Fractional proportional differences with memory

In this paper, we formulate nabla fractional sums and differences and the discrete Laplace transform on the time scale h?. Based on a local type h-proportional difference (without memory), we generate new types of fractional sums and differences with memory in two parameters which are generalizations to the formulated fractional sums and differences. The kernel of the newly defined generalized fractional sum and difference operators contain h-discrete exponential functions. The discrete h-Laplace transform and its convolution theorem are then used to study the newly introduced discrete fractional operators and also used to solve Cauchy linear fractional difference type problems with step 0 < h ≤ 1.     

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.     

Time-Harmonic Gaussian Beams: Exact Solutions of the Helmhotz Equation in Free Space

An exact solution of the Helmholtz equation u _xx + u _yy + u _zz + k²u = 0 is presented, which describes propagation of monochromatic waves in the free space. The solution has the form of a superposition of plane waves with a specific weight function dependent on a certain free parameter a. If ka→∞, the solution is localized in the Gaussian manner in a vicinity of a certain straight line and asymptotically coincides with the famous approximate solution known as the fundamental mode of a paraxial Gaussian beam. The asymptotics of the aforementioned exact solution does not include a backward wave.     

Master integrals for double real radiation emission in heavy-to-light quark decay

We evaluate analytically the master integrals for double real radiation emission in the \(b \rightarrow u W^*\) decay, where b and u are a massive and massless quark, respectively, while \(W^{*}\) is an off-shell charged weak boson. Since the W boson can subsequently decay in a lepton anti-neutrino pair, the results of the present paper constitute a further step toward a fully analytic computation of differential distributions for the semileptonic decay of a b quark at NNLO in QCD. The latter partonic process plays a crucial role in the study of inclusive semileptonic charmless decays of B mesons. Our results are expressed in terms of multiple polylogarithms of maximum weight four.     

Intensity of singlet-triplet transitions in C<Subscript>60</Subscript> fullerene calculated on the basis of the time-dependent density functional theory and taking into account the quadratic response

The singlet-singlet and triplet-triplet absorption spectra of C₆₀ fullerene are calculated using the density functional method and taking into account the theory of linear and quadratic responses. The B3LYP density functional and the 6–31G and 3–21G atomic basis sets are used. The calculations are performed using the D_2h and D_5d symmetry groups, although the real symmetry of the ground state is described by the I _h symmetry group. The matrix elements of the operator of the spin-orbit coupling are calculated and the probabilities of some singlet-triplet transitions are estimated. Taking into account the data in the literature on vibronic interactions of vibrations of the t_1u, t_2u, g _u , and h _u symmetry species, the radiative lifetime of the 1³T_2g → 1¹A _g phosphorescence was estimated to be 45 s. The fact that this time proved to be considerably greater than the experimentally observed total lifetime of the triplet testifies to a fast nonradiative deactivation of the lowest triplet state of C₆₀ fullerene and agrees with a low phosphorescence intensity. The zero-field splitting of some triplets and the intensities of magnetic dipole transitions are discussed.     

Forced snaking: Localized structures in the real Ginzburg-Landau equation with spatially periodic parametric forcing

We study spatial localization in the real subcritical Ginzburg-Landau equation u _t = m ₀ u + Q(x)u + u _xx + d|u|² u ?|u|⁴ u with spatially periodic forcing Q(x). When d>0 and Q ≡ 0 this equation exhibits bistability between the trivial state u = 0 and a homogeneous nontrivial state u = u ₀ with stationary localized structures which accumulate at the Maxwell point m ₀ = ?3d ²/16. When spatial forcing is included its wavelength is imprinted on u ₀ creating conditions favorable to front pinning and hence spatial localization. We use numerical continuation to show that under appropriate conditions such forcing generates a sequence of localized states organized within a snakes-and-ladders structure centered on the Maxwell point, and refer to this phenomenon as forced snaking. We determine the stability properties of these states and show that longer lengthscale forcing leads to stationary trains consisting of a finite number of strongly localized, weakly interacting pulses exhibiting foliated snaking.     

On Cluster Properties of Classical Ferromagnets in an External Magnetic Field

Correlation functions of ferromagnetic spin systems satisfying a Lee-Yang property are studied. It is shown that, for classical systems in a non-vanishing uniform external magnetic field h, the connected correlation functions decay exponentially in the distances between the spins, i.e., the inverse correlation length (“mass gap”), m(h), is strictly positive. Our proof is very short and transparent and is valid for complex values of the external magnetic field h, provided that \(\mathrm {Re}\, h \not = 0\). It implies a mean-field lower bound on m(h), as \(h \searrow 0\), first established by Lebowitz and Penrose for the Ising model. Our arguments also apply to some quantum spin systems.     

