On Probability Domains

Motivated by IF-probability theory (intuitionistic fuzzy), we study n-component probability domains in which each event represents a body of competing components and the range of a state represents a simplex S _n of n-tuples of possible rewards–the sum of the rewards is a number from [0,1]. For n=1 we get fuzzy events, for example a bold algebra, and the corresponding fuzzy probability theory can be developed within the category ID of D-posets (equivalently effect algebras) of fuzzy sets and sequentially continuous D-homomorphisms. For n=2 we get IF-events, i.e., pairs (μ,ν) of fuzzy sets μ,ν∈[0,1] ^X such that μ(x)+ν(x)≤1 for all x∈X, but we order our pairs (events) coordinatewise. Hence the structure of IF-events (where (μ ₁,ν ₁)≤(μ ₂,ν ₂) whenever μ ₁≤μ ₂ and ν ₂≤ν ₁) is different and, consequently, the resulting IF-probability theory models a different principle. The category ID is cogenerated by I=[0,1] (objects of ID are subobjects of powers I ^X ), has nice properties and basic probabilistic notions and constructions are categorical. For example, states are morphisms. We introduce the category S _n D cogenerated by \(S_{n}=\{(x_{1},x_{2},\ldots ,x_{n})\in I^{n};\:\sum_{i=1}^{n}x_{i}\leq 1\}\) carrying the coordinatewise partial order, difference, and sequential convergence and we show how basic probability notions can be defined within S _n D.     

Reciprocal Time Relation of Noncolliding Brownian Motion with Drift

We consider an N-particle system of noncolliding Brownian motion starting from x ₁≤x ₂≤…≤x _N with drift coefficients ν _j , 1≤j≤N satisfying ν ₁≤ν ₂≤…≤ν _N . When all of the initial points are degenerated to be zero, x _j =0, 1≤j≤N, the equivalence is proved between a dilatation with factor 1/t of this drifted process and the noncolliding Brownian motion starting from ν ₁≤ν ₂≤…≤ν _N without drift observed at reciprocal time 1/t, for arbitrary t>0. Using this reciprocal time relation, we study the determinantal property of the noncolliding Brownian motion with drift having finite and infinite numbers of particles.     

Electron paramagnetic resonance in mixed crystals (BaF<Subscript>2</Subscript>)<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>(LaF<Subscript>3</Subscript>)<Subscript>x</Subscript> activated by Ce<Superscript>3+</Superscript> ions

The electron paramagnetic resonance (EPR) spectra of mixed crystals (BaF₂)_{1?x? y}(LaF₃)_x(CeF₃)_y (y = 0.001 = 0.1%, x = 0–0.02) are investigated in a magnetic field H‖C₄ at a frequency of 9.5 GHz. The angular dependence of the EPR spectrum is measured for the sample with x = 0.02. The lines attributed to Ce³⁺ impurity centers with tetragonal symmetry and g factors (g_‖ = 0.75, g_⊥ = 2.4) close to those measured for the KY₃F₁₀: Ce³⁺ compound are separated in the complex EPR spectrum. The assumption is made that the aforementioned impurity centers are cubooctahedral clusters of the La₆F₃₇ type in which one of the La³⁺ ions is replaced by the Ce³⁺ ion.     

Single-crystalline M-type Sr Hexaferrites studied by <Superscript>57</Superscript>Fe Mössbauer spectroscopy

The ⁵⁷Fe Mössbauer spectra of the single crystalline and the finely ground Sr_1?x La _x Fe_12?y Co _y O₁₉ (x = 0 : y = 0, x = 0.192 : y = 0.152 and x = 0.456 : y = 0.225) samples have been measured to investigate the La-Co substitution effects. All observed spectra at 150 K were well fitted using the five subspectra which correspond to the five crystallographical nonequivalent Fe sites in the M-type hexaferrite, indicating that the valence changes to Fe²⁺ ions in the Fe³⁺ ions were not observed in our Sr_1?x La _x Fe_12?y Co _y O₁₉ samples. In SrFe₁₂O₁₉, the relative absorption intensities in the five subspectra show the large anisotropies in the recoilless fractions at the five Fe sites whereas these anisotropies were not observed in Sr_0.544La_0.456Fe_11.775Co_0.225O₁₉. These results indicate the chemical compositional dependence on the anisotropies of the recoilless fractions at the five Fe sites. The substitution of a Co²⁺ ion for the Fe³⁺ ion changes the center shifts of the Fe³⁺ ions near the Co²⁺ ion by the perturbation of the Fe-O-Co hybridizations. Therefore, the Co²⁺ ions occupy the 4f ₁ and the 4f ₂ sites due to the chemical compositional dependences of the refined magnetic hyperfine field and center shifts of the Fe³⁺ ions.     

