Polynomial identities,indices, and duality for theN=1 superconformal modelSM(2, 4v)

We prove polynomial identities for theN=1 superconformal modelSM(2, 4v) which generalize and extend the known Fermi/Bose character identities. Our proof uses theq-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations forq-series on the fermionic side. We use these polynomials to demonstrate a dual relation underqq ^–1 betweenSM(2, 4v) andM(2v–1, 4v). We also introduce a genralization of the Witten index which is expressible in terms of the Rogers false theta functions.Dedicated to the memory of Claude Itzykson.     

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.     

Effects of nitrogen and vanadium on the57Fe hyperfine field of RFe12−x V x N y (R=Y,Nd)

Magnetization and⁵⁷Fe Mössbauer measurements were carried out on RFe_12–x V _x N _y compounds (R=Y and Nd,x=1.7 and 2.2) and the effects of nitrogen and vanadium atoms on the⁵⁷Fe hyperfine fields at the different iron crystallographic sites were investigated. The hyperfine field decreases with increasing number of vanadium neighbour atoms at all the iron sites. The hyperfine field is strongly enhanced in the nitrogen composition withy>1 where the compound tends to transform into an amorphous-like solid. The iron moment deduced from the hyperfine field increases more upon nitrogenation for the 8i-site than for the other sites, and exceeds the moment of bcc iron.     

Magnetic properties of U (Fe x Al1−x )2 and U(Fe y Ni1−y )2 compounds

The results of magnetic measurements and ferromagnetic resonance studies performed on U(Fe _x Al_1–x )₂ and U(Fe _y Ni_1–y )₂ compounds over a large temperature range are reported. The saturation magnetization decreases nearly linearly when substituting Fe by Al or Ni. In the composition range x<0.84 and y<0.81, the compounds are Pauli paramagnets, except in the region with y0.10. For UNi₂ two types of magnetic behaviours are shown. This compound can be both a ferromagnet withT _c =23.5 K and a Pauli paramagnet, depending on the crystal structure. Above the Curie temperatures, the reciprocal susceptibility for the compounds with x>0.84 and y>0.81 obeys a temperature dependence of the formX=X _o+C(T-) ^–1. The effective iron moments decrease when substituting iron by nickel or aluminium. The ferromagnetic resonance measurements show that theg values are not composition-dependent. A linear variation of the mean iron magnetization with the exchange field is observed. Finally, the magnetic behaviour of iron in these compounds is analysed.     

A tunable green alkaline-earth silicon-oxynitride solid solution (Ca1? x Sr x )Si2O2N2:Eu2+ and its application in LED

A series of (Ca_1−x−y Sr _x )Si₂O₂N₂:yEu²⁺ (x=0.0–0.97, y=0.03) phosphors were synthesized by high-temperature solid-state reaction. The XRD patterns confirm the formation of a solid solution of (Ca_1−x−y Sr _x )Si₂O₂N₂:yEu²⁺. An intense tunable green light is observed with the increasing ratio of Sr/Ca. With an increase in x, the excitation and emission spectra show a redshift and blueshift, respectively, due to large centroid shift and small Stokes shift. The temperature dependent luminescence is also investigated in the temperature range of 77–450 K. The Huang–Rhys factor and the thermal-quenching temperature are determined. Intense green LEDs were successfully fabricated based on the (Ca_1−x−y Sr _x )Si₂O₂N₂:yEu²⁺ phosphor and near-ultraviolet (∼395 nm) GaN/blue (460 nm) InGaN chips. All the results indicate that the solid solution (Ca_1−x−y Sr _x )Si₂O₂N₂:yEu²⁺ is a promising phosphor applicable to near-UV and blue LEDs for solid-state lighting.     

Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

We consider a generic classical many particle system described by an autonomous Hamiltonian H(x ¹,…,x ^N+2) which, in addition, has a conserved quantity V(x ¹,…,x ^N+2)=v, so that the Poisson bracket {H,V} vanishes. We derive in detail the microcanonical expressions for entropy and temperature. We show that both of these quantities depend on multidimensional integrals over sub-manifolds given by the intersection of the constant energy hyper-surfaces with those defined by V(x ¹,…,x ^N+2)=v. We show that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. We derive the explicit expression of the function that gives the temperature.     

(Z N×) n−1 generalization of the chiral Potts model

We show that theR-matrix which intertwines twon-by-N ^n–1 state cyclicL-operators related with a generalization ofU _q(sl(n)) algebra can be considered as a Boltzmann weight of four-spin box for a lattice model with two-spin interaction just as theR-matrix of the checkerboard chiral Potts model. The rapidity variables lie on the algebraic curve of the genusg=N ^{2(n–1)((n–1)N-n)+1} defined by 2n–3 independent moduli. This curve is a natural generalization of the curve which appeared in the chiral Potts model. Factorization properties of theL-operator and its connection to the SOS models are also discussed.     

