Vol 201, No 1 (2019)

We explicitly describe solutions of the noncommutative unitary U (1) sigma model that represent finitedimensional perturbations of the identity operator and have only one eigenvalue and the minimum uniton number 3. We also show that the solution set M(e, r, u) of energy e and canonical rank r with the minimum uniton number u = 3 has a complex dimension greater than r for e = 4 n - 1 and r = n+1, where n ≥ 3. This disproves the dimension conjecture that holds in the case u ∈ {1, 2}.

We study solutions of the Volterra lattice satisfying the stationary equation for its nonautonomous symmetry. We show that the dynamics in t and n are governed by the respective continuous and discrete Painlevé equations and describe the class of initial data leading to regular solutions. For the lattice on the half-axis, we express these solutions in terms of the confluent hypergeometric function. We compute the Hankel transform of the coefficients of the corresponding Taylor series based on the Wronskian representation of the solution.

We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes a boson at the instant of separating from a fermionic nucleus. We study the further transformations of boson and fermion states of separation in a small neighborhood of zero pressure and obtain new important parastatistical relations between the temperature and the chemical potential when a nucleon separates from an atomic nucleus. The obtained relations allow constructing a new diagram (an aF diagram) or isotherms of very high temperatures corresponding to nuclear matter. We mathematically prove that the transition of particles from the domain governed by Fermi-Dirac statistics to the domain governed by Bose-Einstein statistics near the zero pressure P occurs in the neutron uncertainty domain or halo domain. We obtain equations for the chemical potential that allow determining the width of the uncertainty domain. Based on the calculated values of the minimum intensivity for Bose particles, the chemical potential, the compressibility factor, and the minimum mean square fluctuation of the chemical potential, we construct a table of stable nuclei of chemical elements, demonstrating a monotonic relation between the nucleus mass number and the other parameters.

We develop an analytic formalism using basic quantum mechanics techniques to successfully solve the multiphoton Jaynes–Cummings and the generalized Dicke Hamiltonians. For this, we split the Hamiltonians of these models into two operators that have the properties of constants of motion for these systems. We then use some well-known operator properties to obtain complete analytic spectra for the considered models.

We consider a p-adic Ising model on the Cayley tree of order k ≥ 2. We completely describe all p-adic-translation-invariant generalized Gibbs measures for k = 3. Moreover, we show the existence of a phase transition for the p-adic Ising model for any k ≥ 3 if p ≡ 1 (mod 4).

Reference [3] on page 1125, 3. A. A. Grib and Yu. V. Pavlov, “Comparison of particle properties in Kerr metric and in rotating coordinates,” Gen. Rel. Grav., 49, 78 (2017); arXiv:1609.04202v2 [gr-qc] (2016).C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisco (1973).

should be split as

3. A. A. Grib and Yu. V. Pavlov, “Comparison of particle properties in Kerr metric and in rotating coordinates,” Gen. Rel. Grav., 49, 78 (2017); arXiv:1609.04202v2 [gr-qc] (2016).

4. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisco (1973).

The subsequent reference numbers should incremented (old numbers 4–20 become 5–21). Then the citations in the text will correctly relate to the list of references.

The editorial staff apologizes to the authors.