A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers

Let {X _n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S _n =X ₁+???+X _n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,

$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$

if and only if

$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$

where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V _n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a _n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).

     

N‐Heterocyclic Carbene‐catalyzed Reactions of o‐Aminonitriles with Carbonyl Compounds Approach to 2,3‐Dihydroquinazolin‐4(1H)‐ones

A green, efficient and convenient N‐heterocyclic carbene‐catalyzed procedure for the synthesis of novel 2,3‐dihydroquinazolin‐4(1H)‐one derivates via condensation of o‐aminonitriles and various carbonyl compounds was described.     

Construction of coaxial liquid centrifugal platform for the extraction of active ingredients from natural products

In the classical natural product extraction and separation process, it is tedious and requires large amounts of reagents and time. In this study, an efficient coaxial liquid centrifugal oil-water-oil triple-liquid-phase system with a simple structure and convenient operation was successfully constructed and used to extract flavonoids from Platycladi Cacumen. The results showed that the coaxial liquid centrifugal platform constructed in this study had good stability and 6 ml was the minimum volume of the middle phase for 1000 rpm to stabilize the system. Besides, it was easy to repeat the operation: the relative standard deviations of the extraction yields of flavonoids and sugar in six parallel operations were all less than 10%. Moreover, it was only one-tenth of the time required for this method as traditional liquid-liquid extraction while reducing the use of volatile organic reagents. Finally, the new method was more selective than the traditional method for the extraction of flavonoids. Therefore, this study provides a possibility for the coaxial liquid centrifugal platform to be used in multi-liquid phase systems to achieve the simultaneous extraction of different parts of natural products by different liquid phases. It is expected to provide a reliable reference for further expansion of small-scale experimental operations to industrial production.     

Structure analysis using acoustically levitated droplets

Synchrotron diffraction with a micrometer-sized X-ray beam permits the efficient characterization of micrometer-sized samples, even in time-resolved experiments, which is important because often the amount of sample available is small and/or the sample is expensive. In this context, we will present acoustic levitation as a useful sample handling method for small solid and liquid samples, which are suspended in a gaseous environment (air) by means of a stationary ultrasonic field. A study of agglomeration and crystallization processes in situ was performed by continuously increasing the concentration of the samples by evaporating the solvent. Absorption and contamination processes on the sample container walls were suppressed strongly by this procedure, and parasitic scattering such as that observed when using glass capillaries was also absent. The samples investigated were either dissolved or dispersed in water droplets with diameters in the range of 1 micrometer to 2 millimeters. Initial results from time-resolved synchrotron small- and wide-angle X-ray scattering measurements of ascorbic acid, acetylsalicylic acid, apoferritin, and colloidal gold are presented.     

Agglomeration of proteins in acoustically levitated droplets

An ultrasonic trap (acoustic levitator) was used as an analytical tool to allow container-free handling of proteins in small sample volumes. This trap was combined for the first time with synchrotron small-angle X-ray scattering (SAXS) for structure analysis of biological macromolecules in a solution. The microfocus beamline at BESSY was used as a source of intense X-ray radiation. Apoferritin (APO) was used as a model protein, and its aggregation behavior in a levitator was followed from a diluted solution to the solid state. Different stages of APO agglomeration were observed without solid container walls, which may influence aggregation behavior and produce a parasitic scattering background. Starting with a volume of 5 microL we analyzed the concentration dependence of APO structure factors in the range from 5 to 1,200 mg/mL (solid protein). The solution was stirred automatically due to convection inside the droplet caused by the ultrasonic field. SAXS data recording of APO was performed in time intervals of 60 s during an aggregation experiment of 30 to 60 min.     

Synthesis and Characterization of Montmorillonite-Graft-Acrylic Acid Superabsorbent by Using Glow-Discharge Electrolysis Plasma

Crosslinking graft polymerization of poly acrylic/montmorillonite superabsorbent composite in aqueous solution was prepared by using glow-discharge electrolysis plasma, in which acrylic acid and acid-activated montmorillonite were used as starting materials, N,N-methylenebisacrylamide as a crosslinking agent. To optimize the synthetic conditions, seven important parameters of the graft-polymerization were examined in detail, such as the discharge voltage, discharge time, the neutralization of acrylic acid, post polymerization temperature, amounts of crosslinking agent, montmorillonite and acrylic acid used in this study. The structure, thermal stability and morphology of product were characterized by Fourier transform infrared spectroscopy, thermogravimetric analysis, scanning electron microscopy, respectively. The absorbency of poly acrylic/montmorillonite superabsorbent was examined to yield the following results: 1,834 g/g for distilled water, 116 g/g for 0.05 mol/L and 75 g/g for 0.15 mol/L sodium chloride solution. Such excellent character could be important to use in many fields, for example, in agricultural and horticultural applications.     

Strong convergence in nonparametric regression with truncated dependent data

In this paper we derive rates of uniform strong convergence for the kernel estimator of the regression function in a left-truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The estimation of the covariate’s density is considered as well. Under the assumption that the lifetime observations are bounded, we show that, by an appropriate choice of the bandwidth, both estimators of the covariate’s density and regression function attain the optimal strong convergence rate known from independent complete samples.     

Sulfone-Embedded Heterocyclic Narrowband Emitters with Strengthened Molecular Rigidity and Suppressed High-Frequency Vibronic Coupling

Sulfone-embedded heterocyclics are of great interest in organic light-emitting diodes (OLEDs), however, exploring highly efficient narrowband emitters based on sulfone-embedded heterocyclics remains challenging. Herein, five emitters with different sulfur valence state and molecular rigidity, namely tP, tCPD, 2tCPD, tPD and tPT, are thoroughly analysed. With restricted twisting of flexible peripheral phenyl by strengthening molecular rigidity, molecular emission spectra can be enormously narrowed. Further, introducing the sulfone group with bending vibration in low-frequency region that suppresses high-frequency vibration, sharp narrow full-widths at half-maximum of 28 and 25 nm are achieved for 2tCPD and tPD, respectively. Maximum external quantum efficiencies of 22.0 % and 27.1 % are successfully realized for 2tCPD- and tPD-based OLED devices. These results offer a novel design strategy for constructing narrowband emitters by introducing sulfone group into a rigid molecular framework.