Existence of many ergodic absolutely continuous invariant measures for piecewise-expandingC 2 chaotic transformations in ℝ2 on a fixed number of partitions

Let Ω be a region in ℝⁿ and letp = P_i ) _i ^1m , be a partition ofΩ into a finite number of closed subsets having piecewise C² boundaries of finite(n - 1 )dimensional measure. Let τ:Ω→Ω be piecewise C² onP where, τ_i = τ|p_i is aC ² diffeomorphism onto its image, and expanding in the sense that there exists α > 1 such that for anyi = 1, 2,...,m ‖Dτ_i ^-1 ‖ < α^-1, where Dτ_i ^-1 is the derivative matrixτ _i ^- 1 and |‖·‖ is the Euclidean matrix norm. By means of an example, we will show that the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, we will construct a piecewise expanding C² transformation on a fixed partition with a finite number of elements in ℝ², but which has an arbitrarily large number of ergodic, absolutely continuous invariant measures     

The Behaviour of Aging Functions in One-Dimensional Bouchaud's Trap Model

Let τ_i be a collection of i.i.d. positive random variables with distribution in the domain of attraction of an α-stable law with α<1. The symmetric Bouchaud's trap model on ℤ is a Markov chain X(t) whose transition rates are given by w_xy=(2τ_x)⁻¹ if x, y are neighbours in ℤ. We study the behaviour of two correlation functions: ℙ[X(t_w+t)=X(t_w)] and It is well known that for any of these correlation functions a time-scale t=f(t_w) such that aging occurs can be found. We study these correlation functions on time-scales different from f(t_w), and we describe more precisely the behaviour of a singular diffusion obtained as the scaling limit of Bouchaud's trap model. Work supported by DFG Research Center Matheon ``Mathematics for key technologies'     

Anomalous light-induced drift of linear molecules

The sudden approximation in energy is used to derive analytic formulas that describe the anomalous light-induced drift (LID) of linear molecules absorbing radiation in the rovibrational transition nJ _i −mJ _f (n and m are the ground and excited vibrational states, and J _α is the rotational quantum number in the vibrational state α=m, n). It is shown that for all linear molecules with moderate values B≲1 cm⁻¹ of the rotational constant, anomalous LID can always by observed under the proper experimental conditions; temperature T, rotational quantum number J _i , and type of transition (P or R). The parameter γ=B[J _i (J _i +1)−J _f (J _f +1)] ν _n /2k _BT (ν _m −ν _n ) is used to derive a condition for observing anomalous LID: γ∼1 (k _B is the Boltzmann constant and ν _α is the transport rate of collisions of molecules in the vibrational state α and buffer particles at moderate molecular velocities , where is the most probable velocity of the buffer particles). For ν _m >ν _n anomalous LID can be observed only in P-transitions, while for ν _m <ν _n it can be observed only in R-transitions. It is shown that anomalous LID is possible for all ratios β=M _b /M of the masses of the buffer particles (M _b ) and of the resonant particles (M) and any absorption-line broadening (Doppler or homogeneous). The optimum conditions for observing anomalous LID are realized when the absorption line is Doppler-broadened in an atmosphere of medium-weight (β∼1) and heavy (β≫1) buffer particles. In this case, anomalous LID can be observed in the same transition within a broad temperature interval ΔT∼T. If the buffer particles are light (β≪1) or if the broadening of the absorption line is homogeneous, anomalous LID in the same transition can be observed only within a narrow temperature range ΔT≪T. Zh. éksp. Teor. Fiz. 115, 1664–1679 (May 1999)     

Fractional Moment Bounds and Disorder Relevance for Pinning Models

We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n ^−α-1 L(n), with α ≥ 0 and L(·) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For α < 1/2 disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [3,28]. The same has been proven also for α = 1/2, but under the assumption that L(·) diverges sufficiently fast at infinity, a hypothesis that is not satisfied in the (1 + 1)-dimensional wetting model considered in [12,17], where L(·) is asymptotically constant. Here we prove that, if 1/2 < α < 1 or α > 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case α = 1/2, under the assumption that L(·) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered in [12,17] is out of our analysis and remains open. The results are achieved by setting the parameters of the model so that the annealed system is localized, but close to criticality, and by first considering a quenched system of size that does not exceed the correlation length of the annealed model. In such a regime we can show that the expectation of the partition function raised to a suitably chosen power is small. We then exploit such an information to prove that the expectation of the same fractional power of the partition function goes to zero with the size of the system, a fact that immediately entails that the quenched system is delocalized.     

