(GL(<Emphasis Type="Italic">n</Emphasis> + 1, ℝ), GL(<Emphasis Type="Italic">n</Emphasis>, ℝ)) is a generalized Gelfand pair

Denote by G = GL(n + 1, ℝ) the group of invertible (n + 1) × (n + 1) matrices with real entries, acting on ℝ ⁿ⁺¹ in the usual way, and let H ₁ = GL(n, ℝ) be the stabilizer of the first unit vector e ₀. Let H ₀ = GL(1, ℝ) and set H = H ₀ × H ₁. It is known that the pair (G,H) is a generalized Gelfand pair. Define a character χ of H by χ(h) = χ(h ₀ h ₁) = χ0(h ₀) where χ0 is a unitary character of H ₀ (h ₀ ∈ H ₀, h ₁ ∈ H ₁). Let σ be the anti-involution on G given by σ(g) = ^t g. In this note, we show that any distribution T on G satisfying T(h ₁ gh ₂) = χ(h ₁ h ₂) T(g) (g ∈ G; h ₁, h ₂ ∈ H) is invariant under the anti-involution σ. This result implies that (G,H ₁) is a generalized Gelfand pair.     

Inverse problem for the Grad-Shafranov equation with affine right-hand side

For a given domain ω ⋐ ℝ² with boundary γ = ∂ω, we study the cardinality of the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) of pairs of numbers (a, b) for which there is a function u = u _(a,b): ω → ℝ such that ∇² u(x) = au(x) + b ⩾ 0 for x ∈ ω, u| _γ = 0, and ||∇u(s)| − Φ(s) ⩽ η for s ∈ γ. Here η ⩾ 0 stands for a very small number, Φ(s) = |∇(s)| / ∫ _γ |∇v| d γ, and v is the solution of the problem ∇² v = a ₀ v + 1 ⩾ 0 on ω with v| _γ = 0, where a ₀ is a given number. The fundamental difference between the case η = 0 and the physically meaningful case η > 0 is proved. Namely, for η = 0, the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) contains only one element (a, b) for a broad class of domains ω, and a = a ₀. On the contrary, for an arbitrarily small η > 0, there is a sequence of pairs (a _j , b _j ) ∈ $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) and the corresponding functions u _j such that ‖f _u ^j+1‖ − ‖f _u ^j ‖ > 1, where ‖f _u ^j = max _x∈ω |f _u ^j (x)| and f _u ^j (x) = a _j u _j (x) + b _j . Here the mappings f _u ^j : ω → ℝ necessarily tend as j → ∞ to the δ-function concentrated on γ.     

Asymptotic behavior of orbits of C 0-semigroups and solutions of linear and semilinear abstract differential equations

In this paper, we study the asymptotic behavior of solutions of semilinear abstract differential equations (*) u′(t) = Au(t) + t ⁿ f(t, u(t)), where A is the generator of a C ₀-semigroup (or group) T(·), f(·, x) ∈ A for each x ∈ X, A is the class of almost periodic, almost automorphic or Levitan almost periodic Banach space valued functions ϕ: ℝ → X and n ∈ {0, 1, 2, ...}. We investigate the linear case when T(·)x is almost periodic for each x ∈ X; and the semilinear case when T(·) is an asymptotically stable C ₀-semigroup, n = 0 and f(·, x) satisfies a Lipschitz condition. Also, in the linear case, we investigate (*) when ϕ belongs to a Stepanov class S ^p-A defined similarly to the case of S ^p-almost periodic functions. Under certain conditions, we show that the solutions of (*) belong to A _u:= A ∩ BUC(ℝ, X) if n = 0 and to t ⁿ A _u ⊕ w _n C ₀ (ℝ, X) if n ∈ ℕ, where w _n(t) = (1 + |t|)ⁿ. The results are new for the case n ∈ ℕ and extend many recent ones in the case n = 0. Dedicated to the memory of B. M. Levitan     

Dynamical Localization of Quantum Walks in Random Environments

We study shock statistics in the scalar conservation law ∂ _t u+∂ _x f(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Propagation of electromagnetic waves generated by moving sources in dispersive media

The propagation of electromagnetic waves issued by modulated moving sources of the form j( t,x ) = a( t )e ^{- iw₀ t} [(x)\dot]₀ ( t )d( x - x₀ ( t ) )j\left( {t,x} \right) = a\left( t \right)e^{ - i\omega _0 t} \dot x_0 \left( t \right)\delta \left( {x - x_0 \left( t \right)} \right) is considered, where j(t, x) stands for the current density vector, x = (x ₁, x ₂, x ₃) ∈ ℝ³ for the space variables, t ∈ ℝ for time, t → x ₀(t) ∈ ℝ³ for the vector function defining the motion of the source, ω ₀ for the eigenfrequency of the source, a(t) for a narrow-band amplitude, and δ for the standard δ function. Suppose that the media under consideration are dispersive. This means that the electric and magnetic permittivity ɛ(ω), μ(ω) depends on the frequency ω. We obtain a representation of electromagnetic fields in the form of time-frequency oscillating integrals whose phase contains a large parameter λ > 0 characterizing the slowness of the change of the amplitude a(t) and the velocity [(x)\dot]₀ ( t )\dot x_0 \left( t \right) and a large distance between positions of the source and the receiver. Applying the two-dimensional stationary phase method to the integrals, we obtain explicit formulas for the electromagnetic field and for the Doppler effects. As an application of our approach, we consider the propagation of electromagnetic waves produced by moving source in a cold nonmagnetized plasma and the Cherenkov radiation in dispersive media.     

