Expanding Domain Limit for Incompressible Fluids in the Plane

The general class of problems we consider is the following: Let Ω ₁ be a bounded domain in \({\mathbb{R}^d}\) for d ≥ 2 and let u ⁰ be a velocity field on all of \({\mathbb{R}^d}\) . Suppose that for all R ≥ 1 we have an operator \({\mathcal{T}_R}\) that projects u ⁰ restricted to RΩ ₁ (Ω ₁ scaled by R) into a function space on RΩ ₁ for which the solution to some initial value problem is well-posed with \({\mathcal{T}_{R}u^0}\) as the initial velocity. Can we show that as R → ∞ the solution to the initial value problem on RΩ ₁ converges to a solution in the whole space? We answer this question when d  =  2 for weak solutions to the Navier-Stokes and Euler equations. For the Navier-Stokes equations we assume the lowest regularity of u ⁰ for which one can obtain adequate control on the pressure. For the Euler equations we assume the lowest feasible regularity of u ⁰ for which uniqueness of solutions to the Euler equations is known (thus, we allow “slightly unbounded” vorticity). In both cases, we obtain strong convergence of the velocity and the vorticity as R → ∞ and, for the Euler equations, the flow. Our approach yields, in principle, a bound on the rates of convergence.     

Some exact results for the three-layer zamolodchikov model

In this talk, we present our recent results on the three-layer Zamolodchikov model. We discuss solutions to the Bethe ansatz equations following from functional relations. We consider two regimes I and II that differ by the signs of the spherical sides (a₁, a₂, a₃) → (?a₁, ?a₂, ?a₃). Also, we accept the two-line hypothesis for regime I and the one-line hypothesis for regime II. In the thermodynamic limit, we derive integral equations for distribution densities and solve them exactly. Using this solution, we calculate the partition function for the three-layer Zamolodchikov model and check the compatibility of this result with functional relations. We also discuss the reasons for the discrepancy with Baxter’s result of 1986.     

The electrical conductance growth of a metallic granular packing

We report on measurements of the electrical conductivity on a two-dimensional packing of metallic disks when a stable current of ~1 mA flows through the system. At low applied currents, the conductance σ is found to increase by a pattern σ(t) = σ _∞ ? Δσ E _α [ ? (t/τ) ^α ], where E _α denotes the Mittag-Leffler function of order α ∈ (0,1). By changing the inclination angle θ of the granular bed from horizontal, we have studied the impact of the effective gravitational acceleration g _eff = gsinθ on the relaxation features of the conductance σ(t). The characteristic timescale τ is found to grow when effective gravity g _eff decreases. By changing both the distance between the electrodes and the number of grains in the packing, we have shown that the long term resistance decay observed in the experiment is related to local micro-contacts rearrangements at each disk. By focusing on the electro-mechanical processes that allow both creation and breakdown of micro-contacts between two disks, we present an approach to granular conduction based on subordination of stochastic processes. In order to imitate, in a very simplified way, the conduction dynamics of granular material at low currents, we impose that the micro-contacts at the interface switch stochastically between two possible states, “on” and “off”, characterizing the conductivity of the micro-contact. We assume that the time intervals between the consecutive changes of state are governed by a certain waiting-time distribution. It is demonstrated how the microscopic random dynamics regarding the micro-contacts leads to the macroscopic observation of slow conductance growth, described by an exact fractional kinetic equations.     

Cosmology in modified <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>)-gravity

In the present paper we propose a further modification of f(R, T)-gravity (where T is trace of the energy-momentum tensor) by introducing higher derivatives matter fields. We discuss stability conditions in the proposed theory and find restrictions for the parameters to prevent appearance of main type of instabilities, such as ghost-like and tachyon-like instabilities. We derive cosmological equations for a few representations of the theory and discuss main differences with conventional f(R, T)-gravity without higher derivatives. It is demonstrated that in the theory presented inflationary scenarios appear quite naturally even in the dust-filled Universe without any additional matter sources. Finally, we construct an inflationary model in one of the simplest representation of the theory, calculate the main inflationary parameters and find that it may be in quite good agreement with observations.     

