Statistical extension of classical Tauberian theorems in the case of logarithmic summability

Let s: [1,∞) → ? be a locally integrable function in Lebesgue’s sense. The logarithmic (also called harmonic) mean of the function s is defined by $$\tau (t): = \frac{1} {{\log t}}\int_1^t {\frac{{s(x)}} {x}dx, t > 1,}$$ where the logarithm is to the natural base e. Besides the ordinary limit lim _x→∞ s(x), we use the notion of the so-called statistical limit of s at ∞, in notation: st-lim _x→∞ s(x) = l, by which we mean that for every ? > 0, $$\mathop {\lim }\limits_{b \to \infty } \frac{1} {b}\left| {\left\{ {x \in (1,b):\left| {s(x) - \ell } \right| > \varepsilon } \right\}} \right| = 0.$$ We also use the ordinary limit limt→∞ τ (t) as well as the statistical limit st-limt→∞ τ (t). We will prove the following Tauberian theorem: Suppose that the real-valued function s is slowly decreasing or the complex-valued s is slowly oscillating. If the statistical limit st-limtt→∞ τ (t) = l exists, then the ordinary limit limx→∞ s (x) = l also exists.     

Measuring the power of soft correlated equilibrium in 2-facility simple non-increasing linear congestion games

The performance of soft correlated equilibrium, a new generalization of Aumann’s correlated equilibrium (Forgó in Math Soc Sci 60:186–190, 2010) is measured in $2$ -facility simple non-increasing linear congestion games. The mediation value and the enforcement value are determined for $2,3$ and $4$ -person games and bounds are computed for the general $n$ -person case.     

LC Enantioseparation of β-Amino Acids on a Crown Ether-Based Stationary Phase

A pentaproline-based chiral stationary phase was prepared and the selectivity of the column was evaluated with 194 racemic compounds in three mobile phase modes: normal-phase mode, polar organic mode and reversed-phase mode. 94 racemates out of 194 were separated and the normal-phase mode proved to be the separation mode of broadest applicability. The column is stable in all common organic solvents and no degradation in column performance was observed in any mode even after more than 1,000 injections. A brief sample loading test was performed on the 250 mm × 4.6 mm column and 13.2 mg of α-methyl-9-anthracenemethanol was baseline separated. Retention behavior in the normal-phase mode and the effect of analyte structure on retention and enantioselectivity are discussed.     

Capillary electrophoresis chips for screening of endotoxin chemotypes from whole-cell lysates

A fast microchip electrophoresis method was developed to analyze and differentiate bacterial endotoxins directly from whole-cell lysates after removal of the proteinaceous components with proteinase K digestion and a precipitation of the endotoxin components. The partially purified endotoxin components were visualized by the interaction with dodecyl sulphate and then a fluorescent dye. The lipopolysaccharide (LPS) profiles can be directly evaluated from digested bacterial cells, and the electrophoresis patterns very closely resembled to those of pure LPSs, and the R and S chemotypes can be used to assign the strains. The method has been found to be useful in the screening of a large number of bacterial mutants and the structural characterization of endotoxins extracted only from 1 ml cultures.     

Fluoroalkylations and Fluoroalkenylations with Iodonium Salts

Synthesis and applications of fluoroalkyl and fluoroalkenyliodonium salts are summarized in this account article, focusing preferably to the reagents designed in our laboratory in the last decade. Among these reagents trifluoroethyl(aryl)iodonium salts have been used most frequently to build carbon-carbon and carbon-heteroatom bonds in simple nucleophilic substitutions and through transition metal catalyzed coupling reactions. Iodonium salts equipped with unsaturated fluorinated function showed diverse reactivity due to their electron deficient character, and these molecular motifs enable cycloadditions and nucleophilic additions to prepare fluorinated carbo- and heterocyclic molecules. Beyond the overview of existing transformations, with the presented collection, we aim to inspire future developments of iodonium reagents and their application in organic synthesis.     

Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ? ^N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ ^N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.     

