The Action of √−Δ on Weighted Sobolev Spaces

The action of – on distributions is examined within the context of weighted Sobolev spaces. The results obtained are as follows: (1) – is a continuous map of R ⁿ ), the space of rapidly decreasing functions, to L ^{2, s} (R ⁿ for any s < n/2 +1; (2) if k R and s > –n/2 – 1, then – is a continuous map from H ^{k, s} (R ⁿ ), the weighted Sobolev space, to H ^{k–1, t} (R ⁿ for some t. The results are optimal in a sense.     

Synthesis and Reactivity of Mg(II)–Fe(II)–Fe(III) Hydroxycarbonates

Green rust-like compounds (GRs) were discovered as natural minerals in various hydromorphic soils, where anoxic conditions allow their stability. They may control some redox processes in aquifers and participate to the transformation of various pollutants. Since Mg(II) cations are present in the fields where GRs were discovered, a partial substitution of Mg(II) to Fe(II) leading to intermediate compounds between GRs and usual Mg(II)–Fe(III) hydroxysalts is suspected. Mg(II)–Fe(II)–Fe(II) hydroxycarbonates can be obtained as intermediate oxidation products of (Mg, Fe)(OH)₂ in carbonate-containing aqueous media obeying to [Fe^II _4(1–x)Mg^II _4x Fe^III ₂(OH)₁₂]²⁺ [CO₃ ^2– nH₂O]^–2. TMS spectra at 12 K are similar to those of GRs, i.e., two quadrupole doublets, one due to Fe(II) with a large isomer shift =1.29 mms^–1 (with respect to -iron at room temperature) and quadrupole splitting E _Q=2.76 mms^–1, the other one due to Fe(III) with smaller hyperfine parameters =0.49 mms^–1 and E _Q=0.44 mms^–1. Fe(II) ions oxidise rapidly into Fe(III) with dissolved O₂. The reactivity is similar to that of Fe(II)–Fe(III) hydroxysalts GR, and thus the potential of Mg(II)–Fe(II)–Fe(III) compounds for reducing pollutants.     

The Debye-Waller factor and the diffusion process for hydrogen in Nb,Ta, and V single crystals investigated by neutron spectroscopy

Quasielastic scattering of slow neutrons on hydrogen diffusing in the-phase of NbH_0.02, TaH_0.13 and VH_0.07 single crystals was investigated in a wide range of temperatures and scattering vectorsQ (0.5Q2.5 Å^–1). The incoherent scattering lawS _d (Q,) for four different diffusion models was consistently compared with the measured lineshapes. At elevated temperatures one had to introduce correlated jumps to describe the experimental data, whereas at room temperature a model with jumps between adjacent sites is sufficient. The integrated quasielastic intensityI(Q) obtained from the fit ofS _d (Q,) with the measured spectra follows an isotropic Debye-Waller factor with mean square amplitudes u ²=0.02–0.04 Ų for H in Ta (20°C–500°C), and u ²=0.03–0.04 Ų for H in Nb (20°C–300°C). For H in V,I(Q) obtained from the analysis of the quasielastic scattering deviates from a normal Debye-Waller factor behaviour. This effect is assumed to be due to the flight process between the interstitial sites. On the other hand, a normal Debye-Waller factor was obtained from theQ-dependence of the inelastic scattering of the band modes, with values of u ²=0.02–0.04 Ų (135°C–500°C). The observed values of u ² were compared with theoretical calculations.     

Spin-1/2 Maxwell Fields

Requiring covariance of Maxwell's equations without a priori imposing charge invariance allows for both spin-1 and spin-1/2 transformations of the complete Maxwell field and current. The spin-1/2 case yields new transformation rules, with new invariants, for all traditional Maxwell field and source quantities. The accompanying spin-1/2 representations of the Lorentz group employ the Minkowski metric, and consequently the primary spin-1/2 Maxwell invariants are also spin-1 invariants; for example, ²–A ², E ²–B ²+2i EB–(₀ +A)². The associated Maxwell Lagrangian density is also the same for both spin-1 and spin-1/2 fields. However, in the spin-1/2 case, standard field and source quantities are complex and both charge and gauge invariance are lost. Requiring the potentials to satisfy the Klein–Gordon equation equates the Maxwell and field-potential equations with two Dirac equations of the Klein–Gordon mass, and thus one complex Klein–Gordon Maxwell field describes either two real vector fields or two Dirac fields, all of the same mass.     

