Extremal spectral properties of Otsuki tori

Otsuki tori form a countable family of immersed minimal two‐dimensional tori in the unitary three‐dimensional sphere. According to the El Soufi‐Ilias theorem, the metrics on the Otsuki tori are extremal for some unknown eigenvalues of the Laplace‐Beltrami operator. Despite the fact that the Otsuki tori are defined in quite an implicit way, we find explicitly the numbers of the corresponding extremal eigenvalues. In particular we provide an extremal metric for the third eigenvalue of the torus.     

Four-body calculations of elastic scattering in H–H̄ collisions

We present a nonadiabatic treatment of the hydrogen-antihydrogen system. The technique used to describe H- H? collisions is based on the coupled rearrangement channels method. Within this approach the total, nonadiabatic wave function of the system is divided into two parts: an inner and an outer one. To describe the inner part a set of square-integrable 4-body functions is used. These functions are obtained by a diagonalization of the total Hamiltonian projected on a chosen L ² subspace, they explicitly contain components of various arrangement channels expressed in terms of corresponding Jacobi coordinates. The outer part of the total wave function reflects its asymptotic character. Our procedure leads to the system of non-local integro-differential equations that are solved iteratively and simultaneously determine both the shape of the outer part of the wave function and the coefficients in the four-body expansion of the inner part. Using this formalism we perform the one-channel calculation of the elastic scattering to obtain the S-matrix and nonadiabatic scattering length.     

Compensation of Magnetic Field Instabilities in Field Cycling NMR by Reference Deconvolution

Performing a nuclear magnetic resonance (NMR) experiment in an unstable magnetic field causes fluctuations in the NMR frequency, leading to a loss of reproducibility and an effective shortening of the free induction decay after data averaging. Reference deconvolution allows the compensation of field fluctuations via simultaneous measurement of an internal or external reference signal. The technique was applied to compensate the effect of field fluctuations in a resistive electromagnet used for fast field cycling NMR. An external sample was chosen as the reference.     

The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger–Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a central element, and can be seen as a deformation of the Lie algebra \({\mathfrak{u}(2)}\) . Two of the symmetry generators, J ₃ and J ₂, are respectively associated to the separation of variables in Cartesian and polar coordinates. Using the parabosonic creation/annihilation operators, two bases for the representations of sd(2), the Cartesian and circular bases, are constructed. In the Cartesian basis, the operator J ₃ is diagonal and the operator J ₂ acts in a tridiagonal fashion. In the circular basis, the operator J ₂ is block upper-triangular with all blocks 2 × 2 and the operator J ₃ acts in a tridiagonal fashion. The expansion coefficients between the two bases are given by the Krawtchouk polynomials. In the general case, the eigenvectors of J ₂ in the circular basis are generated by the Heun polynomials, and their components are expressed in terms of the para-Krawtchouk polynomials. In the fully isotropic case, the eigenvectors of J ₂ are generated by little ?1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J ₂ is diagonal is considered. In this basis, the defining relations of the Schwinger–Dunkl algebra imply that J ₃ acts in a block tridiagonal fashion with all blocks 2 × 2. The matrix elements of J ₃ in this basis are given explicitly.     

Spectral Duality Between Heisenberg Chain and Gaudin Model

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GL _k Heisenberg chain is dual to the special reduced k + 2-points gl _N Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.     

Log-Gamma Polymer Free Energy Fluctuations via a Fredholm Determinant Identity

We prove that under n ^1/3 scaling, the limiting distribution as n → ∞ of the free energy of Seppäläinen’s log-Gamma discrete directed polymer is GUE Tracy-Widom. The main technical innovation we provide is a general identity between a class of n-fold contour integrals and a class of Fredholm determinants. Applying this identity to the integral formula proved in Corwin et al. (Tropical combinatorics and Whittaker functions. http://arxiv.org/abs/1110.3489v3 [math.PR], 2012) for the Laplace transform of the log-Gamma polymer partition function, we arrive at a Fredholm determinant which lends itself to asymptotic analysis (and thus yields the free energy limit theorem). The Fredholm determinant was anticipated in Borodin and Corwin (Macdonald processes. http://arxiv.org/abs/1111.4408v3 [math.PR], 2012) via the formalism of Macdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach.     

Lift zonoid and barycentric representation on a Banach space with a cylinder measure

We show that the lift zonoid concept for a probability measure on $ {{\mathbb{R}}^d} $ , introduced in [G.A. Koshevoy and K. Mosler, Zonoid trimming for multivariate distributions, Ann. Stat., 25(5):1998–2017, 1997], naturally leads to a oneto-one representation of any interior point of the convex hull of the support of a continuous measure as the barycenter w.r.t. this measure of either a half-space or the whole space. We prove an infinite-dimensional generalization of this representation, which is based on the extension of the concept of lift zonoid for a cylindrical probability measure.     

On Using the Join Operation to Define Classes of Algebras

We call a class of algebras a finitary prevariety if the class is closed under the formation of subalgebras and finitary direct products, and contains the one-element algebra. The join of two finitary prevarieties and a concept of a join-irreducible finitary prevariety may be introduced naturally. We develop techniques for proving that a finitary prevariety of semigroups is join-irreducible, and find many examples of join-irreducible finitary prevarieties of semigroups. For example, we prove that if a class of finite semigroups is defined by ω-identities and contains the class J, then it is a join-irreducible finitary prevariety.