Path integral approach for superintegrable potentials on spaces of nonconstant curvature: I. Darboux spaces <Emphasis Type="Italic">D</Emphasis><Subscript>I</Subscript> and <Emphasis Type="Italic">D</Emphasis><Subscript>II</Subscript>

In this paper, the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of nonconstant curvature: these spaces are Darboux spaces D _I and D _II. On D _I, there are three, and on D _II four such potentials. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on D _I in particular show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant curvature zero) and the two-dimensional hyperboloid (constant negative curvature) emerge. The text was submitted by the authors in English.     

<Emphasis Type="Italic">D</Emphasis>-Branes in the Euclidean <Emphasis Type="Italic">AdS</Emphasis><Subscript>3</Subscript> and <Emphasis Type="Italic">T</Emphasis>-Duality

We show that D-branes in the Euclidean AdS ₃ can be naturally associated to the maximally isotropic subgroups of the Lu–Weinstein double of SU(2). This picture makes very transparent the residual loop group symmetry of the D-brane configurations and gives also immediately the D-branes shapes and the σ-model boundary conditions in the de Sitter T-dual of the SL(2,C)/SU(2) WZW model.     

Nondegenerate superintegrable systems in <Emphasis Type="Italic">n</Emphasis>-dimensional Euclidean spaces

We analyze the concept of a nondegenerate superintegrable system in n-dimensional Euclidean space. Attached to this idea is the notion that every such system affords a separation of variables in one of the various types of generic elliptical coordinates that are possible in complex Euclidean space. An analysis of how these coordinates are arrived at in terms of their expression in terms of Cartesian coordinates is presented in detail. The use of well-defined limiting processes illustrates just how all these systems can be obtained from the most general nondegenerate superintegrable system in n-dimensional Euclidean space. Two examples help with the understanding of how the general results are obtained. The text was submitted by the authors in English.     

<Emphasis Type="Italic">SU</Emphasis><Subscript><Emphasis Type="Italic">L</Emphasis></Subscript>(4)×<Emphasis Type="Italic">U</Emphasis>(1) model for electroweak unification and sterile neutrinos

Some of the basic problems in neutrino physics, such as new energy scales, the enormous gap between the neutrino masses and the lightest charged fermion mass, and the possible existence of sterile neutrinos in the eV mass range are studied in the local gauge group SU _L (4)×U(1) for electroweak unification, which does not contain fermions with exotic electric charges. It is shown that the neutrino mass spectrum can be decoupled from that of the other fermions. The further normal seesaw mechanism for neutrinos, with right-handed neutrino Majorana masses of order M≫M _weak as well a new eV-scale can be accommodated. The eV-scale seesaw may manifest itself in experiments like the Liquid Scintillation Neutrino Detector (LSND) and MiniBooNE (MB) experimental results and future neutrino experiments.     

A <Emphasis Type="Italic">CR</Emphasis>-manifold in general position as an {<Emphasis Type="Italic">e</Emphasis>}-structure

In the paper, it is proved that a “generic” CR-structure is reduced to an {e}-structure. Using the invariant frames obtained as a result of this reduction (an analog of the Darboux frame of classical differential geometry), we give a simple proof of an analog of the Vitushkin theorem on the extension of a germ for a rather broad class of real submanifolds of complex spaces. Partially financially supported by the RFBR grants nos. 05-01-0981 and NSh-2040.2003.1.     

Weak annihilation and the effective parameters <Emphasis Type="Italic">a</Emphasis><Subscript>1</Subscript> and <Emphasis Type="Italic">a</Emphasis><Subscript>2</Subscript> in non-leptonic <Emphasis Type="Italic">D</Emphasis> decays

Based on SU(3) flavor symmetry, many of the quark-graph amplitudes in two-body non-leptonic decays of charmed mesons can be extracted from experiment, which enable us to see the relevance and importance of weak annihilation topologies and to determine the complex parameters a₁ and a₂ to test the factorization approach. It is found that a ₂ /a ₁ in and can be different by a factor of 2, indicating that non-factorizable corrections to the latter are far more important than the former. The relative phase between a₁ and a₂ is about 150°. Weak annihilation topologies induced by nearby resonances via final-state rescattering can be described in a model-independent manner. Although the W-exchange contribution in decays is dominated by resonant final-state interactions (FSIs), its amplitude in VP decays (V: vector meson, P: pseudoscalar meson) receives little contributions from FSIs in the quark-antiquark resonance formation. As a consequence, the sign flip of the W-exchange amplitude in and decays, which is needed to explain the relatively real decay amplitudes of , remains unexplained. SU(3) symmetry is badly broken in some Cabibbo-suppressed modes and this can be accounted for by the accumulation of some modest SU(3) violation in individual quark-graph amplitudes. Received: 19 July 2002 / Published online: 20 November 2002     

The determination of <Emphasis Type="Italic">α</Emphasis><Subscript><Emphasis Type="Italic">S</Emphasis></Subscript> from <Emphasis Type="Italic">τ</Emphasis> decays revisited

We revisit the determination of α _S (m _τ ²) using a fit to inclusive τ hadronic spectral moments in light of (1) the recent calculation of the fourth-order perturbative coefficient K ₄ in the expansion of the Adler function, (2) new precision measurements from BABAR of e⁺e⁻ annihilation cross sections, which decrease the uncertainty in the separation of vector and axial-vector spectral functions, and (3) improved results from BABAR and Belle on τ branching fractions involving kaons. We estimate that the fourth-order perturbative prediction reduces the theoretical uncertainty, introduced by the truncation of the series, by 20% with respect to earlier determinations. We discuss to some detail the perturbative prediction of two different methods: fixed-order perturbation theory (FOPT) and contour-improved perturbative theory (CIPT). The corresponding theoretical uncertainties are studied at the τ and Z mass scales. The CIPT method is found to be more stable with respect to the missing higher order contributions and to renormalization scale variations. It is also shown that FOPT suffers from convergence problems along the complex integration contour. Nonperturbative contributions extracted from the most inclusive fit are small, in agreement with earlier determinations. Systematic effects from quark-hadron duality violation are estimated with simple models and found to be within the quoted systematic errors. The fit based on CIPT gives α _S (m _τ ²)=0.344±0.005±0.007, where the first error is experimental and the second theoretical. After evolution to M _Z we obtain α _S (M _Z ²)=0.1212±0.0005±0.0008±0.0005, where the errors are respectively experimental, theoretical and due to the evolution. The result is in agreement with the corresponding N³LO value derived from essentially the Z width in the global electroweak fit. The α _S (M _Z ²) determination from τ decays is the most precise one to date.     

Nuclear reactions in the Pd/PdO:D<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript> and Ti/TiO<Subscript>2</Subscript>:D<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript> systems excited by ionizing radiation

The yield of the products of nuclear reactions from deuterated palladium and titanium irradiated by an electron beam and X rays has been studied. Charged particles have been detected by CR-39 track detectors, which are not sensitive to electronic noise, electrons, and X-ray photons. To identify the type of particles and to estimate their energy, three detectors covered by aluminum and copper foils of various thicknesses have been used. It has been established with reliable statistics that 30-keV electrons and X rays initiate the synthesis of deuterons in the Pd/PdO:D _x and Ti/TiO₂:D _x systems with the yield of 3-MeV protons.     

On the Dirac eigenvalues as observables of the on-shell <Emphasis Type="Italic">N</Emphasis> = 2<Emphasis Type="Italic">D</Emphasis> = 4 Euclidean supergravity

We generalize previous works on the Dirac eigenvalues as dynamical variables of Euclidean gravity and N =1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their value for the Dirac eigenvalues.      

On twist quantizations of <Emphasis Type="Italic">D</Emphasis> = 4 Lorentz and Poincare algebras

We use the decomposition of o(3, 1) = sl(2; ℂ)₁ ⊕sl(2; ℂ)₂ in order to describe nonstandard quantum deformation of o(3, 1) linked with Jordanian deformation of sl(2; ℂ). Using the twist quantization technique, we obtain the deformed coproducts and antipodes, which can be expressed in terms of real physical Lorentz generators. We describe the extension of the considered deformation of D = 4 Lorentz algebra to the twist deformation of D = 4 Poincare algebra with dimensionless deformation parameter. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Magnetic and transport properties of monocrystalline<Emphasis Type="Italic">Fe</Emphasis><Subscript>3</Subscript><Emphasis Type="Italic">O</Emphasis><Subscript>4</Subscript>

A monocrystal ofFe ₃ O ₄ is characterized by resistance, magnetoresistance and magnetic measurements in a temperature range from 4.2 K to 350 K and magnetic field-cycling from −9 T to 9 T. The resistance measurements revealed a metal-insulator Verwey transition (VT) atT _v =123.76 K with activation energy E=92.5 meV at T >T _v and temperature-substitute for the activation energy below the VT,T ₀=E/k _B ≈3800 K within 70 K–110K. The magnetotransport results independently verified the VT at 123.70 K, with discontinuous change in the magnetic moment ΔM≈0.21 ΔM≈0.21μ _B and resistance hysteresis, dependent on the magnetic field in a narrow temperature range of 0.4° around theT _v . The magnetic characterization established self consistentlyT _v as ≈123.67 K, the jump in the magnetization at the VT≈0.25μ _B and confirmed, that the magnetocrystalline anisotropy is the main microscopic mechanism responsible for the magnetization of the monocrystal (88%) with additional natural and imposed defects contributing as 12%.     

Two-Dimensional <Emphasis Type="Italic">D</Emphasis><Superscript>−</Superscript>-States in a Longitudinal Magnetic Field

The quantum-well D(−)-states in the presence of magnetic field longitudinal with respect to the growth axis are considered. Within a model of zero-radius potential, an equation is derived that determines the dependence of the D(−)-state binding energy on the parameters of the potential of the structure, coordinates of the D(−) center, and the magnetic field. The results are compared with the experimental dependence of the D(−)-state binding energy on the magnetic field and the data are shown to be in good agreement with calculations for magnetic fields B < 10 T. A dimension factor is defined in the dependence of the binding energy on coordinates for the 2D → 1D → 0D transition. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 25–29.     

Super <Emphasis Type="Italic">D</Emphasis>-Differentiation for <Emphasis Type="Italic">R</Emphasis><Superscript> ∞ </Superscript>-Supermanifolds

The concept of ‘D-Differentiation’, which, in the context of smooth manifolds, generalises Lie and covariant differentiation, is extended to R ^∞-supermanifolds under the name of ‘Super D-Differentiation’. This is done by defining new (non-linear) mappings, called ‘μ-mappings’ and by relating their non-linearity to the Leibniz rule that a derivation must satisfy when it acts on a tensor product. The resulting axiomatics, which is basis-independent and coordinate-free, is then expressed in a general basis (not necessarily holonomic). Super Lie and Super covariant differentiation are, amongst others, special cases of Super D-Differentiation. In particular, the transformation rules for the connection coefficients and the commutation coefficients of non-holonomic bases are obtained. These special cases are found to be in agreement with the DeWitt Super covariant and Super Lie derivatives.      

Correlation energies in distorted 3<Emphasis Type="Italic">d</Emphasis>-<Emphasis Type="Italic">t</Emphasis><Subscript>2g</Subscript> perovskite oxides

Using an effective low-energy Hamiltonian derived from the first-principles electronic structure calculations for the narrow t _2g bands of YTiO₃, LaTiO₃, YVO₃, and LaVO₃, we evaluate the contributions of the correlation energy E _c to the stability of different magnetic structures that can be realized in these distorted perovskite oxides. We consider two approximations for E _c that are based on regular perturbation theory expansion around a nondegenerate Hartree-Fock ground state. One is the second order of perturbation theory, which allows comparing the effects of local and nonlocal correlations. The other is the local t-matrix approach, which allows treating some higher-order contributions to E _c . The correlation effects systematically improve the agreement with the experimental data and additionally stabilize the experimentally observed G- and C-type antifer-romagnetic (AFM) structures in YVO₃ and LaVO₃, although the absolute magnitude of the stabilization energy is sensitive to the level of approximations and is somewhat smaller in the t-matrix method. The nonlocal correlations additionally stabilize the ferromagnetic ground state in YTiO₃ and the C-type AFM ground state in LaVO₃. Among two inequivalent transition-metal sites in the monoclinic structure, the local correlations are stronger at the sites with the least distorted environment. Limitations of the regular perturbation-theory expansion for LaTiO₃ are also discussed. The text was submitted by the authors in English.     

Passive Advection and the Degenerate Elliptic Operators <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

 We prove estimates for the stationary state n-point functions at zero molecular diffusivity in the Kraichnan model [13]. This is done by proving upper bounds for the heat kernels and Green's functions of the degenerate elliptic operators M _n that occur in the Hopf equations for the n-point functions. Received: 25 August 2001 / Accepted: 30 September 2002 Published online: 20 January 2003 Communicated by A. Kupiainen     

Density functional study of structural and electronic properties of Al<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>P (2 ≤ <Emphasis Type="Italic">n</Emphasis> ≤ 12) clusters

