The strong topology on the algebra of polynomials

It is shown that under a quite general condition on the operator T (unbounded, symmetric) and on the domain $\text{D}$ for the representation x → T of the algebra $\text{P}$(x) on $\text{D}$ in $\text{P}$(T) the strongest locally convex topology τ_∞ coincides with the strong topology σ^$\text{D}$.     

Dynamical gauge fields in four dimensions from σ-models

We calculate the modification of the D = 4 CP(n) σ-model required by the existence of the effective action for composite fields which is finite in all orders in $\text{1}\text{N}$. We obtain the Higgs-like action with “free” bilinear terms for composite scalar and gauge fields. Contrary to the D = 2 case, due to the freedom of finite renormalization, the composite field excitations are not uniquely determined by the dynamics.     

Supersymmetry and the division algebras

We show how spinors in space-times of dimension D = t + s, where t is the time dimension, are associated for s - t = 1, 2, 4, 8 (and if t = 0, 1, 2) with the number systems (division algebras), $\text{|R, C, H, O}$. For t = 1 and s - t = 1, 2, 4 this association is “realized” by the sequence of Lorentz groups S1(2,$\text{|R}$), S1(2;$\text{|C}$), S1(2;$\text{|H}$) for D = 3, 4, 6 respectively. We discuss how octonions may be related to D = 10. For D = 6 we give details of S1(2; $\text{|H}$) spinors and construct supersymmetric models with them. These results explain various “empirical” observations in the literature relating quaternions and supersymmetry.     

Statistical model cross sections for overlapping resonances

The cross section for a reaction dominated by numerous incoherent resonances is studied, as a function of the average resonance width $\text{Λ}$ divided by the spacing D. Both the Bohr and Wheeler formulae for the average cross sections, and the standard Ericson analysis of fluctuations, are found to be valid only for weakly overlapping resonances, $\text{Λ}\text{D}\text{?}\text{n}\text{2π}$, where n is the number of open scattering channels. Two different models are presented which respect unitarity whatever the resonance widths, namely a K-matrix or Weisskopf-Wigner model, and an eikonal type model. The cross sections are found to saturate either the elastic or the inelastic unitarity bound for strongly overlapping resonances, $\text{Λ}\text{D}\text{?}\text{n}\text{2π}$; and the fluctuations in the cross sections are found to have a coherence length of order nD, rather than $\text{Λ}$.     

A slice theorem for the space of solutions of Einstein's equations

A slice for the action of a group G on a manifold X at a point x ? X is, roughly speaking, a submanifold S_x which is transverse to the orbits of G near x. Ebin and Palais proved the existence of a slice for the diffeomorphism group of a compact manifold acting on the space of all Riemannian metrics. We prove a slice theorem for the group $\text{D}$ of diffeomorphisms of spacetime acting on the space $\text{E}$ of spatially compact, globally hyperbolic solutions of Einstein's equations. New difficulties beyond those encountered by Ebin and Palais arise because of the Lorentz signature of the spacetime metrics in $\text{E}$ and because $\text{E}$ is not a smooth manifold- it is known to have conical singularities at each spacetime metric with symmetries. These difficulties are overcome through the use of the dynamic formulation of general relativity as an infinite dimensional Hamiltonian system (ADM formalism) and through the use of constant mean curvature foliations of the spacetimes in $\text{E}$. (We devote considerable space to a review and extension of some special properties of constant mean curvature surfaces and foliations that we need.) The conical singularity structure of $\text{E}$, the sympletic aspects of the ADM formalism, and the uniqueness of constant mean curvature foliations play key roles in the proof of the slice theorem for the action of $\text{D}$ on $\text{E}$. As a consequence of this slice theorem, we find that the space $\text{D}$ = $\text{E}$/$\text{D}$ of gravitational degrees of freedom is a stratified manifold with each stratum being a sympletic manifold. The spaces for homogeneous cosmologies of particular Bianchi types give rise to special finite dimensional symplectic strata in this space $\text{G}$. Our results should extend to such coupled field theories as the Einstein-Yang-Mills equations, since the Yang-Mills system in a given background spacetime admits a slice theorem for the action of the gauge transformation group on the space of Yang-Mills solutions, since there is a satisfactory Hamiltonian treatment of the Einstein-Yang-Mills system, and since the singularity structure of the solution set is known.     

