Hausdorff-summability of power series II

Let H _x a regular Hausdorff method and P(w)=∑ a_k w^k a power series with positive radius of convergence. A theorem of Okada states that P(w) is summable (H _x ) for w in a certain starshaped region G(H _x ,P). We call G=G(H _x ,P) the exact region of summability for P if summability cannot hold for any w \( \in \bar G\) Okada's theorem is said to be sharp for H_x if G(H_x,P) is the exact region of summability for any P. Three items are treated: 1. Criteria for Okada's theorem to be sharp are given in terms of the distribution function X (t) and the Mellin transform \(D(z) = \int\limits_0^1 {t^z d\chi (t)} \) . 2. When is Okada's theorem sharp for product methods? 3. Special classes of functions P(w) are indicated such that G(H_x, P) is the exact region of summability for any H_x. We use the notations of “Hausdorff-Summability of Power Series I” referred as “I”.     

A new proof of the Komlós-Révész-theorem

The Komlós-Révész theorem states: For r.v.s.X _n with X _{n 1}M there exists a subsequenceX _{k n} and a r.v.X with X₁M such that

Effect of disorder on the conductivity of two-phase strongly inhomogeneous highly filled composites

The effect of the “stir” of a structure (small deviations from strict periodicity) on effective conductivity is considered. For determinate and random deviations, concentration and field dependences of the effective conductivity are found. Numerical experiments with determinate deviations are carried out for the cases of linear (with respect to the field) inclusions embedded in both a linear and nonlinear matrix. The numerical results are compared with the effective conductivity calculated analytically.     

Finite-difference approach for the high-precision analysis of rotating-frame diffusion images

A finite-difference approach has been developed for precisely determining diffusion coefficient and T₁ relaxation time in fluid samples analyzed by magnetization-grating rotating-frame imaging (MAGROFI) with either a surface coil or a toroid cavity detector (TCD). This approach avoids shortcomings of phenomenologically based approximations, such as neglect of sample geometries with singularities at the confines of the sample volume, and accounts for the diffusive edge enhancement observed in fluid imaging. Error limits are discussed. The new method has been applied to the determination of the self-diffusion coefficient for MAGROFI experiments using TCDs filled with acetone.     

Summierbarkeit der geometrischen Reihe auf vorgeschriebenen Mengen

In this paper we are concerned with the summability of the geometric series by matrix methods. We prove the following theorem: Suppose M_o:={z:|z|<1}, M₁, M₂, is a collection of countably many Lebesgue measureable, disjoint sets. For k=1,2, let f_k be a prescribed function, analytic on . Then there exists a triangular matrix , such that the V-transform {_n(z)} of the geometric series has the following properties: {_n(z)} converges compactly to on M_o; for k=1,2, there are sets B_k, such that has Lebesgue-measure zero and _n(z)f_k(z) for zB_k; if there is a set B^*, such that B^*M^* has Lebesgue-measure zero and {_n(z)} diverges for zB^*.     

Verschärfung eines Satzes von Borel-Okada über Summierbarkeit von Potenzreihen

Ohne Zusammenfassung

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Fast chemical-shiftT 1 imaging in toroid cavities for the structural analysis of gels and emulsions

Fast chemical-shiftT ₁ imaging in toroid cavity cells (TCCs) is introduced and applications to diagnostic ultrasound gel and skin-care ointment are presented. TCCs are an advancement over previously used toroid cavity detectors because they combine resonator and sample container into one part. Additionally, they are removable from the top of the probe and facilitate convenient probe and sample handling. Radially resolvedT ₁ relaxation times in TCCs are obtained through combination of SR (saturation recovery) experiments with the RIPT (rapid imaging with a pulse train) technique. Because of the strong radialB ₁ gradient in TCCs, only pulse-burst saturation was found satisfactory to generate the most even starting condition for SR experiments. Because RIPT does not resolve chemical-shift information, magnetization andT ₁ profiles of individual components in mixed samples are monitored by double-transient experiments with selective on-resonance saturation, which is achieved by converging trains of sinc pulses. The new techniques were applied to cellulose hydrogel and oil-in-water emulsion both exposed to significant shear stress deformation during the charging of the TCC. In both cases,T ₁ profiles as a function of time reveal structural recovery (thixotropy) that slowly progresses from one or more sample interfaces into the bulk.     