Absorption spectra and magneto-optic figures of merit in the Bi x Sm3-x Fe5-y Ga y O12 system

The absorption (α) and Faraday rotation (θ) spectra of 14 garnets belonging to the series Bi _x Sm_3-x Fe_5-y Ga _y O₁₂ (0<x<1.05, 0.8<y<1.15) have been measured between 15 000 cm⁻¹ and 19 000 cm⁻¹. The figure of merit (θ/α) at 17 850 cm⁻¹ (560 nm) increases linearly with increasing bismuth concentration up tox∼0.6 where it begins to increase less rapidly. For operation of magneto-optic display devices at 17 850 cm⁻¹ there is no advantage in using garnets in this series withx>0.8. The Faraday rotation at 17 850 cm⁻¹ increases linearly with bismuth concentration whereas the absorption coefficient increases more rapidly. The presence of Bi³⁺ increases the intensity of all Fe³⁺ pair transitions in the garnet system as a result of the increased superexchange induced by Bi³⁺. This is in keeping with the observation that the intensity of the⁶A_1g (S)→⁴T_1g (G) transition in (RE)₃Fe₅O₁₂ (RE=Er, Y, Dy, Gd, Eu) increases on traversing the above RE series as do the Curie temperatures of these iron garnets.     

Low-frequency fluctuations in stochastic processes with a 1/<Emphasis Type="Italic">f</Emphasis><Superscript>α</Superscript> spectrum

The results of numerical analysis of the Brownian movement of a particle in the force field of the potential corresponding to interacting subcritical and supercritical phase transitions are considered. If the white noise intensity corresponds to the critical intensity of the noise-induced transition, the system of stochastic differential equations describes random steady-state processes with fluctuation power spectra inversely proportional to frequency f, S(f) ∼ 1/f ^α, where exponent α varies in the interval 0.8 ≤ α ≤ 1.8. Exponent β of distribution function P(τ) ∼ τ^−β for the duration of low-frequency extremal fluctuations, which are analogous to avalanches considered in the models of self-organized criticality in many respects, varies between the same limits. It is shown that exponents α and β are connected through the relation α + β = 2.     

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.     

Effect of Fe substitution on the thermal expansion of the highT c superconductor Y1Ba2Cu3O y

Thermal expansion measurements have been carried out on Fe substituted superconducting compounds Y₁Ba₂(Cu_1−x Fe _x )₃O _y (0<x<4%) using a high resolution dilatometer employing the three terminal capacitance technique. The experimental set up is sensitive enough to detect changes in α of less than 10⁻⁸K⁻¹. Results show that the jump Δα in the coefficient of linear thermal expansion at the superconducting transition temperature,T _c, increases almost linearly with Fe concentration. The normal state thermal expansion coefficient α first decreases, attains a minimum value aroundx=1% and then increases for higher Fe concentrations. The oxygen content per unit formula is almost constant up tox=1% and then increases rapidly withx. It has also been observed that the anomalous behaviour of α around 260 K observed by Meingastet al [22] for Fe concentrationx=5% is due to inadequate annealing of the sample.     

Relaxation of post-illumination electric fields in strongly biased high-resistance structures with a single deep impurity level

This paper examines how an electric field relaxes when a discontinuous large carrier-depleting voltage applied to high-resistance symmetric metal-semiconductor-metal (MSM) and metal-insulator-semiconductor-insulator-metal (MISIM) structures having a single impurity level, and how its energy level ɛ _t=E _c−E _t and the tunneling transparency T _n,p of the metal-semiconductor or metal-insulator boundary affect the relaxation. It is shown that the relaxation of the field and the form of its steady-state distribution depend on the ratio of the time constant t _p in the majority-carrier (hole) region to the ionization time τ _t ⁻¹ =α _n (n _*+n ₁)+α _p (p _*+p ₁) of a deep trap in the bulk. This ratio determines the relative contributions of free ρ _p,n and bound charge dnsities ρ _t (where α _n,p is the coefficient for capture by an impurity, and p _*, n _*, p ₁, n ₁ are equilibrium concentrations and Shockley-Read constants in the bulk). For τ _t ≈(τ _t )_max≫t _p ,ρ _t ≫ it is found that ρ _p,n and ρ _t ≫ρ _p,n , which corresponds to a trap energy close to , independent of the value of T _n,p , decaying oscillations arise in the concentration distribution, bulk charge, and field appear in the bulk. The amplitude of these oscillations reaches a maximum at time t≈0.4τ _t. Decreasing the ratio α _p/α_n causes τ _t to deviate from (τ _t)_max. When this happens, the field no longer oscillates; instead, it increases with positive curvature in the cathode portion of the bulk. The quantity T _n,p determines the behavior of the field in the neighborhood of the anode. The value of (dE/dx)₀ is positive for MSM structures (T _n,p ≈1), and negative for MISIM structures (T _n,p ≈0). For transparencies close to a critical value T _n,p ⁰ , the field in the structure remains almost uniform over an impurity ionization time. Fiz. Tverd. Tela (St. Petersburg) 39, 1775–1782 (October 1997)     

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     

Di-pion emission in heavy quarkonia decays

The dipion spectrum for the ϒ(nS) → ϒ(n′S) transition with n < 4 has the form dw/dq ∼ (phase space) |η − x|², with x = q ² − 4m _π² / (ΔM)² − 4m _π² < q ² ≡ M _ππ², and ΔM = M(nS) − M(n′S). The parameter η is calculated and the spectrum is shown to reproduce the experimental data for all three types of decays: 3 → 1, 2 → 1, and 3 → 2 with η ≈ 0.5, 0, and −3, respectively. The text was submitted by the author in English.     

