Generalized compatibility equations for tensors of high ranks in multidimensional continuum mechanics

Compatibility equations are derived for the components of generalized strains of rank m associated with generalized displacements of rank m ? 1 by analogs of Cauchy kinematic relations in n-dimensional space (multi-dimensional continuous medium) (m ≥ 1, n ≥ 2). These relations can be written in the form of equating to zero all components of the incompatibility tensor of rank m(n ? 2) or its dual generalized Riemann–Christoffel tensor of rank 2m. The number of independent components of these tensors is found; this number coincides with that of compatibility equations in terms of generalized strains or stresses. The inequivalence of the full system of compatibility equations to any of its weakened subsystems is discussed, together with diverse formulations of boundary value problems in generalized stresses in which the number of equations in a domain can exceed the number of unknowns.     

Microscopic theory of second-order and third-order elastic constants for an NaCl crystal

Relations between the second-order and third-order symmetry-independent elastic constants and the energy of interatomic interactions dependent on the mutual arrangement of pairs and triplets of atoms are obtained for crystals belonging to the crystal class O _h. The derived relations and experimental data on the elastic constants are used to calculate four third-order elastic constants and the temperature dependence of the elastic anisotropy factor a(T) for an NaCl crystal. The calculated dependence a(T) is in qualitative agreement with the experimental dependence a _exp(T).     

Almost Commutative Riemannian Geometry: Wave Operators

Associated to any (pseudo)-Riemannian manifold M of dimension n is an n + 1-dimensional noncommutative differential structure (Ω¹, d) on the manifold, with the extra dimension encoding the classical Laplacian as a noncommutative ‘vector field’. We use the classical connection, Ricci tensor and Hodge Laplacian to construct (Ω², d) and a natural noncommutative torsion free connection \({(\nabla,\sigma)}\) on Ω¹. We show that its generalised braiding \({\sigma:\Omega^1\otimes\Omega^1\to \Omega^1\otimes\Omega^1}\) obeys the quantum Yang-Baxter or braid relations only when the original M is flat, i.e. their failure is governed by the Riemann curvature, and that σ ² = id only when M is Einstein. We show that if M has a conformal Killing vector field τ then the cross product algebra \({C(M)\rtimes_\tau\mathbb{R}}\) viewed as a noncommutative analogue of \({M\times\mathbb{R}}\) has a natural n + 2-dimensional calculus extending Ω¹ and a natural spacetime Laplacian now directly defined by the extra dimension. The case \({M=\mathbb{R}^3}\) recovers the Majid-Ruegg bicrossproduct flat spacetime model and the wave-operator used in its variable speed of light prediction, but now as an example of a general construction. As an application we construct the wave operator on a noncommutative Schwarzschild black hole and take a first look at its features. It appears that the infinite classical redshift/time dilation factor at the event horizon is made finite.     

Second-order superintegrable quantum systems

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n ? 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrödinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.     

Brownian motion on a manifold as a limit of Brownian motions with drift

Let ?ⁿ be n-dimensional Euclidean space and let M ? ?ⁿ be a smooth compact m-dimensional Riemannian manifold (without boundary) embedded in ?ⁿ. By a Brownian motion on M we mean a Markovian process whose transition semigroup is defined by the generator ?½Δ_M, where Δ_M stands for the Laplace-Beltrami operator on M (see, e.g., [2]). This note extends a series of papers in which a measure generated by a Brownian motion on M on the space of trajectories (with values in M) can be represented as the weak limit of measures on the space of trajectories in the ambient space ?ⁿ (see [7–10]). Namely, we claim that a sequence of diffusion processes on ?n which are Brownian motions with drift (in the direction of the manifold) with infinitely increasing modulus converges in distribution to a Brownian motion on the manifold.     

A generalized scheme for designing multistable continuous dynamical systems

In this paper, a generalized scheme is proposed for designing multistable continuous dynamical systems. The scheme is based on the concept of partial synchronization of states and the concept of constants of motion. The most important observation is that by coupling two m-dimensional dynamical systems, multistable nature can be obtained if i number of variables of the two systems are completely synchronized and j number of variables keep a constant difference between them i.e., their differences are constants of motion, where i + j = m and 1 ≤ i, j ≤ m?1. The proposed scheme is illustrated by taking coupled Lorenz systems and coupled chaotic Lorenz-like systems. According to the scheme, two coupled systems reduce to single modified system with some initial condition-dependent parameters. Time evolution plots, phase diagrams, variation of maximum Lyapunov exponent and bifurcation diagrams of the systems are presented to show the multistable nature of the coupled systems.     

