Continuous measures in one dimension

Families of unimodal maps satisfying

1. T _λ: [?1,1]?[?1,1] withT(±1)=?1 and |T _λ ^′ (1)|>1.
2. T _λ(x) isC ² inx ² and λ, and symmetric inx.
3. T ₀(0)=0,T ₁(0)=1 with \(\frac{d}{{d\lambda }}\) T _λ(0)>0

are considered. The results of Guckenheimer (1982) are extended to show that there is a positive measure of λ for whichT _λ has a finite absolutely continuous invariant measure. The appendix contains general theorems for the existence of such measures for some markov maps of the interval.     

Bounds on the number of eigenvalues of the Schrödinger operator

Inequalities on eigenvalues of the Schrödinger operator are re-examined in the case of spherically symmetric potentials. In particular, we obtain:

1. A connection between the moments of order (n ? 1)/2 of the eigenvalues of a one-dimensional problem and the total number of bound statesN _n, inn space dimensions;
2. optimal bounds on the total number of bound states below a given energy in one dimension;
3. alower bound onN ₂;
4. a self-contained proof of the inequality for α ≧ 0,n ≧ 3, leading to the optimalC ₀₄,C ₃;
5. solutions of non-linear variation equations which lead, forn ≧ 7, to counter examples to the conjecture thatC _0n is given either by the one-bound state case or by the classic limit; at the same time a conjecture on the nodal structure of the wave functions is disproved.

     

A proof of the unitarity ofS-matrix in a nonlocal quantum field theory

Using the formfactors which are entire analytic functions in a momentum space, nonlocality is introduced for a wide class of interaction Lagrangians in the quantum theory of one-component scalar field φ(x). We point out a regularization procedure which possesses the following features:

1. The regularizedS ^δ matrix is defined and there exists the limit $$\mathop {\lim }\limits_{\delta \to 0} S^\delta = S.$$
2. The Green positive-frequency functions which determine the operation of multiplication in \(S \cdot S^ + \mathop = \limits_{Df} S \circledast S^ + \) can be also regularized ?^δ and there exists the limit $$\mathop {\lim }\limits_{\delta \to 0} \circledast ^\delta = \circledast \equiv .$$
3. The operator \(J(\delta _1 ,\delta _2 ,\delta _3 ) = S^{\delta _1 } \circledast ^{\delta _2 } S^{\delta _3 + } \) is continuous at the point δ₁=δ₂=δ₃=0.
4. $$S^\delta \circledast ^\delta S^{\delta + } \equiv 1at\delta > 0.$$ Consequently, theS-matrix is unitary, i.e. $$S \circledast S^ + = S \cdot S^ + = 1.$$

     

A phase cell cluster expansion for a hierarchical Ø 3 4 model

The formalism developed in a previous paper is applied to yield a phase cell cluster expansion for a hierarchical ø ₃ ⁴ model. The field is expanded into modes with specific renormalization group scaling properties. The present cluster expansion for a vacuum expectation value is formally the natural factorization of each term in the perturbation expansion into the contribution of modes connected to the variables in the expectation via interactions, and that of the complementary set. The expectation value is thus realized as a sum of contributions due tofinite subsets of the modes. We emphasize the following additional features:

1. Partitions of unity are not used.
2. There areessentially no cut-offs.
3. The expansion is developed directly, without an initial need to prove an ultraviolet stability bound, the most difficult part of the traditional approach.

Our main interest in the present phase cell cluster expansion is founded in the belief that it may be the right vehicle for proving the existence of a nontrivial four-dimensional field theory.     