Anisotropic impurity scattering in palladium silver alloys

Measurements have been made of the Hall coefficientR of some alloys of silver in palladium over the temperature range 1°K to 120°K. The alloys contain between ∼1 and ∼10 at.-% silver. Values ofR were also obtained at room temperature and these were in good agreement with earlier published work. The values ofR are negative in all the alloys, and |R| increases both on reducing the temperature and increasing the silver concentration,c. Below ∼10°K, |R| becomes independent of temperature but shows a linear dependence onc, increasing by a factor of 2.5 over the concentration range measured. This increase is too great to be accounted for in terms of band structure changes alone, so we have examined the effects of anisotropic impurity scattering. To a first approximation it can be shown thatR is proportional to an anisotropy parameterA, defined asA=〈τ ²(k)〉/〈τ(k)〉², whereτ(k) represents the relaxation time of an electron in a statek, and 〈〉 is an average over the Fermi surface. In palladium we assume that the majority of the current is carried by the s-electrons. In the presence of silver impurities these electrons can be scattered into s-states or d-states with relaxation times given byτ _ss α1/c(1−c) andτ _sd α1/c ²(1−c) respectively. FollowingPlate we have assumed thatτ _ss is isotropic and thatτ _sd is anisotropic, leading to an overall anisotropic relaxation time for impurity scattering. We then find the parameterA increases approximately linearly with silver content, in accordance with our experimental results.     

On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum

We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E _n ∼n ^α , with 0<α<1. In particular, the gaps between successive eigenvalues decay as n ^α−1. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate ‖V(t) _m,n ‖≤ε|m−n|^−p max {m,n}^−2γ for m≠n, where ε>0, p≥1 and γ=(1−α)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and ε is small enough. More precisely, for any initial condition Ψ∈Dom(H ^1/2), the diffusion of energy is bounded from above as 〈H〉 _Ψ (t)=O(t ^σ ), where . As an application we consider the Hamiltonian H(t)=|p| ^α +ε v(θ,t) on L ²(S ¹,dθ) which was discussed earlier in the literature by Howland.     

Periodic Orbits Contribution to the 2-Point Correlation Form Factor for Pseudo-Integrable Systems

Using heuristic arguments based on the trace formulas we relate the 2-point correlation form factor, K ₂(τ), at small values of τ with sums over classical periodic orbits for typical examples of pseudo-integrable systems. The later sums have been explicitly calculated for the following models: (i) plane billiards in the form of right triangles with one angle π/n and (ii) rectangular billiards with the Aharonov-Bohm flux line. In the first model, using the properties of the Veech structure, it is shown that K ₂(0)=(n+ε(n))/(3(n−2)), where ε(n)= 0 for odd n, ε(n)= 2 for even n not divisible by 3, and ε(n)=6 for even n divisible by 3. For completeness we also recall informally the main features of the Veech construction. In the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle. When these ratios are non-commensurable irrational numbers, K ₂(0)=1−3 , where is the fractional part of the flux through the rectangle when and it is symmetric with respect to the line when . The comparison of these results with numerical calculations of the form factor is discussed in detail. The above values of K ₂(0) differ from all known examples of spectral statistics, thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo-integrable systems. Received: 10 January 2000 / Accepted: 18 May 2001     

Electrical and magnetic properties of [(CoFeZr)x(Al2O3)1 ? x/(α-SiH)]n multilayer structures

The concentration dependences of the electrical resistivity and complex permeability of [“(Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} ”/“α-Si: H”] _n multilayer structures and (Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} composites have been studied. It has been established that introduction of a semiconductor interlayer into the (Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} composites substantially decreases the electrical resistivity of [“(Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} ”/“α-Si: H”] _n multilayer structures. The concentration dependences of the real and imaginary parts of the complex permeability of the [“(Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} ”/“α-Si: H”] _n nanomultilayer structures substantially differ from those of the (Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} composites. The real part of the complex permeability of the [“(Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} ”/“α-Si: H”] _n nanomultilayer structures follows the curve with a minimum near the percolation threshold of the composite, and the imaginary part smoothly decreases as the ferromagnetic phase concentration increases. The results obtained are explained by the increase in the bifurcation temperature due to the conduction electrons of the semiconductor interlayer, which favor magnetic ordering of ferromagnetic grains.     