The cooperative Jahn-Teller effect in uranium (4+) hydroxy-sulphate

The magnetic susceptibility measurements of orthorhombic U(OH)₂SO₄ within the temperature range 4.2–300 K have revealed a magnetic anomaly at T_D = 21 K associated with crystallographic transition induced by the cooperative Jahn-Teller effect. Above 21 K the magnetic susceptibility of the uranium (4+) ion corresponds to the electronic ground doublet ${}_{\mathrm{¡}}\text{M}{}_{\mathrm{J}}\text{= ± 2〉}$ confirming thus the antiprismatic symmetry (D_4d) of the crystal field at the uranium site. Below T_D the system of two singlets ($\text{1}\text{√2}$)|2〉 ± ($\text{1}\text{√2}$)|$\text{2}$〉 separated by δ(T) is the ground state of the uranium ion.     

Non-cubic Mn2+-centers in BaF2

The EPR spectra of thermally treated BaF₂: Mn samples is reported. After thermal annealing at 900 K a trigonal Mn²⁺ center with g=2.000±0.005, |D|=2725±40MHz, |A|=265±10MHz, $\text{D}\text{A}\text{>0}$, is observed. Annealing at 1200 K produces an orthorhombic Mn²⁺ center with g=2.00±0.01, |D|=2430±40MHz, |E|=570±20MHz, |A|=265±10MHz, $\text{D}\text{A}\text{<0}$. The superhyperfine (SHF) structures due to interactions with the neighbouring fluorines indicates that the trigonal manganese interacts with four fluorines, three of them equivalent. The orthorhombic Mn²⁺ shows interaction with four equivalent fluorine nuclei.     

Stark mixing spectroscopy in cesium

We present a new technique for selectively populating excited states which are inaccessible by dipole excitation from the ground state. The method uses a static electric field to introduce a component of a dipole-allowed state into the state of interest. We have applied the method to cesium to measure lifetimes and a Stark mixing coefficient. The results are $\text{τ(6}{}^2\text{D}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{)=64(2) ns}$, $\text{τ(7}{}^2\text{D}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{)=92.5(15) ns}$, and <6²D_{$\text{5}\text{2}$}|;ez |7²P_{$\text{3}\text{2}$}>/(E_7P?E_6D)=0.7(3)×10^?3 where is in kV/cm. 141     

Etude comparative des structures des alumines β stoechiometrique et non stoechiometrique: Transition de phase dans l'alumine β stoechiometrique (II)

The usual preparation methods of β alumina lead to a non stoichiometric compound $\text{(β“N.S.”)}$ of formula 11Al₂O₃?(1 + x) B₂O with x ≈-0.3; a metastable phase with a composition close to stoichiometry $\text{(β“S”}\text{and}\text{x ? 0)}$ can however also be obtained. X-Ray diffuse scattering studies of this stoichiometric form of silver β alumina reveals a sharp order—disorder phase transition at about 307 K. The low temperature ordered state of the silver ions is found to correspond to a 3D hexagonal superstructure with the lattice constants $\text{a}\text{3}\text{, a}\text{3}\text{, c}$. Above the transition temperature 3D short range order is observed up to about 315 K, where a cross over occurs towards a higher temperature 2D short range state, similar to that previously observed at low temperature in $\text{β}{}_{\mathrm{<mtext>Ag</mtext>}}\text{“N.S.”}$. Above 500 K the conducting silver ions are found to be in a 2D quasi liquid state. A similar type of order—disorder phase transition seems to occur in stoichiometric sodium β alumina at lower temperature. It is concluded that the very particular behaviour namely the absence of phase transition in the usual forms of β alumina is a direct consequence of non stoichiometry.     

