Inhomogeneous contact processes on trees

We consider an inhomogeneous contact process on a tree of degreek, where the infection rate at any site isλ, the death rate at any site in isδ (with 0 <δ ⩽ 1) and that at any site in is 1. Denote by the critical value for thehomogeneous model (i.e.,δ=1) on and byϑ(δ, λ) the survival probability of the inhomogeneous model on . We prove that whenk > 4, if , a subtree embedded in , with 1 ⩽σ ⩽ √k, then three existsδ _c ^σ strictly between ( ) and 1 such that ( ) whenδ >δ _c ^σ andϑ(δ, λ _c( ) > 0 whenδ <δ _c ^σ ; ifS={o}, the origin of , then for anyδ ε (0, 1).     

A study of the nuclear-medium influence on neutral-strange-particle production in deep-inelastic neutrino scattering

The influence of nuclear effects on the production of neutral strange particles (V ⁰) is investigated using the data obtained with SKAT propane-freon bubble chamber irradiated in the neutrino beam (with E _v =3−30 GeV) at the Serpukhov accelerator. The mean multiplicity of V ⁰ particles in nuclear interactions, , is found to exceed significantly that in “quasideuteron” interactions, . The ratio of is larger than that for π⁻ mesons, . It is shown that the multiplicity gain of V ⁰ particles can be explained by intranuclear interactions of produced pions. The text was submitted by the authors in English.     

Reflectivity and dynamical conductivity ofn-type HgCr2Se4

Summary Reflectivity measurements have been done on dopedn-type HgCr₂Se₄ (n≈(10¹⁵÷10¹⁹) cm⁻³), at various temperatures ((4÷300) K) in the infra-red range . The spectra show the structure of a plasma edge and reststrahlbands. Data have been analysed by a Kramers-Kronig procedure. The deduced dynamical conductivity shows an anomaly near the plasma frequency ω_p. The transmitivity measurements show a strong temperature dependence of the conduction band. Paper presented at the ?V International Conference on Ternary and Multinary Compounds?, held in Cagliari, September 14–16, 1982.     

Classification of Simple Linearly Compact n-Lie Superalgebras

We classify simple linearly compact n-Lie superalgebras with n > 2 over a field ${\mathbb{F}}We classify simple linearly compact n-Lie superalgebras with n > 2 over a field \mathbbF{\mathbb{F}} of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive \mathbbZ{\mathbb{Z}}-graded Lie superalgebras of the form L=?_j=-1^n-1 L_j{L=\oplus_{j=-1}^{n-1} L_j}, where dim L _n−1 = 1, L ₋₁ and L _n−1 generate L, and [L _j , L _n−j−1] = 0 for all j, thereby reducing it to the known classification of simple linearly compact Lie superalgebras and their \mathbbZ{\mathbb{Z}}-gradings. The list consists of four examples, one of them being the n + 1-dimensional vector product n-Lie algebra, and the remaining three infinite-dimensional n-Lie algebras.     

L p -Boundedness of the Wave Operator for the One Dimensional Schrödinger Operator

Given a one dimensional perturbed Schrödinger operator H =  ? d ²/dx ² + V(x), we consider the associated wave operators W  _± , defined as the strong L ² limits $\lim_{s\to\pm\infty}e^{isH}e^{-isH_{0}}Given a one dimensional perturbed Schr?dinger operator H = − d ²/dx ² + V(x), we consider the associated wave operators W _± , defined as the strong L ² limits . We prove that W _± are bounded operators on L ^p for all 1 < p < ∞, provided , or else and 0 is not a resonance. For p = ∞ we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.     

Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators

We consider the class of matrix h-pseudodifferential operators Op _h (a) with symbols a = (a _ij ) _i,j=1 ^N , where the coefficients a _ij ∈ C ^∞(? _x ⁿ × ? _ξ ⁿ ? C(0, 1] satisfy the estimates |? _x ^β g6 _ξ ^α α _ij (x, ξ, h)| ? C _αβ 〈ξ〉 ^m and 〈ξ〉 = (1 + |ξ|²)^1/2 for every multi-indices α, β. We also assume that a _ij (x, ξ) is analytically continued with respect to ξ to a tube domain ? ⁿ + i $ \mathcal{B} We consider the class of matrix h-pseudodifferential operators Op _h (a) with symbols a = (a _ij ) _i,j=1^N, where the coefficients a _ij ∈ C ^∞(ℝ _x ⁿ × ℝ _ξ ⁿ ⊗ C(0, 1] satisfy the estimates |ϖ _x ^β g6 _ξ ^α α _ij (x, ξ, h)| ⩽ C _αβ 〈ξ〉 ^m and 〈ξ〉 = (1 + |ξ|²)^1/2 for every multi-indices α, β. We also assume that a _ij (x, ξ) is analytically continued with respect to ξ to a tube domain ℝ ⁿ + i , where is a bounded domain in ℝ ⁿ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let H _h ^s (ℝ ⁿ , ℂ ^N ) be the space of distributions with values in ℂ ^N which is equipped with the norm , let Ω ⊂ ℝ ⁿ be a bounded open set, let v ∈ C ^∞(ℝ ⁿ ), let ▿v(x) ∈ for any x ∈ Ω, and let . Let u _h (∈ H _h ^s (ℝ ⁿ ,‒ ^N )) be a solution of the equation Op _h (α)u = 0. In this case, for every ϕ ∈ C ₀^∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h ₀ > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h ₀]:

((1))

We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schr?dinger, Dirac, and square-root Klein-Gordon operators. To the memory of Professor V. A. Borovikov     

Bounds on neutrino mixing with exotic singlet neutrinos

We examine the effects of mixing induced non-diagonal light-heavy neutrino weak neutral currents on the amplitude for the process (with a=e, μ or τ). By imposing constraint that the amplitude should not exceed the perturbative unitarity limit at high energy , we obtain bounds on light-heavy neutrino mixing parameter sin² where is the mixing angle. In the case of one heavy neutrino (mass mξ) or mass degenerate heavy neutrinos, for Λ=1 TeV, no bound is obtained for mξ<0.50 TeV. However, sin² ≤3.8 × 10⁻⁶ for mξ=5 TeV and sin ≤6.0 × 10⁻⁸ for mξ=10 TeV. For Λ=∞, no constraint is obtained for mξ<0.99 TeV and sin² ≤3.8 × 10⁻² (for mξ=5 TeV) and sin² ≤9.6 × 10⁻³ (for mξ=10 TeV).     

Neutron diffraction studies of the negative thermal expansion in a layered indium selenide crystal

The neutron diffraction patterns have been analyzed for a layered single crystal and a powder of the γ-polytype of indium selenide in the temperature range 10–300 K. In the temperature range 10–50 K, the excitation of bending vibrations due to the charge density waves changes the phonon spectrum and gives rise to a negative thermal expansion in the plane of layers, i.e., α_‖c = −2.2 × 10⁻⁶ K⁻¹, which is characteristic of two-dimensional structures. The average (over the range T = 50–300 K) coefficients of thermal expansion along the principal crystallographic directions have been calculated: $ \bar \alpha _{ \bot c} $ \bar \alpha _{ \bot c} = 10.48 × 10⁻⁶ K⁻¹ and $ \bar \alpha _{\parallel c} $ \bar \alpha _{\parallel c} = 12.97 × 10⁻⁶ K⁻¹, which agree with the X-ray diffraction data previously obtained by the authors at T = 290 K.     

On the Schrödinger equation and the eigenvalue problem

If _k is thek ^th eigenvalue for the Dirichlet boundary problem on a bounded domain in ⁿ , H. Weyl's asymptotic formula asserts that , hence . We prove that for any domain and for all . A simple proof for the upper bound of the number of eigenvalues less than or equal to - for the operator –V(x) defined on ⁿ (n3) in terms of is also provided.Research partially supported by a Sloan Fellowship and NSF Grant No. 81-07911-A1     

A nonlinear instability for 3×3 systems of conservation laws

The phenomenon of nonlinear resonance provides a mechanism for the unbounded amplification of small solutions of systems of conservation laws. We construct spatially 2-periodic solutionsu ^N C ([0,t _N ] × witht _N bounded, satisfying

Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term that depends on theS 2−P 2 invariant