Measurement of the τ lepton mass at the KEDR detector

A precise τ lepton mass measurement performed at the VEPP-4M collider with the KEDR detector is reported. The mass value is evaluated from the τ⁺τ⁻ cross section behavior around the production threshold. The result based on 6.7 pb⁻¹ of data is m _τ = 1776.81 _−0.23 ^+0.25 ± 0.15 MeV. Using 0.8 pb⁻¹ of data collected at the ψ′ peak, we have also determined that Γ _ee B _ττ(ψ′) = 9.0 ± 2.6 eV. The text was submitted by the authors in English.     

Investigation of thermal expansion,phase diagrams,and barocaloric effect in the (NH4)2WO2F4 and (NH4)2MoO2F4 oxyfluorides

The thermal expansion along the principal crystallographic axes of the (NH₄)₂WO₂F₄ and (NH₄)₂MoO₂F₄ oxyfluorides has been studied. The anomalous behavior of α _i (T) due to the phase transitions has been revealed at T ₁ = 271.4 K and T ₂ ≈ 180 K for the molybdate and at T ₁ = 201.5 K and T ₂ ≈ 161 K for the tungstate. The quantities dT/dp and dT/dσ _i , which characterize the dependence of the phase transition temperatures on the hydrostatic and uniaxial pressures, have been determined from analyzing the results of studies of the thermal expansion and heat capacity with the use of the Pippard relations. The p-T and α _i -T phase diagrams reflect different characters of the influence of the pressure on the stability of the initial and distorted phases of the oxyfluorides. The magnitudes of the extensive and intensive barocaloric effects determined in the vicinity of the structural phase transitions are as follows: ΔS _BCE varies from approximately −10 to −17 J/mol K and ΔT _AD ≈ 8−17 K for the molybdate and ΔS _BCE varies from approximately −10 to −17 J/mol K and ΔT _AD ≈ 8−13 K for the tungstate.     

Measurement and analysis of alpha particle induced reactions on gold

Stacked foil activation technique and Ge(Li) gamma ray spectroscopy have been used for the measurement of excitation functions of¹⁹⁷Au(α,xn) (x=1−3),¹⁹⁷Au(α,2pn) and¹⁹⁷Au(α,αn) reactions up to 50 MeV. The experimental cross-sections were compared with the predictions of pre-equilibrium hybrid model, as well as with the more recent index model. A general agreement was found in all reactions using initial exciton numbern ₀=4(4p0h) except for¹⁹⁷Au(α,n) reaction, where index model gives fairly good agreement withn ₀=5(5p0h).     

KdV-soliton dynamics in a random field

We consider the interaction between soliton and a spatially uniform external random field within the framework of the forced Korteweg-de Vries equation. In the general case, the averaged soliton field is transformed to a Gaussian pulse whose amplitude falls off with time as t^−α, while its width increases as t^α, where the parameter α is characterized by the statistical properties of the external force. We obtain an analytical solution for α = 2, which corresponds to the limiting case of an infinitely long correlation time (τ₀ → ∞). The obtained solution is compared with the well-known Wadati solution for the case of a delta-correlated external force (τ₀ → 0) where the soliton is transformed to a Gaussian pulse with amplitude falling off at a lower rate α = 3/2. The numerical solutions of the forced Korteweg-de Vries equation, which demonstrate an increase in the parameter α from 3/2 to 2 with increasing correlation time, are given for the intermediate case corresponding to 0 < τ₀ < ∞. It is shown that the amplitude of the averaged soliton in a periodic random field falls off as t⁻¹ for the long times t. In this case, two pulses propagating in different directions are formed. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 7, pp. 599–606, July 2006.     

Confrontation of nucleus deformation and multipole mixture parameters in (2+0 n ?2+01) transitions with rotational band structure for K π = 0 2+ and 0 3+

The difference of the energies of levels Δ _n = E _lev(2⁺0 _n ) − E _lev(0⁺0 _n ) at n = 1, 2, and 3 and the multipole-mixture parameter δ for (2⁺0₂−2⁺0₁) and (2⁺0₃−2⁺0₁) transitions are contrasted against the structure of the K ^π = 0₂⁺ and 0₃⁺ rotational bands that was calculated on the basis of the quasiparticle-phonon model. The values of (Δ₂ − Δ₁), (Δ₃ − Δ₁), and (Δ₂ − Δ₃) are found to correlate with the sign of the parameter δ and with the calculated structure of the K ^π = 0₂⁺ and 0₃⁺ bands. Original Russian Text ? A.M. Demidov, L.I. Govor, V.A. Kurkin, I.V. Mikhailov, 2009, published in Yadernaya Fizika, 2009, Vol. 72, No. 2, pp. 236–240.