Construction of a dynamical matrix for an HCP crystal using phonon data at the symmetry points of the Brillouin zone and elastic constants

General analytical expressions are obtained for the dynamical matrix D(k) and the elastic constants C _ik in an HCP crystal in terms of the Born-von Karman (BvK) parameters. An analytical method is proposed for constructing D(k) on the basis of data about the phonon frequencies ω _i (N) at the symmetry points of the Brillouin zone and the elastic constants C _ik . A number of relations between the values of ω _i (N) and C _ik are presented for conventional interaction models. It is shown that the standard method for determining BvK parameters by fitting them to experimental phonon spectra in HCP lattices is, as a rule, ambiguous, whereas the analytical method proposed allows one to find all the solutions of the problem. The methods developed are illustrated by the construction of dynamical matrices for Tb, Sc, Ti, and Co.     

Effect of pressure on the elastic properties of silicon carbide

The pressure dependences of the second-order elastic constants C _ij and the velocity of sound in 3C-SiC and 2H-SiC crystals are calculated in the framework of the Keating model. The third-order elastic constants C _ijk for 3C-SiC are determined from the dependences of the second-order elastic constants C _ij on the pressure p.     

Lattice dynamics of BiFeO<Subscript>3</Subscript> under hydrostatic pressure

The vibrational frequencies of the BiFeO₃ crystal lattice in the cubic phase (Pm3m) and the rhombohedral paraelectric phase (R3c) are calculated in terms of the ab initio model of an ionic crystal with the inclusion of the dipole and quadrupole polarizabilities. In the ferroelectric phase with the symmetry R3c, the calculated spontaneous polarization of 136 μC cm^?2 agrees well with the experimental data. The dependences of the unit cell volume, the elastic modulus, and the vibrational frequencies on the pressure are calculated. It is found that the frequency of an unstable ferroelectric mode in both the cubic (Pm3m) and rhombohedral (R3c) phases are almost independent of the applied pressure, in contrast to classical ferroelectrics with a perovskite structure, where the ferroelectric instability is very sensitive to a variation in the pressure.     

The effect of non-linear capacitances in the localization properties of aperiodic dual electric transmission lines

This study investigates the localization properties of dual electric transmission lines with non-linear capacitances. The V_C,n voltage across each capacitor is selected as a non-linear function of the electric charge q_n, i.e., V_C,n = q_n(1/C_n -ε_n|q_n|²)where C_n is the linear part of the capacitance and ε_n the amplitude of the non-linear term. We follow a binary distribution of values of ε_n, according to the Thue-Morse m-tupling sequence. The localization behavior of this non-linear case indicates that the case m = 2 does not belong to the m ≥ 3, family because when m changes from m = 2 to m = 3, the number of extended states diminishes dramatically. This proves the topological difference of the m = 2 and m = 3 families. However, by increasing m values, localization behavior of the m-tupling family resembles that of the m = 2, case because the system begins to regain its extended states. The exact same result was obtained recently in the study of linear direct transmission lines with m-tupling distribution of inductances. Consequently, we state that the localization behavior of the m-tupling family as a function of the m value is independent of both the linear and the non-linear system under study, but independent of the kind of transmission line (dual or direct). This is curious behavior of the m-tupling family and thus deserves more scholarly attention.     

Boundary Conditions for Topological Quantum Field Theories,Anomalies and Projective Modular Functors

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level m, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for n-characters on ∞-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for n + 1-dimensional TQFTs produce n-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.     

