An expansion of Hartree-Fock and correlation energies,relativistic corrections and the dipole matrix element in the powers of of 1/Z

Feynman diagrammatic technique was used for the calculation of Hartree-Fock and correlation energies, relativistic corrections, dipole matrix element. The whole energy of atomic system was defined as a polen-electron Green function. Breit operator was used for the calculation of relativistic corrections. The Feynman diagrammatic technique was developed for 〈H_B>. Analytical expressions for the contributions from diagrams were received. The calculations were carried out for the terms of such configurations as 1s² 2sⁿ¹ 2pⁿ² (2 ≧n₁≧ 0, 6≧ n₂ ≧ 0). Numerical results are presented for the energies of the terms in the form $$E = E_0 Z^2 + \Delta {\rm E}_2 + \frac{1}{Z}\Delta {\rm E}_3 + \frac{{\alpha ^2 }}{4}(E_0^r + \Delta {\rm E}_1^r Z^3 )$$ and for fine structure of the terms in the form $$\begin{gathered} \left\langle {1s^2 2s^{n_1 } 2p^{n_2 } LSJ|H_B |1s^2 2s^{n_1 \prime } 2p^{n_2 \prime } L\prime S\prime J} \right\rangle = \hfill \\ = ( - 1)^{\alpha + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 1} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 [E^{(0)} (Z - B) + \varepsilon _{co} ] + \hfill \\ + ( - 1)^{L + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 2} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 \varepsilon _{cc} . \hfill \\ \end{gathered} $$ Dipole matrix elements are necessary for calculations of oscillator strengths and transition probabilities. For dipole matrix elements two members of expansion by 1/Z have been obtained. Numerical results were presented in the form P(a,a′) = a/Z(1+τ/Z).     

Study of hypercharge exchange reactions of the type 0−1/2+→2+1/2+ at 4GeV/c

Results are presented for the quasi two-body hypercharge exchange reactions of the type 0^?1/2⁺→2⁺1/2⁺: $$\begin{gathered} \pi ^ - p \to K^0 (1420)(\Lambda ,\sum ^0 ), \hfill \\ K^ - p \to f^0 (1270)(\Lambda ,\sum ^0 ), \hfill \\ K^ - p \to f\prime (1515)(\Lambda ,\sum ^0 ), \hfill \\ \end{gathered} $$ using data from two high statistics bubble chamber experiments. The total and differential cross sections are presented and compared with those obtained for the corresponding vector meson reactions and with simple phenomenological ideas.     

Gyromagnetic ratios of collective states of166Er

The static hyperfine field ofB _hf ^4.2k (ErHo) = 739(18)T of a ferromagnetic holmium single crystal polarized in an external magnetic field of ± 0.48T at ～4.2K was used for integral perturbed γ-γ angular correlation (IPAQ measurements of the g-factors of collective states of¹⁶⁶Er. The 1,200y ^166m Ho activity was used which populates the ground state band and the γ vibrational band up to high spins. The results: $$\begin{gathered} g(4_g^ + ) = + 0.315(16) \hfill \\ g(6_g^ + ) = + 0.258(11) \hfill \\ g(8_g^ + ) = + 0.262(47)and \hfill \\ g(6_\gamma ^ + ) = + 0.254(32) \hfill \\ \end{gathered}$$ exhibit a significant reduction of the g-factors with increasing rotational angular momentum. The followingE2/M1 mixing ratios of interband transitions were derived from the angular correlation coefficients: $$\begin{gathered} 5_\gamma ^ + \Rightarrow 4_g^ + :\delta (810keV) = - (36_{ - 7}^{ + 11} ) \hfill \\ 7_\gamma ^ + \Rightarrow 6_g^ + :\delta (831keV) = - (18_{ - 2}^{ + 3} )and \hfill \\ 7_\gamma ^ + \Rightarrow 8_g^ + :\delta (465keV) = - (63_{ - 12}^{ + 19} ). \hfill \\ \end{gathered}$$ The results are discussed and compared with theoretical predictions.     

