The Boundary Control Approach to the Titchmarsh-Weyl <Emphasis Type="Italic">m</Emphasis>–Function. I. The Response Operator and the <Emphasis Type="Italic">A</Emphasis>–Amplitude

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the A?amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl m?function associated with the Schrödinger operator H = ?? _x ²  + q(x) on L ₂(0, ∞) with Dirichlet boundary condition at x = 0.     

Erratum to “Lower bounds for the eigenvalues of the basic Dirac operator on a Riemannian foliation” [J. Geom. Phys. 51 (2004) 166–182]

We correct some statements in the paper mentioned in the title.     

On the Non-Lorentz-Invariance of <Emphasis Type="SmallCaps">M.W. Evans’</Emphasis><Emphasis Type="Italic">O</Emphasis>(3)-Symmetry Law

In 1992 M.W. Evans proposed the O(3) symmetry of electromagnetic fields by adding a constant longitudinal magnetic field to the well-known transverse electric and magnetic fields of circularly polarized plane waves, such that certain cyclic relations of a so-called O(3) symmetry are fulfilled. Since then M.W. Evans has elevated this O(3) symmetry to the status of a new law of electromagnetics. As a law of physics must be invariant under admissible coordinate transforms, namely Lorentz transforms, in 2000 he published a proof of the Lorentz invariance of O(3) symmetry of electromagnetic fields. As will be shown below this proof is incorrect; more, after simple correction it will turn out here that the O(3) symmetry cannot be Lorentz invariant.     

x-dependence of the quark distribution functions in the χCQM<Subscript>config</Subscript>

Chiral constituent quark model with configuration mixing (χCQM_config) is known to provide a satisfactory explanation of the “proton spin problem” and related issues. In order to enlarge the scope of χCQM_config, we have attempted to phenomenologically incorporate x-dependence in the quark distribution functions. In particular, apart from calculating valence and sea quark distributions q_val(x) and q̄(x), we have carried out a detailed analysis to estimate the sea quark asymmetries d̄(x)-ū(x), d̄(x)/ū(x) and as well as spin independent structure functions F₂ ^p(x)-F₂ ⁿ(x) and F₂ ⁿ(x)/F₂ ^p(x) as functions of x. We are able to achieve a satisfactory fit for all the above mentioned quantities simultaneously. The inclusion of effects due to configuration mixing have also been examined in the case F₂ ^p(x)-F₂ ⁿ(x) and F₂ ⁿ(x)/F₂ ^p(x) where the valence quark distributions dominate and it is found that it leads to considerable improvement in the results. Further, the valence quark structure has also be tested by extrapolating the predictions of our model in the limit x→1 where data is not available.     

Origin of the E.S.R. hyperfine anisotropy and the cos2 θ rule of the β-fluorine couplings

The origin of the hyperfine couplings of β-fluorine nuclei in free radicals has been studied using the hyperfine tensors determined for and in single crystals. The direction of the maximum principal element of the β-fluorine coupling tensor in was found to be very close to the direction of the maximum overlap of the fluorine 2p π orbital with the half-filled carbon 2p π orbital. This means that the spin density in the fluorine 2p π orbital is mainly due to the direct overlap with the carbon 2p π orbital and the contributions from the hyperconjugation and the spin polarization mechanisms are less important. The cos²θ rule, B ₀ + B cos²θ, for the β-fluorine coupling tensor was also derived from the conformational dependence of the overlap integral. Using the ratio of B/B ₀ calculated from the overlap integral together with the observed hyperfine tensor for , the absolute values of B ₂ and B were determined and results were applied to the β-fluorine coupling tensor of the rotating CF₃ group in . The calculated tensor is in very good agreement with the one observed.     

Measurements of the isotopic composition of UF<Subscript>6</Subscript> according to the fine structure of the IR absorption spectrum in the ν<Subscript>1</Subscript> + ν<Subscript>3</Subscript> band

Absorption spectra of the Q-branch of the ν₁ + ν₃ vibrational–rotational band of uranium hexafluoride (UF₆) recorded in a range of 1290.0–1292.5 cm^–1 using a laser spectrometer based on a quantum cascade laser have been studied. The spectra of samples with a natural isotopic composition (0.7% U²³⁵), an enriched sample (90% U²³⁵), and their gas mixtures (2, 5, and 20% U²³⁵) in a pressure range of 10–70 Torr at a temperature of T = 296 K have been analyzed. The experiments have revealed a highly reproducible fine structure of the recorded spectra. Periodic singularities in the fine-structure spectra have been interpreted as a manifestation of hot band transitions near the Q-branch. Anharmonicity constants X ₂₁, X ₃₁, and X ₃₂ and their combinations X _i1 + X _i3 (i = 4, 5, 6) have been determined. The characteristic features in the fine-structure spectra and the initial spectrum have been used to determine the isotopic composition of enriched UF₆ samples.     

Measurement of the Dalitz-decay branching ratio of the π<Superscript>0</Superscript>

The ratio is measured by comparing the production rates of the π⁰ in the two decay channels. The analysis is based on approximately four million hadronic decays of the Z boson recorded by the ALEPH detector in the years 1991 to 1995. The value of is determined to be . A recalculation of the Particle Data Group world average gives rise to a slightly improved error, and moves the world average closer to the theoretical value.     