Exact plane-symmetric solutions of the spinor-field equation with zero mass parameter and nonlinear term that depends arbitrarily on the S²−P² invariant are derived with consideration of an intrinsic gravitational field. The existence of regular solutions with localized energy density among the solutions obtained is investigated. Equations with powerlaw and polynomial nonlinearity types are examined in detail. For the power-law nonlinearity, when the nonlinear term entering into the Lagrangian has the form L_N=γIⁿ, where γ is the nonlinearity parameter and n=const, it is shown that the initial system of Einstein and spinor-field equations has regular solutions with localized energy density only under the conditions λ=−Λ² < 0, n > 1. In this case, the examined field configuration posesses a negative energy. In the case of polynomial nonlinearity, regular solutions with localized energy density T ₀ ⁰ (x), positive energy (upon integration over y and z between finite limits), and an everywhere regular metric that transforms into a two-dimensional space-time metric at spatial infinity are obtained. It is shown that the initial nonlinear spinor-field equations in two-dimensional space-time have no solutions with localized energy density. Thus, it is established that the intrinsic gravitational field plays a regularizing role in the frmation of regular localized solutions to the examined nonlinear spinor-field equations. Russian University of People's Friendship. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 12–19, November, 1999.     

A sharp decrease of resistivity in La0.7Ca0.3Mn0.96Cu0.04O3: Evidence for Cu-assisted coherent tunneling of spin Polarons

Nearly a 50% decrease of the resistivity ρ(T, x) is observed upon just 4% Cu doping at the Mn site of La_2/3Ca_1/3Mn_1−x Cu_xO₃. When the observed phenomenon is attributed to a decrease of the spin-polaron energy E _σ(x) below T _C (x), all of the data are found to be well fitted by the nonthermal coherent tunneling expression , assuming that the magnetization in the ferromagnetic state is given by the expression . The best fits through all the data points suggest and E _σ(x)≃E _σ(0)(1−x)⁴ for the explicit x dependence of the Cu-induced modifications of the Mn-spin-dominated zero-temperature spontaneous magnetization, residual paramagnetic contribution, and spin-polaron tunneling energy, respectively, with E _σ(0)=0.12 eV. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 11, 812–815 (10 June 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

A Note on the Extreme Points of Positive Quantum Operations

In this paper, an error in the proof of Theorem 4.9 in Gudder’s paper (Int. J. Theor. Phys. 47(1):268–279, 2008) is pointed out and it is proved that if such that E _i ∈ℂI∖{0} and E _j ∉ℂI for some i,j in {1,2,…,n}, then . This subject is supported by the NNSF of China (No. 10571113, 10871224).     

On the Best Constant in the Moser-Onofri-Aubin Inequality

Let S ² be the 2-dimensional unit sphere and let J _α denote the nonlinear functional on the Sobolev space H ¹(S ²) defined by

Brownian motion in a convex ring and quasi-concavity

LetX be the Brownian motion in ? ⁿ and denote by τ _M the first hitting time ofM?? ⁿ . Given convex setsK?L?? ⁿ we prove that all the level sets $$\{ \left( {x,t} \right) \in \mathbb{R}^n \times [0, + \infty [;P_x [\tau _K \leqq t \wedge \tau _{L^c } ] \geqq \lambda \} ,\lambda \in \mathbb{R}$$ are convex.     

NMR dynamics in disordered magnets

The excitation dynamics of site diluted magnets can be described at low energies (long length scales) by magnons, and above a crossover frequency, ω_c, (short length scales) by fractons. The density of fracton states is given by , where is the fracton dimensionality. Dilution gives rise to a characteristic length ξ∝(p−p _c)^ν, wherep _c is the critical concentration for (magnetic) percolation. The crossover frequency ω_c is proportional to ξ^-1[1+(θ/2)], where θ is the rate at which the diffusion constant decays with distance for diffusion on an equivalent network. A fractal dimensionD describes the density of magnetic sites on the infinite network, and . For percolating networks, for all dimensions ≥2. Neutron scattering structure factor measurements by Uemura and Birgeneau compare well with calculations using fracton concepts. Magnons are extended at low energies, while the fracton states are geometrically localized, with a wave function envelope proportional to exp . Here, is the fracton length scale at frequency ω. The exponentd _ϕ lies between 1 andd _min, the chemical length index (of the order of 1.6 in three dimensions). The localization of the magnetic excitations causes a spread in the NMR relaxation rates. A given nuclear moment will experience only a limited set of fracton excitations, resulting in an overall non-exponential decay of the NMR relaxation signal. When strong cross-relaxation is present, the relaxation will be exponential, but the temperature dependence will be strongly altered from the concentrated result.