Parafermionic algebras, their modules and cohomologies

We explore the Fock spaces of the parafermionic algebra introduced by H.S. Green. Each parafermionic Fock space allows for a free minimal resolution by graded modules of the graded 2-step nilpotent subalgebra of the parafermionic creation operators. Such a free resolution is constructed with the help of a classical Kostant's theorem computing Lie algebra cohomologies of the nilpotent subalgebra with values in the parafermionic Fock space. The Euler-Poincar\'e characteristics of the parafermionic Fock space free resolution yields some interesting identities between Schur polynomials. Finally we briefly comment on parabosonic and general parastatistics Fock spaces.Comment: 10 pages, talk presented at the International Workshop "Lie theory and its applications in Physics" (17-23 June 2013, Varna, Bulgaria

Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions

We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the occasion of his 65th birthday; v2: fixed some typos and added reference

Schur Q-functions and degeneracy locus formulas for morphisms with symmetries

We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle, and is based on a certain push-forward formula for these polynomials in a Grassmann bundle.Comment: 22 pages, AMSTEX, misprints corrected, exposition improved. to appear in the Proceedings of Intersection Theory Conference in Bologna, "Progress in Mathematics", Birkhause

Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure

Artificial neural network (ANN) approach for modelling of pile settlement of open-ended steel piles subjected to compression load

This study was devoted to examine pile bearing capacity and to provide a reliable model to simulate pile load-settlement behaviour using a new artificial neural network (ANN) method. To achieve the planned aim, experimental pile load test were carried out on model open-ended steel piles, with pile aspect ratios of 12, 17, and 25. An optimised second-order Levenberg-Marquardt (LM) training algorithm has been used in this process. The piles were driven in three sand densities; dense, medium, and loose. A statistical analysis test was conducted to explore the relative importance and the statistical contribution (Beta and Sig) values of the independent variables on the model output. Pile effective length, pile flexural rigidity, applied load, sand-pile friction angle and pile aspect ratio have been identified to be the most effective parameters on model output. To demonstrate the effectiveness of the proposed algorithm, a graphical comparison was performed between the implemented algorithm and the most conventional pile capacity design approaches. The proficiency metric indicators demonstrated an outstanding agreement between the measured and predicted pile-load settlement, thus yielding a correlation coefficient (R) and root mean square error (RMSE) of 0.99, 0.043 respectively, with a relatively insignificant mean square error level (MSE) of 0.0019. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group

O pewnej hipotezie L. Jeśmanowicza dotyczącej liczb pitagorejskich

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Industrial Briquetting Trials for the Waste Management of Bearing Grinding

The paper presents the results of research on briquetting process carried out at production scale for the management of troublesome solid waste. These trials enabled to develop a method for continuous processing of large quantities of waste and implement a production line of iron containing briquettes. The material was collected from the landfill site located on the utilization company of waste taken from Bearing Factory – Kraśnik S.A. The following binders were added to the homogenized mixture of waste: molasses, starch and two-component adhesive – dry hydrated lime with molasses. Briquetting process was carried out in an industrial production line equipped with a crusher, screen, conveyor belt, gravity storage tank and roller-briquetting machine. Briquettes with the approximate dimensions of 50×20 mm were obtained. Their mechanical properties were determined in gravitational dump tests. The best briquettes were made of a mixture of waste with starch as binder in the mass fraction of 3%, wherein the moisture content was approx. 8%. The mechanical strength of briquettes with starch meets the established requirements at the moment of production; therefore, the time-consuming seasoning before shipment to the customer was unnecessary. We concluded that the briquettes meet the criteria for metallurgical processes, and can be added – together with scrap – as feedstock for melting in a steelmaking furnace