On the asymptotic behaviour of linear learning processes with partial forgetting

For linear learning processes with partial forgetting a recursion of the form $$X_{n + 1} = \frac{1}{n}\sum\limits_{k = j_n }^n {(B_k X_k + b_k ) with \frac{{j_n }}{n}} \to \alpha \in [0,1)$$ in a real separable Banach spaceB is investigated. Results on a.s. convergence and rate of convergence of (X _n ) (loglog and distributional invariance principle) are obtained under corresponding assumptions on the sequences of arithmetic means of the random elementsb _n andB _n inB andL(B), resp., together with suitable spectral conditions on the (deterministic) Cesàro limit of (B _n ). By the special casej _n ≡1 results on stochastic approximation are comprehended. The limit process in the invariance principles satisfies a stochastic differential equation with delay.     

N2O influence on isotopic measurements of atmospheric CO2

In spite of extensive efforts, even the most experienced laboratories dealing with isotopic measurements of atmospheric CO2 still suffer from poor inter-laboratory consistency. One of the complicating factors of these isotope measurements is the presence of N2O, giving rise to mass overlap in the isotope ratio mass spectrometer (IRMS). The aim of the experiment reported here has been twofold: first, the re-establishment of the correction for 'mechanical' interference of N2O in the IRMS, along with its variability and drift, and the best way to quantitatively determine the correction factors. Second, an investigation into secondary effects, i.e. the influence of N2O admitted with the CO2 sample on the "cross contamination" between sample and (pure CO2) working gas. To make the suspected effects better detectable, isotopically enriched CO2 gas with different concentrations of N2O has been measured for the first time. No evidence of secondary effects was observed, from which we conclude that N2O is not a major player in the inter-laboratory consistency problems. Still, we also found that the determination of the 'mechanical' N2O correction needs to be very carefully determined for each individual IRMS, and should be periodically re-determined. We show that the determination of the correction should be performed using CO2/N2O mixtures with concentration ratios around that of the atmosphere, as the extrapolation from pure gas end member behaviour will give erroneous results due to non-linearities. For our IRMS, a VG SIRA series II, we find a correction of 0.23 per thousand for delta45CO2 and 0.30 per thousand for delta46CO2 of atmospheric samples, (with 0.85 per thousand mixing ratio). This implies that the relative ionisation efficiency (E) value associated with this machine is 0.75.     

Spectral,thermal and magnetic studies of 2-chloro-5-nitrobenzoates of rare earth elements(iii)

The conditions of the formation of yttrium(III) and lanthanide(III) 2-chloro-5-nitrobenzoates were studied and their quantitative composition and solubilities in water at 298 K were determined. The compounds are dihydrates and their solubilities are of the order of 10^-3 mol dm^-3. The FTIR, FIR and X-ray spectra for the complexes were recorded. All complexes are crystalline compounds. Their thermal decomposition was studied only in the range 293-523 K since on heating above 523 K the complexes decompose explosively. Hydrated 2-chloro-5-nitrobenzoates of rare earth elements lose two crystallization water molecules in one step. The enthalpy of dehydration process was established and the magnetic moments of the complexes were determined in the range 77-300 K. The compounds obey the Curie-Weiss law. The 4f electrons do not participate in the formation of the Ln-O bonds. This revised version was published online in August 2006 with corrections to the Cover Date.     

Thermal and spectral studies of 2,4,5-trimethoxybenzoates of heavy lanthanides(III) and yttrium(III)

2,4,5-Trimethoxybenzoates of Tb(III), Dy(III), Ho(III), Er(III), Tm(III), Yb(III), Lu(III) and Y(III) are crystalline, hydrated salts with colours typical for M(III) ions. The carboxylate group is a bidenate, chelating ligand. The complexes of Tb(III), Dy(III) and Ho(III) are dihydrates while those of Er(III), Tm(III), Yb(III), Lu(III) and Y(III) are trihydrates. These compounds are characterized by low symmetry. On heating in air to 1273 K, the 2,4,5-trimethoxybenzoates of heavy lanthanides(III) and yttrium(III) decompose in two steps. At first they dehydrate to form anhydrous salts which next are decomposed to the oxides of the respective metals. The values of the enthalpy of dehydration process were determined. The solubility in water at 293 K for all heavy lanthanides(III) and yttrium(III) are in the orders of 10^-3-10^-4 mol dm^-3. The magnetic moments of the complexes were determined in the temperature range 77-300 K.     

Thermal,spectral and magnetic properties of 3,5-dimethoxybenzoates of Co(II), Ni(II) and Cu(II)

The complexes of 3,5-dimethoxybenzoates of Co(II), Ni(II) and Cu(II) have been synthesized as hydrated polycrystalline solids and characterized by elemental analysis, IR, FIR and electronic spectroscopy, magnetic studies and X-ray diffraction measurements. They possess colours typical of the M(II) ions: Cu-blue, Ni-green, Co-pink. The carboxylate groups bind as monodentate or a symmetrical, bidentate chelating or bridging ligands. The thermal stabilities were determined in air. When heated they dehydrate to form anhydrous salts which are decomposed to the oxides of respective metals. The magnetic susceptibilities of the complexes were measured over the range 77-300 K and the magnetic moments were calculated. The results reveal the complexes of Ni(II) and Co(II) to be high-spin complexes and that of Cu(II) to form dimer.     