Search for rare radiative decays ofΦ-meson at VEPP-2M

Results of the search for rare radiative decay modes of the ?-meson performed with the Neutral Detector at the VEPP-2M collider are presented. For the first time upper limits for the branching ratios of the following decay modes have been placed at 90% confidence level: $$\begin{gathered} B(\phi \to \eta '\gamma )< 4 \cdot 10^{ - 4} , \hfill \\ B(\phi \to \pi ^0 \pi ^0 \gamma )< 10^{ - 3} , \hfill \\ B(\phi \to f_0 (975)\gamma )< 2 \cdot 10^{ - 3} , \hfill \\ B(\phi \to H\gamma )< 3 \cdot 10^{ - 4} , \hfill \\ \end{gathered} $$ whereH is a scalar (Higgs) boson with a mass 600 MeV<m _H <1000 MeV, the real measurement isB(φ→H γ)·B(H→2π⁰)<0.8·10^-4, the quoted result is model dependent, as explained in the text, $$\begin{gathered} B(\phi \to a\gamma ) \cdot B(a \to e^ + e^ - )< 5 \cdot 10^{ - 5} , \hfill \\ B(\phi \to a\gamma ) \cdot B(a \to \gamma \gamma )< 2 \cdot 10^{ - 3} , \hfill \\ \end{gathered} $$ wherea is a particle with a low mass and a short lifetime, $$B(\phi \to a\gamma )< 0.7 \cdot 10^{ - 5} ,$$ wherea is a particle with a low mass not observed in the detector.     

Determination of hyperfine interactions from partly resolved moessbauer spectra

Moessbauer spectra with different sets of parameters were calculated. A fit with a superposition of Lorentzians to these theoretical spectra showed, that systematic errors must be expected if the hyperfine structure of the spectrum is only partly resolved. Correction factors for some simple cases are given. Experiments to test the calculations were performed with¹³³Cs (81 keV transition),¹⁶⁵Ho (94.7 keV transition) and¹⁷⁸Hf (93 keV transition). In all cases fits using the transmission integral and superpositions of Lorentzians showed the expected trends. We get the following results: $$\begin{gathered} ^{133} Cs:\frac{{g_{ex} }}{{g_{gr} }} = 1.90\left( 4 \right) \hfill \\ ^{165} Ho:\tau \left( {94.7keVlevel} \right) = 32\left( 1 \right)ps \hfill \\ \frac{{g_{ex} }}{{g_{gr} }} = 0.77\left( 3 \right) \hfill \\ ^{178} Hf:|H_{eff} \left( {4K,in iron} \right)| = 633\left( {40} \right)KG \hfill \\ |H_{eff} \left( {77K,in iron} \right)| = 630\left( {41} \right)KG. \hfill \\ \end{gathered}$$      

Three-loop on-shell charge renormalization without integration:\Lambda _{QED}^{\overline {MS} } to four loops

Using algebraic methods, we find the three-loop relation between the bare and physical couplings of one-flavourD-dimensional QED, in terms of Γ functions and a singleF ₃₂ series, whose expansion nearD=4 is obtained, by wreath-product transformations, to the order required for five-loop calculations. Taking the limitD→4, we find that the \(\overline {MS} \) coupling \(\bar \alpha (\mu )\) satisfies the boundary condition $$\begin{gathered} \frac{{\bar \alpha (m)}}{\pi } = \frac{\alpha }{\pi } + \frac{{15}}{{16}}\frac{{\alpha ^3 }}{{\pi ^3 }} + \left\{ {\frac{{11}}{{96}}\zeta (3) - \frac{1}{3}\pi ^2 \log 2} \right. \hfill \\ \left. { + \frac{{23}}{{72}}\pi ^2 - \frac{{4867}}{{5184}}} \right\}\frac{{\alpha ^4 }}{{\pi ^4 }} + \mathcal{O}(\alpha ^5 ), \hfill \\ \end{gathered} $$ wherem is the physical lepton mass and α is the physical fine structure constant. Combining this new result for the finite part of three-loop on-shell charge renormalization with the recently revised four-loop term in the \(\overline {MS} \) β-function, we obtain $$\begin{gathered} \Lambda _{QED}^{\overline {MS} } \approx \frac{{me^{3\pi /2\alpha } }}{{(3\pi /\alpha )^{9/8} }}\left( {1 - \frac{{175}}{{64}}\frac{\alpha }{\pi } + \left\{ { - \frac{{63}}{{64}}\zeta (3)} \right.} \right. \hfill \\ \left. { + \frac{1}{2}\pi ^2 \log 2 - \frac{{23}}{{48}}\pi ^2 + \frac{{492473}}{{73728}}} \right\}\left. {\frac{{\alpha ^2 }}{{\pi ^2 }}} \right), \hfill \\ \end{gathered} $$ at the four-loop level of one-flavour QED.     

