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     

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.     

A very strong difference property for semisimple compact connected lie groups

Let G be a topological group. For a function f: G → ℝ and h ∈ G, the difference function Δ _h f is defined by the rule Δ _h f(x) = f(xh) − f(x) (x ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H(x + y) = H(x) + H(y) for every x, y ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f: G → ℝ satisfying Δ _h f ∈ F for each h ∈ G, there is an additive function H such that f − H ∈ F. Erdős’ conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by N. G. de Bruijn; later on, F. W. Carroll and F. S. Koehl obtained a similar result for compact Abelian groups and, under the additional assumption that the other one-sided difference function ∇ _h f defined by ∇ _h f(x) = f(xh) − f(x) (x ∈ G, h ∈ G) is measurable for any h ∈ G, also for noncommutative compact metric groups. In the present paper, we consider a narrower class of groups, namely, the family of semisimple compact connected Lie groups. It turns out that these groups admit a significantly stronger difference property. Namely, if a function f: G → ℝ on a semisimple compact connected Lie group has continuous difference functions Δ _h f for any h ∈ G (without the additional assumption concerning the measurability of the functions of the form ∇ _h f), then f is automatically continuous, and no nontrivial additive function of the form H is needed. Some applications are indicated, including difference theorems for homogeneous spaces of compact connected Lie groups.     

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.     

A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces

We study integrable cocycles u(n,x) over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y, e.g. a Cartan–Hadamard space or a uniformly convex Banach space. It is proved that for any y∈Y and almost all x, there exist A≥ 0 and a unique geodesic ray γ (t,x) in Y starting at y such that

Lattice Dislocations in a 1-Dimensional Model

The spectral properties of the Schr?dinger operator T(t)=−d ²/dx ²+q(x,t) in L ²(ℝ) are studied, where the potential q is defined by q=p(x+t), x>0, and q=p(x), x<0; p is a 1-periodic potential and t∈ℝ is the dislocation parameter. For each t the absolutely continuous spectrum σ _ac (T(t))=σ _ac (T(0)) consists of intervals, which are separated by the gaps γ _n (T(t))=γ _n (T(0))=(α _n ⁻,α _n ⁺), n≥1. We prove: in each gap γ _n ≠?, n≥ 1 there exist two unique “states” (an eigenvalue and a resonance) λ _n ^±(t) of the dislocation operator, such that λ _n ^±(0)=α _n ^± and the point λ _n ^±(t) runs clockwise around the gap γ _n changing the energy sheet whenever it hits α _n ^±, making n/2 complete revolutions in unit time. On the first sheet λ _n ^±(t) is an eigenvalue and on the second sheet λ _n ^±(t) is a resonance. In general, these motions are not monotonic. There exists a unique state λ₀(t) in the basic gap γ₀(T(t))=γ₀(T(0))=(−∞ ,α₀ ⁺). The asymptotics of λ _n ^±(t) as n→∞ is determined. Received: 5 April 1999 / Accepted: 3 March 2000     

Super-Diffusivity in a Shear Flow Model¶from Perpetual Homogenization

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dy _t =dω _t −∇Γ(y _t ) dt, y ₀=0 and d=2. Γ is a 2 &\times; 2 skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales Γ₁₂=−Γ₂₁=h(x ₁), with h(x ₁)=∑ _{n =0} ^∞γ _n h ⁿ (x ₁/R _n ), where h ⁿ are smooth functions of period 1, h ⁿ (0)=0, γ _n and R _n grow exponentially fast with n. We can show that y ^t has an anomalous fast behavior (?[|y ^t |²]∼t ^1+ν with ν > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Received: 1 June 2001 / Accepted: 11 January 2002     

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 Measurement of the ψ′ to J/ψ production ratio in 920?GeV proton-nucleus interactions

Ratios of the ψ^′ over the J/ψ production cross sections in the dilepton channel for C, Ti and W targets have been measured in 920 GeV proton-nucleus interactions with the HERA-B detector at the HERA storage ring. The ψ^′ and J/ψ states were reconstructed in both the μ⁺μ^- and the e⁺e^- decay modes. The measurements covered the kinematic range -0.35≤x_F≤0.1 with transverse momentum p_T≤4.5 GeV/c. The angular dependence of the ratio has been used to measure the difference of the ψ^′ and J/ψ polarization. All results for the muon and electron decay channels are in good agreement: their ratio, averaged over all events, is R_ψ′(μ)/R_ψ′(e)=1.00±0.08±0.04. This result constitutes a new, direct experimental constraint on the double ratio of branching fractions, (B^′(μ)B(e))/(B(μ)B^′(e)), of ψ^′ and J/ψ in the two channels. The ψ^′ to J/ψ production ratio is almost constant in the covered x_F range and shows a slow increase with p_T.     