Relativistic dynamics of interacting point particles: Central position of the Wheeler-Feynman scheme

The Wheeler-Feynman (WF) relativistic theory of interacting point particles, generalized by acceptance of an arbitrary spacelike interaction, is shown to possess a privileged status, reminiscent of the central force interactions occurring in Newtonian mechanics. This scheme is shown to be isomorphic to the classical one of the statics of interacting flexible current-carrying wires obeying the Ampère-Laplace (AL) formulas: to the tensionT (T ² =const) of the wire corresponds the momentum-energy pⁱ (p_ipⁱ=–c²m²) of the particle; to the Laplace linear force density –iH×dr corresponds the Lorentz force QH^ij dr_j; to the Laplace potential ir^–1 dr corresponds the WF potential Q(r²) drⁱ, etc. Among the differences, there is self-action in the AL scheme and no self-action in the WF scheme. A stationary energy principle in the AL scheme is isomorphic to Fokker's stationary action principle in the WF scheme.     

Scaling function for energy-density correlations: Dispersion theory andε-expansion forT>T c

A dispersion representation for the static energy-density correlation function ² (q) ²(–q) _c =C(q,T)=A+Bt ^–h(z ²), wherez=q , t=(T—T)_c/T _c and is the correlation length, is discussed.h(z ²) is calculated to order ² in the zero-field critical region (T>T _c) for the standard isotropicn-component ⁴Ginzburg-Landau-Wilson model. Utilizing a procedure similar to that introduced by Bray for the two-point correlation function, the-expansion results are used in conjunction with an approximant for the spectral functionF(z/2) Imh(—z ²) based on the asymptotically exact short-distance expansion resulth ^–1(z ²)z ^/v[D ₀+D ₁ z ^{–(1 —)/v} +D ₂ z ^–1/v ] to predict quantitatively the full momentum dependence ofC(q,T) forT>T _c. In contrast to the two-point correlation function,C(q,T) is found to be a monotonic function as the critical temperature is approached at fixedq (forT>T _c).     

The Dynamics of Intramolecular Excited State Relaxation of <Emphasis Type="Italic">N</Emphasis>-Anthryl Substituted Pyridinium Cations

N-(1-Anthryl)-2,4,6-trimethyl-pyridinium (I), N-(2-anthryl)-2,4,6-trimethyl-pyridinium (II) and 10-(1-anthryl)-1,2,3,4,5,6,7,8-octahydro-acridinium cations (III) with anomalously high fluorescence Stokes shift have been investigated. Fluorescence kinetics analysis at various temperatures showed that in the range 293–77 K, the radiative deactivation rate constants (k_f) increase by 5.5 to 30 times. The low-temperature time-resolved emission spectra of I–III were found to be consistent with the model: A A^* B^* where A^* is the local excited twisted form and B^* is the relaxed more planar, bent conformer of the molecule. The rate constants of the excited relaxed state formation (k₁) and back reaction (k_–1) of compounds studied were estimated.     

Supertransferred hyperfine interaction in magnetic insulators (II)

The important mechanisms of supertransferred hyperfine (STHF) interactions in N–O–M chains are briefly discussed: (i) spin polarization ofns states in the N-ion due to the s-d exchange interaction,H _STHF ^sd ; (ii) contributions of spin-polarized states of the intervening O-ion,H _STHF ^II ; (iii) transfer of d-electrons of the M-ion to emptyns states in the N-ion,H _STHF ^III . The dependence ofH _STHF upon the N–O–M bond configuration, electronic structure, and orbital state of the M-ion is presented in a convenient form. The STHF interactions in the chains Sn⁴⁺–O^2––Fe³⁺, Cr³⁺ in compounds with slightly distorted Perovskite structure are considered. The STHF field in the chain Sn⁴⁺–O^2––Cr³⁺ is shown to change the sign within the range of angles near 170°. This conclusion is in line with published data on the isoelectronic chain Sn⁴⁺–O²–Mn⁴⁺ in the compounds Ca_1–x Sr _x MnO₃. The results obtained for the N–O–Fe³⁺ chain are rationalized by the predicted angular dependence ofH _STHF=+ cos + cos². Features of the STHF interactions in N–O–M chains with an M-ion in an orbital degenerate state are examplified by a preliminary analysis of N–O^2––V³⁺ chains in orthovanadites.     