Centrifugal distortion constants of pentafluorobenzene from its pure rotation spectrum

The centrifugal distortion analysis of the pure rotational spectrum of pentafluorobenzene in the frequency region of 8 to 18 GHz involvingJ upto 54 has yielded the following rotational and quartic centrifugal distortion constants: $$\begin{gathered} A'' = 1480 \cdot 8665 \pm 0 \cdot 0026 MHz, \tau = - 1 \cdot 751 \pm 0 \cdot 20 kHz, \hfill \\ B'' = 1030 \cdot 0782 \pm 0 \cdot 0025 MHz, \tau _2 = - 0 \cdot 567 \pm 0 \cdot 066 kHz, \hfill \\ C'' = 607 \cdot 5152 \pm 0 \cdot 0026 MHz, \tau _{aaaa} = - 0 \cdot 765 \pm 0 \cdot 068 kHz, \hfill \\ \tau _{bbbb} = - 0 \cdot 612 \pm 0 \cdot 065 kHz, \hfill \\ \tau _{cccc} = - 0 \cdot 547 \pm 0 \cdot 068 kHz. \hfill \\ \end{gathered} $$      

Rigorous bounds on quark masses within a nonrelativistic approach

By applying the Feynman-Hellmann theorem to \(q\bar q\) systems we find the following bounds on quark mass differences from the spectrum ofall quarkonium states $$\begin{gathered} 0.27 \leqq m_s - m_u \leqq 0.45GeV \hfill \\ 1.23 \leqq m_c - m_s \leqq 1.46GeV \hfill \\ 3.30 \leqq m_b - m_c \leqq 3.55GeV. \hfill \\ \end{gathered}$$ As best values we derive $$\begin{gathered} m_u = m_d = 0.31GeV,m_s = 0.62GeV, \hfill \\ m_c = 1.91GeV,m_b = 5.27GeV. \hfill \\ \end{gathered}$$      

BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one

We show that for most non-scalar systems of conservation laws in dimension greater than one, one does not have BV estimates of the form $$\begin{gathered} \parallel \overline V u(\overline t )\parallel _{T.V.} \leqq F(\parallel \overline V u(0)\parallel _{T.V.} ), \hfill \\ F \in C(\mathbb{R}),F(0) = 0,F Lipshitzean at 0, \hfill \\ \end{gathered} $$ even for smooth solutions close to constants. Analogous estimates forL ^p norms $$\parallel u(\overline t ) - \overline u \parallel _{L^p } \leqq F(\parallel u(0) - \overline u \parallel _{L^p } ),p \ne 2$$ withF as above are also false. In one dimension such estimates are the backbone of the existing theory.     

Nuclear orientation of105Rh and95Tc in iron

The temperature-dependent anisotropy of γ-rays following the decay of oriented⁹⁵Tc and¹⁰⁵Rh nuclei was studied with a Ge(Li) detector. Mixing coefficients of some γ-and preceding β-transitions, the spins of two intermediate levels, and the magnetic hyperfine splitting of the⁹⁵Tc and¹⁰⁵Rh ground states in an Fe host were measured. From the known hyperfine fields the following magnetic moments were deduced: $$\begin{gathered} \mu \left( {^{105} Rh,\tfrac{{7 + }}{2}} \right) = 4.34\left( {12} \right) n.m.; \hfill \\ \mu \left( {^{95} Tc,\tfrac{{9 + }}{2}} \right) = 5.82\left( {12} \right)n.m. \hfill \\ \end{gathered}$$      

Nuclear orientation and NMR/ON of205,207Po

^205,207Po have keen implanted with an isotope separator on-line into cold host matrices of Fe, Ni, Zn and Be. Nuclear magnetic resonance of oriented²⁰⁷Po has been observed in Fe and Ni, of²⁰⁵Po in Fe. The resonance frequencies for zero external field are $$\begin{gathered} v_L (^{207} Po\underline {Fe} ) = 575.08(20)MHz \hfill \\ v_L (^{207} Po\underline {Ni} ) = 160.1(8)MHz \hfill \\ v_L (^{205} Po\underline {Fe} ) = 551.7(8)MHz. \hfill \\ \end{gathered} $$ From the dependence of the resonance frequency on external magnetic field theg-factor of²⁰⁷Po was derived as $$g(^{207} Po) = + 0.31(22).$$ Using this value the magnetic hyperfine fields of Po in Fe and Ni were obtained as $$\begin{gathered} B_{hf} (Po\underline {Fe} ) = + 238(16)T \hfill \\ B_{hf} (Po\underline {Ni} ) = 66.3(4.6)T. \hfill \\ \end{gathered}$$ Theg-factor of²⁰⁵Po follows as $$g(^{205} Po) = + 0.304(22).$$ From the temperature dependence of the anisotropies ofγ-lines in the decay of^205,207Po the multipole mixing of several transitions was derived. The electric interaction frequenciesv _Q=eQV_zz/h in the hosts Zn and Be were measured as $$\begin{gathered} v_Q (^{207} Po\underline {Zn} ) = + 42(3)MHz \hfill \\ v_Q (^{207} Po\underline {Be} ) = - 70(20)MHz \hfill \\ v_Q (^{205} Po\underline {Be} ) = - 42(17)MHz. \hfill \\ \end{gathered}$$      