Minimum Polynomial for Single-Mode Parabose Parasupercharge and Hamiltonian

We calculate the minimum polynomial φ(x,y) of parasupercharge Q and Hamiltonian H for single-mode parabose parasupersymmetry (P-PSUSY). Suppose that φ(x,y) satisfies the homogeneity ∀ λ∈ℝ,φ(λ x,λ ² y)=λ ⁿ φ(x,y), then the parafermionic order p _f is restricted to either 1, 2, or 4. Under the P-PSUSY, the homogeneity of the φ(x,y) is equivalent to the parasuperconformality of Q and H. The physical meaning of the parasuperconformality is discussed, in connection with the spin of the elementary particle.     

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     

Harness Processes and Non-Homogeneous Crystals

We consider the Harmonic crystal, a measure on with Hamiltonian H(x)=∑ _i,j J _i,j (x(i)−x(j))²+h∑ _i (x(i)−d(i))², where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤ ^d . The configuration d is given and considered as observations. The ‘couplings’ J _i,j are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration m corresponding to the Hamiltonian.     

Approximate Homomorphisms of Ternary Semigroups

A mapping f : (G ₁,[ ]₁)→ (G ₂,[ ]₂) between ternary semigroups will be called a ternary homomorphism if f([xyz]₁)=[f(x)f(y)f(z)]₂. In this paper, we prove the generalized Hyers–Ulam–Rassias stability of mappings of commutative semigroups into Banach spaces. In addition, we establish the superstability of ternary homomorphisms into Banach algebras endowed with multiplicative normsMathematics Subject Classifications (2000). Primary 39B52, Secondary 39B82, 46B99, 17A40     

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

The Asymptotic Limits of Zero Modes of Massless Dirac Operators

Asymptotic behaviors of zero modes of the massless Dirac operator H = α · D + Q(x) are discussed, where α = (α₁, α₂, α₃) is the triple of 4 × 4 Dirac matrices, , and Q(x) = (q _jk (x)) is a 4 × 4 Hermitian matrix-valued function with | q _jk (x) | ≤ C 〈x〉^−ρ, ρ > 1. We shall show that for every zero mode f, the asymptotic limit of |x|² f (x) as |x| → + ∞ exists. The limit is expressed in terms of the Dirac matrices and an integral of Q(x) f (x).      

Spherical Symmetric Solution in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) Model Around Charged Black Hole

A static, asymptotically flat, spherically symmetric solutions is investigated in f(R) theories of gravity for a charged black hole. We have studied the weak field limit of f(R) gravity for the some f(R) model such as f(R)=R+ε h(R). In particular, we consider the case lim  _R→0 h(R)/h′(R)→0 and find the space time metric for f(R)=R+[(m⁴)/(R)]f(R)=R+{\mu^{4}\over R} and f(R)=R ^1+ε theories of gravity far away a charged mass point.     

Method of Quantum Characters in Equivariant Quantization

 Let G be a reductive Lie group, g its Lie algebra, and M a G-manifold. Suppose 𝔸 _h (M) is a 𝕌 _h (g)-equivariant quantization of the function algebra 𝔸(M) on M. We develop a method of building 𝕌 _h (g)-equivariant quantization on G-orbits in M as quotients of 𝔸 _h (M). We are concerned with those quantizations that may be simultaneously represented as subalgebras in 𝕌^* _h (g) and quotients of 𝔸 _h (M). It turns out that they are in one-to-one correspondence with characters of the algebra 𝔸 _h (M). We specialize our approach to the situation g=gl(n,ℂ), M=End(ℂ ⁿ ), and 𝔸 _h (M) the so-called reflection equation algebra associated with the representation of 𝕌 _h (g) on ℂ ⁿ . For this particular case, we present in an explicit form all possible quantizations of this type; they cover symmetric and bisymmetric orbits. We build a two-parameter deformation family and obtain, as a limit case, the 𝕌(g)-equivariant quantization of the Kirillov-Kostant-Souriau bracket on symmetric orbits. Received: 28 April 2002 / Accepted: 3 October 2002 Published online: 24 January 2003 RID="*" ID="*" This research is partially supported by the Israel Academy of Sciences grant no. 8007/99-01. Communicated by L. Takhtajan     

Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order <Emphasis Type="Italic">α</Emphasis>

In a first stage, the paper deals with the derivation and the solution of the equation of the probability density function of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E _α(h^αD _x ^α)f(x) where E _α() denotes the Mittag-Leffler function, and D _x ^α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value of the function under consideration. The corresponding fractional linear partial differential equation is solved by using a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example, one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.      

