On deformation and classification of V-systems

The V-systems are special finite sets of covectors which appeared in the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known but their classification is an open problem. We derive the relations describing the infinitesimal deformations of V-systems and use them to study the classification problem for V-systems in dimension 3. We discuss also possible matroidal structures of V-systems in relation with projective geometry and give the catalogue of all known irreducible rank 3 V-systems.Comment: Slightly revised version, one of the figures correcte

Discrete Lagrangian systems on the Virasoro group and Camassa-Holm family

We show that the continuous limit of a wide natural class of the right-invariant discrete Lagrangian systems on the Virasoro group gives the family of integrable PDE's containing Camassa-Holm, Hunter-Saxton and Korteweg-de Vries equations. This family has been recently derived by Khesin and Misiolek as Euler equations on the Virasoro algebra for $H^1_{\alpha,\beta}$-metrics. Our result demonstrates a universal nature of these equations.Comment: 6 pages, no figures, AMS-LaTeX. Version 2: minor changes. Version 3: minor change

Elliptic Dunkl operators, root systems, and functional equations

We consider generalizations of Dunkl's differential-difference operators associated with groups generated by reflections. The commutativity condition is equivalent to certain functional equations. These equations are solved in many cases. In particular, solutions associated with elliptic curves are constructed. In the $A_{n-1}$ case, we discuss the relation with elliptic Calogero-Moser integrable $n$-body problems, and discuss the quantization ($q$-analogue) of our construction.Comment: 30 page

In search for a perfect shape of polyhedra: Buffon transformation

For an arbitrary polygon consider a new one by joining the centres of consecutive edges. Iteration of this procedure leads to a shape which is affine equivalent to a regular polygon. This regularisation effect is usually ascribed to Count Buffon (1707-1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine $B$-regular polyhedra. The main result is the proof of existence of star-shaped affine $B$-regular polyhedra with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdiere and Lovasz.Comment: Slightly revised version with added example of pentakis dodecahedro

Canonically conjugate variables for the periodic Camassa-Holm equation

The Camassa-Holm shallow water equation is known to be Hamiltonian with respect to two compatible Poisson brackets. A set of conjugate variables is constructed for both brackets using spectral theory.Comment: 10 pages, no figures, LaTeX; v. 2,3: references updated, minor change

Yang-Baxter maps and multi-field integrable lattice equations

A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice equations introduced by Adler and Yamilov and which are related to the nonlinear superposition formulae for the B\"acklund transformations of the nonlinear Schr\"odinger system and specific ferromagnetic models.Comment: 16 pages, 4 figures, corrected versio

Problems of Residential Building Ensuring Safe Operation with Wall Panels Made of the Autoclaved Aerated Concrete

AbstractThe article ‘Problems of Residential Building Ensuring Safe Operation with Wall Panels Made of the Autoclaved Aerated Concrete’ presents research methods for the estimation of the wear degree for lightweight aggregate concrete panels