SIMS and flash desorption studies of nickel oxygen interaction

O₂ exposure of polycrystalline nickel at 300 K results in characteristics changes of secondary ion emission. These can be described by a model which is in good agreement with corresponding LEED, AES, XPS, and ΔΦ results of other authors. According to this model, oxygen can be bonded on Ni in at least five different phases:

1. chemisorption, indicated by a rapid increase of Ni⁺, Ni ₂ ⁺ , and Ni₂O⁺ (≦5 L);
2. a rearranged chemisorption layer, characterized by a drastic decrease of Ni⁺, Ni ₂ ⁺ , and Ni₂O⁺ (5–15 L);
3. nickel oxide (NiO) responsible for a strong NiO^?- and NiO ₂ ^? -emission (≦40 L);
4. oxygen on top of this NiO layer, producing a final increase of Ni⁺ and NiO⁺ and a O₂-flash signal at 400 K (>40 L);
5. bulk dissolved oxygen in thermal equilibrium with a chemisorption layer (after several exposure/heating cycles).

During ion bombardment of a 100 L O₂ exposed Ni surface these different binding states occur in a reversed order of succession. O₂-flash signals at 400 and 1100 K, related to drastic changes in secondary ion emission at 400, 700, and 1100 K, reflect the disappearance of various oxygen binding states. The exchange between different oxygen phases was studied by¹⁶O₂/¹⁸O₂ isotope experiments.     

Small-angle X-ray scattering on spinodal,nucleotic or coarsening processes in two-phase systems containing spherical,fibrillar or lamellar inhomogeneities

We study the effects of some of the most important and typical structural changes in two-phase systems on selected structural parameters obtained from small-angle x-ray scattering (SAXS) measurements. To limit the present study, it was assumed that the Phase, 1, embedded in the matrix

1. is monodispersed and homogeneous,
2. possesses one of the three most extreme shapes (spherical, fibrillar or lamellar) and
3. changes its behaviour

1. through type change (spinodal or nucleotic or coarsening), without changing the shape,
2. through a change of the shape only, or
3. through a) (type change) and b) (shape change) simultaneously.

To find the type of change for three basically different shapes of Phase 1 and to calculate its intensity (amount of the change) the following three SAXS parameters must be compared before and after the treatment of the system:

1. chord lengthl ₁ (and/or radius of gyrationR),
2. volume partw ₁ of the Phase 1, and
3. relative inner surfaceS _v of the system.

It is shown by this comparison that by the pure type change in the case of

1. spinodal change, all three SAXRS parameters are increasing or decreasing simultaneously and proportional to a power of the intensity of the change,
2. nucleotic change,l ₁ (and/orR) is unchanged, the other two (w ₁ andS _v ) are increasing or decreasing simultaneously and directly proportional to the intensity,
3. coarsening change,w ₁ is unchanged and anincreasing ofl ₁ is always accompanied by adecreasing ofS _v and vice versa.

In addition to this type change, the cases of mere changes of the shape (“shape change”) and finally of simultaneous type and/or shape change are studied. For the case of pure shape change the alteration of the dimensions (chord lengthl ₁ and/or radius of gyrationR) must be followed. The limitations of the analyses of the simultaneous type and/or shape change are discussed in detail.     

Neutron-rich hypernuclei in the chiral soliton model

The binding energies of neutron-rich strangeness S = ?1 hypernuclei are estimated in the chiral soliton approach using the bound state rigid oscillator version of the SU(3) quantization model. Additional binding of strange hypernuclei in comparison with nonstrange neutron-rich nuclei takes place at not large values of atomic (baryon) numbers, A = B ?? ??10. This effect becomes stronger with increasing isospin of nuclides, and for the ??nuclear variant?? of the model with rescaled Skyrme constant e. Binding energies of _?? ⁸ He and recently discovered _?? ⁶ H satisfactorily agree with data. Hypernuclei _?? ⁷ H, _?? ⁹ He are predicted to be bound stronger in comparison with their nonstrange analogues ⁷H, ⁹He; hypernuclei _?? ¹⁰ Li, _?? ¹¹ LI, _?? ¹² Be, _?? ¹³ Be, etc. are bound stronger in the nuclear variant of the model.     

