Topological characteristics of the spectrum of the Schrödinger operator in a magnetic field and in a weak potential

Conclusion For a general two-dimensional Schrödinger operator with constant magnetic field with rational flux and electric field with periodic potential there is a countable number of dispersion laws E _j (p₁, p₂) for the magnetic-Bloch functions. These dispersion laws form one-dimensional bundles over the torus T² of quasimomenta with fiber C¹(p₁, p₂) and have arbitrary quantum numbers in no way related to each other or to the flux of the external magnetic field for sufficiently high energy levels, where the doubly periodic potential V(x, y) produces only a small perturbation of the Landau levels.All-Union Correspondence Electrotechnical Communications Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol.65, No.3, pp.368–378, December, 1985.     

The Vlasov‐Poisson‐Boltzmann system near Maxwellians

The dynamics of dilute electrons can be modeled by the Vlasov‐Poisson‐Boltz‐mann system, where electrons interact with themselves through collisions and with their self‐consistent electric field. It is shown that any smooth, periodic initial perturbation of a given global Maxwellian that preserves the same mass, momentum, and total energy (including both kinetic and electric energy), leads to a unique global‐in‐time classical solution. The construction of global solutions is based on an energy method with a new estimate of dissipation from the collision: ∫₀^t〈Lf(s), f(s)〉ds is positive definite for solution f(t,x,v) with small amplitude to the Vlasov‐Poisson‐Boltzmann system (1.4). © 2002 Wiley Periodicals, Inc.     

Completeness of Averaged Scattering Solutions and Inverse Scattering at a Fixed Energy

We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials (V, A) where V(x) = O(|x|^?ρ), A(x) = O(|x|^?ρ), |x| → ∞, ρ > 1, are dense in the set of all solutions to the Schrödinger equation that are in L ²(K) where K is any connected bounded open set in ? ⁿ ,n ≥ 2, with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials (V ₁, A ₁) and (V ₂, A ₂) in ? ⁿ , n ≥ 3, have the same scattering matrix at a fixed positive energy and if the electric potentials V _j and the magnetic fields F _j : = curl A _j , j = 1, 2, coincide outside of some ball they necessarily coincide everywhere. In a previous paper of Weder and Yafaev the case of electric potentials and magnetic fields that are asymptotic sums of homogeneous terms at infinity was studied. It was proven that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at a fixed positive energy. The combination of the new uniqueness result of this paper and the result of Weder and Yafaev implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field.     

λφ4model and Higgs mass in standard model calculated by Gaussian effective potential approach with a new regularization-renormalization method

Based on a new regularization-renormalization method, the λφ⁴ model used in standard model (SM) is studied both perturbatively and nonperturbatively by Gaussian effective potential (GEP). The invariant property of two mass scales is stressed and the existence of a (Landau) pole is emphasized. Then after coupling with theSU(2) ×U(1) gauge fields, the Higgs mass in standard model (SM) can be calculated to bem _H≈138 GeV. The critical temperature (T _c ) for restoration of symmetry of Higgs field, the critical energy scale (μ_max, the maximum energy scale under which the lower excitation sector of the GEP is valid) and the maximum energy scale (μ_max, at which the symmetry of the Higgs field is restored) in the SM areT _c ≈476 GeV, μ_c≈0.547 × 10¹⁵ and μ_max≈0.873 × 10¹⁵, respectively. Project supported in part by the National Natural Science Foundation of China.     

New energy identities and super convergence analysis of the energy conserved splitting FDTD methods for 3D Maxwell's equations

The energy‐conserved splitting finite‐difference time‐domain (EC‐S‐FDTD) method has recently been proposed to solve the Maxwell equations with second order accuracy while numerically keep the L² energy conservation laws of the equations. In this paper, the EC‐S‐FDTD scheme for the 3D Maxwell equations is proved to be energy‐conserved and unconditionally stable in the discrete H¹ norm. The EC‐S‐FDTD scheme is of second‐order accuracy both in time step and spatial steps, which suggests the super‐convergence of this scheme in the discrete H¹ norm. And the divergence of the electric field of the EC‐S‐FDTD scheme in the discrete L² norm is second‐order accurate. Numerical experiments confirm our theoretical analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Spin depolarization decay rates in α-symmetric stable fields on cubic lattices

We study the asymptotic, long-time behavior of the energy function where {X_s : 0 ≤ s < ∞} is the standard random walk on the d-dimensional lattice Z^d, 1 < α ≤ 2, and f:R⁺ → R⁺ is any nondecreasing concave function. In the special case f(x) = x, our setting represents a lattice model for the study of transverse magnetization of spins diffusing in a homogeneous, α-stable, i.i.d., random, longitudinal field {λV(x) : x ∈ Z^d} with common marginal distribution, the standard α-symmetric stable distribution; the parameter λ describes the intensity of the field. Using large-deviation techniques, we show that S_c(λ α f) = lim_t→∞ E(t; λ f) exists. Moreover, we obtain a variational formula for this decay rate S_c. Finally, we analyze the behavior S_c(λ α f) as λ → 0 when f(x) = x^β for all 1 ≥ β > 0. Consequently, several physical conjectures with respect to lattice models of transverse magnetization are resolved by setting β = 1 in our results. We show that S_c(λ, α, 1) ≈ λ^α for d ≥ 3, λ^agr;(ln 1/λ)^α−1 in d = 2, and in d = 1. © 1996 John Wiley & Sons, Inc.     

