Invariant Tensors Formulae via Chord Diagrams

We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra $sl(n)$, as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint representations of the orthogonal and symplectic Lie algebras $so(n)$ and $sp(n)$.Comment: 18 pages, 8 figures. To appear in a special issue of Journal of Mathematical Science

Photography principle, data transmission, and invariants of manifolds

In the present paper we develop the techniques suggested in \cite{ManturovNikonov} and the photography principle \cite{ManturovWan} to open a very broad path for constructing invariants for manifolds of dimensions greater than or equal to 4

Describing semigroups with defining relations of the form xy=yz xy and yx=zy and connections with knot theory

We introduce a knot semigroup as a cancellative semigroup whose defining relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we define it is closely related to such tools of knot theory as the twofold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T(2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally defined factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research

Erhöhung der Zuverlässigkeit der Bestimmung der Neutronenbelastung von WWER-Reaktorkomponenten zwecks Ableitung von Vorschlägen für eine sicherere Betriebsführung von WWER-Reaktoren

The results of a project sponsored by the German Bundesministerium fuer Bildung, Wissenschaft, Forschung und Technologie are presented. The Project aimed to improve the safety against embrittlement of VVER-1000 type reactors by a more reliable and accurate determination of the neutron load of reactor pressure vessels. Therefore, six scientist from three Russian research institutions were sponsored to support with their work another BMBF project of the FZR aimed at the same goal. By providing reliable data for the evaluation of ex-vessel neutron activation experiments at two VVER-1000 and formulating the corresponding reactor models a basis has been established for further investigations as well in the FZR as well as in several Russian and Western research institutions. The leading Russian nuclear data library ABBN/MULTIC has been improved and tested. The uncertainties affecting the calculations of the fluence spectrum at the outer boundary of the pressure vessel have been analysed and a spectrum covariance matrix has been derived. The methodologies for the experimental determination of activation rates and for calculations of fluence spectra and activation rates have been further developed and tested by interlaboratoy comparisons. Measurements of different laboratories were compared with each other, as well as the corresponding calculations. Moreover, measurements and calculations were compared against each other, partly with participation of further Russian, Czech and Western institutes. The results of the Intercomparisons have been evaluated by the "International Workshop on the Balakovo-3 Interlaboratory Dosimetry Experiment" in September 1997 in Rossendorf. As a result of these works a better evaluation of the reached accuracies was possible and proposals for an improvement of the used methods could be derived

Knotting probabilities after a local strand passage in unknotted self-avoiding polygons

We investigate the knotting probability after a local strand passage is performed in an unknotted self-avoiding polygon on the simple cubic lattice. We assume that two polygon segments have already been brought close together for the purpose of performing a strand passage, and model this using Theta-SAPs, polygons that contain the pattern Theta at a fixed location. It is proved that the number of n-edge Theta-SAPs grows exponentially (with n) at the same rate as the total number of n-edge unknotted self-avoiding polygons, and that the same holds for subsets of n-edge Theta-SAPs that yield a specific after-strand-passage knot-type. Thus the probability of a given after-strand-passage knot-type does not grow (or decay) exponentially with n, and we conjecture that instead it approaches a knot-type dependent amplitude ratio lying strictly between 0 and 1. This is supported by critical exponent estimates obtained from a new maximum likelihood method for Theta-SAPs that are generated by a composite (aka multiple) Markov Chain Monte Carlo BFACF algorithm. We also give strong numerical evidence that the after-strand-passage knotting probability depends on the local structure around the strand passage site. Considering both the local structure and the crossing-sign at the strand passage site, we observe that the more "compact" the local structure, the less likely the after-strand-passage polygon is to be knotted. This trend is consistent with results from other strand-passage models, however, we are the first to note the influence of the crossing-sign information. Two measures of "compactness" are used: the size of a smallest polygon that contains the structure and the structure's "opening" angle. The opening angle definition is consistent with one that is measurable from single molecule DNA experiments.Comment: 31 pages, 12 figures, submitted to Journal of Physics