Radial anomalous diffusion in an annulus

In this study, time fractional radial diffusion has been modeled in cylindrical coordinates in order to analyze the anomalous diffusion in an annulus. By using an integral transform technique, the analytical solution of the concentration distribution formula is obtained. The establishing of the concentration distribution is found to be controlled by the fractional derivative α, and the influences of α on the concentration field, the total amount diffused and the quantity of mass passing through the inner wall are presented graphically and studied in detail. Asymptotic expressions for the exact solutions are developed in order to explain the numerical results at small and large time, respectively, and the physical mechanism explanation for the paradoxical behavior shown in the numerical results is given.     

Jack polynomial fractional quantum Hall states and their generalizations

In the study of fractional quantum Hall states, a certain clustering condition involving up to four integers has been identified. We give a simple proof that particular Jack polynomials with α=−(r−1)/(k+1)$\alpha =-(r-1)/(k+1)$, (r−1)$(r-1)$ and (k+1)$(k+1)$ relatively prime, and with partition given in terms of its frequencies by [n₀^0(r−1)sk^0r−1k^0r−1k?^0r−1m]$[n_0{0}^{(r-1)s}k{0}^{r-1}k{0}^{r-1}k?0^{r-1}m]$ satisfy this clustering condition. Our proof makes essential use of the fact that these Jack polynomials are translationally invariant. We also consider nonsymmetric Jack polynomials, symmetric and nonsymmetric generalized Hermite and Laguerre polynomials, and Macdonald polynomials from the viewpoint of the clustering.     

Alterations in diffusion properties of white matter in Williams syndrome

Diffusion tensor imaging (DTI) was used to investigate the involvement of brain white matter in Williams syndrome (WS), a genetic neurodevelopmental disorder. Whole-brain DTIs were obtained from 16 young adults with WS and 16 normal controls. A voxel-based analysis was performed to compare fractional anisotropy (FA) values between the two groups. A tract-based analysis was also performed to compare FA values between the two groups along two major white matter tracts that pass through the external capsule: the uncinate and inferior fronto-occipital fasciculi. Several regions of both increased and decreased FA were found within major white matter tracts that connect functional regions that have previously been implicated in the cognitive and neurological symptoms of the syndrome. The tract-based analysis provided additional insight into the involvement of specific white matter tracts implicated in the voxel-based analysis within the external capsule. The results from this study support previously reported changes in white matter diffusion properties in WS and demonstrate the potential usefulness for tract-based analysis in future studies of the disorder.     

Moderating registration misalignment in voxelwise comparisons of DTI data: a performance evaluation of skeleton projection

An evaluative methodology and five accompanying performance measures were developed to quantitatively assess the performance of the skeleton projection algorithm constituting the heart of tract-based spatial statistics (TBSS). The performance measures were designed to quantify the accuracy of skeleton projection in its indented task of alleviating any residual misalignment that may remain after image registration. A ground truth fractional anisotropy (FA) image was slightly warped using a realistic warp field that served to model post-registration residual misalignment of varying magnitudes. Skeleton projection was then used to register the warped FA image to the ground truth. Performing skeleton projection was found to yield up to 50% better correspondence between the values of FA compared to smoothing, despite the fact that less than 10% of post-registration misalignment was corrected. The align-max-with-max strategy underlying TBSS was posited as a potential explanation for this high correspondence in the values of FA, at the expense of lesser alignment between anatomically concordant voxels.     

蛋白质构型变化的二维广义郎之万方程描述

采用二维广义郎之万方程描述蛋白质构型的随机变化,并与电子输运过程的原子密度填充模型相协调. 假设通过键和通过空间的两类电子输运路径分别受到高斯分数噪声和高斯白噪声的影响. 推导了电子转移给体-受体距离涨落和荧光寿命涨落自相关函数的一般表达式. 采用数值拉普拉斯反变换计算了蛋白质构型涨落动力学,并详尽讨论了长时间和短时间行为的渐近解析. 最后,明确了基于二维广义郎之万方程的蛋白质构型描述与一维描述之间的关系.     

On a Nirenberg-type problem involving the square root of the Laplacian

Through an Euler–Hopf-type formula, we establish existence result to a Nirenberg-type problem involving the square root of the Laplacian in sphere S²$S^2$.     

Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces

In this article, a recent nonlinear alternative for contraction maps in Fréchet spaces due to Frigon and Granas [1998, Résultats de type Leray-Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22, 161–168] is used to investigate the existence and uniqueness of solutions for fractional order functional differential equations with infinite delay.     

Four-digit numbers which are squared sums

There is a very natural way to divide a four-digit number into 2 two-digit numbers. Applying an algorithm to this pair of numbers, determine how often the original four-digit number reappears.     

Three cubes in one number

Separate a three-digit number into its component digits. After raising each digit to the third power and computing the sum of the cubes, determine how often the original number reappears. Modular arithmetic is used to reduce the number of potential solutions to a more manageable quantity.