On the covering by small random intervals

Consider the random intervals In = ωn+ (0, n) (modulo 1) with their left points ωn independently and uniformly distributed over the interval [0,1)=R/Z and with their lengths decreasing to zero. We prove that the Hausdorff dimension of the set limnIn of points covered infinitely often is almost surely equal to 1/α when n = a/nα for some a> 0 and α> 1

On the fast Khintchine spectrum in continued fractions

For $x\in [0,1)$, let $x=[a_1(x), a_2(x),...]$ be its continued fraction expansion with partial quotients ${a_n(x), n\ge 1}$. Let $\psi : \mathbb{N} \rightarrow \mathbb{N}$ be a function with $\psi(n)/n\to \infty$ as $n\to \infty$. In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set E(\psi):=\Big{x\in [0,1): \lim_{n\to\infty}\frac{1}{\psi(n)}\sum_{j=1}^n\log a_j(x)=1\Big} is completely determined without any extra condition on $\psi$.Comment: 10 page

Effect of sodium hyaluronate in treating fungal corneal ulcer

AIM: To retrospectively analyze the effects of sodium hyaluronate in treating fungal corneal ulcer.<p>METHODS: Since June, 2006, there were 178 patients(178 eyes)with fungal corneal ulcer receiving medical treatment in our hospital. Among them, 81 patients(81 eyes)as the control group received the traditional antifungal treatment with the natamycin and fluconazole being the major medicine, from June 2006 to June 2008. While, 97 patients(97 eyes)as the treatment group received sodium hyaluronate treatment based on traditional antifungal treatment during the period of June 2008 to March 2010. Effects of two therapeutic methods were compared and analyzed. <p>RESULTS: Of the 97 cases in the treatment group, the average hospital stay was: 14.15±4.23d, with 90 cases(92.8%)cured, 5 cases(5.2%)improved, 2 cases(2.1%)ineffective, the final visual acuity of 51.6% patients better than 0.3. Of the 81 cases in the control group, average hospital stay was: 17.26±6.23d, with 69 cases(85.2%)cured, 7 cases(8.6%)improved, 5 cases(6.2%)ineffective, the final visual acuity of 39.5% patients better than 0.3. After statistical analysis, the average hospital stay, the cure rate, the effective rate and the final visual acuity in both groups showed statistically significant difference(<i>P</i><0.05). The average hospital stay of the treatment group was shorter than that of the control group, while the cure rate and effective rate and the final visual acuity was better than that of the control group.<p>CONCLUSION: Sodium hyaluronate can promote fungal corneal ulcer healing, improve the cure rate and reduce the formation of corneal scar

Uncertainty Relation of Functions for Kernel-based Transforms Obtained in Quantum Mechanics

In this paper, the generic uncertainty relation (UR) for kernel-based transformations (KT) of signals is derived. Instead of using the statistics approach as shown in the literature before, here we employ quantum mechanical operator approach for directly deriving the UR for KT's. We are able to do this because we have found the quantum operator realization of KT. Our new method is concise and applicable to any kinds of KT's with continuous and discrete parameters and variables. An explicit result of UR for a family of KT's including FrFT, generalized fractional transformation (GFrT) and linear canonical transformation (LCT) is provided as an application of our new method

Effect of mesenchymal stem cell transplantation on behavior and structural changes in myelin in experimental rat autoimmune encephalomyelitis

Purpose: To investigate the effect of bone marrow mesenchymal stem cell (BM-MSC) transplantation on behavior and structural changes in myelin of experimental autoimmune encephalomyelitis (EAE) in rats.Methods: Wistar rats were randomly assigned to normal control, EAE, placebo injection treatment, and MSCs treatment. EAE, placebo injection treatment and MSCs treatment groups were further divided into six groups, i.e., 1-day, 3-day, 7-day, 14-day, 21-day, and 28-day post-onset. Changes in the diseasestatus of the rats and structural changes in myelin at different time points were assessed with silver staining.Results: Behavioral changes peaked between 13 to 17 days post-immunization (71.90 % incidence), while disease symptoms peaked between 3 and 5 days after onset, were sustained for about 7 days, and then eased gradually thereafter. The highest therapeutic scores and the entire course of the disease in EAE and placebo treatment groups were not significantly different (p &gt; 0.05). However, in MSC treatment group, these parameters were significantly lower than in the above two groups (p &lt; 0.05). In EAE and placebo treatment groups, myelin sheath lesions were obvious from day 3 to 7 but on day 14, the number of myelin sheath fragments decreased significantly. Recovery at different time points was also better than those in EAE and placebo treatment groups.Conclusion: MSC transplantation shortens the course of EAE, and also reduces its severity. Thus, it has some prospects for use in the management of EAE.Keywords: Bone marrow mesenchymal stem cells, Autoimmune encephalomyelitis, Transplantation, Myeli