Simple quantum cosmology: vacuum energy and initial state

A static non-singular 10-dimensional closed Friedmann universe of Planck size, filled with a perfect fluid with equation of state p = – , can arise spontaneously by a quantum fluctuation from nothing in 11-dimensional spacetime. A quantum transition from this state can initiate the inflationary quantum cosmology outlined in [Gen. Relativ. Gravit. 33, 1415 (2001)]. With no fine-tuning, that cosmology predicts about 60 e-folds of inflation and a vacuum energy density depending only on the number of extra space dimensions (seven), G,, c and the ratio between the strength of gravity and the strength of the strong force. The fraction of the total energy in the universe represented by this vacuum energy depends on the Hubble constant H₀. Estimates of H₀ from WMAP, SDSS, the Hubble Key Project, and Sunyaev-Zeldovich and X-ray flux measurements range from 60 to 72 km s^–1 Mpc^–1. Using a mid-range value of H₀ = 65 km s^–1 Mpc^–1, the model in [Gen. Relativ. Gravit. 33, 1415 (2001)] predicts 0.7.     

High-frequency expansion of relativistic classical plasma dielectric tensor

A high-frequency sum-rule expansion is derived for the transverse elements of the relativistic classical plasma dielectric tensor in an isotropic system. The relativistic results are different from the nonrelativistic ones by a factor of ^–1(1 – ²/3c ²) for ₂ (k) and longitudinal plasmon and transverse photon frequencies, and by ^–2(1–2 ²/3c ²) for ₂ (k).     

Influence of magnetic field on cesium thermionic energy converter

This article deals with the calculation of the influence of the magnetic field upon the electric current of a thermionic converter presupposing the approach to conditions in a low-pressure cesium converter. The distribution of the starting velocities of the emitted electrons is considered firstly as independent of the angle from the perpendicular to the emitter plane, and secondly according to the cosine law.The magnetic field effect from the converter current is calculated and compared with the calculations in the papers by Schock [1] and Block [2]; the effect of the external magnetic field is verified by measurements on a solar thermionic converter prototype.Symbols F=I/I ₀ factor of current reduction from magnetic field effect - ¯F value of factorF (when the magnetic field is not constant) - I [A/m²] density of collector current (real current influenced by magnetic field) - I ₀ [A/m²] theoretical density of collector current (in ideal case equals electron emission current) - T _e [°K] electron gas temperature; assumed equal to emitter temperatureT _E [°K] - B[Wb/m²] magnetic induction (field) - D[m] distance from emitter to collector - R[m] radius of electrodes, emitter and collector - r[m] variable radius in the limits 0 toR - V [m/s] random velocity of electron - v _xz [m/s] component of the vectorV inx-z plane - v _m =2kT _E /m most probable velocity in the velocity distribution according to Maxwell and Boltzmann - w-v _xz /v _m relatively expressed electron velocityv _xz - the angle of any vectorV - [m] radius of circular electron path - n [m^–3] number (density) of electrons with certain value of random velocity - n ₀ [m^–3] total electron number (density) - n ₁ [m^–3] number of electrons returned to emitter by means of magnetic field - N ₀ [m^–2s^–1] total flow of thermionic electrons emitted from a unit surface - N ₁ [m^–2s^–1] partial flow of electrons returned to emitter - P=N ₁/N₀ relatively expressed flow of electrons returned to emitter (whenB = const.) - ¯P mean value ofP (whenB const.) - F _cos, ,P _cos, values asF,¯F,P,¯P in case of velocity distribution according to cosine law - m=9·107×10^–31 [gk] electron mass - e=1·60×10^–19 [C] electron charge - k×1·38×10^–23 [J/grad] Boltzmann's constant - ₀ 1·257×10^–6 [H/Vs] permeability of vacuum     