Conductance distribution near the Anderson transition

Using a modification of the Shapiro approach, we introduce the two-parameter family of conductance distributions W(g), defined by simple differential equations, which are in the one-to-one correspondence with conductance distributions for quasi-one-dimensional systems of size L ^d–1 × L _z , characterizing by parameters L/ξ and L _z /L (ξ is the correlation length, d is the dimension of space). This family contains the Gaussian and log-normal distributions, typical for the metallic and localized phases. For a certain choice of parameters, we reproduce the results for the cumulants of conductance in the space dimension d = 2 + ? obtained in the framework of the σ-model approach. The universal property of distributions is existence of two asymptotic regimes, log-normal for small g and exponential for large g. In the metallic phase they refer to remote tails, in the critical region they determine practically all distribution, in the localized phase the former asymptotics forces out the latter. A singularity at g = 1, discovered in numerical experiments, is admissible in the framework of their calculational scheme, but related with a deficient definition of conductance. Apart of this singularity, the critical distribution for d = 3 is well described by the present theory. One-parameter scaling for the whole distribution takes place under condition, that two independent parameters characterizing this distribution are functions of the ratio L/ξ.     

Efficient Quantum Algorithm for the Parity Problem of a Certain Function

Based on a particular mathematical structure of a certain function f(x) under our attention, we present a novel quantum algorithm. The algorithm allows one to determine the property of a certain function. In our study, it is f(x) = f(?x). Therefore, there would be a question here, “How fast can we succeed in this?” All we need to do is only the evaluation of a single quantum state \(|\overbrace {0,0,\ldots ,0,1}^{N}\rangle \) (N ≥?2). Only using that with a little amount of information, we can derive the global property f(x) = f(?x). Our quantum algorithm overcomes a classical counterpart by a factor of the order of 2^N.     

Hypersurfaces Satisfying <Emphasis Type="BoldItalic">τ</Emphasis><Subscript><Emphasis Type="BoldItalic">2</Emphasis></Subscript><Emphasis Type="BoldItalic">(ϕ) = ητ(ϕ)</Emphasis> in Pseudo-Riemannian Space Forms

In this paper, we derived the equations for the hypersurface \({M^{n}_{r}}\) of a pseudo-Riemannian space form \(N^{n+1}_{q}(c)\) to satisfy τ ₂(?) = η τ(?) (η a constant) with τ(?) and τ ₂(?) be the tension and bitension fields of \({M^{n}_{r}}\). As applications, we prove that a hypersurface \({M^{n}_{r}}\) satisfying τ ₂(?) = η τ(?) in \(N^{n+1}_{q}(c)\) has constant mean curvature, under the assumption that \({M^{n}_{r}}\) has diagonalizable shape operator with at most three distinct principal curvatures. Then, using this result, we classify partially such hypersurface. We also make a preliminary study of hypersurfaces satisfying τ ₂(?) = f τ(?) with f be function.     

Regularisierende Deltafunktionen

In the Laurent Schwartz theory of distributions the integral∫ δ(r)f(r)dτ _r is only defined for a certain class of test functions. Unfortunately, in physics, we obtain test functionsf(r) likee ^ikr /r, e ^?kr /r, ?(e ^ikr /r) and so on, which are not belonging to the Laurent Schwartz class (§1). Here, we want to extend the class of test functions so that it includes physically meaningful functions with poles of finte order inr=0. For this purpose we replaceLaurent Schwartz's axiomatic method defining theδ-distribution by a constructive one consideringδ(r) as a given set of sequences of functions (§2). First we prove that the redefinedδ-function satisfy the equations, axiomatically assumed byLaurent Schwartz (§3). Then we obtain well defined and finite results even in the case of test functions with poles atr=0 (§4). The Fourier components of the newδ-function are given (§5). Finally we show why theδ-function is Lorentz invariant (§6).     

Quark-antiquark composite systems: The Bethe-Salpeter equation in the spectral-integration technique

The Bethe-Salpeter equations for the quark-antiquark composite systems, q\(\bar q\), are written in terms of spectral integrals. For the q\(\bar q\) mesons characterized by the mass M, spin J, and radial quantum number n, the equations are presented for the following (n, M²) trajectories: π _J , η _J , a _J , f _J , ρ _J , ω _J , h _J , and b _J .     

New Analytical Solution of the Equilibrium Ampere’s Law Using the Walker’s Method: a Didactic Example

This work aims to demonstrate the analytical solution of the Grad-Shafranov (GS) equation or generalized Ampere’s law, which is important in the studies of self-consistent 2.5-D solution for current sheet structures. A detailed mathematical development is presented to obtain the generating function as shown by Walker (RSPSA 91, 410, 1915). Therefore, we study the general solution of the GS equation in terms of the Walker’s generating function in details without omitting any step. The Walker’s generating function g(ζ) is written in a new way as the tangent of an unspecified function K(ζ). In this trend, the general solution of the GS equation is expressed as exp(??2Ψ) =?4|K ^′(ζ)|²/cos²[K(ζ) ? K(ζ ^?)]. In order to investigate whether our proposal would simplify the mathematical effort to find new generating functions, we use Harris’s solution as a test, in this case K(ζ) = arctan(exp(i ζ)). In summary, one of the article purposes is to present a review of the Harris’s solution. In an attempt to find a simplified solution, we propose a new way to write the GS solution using g(ζ) = tan(K(ζ)). We also present a new analytical solution to the equilibrium Ampere’s law using g(ζ) = cosh(b ζ), which includes a generalization of the Harris model and presents isolated magnetic islands.     