Essentially different distributions of current in the inverse problem for the Grad-Shafranov equation

The paper answers a question debated by physicists for many years. It is proved that, for almost equal gradients of the magnetic flux u at its zero-level curve ∂ω, which is the piecewise smooth boundary of a simply-connected domain ω ⋐ ℝ², the inverse problem for the Grad-Shafranov equation of plasma equilibrium in a tokamak (in the cylindrical approximation) admits essentially different profiles of distributions f _u : ω ∋ (x, y) ↦ f(u(x, y)) = u _xx (x, y) + u _yy (x, y) ⩾ 0 in the class of third-order polynomials f(u) = Σ _m=0³ a _m u ^m .     

Structural studies and T c dependence in La2−x Dy x Ca y Ba2Cu4+y O z type mixed oxide superconductors

A new series of mixed oxide superconductors with the stoichiometric composition La_2−x Dy _x Ca _y Ba₂Cu_4+y O _z (x=0.0 − 0.5, y=2x) has been studied for structural and superconductiong properties. Our earlier studies on La_2−x (Y/Er) _x Ca _y Ba₂Cu_4+y O _z series, show a strong dependence of T _c on hole concentration (p _sh). In the present work, the results of the analysis of the neutron diffraction measurements at room temprerature on x=0.3 and 0.5 samples are reported. It is interesting to know that Ca substitutes for both La and Ba site with concomitant displacement of La onto Ba site. Superconductivity studies show that maximum T _c is obtained for x=0.5, y=1.0 sample (T _c ∼ 75 K), for La_1.5Dy_0.5Ca₁Ba₂Cu₅O _z (La-2125).     

Influence of thermal treatment in oxidizing and reducing atmospheres on the composition and properties of solid solutions based on bismuth vanadate: (Bi,La)4(V,Me)2O11?y (Me = Ga, Zr, Cu)

Ceramic solid solutions (Bi_{1 − y} La _y )₄(V_{1 − x} Me _x )₂O_{11 − z} (x, y < 0.2: Me = Zr, Ga, Fe, Cu) were prepared by solid-state reaction. It was shown that the annealing (973 K, reducing atmosphere H₂/Ar (20/80)) of the samples whose compositions belong to the stability domains of α, β, and γ′ polymorphs increases their electronic conductivity by six orders of magnitude. The samples with low concentrations of dopant cations exhibited good compositional stability and a reversible change in their structure parameters. At the same time, the solid solutions with a high concentration of lanthanum cations and/or copper cations (y, x ∼ 0.1) underwent partial decomposition.     

Dependence of the critical temperature of high-temperature cuprate superconductors on hoppings and spin correlations between CuO<Subscript>2</Subscript> planes

The influence of interlayer hoppings on the superconducting transition temperature (T _c) in bilayer cuprates has been studied. The parameter of hopping between layers is expressed as t _⊥(k) = t _⊥(cos(k _x ) − cos(k _y ))² and treated as a small perturbation for the states of two CuO₂ planes described by the t-t′-t″-J* model. In the generalized mean field approximation for d_{x² - y²}{d_{{x^2} - {y^2}}} symmetry of the superconducting gap, neither the interlayer hopping or exchange interaction, nor the pair hopping between CuO₂ layers provides an additional mechanism of Cooper pair formation or an increase in T _c. In the concentration dependence of T _c, the bilayer splitting of the upper Hubbard band of quasi-holes is manifested as two peaks with temperatures slightly lower than the maximum T _c for a single-layer cuprate. Interlayer antiferromagnetic spin correlations suppress bilayer splitting.     

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.     