Unitarity and the energy averaged S-matrix

A unitarity relation between mean width $\text{γ}$, level distance D and the absorptive part of the energy averaged S-matrix is proved and discussed. Consequences for the determination of $\text{γ}\text{D}$ are pointed out.     

Effect of single-ion anisotropy on the critical-point properties of the exchange interaction model of ferromagnetism

For the spin S exchange interaction model it is shown that both magnetic field and single-ion anisotropy (D) can be exactly treated. When D → ± ∞ the critical-point parameters of this model are exactly the same as those of the $\text{S}\text{=}\text{1}\text{2}$ Heisenberg model.     

Sodium resonances in fluorescence from wide-range laser scan in H2-O2-Ar flame

Several resonances and underlying continua were observed in the excitation profiles of sodium atoms in an H₂-O₂-Ar flame, irradiated by the focused beam of a tunable, pulsed dye layer. These profiles are obtained by tuning the laser through the spectral range of Rhodamine 6G and detecting the ensuing fluorescence signals from the 3D-3P and 4D-3P transitions. We observed resonances corresponding to the 3S_{$\text{1}\text{2}$}-3P_{$\text{1}\text{2}$}, 3S_{$\text{1}\text{2}$}-3P_{$\text{3}\text{2}$}, 3P_{$\text{1}\text{2}$}-5S_{$\text{1}\text{2}$}, 3P_{$\text{3}\text{2}$}-5S_{$\text{1}\text{2}$} one-photon transitions and resonances corresponding to the 3S_{$\text{1}\text{2}$}-5S_{$\text{1}\text{2}$}, 3S_{$\text{1}\text{2}$}-4D_{$\text{3}\text{2}$}, _{$\text{5}\text{2}$} two-photon transitions. The observed resonances result from different sequences of radiative and collisional excitation. The presence of the continua is explained by consecutive absorption of laser photons in the collisionally broadened wings of absorption lines. The spatial inhomogeneity of the laser beam and hence of the saturation causes submerging of some resonances into continua.     

Effects of the uniaxial anisotropy on the Curie temperature of the anisotropic Heisenberg ferromagnet with mixed spins S = 12 and S = 1

With the use of the pair model approximation, the effects of the D-term on the Curie temperature of the anisotropic Heisenberg ferromagnet with mixed spins $\text{S =}\text{1}\text{2}$ have been investigated and shown to be significant for D > 0.     

Energies of the n2S12 and n2D32,52 states of Cs

Using the technique of Doppler-free two-photon spectroscopy the energies of the n²S_{$\text{1}\text{2}$} (12 ? n ? 35) and n²D_{$\text{3}\text{2}$,$\text{5}\text{2}$} (11 ? n ? 48) states of CsI have been measured with a thermionic detector. The absorption spectrum of molecular iodine was used as a reference giving us a total estimated accuracy of about 2 × 10^-7. Taking also into account the energies of the 7–11 ²S_{$\text{1}\text{2}$} and 5 and 6 ²D_{$\text{3}\text{2}$,$\text{5}\text{2}$} states measured by other authors and using an extended Ritz-formula we found the ionization limit to be E_i = 31406.468 ± 0.006 cm^-1.     

Small angle neutron scattering and structural properties in electrolytic solutions

A new direct indication concerning the structural properties of liquid systems is obtained for the first time. Small angle neutron scattering (SANS) measurements have been performed on solutions of ZnCl₂, NiCl₂, CuBr₂ in D₂O and in non aqueous solvent (EtOAc). The spectra, taken at room temperature, as a function of concentration, in the range $\text{0.019 ? K ? 0.25 (}\text{A}\text{?}{}^{\mathrm{?1}}\text{)}$ show in aqueous solvent an “abnormal” k-dependent small angle coherent cross section. This anomalous contribution clearly indicates the existence in the systems of correlated regions (～ 30 Å) perhaps connected through a preferred distance of ～ 50 Å. Also in pure water the results seem to indicate the existence of correlated regions of finite size.     

