Gliding Down the QCD Transition Line, from Nf = 2 till the Onset of Conformality

We review the hot QCD transition with varying number of flavours, from two till the onset of the conformal window. We discuss the universality class for Nf=2, along the critical line for two massless light flavours, and a third flavour whose mass serves as an interpolator between Nf=2 and Nf=3. We identify a possible scaling window for the 3D O(4) universality class transition, and its crossover to a mean field behaviour. We follow the transition from Nf=3 to larger Nf, when it remains of first order, with an increasing coupling strength; we summarise its known properties, including possible cosmological applications as a model for a strong electroweak transition. The first order transition, and its accompanying second order endpoint, finally morphs into the essential singularity at the onset of the conformal window, following the singular behaviour predicted by the functional renormalisation group.

Determination of the stress concentration in the corner point of the wedge-shaped region reinforced by a more rigid thin coating

We analysed the problem of determining the exponents in the asymptotic solution of the isotropic theory of elasticity problem at the top of the wedge-shaped region where its sides (or one of them) are supported by a thin coating and lean without friction on the rigid bases. On the other side of the wedge-shaped region, it is assumed that there are various boundary conditions, including when there is a thin coating. Mathematically, the problem reduces to the problem of determining the roots of transcendental characteristic equations arising from the condition for the existence of a nontrivial solution of a system of the linear homogeneous equations. The characteristics of the stress tensor components have been determined for the various combinations of boundary conditions and physical and geometric parameters. The qualitative conclusions are made. In particular, we have established the combinations of the values of these parameters at which the singular behaviour of stresses arises.

FEATURES OF STRESSES AT THE APEX OF AN ELASTIC WEDGE, SUPPORTED BY A THIN FLEXIBLE COATING ON THE SIDES

Objectives To study the problem of determining the degree of stress at the apex of a wedge-shaped area in cases where the sides (or one of them) are covered with a thin flexible coating.Method It is assumed that the coating is not stretchable. On the other side of the wedge-shaped area, the same coating is assumed to be present; it is either fixed, stress-free or in smooth contact with a rigid base. Mathematically, the problem is reduced to the task of determining the roots of characteristic transcendental equations arising from the existence of a nontrivial solution to the system of linear homogeneous equations.Results Values for the specific characteristics of the radial component of a stress tensor are determined for different combinations of boundary conditions and solution angles. In particular, the angles at which the singular behaviour of stresses occurs are determined. The case is considered when a special boundary condition is given on the edge surface, simulating the overlay. Characteristic equations are obtained to determine the index of the degree dependency of the asymptotic solution in its vicinity for four variants of boundary conditions. In two cases, transcendental equations are obtained, which are solved numerically.Conclusion Calculations of the first positive roots of the equations depending on the angle of the edge solution and Poisson's ratio are presented. The values of the angles, at which the singular behaviour of stresses occurs, are determined. In the case of a combination of boundary conditions (III – IV), the singular stress behaviour is observed for the angle ???? = ????/8, while in the case of (III – III) this value is equal to ????/4.

Electrostatic stability of electron–positron plasmas in dipole geometry

The electrostatic stability of electron–positron plasmas is investigated in the point-dipole and Z-pinch limits of dipole geometry. The kinetic dispersion relation for sub-bounce-frequency instabilities is derived and solved. For the zero-Debye-length case, the stability diagram is found to exhibit singular behaviour. However, when the Debye length is non-zero, a fluid mode appears, which resolves the observed singularity, and also demonstrates that both the temperature and density gradients can drive instability. It is concluded that a finite Debye length is necessary to determine the stability boundaries in parameter space. Landau damping is investigated at scales sufficiently smaller than the Debye length, where instability is absent.

On the homogenization of the Helmholtz problem with thin perforated walls of finite length

In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in ℝ2 with a thin and periodic layer of finite length. The layer may consists of a periodic pertubation of the material coefficients or it is a wall modelled by boundary conditions with an periodic array of small perforations. We consider the periodicity in the layer as the small variable δ and the thickness of the layer to be at the same order. Moreover we assume the thin layer to terminate at re-entrant corners leading to a singular behaviour in the asymptotic expansion of the solution representation. This singular behaviour becomes visible in the asymptotic expansion in powers of δ where the powers depend on the opening angle. We construct the asymptotic expansion order by order. It consists of a macroscopic representation away from the layer, a boundary layer corrector in the vicinity of the layer, and a near field corrector in the vicinity of the end-points. The boundary layer correctors and the near field correctors are obtained by the solution of canonical problems based, respectively, on the method of periodic surface homogenization and on the method of matched asymptotic expansions. This will lead to transmission conditions for the macroscopic part of the solution on an infinitely thin interface and corner conditions to fix the unbounded singular behaviour at its end-points. Finally, theoretical justifications of the second order expansion are given and illustrated by numerical experiments. The solution representation introduced in this article can be used to compute a highly accurate approximation of the solution with a computational effort independent of the small periodicity δ.