Anisotropy of rotation of nitroxide probes in nematogenic liquid crystals

An electron paramagnetic resonance (EPR) line shape simulation for nitroxide spin probes in the motional narrowing region was carried out assuming axially symmetricg andA tensors and with different anisotropies of rotationN (=R _∥/R _⊥) whereR _∥ andR _⊥ are, respectively, elements of the diffusion tensor along and perpendicular to its principal axisz′. In addition, it was assumed that the principal axes of the diffusion tensor coincide with the molecular axes. Each of three casesz′=x,z′=y andz′=z, which result from cyclic permutations of the molecular axesx, y andz with thez′,y′ andx′ axes of the diffusion tensor, yields its typical EPR spectrum characterized by the relative intensities of the low-, center- and high-field lines. The parameter δ defined by and calculable from the intensities of the three lines was found to vary linearly withN for thez′=x andz′=y cases and, as anticipated, to be practically constant at a value of 1 for thez′=z case. This suggested a method for estimatingN for a probe from its EPR spectrum. Experimental spectra over a narrow temperature range (1°C) in the vicinity of the nematic-to-isotropic transition (about 34.6°C) ofN-(4-n-butylbenzilidene)-4-amino-2,2,6,6-tetramethylpiperidine-1-oxide at a mole fraction of 1·10⁻³ in 4-n-pentyl-4′-cyanobiphenyl showed a pattern of peak heights characteristic of thez′=x case with δ values that gave, neglecting effects of the mean field, higher and lowerN values in the nematic and isotropic regions, respectively. Analysis of other similar systems in the literature gave similar results.     

Specific heat of the harmonic oscillator within generalized equilibrium statistics

Summary Within the generalized equilibrium statistics recently introduced by Tsallis (p _n ∝[1−β(q−-1) _εn ]^1/(q−)), we calculate the thermal dependence of the specific heat corresponding to a harmonic-oscillator-like spectrum, namely ε _n ω(n−α) (∀ω>0,n=0,1,2,...). The influences ofq and α are exhibited. Physically inaccessible and/or thermally frozen gaps are obtained in the low-temperature region, and, forq>1, oscillations are observed in the high-temperature region. The specific heat of the two-level system is also shown.     

Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ² subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H ₀+εP(ωt) for ε small. Here H ₀ is the one-dimensional Schr?dinger operator p ²+V, V(x)∼|x|^α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|^β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients. Received: 3 October 2000 / Accepted: 20 December 2000     

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} .     

Effects of time delay on the statistical fluctuations for a bistable system driven by cross-correlated noises

We study the effects of time delay on the normalized correlation function C(s) and the associated relaxation time T _c for a bistable system with correlations between multiplicative and additive white noises under the condition of small time delay. Using the projection operator method, the expressions of T _c and C(s) are obtained. Based on numerical computations, it is found that the delay time τ slows down the rate of fluctuation decay of dynamical variable for the presence of positive feedback intensity (∈ > 0), while speeds up the rate of fluctuation decay of dynamical variable for the presence of negative feedback intensity (∈ < 0). The effects of the delay time τ on the T _c and C(s) are entirely opposite for ∈ 〉 0 and ∈ < 0.      

Overall values of conventional discrepancy indices for crystals with heavy atoms. Part II. Results for monoclinic and orthorhombic crystals when the heavy-atom part alone constitutes the model

Theoretical expressions of 〈y _N〉, 〈|y _N − σ₁ y _P ^c |〉 and 〈|y _N ² −σ ₁ ² (y _P ^c )²|〉 (wherey _N andy _P ^c are the normalized structure amplitudes of the structure and the model respectively) are derived in terms of the heavy atom contributionσ ₁ ² for monoclinic and orthorhombic crystals containing a few (i.e., 1 or 2) heavy atoms of the same kind per asymmetric unit by taking the heavy atom part alone as the model. Results are obtained for both the related and unrelated cases. The local values of 〈y _N〉 and 〈|y _N ⁿ − σ ₁ ⁿ (y _P ^c ) ⁿ |〉, (n=1, 2) calculated from these expressions can be used to calculate the overall values of the conventionalR-indicesR(F) andR(I) for the related and unrelated cases. These overall values could be used to check the correctness of heavy atoms located in the structure. Contribution No. 550