Experimental study of the inclusiveη-spectrum fromp \bar p annihilations at rest in liquid hydrogen

The inclusive η-momentum spectrum from \(\bar p\) annihilations at rest in liquid hydrogen was measured at LEAR. Branching ratios were obtained for $$\begin{gathered} p\bar p \to \eta \omega \left( {1.04_{ - 0.10}^{ + 0.09} } \right)\% ,\eta \rho ^0 \left( {0.53_{ - 0.08}^{ + 0.20} } \right)\% , \hfill \\ \pi a_2 \left( {8.49_{ - 1.10}^{ + 1.05} } \right)\% ,\eta \pi ^0 \left( {1.33 \pm 0.27} \right) \times 10^{ - 4} , \hfill \\ \end{gathered} $$ , and ηη(8.1±3.1)×10^?5. An upper limit for \(p\bar p \to \eta \eta '\) of 1.8×10^?4 at 95% CL was found. The ratio of the branching ratios is BR(η?)/BR(ηω)=0.51 _?0.06 ^+0.20 . For the ratio of branching ratios into two pseudoscalar mesons, we have BR(ηπ⁰)/BR(π⁰π⁰)=0.65±0.14, BR(ηη)/BR(π⁰π⁰), BR(η η ^′ )/BR(π⁰π⁰) at 95% CL, and BR(ηη)/BR(ηπ⁰).     

Hyperfine structure,g j factors and lifetimes of excited2 P 1/2 states of Rb

The hyperfine structure of the 6² P _1/2 and 7² P _1/2 state of⁸⁵Rb and⁸⁷Rb and of the 6² P _3/2 state of⁸⁷Rb has been investigated with optical double resonance at intermediate magnetic fields. The magnetic interaction constants,g _j factors and lifetimes are: $$\begin{gathered} 6^2 P_{1/2} state: A\left( {^{85} Rb} \right) = 39.11\left( 3 \right) MHz,A\left( {^{87} Rb} \right) = 132.56 \left( 3 \right)MHz, \hfill \\ g_j = 0.6659\left( 3 \right), \tau = 1.14\left( {13} \right) \cdot 10^{ - 7} \sec , \hfill \\ 7^2 P_{1/2} state: A\left( {^{85} Rb} \right) = 17.68\left( 8 \right)MHz,A\left( {^{87} Rb} \right) = 59.92\left( 9 \right)MHz, \hfill \\ g_j = 0.6655\left( 5 \right), \hfill \\ 6^2 P_{3/2} state: g_j = 1.3337\left( {10} \right), \tau = 1.12\left( 8 \right) \cdot 10^{ - 7} \sec for ^{87} Rb. \hfill \\ \end{gathered} $$ From the hfs coupling constants of then ² P multiplets a 11.5% core polarization contribution to the magnetic hfs of then ² P _3/2 states is obtained, which is found to be independent from the main quantum numbern. The expectation values <r ^?3> _j for thenp valence electrons corrected for core polarization are compared with those derived from the² P fine structure separation. Good agreement is achieved for allnp levels with the choice ofZ _i =Z?3=34 for the effective nuclear charge number. The nuclear quadrupole moments of⁸⁵Rb and⁸⁷Rb are rederived on the basis of this more improved treatment for thep-electron-nucleus interaction yielding $$\begin{gathered} Q_N \left( {^{85} Rb} \right) = + 0.274\left( 2 \right) \cdot 10^{ - 24} cm^2 \hfill \\ Q_N \left( {^{85} Rb} \right) = + 0.132\left( 1 \right) \cdot 10^{ - 24} cm^2 \hfill \\ \end{gathered} $$ where the error does not include the remaining theoretical uncertainty of about 10%.     