Fluctuations for the Ginzburg-Landau $${\nabla \phi}$$ Interface Model on a Bounded Domain

We study the massless field on \({D_n = D \cap \tfrac{1}{n} \mathbf{Z}^2}\), where \({D \subseteq \mathbf{R}^2}\) is a bounded domain with smooth boundary, with Hamiltonian \({\mathcal {H}(h) = \sum_{x \sim y} \mathcal {V}(h(x) - h(y))}\). The interaction \({\mathcal {V}}\) is assumed to be symmetric and uniformly convex. This is a general model for a (2 + 1)-dimensional effective interface where h represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt: h(x) = n x · u + f(x) for \({x \in \partial D_n,\,u \in \mathbf{R}^2}\), and f : R ² → R continuous. We prove that the fluctuations of linear functionals of h(x) about the tilt converge in the limit to a Gaussian free field on D, the standard Gaussian with respect to the weighted Dirichlet inner product \({(f,g)_\nabla^\beta = \int_D \sum_i \beta_i \partial_i f_i \partial_i g_i}\) for some explicit β = β(u). In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of h are asymptotically described by SLE(4), a conformally invariant random curve.     

Maslov index for power-law potentials

The Maslov index in the semiclassical Bohr–Sommerfeld quantization rule is calculated for one-dimensional power-law potentials V (x) = ?V ₀/x ^s, x > 0, 0 < s < 2 The result for the potential V(x)=-V ₀/x ^1/2 is compared with the recently reported exact solution. The case of a spherically symmetric power-law potential is also considered.     

Analytic properties of the effective dielectric constant of a two-dimensional Rayleigh model

Analytic properties of the dimensionless static effective dielectric constant f(p, h) of a two-dimensional Rayleigh model (p is the concentration and h is the ratio of the dielectric constants of components) are considered as a function of the complex variable h. It is shown that the only singularities of the function f(p, h) are first-order poles for real h = h _n < 0 (n = 1, 2, ...) with the condensation point h = ?1, which form an infinite discrete (countable) set. The positions of the first ten poles of the function f(p, h) and the residues at these points are calculated and represented graphically versus the concentration. Based on the results obtained, a pole-type approximate formula is proposed that describes the behavior of the function f(p, h) over a wide range of p and complex h.     

Quark diagram analysis of bottom meson decays emitting pseudoscalar and vector mesons

This paper presents the two body weak nonleptonic decays of B mesons emitting pseudoscalar (P) and vector (V) mesons within the framework of the diagrammatic approache at flavor SU(3) symmetry level. Using the decay amplitudes, we are able to relate the branching fractions of B → PV decays induced by both b → c and b → u transitions, which are found to be well consistent with the measured data. We also make predictions for some decays, which can be tested in future experiments.     

Mehrfache Dispersionsrelationen für kurzreichweitige Potentiale,die sich nicht durch eine Überlagerung von Yukawa-Potentialen darstellen lassen

A triple dispersion relation is derived for the functionf(s, t, u) which becomes the scattering amplitudef(s, t) foru=4s?t. Besides the usual conditions which are needed for deriving a dispersion relation ins, the potential must decrease faster than exponentially at infinity. For this class of potentialsf(s, t, u) has essential singularities fort→∞ andu→∞. It is shown thatf(s, t, u) is bounded in the physical sheets of two independent Riemannian surfaces which are constructed by conformal mappings of thet- and theu-plane. In the new variables the conditions for the existence of dispersion relations are fulfilled.     

Electromagnetic radiation in a medium with a velocity gradient

A relativistic expression has been obtained for the curvature of trajectory of the wave vector of an electromagnetic wave in a moving optically transparent medium. It has been shown that the curvature of the trajectory and angular deviation of rays appear in a homogeneous isotropic medium if the gradient of the velocity field in the medium is nonzero. The bending of the trajectory in the medium with the velocity gradient is a firstorder effect in the ratio u/c.     

Crossover between the thermodynamic and nonequilibrium scenarios of structural transformations of H<Subscript>2</Subscript>O I<Emphasis Type="Italic">h</Emphasis> ice during compression

A detailed investigation of different scenarios of structural transformations of H₂O Ih ice during compression to a pressure of 2 GPa in the temperature range from 77 to 200 K is performed. In the range of temperatures and pressures being treated, detailed data are obtained for the density and the shear modulus for different phases of ice including the hda, IX, and XII phases. The experimentally obtained correlations for the density and ultrasonic velocities, with due regard for the available data of structural investigations, are used to identify the transformation sequences Ih→hda (below 135 K), Ih→II→VI (above 165 K), and Ih→IX→VI (from 155 to 180 K). In the vicinity of the crystallization temperature of amorphous ice, i.e., at about 140 K, an anomalous transformation pattern is observed, which is interpreted as predominantly the Ih→XII phase transition. The temperature crossover is discussed between the mode of solid-phase amorphization (Ih→hda) and crystal-crystal transitions, as well as the metastable nature of IX ice and the mechanism of solid-phase amorphization.     