The Eight-Vertex Model and Lattice Supersymmetry

We show that the XYZ spin chain along the special line of couplings J _x J _y +J _x J _z +J _y J _z =0 possesses a hidden N=(2,2)\mathcal{N}=(2,2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter’s Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain.     

The Fixed Point of a Generalization of the Renormalization Group Maps for Self-Avoiding Paths on Gaskets

Let W(x,y) = ax ³+ bx ⁴+ f ₅ x ⁵+ f ₆ x ⁶+ (3 ax ²)² y+ g ₅ x ⁵ y + h ₃ x ³ y ² + h ₄ x ⁴ y ² + n ₃ x ³ y ³+a ₂₄ x ² y ⁴+a ₀₅ y ⁵+a ₁₅ xy ⁵+a ₀₆ y ⁶, and X = , , where the coefficients are non-negative constants, with a > 0, such that X ²(x,x ²)−Y(x,x ²) is a polynomial of x with non-negative coefficients. Examples of the 2 dimensional map Φ: (x,y)↦ (X(x,y),Y(x,y)) satisfying the conditions are the renormalization group (RG) maps (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point (x _f ,y _f ) of Φ in the invariant set . 2000 Mathematics Subject Classification Numbers: 82B28; 60G99; 81T17; 82C41.     

Fourier Transform Infrared Emission Spectroscopy of thebΠ–aΣSystem of BN

The emission spectrum of BN has been investigated in the 1800–9000 cm⁻¹region using a Fourier transform spectrometer. BN was formed in a microwave discharge of He with a trace of BCl₃and N₂. The bands observed in the 3000–7800 cm⁻¹interval have been assigned as theb¹Π–a¹Σ⁺transition, with the 0–0 band at 3513.99040(43) cm⁻¹. This transition is analogous to theA¹Π_u–X¹Σ⁺_g(Phillips) system of the isoelectronic C₂molecule. The rotational analysis of the 0–0, 1–1, 1–0, 2–1, 3–2, 2–0, 3–1, 4–2, and 4–1 bands has been obtained and the molecular constants for theb¹Π anda¹Σ⁺states have been determined. A local perturbation has been observed in thev= 1 vibrational level of theb¹Π state nearJ= 18 caused by the interaction with thev= 3 vibrational level of thea¹Σ⁺state. The principal equilibrium constants for thea¹Σ⁺state are: ω_e= 1705.4032(11) cm⁻¹, ω_ex_e= 10.55338(52) cm⁻¹,B_e= 1.683771(10), α_e= 0.013857(16) cm⁻¹, andr_e= 1.2745081(37) Å. Although theb¹Π–a¹Σ⁺transition has recently been seen in emission from boron nitride trapped in solid neon matrices [J. Chem. Phys.104,3143–3146 (1996)], our work represents the first observation of this transition of BN in the gas phase.     

Energy levels of the Schrödinger equation for some rational potentials in two-dimensional space using the inner product technique

The energy levels of a two-dimensional system are calculated for the rational potential,V(x, y; λ, g)=x ²+y ²+λ[x ²/(1+gx ²)+y ²/(1+gy ²)+a _xxx⁴+a _xyx² y ²+a _yyy⁴] using the inner product technique over a wide range of values of the perturbation parameters (g, λ) and for various eigenstates.     

Numerical Range for Orbits Under a Central Force

We present an explicit form for the central force that describes the orbit of some roulette curve, and interpret the orbit of the roulette curve as an algebraic curve F(1, x, y) = 0 associated to the homogeneous polynomial F(t, x, y) of a matrix A. The hodograph of the orbit is obtained as the boundary generating curve of the numerical range of A.     

Differential geometry of the Fermat quartic and theta functions

The universal curve over a finite cover of the moduli space of elliptic curves with level four structure is embedded in CP³ as the Fermat quartic and is parametrized via the four Jacobi theta functions. Constructions from completely integrable systems have shown the importance of looking at the curvature of certain spaces and here we compute sectional curvatures. For our computations, we choose the ambient Fubini-Study metric of CP³. We also derive several theta identities which arise from the quartic’s holomorphic two-form.     

Structure of28Si from the spectroscopy of high-spin states

A search for high-spin states in²⁸Si has been performed byn?y coincidence measurements in the²⁵Mg(α,nγy) reaction atE _α=14 and 15.5 MeV. Spin-parity assignments of the observed levels were obtained fromn?γ angular correlation and lifetime measurements atE _α=14.5 MeV. Theγ-decay of the 9,164 keV level was investigated separately with the²⁷Al(p, γ) reaction at theE _p=2,160 and 2,312 keV resonances. Rotational bands withK ^π=3^? (comprising levels atE _x=6,879, 8,413, 10,188 and 12,204 keV),K ^π =5^? (comprising levels atE _x=9,702, 11,577 and 13,741 keV) andK ^π=0⁺ (comprising levels atE _x=6,691, 7,381, 9,164 and 11,509 keV) were observed. The finding of the latter band supports the idea of coexisting oblate and prolate shapes in²⁸Si. A level at 14,643 keV excitation energy has the properties of theI ^π=8⁺ member of the ground state band. There are additional positive-parity high-spin states which do not fit into rotational bands. All types of positive-parity states are well accounted for by shell model calculations.     

