Chebyshev polynomials on symmetric matrices

In this paper we evaluate Chebyshev polynomials of the second kind on a class of symmetric integer matrices, namely on adjacency matrices of simply laced Dynkin and extended Dynkin diagrams. As an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type.     

Diagrams and torsors

Suppose that A is a presheaf of categories enriched in simplicial sets on a small Grothendieck site. It is shown that the homotopy theory of enriched A-diagrams of equivalences taking values in simplicial sets can be identified with the homotopy theory of simplicial presheaves fibred over the diagonalized nerve dBA of A. The set [*,dBA] of morphisms in the simplicial presheaf homotopy category is identified with the set of path components of a category of A-torsors, for a suitable definition of A-torsor. These statements are special cases of localized results which hold when the corresponding localized model structures are proper. Examples of the latter include the motivic homotopy category, and so these results lead to a theory of motivic A-torsors which is classifiable up to equivalence by a family of morphisms in the motivic homotopy category. This research was supported by NSERC. Received: March 2006     

Phase Diagrams and Synthesis of Diamond

For reasons of phase equilibria, the lowest temperatures T min , above which at high pressures the diamond crystallization from melt solutions is allowable in terms of thermodynamics, have been found for a number of metal-carbon systems. In the Ta-C and Nb-C systems, the diamond synthesis is possible at temperatures below T min , while to synthesize diamond in the Mg-Zn-C system, the temperatures much higher than T min , are required because of the necessity to overcome the kinetic difficulties.     

QSARs for identifying and prioritizing substances with persistence and bioconcentration potential

From the 8511 chemicals with 1998 production volumes reported to the U.S. Environmental Protection Agency (U.S. EPA), the TSCA Interagency Testing Committee's (ITC's) Degradation Effects Bioconcentration Information Testing Strategies (DEBITS) was used to identify 56 chemicals. The DEBITS Quantitative Structure-Activity Relationships (QSARs) and the U.S. EPA's PBT profiler QSARs were used to predict the persistence and bioconcentration factors of these 56 chemicals. Partial order ranking was used to prioritise the chemicals based on persistence and bioconcentration potential.     

Minimal Cantor Systems and Unimodal Maps

We consider the restriction of unimodal maps f to the omega-limit set y ( c ) of the critical point for certain cases where y ( c ) is a Minimal Cantor set. We investigate the relation of these minimal systems to enumeration scales (generalized adding machines), to Vershik adic transformations on ordered Bratelli diagrams and to substitution shifts. Sufficient conditions are given for ( y ( c ), f ) to be uniquely ergodic.     

Zur thermischen Zersetzung von TeSeO4 und Te3SeO8 – Koexistenzbeziehungen im ternären System TeO2/SeO2/Bi2SeO5

Thermal Decomposition of TeSeO₄ and Te₃SeO₈ – Phase Relations in the Ternary System TeO₂/SeO₂/Bi₂SeO₅ The saturation decomposition pressure of TeSeO₄ and Te₃SeO₈ were determined in a membran zero manometer. From the equilibrium data are derived the Enthalpies of formation and the Standard Entropies: Data see Inhaltsübersicht. Thus the coexistence ranges in the ternary triangle TeO₂/SeO₂/Bi₂SeO₅ can be determined. Informations about the melting diagram obtained by thermal analysis of the condensed compositions TeO₂/SeO₂.     

Improvements to Variable Elimination and Symbolic Probabilistic Inference for evaluating Influence Diagrams

An Influence Diagram is a probabilistic graphical model used to represent and solve decision problems under uncertainty. Its evaluation requires performing several combinations and marginalizations on the potentials attached to the Influence Diagram. Finding an optimal order for these operations, which is NP-hard, is an element of crucial importance for the efficiency of the evaluation. In this paper, two methods for optimizing this order are proposed. The first one is an improvement of the Variable Elimination algorithm while the second is the adaptation of the Symbolic Probabilistic Inference for evaluating Influence Diagrams. Both algorithms can be used for the direct evaluation of IDs but also for the computation of clique-to-clique messages in Lazy Evaluation of Influence Diagrams. In the experimental work, the efficiency of these algorithms is tested with several Influence Diagrams from the literature.     

Ein neues Oxoberyllat: K2Na4[O2BeOBeO2]

On K₂Na₄[O₂BeOBeO₂] For the first time colourless single crystals of K₂Na₄[Be₂O₅] which are isotypic with K₂Na₄[Co₂O₅] [2] and Rb₂Na₄[Co₂O₅] have been prepared by heating e.g. a well ground mixture of K₂O, Na₂O and BeO (K:Na:Be = 2.2:4.4:2; 750°C; 21 d; Ni-tube). The crystal structure was solved by four-circle diffractometer data [Siemens AED 2; 389 I_o(hkl); space group P4₂/mnm; Z = 2; a = 595.17(4) pm, c = 1 002.66(9) pm; R = 4.1%; R_w = 2.3%]. Mean Fictive Ionic Radii, MEFIR, Effective Coordination Numbers, ECoN, the Madelung Part of Lattice Energy, MAPLE, and the charge distribution are calculated.     

Diagrams,Tensors and Geometric Reasoning

Geometry and in particular projective geometry (and its corresponding invariant theory) deals a lot with structural properties of geometric objects and their interrelations. This papers describes how concepts of tensor calculus can be used to express geometric invariants and how, in particular, diagrammatic notation can be used to deal with invariants in a highly intuitive way. In particular we explain how geometries like euclidean or spherical geometry can be dealt with in this framework. Dedicated to the memory of Victor Klee, and in particular to his striving for conceptual simplicity     

Blob algebra and two-color Soergel calculus

In 2003, Martin and Woodcock noticed a connection between the representation theory of the blob algebra and the Kazhdan–Lusztig polynomials associated with the infinite dihedral group. However, no conceptual explanation for this coincidence has yet been provided. In this study, a possible explanation of this phenomenon is suggested by enunciating a conjecture that relates the endomorphism algebra of Bott–Samelson bimodules to certain subalgebras of the blob algebra obtained by idempotent truncation. Evidence supporting this conjecture is provided.