An improvement of the Griffiths-Hurst-Sherman inequality for the Ising ferromagnet

We prove the following inequality for the truncated correlation in the Ising model in zero external field: $$\begin{gathered} \langle \sigma _i \sigma _j \sigma _k \sigma _l \rangle - \langle \sigma _i \sigma _j \rangle \langle \sigma _k \sigma _l \rangle - \langle \sigma _i \sigma _k \rangle \langle \sigma _j \sigma _l \rangle - \langle \sigma _i \sigma _l \rangle \langle \sigma _j \sigma _k \rangle \hfill \\ \leqslant - 2\langle \sigma _i \sigma _m \rangle \langle \sigma _j \sigma _m \rangle \langle \sigma _k \sigma _m \rangle \langle \sigma _l \sigma _m \rangle \hfill \\ - 2 \left( {\langle \sigma _i \sigma _k \rangle - \langle \sigma _i \sigma _m \rangle \langle \sigma _m \sigma _k \rangle } \right) \left( {\langle \sigma _j \sigma _k \rangle - \langle \sigma _j \sigma _m \rangle \langle \sigma _m \sigma _k \rangle } \right)\langle \sigma _k \sigma _l \rangle \hfill \\ - 2 \langle \sigma _i \sigma _m \rangle \langle \sigma _j \sigma _m \rangle \left( {\langle \sigma _i \sigma _k \rangle - \langle \sigma _i \sigma _m \rangle \langle \sigma _m \sigma _k \rangle } \right)\left( {\langle \sigma _i \sigma _l \rangle - \langle \sigma _i \sigma _m \rangle \langle \sigma _m \sigma _l \rangle } \right) \hfill \\ \end{gathered} $$ This inequality is a strengthening of the Lebowitz inequality for the four-point function and implies the following improvement of the GHS inequality: $$\langle \sigma _i ;\sigma ;_j \sigma _k \rangle ^T \leqslant - 2\langle \sigma _i ;\sigma _k \rangle ^T \langle \sigma _j ;\sigma _k \rangle ^T \langle \sigma _k \rangle $$ This in turn implies the critical exponent inequality $$\Delta '_3 \geqslant \gamma ' - \beta $$      

Conformal properties of nonpeeling vacuum space-times

In a previous paper we investigated a class ofnonpeeling asymptotic vacuum solutions which were shown to admit finite expressions for the Winicour-Tamburino energy-momentum and angular momentum integrals. These solutions have the property that $$\psi _0 = O(r^{ - 3 - \in _0 } ), \in _0 \leqslant 2$$ and $$\psi _1 = O(r^{ - 3 - \in _1 } ), \in _1< \in _0 and \in _1< 1$$ withψ ₂,ψ ₃, andψ ₄ having the same asymptotic behavior as they do for peeling solutions. The above investigation was carried out in the physical space-time. In this paper we examine the conformal properties of these solutions, as well as the more general Couch-Torrence solutions, which include them as a subclass. For the Couch-Torrence solutions $$\begin{gathered} \psi _0 = O(r^{ - 2 - \in _0 } ) \hfill \\ \psi _1 = O(r^{ - 2 - \in _1 } ), \in _1< \in _0 {\text{ }}and \in _1 \leqslant 2 \hfill \\ \end{gathered} $$ and , $$\psi _2 = O(r^{ - 2 - \in _2 } ),{\text{ }} \in _2< \in _1 {\text{ }}and \in _2 \leqslant 1$$ withψ ₃ andψ ₄ behaving as they do for peeling solutions. It is our purpose to determine how much of the structure generally associated with peeling space-times is preserved by the nonpeeling solutions. We find that, in general, a three-dimensional null boundary (?⁺) can be defined and that the BMS group remains the asymptotic symmetry group. For the general Couch-Torrence solutions several physically and/or geometrically interesting quantities     

Hyperfine structure measurements in the ground states4 F 3/2,4 F 5/2 and4 F 7/2 of Ta181 with the atomic beam magnetic resonance method