Quantum normal form and the harmonic oscillator withx 6 perturbation

The eigenvalue problem of the harmonic oscillator with thex ⁶ perturbation,H=1/2(p ²+x ²+x ⁶), is investigated using the method of quantum normal form. The energy eigenvalues are found to be in good agreement with the WKB results.     

Structure, electrical properties and temperature characteristics of?Bi0.5Na0.5TiO3–Bi0.5K0.5TiO3–Bi0.5Li0.5TiO3 lead-free piezoelectric ceramics

(1−x−y)Bi_0.5Na_0.5TiO₃–xBi_0.5K_0.5TiO₃– yBi_0.5Li_0.5TiO₃ lead-free piezoelectric ceramics have been prepared by an ordinary sintering technique, and their structure, electrical properties, and temperature characteristics have been studied systematically. The ceramics can be well-sintered at 1050–1150 °C. The increase in K⁺ concentration decreases the grain-growth rate and promotes the formation of grains with a cubic shape, while the addition of Li⁺ decreases greatly the sintering temperature and assists in the densification of BNT-based ceramics. The results of XRD diffraction show that K⁺ and Li⁺ diffuse into the Bi_0.5Na_0.5TiO₃ lattices to form a solid solution with a pure perovskite structure. As x increases from 0.05 to 0.50, the ceramics transform gradually from rhombohedral phase to tetragonal phase and consequently a morphotropic phase boundary (MPB) is formed at 0.15≤x≤0.25. The concentration y of Li⁺ has no obvious influence on the crystal structure of the ceramics. Compared with pure Bi_0.5Na_0.5TiO₃, the partial substitution of K⁺ and Li⁺ for Na⁺ lowers greatly the coercive field E _c and increases the remanent polarization P _r of the ceramics. Because of the MPB, lower E _c and large P _r, the piezoelectricity of the ceramics is improved significantly. For the ceramics with the compositions near the MPB (x=0.15–0.25 and y=0.05–0.10), the piezoelectric properties become optimum: piezoelectric coefficient d ₃₃=147–231 pC/N and planar electromechanical coupling factor k _P=20.2–41.0%. In addition, the ceramics exhibit relaxor characteristic, which probably results from the cation disordering in the 12-fold coordination sites. The depolarization temperature T _d shows a strong dependence on the concentration x of K⁺ and reaches the lowest values at the MPB. The temperature dependences of the ferroelectric and dielectric properties at high temperatures may imply that the ceramics may contain both the polar and non-polar regions at temperatures above T _d.     

Sensitivity ofn-p scattering observables to charge-independence breaking of the3 P N-N phase shifts

The sensitivity of neutron-proton scattering observables to the recently reported charge-independence breaking of the isospinT=1³ P nucleon-nucleon interaction has been calculated. It is shown that only the analyzing powerA _y () at extreme forward angles and at =90° c.m. and the spin-correlation coefficientA _zx () at 90° c.m. exhibit a measurable sensitivity. It is concluded that the determination ofA _y at 90° c.m. is currently the most accurate approach to verify independently charge-independence breaking in the³ P nucleon-nucleon phase shifts, considering the status of present experimental techniques.     

A two-dimensional mapping with a strange attractor

Lorenz (1963) has investigated a system of three first-order differential equations, whose solutions tend toward a strange attractor. We show that the same properties can be observed in a simple mapping of the plane defined by:x _i+1=y _i +1–ax _i ² ,y _i+1=bx _i . Numerical experiments are carried out fora=1.4,b=0.3. Depending on the initial point (x ₀,y ₀), the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor, which appears to be the product of a one-dimensional manifold by a Cantor set.     

Some Questions of the Nevanlinna Theory for the Complex Half-Plane

In this paper we give variants of the logarithmic derivative lemma and the second main theorem for the complex half-plane. Here the second main theorem is a special case of a theorem concerning polynomial functions of the coordinates of a holomorphic curve f : C ₊ P ⁿ . This theorem is a variant for C ₊ of a theorem of A. Eremenko and M. Sodin, who considered an entire curve f : C ₊ P ⁿ . The proofs are realized because of the introduction of natural classes of subharmonic, -subharmonic, holomorphic, meromorphic functions in C ₊ . Such functions we shall call just functions.