Theoretical evidence for di-molecular rotational behaviour in the hypernuclei ∑ 8 Li and ∑ 8 Be

We have performed microscopic cluster studies of light∑-hypernuclei based on effective nucleon-nucleon and two different sigma-nucleon interactions. Our calculation confirms a bound 0⁺ state in _∑ ⁴ He and _∑ ⁴ H, but reveals no evidence for levels in other partial waves. We have investigated the _∑ ⁸ Li and _∑ ⁸ Be hypernuclei in a microscopic 3-cluster approach. For both hypernuclei, our calculation shows that the lowest levels in the partial wavesJ=0–4 are bound or quasibound states of collective nature and show a strong α _∑ ⁴ H and α + _∑ ⁴ He clustering. These levels form a sequence of states which can be well interpreted as arising from a rotatingα+ _∑ ⁴ H orα+ _∑ ⁴ He di-molecule. Our results are compared to those of similar studies for lightΛ-hypernuclei which do not show evidence for this kind of collective degrees of freedom.     

On the signatures of the canonical and the metrical energy densities of vector fields

The energy-density components Θ ₀ ⁰ andT ₀ ⁰ of the canonical and of the metrical energy-momentum tensors Θ _k ⁱ andT _k ⁱ for a statical field of vector mesons have opposite signatures: Θ ₀ ⁰ =H=?T ₀ ⁰ =?L. From this property some relativistic and field-theoretical theorems can be deduced in an elementary way.     

The use of unitarity bounds for a stable extrapolation of low energy data

The properties of the scattering amplitude allow to define a functionf(z) satisfying the following conditions:

1. f(z) is holomorphic in a simply connected domain , which can be mapped conformally onto the unit disk;
2. |Imf(z)| is bounded by some constantM in ;
3. |Ref(z)| is known not to exceed some constantm on a certain part Γ₁ of the boundary Γ of ;f(z) is continuously extensible onto Γ.

Using these properties, constraints are derived on the real part off(z) valid at any point . The result is used for performing a stable extrapolation of low energy pion-pion scattering data to any finite energy. We derive a bound on energy averaged values of the real part of the scattering amplitude. The bound depends onm, M, on the energy variabless and on the energy average intervals ₂?s ₁. Generalizations of the method are discussed.     

Berechnung der Signalform bei Doppelresonanzexperimenten an2 P 3/2-Zuständen im starken Magnetfeld

A computer program has been set up, which permits to calculate the signal shape for optical double resonance experiments in² P _3/2-states of alkali atoms in a strong magnetic field. The application to the 5² P _3/2-state of³⁹K, where recently large discrepancies have been reported between early double resonance measurements and new level crossing experiments, has the following results:

1. The measured strong field spectra can be reproduced satisfactorily, if the level crossing data for the hyperfine structure coupling constants are used.
2. The computed line shifts are in good agreement with a formula recently deduced by means of a new approximation methode.

     

ΛΛ 6 He and Λ 9 Be systems in the three-body cluster model treated on the basis of differential Faddeev equations

The _ΛΛ ⁶ He and _Λ ⁹ Be hypernuclei are treated as the S=0, T=0 (for the former) and S=1/2, T=0 (for the latter) bound states of the three-cluster systems ΛΛα and Λαα, respectively. The cluster-reduction method is used to solve the s-wave differential Faddeev equations for these systems. On the basis of the MT I–III model, the ΛΛ interaction potential is specified in the form $V_{\Lambda \Lambda } = \frac{2}{3}V_{NN} $ . Phenomenological potentials are used to describe Λα and αα interactions. The binding energies of the _ΛΛ ⁶ He and _Λ ⁹ Be hypernuclei and the parameters of low-energy Λ-hyperon and α-particle scattering on a _Λ ⁵ He hypernucleus are calculated. It is shown that the proposed ΛΛ interaction potential makes it possible to reproduce faithfully the binding energy of the _ΛΛ ⁶ He hypernucleus and that scattering in the Λ _Λ ⁵ He system is similar to neutron scattering on a deuteron.     