Applying a recently developed evaporation technique for refractory elements the following results have been obtained for Ta¹⁸¹ in an atomic beam magnetic resonance experiment studying the hyperfine structure of 3 levels of the ground state multiplet⁴ F: $$\begin{gathered} g_J (^4 F_{3/2} ) = 0.45024 (4) \hfill \\ \Delta v (^4 F_{3/2} ;F = 5 \leftrightarrow F = 4) = 1822.389 (6) MHz \hfill \\ \Delta v (^4 F_{3/2} ;F = 4 \leftrightarrow F = 3) = 2325.537 (2) MHz \hfill \\ \Delta v (^4 F_{5/2} ;F = 6 \leftrightarrow F = 5) = 1451.476 (7) MHz \hfill \\ \Delta v (^4 F_{5/2} ;F = 5 \leftrightarrow F = 4) = 1537.530 (8) MHz \hfill \\ \Delta v (^4 F_{5/2} ;F = 4 \leftrightarrow F = 3) = 1444.685 (2) MHz \hfill \\ \Delta v (^4 F_{7/2} ;F = 4 \leftrightarrow F = 3) = 1218.372 (2) MHz. \hfill \\ \end{gathered}$$ From these measurements the following constants of the magnetic dipole interaction (A) and the electric quadrupole interaction (B) have been derived: $$\begin{gathered} A (^4 F_{3/2} ) = 509.0801 (8) MHz \hfill \\ B (^4 F_{3/2} ) = - 1012.251 (8) MHz \hfill \\ A (^4 F_{5/2} ) = 313.4681 (8) MHz \hfill \\ B (^4 F_{5/2} ) = - 834.820 (12) MHz. \hfill \\ \end{gathered}$$      

Radiative decays of ρ and ω mesons

The results of the measurements of radiative decays of ρ and ω mesons with the Neutral Detector at thee ⁺ e ^? collider VEPP-2M are presented. The branching ratio of the decay ω→π ⁰γ was measured with higher than in previous experiments accuracy: $${\rm B}(\omega \to \pi ^0 \gamma ) = 0.0888 \pm 0.0062$$ . The ρ⁰→π ⁰ γ branching ratio was measured for the first time: $$B(\rho ^0 \to \pi ^0 \gamma ) = (7.9 \pm 2.0) \cdot 10^{ - 4} $$ . The decays ρ, ω→ηγ were studied. Their branching ratios with the assumption of constructive ρ?ω interference are: $$\begin{gathered} B(\omega \to \eta \gamma ) = (7.3 \pm 2.9) \cdot 10^{ - 4} , \hfill \\ B(\rho \to \eta \gamma ) = (4.0 \pm 1.1) \cdot 10^{ - 4} \hfill \\ \end{gathered} $$ . The branching ratios of ρ, ω→ηγ and ω→e ⁺ e ^? decays were also measured: $$\begin{gathered} B(\omega \to \pi ^ + \pi ^ - \pi ^0 ) = 0.8942 \pm 0.0062, \hfill \\ B(\omega \to e^ + e^ - ) = (7.14 \pm 0.36) \cdot 10^{ - 5} \hfill \\ \end{gathered} $$ . The upper limit for the ω→π ⁰ π ⁰ γ branching ratio was placed: B(ω→π ⁰ π ⁰ γ)<4·10^?4 at 90% confidence level.     

Analyticity and global existence of small solutions to some nonlinear Schrödinger equations

In this paper we will study the nonlinear Schrödinger equations: $$\begin{gathered} i\partial _t u + \tfrac{1}{2}\Delta u = \left| u \right|^2 u, (t,x) \in \mathbb{R} \times \mathbb{R}_x^n , \hfill \\ u(0,x) = \phi (x), x \in \mathbb{R}_x^n \hfill \\ \end{gathered} $$ . It is shown that the solutions of (*) exist and are analytic in space variables fort∈??{0} if φ(x) (∈H ^2n+1,2(? _x ⁿ )) decay exponentially as |x|→∞ andn≧2.     

Theg-factor of the 181 keV level of99Tc and hyperfine fields of Tc in Fe,Co, Ni

Theg-factor of the 181 keV-level of⁹⁹Tc has been redetermined by the spin rotation method. Measurements with polycrystalline sources of Tc in Fe, Co, and Ni yielded values of the hyperfine fields at the Tc nucleus. $$\begin{gathered} g = + 1.310(25) \hfill \\ H_{hf} (Tc{\mathbf{ }}in{\mathbf{ }}Fe) = ( - )290(15)kOe \hfill \\ H_{hf} (Tc{\mathbf{ }}in{\mathbf{ }}Co) = ( - )170(5)kOe \hfill \\ H_{hf} (Tc{\mathbf{ }}in{\mathbf{ }}Ni) = - 47.8(1.5)kOe. \hfill \\ \end{gathered} $$