On Close Relationship Between Classical Time-Dependent Harmonic Oscillator and Non-relativistic Quantum Mechanics in One Dimension

In this paper, I present a mapping between representation of some quantum phenomena in one dimension and behavior of a classical time-dependent harmonic oscillator. For the first time, it is demonstrated that quantum tunneling can be described in terms of classical physics without invoking violations of the energy conservation law at any time instance. A formula is presented that generates a wide class of potential barrier shapes with the desirable reflection (transmission) coefficient and transmission phase shift along with the corresponding exact solutions of the time-independent Schrödinger’s equation. These results, with support from numerical simulations, strongly suggest that two uncoupled classical harmonic oscillators, which initially have a 90° relative phase shift and then are simultaneously disturbed by the same parametric perturbation of a finite duration, manifest behavior which can be mapped to that of a single quantum particle, with classical ‘range relations’ analogous to the uncertainty relations of quantum physics.     

The Coulomb-oscillator relation on <Emphasis Type="Italic">n</Emphasis>-dimensional spheres and hyperboloids

We establish a relation between Coulomb and oscillator systems on n-dimensional spheres and hyperboloids for n≥2. We show that, as in Euclidean space, the quasiradial equation for the (n+1)-dimensional Coulomb problem coincides with the 2n-dimensional quasiradial oscillator equation on spheres and hyperboloids. Using the solution of the Schrödinger equation for the oscillator system, we construct the energy spectrum and wave functions for the Coulomb problem.     

Phase transition in the isotropic <Emphasis Type="Italic">m</Emphasis>-vector model on a random graph

The thermodynamics of a phase transition is investigated within an isotropic m-vector model on a graph with sparse random connections. This model adequately describes superconductivity and superfluidity (m=2), Heisenberg magnets (m=3), and some structural transitions in different systems with macroscopic disorder (gels, composites, and porous media). It is demonstrated that the phase transition is characterized by classical effective-field anomalies. The thermodynamics of the phase transition is also analyzed in terms the proposed model at low temperatures. The dependences of the thermodynamic parameters on the external field, the mean coordination number, and the dimension of the order parameters are determined.     

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

The Rényi entropies R_p [ ρ], p> 0,≠ 1 of the highly-excited quantum states of the D-dimensional isotropicharmonic oscillator are analytically determined by use of the strong asymptotics of theorthogonal polynomials which control the wavefunctions of these states, the Laguerrepolynomials. This Rydberg energetic region is where the transition from classical toquantum correspondence takes place. We first realize that these entropies are closelyconnected to the entropic moments of the quantum-mechanical probability ρ_n(r)density of the Rydberg wavefunctions Ψ_{n,l, { μ}}(r); so, to the?_p-norms of the associated Laguerrepolynomials. Then, we determine the asymptotics n → ∞ of these norms by use of modern techniques ofapproximation theory based on the strong Laguerre asymptotics. Finally, we determine thedominant term of the Rényi entropies of the Rydberg states explicitly in terms of thehyperquantum numbers (n,l), the parameter order p and the universedimensionality D for all possible cases D ≥ 1. We find that (a) theRényi entropy power decreases monotonically as the order p is increasing and (b) thedisequilibrium (closely related to the second order Rényi entropy), which quantifies theseparation of the electron distribution from equiprobability, has a quasi-Gaussianbehavior in terms of D.     

Congratulations to Akusticheskii Zhurnal on the occasion of its 50th anniversary. Vibrations of spherical inclusions in elastic solids

Variational principles are derived for the analysis of dynamical phenomena associated with spherical inclusions embedded in homogeneous isotropic elastic solids. The starting point is Hamilton’s principle, with the material properties assumed to vary only with the radial distance r from the origin. Attention is restricted to disturbances that are symmetric about the polar (z) axis, such that the nonzero displacement components in spherical coordinates, u _r and u_θ, are independent of the polar coordinate φ. The symmetry allows for a decoupling of the polar components, the nth of which is described by U _{r, n} (r, t)P _n (cosθ) and U_{θ, n}(r, t)dP _n /dθ. A variational principle is subsequently derived for the field quantities U _{r, n} and U_{θ, n}. Concepts analogous to those of the theory of matched asymptotic expansions are used to embellish the principle in order to allow for the damping associated with the outward radiation of elastic waves. Examples illustrating the use of the variational principle for formulating plausible lumped-parameter models are given for the cases of n = 0 and n = 1.     

Influence of the dimension of a polycrystalline film and the optical anisotropy of crystallites on the effective dielectric constant of the film

The dimension D of a polycrystalline film and the optical anisotropy m = ε_z/ε_x of uniaxial crystallites with the principal components ε_x = ε_y and ε_z of the tensor of the dielectric constant have been shown to produce a strong influence on the effective dielectric constant ε_D^* and the effective refractive index n_D^* = (ε_D^*)^1/2 of the film in the optical transparency region, as well as on the boundaries of the intervals B_Dl ≤ ε_D^* ≤ B_Du. The intervals Δ₂(m) = B_2l–B_2u and Δ₃(m) = B_3l–B_3u are separated by a gap for m in the range 1 < m < 2, whereas the theoretical dependence ε₂^*(m) is separated by a gap from the interval Δ₃(m) for m in the range 1 < m < 4. This is confirmed by a comparison of the experimental (n_oP) and theoretical (n_D^*) ordinary refractive indices for uniaxial polycrystalline films of the conjugated polymer poly(p-phenylene vinylene) (PPV) with uniaxial crystallites and appropriate values of m. In the visible transparency region of the PPV films with a change in m(λ) in the range 2 < m(λ) < 3 due to the dependence of the components ε_x,z(λ) on the light wavelength λ, the refractive indices n_oP²(λ) = ε_oP(λ) are consistent with the theoretical values of ε₂^*(λ) and lie outside the interval Δ₃(m). For m(λ) > 3 near the electronic absorption band of the crystallites, the values of ε_oP(λ) lie in the region of the overlap of the intervals Δ₂(m) and Δ₃(m). The boundaries mc of the range 1 < m < m_c are determined, for which the interval Δ₂(m) is separated by a gap from the dependences ε₃^*(m) corresponding to the effective medium theory with spherical crystallites and hierarchical models of a polycrystal, as well as from the proposed new dependence ε₃^*(m).     

Multiple exchanges in lepton pair production in high-energy heavy ion collisions

As an archetype reaction for pQCD multigluon hard processes in collisions of ultrarelativistic nuclei, we analyze generic features of lepton pair production via multiphoton processes in peripheral heavy ion scattering. We report explicit results for collisions of two photons from one nucleus with two photons from the other nucleus, 2γ + 2γ → l⁺l^?. The results suggest that the familiar eikonalization of Coulomb distortions breaks down for oppositely moving Coulomb centers. The breaking of eikonalization in QED suggests that multigluon pQCD processes cannot be described in terms of collective nuclear gluon distributions. We discuss a logarithmic enhancement of the contribution from the 2γ + 2γ → l⁺l^? process to production of lepton pairs with large transverse momentum; similar enhancement is absent for the nγ + mγ → l⁺l^? processes with m, n > 2. We comment on the general structure of multiphoton collisions and properties of higher-order terms that cannot be eikonalized.     

Early stages of generation of two-dimensional structures by the Hastings-Levitov method of conformal mapping dynamics

Two-dimensional structures obtained by the Hastings-Levitov conformal mapping were studied for a relatively small number of mappings n. The fractal dimension D of these structures is computed by the recent Davidovitch-Procaccia technique [6] as a function of n. For small n < n₀ (where n₀ is the number of particles at the first layer), D exponentially decreases, which should have supported the conclusion made in [6] about the possibility of determining the fractal dimension with an arbitrary accuracy using a relatively small number of mappings n≈ n₀. On the other hand, it turned out that D irregularly deviates from a certain quantity D₀ depending on the initial size of the bump \(\sqrt {\lambda _0 } \), which contradicts the main assertion of     

Monogamy Relations in Terms of Tsallis-Q Entropy in any Dimensional System

Monogamy of entanglement is a fundamental property of multipartite entangled states. In this article, due to the convexity of Trρ^q with respect to q when q ≥ 1, we give a monogamy-like relation in terms of Tsallis-q entanglement entropy of assistance (TqEEA) for pure states over an n- partite any dimensional system and monogamy-like relations in terms of Tsallis-q entanglement entropy (TqEE) for mixed states for any dimensional system, we also give a lower bound for the TqEE of a four-partite pure state. At last, we show that the generalized W-class states satisfy the polygamy relation in terms of TqEE when q = 2.