Precise rates in the law of logarithm for i.i.d. random variables

AbstractLet {X, Xn; n ≥ 1} be a sequence of i.i.d. random variables. Set Sn = X1 + X2 + … + Xn and Mn = maxk≤n |Sk|, n ≥ 1. By using the strong approximation method, we obtain that for any −1 < b ≤ 1, lim⁡ε↘0ε2b+2∑n=1∞(log⁡n)bnP(Mn≥εσnlog⁡n)=2E|N|(2b+2)b+1∑k=0∞(−1)k(2k+1)2b+2 if and only if Ex = 0 and Ex2 < ∞, which strengthen and extend the result of Gut and Spǎtaru [1], where N is the standard normal random variable. Furthermore, L2 convergence and a.s. convergence are also discussed

Comparison of form-deprived myopia and lens-induced myopia in guinea pigs

<b>AIM:</b> To study the efficacy difference between form-deprived myopia (FDM) and lens-induced myopia (LIM), the degree of myopia, axial length and pathological changes of the posterior sclera from guinea pigs were evaluated.<b>METHODS:</b> Four-week pigmented guinea pigs were randomly assigned into 3 groups, including normal control (<i>n</i>=6), FDM group with monocular cover (<i>n</i>=11) and LIM group with monocular -7D lens treatment (<i>n</i>=11). FDM group was form-deprived while LIM group was lens-induced for 14 d. Refractive error and axial length were measured prior to and post treatment, respectively. Morphological changes of sclera were examined using both light and electronic microscopes.<b>RESULTS:</b> After 14d treatment, refractive errors for FDM group and LIM group were -3.05±0.71D and -2.12±1.29D, respectively, which were significantly more myopic than that of normal controls and fellow control eyes (<i>P</i>&lt;0.01). As for axial length, it was 7.93±0.03 mm for FDM group and 7.89±0.06 mm for LIM group, which were significantly longer than both normal and fellow controls (<i>P</i>&lt;0.01). With respect to both refractory error and axial length, the differences between FDM group and LIM group were not significant (<i>P</i>&gt;0.05). Under light microscope, both FDM group and LIM group showed thinned sclera, disarrangement of fibrosis and enlarged disassociation between fibers. Consistently, ultrastructural examination showed degenerated fibroblasts and thinned fibers in posterior sclera.<b>CONCLUSION:</b>Following two weeks of myopia induction in guinea pigs, with regard to the degree of myopia, axial length and pathological alterations, there was no significant difference between FDM and LIM models. Therefore, FDM and LIM are equally effective and useful as a model of experimental myopia and guinea pigs are ideal animals for induction of experimental myopia because their high sensitivity to both form-deprivation and lens-induction

Pretreatment of microcrystalline cellulose in organic electrolyte solutions for enzymatic hydrolysis

<p>Abstract</p> <p>Background</p> <p>Previous studies have shown that the crystalline structure of cellulose is negatively correlated with enzymatic digestibility, therefore, pretreatment is required to break down the highly ordered crystalline structure in cellulose, and to increase the porosity of its surface. In the present study, an organic electrolyte solution (OES) composed of an ionic liquid (1-allyl-3-methylimidazolium chloride ([AMIM]Cl)) and an organic solvent (dimethyl sulfoxide; DMSO) was prepared, and used to pretreat microcrystalline cellulose for subsequent enzymatic hydrolysis; to our knowledge, this is the first time that this method has been used.</p> <p>Results</p> <p>Microcrystalline cellulose (5 wt%) rapidly dispersed and then completely dissolved in an OES with a molar fraction of [AMIM]Cl per OES (χ _[AMIM]Cl) of greater than or equal to 0.2 at 110°C within 10 minutes. The cellulose was regenerated from the OES by precipitation with hot water, and enzymatically hydrolyzed. As the χ _[AMIM]Clof the OES increased from 0.1 to 0.9, both the hydrolysis yield and initial hydrolysis rate of the regenerated cellulose also increased gradually. After treatment using OES with χ _[AMIM]Clof 0.7, the glucose yield (54.1%) was 7.2 times that of untreated cellulose. This promotion of hydrolysis yield was mainly due to the decrease in the degree of crystallinity (that is, the crystallinity index of cellulose I).</p> <p>Conclusions</p> <p>An OES of [AMIM]Cl and DMSO with χ _[AMIM]Clof 0.7 was chosen for cellulose pretreatment because it dissolved cellulose rapidly to achieve a high glucose yield (54.1%), which was only slightly lower than the value (59.6%) obtained using pure [AMIM]Cl. OES pretreatment is a cost-effective and environmentally friendly technique for hydrolysis, because it 1) uses the less expensive OES instead of pure ionic liquids, 2) shortens dissolution time, 3) requires lower energy for stirring and transporting, and 4) is recyclable.</p

(Z)-1-(2,4-Difluoro­phen­yl)-2-(1H-1,2,4-triazol-1-yl)ethanone oxime

In the title compound, C10H8F2N4O, the dihedral angle between the rings is 65.4 (1)°. In the crystal, inter­molecular O—H⋯N and C—H⋯F hydrogen bonds link the mol­ecules in a stacked arrangement along the a and c axes, respectively

Classical-driving-assisted quantum synchronization in non-Markovian environments

We study the quantum phase synchronization of a driven two-level system (TLS) coupled to a structured environment and demonstrate that quantum synchronization can be enhanced by the classical driving field. We use the Husimi $Q$-function to characterize the phase preference and find the in-phase and anti-phase locking phenomenon in the phase diagram. Remarkably, we show that the classical driving enables a TLS to reach stable anti-phase locking in the Markovian regime. However, we find that the synergistic action of classical driving and non-Markovian effects significantly enhances the in-phase locking. By introducing the $S$-function and its maximal value to quantify the strength of synchronization and sketch the synchronization regions, we observe the typical signatures of the hollowed Arnold tongue in the parameter regions of synchronization. In the hollowed Arnold tongue, the synchronization regions exist both inside and outside the tongue while unsynchronized regions only lie on the boundary line. We also provide an intuitive interpretation of the above results by using the quasimode theory.Comment: 10 pages, 5 figures, revised versio