119Sn-mössbauer,IR, and DSC study of tin oxyfluoride glasses

The¹¹⁹Sn-Mössbauer spectra ofxSnO·(70–x)SnF₂·30P₂O₅ glasses (0 x70) measured at 78 K comprised a doublet due to Sn²⁺ (=3.30–3.36 mm s^–1, = 1.70–1.72 mm s^–1) and a weak singlet due to Sn⁴⁺ located at –0.23 mm s^–1 with respect to BaSnO₃. The and of Sn²⁺ were comparable to those of Sn₂P₂O₇. Small Debye temperatures (146 and 155 K) were obtained from the low-temperature measurements. These results indicate that Sn²⁺ and Sn₄₊ occupied interstitial sites, being loosely and ionically bonded to distorted PO_4–x F _x tetrahedra.     

Trace norm convergence of exponential product formula

LetH andK be lower-bounded self-adjoint operators whose form sum is denoted byH K. We show the norm inequality (e^–rH/2 e^–rK e^–rH/2)^1/r forr > 0 and any symmetric norm . WhenH +K is essentially self-adjoint and e^–K is of trace class, we prove that (e^–rH/2e^–rK e^–rH/2)^1/r converges asr 0 to e^–(H+K) in the trace norm.     

ADDENDUM TO “THE RELATIVISTIC COVARIANCE OF THE B-CYCLIC RELATIONS”

In a previous paper [Found. Phys. Lett. 10, 383–391 (1997)] we proved that the Evans–Vigier definitions of B ⁽⁰⁾ and B ⁽³⁾ may be related not with magnetic fields but with a 4-vector field. In the present Addendum it is shown that the terms used in the B-cyclic theorem proposed by M. Evans and J.-P. Vigier may have various transformation properties with respect to Lorentz transformations. The fact whether theB ⁽³⁾ field is a part of a bivector (which is equivalent to antisymmetric second-rank tensor) or a part of a 4-vector depends on the phase factors in the definition of positive- and negative-frequency solutions of the (B, E) transverse field. This is closely connected to our considerations of the Bargmann–Wightman–Wigner (Gelfand–Tsetlin–Sokolik) constructs and with Ahluwalia's recent consideration of the phase factor related to gravity. The physical relevance of proposed constructs is discussed.     

Synthesis, Characterization, and Luminescence Spectroscopic Accessibility Studies of tris(2,2′-Bipyridine)Ruthenium(II)-Labeled Inorganic-Organic Hybrid Polymers

The modified tris(2,2-bipyridine)ruthenium(II) complex 2 was anchored via a sol-gel process to different polysiloxane matrices to give a series of novel inorganic–organic hybrid polymers. One of the bipyridine ligands of 2 was provided with a long-chain spacer carrying a triethoxysilyl function (T group) at the end, which enables almost free mobility of the transition metal center. The polymers were swollen in several organic solvents of different polarity to investigate the luminescence behavior in the presence of quencher molecules. The luminescence of 2 was quenched using anthracene and atmospheric oxygen in appropriate concentrations of 5 · 10^–4 to 5 · 10^–3 and 3 · 10^–3 M, respectively. The luminescence behavior of 2 was determined by steady-state and time-resolved luminescence experiments. Translational mobilities of molecular species dissolved in the liquid phase were investigated by the kinetics of luminescence quenching after laser excitation. Both the translational mobility of anthracene and atmospheric oxygen, specified with the rate constant k₂ and the percentage of accessible luminescent centers were determined. Translational mobilities and the accessibility for anthracene and atmospheric oxygen in hybrid materials are significantly higher than in conventional Q type polysiloxanes.     

Ensemble average of an arbitrary number of pairs of different eigenvalues using Grassmann integration

An identity satisfied by the eigenvalues of a real-symmetric matrix and an integral representation of a determinant using Grassmann variables are used to show that the ensemble average ofS different pairs of eigenvalues of a GOE is given by (–1) ^S 2^{–S –1/2}(S+1/2).     

