Yang-Baxter equation, parameter permutations, and the elliptic beta integral

We construct an infinite-dimensional solution of the Yang-Baxter equation (YBE) of rank 1 which is represented as an integral operator with an elliptic hypergeometric kernel acting in the space of functions of two complex variables. This R-operator intertwines the product of two standard L-operators associated with the Sklyanin algebra, an elliptic deformation of sl(2)-algebra. It is built from three basic operators $\mathrm{S}_1, \mathrm{S}_2$, and $\mathrm{S}_3$ generating the permutation group of four parameters $\mathfrak{S}_4$. Validity of the key Coxeter relations (including the star-triangle relation) is based on the elliptic beta integral evaluation formula and the Bailey lemma associated with an elliptic Fourier transformation. The operators $\mathrm{S}_j$ are determined uniquely with the help of the elliptic modular double.Comment: 43 pp., to appear in Russian Math. Survey

Finite-dimensional representations of the elliptic modular double

We investigate the kernel space of an integral operator M(g) depending on the "spin" g and describing an elliptic Fourier transformation. The operator M(g) is an intertwiner for the elliptic modular double formed from a pair of Sklyanin algebras with the parameters $\eta$ and $\tau$, Im$\tau>0$, Im$\eta>0$. For two-dimensional lattices $g=n\eta + m\tau/2$ and $g=1/2+n\eta + m\tau/2$ with incommensurate $1, 2\eta,\tau$ and integers $n,m>0$, the operator M(g) has a finite-dimensional kernel that consists of the products of theta functions with two different modular parameters and is invariant under the action of generators of the elliptic modular double.Comment: 25 pp., published versio

New elliptic solutions of the Yang-Baxter equation

We consider finite-dimensional reductions of an integral operator with the elliptic hypergeometric kernel describing the most general known solution of the Yang-Baxter equation with a rank 1 symmetry algebra. The reduced R-operators reproduce at their bottom the standard Baxter's R-matrix for the 8-vertex model and Sklyanin's L-operator. The general formula has a remarkably compact form and yields new elliptic solutions of the Yang-Baxter equation based on the finite-dimensional representations of the elliptic modular double. The same result is also derived using the fusion formalism.Comment: 34 pages, to appear in Commun. Math. Phy

From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

We start from known solutions of the Yang-Baxter equation with a spectral parameter defined on the tensor product of two infinite-dimensional principal series representations of the group $\mathrm{SL}(2,\mathbb{C})$ or Faddeev's modular double. Then we describe its restriction to an irreducible finite-dimensional representation in one or both spaces. In this way we obtain very simple explicit formulas embracing rational and trigonometric finite-dimensional solutions of the Yang-Baxter equation. Finally, we construct these finite-dimensional solutions by means of the fusion procedure and find a nice agreement between two approaches

Reconstruction of dielectric constants of multi-layered optical fibers using propagation constants measurements

We present new method for the numerical reconstruction of the variable refractive index of multi-layered circular weakly guiding dielectric waveguides using the measurements of the propagation constants of their eigenwaves. Our numerical examples show stable reconstruction of the dielectric permittivity function $\varepsilon$ for random noise level using these measurements

Discrete Darboux transformation for discrete polynomials of hypergeometric type

Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn)

Determinants of elliptic hypergeometric integrals

We start from an interpretation of the BC_(2)-symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation and then generalize this construction to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding formulas for the elliptic beta integral and symmetry transformation in a new way, by proving that both sides satisfy the same difference equations and that these difference equations satisfy a needed Galois-theoretic condition ensuring the uniqueness of the simultaneous solution

Modeling of pipe-drawing tool for drawing the multifaceted pipes of nonferrous metals on an immediate arbor

A method of mathematical modeling of a pipe-drawing tool for drawing the multifaceted pipes of nonferrous metals and alloys using the vector-matrix apparatus, which can be applied for the analytical description of the bulk deformation region, is presented. Arbors with various geometries of the reduction zone are considered. As a result of modeling the deformation region, which appears when manufacturing the profiled multifaceted pipes by arbor drawing using all types of considered arbors, it is established that the best result with the smallest rounding radii is attained for arbors with a pyramidal input into the reduction zone. © 2013 Allerton Press, Inc

Molecular taxonomic study of Trichinella spp. from mammals of Russian Arctic and subarctic areas

Analysis of taxonomic affiliation of Trichinella species circulating in the Chukotka Autonomous Region and some subarctic areas of the Russian Federation showed that the representatives of T. spiralis and the Arctic trichinellas - T. nativa (genotype T2) and Trichinella sp. (genotype T6) can be found there. The partial sequences of Coxb (704 bp) of these Arctic Trichinella spp. from Russia differ from Coxb sequences of those genotypes (T2 and T6) deposited in NCBI GenBank (1-3 bp). The cultivated larvae of Trichinella sp., which were established from muscular tissue sample of stray cat (shot on the fur farm in Chukotka peninslula) differ at molecular level (Coxb) even more significantly; 21-24 bp difference between Trichinella sp. and T. nativa and 46-47 bp difference between the same isolate and T. spiralis were recorded