Exact travelling solutions for some nonlinear physical models by ( <Emphasis Type="Italic">G</Emphasis>′/ <Emphasis Type="Italic">G</Emphasis>)-expansion method

In this paper, we establish exact solutions for some special nonlinear partial differential equations. The (G′/G)-expansion method is used to construct travelling wave solutions of the two-dimensional sine-Gordon equation, Dodd–Bullough–Mikhailov and Schrödinger–KdV equations, which appear in many fields such as, solid-state physics, nonlinear optics, fluid dynamics, fluid flow, quantum field theory, electromagnetic waves and so on. In this method we take the advantage of general solutions of second-order linear ordinary differential equation (LODE) to solve many nonlinear evolution equations effectively. The (G′/G)-expansion method is direct, concise and elementary and can be used with a wider applicability for handling many nonlinear wave equations.     

Global transformations preserving Sturm–Liouville spectral data

We show the existence of a real analytic isomorphism between the space of the impedance function ρ of the Sturm–Liouville problem ?ρ ^?2(ρ ² f′)′ +uf on (0, 1), where u is a function of ρ, ρ′, ρ″, and that of potential p of the Schrödinger equation ?y″ +py on (0, 1), keeping their boundary conditions and spectral data. This mapping is associated with the classical Liouville transformation f → ρf, and yields a global isomorphism between solutions of inverse problems for the Sturm–Liouville equations of the impedance form and those of the Schrödinger equations.     

The trinification model <Emphasis Type="Italic">SU</Emphasis>(3)<Superscript>3</Superscript> from orbifolds for fuzzy spheres

In this review, we consider an N = 4 supersymmetric SU(3N) gauge theory defined on the Minkowski spacetime. Then we apply an orbifold projection leading to an N = 1 supersymmetric SU(N)³ model, with a truncated particle spectrum. Then, we present the dynamical generation of (twisted) fuzzy spheres as vacuum solutions of the projected field theory, breaking the SU(N)³ spontaneously to a chiral effective theory with unbroken gauge group the trinification group, SU(3)³.     

Late-time cosmological approach in mimetic <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) gravity

In this paper, we investigate the late-time cosmic acceleration in mimetic f(R, T) gravity with the Lagrange multiplier and potential in a Universe containing, besides radiation and dark energy, a self-interacting (collisional) matter. We obtain through the modified Friedmann equations the main equation that can describe the cosmological evolution. Then, with several models from \(\mathcal {Q}(z)\) and the well-known particular model f(R, T), we perform an analysis of the late-time evolution. We examine the behavior of the Hubble parameter, the dark energy equation of state and the total effective equation of state and in each case we compare the resulting picture with the non-collisional matter (assumed as dust) and also with the collisional matter in mimetic f(R, T) gravity. The results obtained are in good agreement with the observational data and show that in the presence of the collisional matter the dark energy oscillations in mimetic f(R, T) gravity can be damped.     

The Padé interpolation method applied to <Emphasis Type="Italic">q</Emphasis>-Painlevé equations II (differential grid version)

Recently, we studied Padé interpolation problems of q-grid, related to q-Painlevé equations of type \(E_7^{(1)}\), \(E_6^{(1)}\), \(D_5^{(1)}\), \(A_4^{(1)}\) and \((A_2+A_1)^{(1)}\). By solving those problems, we could derive evolution equations, scalar Lax pairs and determinant formulae of special solutions for the corresponding q-Painlevé equations. It is natural that the q-Painlevé equations were derived by the interpolation method of q-grid, but it may be interesting in terms of differential grid that the Padé interpolation method of differential grid (i.e. Padé approximation method) has been applied to the q-Painlevé equation of type \(D_5^{(1)}\) by Ikawa. In this paper, we continue the above study and apply the Padé approximation method to the q-Painlevé equations of type \(E_6^{(1)}\), \(D_5^{(1)}\), \(A_4^{(1)}\) and \((A_2+A_1)^{(1)}\). Moreover, determinant formulae of the special solutions for q-Painlevé equation of type \(E_6^{(1)}\) are given in terms of the terminating q-Appell Lauricella function.     