The Sixth-Moment Sum Rule for the Pair Correlations of the Two-Dimensional One-Component Plasma: Exact Result

The system under consideration is a two-dimensional one-component plasma in the fluid regime, at density n and arbitrary coupling =e ² (e=unit charge, =inverse temperature). The Helmholtz free energy of the model, as the generating functional for the direct pair correlation c, is treated in terms of a convergent renormalized Mayer diagrammatic expansion in density. Using specific topological transformations within the bond-renormalized Mayer expansion, we prove that the nonzero contributions to the regular part of the Fourier component of c up to the k ²-term originate exclusively from the ring diagrams (unable to undertake the bond-renormalization procedure) of the Helmholtz free energy. In particular, (k)=–/k ²+/(8n)–k ²/[96(n)²]+O(k ⁴). This result fixes via the Ornstein–Zernike relation, besides the well-known zeroth-, second-, and fourth-moment sum rules, the new sixth-moment condition for the truncated pair correlation h, n(n/2)³ r ⁶ h(r) d r=3(–6)(8–3)/4.     

On First-order Corrections to the LSW Theory II: Finite Systems

We consider first–order corrections to the classical theory by Lifshitz, Slyozov and Wagner (LSW) for systems with a finite number of particles. Numerical simulations in V. E Fradkov etal. [Phys. Rev. E 53:3925–3932 (1996)] show a cross–over in the scaling of the correction term from ^1/3 to ^1/2 ( is the volume fraction of particles), when the system size becomes larger than the screening length. We rigorously derive this cross–over for the rate of change of the energy, starting from the monopole approximation. The proof exploits the fact that the rate of change of energy has a variational characterization.     

Low temperature magnetic ordering of a series of structurally characterized chloroferrates

Iron-57 Mössbauer spectroscopic results for a new series of related chloroferrate salts of complex ammonium cations are presented. In particular, spectra of materials based on FeCl ₄ ^– , FeCl ₅ ^2– , FeCl ₆ ^– , [FeCl₅(H₂O)]^2–, and [FeCl₅(CH₃OH)]^2– anions are discussed relative to the systematics of their isomer shifts, coordination numbers, and low temperature 3D critical ordering temperatures. The following specific systems have been studied by Mössbauer spectroscopy and definitively characterized by single-crystal X-ray analysis: [CH₃NH ₃ ⁺ ]₄[FeCl₆]^3–[Cl^–](A), [Me₂N(-CH₂CH₂)₂NMe ₂ ⁺ ][FeCl ₅ ^2– ] (B), [H₃NCH₂CH₂NH ₃ ²⁺ ][FeCl₅·H₂O^2–], (C), and [NH₃(CH₂)₆NH ₃ ²⁺ ]₄[(FeCl ₆ ^3– )(FeCl ₄ ^– )ClCl ₄ ^– ] (D). The spectral data for these complexes are considered in the light of results of previous studies for systems such as [K₂FeCl₅·H₂O], [Co(NH₃)₆][FeCl₆], diamethyltriethylenediammoniumpentachloroferrate (E), and the related [FeCl₅·CH₃OH]^2– salt (F). The critical 3D ordering temperatures are 1.45, 6.80, 3.45, and 1.22K forA, B, C, andD, respectively.     

Energieverlust und Lichtausbeute bei axialem Channeling von α-Teilchen in Anthrazen

In an atomic beam experiment we measured the hyperfine structure (hfs) of the metastable states 3d ² 4s ⁴ F _3/2,9/2 of Scandium, which were populated by electron impact. A single mode cw dye laser was used to create a population difference between neighbouring hfs states by optical pumping. Rf transitions between these states, equalizing the population numbers, could then be detected by repeating the optical pumping experiment and looking for the laser induced fluorescent light. The following constants of the magnetic dipole and electric quadrupole interactions have been obtained by this method:A _3/2=–158.513(3)MHzB _3/2=–5.19(2)MHzA _9/2=285.967(9)MHzB _9/2=–15.46(70)MHzFrom these constants and the corresponding values forJ=5/2, 7/2, as measured by [1], the following parameters can be calculated: r ^–3 _d ⁰¹ =0.911(4)a ₀ ^–3 r ^–3 _d ¹² =2.395 (62)a ₀ ^–3 and:=r ^–3 _d ⁰¹ /r ^–3 _d ¹² =0.38(1).