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## The Inverse-Square Interaction Phase Diagram: Unitarity in the Bosonic Ground State

Ground-state properties of bosons interacting via inverse square potential (three dimensional Calogero-Sutherland model) are analyzed. A number of quantities scale with the density and can be naturally expressed in units of the Fermi energy and Fermi momentum multiplied by a dimensionless constant (Bertsch parameter). Two analytical approaches are developed: the Bogoliubov theory for weak and the harmonic approximation (HA) for strong interactions. Diffusion Monte Carlo method is used to obtain the ground-state properties in a non-perturbative manner. We report the dependence of the Bertsch parameter on the interaction strength and construct a Padé approximant which fits the numerical data and reproduces correctly the asymptotic limits of weak and strong interactions. We find good agreement with beyond-mean field theory for the energy and the condensate fraction. The pair distribution function and the static structure factor are reported for a number of characteristic interactions. We demonstrate that the system experiences a gas-solid phase transition as a function of the dimensionless interaction strength. A peculiarity of the system is that by changing the density it is not possible to induce the phase transition. We show that the low-lying excitation spectrum contains plasmons in both phases, in agreement with the Bogoliubov and HA theories. Finally, we argue that this model can be interpreted as a realization of the unitary limit of a Bose system with the advantage that the system stays in the genuine ground state contrarily to the metastable state realized in experiments with short-range Bose gases.

We study the planar matching problem, defined by a symmetric random matrix with independent identically distributed entries, taking values 0 and 1. We show that the existence of a perfect planar matching structure is possible only above a certain critical density of allowed contacts, $p_{c}$. This problem has an important application for the prediction of the optimal folding of RNA-type polymers. Using an alternative formulation of the problem in terms of Dyck paths and a matrix model of planar contact structures, we provide an analytical estimation for the value of the transition point, $p_{c}$, in the thermodynamic limit. This estimation is close to the critical value, $p_{c}\approx 0.38$, obtained in numerical simulations based on an exact dynamic-programming algorithm. We characterize the corresponding critical behavior of the model and discuss the relation of the perfect-imperfect matching transition to the known molten-glass transition in the context of random RNA secondary structure's formation. In particular, we provide strong evidence supporting the conjecture that the molten-glass transition at $T=0$ occurs at $p_{c}$

We study the fraction f of nucleotides involved in the formation of a cactuslike secondary structure of random heteropolymer RNA-like molecules. In the low-temperature limit, we study this fraction as a function of the number c of different nucleotide species. We show, that with changing c, the secondary structures of random RNAs undergo a morphological transition:f(c)→1 for c≤ccr as the chain length n goes to infinity, signaling the formation of a virtually perfect gapless secondary structure; while f(c)<1 for c>ccr, which means that a nonperfect structure with gaps is formed. The strict upper and lower bounds 2≤ccr≤4 are proven, and the numerical evidence for ccr is presented. The relevance of the transition from the evolutional point of view is discussed.

A simple sociophysical model is proposed to describe the transition between a chaotic and a coherent state of a microblogging social network. The model is based on the equations of evolution of the order parameter, the conjugated field, and the control parameter. The self-consistent evolution of the networks is presented by equations in which the correlation function between the incoming information and the subsequent change of the number of microposts plays the role of the order parameter; the conjugate field is equal to the existing information; and the control parameter is given by the number of strategically oriented users. Analysis of the adiabatic approximation shows that the second-order phase transition, which means following a definite strategy by the network users, occurs when their initial number exceeds a critical value equal to the geometric mean of the total and critical number of users.

In this work, we explore the properties of antiferromagnetic cycloid and the phase transitions between commensurate and incommensurate magnetic states in epitaxial BiFeO3 film. Additional magnetic anisotropy induced by strain effects in the films allocates cycloids with the definite directions of spin rotation. Peculiar feature of the cycloids propagating in the films whose symmetry is different from the single crystals is the orientation of spin rotational plane that does not contain electric polarization in contrast with the bulk materials. We construct a diagram of phase transitions induced by magnetic field applied along normal to the surface and show considerable decrease of the strength of magnetic field destroying cycloid in films compared with the bulk.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.