Near and far infrared vibrational absorption ofH i − andD i − centres in KBr

Near infrared absorption measurements by Gross and Bron led to the suggestion that the threefold degeneracy of the localized vibration ofH _i ^? andD _i ^? centres in KI is completely lifted in KBr. Using thermal annealing procedures,H _i ^? -D _i ^? substitution and far infrared techniques we found that the lines at 794, 840, and 896 cm^?1 cannot be interpreted as a splitting of the main line. Only the 794 cm^?1 line can be correlated with the direct excitation of theH _i ^? oscillator. The absence of any splitting in this line (its halfwidth at 9 °K is only 1.5 cm^?1) indicates that also in KBrH _i ^? centres have at most only slightly perturbedT _d symmetry.     

g-factors of the lowest 2+ and 4+ states in198, 200, 202Hg

Theg-factors of the 2 ₁ ⁺ and 4 ₁ ⁺ states in^198,200,202Hg were simultaneously measured by means of the transient-field perturbed angular correlation method in Coulomb excitation using multi-isotopic targets and thin polarized Gd foils as ferromagnetic host. Theg(2 ₁ ⁺ ) andg(4 ₁ ⁺ ) were found identical within errors in^{198, 200}Hg, while in²⁰⁰Hg lowerg-factor values have been determined. The experimentalg-factors were compared with the predictions of the pairing-plus-quadrupole, dynamical deformation and interacting boson models.     

On the free energy of the hopfield model

The general theory of inhomogeneous mean-field systems of Raggio and Werner provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model $$H_{N,p}^{\{ \xi \} } (S) = - \frac{1}{{2N}}\sum\limits_{i,j = 1}^N {\sum\limits_{\mu = 1}^N {\xi _i^\mu \xi _j^\mu S_i S_j } } $$ for Ising spinsS _i andp random patterns ξ^μ=(ξ ₁ ^μ ,ξ ₂ ^μ ,...,ξ _N ^μ ) under the assumption that $$\mathop {\lim }\limits_{N \to \gamma } N^{ - 1} \sum\limits_{i = 1}^N {\delta _{\xi _i } = \lambda ,} \xi _i = (\xi _i^1 ,\xi _i^2 ,...,\xi _i^p )$$ exists (almost surely) in the space of probability measures overp copies of {?1, 1}. Including an “external field” term ?ξ _μ ^p h^μμξ _i=1 ^N ξ _i ^μ S_i, we give a number of general properties of the free-energy density and compute it for (a)p=2 in general and (b)p arbitrary when λ is uniform and at most the two componentsh ^μ1 andh ^μ2 are nonzero, obtaining the (almost sure) formula $$f(\beta ,h) = \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } + h^{\mu _2 } ) + \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } - h^{\mu _2 } )$$ for the free energy, wheref ^cw denotes the limiting free energy density of the Curie-Weiss model with unit interaction constant. In both cases, we obtain explicit formulas for the limiting (almost sure) values of the so-called overlap parameters $$m_N^\mu (\beta ,h) = N^{ - 1} \sum\limits_{i = 1}^N {\xi _i^\mu \left\langle {S_i } \right\rangle } $$ in terms of the Curie-Weiss magnetizations. For the general i.i.d. case with Prob {ξ _i ^μ =±1}=(1/2)±?, we obtain the lower bound 1+4?²(p?1) for the temperatureT _c separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero. This lower bound is exact forp=2, or ε=0, or ε=±1/2. Forp=2 we identify an intermediate temperature region between T_*=1?4?² and T_c=1+4?² where the overlap vector is homogeneous (i.e., all its components are equal) and nonzero.T _* marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct. We conjecture that the homogeneous nonzero regime exists forp≥3 and that T_*=max{1?4?²(p?1),0}.     