Upper bound on the coarsening rate for an epitaxial growth model

We study a specific example of energy‐driven coarsening in two space dimensions. The energy is ∫|??u|² + (1 ‐ | ?u|²)²; the evolution is the fourth‐order PDE representing steepest descent. This equation has been proposed as a model of epitaxial growth for systems with slope selection. Numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t^1/3, and the energy per unit area decays like t^‐1/3. We prove a weak, one‐sided version of the latter statement: The time‐averaged energy per unit area decays no faster than t^‐1/3. Our argument follows a strategy introduced by Kohn and Otto in the context of phase separation, combining (i) a dissipation relation, (ii) an isoperimetric inequality, and (iii) an ODE lemma. The interpolation inequality is new and rather subtle; our proof is by contradiction, relying on recent compactness results for the Aviles‐Giga energy. © 2003 Wiley Periodicals, Inc.     

Novel energy stable schemes for Swift-Hohenberg model with quadratic-cubic nonlinearity based on the H−1-gradient flow approach

The Swift-Hohenberg model is a very important phase field crystal model which can be described many crystal phenomena. This model with quadratic-cubic nonlinearity based on the H^??1-gradient flow approach is a sixth-order system which satisfies mass conservation and energy dissipation law. The negative energy of this model will bring huge difficulties to energy stability for many existing approaches. In this paper, we consider two linear, second-order and unconditionally energy stable schemes by linear invariant energy quadratization (LIEQ) and modified scalar auxiliary variable (MSAV) approaches. These two schemes will be effective for all negative E₁. Furthermore, we proved that all the semi-discrete schemes are unconditionally energy stable with respect to a modified energy. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.

     

On generalized statistical convergence of double sequences via ideals

The concept of statistical convergence is one of the most active area of research in the field of summability. Most of the new summability methods have relation with this popular method. In this paper we generalize the notions of statistical convergence, (λ, μ)-statistical convergence, (V, λ, μ) summability and (C, 1, 1) summability for a double sequence x = (x _jk ) via ideals. We also establish the relation between our new methods.     

Homogenization of the stationary periodic Maxwell system in the case of constant permeability

The homogenization problem in the small period limit for the stationary periodic Maxwell system in ℝ³ is considered. It is assumed that the permittivity η^ε(x)=η(ε⁻x), ε > 0, is a rapidly oscillating positive matrix function and the permeability μ₀ is a constant positive matrix. For all four physical fields (the electric and magnetic field intensities, the electric displacement field, and the magnetic flux density), we obtain uniform approximations in the L ₂(ℝ³)-norm with order-sharp remainder estimates. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 41, No. 2, pp. 3–23, 2007 Original Russian Text Copyright ? by M. Sh. Birman and T. A. Suslina Dedicated to the memory of the great mathematician Mark Grigor’evich Krein Supported by RFBR grants no. 05-01-01076-a, 05-01-02944-YaF-a.     

The Evolution of the Weyl and Maxwell Fields in Curved Space—Times

The covariant Weyl (spin s = 1/2) and Maxwell (s = 1) equations in certain local charts (u, φ) of a space-time (M, g) are considered. It is shown that the condition g⁰⁰(x) > 0 for all x ε u is necessary and sufficient to rewrite them in a unified manner as evolution equations δ_tφ = L_(s)φ. Here L_(s) is a linear first order differential operator on the pre—Hilbert space (C (U_t, ^2s+1). (…)), where U_t ? IR³ is the image of the coordinate map of the spacelike hyper-surface t = const, and (φ, C) = ?U_t ? *Q d⁽³⁾x with a suitable Hermitian n × n- matrix Q = Q(t,x). The total energy of the spinor field ? with respect to U_t is then simply given by E = 〈?,?〉. In this way inequalities for the energy change rate with respect to time, δ_t|?|² = 2Re (?, L_(s)?) are obtained. As an application, the Kerr—Newman black hole is studied, yielding quantitative estimates for the energy change rate. These estimates especially confirm the energy conservation of the Weyl field and the well—known superradiance of electromagnetic waves.     

