Symmetrisierte und zentrierte Folgen von Zufallsvariablen

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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 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.     

Strong Laws of Large Numbers by Elementary Tauberian Arguments

For Kolmogorovs strong law of large numbers an alternative short proof is given which weakens Etemadis condition of pairwise independence. The argument uses the known – and elementary – equivalence of (Cesàro) C₁- and C₂-summability for one-sided bounded sequences. Also other strong laws of large numbers are established, in part via Borel summability.     

Wachstumsverhalten zufÄlliger Potenzreihen

Summary Using the theorems collected in Chapter 2 the paper discusses growth and divergence properties of random power series (rps's) with — generally — independent coefficients a _n. In Chapter 3 assertions about rps's with a _n symmetric relative to 0 are transferred to rps's for which the medians a _n of a _n are 0. Chapter 4 treats analogies between rps's with symmetrized coefficients a _s ⁿ and rps's with centered coefficients a _n -a _n ; they concern growth of random entire functions defined by rps's with arbitrarily dependent coefficients, and local behaviour near the circle of convergence. For some kinds of singular behaviour, e.g. unboundedness of characteristic in every sector of the circle of convergence, Chapter 5 gives necessary and sufficient conditions in connection with the notion of essential divergence. The theorems hold also for -quantiles a _n ( fixed, 0<<1) instead of medians a _n .Herrn Professor Dr. F. Lösch zum 65. Geburtstag gewidmet.     

Strong universal consistency of smooth kernel regression estimates

The paper deals with kernel estimates of Nadaraya-Watson type for a regression function with square integrable response variable. For usual bandwidth sequences and smooth nonnegative kernels, e.g., Gaussian and quartic kernels, strongL ₂-consistency is shown without any further condition on the underlying distribution. The proof uses a Tauberian theorem for Cesàro summability.     

Thermal and Spectral Features of Yttrium and Heavy Lanthanide Complexes with 2,4-dimethoxybenzoic Acid

The complexes of yttrium and heavy lanthanides with 2,4-dimethoxybenzoic acid of the formula: Ln(C₉H₉O₄)₃×nH₂O, where Ln=Tb(III), Dy(III), Ho(III), Er(III), Tm(III), Yb(III), Lu(III) and Y(III), n=2 for Tb(III), Dy(III), Ho(III), Er(III), Tm(III) and Y(III), and n=0 for Yb(III) and Lu(III), have been synthesized and characterized by elemental analysis, IR spectroscopy, themogravimetric studies, as well as X–ray and magnetic susceptibility measurements. The complexes have a colour typical of Ln ³⁺ salts (Tb, Dy, Tm, Yb, Lu, Y – white, Ho – cream, Er – pink). The carboxylate group in these complexes is a bidentate, chelating ligand. The compounds form crystals of various symmetry. 2,4-Dimethoxybenzoates of Yb(III) and Lu(III) are isostructural. 2,4-Dimethoxybenzoates of yttrium and heavy lanthanides decompose in various ways on heating in air to 1173 K. The hydrated complexes first lose water to form anhydrous salts and then decompose to the oxides of respective metals. The ytterbium and lutetium 2,4-dimethoxybenzoates decompose in one step to form Yb₂O₃ and Lu₂O₃. The solubilities of the 2,4-dimethoxybenzoates of yttrium and heavy lanthanides in water and ethanol at 293 K are of the order of: ^10–3 and 10^–3 –10^–2 mol dm^–3, respectively. The magnetic moments for the complexes were determined over the range of 77–298 K. They obey the Curie–Weiss law. The results show that there is no influence of the ligand field on the 4f electrons of lanthanide ions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Eine Bemerkung zur Integrierbarkeit eines Produktes von zwei Funktionen in Sobolewschen Räumen

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