Integral characteristics of vibronic-spin-orbit interactions in a phosphorus-containing analogue of the fluorene molecule

The strengths (P ₀₀ ^s )² and F _VSO ² of the transitions from the triplet sublevels s = z, y, and x of the electronic state ³A″ of the phenyldibenzophosphole (DB(P-Ph)) molecule are calculated taking into account the intramolecular spin-orbit (SO) and joint vibronic-spin-orbit (VSO) interactions. The contributions to the vibronic transition strengths from the SO interactions in different structural elements of the molecule (the C atoms of the dibenzene framework, the P atom, and the Ph substituent) are determined. The effect of the nonplanar nuclear configuration of the DB(P-Ph) molecule on the values of F _VSO ^s is investigated. The radiative deactivation rate constants of the k _rad ^s triplet sublevels T ^s are estimated. It is found that the vibrations of the A′(B₁) symmetry in the fine-structure phosphorescence spectrum of DB(P-Ph) occur due to both the SO coupling exclusively in the P atom and the T ^x → S₀ transition (the x axis is perpendicular to the planar dibenzene framework of the molecule) with a high (preferential) population of this triplet sublevel.     

Computation of the electronic structure and direct-gap absorption spectra in Ge-rich Si<Subscript>1−x</Subscript> Ge<Subscript>x</Subscript>/Ge/Si<Subscript>1−x</Subscript>Ge<Subscript>x</Subscript> type-I quantum wells

This paper presents the results of conduction band discontinuities calculation for strained/relaxed Si_1?x Ge _x /Si_1?y Ge _y heterointerfaces in Γ _15C , Γ _2′C and L upper bands minima, as well as the room-temperature strained (vs. relaxed) band gaps deduced from the classical model-solid theory. Based upon the obtained data, we propose a type-I W-like Si_1?y Ge _y /Si_1?x Ge _x /Ge/Si_1?x Ge _x /Si_1?y Ge _y quantum wells heterostructure optimized in terms of compositions and thicknesses. Electronic states and wave functions are found by solving Schrödinger equation without and under applied bias voltage. An accurate investigation of the optical properties of this heterostructure is done by calculating the energies of the interband transitions and their oscillator strengths. Moreover, a detailed computation of the bias-voltage evolution of the absorption spectra is presented. These calculations prove the existence of type-I band alignment at Γ _2′C point in compressively strained Ge quantum wells grown on relaxed Ge-rich Si_1?y Ge _y buffers. The strong absorption coefficient (> 8 × 10³ cm^-1) and the large Stark effect (0.1 eV @ 2 V) of the Γ _2′C transitions thresholds open up perspectives for application of these heterostructures for near-infrared optical modulators.     

Random time averaged diffusivities for Lévy walks

We investigate a Lévy walk alternating between velocities ±v ₀ with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at large times is ?x ²? ∝ t ², the latter to enhanced diffusion with ?x ²? ∝ t ^ν, 1 < ν < 2. The correlation function and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSD from a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function. In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yet very slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averaged MSDs.     

Stabilizer of a function in the gage group

It is proved that, for the dimension d of the stabilizer of an analytic function z(x, y) in the gage pseudogroup G = {z(x, y) → c(z(a(x), b(y))}, there are precisely four possibilities: (1) d = ∞ and the complexity of z is zero, (2) d = 3 and the complexity of z is equal to one, (3) d = 1 and z is equivalent the function r(x + y) ? x of complexity two, (4) d = 0 in all remaining cases.     