On K(3)4 of curves over number fields

In this paper we consider the group K ₄ ⁽³⁾ (F) of a field, in case F is the function field of a smooth geometrically irreducible curve over a number field. We do this using the complexes constructed in [4], together with an auxiliary complex. On the image in K ₄ ⁽³⁾ (F) of those complexes, we derive a formula for the Beilinson regulator, and compute an approximation of the boundary map at the closed points of the curve in the localization sequence. We give a way of finding examples of elliptic curves E with elements in K ₄ ⁽³⁾ (E), and in some cases use computer calculations to check numerically the relation between the regulator and the L-function, as conjectured by Beilinson. Oblatum 10-IX-1995 & 8-I-1996     

High-velocity gas associated ultracompact HII regions

We present the results of a survey for high-velocity¹²CO (1-0) emission associated H₂O masers and ultracompact (UC) HII regions. The aim is to investigate the relationship between H₂O masers, CO high-velocity gas (HVG) and their associated infrared sources. Our sample satisfies Wood & Churchwell criterion. Almost 70 % of the sources have full widths (FWs) greater than 15 km · s^-1 atT* _a = 100 mK and 15 % have FWs greater than 30 km · s^-1 In most of our objects there is excess high velocity emission in the beam. There is a clear correlation between CO line FWs and far-infrared luminosities: the FW increases with the FIR luminosity. The relation suggests that more luminous sources are likely to be more energetic and able to inject more energy into their surroundings. As a result, larger FW of the CO line could be produced. In most of our sources, the velocities of peak of the H₂O emission are in agreement with those of the CO cloud, but a number of them have a large blueshift with respect to the CO peak. These masers might stem from the amplifications of a background source, which may amplify some unobservable weak masers to an observable level.     

A kinetic model for polytropic gases with internal energy

We consider a mixture of N ideal, polytropic gases. Each species is described by a distribution function f_i(t, x, v, I) ≥ 0, 1 ≤ i ≤ N, defined on , and its evolution is governed by a Boltzmann-type equation. In order to recover the energy law of polytropic gases, the authors of [4] proposed a kinetic model in the framework of a weighted L¹ space. Another approach has been developed in [3] in the context of polyatomic gases. Following this previous lead, our model provides a L² framework in both variables v and I, to eventually perform a mathematical study of the diffusion asymptotics, as it was done in [2] for a model without energy exchange. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Embedding results pertaining to strong approximation of Fourier series. IV

Summary We establish an improvement of a recent theorem of S. M. Mazhar which is a generalization of our result and studies the embedding relation between the class W^r H_S ^ω, including only odd functions and a set of functions defined via the strong means of Fourier series of odd continuous functions (see the precise definitions below).     

Binding energy for hydrogen-like atoms in the Nelson model without cutoffs

In the Nelson model particles interact through a scalar massless field. For hydrogen-like atoms there is a nucleus of infinite mass and charge Ze, Z>0, fixed at the origin and an electron of mass m and charge e. This system forms a bound state with binding energy E_bin=me⁴Z²/8π² to leading order in e. We investigate the radiative corrections to the binding energy and prove upper and lower bounds which imply that with explicit coefficient c₀ and independent of the ultraviolet cutoff. c₀ can be computed by perturbation theory, which however is only formal since for the Nelson Hamiltonian the smallest eigenvalue sits exactly at the bottom of the continuous spectrum.     

Asymptotic ordering of distribution functions and convolution semigroups

It is well-known that the setH of distribution functions on [0,∞] supplied with the convolution product * is a semigroup. We introduce a preorder on (H,*) with attractive algebraic properties. The corresponding equivalence relation ≈^W extends the concept of tail-equivalence of distribution functions. We show that the idempotents of the factorsemigroupH _W form a subsemigroup ofH _W.     

Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

In this paper, the weak Galerkin finite element method (WG-FEM) is applied to a pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. A fitted WG-FEM is proposed to approximate the voltage of the pulsed electric model across the physical media involving an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Optimal pointwise-in-time error estimates in L²-norm and H¹-norm are shown to hold for the semidiscrete scheme even if the regularity of the solution is low on the whole domain. Furthermore, a fully discrete approximation based on backward Euler scheme is analyzed and related optimal error estimates are derived.     

Wavelets on the 2-Sphere: A Group-Theoretical Approach

We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the 2-sphere S², based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and ⁺_* for dilations on S², which are embedded into the Lorentz group SO₀(3, 1) via the Iwasawa decomposition, so that X SO₀(3, 1) M Y S O L N, where N . We select an appropriate unitary representation of SO₀(3, 1) acting in the space L²(S², d μ) of finite energy signals on S². This representation is square integrable over X; thus it yields immediately the wavelets on S² and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S² is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R → ∞. Then the parameter space goes into the similitude group of ² and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility.