Differential Invariants of Immersions of Manifolds with Metric Fields

The problem of finding all r^th order differential invariants of immersions of manifolds with metric fields, with values in a left (G¹_m×G¹_n)-manifold is formulated. For obtaining the basis of higher order differential invariants the orbit reduction method is used. As a new result it appears that r^th order differential invariants depending on an immersion f:M N of smooth manifolds M and N and metric fields on them can be factorized through metrics, curvature tensors and their covariant differentials up to the order (r–2), and covariant differentials of the tangent mapping Tf up to the order r. The concept of a covariant differential of Tf is also introduced in this paper. The obtained results are geometrically interpreted as well.This research is supported by grants GAR 201/03/0512 and MSM 143100006.     

Gauge Theoretical Construction of Non-compact Calabi-Yau Manifolds

We construct the non-compact Calabi-Yau manifolds interpreted as the complex line bundles over the Hermitian symmetric spaces. These manifolds are the various generalizations of the complex line bundle over CP^N−1. Imposing an F-term constraint on the line bundle over CP^N−1, we obtain the line bundle over the complex quadric surface Q^N−2. On the other hand, when we promote the U(1) gauge symmetry in CP^N−1 to the non-abelian gauge group U(M), the line bundle over the Grassmann manifold is obtained. We construct the non-compact Calabi-Yau manifolds with isometries of exceptional groups, which we have not discussed in the previous papers. Each of these manifolds contains the resolution parameter which controls the size of the base manifold, and the conical singularity appears when the parameter vanishes.     

Geometry of the Aharonov–Bohm Effect

We show that the connection responsible for any Abelian or non-Abelian Aharonov–Bohm effect with n parallel “magnetic” flux lines in ℝ³, lies in a trivial G-principal bundle P→M, i.e. P is isomorphic to the product M×G, where G is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering space , where path integrals are computed, and the associated bundle P× _G ℂ ^m →M, where the wave function and its covariant derivative are sections.     

Lifting classes of principal bundles

Given a principal G-bundle P over X, we define a particularly suitable equivalence relation between liftings of P with respect to a group morphism σ:M→G. Supposing that σ has a central kernel C, we obtain a free operation of H¹(X$\text{C}$) (with coefficients in the sheaf of C-valued functions) on the set of lifting classes of P, which is natural under change of groups and base spaces. It is simply transitive, if in addition σ is an epimorphism; otherwise we classify its orbits by sections in the associated bundle P × _G(G/σM).For $\text{C=}\text{Z}{}_{\mathrm{n}}$ we relate the lifting classes to similar classes of n-th roots of associated line bundles. In the differentiable case and for an epimorphism σ with discrete kernel, there is a natural lifting of partial principal connections in each of these lifting classes. Finally, we indicate some applications to geometric quantization.     

Electromagnetic form factors of nucleons in a relativistic square-root potential model of independent quarks

The nucleon electromagnetic form factorsG _E ^P (q²),G _M ^P (q²) and the axial-vector form factor G_A(q²) are studied in a relativistic model of independent quarks confined by an equally mixed scalar-vector square root potentialV _q(r)=1/2(1+γ ⁰)(ar ^1/2+ν ₀) taking into account the appropriate centre-of-mass corrections. The respective root-mean-square radii associated withG _E ^P (q²) and G _A (q²) come out as [〈r ²〉 _E ^P ]^1/2=0.86 fm and 〈r _A ² 〉^1/2=0.88 fm. Restoration of chiral symmetry in this model is discussed to derive the pion-nucleon form factorG _πNN(q²) and consequently the pion-nucleon coupling constant is obtained asg _πNN(q²)=12.81 as compared tog _πNN(q²)_exp⋍13.     

Clusters of cycles

A cluster of cycles (or (r,q)-polycycle) is a simple planar 2-connected finite or countable graph G of girth r and maximal vertex-degree q, which admits an (r,q)-polycyclic realization P(G) on the plane. An (r,q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q; (ii) all interior faces (denote their number by p_r) are combinatorial r-gons; (iii) all vertices, edges and interior faces form a cell-complex.An example of (r,q)-polycycle is the skeleton of (r^q), i.e. of the q-valent partition of the sphere, Euclidean plane or hyperbolic plane by regular r-gons. Call spheric pairs (r,q)=(3,3),(4,3),(3,4),(5,3),(3,5). Only for those five pairs, P((r^q)) is (r^q) without exterior face; otherwise, P((r^q))=(r^q).Here we give a compact survey of results on (r,q)-polycycles. We start with the following general results for any (r,q)-polycycle G: (i) P(G) is unique, except of (easy) case when G is the skeleton of one of the five Platonic polyhedra; (ii) P(G) admits a cell-homomorphism f into (r^q); (iii) a polynomial criterion to decide if given finite graph is a polycycle, is presented.Call a polycycle proper if it is a partial subgraph of (r^q) and a helicene, otherwise. In [ARS Comb. A 29 (1990) 5], all proper spheric polycycles are given. An (r,q)-helicene exists if and only if p_r>(q−2)(r−1) and (r,q)≠(3,3). We list the (4,3)-, (3,4)-helicenes and the number of (5,3)-, (3,5)-helicenes for first interesting p_r. Any outerplanar (r,q)-polycycle G is a proper (r,2q−2)-polycycle and its projection f(P(G)) into (r^2q−2) is convex. Any outerplanar (3,q)-polycycle G is a proper (3,q+2)-polycycle.The symmetry group Aut(G) (equal to Aut(P(G)), except of Platonic case) of an (r,q)-polycycle G is a subgroup of Aut((r^q)) if it is proper and an extension of Aut(f(P(G))), otherwise. Aut(G) consists only of rotations and mirrors if G is finite, so its order divides one of the numbers 2r, 4 or 2q. Almost all polycycles G have trivial AutG.Call a polycycle G isotoxal (or isogonal, or isohedral) if AutG is transitive on edges (or vertices, or interior faces); use notation IT (or IG, or IH), for short. Only r-gons and non-spheric (r^q) are isotoxal. Let T^*(l,m,n) denote Coxeter’s triangle group of a triangle on S², E² or H² with angles π/l, π/m, π/n and let T(l,m,n) denote its subgroup of index 2, excluding motions of 2nd kind. We list all IG- or IH-polycycles for spheric (r,q) and construct many examples of IH-polycycles for general case (with AutG being above two groups for some parameters, including strip and modular groups). Any IG-, but not IT-polycycle is infinite, outerplanar and with same vertex-degree, we present two IG-, but not IH-polycycles with (r,q)=(3,5),(4,4) and AutG=T(2,3,∞)PSL(2,Z), T^*(2,4,∞). Any IH-polycycle has the same number of boundary edges for each its r-gon. For any r≥5, there exists a continuum of quasi-IH-polycycles, i.e. not isohedral, but all r-gons have the same 1-corona.On two notions of extremal polycycles:

1. We found for the spheric (r,q) the maximal number n_int of interior points for an (r,q)-polycycle with given p_r; in general case, (p_r/q)≤n_int<(rp_r/q) if any r-gon contains an interior point.

2. All non-extendible (r,q)-polycycles (i.e. not a proper subgraph of another (r,q)-polycycle) are (r^q), four special ones, (possibly, but we conjecture their non-existence) some other finite (3,5)-polycycles, and, for any (r,q)≠(3,3),(3,4),(4,3), a continuum of infinite ones.

On isometric embedding of polycycles into hypercubes Q_m, half-hypercubes and, if infinite, into cubic lattices Z_m, : for (r,q)≠(5,3),(3,5), there are exactly three non-embeddable polycycles (including (4³)−e, (3⁴)−e); all non-embeddable (5,3)-polycycles are characterized by two forbidden sub-polycycles with p₅=6.     

Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons

We consider Lie(G)-valued G-invariant connections on bundles over spaces ${G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}We give a geometric construction of the ${\mathcal{W}_{1+\infty}}We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, \mathbbR×G/H and \mathbbR²×G/H{G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}, where G/H is a compact nearly K?hler six-dimensional homogeneous space, and the manifolds \mathbbR×G/H{\mathbb{R}\times G/H} and \mathbbR²×G/H{\mathbb{R}^2\times G/H} carry G ₂- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on \mathbbR×G/H{\mathbb{R}\times G/H} is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G ₂-structures on \mathbbR×G/H{\mathbb{R}\times G/H}. It is shown that both G ₂-instanton equations can be obtained from a single Spin(7)-instanton equation on \mathbbR²×G/H{\mathbb{R}^2\times G/H}.     

Temperature dependence of magnetic anisotropy constant in cobalt ferrite nanoparticles

The temperature dependence of the effective magnetic anisotropy constant K(T) of CoFe₂O₄ nanoparticles is obtained based on the SQUID magnetometry measurements and Mössbauer spectroscopy. The variation of the blocking temperature T_B as a function of particle radius r is first determined by associating the particle size distribution and the anisotropy energy barrier distribution deduced from the hysteresis curve and the magnetization decay curve, respectively. Finally, the magnetic anisotropy constant at each temperature is calculated from the relation between r and T_B. The resultant effective magnetic anisotropy constant K(T) decreases markedly with increasing temperature from 1.1×10⁷ J/m³ at 5 K to 0.6×10⁵ J/m³ at 280 K. The attempt time τ₀ is also determined to be 6.1×10⁻¹² s which together with the K(T) best explains the temperature dependence of superparamagnetic fraction in Mössbauer spectra.     

The extended phase space of the BRS approach

The origin of the classical BRS symmetry is discussed for the case of a first class constrained system consisting of a 2n-dimensional phase spaceS with free action of a Lie gauge groupG of dimensionm. The extended phase spaceS _ext of the Fradkin-Vilkovisky approach is identified with a globally trivial vector bundle overS with fibreL*(G)L(G), whereL(G) is the Lie algebra ofG andL*(G) its dual. It is shown that the structure group of the frame bundle of the supermanifoldS _ext is the orthosymplectic group OSp(m,m; 2n), which is the natural invariance group of the super Poisson bracket structure on the function spaceC (S _ext). The action of the BRS operator is analyzed for the caseS=R ²ⁿ with constraints given by pure momenta. The breaking of the osp(m,m; 2n)-invariance down to sp(2n–2m) occurs via an intermediate osp(m; 2n–m). Starting from a (2n+2m)-dimensional system with orthosymplectic invariance, different choices for the BRS operator correspond to choosing different 2n-dimensional constraint supermanifolds inS _ext, which in general characterize different constrained systems. There is a whole family of physically equivalent BRS operators which can be used to describe a particular constrained system.     

Illumination dependence of I-V and C-V characterization of Au/InSb/InP(1 0 0) Schottky structure

The effects of surface preparation and illumination on electric parameters of Au/InSb/InP(100) Schottky diode were investigated, in the later diode InSb forms a fine restructuration layer allowing to block In atoms migration to surface. In order to study the electric characteristics under illumination, we make use of an He-Ne laser of 1 mW power and 632.8 nm wavelength. The current-voltage I(V_G), the capacitance-voltage C(V_G) measurements were plotted and analysed. The saturation current I_s, the serial resistance R_s and the mean ideality factor n are, respectively, equal to 2.03 × 10⁻⁵ A, 85 Ω, 1.7 under dark and to 3.97 × 10⁻⁵ A, 67 Ω, 1.59 under illumination. The analysis of I(V_G) and C(V_G) characteristics allows us to determine the mean interfacial state density N_ss and the transmission coefficient θ_n equal, respectively, to 4.33 × 10¹² eV⁻¹ cm⁻², 4.08 × 10⁻³ under dark and 3.79 × 10¹² eV⁻¹ cm⁻² and 5.65 × 10⁻³ under illumination. The deep discrete donor levels presence in the semiconductor bulk under dark and under illumination are responsible for the non-linearity of the C⁻²(V_G) characteristic.     

Quantum Yang-Mills on a Riemann surface

We obtain the quantum expectations of gauge-invariant functions of the connection on aG=SU(N) product bundle over a Riemann surface of genusg. We show that the spaceA/G _m of connections modulo those gauge transformations which are the identity at one point is itself a principal bundle with affine linear fiber. The base space Path^2g G consists of 2g-tuples of paths inG subject to a relation on their endpoint values. Quantum expectations are iterated path integrals over first the fiber then over Path^2g G, each with respect to the push-forward toA/G _m of the measuree ^–S(A) D A. Here,S(A) denotes the Yang-Mills action onA. We exhibit a global section ofA/G _m to define a choice of origin in each fiber, relative to which the measure on the fiber is Gaussian. The induced measure on Path^2g G is the product of Wiener measures on the component paths, conditioned to preserve the endopoint relation. Conformal transformations of the metric onM act by reparametrizing these paths. We explicitly compute the partition function in the general case and the expectations of functions of certain products of Wilson loops in the caseg=1.Research supported in part by DOE grant DE-FGO2-88ER25066     

Optical absorption, fluorescence and decay properties of Pr-doped PbO-H3BO3-TiO2-AlF3 glasses

Spectroscopic and laser properties of PbO-H₃BO₃-TiO₂-AlF₃ glasses doped with 0.1, 0.5, 1.0 and 2.0 mol% of Pr₆O₁₁ have been studied. Optical absorption spectra were recorded in the UV-vis-NIR regions and the observed absorption bands were assigned to different electronic transitions from ³H₄ ground state of 4f² configuration. The three phenomenological Judd-Ofelt (J-O) parameters Ω_t (t=2, 4, 6) were determined from the measured oscillator strengths by including as well as excluding the ³H₄→³P₂ hypersensitive transition in J-O analysis. The emission characteristics such as stimulated emission cross-sections (σ_e), measured branching ratios (β_m), measured lifetimes (τ_m), quantum efficiencies (η) and gain parameters (σ_e×τ_m) have been evaluated for the principal intermanifold transitions of Pr³⁺ from the ³P₀ and ¹D₂ states to the lower lying manifolds in the visible region. From the emission and decay measurements, the effect of Pr³⁺ ion concentration on the quenching of the ¹D₂ measured lifetimes has been discussed.     

Fluorescence decay dynamics and anneal effect on praseodymium ions doped in lead tungstate single crystal

Fluorescence decay dynamics of main emission manifolds of trivalent praseodymium ions doped in the as-grown and annealed PbWO₄ crystals were analyzed in detail. The fluorescence lifetimes of the ³P₀ and ¹D₂ manifolds were obtained from the fluorescence decay curves measured at room temperature. The main mechanism of fluorescence quenching of the ³P₀ manifold is the non-radiative relaxation to the defect centers coming into being during crystal growth. The dominant non-radiative relaxation for the ¹D₂ manifold is the cross-relaxation energy transfer involving the ¹G₄ and ³F₄ manifolds. The Inokuti–Hirayama model was used to analyze the decay dynamics of the ¹D₂ manifold, and the fluorescence lifetime τ₀ of the as-grown and annealed Pr³⁺:PbWO₄ crystals in absence of cross-relaxation energy transfer are derived as 35.2 μs and 37.9 μs, respectively. The annealing treatment can eliminate most defect centers except for those related to the lead vacancies.     

Interrelationships between 3-T-MRI-derived cortical and trabecular bone structure parameters and quantitative-computed-tomography-derivedbone mineral density

Recently, 3-T magnetic resonance imaging (MRI) has been introduced for bone imaging. Through higher signal-to-noise ratios, as compared to 1.5-T MRI, it promises to be a more powerful tool for the assessment of cortical and trabecular bone measures. The goal of our study was to compare MRI-derived cortical and trabecular bone measures to quantitative computed tomography (QCT)-derived bone mineral density (BMD). Using 3-T MRI in 51 postmenopausal women, apparent (app.) measures of bone volume/total volume, trabecular number (Tb.N), trabecular thickness (Tb.Th) and trabecular separation were derived at the distal radius, distal tibia and calcaneus. Cortical thickness (Ct.Th) was calculated at the distal radius and distal tibia. These measures were compared to QCT-derived BMD of the spine, hip and radius. Significant correlations (^?P<.05; ^??P<.001; ^???P<.0001) were found between spine BMD- and MRI-derived Ct.Th (r_radius=.55, ^?P<.05; r_tibia=.67, ^???P<.0001) and app. Tb.N (r_radius=.33, ^?P<.05; r_tibia=.35, ^?P<.05) at the radius and tibia. Furthermore, within the first 10 mm at the radius, an inverse correlation for Ct.Th and app. BV/TV (r_6mm=−.56, P<.001; r_10mm=−.36, P<.05) and app. Tb.Th (r_6mm=−.54, P<.001; r_10mm=−.41, P<.05) was found.     

On generalized Clifford groups II

The earlier study of the irreducible representations of the generalized Clifford groups G^m_n in the case where m is a prime number, is now extended to the case where m is any integer. The analysis of class structure and hence the construction of the irreducible representations of G^m_n for a non-prime integer m is found to be more complicated. This investigation also requires the properties of the generalized Clifford algebras C^m_n(I) which are studied in Section 2 of the paper. The case of infinite generalized Clifford group, i.e. G^∞_n involving the infinite- order root of unity as well as the physical relevance of the generalized Clifford groups are briefly dealt with.     

Integrable Fredholm Operators and Dual Isomonodromic Deformations

The Fredholm determinants of a special class of integrable integral operators K supported on the union of m curve segments in the complex λ-plane are shown to be the τ-functions of an isomonodromic family of meromorphic covariant derivative operators , having regular singular points at the 2m endpoints of the curve segments, and a singular point of Poincaré index 1 at infinity. The rank r of the corresponding vector bundle over the Riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel. The matrix Riemann–Hilbert problem method is used to deduce an identification of the Fredholm determinant as a τ-function in the sense of Segal–Wilson and Sato, i.e., in terms of abelian group actions on the determinant line bundle over a loop space Grassmannian. An associated dual isomonodromic family of covariant derivative operators , having rank n= 2m, and r finite regular singular points located at the values of the exponents defining the kernel of K is derived. The deformation equations for this family are shown to follow from an associated dual set of Riemann–Hilbert data, in which the r?les of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged. The operators are analogously associated to an integral operator whose Fredholm determinant is equal to that of K. Received: 10 June 1997 / Received revised: 16 February 2001 / Accepted: 27 November 2001     

The group of automorphisms of the galilei group

The group of automorphisms of the Galilei groupG: Aut(G) is calculated. It is shown that Aut(G) has the structure of a semi-direct product byG of the group ? _m ^* ×?_m where ?_m is the group of reals noted multiplicatively and ? _m ^* m is the subgroup of positive reals.     

Ferroelectric properties of neodymium-doped Sr2Bi4Ti5O18 thin film prepared by solgel route

Sr₂Bi₄Ti₅O₁₈ (SBTi) and Nd-modified SBTi (SBNT) thin films were deposited on Pt/Ti/SiO₂/Si (1 0 0) substrates using a sol-gel method. Structure, morphology and electric properties were investigated systematically. These films were randomly oriented and composed of rod-like grains. The remanent polarization (2P_r) and coercive field (E_c) of SBNT films were 30 μC/cm² and 55 kV/cm, respectively. This value of 2P_r was much higher than the reported value of SBTi prepared by pulsed-laser deposition. More importantly, the SBNT films showed high fatigue resistance against continuous switching up to 3×10⁹ cycles and excellent charge-retaining ability up to 3×10⁴ s.