Spectroscopy of optical centers of Eu<Superscript>3+</Superscript> ions in partially stabilized and stabilized zirconium crystals

The structure of the optical centers of Eu³⁺ ions in tetragonal (ZrO₂)_1–x–y (Y₂O₃) _x (Eu₂O₃) _y (х = 2.7–3.6; y = 0.1) and cubic (ZrO₂)_1–x–y (Y₂O₃) _x (Eu₂O₃) _y (х = 8–38; y = 0.1–0.5) crystals of solid solutions on the basis of zirconium dioxide is studied using the methods of optical and Raman-scattering spectroscopy. Characteristic optical centers of Eu³⁺ ions with different crystalline environments are revealed in the above compounds.     

The Partition Formalism and New Entropic-Information Inequalities for Real Numbers on an Example of Clebsch–Gordan Coefficients

We discuss the procedure of different partitions in the finite set of N integer numbers and construct generic formulas for a bijective map of real numbers s _y , where y = 1, 2,…, N, N = \( \underset{k=1}{\overset{n}{\varPi}}{X}_k, \) and X _k are positive integers, onto the set of numbers s(y(x ₁, x ₂,…, x _n )). We give the functions used to present the bijective map, namely, y(x ₁, x ₂, …, x _n ) and x _k (y) in an explicit form and call them the functions detecting the hidden correlations in the system. The idea to introduce and employ the notion of “hidden gates” for a single qudit is proposed. We obtain the entropic-information inequalities for an arbitrary finite set of real numbers and consider the inequalities for arbitrary Clebsch–Gordan coefficients as an example of the found relations for real numbers.     

Study of collective flow effects in CC collisions at a momentum of 4.2 GeV/<Emphasis Type="Italic">c</Emphasis> per nucleon

Directed (in-plane) flows of protons, pions, and projectile light fragments (d, t, ³He, ⁴He) have been observed by investigating the dependence of the mean transverse momentum in the reaction plane 〈p _x 〉 on the rapidity y in the c.m. system for CC collisions at a momentum of 4.2 GeV/c per nucleon. The comparison of our in-plane-flow results of protons with flow data for various projectile/target configurations was made using the scaled flow F _s = F/(A _P ^1/3 + A _T ^1/3 ). F _s demonstrates a common scaling behavior for flow values from different systems. From azimuthal distributions of protons and π^? mesons, out-of-plane (squeeze-out) flow effects have been observed and the parameter a₂ (the measure of the anisotropic emission strength) has been extracted. The quark-gluon string model reproduces the experimental results quite well.     

On rational functions of first-class complexity

It is proved that, for every rational function of two variables P(x, y) of analytic complexity one, there is either a representation of the form f(a(x) + b(y)) or a representation of the form f(a(x)b(y)), where f(x), a(x), b(x) are nonconstant rational functions of a single variable. Here, if P(x, y) is a polynomial, then f(x), a(x), and b(x) are nonconstant polynomials of a single variable.     

Polar Phase of Superfluid <Superscript>3</Superscript>He: Dirac Lines in the Parameter and Momentum Spaces

The time reversal symmetric polar phase of the spin-triplet superfluid ³He has two types of Dirac nodal lines. In addition to the Dirac loop in the spectrum of the fermionic Bogoliubov quasiparticles in the momentum space (p _x , p _y , p _z ), the spectrum of bosons (magnons) has Dirac loop in the 3D space of parameters—the components of magnetic field (H _x ,H _y ,H _z ). The bosonic Dirac system lives on the border between the type-I and type-II.     

Hamilton’s equations of motion of a vortex filament in the rotating Bose–Einstein condensate and their “soliton” solutions

The equation of motion of a quantized vortex filament in a trapped Bose–Einstein condensate [A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A 62, 063617 (2000)] has been generalized to the case of an arbitrary anharmonic anisotropic rotating trap and presented in the variational form. For condensate density profiles of the form ρ = f(x² + y² + ReΨ(x + iy)) in the presence of the plane of symmetry y = 0, the solutions x(z) describing stationary vortices of U and S types coming to the surface and solitary waves have been found in quadratures. Analogous three-dimensional configurations of the vortex filament uniformly moving along the z axis have also been found in strictly cylindrical geometry. The dependence of solutions on the form of the function f(q) has been analyzed.     