Erratum to: “Quantum dissipative dynamics using the Doebner–Goldin equation” [Phys. Lett. A 376 (2012) 3033]

A few misprints appearing in the Letter [Phys. Lett. A 376 (2012) 3033] are corrected.     

Using of the χ<Superscript>2</Superscript> criterion to evaluate the quality of the placement of transitions in the level scheme

The results from a statistical analysis of the evaluated nuclear structure data file (ENSDF) are presented. The article deals with the procedures for evaluating of the quality of the placement of transitions in the level scheme.     

On the nature of the 17 μs isomer of the <Superscript>133</Superscript>Sb valence nucleus

The decay of the 17 μs isomer of ¹³³Sb was re-investigated experimentally. It was produced by thermal neutron induced fission of ²⁴¹Pu. Its detection is based on time correlation between fission fragments selected by the LOHENGRIN spectrometer at ILL (Grenoble), and the γ-rays, and conversion electrons from the isomer. The interpretation of the level scheme is based on shell model calculations, where empirical two-body matrix elements were employed. The good agreement between theory and experiment suggests that the isomer is the 21/2⁺ member of the 2p-1h πg _7/2ν(f _7/2 h _11/2 ⁻¹) configuration. Received: 20 January 2000 / Revised version: 28 February 2000     

Formulation of the Hellmann–Feynman theorem for the “second choice” version of Tsallis’ thermostatistics

An approach to formulating the Hellmann–Feynman theorem within the “second choice” formalism of non-extensive statistical mechanics is considered. For the state of thermal equilibrium, we derive a relation of Hellmann–Feynman type between the derivative of the non-extensive free energy with respect to the external parameter and the quantum statistical q$q$-average of the derivative of the Hamilton operator. We also give a proper extension for an arbitrary observable commuting with the Hamiltonian. Some reasons for the usefulness of new formulas are discussed.     

E.S.R. spectra of the radicals produced in γ-irradiated diborane in the solid state

The hopping and Coulomb repulsion integrals of the Hubbard model, as applied to the NMP-TCNQ charge-transfer crystal, are found by an INDO calculation of the neutral and charged monomer and dimer of TCNQ. The ratio of these integrals is in satisfactory agreement with experiment, although the calculated integrals themselves are very different from the effective ones for electronic transport, due to renormalization, caused largely by counter-ion polarizability.     

Effect of the Composition and the Crystal Structure on the Electrophysical Properties of the Ni<Subscript>1–<Emphasis Type="Italic">x</Emphasis></Subscript>W<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript> System at Low Temperatures

The problem of establishing the correlation between, on the one hand, the chemical and phase compositions of Ni_1–xW_x alloys (0 ≤ x ≤ 0.5) and, on the other hand, the character of the temperature dependences of the electrical resistivity, is considered. Based on the experimental ρ(T) curves, the concentration dependences of are reconstructed in the wide temperature range (50 K ≤ T ≤ 273 K). The ρ(x) curves have features related to a change in the crystal structures of the alloys (concentration fcc–bcc phase transition), their magnetic structures and percolation processes occurring in the two-phase fcc + bcc medium.     

NMR study of the displacement of vortices in the superconductor Tl<Subscript>2</Subscript>Ba<Subscript>2</Subscript>CuO<Subscript>6+δ</Subscript> under the action of a magnetic field pulse

A transition from irreversible to reversible displacement of vortices in the high-temperature super-conductor Tl₂Ba₂CuO_6+δ (T _c =78 K) is investigated by the NMR method combined with pulse modulation of the magnetic field in the temperature range from 20 to 35 K. The results also make it possible to estimate the distance between pinning centers in the sample, which lies in the interval from 4 to 10 nm at a temperature of 20 K.     

Assessment of the relaxation of the autocorrelation function for the <Emphasis Type="Italic">Ã</Emphasis><Superscript>2</Superscript><Emphasis Type="Italic">B</Emphasis><Subscript>2</Subscript> excited state of the NO<Subscript>2</Subscript> molecule

Based on the Bixon-Jortner model of two coupled vibrational ensembles, an analytical expression for the autocorrelation function |P(t)|² of the à ² B ₂ of the term of the NO₂ molecule is derived. Proceeding from the known relaxation parameters of the autocorrelator, a fast method for estimating the nonadiabatic matrix element of the \(\tilde A^2 B_2 - \tilde X^2 A_1\) transition in NO₂ is proposed.     

Specific features of the magneto-optical properties of α-Al<Subscript>2</Subscript>O<Subscript>3</Subscript> in the IR region

The spectra of reflection and magnetoreflection of light from the crystalline insulator α-Al₂O₃ in the IR spectral region (λ = 2.5–25 μm) are investigated. Some features in the magnetoreflection spectra in the wavelength range corresponding to the excitation of optical phonon modes in α-Al₂O₃ are found. A significant increase in magnetoreflection is observed near these wavelengths. The amplitude and shape of the magnetore-flection spectra for the p and s polarizations of probe light are determined. It is shown that the optical and magneto-optical properties of α-Al₂O₃ in the IR region can be described on the basis of the theory of polaron excitation. A satisfactory correlation between the theoretical and experimental spectra is obtained, which indicates that polarons play an important role in determining the optical characteristics of nonmagnetic insulators and make the dominant contribution to the magnetoreflection spectra.