Dark Energy Models in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) Theory with Variable Deceleration Parameter

In this communication we have investigated Bianchi type-II dark energy (DE) cosmological models with and without presence of magnetic field in modified f(R, T) gravity theory as proposed by Harko et al. (Phys. Rev. D, 84, 024020, 2011). The exact solution of the field equations is obtained by setting the deceleration parameter q as a time function along with suitable assumption the scale factor \(a(t)= [sinh(\alpha t)]^{\frac {1}{n}}\), α and n are positive constant. We have obtained a class of accelerating and decelerating DE cosmological models for different values of n and α. The present study believes that the mysterious dark energy is the main responsible force for accelerating expansion of the universe. For our constructed models the DE candidates cosmological constant (Λ) and the EoS parameter (ω) both are found to be time varying quantities. The cosmological constant Λ is very large at early time and approaches to a small positive value at late time whereas the EoS parameters is found small negative at present time. Physical and kinematical properties of the models are discussed with the help of pictorial representations of the parameters. We have observed that our constructed models are compatible with recent cosmological observations.     

On Quantization in Light-cone Variables Compatible with Wavelet Transform

Canonical quantization of quantum field theory models is inherently related to the Lorentz invariant partition of classical fields into the positive and the negative frequency parts u(x) = u⁺(x) + u^?(x), performed with the help of Fourier transform in Minkowski space. That is the commutation relations are being established between nonlocalized solutions of field equations. At the same time the construction of divergence free physical theory requires the separation of the contributions of different space-time scales. In present paper, using the light-cone variables, we propose a quantization procedure which is compatible with separation of scales using continuous wavelet transform, as described in our previous paper (Altaisky, M.V., Kaputkina, N.E.: Phys. Rev. D 88, 025015 2013).     

A continuous time random walk (CTRW) integro-differential equation with chemical interaction

A nonlocal-in-time integro-differential equation is introduced that accounts for close coupling between transport and chemical reaction terms. The structure of the equation contains these terms in a single convolution with a memory function M?(t), which includes the source of non-Fickian (anomalous) behavior, within the framework of a continuous time random walk (CTRW). The interaction is non-linear and second-order, relevant for a bimolecular reaction A + B → C. The interaction term ΓP _A ?(s, t)?P _B ?(s, t) is symmetric in the concentrations of A and B (i.e. P _A and P _B ); thus the source terms in the equations for A, B and C are similar, but with a change in sign for that of C. Here, the chemical rate coefficient, Γ, is constant. The fully coupled equations are solved numerically using a finite element method (FEM) with a judicious representation of M?(t) that eschews the need for the entire time history, instead using only values at the former time step. To begin to validate the equations, the FEM solution is compared, in lieu of experimental data, to a particle tracking method (CTRW-PT); the results from the two approaches, particularly for the C profiles, are in agreement. The FEM solution, for a range of initial and boundary conditions, can provide a good model for reactive transport in disordered media.     

Spherical Symmetric Perfect Fluid Collapse in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) Gravity

This paper contains the study of spherically symmetric perfect fluid collapse in the frame work of f(R, T) modified theory of gravity. We proceed our work by considering the non-static spherically symmetric background in the interior and static spherically symmetric background in the exterior regions of the star. The junction conditions between exterior and interior regions are presented by matching the exterior and interior regions. The field equations are solved by taking the assumptions that the Ricci scalar as well as the trace of energy-momentum tensor are to be constant, for a particular f(R, T) model. By inserting the solution of the field equations in junction conditions, we evaluate the gravitational mass of the collapsing system. Also, we discuss the apparent horizons and their time formation for different possible cases. It is concluded that the term f(R ₀, T ₀) behaves as a source of repulsive force and that’s why it slowdowns the collapse of the matter.     

Zur Supraleitung von Schichtpaketen aus Normal- und Supraleitern

Superimposed films of a super- and a normalconducting metal are produced on a quartz plate by evaporation. In most cases lead and occasionally also tin is used as superconductor. The normal metals are copper, manganese and chromium. For more than 50 samples of different combinations the transitiontemperatures are measured. The mean free path of the electrons for the metals can be varied by the conditions of condensation. By means of a phenomenological theory, which we owe in principle to a discussion with ProfessorFröhlich, Liverpool, we succeed in formulating expressions fort=f(D _s ,D _n ,l _s ,l _n ,ξ₀).D,l andξ ₀ are the thickness, mean free path and the coherence length of the supra- or the normal metals andt is the reduced transitiontemperature. The developed equations can be well fitted to the measured data also to those of other authors.