Thermal, spectral and magnetic characterization of Co(II), Ni(II) AND Cu(II) 4-chloro-2-nitrobenzoates

Physico-chemical properties of 4-chloro-2-nitrobenzoates of Co(II), Ni(II), and Cu(II) were studied. The complexes were obtained as mono- and trihydrates with a metal ion to ligand ratio of 1:2. All analysed 4-chloro-2-nitrobenzoates are polycrystalline compounds with colours depending on the central ions: pink for Co(II), green for Ni(II), and blue for Cu(II) complexes. Their thermal decomposition was studied only in the range of 293–523 K, because it was found that on heating in air above 523 K 4-chloro-2-nitrobenzoates decompose explosively. Hydrated complexes lose crystallization water molecules in one step and anhydrous compounds are formed. The final products of their decomposition are the oxides of the respective transition metals. From the results it appears that during dehydration process no transformation of nitro group to nitrite takes place. The solubilities of analysed complexes in water at 293 K are of the order of 10^–4–10^–2 mol dm^–3. The magnetic moment values of Co²⁺, Ni²⁺ and Cu²⁺ ions in 4-chloro-2-nitrobenzoates experimentally determined at 76–303 K change from 3.89 to 4.82 μ_B for Co(II) complex, from 2.25 to 2.98 μ_B for Ni(II) 4-chloro-2-nitrobenzoate, and from 0.27 to 1.44 μ_B for Cu(II) complex. 4-chloro-2-nitrobenzoates of Co(II), and Ni(II) follow the Curie–Weiss law. Complex of Cu(II) forms dimer.     

Inverspositive Operatoren und Taubersätze in der Erneuerungstheorie

A generalization of ordinary renewal processes has been treated byHinderer and the author: for the stochastic process with values in ₊ the aim is (a,a+) with 0a<, 0< instead of (a, ), and a possible passing over the aim is replaced by a phase of stagnation. In the present paper for the open case of an infinite mean length of the undisturbed steps the relationV(a)/a0(a), with fixed, for the mean waiting timeV(a) until reaching the aim is obtained. The proof uses the theory of inversepositive operators ofCollatz andSchröder. This concept also yields an elementary proof of the so-called elementary renewal theorem. Finally the Tauberian theorem ofIkehara inAgmon's version is generalized; it yields the so-called general renewal theorem in the case of finite mean length of the steps, which is also a corollary of a general Tauberian theorem ofBene.

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Herrn Professor Collatz zum 65. Geburtstag     

Hydrogen maser frequency shifts due to coherently excited Δm F =±1 transitions betweenF=1 levels of the atomic hydrogen ground state

Hydrogen maser frequency shifts, caused by the multiple quantum transition nonlinearities of a resonant multiple frequency excitation of the atomic hydrogen four level ground state system have been investigated. The oscillation characteristics of hydrogen maser operation with simultaneously excited, low frequencyΔm _F =±1 transitions between theF=1 states of the atomic hydrogen ground state have been analysed theoretically and explicit formulas for hydrogen maser frequency shifts and amplitude response have been derived for arbitrary maser oscillation amplitude and a small signal approximation for theΔm _F =±1 “Zeeman” transitions. The comparison with experimentally observed hydrogen maser frequency shifts was specialized to small magnetic fields, for which the difference between the resonance frequencies of the two low frequency,Δm _F =±1 Zeeman transitions is small compared to the linewidth. Special emphasis was placed on the evaluation of frequency pulling effects for a Zeeman transition excitation at off-resonance conditions. For this case the theoretical formulation of frequency pulling effects becomes insensitive against simplifying assumptions about the radiation damping phenomena and a particular good agreement between experiment and theory can therefore be expected. Experimental conditions have been specified, for which the uncertainty of hydrogen maser frequency due to Zeeman transition induced frequency shifts does not restrict the present frequency stability of a hydrogen maser frequency standard.     

Hydrogen maser atomic beam resonances

Au^? ions at anionic places are formed in gold doped crystals by a reducing treatment withF centers. The ultraviolet absorption consists of 4 bands, which are namedA, B, C, andD in analogy to the isoelectronic centers of the s² type, like Tl⁺. TheB band oszillator strength strongly increases with temperature in accordance with a phonon allowed transition. The ratio of the dipole strength of theC band to that of theA band as a function of the relative position of theB band is compared with Suganos prediction. Zero phonon lines are found at helium temperatures for theA band in NaCl (2,985 Å), KCl (3,068 Å), and KBr (3,145 Å) and for theC band in KCl (2,329 Å). In KCl the Huang-Rhys factor isg=3.4 for theA band. The vibronic structure comes from the relatively large radius 6s ² state of the negative ion. Uniaxial stress splits the zero phonon line. The results definitely agree with the stress splitting behaviour of a degenerateΓ ₁→Γ ₄ transition. Inversion symmetry of the center is confirmed by the absence of a linear Stark effect.