Generalized holographic and Ricci dark energy models

In this paper, we consider generalized holographic and Ricci dark energy models where the energy densities are given as ρ _R =3c ² M _pl² Rf(H ²/R) and ρ _h =3c ² M _pl² H ² g(R/H ²), respectively; here f(x), g(y) are positive defined functions of the dimensionless variables H ²/R or R/H ². It is interesting that holographic and Ricci dark energy densities are recovered or recovered interchangeably when the function f(x)=g(y)≡1 or f(x)=Id and g(y)=Id are taken, respectively (for example f(x),g(x)=1−ε(1−x), ε=0or1, respectively). Also, when f(x)≡xg(1/x) is taken, the Ricci and holographic dark energy models are equivalent to a generalized one. When the simple forms f(x)=1−ε(1−x) and g(y)=1−η(1−y) are taken as examples, by using current cosmic observational data, generalized dark energy models are considered. As expected, in these cases, the results show that they are equivalent (ε=1−η=1.312), and Ricci-like dark energy is more favored relative to the holographic one where the Hubble horizon was taken as an IR cut-off. And the suggested combination of holographic and Ricci dark energy components would be 1.312R−0.312H ², which is 2.312H²+1.312[(H)\dot]2.312H^{2}+1.312\dot{H} in terms of H ² and [(H)\dot]\dot{H} .     

No Phase Transition for Gaussian Fields with Bounded Spins

Let a<b, and H be the (formal) Hamiltonian defined on Ω by

Entropy for Zero-Temperature Limits of Gibbs-Equilibrium States for Countable-Alphabet Subshifts of Finite Type

Let Σ _A be a finitely primitive subshift of finite type over a countable alphabet. For suitable potentials f : Σ _A → ℝ we can associate an invariant Gibbs equilibrium state μ _tf to the potential tf for each t ≥ 1. In this note, we show that the entropy h(μ _tf ) converges in the limit t→ ∞ to the maximum entropy of those invariant measures which maximize ∫ f dμ. We further show that every weak-* accumulation point of the family of measures μ _tf has entropy equal to this value. This answers a pair of questions posed by O. Jenkinson, R. D. Mauldin and M. Urbański.     

On the mechanism of phase transformation in Ag2Se

A differential thermal analysis ΔT _y (T) in vacuum has been performed, and the temperature gradient ΔT _x (T) along the Ag₂Se sample during the transition α → β has been studied. It has been shown that the transitions α → α′ and β′ → β are displacive transitions and that the transition α′ → β′ is a reconstructive transition. It has been found that the temperature gradient along the sample during the transition α′ → β′ passes through a deep minimum due to a strong increase in the specific heat capacity.     

A regular potential which is nowhere in L 1

A nonnegative potential V: ℝ^v→ℝ is constructed for which V ∉ L _q (G) for any nonempty open G⊂→^v, q>0, and for which nevertheless W _inf2 ^sup1 ∩ Q(V) is dense in W _inf2 ^sup1 , i.e., is a form core for −1/2Δ in L ₂.     

Effective nonlinear optical properties of shape distributed composite media

The effective linear and nonlinear optical properties of metal/dielectric composite media, in which ellipsoidal metal inclusions are distributed in shape, are investigated. The shape distribution function P(L _x, L _y) is assumed to be 2Δ^-2θ(L _x - 1/3 + Δ/3)θ(L _y - 1/3 + Δ/3)θ(2/3 + Δ/3 - L _x - L _y), where θ( ^{. . .} ) is the Heaviside function, Δ is the shape variance and L_i are the depolarization factors of the ellipsoidal inclusions along i-symmetric axes (i = x, y). Within the spectral representation, we adopt Maxwell-Garnett type approximation to study the effect of shape variance Δ on the effective nonlinear optical properties. Numerical results show that both the effective linear optical absorption α ∼ ωIm() and the modulus of the effective third-order optical nonlinearity enhancement |χ⁽³⁾ _e|/χ⁽³⁾ ₁ exhibit the nonmonotonic behavior with Δ. Moreover, with increasing Δ, the optical absorption and the nonlinearity enhancement bands become broad, accompanied with the decrease of their peaks. The adjustment of Δ from 0 to 1 allows us to examine the crossover behavior from no separation to large separation between optical absorption and nonlinearity enhancement peaks. As Δ → 0, i.e., the ellipsoidal shape deviates slightly from the spherical one, the dependence of |χ⁽³⁾ _e|/χ⁽³⁾ ₁ on Δ becomes strong first and then weak with increasing the imaginary part of inclusions' dielectric constant. In the dilute limit, the exact formula for the effective optical nonlinearity is derived, and the present approximation characterizes the exact results better than old mean field one does. Received 10 December 2002 Published online 4 June 2003 RID="a" ID="a"e-mail: lgaophys@pub.sz.jsinfo.net