Low-lying equilibrium geometric structures of Phosphorus-doped aluminum cluster Al _n P (n = 2–12) clusters obtained by an all-electron linear combination of atomic orbital approach, within spin-polarized density functional theory, are reported. The binding energy, dissociation energy, and stability of these clusters are studied within the local spin density approximation (LSDA) and the three-parameter hybrid generalized gradient approximation (GGA) due to Becke-Lee-Yang-Parr (B3LYP). Ionization potentials, electron affinities, hardness, and static polarizabilities are calculated for the ground-state structures within the GGA. It is observed that symmetric structures with the P atom occupying a peripheral position are lowest-energy geometries of Al _n P (n = 2, 4–11), while the P impurities of Al₃P and Al₁₂P prefer to occupy internal sites in the aluminum clusters. Generalized gradient approximation extends bond lengths as compared to the LSDA lengths. The odd-even oscillations in the dissociation energy, the second differences in energy, the HOMO–LUMO gaps, the ionization potential, the electron affinity, and the hardness are more pronounced within both GGA and LSDA. The stability analysis based on the energies clearly shows the clusters with an even number of valence electrons are more stable than clusters with odd number of valence electrons.     

Betti Numbers of a Class of Barely <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> Manifolds

We calculate explicitly the Betti numbers of a class of barely G ₂ manifolds - that is, G ₂ manifolds that are realised as a product of a Calabi-Yau manifold and a circle, modulo an involution. The particular class which we consider are those spaces where the Calabi-Yau manifolds are complete intersections of hypersurfaces in products of complex projective spaces from which they inherit all their (1, 1)-cohomology and the involutions are free acting.     

Investigation of Rb <Emphasis Type="Italic">D</Emphasis><Subscript>1</Subscript> atomic lines in strong magnetic fields by fluorescence from a half-wave-thick cell

It has been experimentally demonstrated that the use of the effect of significant narrowing of the fluorescence spectrum from a nanocell that contains a column of atomic Rb vapor with a thickness of L = 0.5λ (where λ = 794 nm is the wavelength of laser radiation, whose frequency is resonant with the atomic transition of the D ₁ line of Rb) and the application of narrowband diode lasers allow the spectral separation and investigation of changes in probabilities of optical atomic transitions between levels of the hyperfine structure of the D ₁ line of ⁸⁷Rb and ⁸⁵Rb atoms in external magnetic fields of 10–2500 Gs (for example, for one of transitions, the probability increases ∼17 times). Small column thicknesses (∼390 nm) allow the application of permanent magnets, which facilitates significantly the creation of strong magnetic fields. Experimental results are in a good agreement with the theoretical values. The advantages of this method over other existing methods are noted. The results obtained show that a magnetometer with a local spatial resolution of ∼390 nm can be created based on a nanocell with the column thickness L = 0.5λ. This result is important for mapping strongly inhomogeneous magnetic fields.     

Thermal capacity,diffusion, and conductivity of Nd<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>Sr<Subscript>x</Subscript>Mno<Subscript>3</Subscript> (<Emphasis Type="Italic">x</Emphasis> = 0.45 and 0.5) manganites

Temperature dependences (77–300 K) of the thermal capacity, diffusion, and conductivity are investigated for the Nd _0.5 Sr _0.5 MnO ₃ and Nd _0.55 Sr _0.45 MnO ₃ polycrystalline samples. The examined characteristics show anomalous behavior in the magnetic phase transition and transition to the charge-ordered state. It is demonstrated that the main reason for a sharp decrease in the thermal conductivity during Nd _0.5 Sr _0.5 MnO ₃ transition into the antiferromagnetic charge-ordered state is a change in the phonon spectrum caused by the lattice compression. A temperature dependence of the free phonon path is calculated for the examined temperature interval based on the thermal diffusion obtained and the literature data on the sound propagation velocity. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 72–75, April, 2007.     

Dual properties of orthogonal polynomials of discrete variables associated with the quantum algebra <Emphasis Type="Italic">U</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(<Emphasis Type="Italic">su</Emphasis>(2))

We show that for every set of discrete polynomials y _n (x(s)) on the lattice x(s), defined on a finite interval (a, b), it is possible to construct two sets of dual polynomials z _k (ξ(t)) of degrees k = s-a and k = b-s-1. Here we do this for the classical and alternative Hahn and Racah polynomials as well as for their q-analogs. Also we establish the connection between classical and alternative families. This allows us to obtain new expressions for the Clerbsch-Gordan and Racah coefficients of the quantum algebra U _q (su(2)) in terms of various Hahn and Racah q-polynomials. Dedicated to the memory of our teacher and friend Arnold F. Nikiforov (18.11.1930–27.12.2005).