Structure and conformational properties of carbonylbisimidosulfuryl fluoride, O=C(N=S(O)F2)2

The molecular structure and conformational properties of O=C(N=S(O)F₂)₂ (carbonylbisimidosulfuryl fluoride) were determined by gas electron diffraction (GED) and quantumchemical calculations (HF/3-21G* and B3LYP/6-31G*). The analysis of the GED intensities resulted in a mixture of 76(12)% syn–syn and 24(12)% syn–anti conformer (ΔH⁰=H⁰(syn–anti)−H⁰(syn–syn)=1.11(32) kcal mol⁻¹) which is in agreement with the interpretation of the IR spectra (68(5)% syn–syn and 32(5)% syn–anti, ΔH⁰=0.87(11) kcal mol⁻¹). syn and anti describe the orientation of the S=N bonds relative to the C=O bond. In both conformers the S=O bonds of the two N=S(O)F₂ groups are trans to the C–N bonds. According to the theoretical calculations, structures with cis orientation of an S=O bond with respect to a C–N bond do not correspond to minima on the energy hyperface. The HF/3-21G* approximation predicts preference of the syn–anti structure (ΔE=−0.11 kcal mol⁻¹) and the B3LYP/6-31G* method results in an energy difference (ΔE=1.85 kcal mol⁻¹) which is slightly larger than the experimental values. The following geometric parameters for the O=C(N=S)₂ skeleton were derived (r_a values with 3σ uncertainties): C=O 1.193 (9) Å, C–N 1.365 (9) Å, S=N 1.466 (5) Å, O=C–N 125.1 (6)° and C–N=S 125.3 (10)°. The geometric parameters are reproduced satisfactorily by the HF/3-21G* approximation, except for the C–N=S angle which is too large by ca. 6°. The B3LYP method predicts all bonds to be too long by 0.02–0.05 Å and the C–N=S angle to be too small by ca. 4°.     

((Fluoroformyl)imido)(trifluoromethyl)sulfur fluoride, FC(O)N = S(F)CF3: unexpected conformational properties

The geometric structure and conformational properties of ((fluoroformyl)imido)(trifluoromethyl)sulfur fluoride, FC(O)N = S(F)CF3, are investigated by gas electron diffraction (GED) experiments, IR (gas) spectroscopy, and quantum chemical calculations (HF, MP2, and B3LYP with 6-31G* basis sets). The GED intensities are reproduced best with a mixture of 79(12)% trans-syn and 21(12)% cis-syn conformers. "Trans/cis" describes the orientation around the S=N double bond (FC(O) group relative to sulfur substituents), and "syn" refers to the orientation of the C=O bond relative to the S=N bond. From the intensities of the C=O bands in the IR (gas) spectrum, a composition of 86(8)%:14(8)% is derived. These ratios correspond to delta G0(GED) = 0.79(36) and delta G0(IR) = 1.09(35) kcal mol-1. The preference of a trans structure, around the S=N double bond is unexpected, since all imidosulfur compounds studied thus far possess a cis configuration. The conformational properties are reproduced qualitatively correctly by all theoretical calculations. The predicted energy differences delta E(HF) = 2.41, delta E(MP2) = 0.64, and delta E(B3LYP) = 0.28 kcal mol-1 are larger or slightly smaller than the experimental values. Additional theoretical calculations (B3LYP) for several imidosulfur compounds reveal that only FC(O)N=S(F)CF3, with mixed substitution at sulfur and the FC(O) group bonded to nitrogen, prefers the trans structure.