Oscillator strengths of the first forbidden lines of rubidium

The oscillator strengths of the 5s-4d first forbidden lines of rubidium at 5165 Å were determined by performing CW tunable dye laser absorption measurements. Measured oscillator strengths of 8.06±0.48×10^-7 and 5.38±0.31×10^-7 for the ²D_{$\text{5}\text{2}$} and ²D_{$\text{3}\text{2}$} states, respectively, are compared with previously published values.     

Modal content of noise generated by a coaxial jet in a pipe

The problem investigated was that of noise generated by air flow through a coaxial obstruction in a long, straight pipe of inside diameter, D = 97 mm. Downstream modal pressure spectra in the 200–6000 Hz frequency range were measured by a new technique [1] for orifices and nozzles of diameter d where $\text{0·03 ? (}\text{d}\text{D}\text{) ? 0·52}$. The Mach numbers of the flow through the restrictions ranged from 0·15 to choked conditions. The shape of the modal frequency spectrum was found to be determined by the frequency ratio $\text{f}{}_{\mathrm{r}}\text{=}\text{He}\text{St}\text{=}\text{U}{}_{\mathrm{i}}\text{D}\text{a}{}_0\text{d}$, where U_i is the jet velocity and a₀ is the speed of sound in the gas downstream of the restriction. This parameter is the ratio of two non-dimensional frequencies: namely, He, which controls acoustic propagation inside circular ducts, and St, which scales the jet noise spectrum shape. At low f_r(<3) the higher modes dominate the noise spectrum above their cut-off frequencies, while for higher f_r all modes are approximately of equal amplitude. The nature of large scale turbulence structures in the region of the jet near the nozzle exit may be used to explain these phenomena. The measured modal pressure spectra were converted to modal power spectra and integrated over the frequency range 200–6000 Hz. The acoustic efficiency levels (acoustic power normalized by jet kinetic energy flow), when plotted vs. jet Mach number, depend strongly on the ratio of restriction diameter to pipe diameter ($\text{d}\text{D}$). Dividing the efficiency levels by the area ratio, $\text{(}\text{d}\text{D}\text{)}{}^2$, correlated the results over a moderate range of ($\text{d}\text{D}$).     

Spectral properties of cones of approximately representable reduced density matrices

We introduce a definition of the approximate spectrum of an operator which is useful in reduced density matrix theory. With precise knowledge of $\text{D}$² (the cone of representable reduced 2-densities) our approximate spectrum of any Hermitian, symmetric, one-body operator, A, agrees with the usual spectrum of operators on Fock space. The virtue of our definition is that with only approximate knowledge of $\text{D}$² we can compute the approximate spectrum for A. The approximate spectrum turns out to be a sensitive tool in assessing the quality of cones of approximately representable reduced 2-densities. Using this notion, we are able to point out a dramatic failure of one cone of approximately representable reduced 2-densities often referred to in the literature. In addition we show how reduced density matrices for excited states of systems with two-body interactions can be computed, given $\text{D}$⁴ the cone of reduced 4-densities.     

Pionic and radiative decay widths of charmed hadrons in a badly broken SU(4) symmetry

The width of ψ(4.414) is surprisingly small for a c$\text{c}$ state far above the charm threshold, Γ_exp ≈ 33 MeV. This fact can be qualitatively understood if one makes the conservative assignment 4 ³S₁. The nodes of the wave function reflect as zero's of the decay amplitude in the momentum of the decay products, and are responsible for the suppression of the couplings to the ground state charmed mesons, Γ(ψ(4.414)→D$\text{D}$+ D$\text{D}$¹ + $\text{D}$D¹ + D¹$\text{D}$¹ + F$\text{F}$ + F$\text{F}$¹ + $\text{F}$F¹ + F¹$\text{F}$¹) ≈ 15 MeV. All the modes are suppressed by factors of the order of 10³ with respect to the k³ spin counting prediction. The dominant modes are predicted to be D¹$\text{D}$¹, F$\text{F}$¹ + $\text{F}$F¹ and D$\text{D}$. Moreover, we show that the contributions from P-wave charmed mesons are reasonably expected to be suppressed either by phase space or by the nodes if their masses lie within certain range. We emphasize however that the smallness of the photon coupling remains to be understood.