Das Kernquadrupolmoment des Mn55

With a recording photoelectric Fabry-Perot spectrometer and an atomic-beam light source the hyperfine structure of the Mn I-resonance linesλ=4031 Å,λ=4033 Å,λ=4034 Å (3d ⁵4s ² a ⁶ S _5/2?3d ⁵4s4p z ⁶ P _7/2,5/2,3/2 ⁰)and of the inter-combination linesλ=5395 Å andλ=5433 Å (3d ⁵4s ² a ⁶ S _5/2?3d ⁵4s4p z ⁸ P _7/2,5/2 ⁰) was measured. Furthermore the resonance lines have been measured with a pulsed atomic-beam in absorption. In this case the quotient (I ₀(ν)?I(ν))/I ₀(ν) was recorded, whereI(ν)=I ₀(ν) exp(?α(ν)d) is the observed intensity with absorption andI ₀(ν) the intensity of the light source. From the hyperfine structure splitting the value of the electric quadrupole moment of Mn⁵⁵ was derived to be:Q(Mn⁵⁵)=+(0.35±0.05)·10^?24 cm².     

Signature splitting in two quasiparticle rotational bands of <Superscript>180,182</Superscript>Ta

The signature splittings in K^π = 1 ⁺: 7 /2[404] _π?9 /2[624] _ν, K^π = 0^?: 9 /2[514] _π?9 /2[624] _ν bands of ¹⁸⁰Ta and K^π = 0^?: 7 /2[404] _π?7 /2[503] _ν, K^π = 1^?: 5 /2[402] _π?3 /2[512] _ν, K^π = 1⁺: 7 /2[404] _π?9 /2[624] _ν bands of ¹⁸²Ta are analysed within the framework of two-quasiparticle rotor model. The phase as well as magnitude of the experimentally observed signature splitting in K^π = 1⁺ band of ¹⁸⁰Ta, which could not be explained in earlier calculations, is successfully reproduced. The conflict regarding placement of a 12 ⁺ level in K^π = 1 ⁺: 7 /2 ⁺[404] _π?9 /2 ⁺[624] _ν ground-state rotational band of ¹⁸⁰Ta is resolved and tentative nature of K^π = 0^?: 7 /2[404] _π?7 /2[503] _ν, K^π = 1⁺: 7 /2[404] _π?9 /2[624] _ν bands observed in ¹⁸²Ta is confirmed. As a future prediction for experimentalists, these two-quasiparticle structures observed in ¹⁸⁰Ta and ¹⁸²Ta are extended to higher spins.     

Hyperfeinstruktur-Untersuchungen im Rhenium-I-Spektrum

By optical means (i.e. hollow cathode, Fabry-Perot-interferometer and spectrograph) the hyperfine structure of 15 transitions in the Re-I spectrum was investigated. This permits to state the hfs coupling constantsA, B for 15 levels belonging to the configurations (5d ⁵) 6s ², (5d ⁵) 6s 6p, (5d ⁵) 6s 7s, (5d ⁶) 6s, (5d ⁶) 6p and (5d ⁴) 6s ² 6p. It was found that the (5d ⁵) 6s ^{2 6} S _5/2 ground state, too, shows a small hfs. Of its magnetic splitting both size and sign could be determined (A _0,185,187=?2,2±0,4 mK); for a possible quadrupole splitting upper limits were obtained. Estimates of the quadrupole moments resulted inQ _185,187?2,9b. — In the case of two levels (x ⁶ P _7/2 andy ⁶ F _11/2) the hfs analysis raises doubts as to their classification.     

Spin-spin interaction and the EPR spectrum of KDy(WO<Subscript>4</Subscript>)<Subscript>2</Subscript>

The EPR spectrum of a KDy(WO₄)₂ monoclinic crystal is investigated. It is found that the EPR spectrum of magnetically concentrated materials at a low frequency (9.2 GHz) undergoes a substantial transformation in addition to the well-known broadening of the EPR lines. At low Dy³⁺ concentrations (x<10^?2), the EPR spectrum of an isomorphic crystal, namely, KY_(1?x)Dy_x(WO₄)₂, is characterized by the parameters g_x=0, g_y=1.54, and g_z=14.6. For a magnetically concentrated crystal KDy(WO₄)₂, the g values are as follows: g_x=0, g_y=0.82, and g_z=2.52. It is demonstrated that the difference in the parameters is associated with the specific spin-spin interaction between Dy³⁺ ions, including the Dzyaloshinski interaction, which is not observed at high frequencies.     

Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

The parabolic Anderson model is defined as the partial differential equation ? u(x, t)/? t = κ Δ u(x, t) + ξ(x, t)u(x, t), x ∈ ? ^d , t ≥ 0, where κ ∈ [0, ∞) is the diffusion constant, Δ is the discrete Laplacian, and ξ is a dynamic random environment that drives the equation. The initial condition u(x, 0) = u ₀(x), x ∈ ? ^d , is typically taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d κ, split into two at rate ξ ∨ 0, and die at rate (?ξ) ∨ 0. In earlier work we looked at the Lyapunov exponents

$$ \lambda _{p}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t} \log \mathbb {E} ([u(0,t)]^{p})^{1/p}, \quad p \in \mathbb{N} , \qquad \lambda _{0}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t}\log u(0,t). $$

For the former we derived quantitative results on the κ-dependence for four choices of ξ : space-time white noise, independent simple random walks, the exclusion process and the voter model. For the latter we obtained qualitative results under certain space-time mixing conditions on ξ. In the present paper we investigate what happens when κΔ is replaced by Δ^??, where ?? = {??(x, y) : x, y ∈ ? _d , x ～ y} is a collection of random conductances between neighbouring sites replacing the constant conductances κ in the homogeneous model. We show that the associated annealed Lyapunov exponents λ _p (??), p ∈ ?, are given by the formula

$$ \lambda _{p}(\mathcal{K} ) = \text{sup} \{\lambda _{p}(\kappa ) : \, \kappa \in \text{Supp} (\mathcal{K} )\}, $$

where, for a fixed realisation of ??, Supp(??) is the set of values taken by the ??-field. We also show that for the associated quenched Lyapunov exponent λ ₀(??) this formula only provides a lower bound, and we conjecture that an upper bound holds when Supp(??) is replaced by its convex hull. Our proof is valid for three classes of reversible ξ, and for all ?? satisfying a certain clustering property, namely, there are arbitrarily large balls where ?? is almost constant and close to any value in Supp(??). What our result says is that the annealed Lyapunov exponents are controlled by those pockets of ?? where the conductances are close to the value that maximises the growth in the homogeneous setting. In contrast our conjecture says that the quenched Lyapunov exponent is controlled by a mixture of pockets of ?? where the conductances are nearly constant. Our proof is based on variational representations and confinement arguments.

     

The Nernst-Ettingshausen coefficient in the normal phase of doped HTSCs of the YBa<Subscript>2</Subscript>Cu<Subscript>3</Subscript>O<Subscript>y</Subscript> system

The temperature dependence of the Nernst-Ettingshausen coefficient Q(T) in the normal phase of doped HTSCs of the yttrium system was studied. The main features characterizing the behavior of this coefficient were revealed, and the character and mechanism of the effect that various nonisovalent substituents exert on the Q(T) dependence were analyzed. It is shown that the narrow-band model permits one not only to describe all the specific features observed in the Q(T) curves but also to perform a simultaneous quantitative analysis of the temperature dependences of four kinetic coefficients (the electrical resistivity and the Seebeck, Hall, and Nernst-Ettingshausen coefficients) with the use of a common set of model parameters characterizing the band structure and carrier system in the normal phase of an HTSC. This approach was employed to determine the carrier mobilities and the asymmetry of the dispersion curve in the systems studied (YBa₂Cu₃O_y, y = 6.37–6.91; YBa₂Cu_3?xCo_xO_y, x = 0–0.3; Y_1?xCa_xBa₂Cu₃O_y, x = 0–0.25; Y_1?xCa_xBa_2?xLa_xCu₃O_y, x = 0–0.5) and to analyze the effect of the substitutions involved on the variation of these parameters.