Development of a Novel Acoustic Spectroscopy Method for Detection of Eggshell Cracks

Non-destructive testing (NDT) for eggshell faults is highly important for the egg industry, as cracked eggs account for around 3% of total production. The most commonly used method at present, candling, is labor intensive, while computer vision systems are expensive and complicated. In this paper, we present a simple, yet efficient, novel method for eggshell crack detection by acoustic spectroscopy. Altogether, 693 sound recordings were evaluated by different classification methods. The results show a cross-validated 2.1% total classification error, with only 0.87% false positive rate, which is the crucial metric for fresh eggs. Adapting the developed method to an industrial setting may lead to a reliable, fast and cost-effective detection method.     

Sampling strategies for capillary isoelectric focusing with electroosmotic zone mobilization assessed by high-resolution dynamic computer simulation

The impact of initial sample distribution on separation and focusing of analytes in a pH 3–11 gradient formed by 101 biprotic carrier ampholytes under concomitant electroosmotic displacement was studied by dynamic high-resolution computer simulation. Data obtained with application of the analytes mixed with the carrier ampholytes (as is customarily done), as a short zone within the initial carrier ampholyte zone, sandwiched between zones of carrier ampholytes, or introduced before or after the initial carrier ampholyte zone were compared. With sampling as a short zone within or adjacent to the carrier ampholytes, separation and focusing of analytes is shown to proceed as a cationic, anionic, or mixed process and separation of the analytes is predicted to be much faster than the separation of the carrier components. Thus, after the initial separation, analytes continue to separate and eventually reach their focusing locations. This is different to the double-peak approach to equilibrium that takes place when analytes and carrier ampholytes are applied as a homogenous mixture. Simulation data reveal that sample application between two zones of carrier ampholytes results in the formation of a pH gradient disturbance as the concentration of the carrier ampholytes within the fluid element initially occupied by the sample will be lower compared to the other parts of the gradient. As a consequence thereof, the properties of this region are sample matrix dependent, the pH gradient is flatter, and the region is likely to represent a conductance gap (hot spot). Simulation data suggest that sample placed at the anodic side or at the anodic end of the initial carrier ampholyte zone are the favorable configurations for capillary isoelectric focusing with electroosmotic zone mobilization.     

Highly efficient thermal cyclization reactions of alkylidene esters in continuous flow to give aromatic/heteroaromatic derivatives

Intramolecular thermal cyclization and benzannulation reactions of the Gould–Jacobs and Conrad–Limpach types were performed in a designed continuous flow reactor system at temperatures in the range of 300–360 °C and under high pressure conditions (100–160 bar) with very short residence times (0.45–4.5 min) in tetrahydrofuran as a low-boiling point solvent. Substituted heteroaromatic compounds including pyridopyrimidinones and hydroxyquinolines were synthesized in moderate to high yields. Application of the reaction conditions also allows the synthesis of naphthol and biphenyl derivatives. The procedure involves an easy work-up and the non-batchwise preparative synthesis method is suitable for automation.     

Synthesis and conformational analysis of tetrahydroisoquinoline- and piperidine-fused 1,3,4,2-oxadiazaphosphinanes, new ring systems

Through cyclization of tetrahydroisoquinoline and piperidine 1,2-hydrazino alcohols with phenylphosphonic dichloride and phenyl dichlorophosphate, P-epimeric diastereomers of 1,6,7,11b-tetrahydro-4H-1,3,4,2-oxadiazaphosphino[5,4-a]isoquinoline-3-oxides (13 and 14), 1,6,11,11a-tetrahydro-4H-1,3,4,2-oxadiazaphosphino[4,5-b]isoquinoline-3-oxides (15 and 16) and 1,6,7,8,9,9a-hexahydro-4H-pyrido[1,2-d][1,3,4,2]oxadiazaphosphinane-3-oxides (17 and 18), the first representatives of these ring systems, were prepared. NMR and X-ray diffraction studies revealed that, independently of the P-substituent and the relative configuration of the phosphorus atom, 13, 14, 17 and 18 could be characterized by trans-connected hetero rings and the chair conformation of the 1,3,4,2-oxadiazaphosphinane moiety, while the stereochemistry of the connection of the hetero rings in the 1,3,4,2-oxadiazaphosphinanes linearly fused to tetrahydroisoquinoline (15 and 16) was found to be dependent on the P-configuration relative to that of the carbon at the annelation.