vibrational spectra of two BEDT-TTF-based organic conductors: charge order

Infrared and Raman investigations of two phases of bis(ethylenedithio)tetrathiafulvalene (BEDT-TTF) based organic conductors with the same CF₃CF₂SO₃⁻ anion: β′-(BEDT-TTF)₂CF₃CF₂SO₃ and δ′-(BEDT-TTF)₂CF₃CF₂SO₃, are shortly reviewed and compared with the most typical infrared properties of the family of (BEDT-TTF)₂RR′SO₃ organic conductors, where R = SF₅, CF₃, and R′ are CH₂, CF₂, CHF, CHFCF₂, and CH₂CF₂. The role of the molecular structur and spatial organization of the counterions is discussed. Presented at 2-nd International Conference on Functional Materials and Devices, ICFMD 2008, June 16–19, 2008, Kuala Lumpur, Malaysia     

A Note on the Game Chromatic Index of Graphs

In this note we prove that the game chromatic index χ _g ^′(G) of a graph G of arboricity k is at most Δ + 3k − 1. This improves a bound obtained by Cai and Zhu [J. Graph Theory 36 (2001), 144–155] for k-degenerate graphs. Tomasz Bartnicki: Research of the first author is supported by a PhD grant from Polish Ministry of Science and Higher Education N201 2128 33. Received: November 1, 2006. Final version received: December 22, 2007.     

Cost estimation algorithm and decision-making model for curriculum modification in educational organization

The decision about curriculum modification usually takes place at the knowledge level, mainly with consideration of individual academic staff competences and qualifications. However, traditional approaches to cost estimation of curriculum modification are focused on material resources only. In this paper we present a cost estimation method and decision model for curriculum modification in educational organizations. The proposed method works at the knowledge level and employs competence sets as knowledge representation models in educational organizations. Authors used the theory of hierarchical, multilevel systems in order to define the model of the decision-making process of curriculum modification and its dimension. Basing on this and also using a fuzzy competence model the cost estimation algorithm in the form of a group competences expansion algorithm is proposed. The algorithm focuses on the cost of staff competence expansion caused by the knowledge development process.     

Phosphorus complexes of N-fused porphyrin and its reduced derivatives: new isomers of porphyrin stabilized via coordination

N-fused isophlorin 3 and its tautomeric phlorin forms 4 and 5, the new constitutional isomers of porphyrin which preserve the basic skeleton of their maternal N-fused porphyrin, have been identified in the course of investigation of phosphorus insertion into N-fused porphyrin 2. N-fused porphyrin reacts with PCl3 in toluene yielding phosphorus(V) N-fused isophlorin 3-P wherein the macrocycle acts as a trianionic tridentate ligand. The identical product has been formed in the reaction of N-confused porphyrin 1 and POCl3 or PCl3. The coordinating environment of phosphorus(V) in 3-P as determined by X-ray crystallography resembles a distorted trigonal pyramid with the nitrogen atoms occupying equatorial positions with the oxygen atom lying at the unique apex. Phosphorus(V) is significantly displaced by 0.732(1) A from the N3 plane. The P-N distances are as follows P-N(22) 1.664(2), P-N(23) 1.645(2), and P-N(24) 1.672(2). All P-N(pyrrolic) bond lengths are markedly shorter than the P-N distances in phosphorus porphyrins. 3-P is susceptible to proton addition at the inner C(9) carbon atom, yielding aromatic 4-P. The modified macrocycle acts as a dianionic ligand and allows the efficient 18 pi-electron delocalization pathway. Two stereoisomers affording the syn (4-P syn) and anti (4-P anti) location of the H(9) atom with respect to the oxygen atom of the PO unit have been identified by (1)H NMR. A regioselective reduction of free base N-fused porphyrin 2 with NaBH4 yielded a nonaromatic isomer of 4, that is, N-fused phlorin 5 due to an addition of a hydride to the C(15) carbon and a proton to one of the pyrrolic nitrogens. The isomer 5 reacts with PCl 3 yielding phosphorus(V) fused isophlorin 3-P. Density functional theory has been applied to model the molecular and electronic structure of porphyrin isomers 3, 4, and 5 and their phosphorus(V) complexes.     

Rime samples characterization and comparison using classical and fuzzy principal components analysis

The main objective of this paper is to introduce principal component analysis and two robust fuzzy principal component algorithms as useful tools in characterizing and comparing rime samples collected in different locations in Poland (2004–2007). The efficiency of the applied procedures was illustrated on a data set containing 108 rime samples and concentration of anions, cations, HCHO, as well as pH and conductivity. The fuzzy principal component algorithms achieved better results mainly because they are more compressible than classical PCA and very robust to outliers. For example, a three component model, fuzzy principal component analysis-first component (FPCA-1) accounts for 62.37% of the total variance and fuzzy principal component analysis-orthogonal (FPCA-o) 90.11%; PCA accounts only for 58.30%. The first two principal components explain 51.41% of the total variance in the case of FPCA-1 and 79.59% in the case of FPCA-o as compared to only 47.55% for PCA. As a direct consequence, PCA showed only a partial differentiation of rime samples onto the plane or in the space described by different combination of two or three principal components, whereas a much sharper differentiation of the samples, regarding their origin and location, is observed when FPCAs are applied.      

Basis set and method dependence in quantum theory of atoms in molecules calculations for covalent bonds

The influence of various small- and medium-size basis sets used in Hartree-Fock (HF) and density functional theory (DFT)/B3LYP calculations on results of quantum theory of atoms in molecules based (QTAIM-based) analysis of bond parameters is investigated for several single, double, and triple covalent bonds. It is shown that, in general, HF and DFT/B3LYP methods give very similar QTAIM results with respect to the basis set. The smallest 6-31G basis set and DZ-quality basis sets of Dunning type lead to poor results in comparison to those obtained by the most reliable aug-cc-pVTZ. On the contrary, 6-311++G(2df,2pd) and in a somewhat lesser extent 6-311++G(3df,3pd) basis sets give satisfactory values of QTAIM parameters. It is also demonstrated that QTAIM calculations may be sensitive for the method and basis set in the case of multiple and more polarized bonds.