Mößbauereffekt in157Gd und155Gd

Using the Mössbauer technique electric hyperfine interactions of the first excited non-rotational states at 64 keV in¹⁵⁷Gd and at 86.5 keV in¹⁵⁵Gd have been determined in GdF₃ and GdCl₃· 6H₂O. For the ratio of quadrupolmoments the ratiosQ ₆₄ ¹⁵⁷ /Q _g ¹⁵⁷ =1.74±0.04,Q _86.5 ¹⁵⁵ /Q _g ¹⁵⁷ =?0.07 ± 0.21 andQ _g ¹⁵⁵ /Q _g ¹⁵⁷ =0.78 ± 0.06 were found. In addition isomer shifts were observed from which a ratio δ〈r²〉 ₆₄ ¹⁵⁷ /δ〈r²〉 _86.5 ¹⁵⁵ =?2.6±0.15 can be inferred.     

On the yrast levels in204Pb

High-spin states in²⁰⁴Pb were populated in the²⁰⁴Hg(α,4n) reaction using α-particles in the energy region 42–51 MeV. Prompt and delayedγ-rays as well as conversion electrons were studied in addition to excitation functions, angular distributions andγ-γ coincidences. In this way a stretched cascade ofγ-rays from a level at 8125.9 keV was found to feed the previously known isomeric 9^? level at 2185.7 keV. Spins and parities were established for levels up to and including a 19^? level at 6098.0 keV. The levels with c= 17^2212; and 19^? at excitation energies of 5664.3 and 6098.0 keV are likely to be due to the simplep ₁ ^2/s-1 i ₁₃ ^2/?3 andf ₅ ^2/?1 i ₁₃ ^2/?3 configurations. The agreement between calculated and experimental energies for all observed levels in the regionJ=9–19 is very good in cases where the empirical two-particle interactions used are satisfactorily well known. Above the 19^? level there are three weakly populated levels at 7402.1, 7849.2 and 8125.9 keV, which are likely to haveJ≥20. None of these energies agrees with the calculated value 7695±20 keV for the 20⁺ state of thei ₁₃ ^2/?4 configuration which has the highest angular momentum produced by the four valence neutron holes. This apparent anomaly can be understood if the yrast levels withJ≥20 have angular momentum contribution from the core. It seems likely that the states at 7402.1, 7849.2 and 8125.9 keV are due to proton core excited states of the typeπh _9/2 h ₁₁ ^2/?1 ×νp ₁ ^2/?2 i ₁₃ ^2/?2 withJ ^π=20⁺ andJ ^π=21⁺ andπh _9/2 h ₁₁ ^2/?1 ×νp ₁ ^2/?1 f ₅ ^2?1 i ₁₃ ^2/?2 withJ ^π=22⁺ or 23⁺, respectively. The state at 8126 keV has the highest energy so far directly observed in a stretched cascade ofγ-rays from the decay of a heavy nucleus produced in (α, xn) reactions.     

Voltage/current characteristics of high-current vacuum arc at a small gap length

Measurements of arc voltage for semiwave currents of the amplitude up to 13 kA and frequencies 0.9 kHz and 6.0 kHz were done for CuCr electrodes whose separtion was 2 mm. The obtained data were used to determine voltage/current and voltage-time characteristics of high-current vacuum arc. They enabled to state as:

1. The shape and values of arc voltage during a sinusoidal current semiwave flow depend on the current amplitude and frequency. As the current amplitude increases the arc voltage rises, and its shape changes from relatively low and quiet to high with considerable fluctuations of high frequency and amplitude (HAF).
2. Arc voltage at a given current instantaneous valuei _a1 depends on the current waveform before reachingi _a1 . The arc voltage beyond the range of HAF occurrence, at the same instantaneous value of the sinusoidal current, is higher during the current increase than during its decrease and higher for higher current frequency.
3. Mean arc voltage and current value at the moment of HAF onset, rise with the increase of current peak and frequency.

This article presents qualitative correlations between the measured arc voltage and an expected discharge mode in the intercontact gap.