Top-quark <Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">T</Emphasis></Subscript>-spectra at CMS and flavor independence of <Emphasis Type="Italic">z</Emphasis>-scaling

We present new results of analysis of top-quark differential cross sections obtained by the CMS Collaboration in pp collisions in the framework of the z-scaling approach. The spectra are measured over a wide range of collision energy \(\sqrt s = 7,8,13TeV\) and transverse momentum p _T = 30?500 GeV/c of top-quark using leptonic and jet decay modes. Flavor independence of the scaling function ψ(z) is verified in the new kinematic range. The results of analysis of the top-quark spectra obtained at the LHC are compared with similar spectra measured in \(\overline p p\) collisions at the Tevatron energy \(\sqrt s = 1.96TeV\). A tendency to saturation of ψ(z) for the process at low z and a power-law behavior of ψ(z) at high z is observed. The measurements of high-p _T is observed. The measurements of highspectra of the top-quark production at highest LHC energy is of interest for verification of self-similarity of particle production, understanding flavor origin and search for new physics symmetries with top-quark probe.     

Long-range spin correlations in a honeycomb spin model with a magnetic field

We consider the spin-1/2 model on the honeycomb lattice [A. Kitaev, Ann. Phys. 321, 2 (2006)] in the presence of a weak magnetic field h _α ? J. Such a perturbation treated in the lowest nonvanishing order over h _α leads [K.S. Tikhonov, M.V. Feigel’man, and A.Yu. Kitaev, Phys. Rev. Lett. 106, 067203 (2011)] to a powerlaw decay of irreducible spin correlations 《s ^z (t, r)s ^z (0, 0)》 ∝ h _z ² f(t, r), where f(t, r) ∝ [max(t, Jr)]^–4. We have studied the effects of the next order of perturbation in h _z and found an additional term of the order h _z ⁴ in the correlation function 《s ^z (t, r)s ^z (0, 0)》 which scales as h _z ⁴ cosγ/r ³ at Jt? r, where γ is the polar angle in the 2D plane. We demonstrate that such a contribution can be understood as a result of a perturbation of the effective Majorana Hamiltonian by the weak imaginary vector potential A _x ∝ i h _z ² .     

Role of initial correlation in coarsening of a ferromagnet

We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of theIsing model, employing the Glauber spin-flip mechanism, in space dimensionsd = 2 and3, on square and simplecubic lattices. Results for the persistence probability and the domain growth arediscussed for quenches to various temperatures (T _f ) below the criticalone (T _c ), from differentinitial temperatures T _i ≥T _c . In long timelimit, for T _i >T _c ,the persistence probability exhibits power-law decay with exponents θ ? 0.22 and? 0.18 in d = 2 and 3, respectively. For finite T _i , the early timebehavior is a different power-law whose life-time diverges and exponent decreases asT _i →T _c . The two steps areconnected via power-law as a function of domain length and the crossover to the secondstep occurs when this characteristic length exceeds the equilibrium correlation length atT =T _i . T _i =T _c is expected toprovide a new universality class for which we obtain θ ≡θ _c ? 0.035 ind = 2 and?0.105 in d = 3. The time dependenceof the average domain size ?, however, is observed to be rather insensitive tothe choice of T _i .     

Quantum Fisher Information for <Emphasis Type="Italic">s</Emphasis><Emphasis Type="Italic">u</Emphasis>(2) Atomic Coherent States and <Emphasis Type="Italic">s</Emphasis><Emphasis Type="Italic">u</Emphasis>(1, 1) Coherent States

We investigate quantum Fisher information (QFI) for s u(2) atomic coherent states and s u(1, 1) coherent states. In this work, we find that for s u(2) atomic coherent states, the QFI with respect to \(\vartheta ~(\mathcal {F}_{\vartheta })\) is independent of φ, the QFI with respect to \(\varphi (\mathcal {F}_{\varphi })\) is governed by ??. Analogously, for s u(1,1) coherent states, \(\mathcal {F}_{\tau }\) is independent of φ, and \(\mathcal {F}_{\varphi }\) is determined by τ. Particularly, our results show that \(\mathcal {F}_{\varphi }\) is symmetric with respect to ?? = π/2 for s u(2) atomic coherent states. And for s u(1,1) coherent states, \(\mathcal {F}_{\varphi }\) also possesses symmetry with respect to τ = 0.