The star-triangle relation and 3d superconformal indices

Superconformal indices of 3d N=2 supersymmetric field theories are investigated from the Yang-Baxter equation point of view. Solutions of the star-triangle relation, vertex and IRF Yang-Baxter equations are expressed in terms of the q-special functions associated with these 3d indices. For a two-dimensional monopole-spin system on the square lattice a free energy per spin is explicitly determined. Similar to the partition functions, superconformal indices of 3d theories with the chiral symmetry breaking reduce to Dirac delta functions with the support on chemical potentials of the preserved flavor groups.Comment: 20 pages, v2: minor corrections, comments and refs. adde

Meromorphic Solutions to a Differential--Difference Equation Describing Certain Self-Similar Potentials

In this paper we prove the existence of meromorphic solutions to a nonlinear differential difference equation that describe certain self-similar potentials for the Schroedinger operator.Comment: 10 pages, LaTeX, uses additional package

Quasi-exactly solvable problems and the dual (q-)Hahn polynomials

A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little q-)Jacobi polynomials, and implications of this for quasi-exactly solvable problems are studied. A connection with the Azbel-Hofstadter problem is indicated.Comment: 15 pages, LaTex; final version, presentation improved, title changed, to appear in J.Math.Phy

Infinite elliptic hypergeometric series: convergence and difference equations

We derive finite difference equations of infinite order for theta hypergeometric series and investigate the space of their solutions. In general, such infinite series diverge, we describe some constraints on the parameters when they do converge. In particular, we lift the Hardy-Littlewood criterion on the convergence of $q$-hypergeometric series for $|q|=1, \, q^n\neq 1$, to the elliptic level and prove convergence of the infinite ${}_{r+1}V_r$ very-well poised elliptic hypergeometric series for restricted values of $q$.Comment: 24 pp, to appear in "Sbornik: Mathematics

SUBSTANTIATION OF THE METHOD OF MEAT SAMPLE PREPARATION FOR INSTRUMENTAL DETERMINATION OF CONSISTENCY

In recent years, there has been a transformation in the choice and lifestyle of Russian citizens. An increasing part of the population makes a more reasonable choice, taking into account all aspects, including the most important for meat products — these are organoleptic characteristics, in particular the consistency of the product. Consumers prefer products with the delicate, soft texture, expect good «biteness» and «cheweness». Therefore, preference is given to tender, juicy meat with a low content of connective tissue. The most commonly used method for testing meat consistency in laboratories around the world is a method that uses strength testing machines with a Warner-Bratzler blade (WB blade). In this work, the Shimadzu AGS-1kN universal testing machine (Japan) was chosen for research. Samples obtained from l. dorsi of pork and beef were selected as meat raw materials. To determine the optimal and reproducible method of sample preparation, some of them were subjected to heat treatment before analysis. In the study of samples without heat treatment, deviations from the average were more than 11%. After meat was cooked, a decrease in the relative standard deviation of the maximum shear stress from the average was achieved: from 11% in raw meat to 5% in a pork sample and 5.3% in a beef sample. The heat treatment of pre-cut samples led to a change in their geometric shape, which created additional difficulties for obtaining correct results, and also negatively affected the increase in the relative deviation to 15.5% for beef.In recent years, there has been a transformation in the choice and lifestyle of Russian citizens. An increasing part of the population makes a more reasonable choice, taking into account all aspects, including the most important for meat products — these are organoleptic characteristics, in particular the consistency of the product. Consumers prefer products with the delicate, soft texture, expect good «biteness» and «cheweness». Therefore, preference is given to tender, juicy meat with a low content of connective tissue. The most commonly used method for testing meat consistency in laboratories around the world is a method that uses strength testing machines with a Warner-Bratzler blade (WB blade). In this work, the Shimadzu AGS-1kN universal testing machine (Japan) was chosen for research. Samples obtained from l. dorsi of pork and beef were selected as meat raw materials. To determine the optimal and reproducible method of sample preparation, some of them were subjected to heat treatment before analysis. In the study of samples without heat treatment, deviations from the average were more than 11%. After meat was cooked, a decrease in the relative standard deviation of the maximum shear stress from the average was achieved: from 11% in raw meat to 5% in a pork sample and 5.3% in a beef sample. The heat treatment of pre-cut samples led to a change in their geometric shape, which created additional difficulties for obtaining correct results, and also negatively affected the increase in the relative deviation to 15.5% for beef

Solitons and Normal Random Matrices

We discuss a general relation between the solitons and statistical mechanics and show that the partition function of the normal random matrix model can be obtained from the multi-soliton solutions of the two-dimensional Toda lattice hierarchy in a special limit

Koszul-Tate Cohomology For an Sp(2)-Covariant Quantization of Gauge Theories with Linearly Dependent Generators

The anti-BRST transformation, in its Sp(2)-symmetric version, for the general case of any stage-reducible gauge theories is implemented in the usual BV approach. This task is accomplished not by duplicating the gauge symmetries but rather by duplicating all fields and antifields of the theory and by imposing the acyclicity of the Koszul-Tate differential. In this way the Sp(2)-covariant quantization can be realised in the standard BV approach and its equivalence with BLT quantization can be proven by a special gauge fixing procedure.Comment: 13 pages, Latex, To Be Published in International Journal of Modern Physics

Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework

In the context of a two-parameter $(\alpha, \beta)$ deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using techniques of supersymmetric quantum mechanics combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters $\alpha$, $\beta$ vanishes, in which case shape invariance under parameter translation occurs. In the special case where $\alpha = \beta \ne 0$, the oscillator Hamiltonian is shown to coincide with that of the q-deformed oscillator with $q > 1$ and its eigenvectors are therefore $n$-$q$-boson states. In the general case where $0 \ne \alpha \ne \beta \ne 0$, the eigenvectors are constructed as linear combinations of $n$-$q$-boson states by resorting to a Bargmann representation of the latter and to $q$-differential calculus. They are finally expressed in terms of a $q$-exponential and little $q$-Jacobi polynomials.Comment: LaTeX, 24 pages, no figure, minor changes, additional references, final version to be published in JP

Localization of N=4 Superconformal Field Theory on S^1 x S^3 and Index

We provide the geometrical meaning of the ${\cal N}=4$ superconformal index. With this interpretation, the ${\cal N}=4$ superconformal index can be realized as the partition function on a Scherk-Schwarz deformed background. We apply the localization method in TQFT to compute the deformed partition function since the deformed action can be written as a $\delta_\epsilon$-exact form. The critical points of the deformed action turn out to be the space of flat connections which are, in fact, zero modes of the gauge field. The one-loop evaluation over the space of flat connections reduces to the matrix integral by which the ${\cal N}=4$ superconformal index is expressed.Comment: 42+1 pages, 2 figures, JHEP style: v1.2.3 minor corrections, v4 major revision, conclusions essentially unchanged, v5 published versio