Population stratification is always a concern in association
analysis. There is a debate on the extent of the problem in less extreme
situations (Thomas and Witte[1], Wacholder et al.[2]). Wacholder et al.[3] and Ardlie et al.[4] showed that hidden population structure is not a serious threat to case-control designs. We propose a method of assessing the seriousness of the
population stratification before designing association studies. If population stratification
is not a serious problem, one may consider using case-control study instead of
family-based design to get more power. In a case-control design, we compare
chi-square statistics from a structured population (a union of two
subpopulations) and a homogeneous population with the same prevalence and
allele frequencies. We provide an explicit formula to calculate the chi-square
statistics from 17 parameters, such as proportions of subpopulation, allele
frequencies in subpopulations, etc. We choose these factors because they have
potential to cause false associations. Each parameter takes a random value in a
chosen range. We then calculate the likelihood of getting opposite conclusions
in the structured and the homogeneous populations. This is the likelihood of
having false positives caused by population stratification. The advantage of
this method is to provide a cost effective way to choose between using
case-control data and using family data before
actually collecting those data. We conclude
that sample sizes have a significant effect on the likelihood of false positive
caused by population stratification. The larger the sample size is, the more
likely to have false positive if the population structure is ignored. If the
sample size will be smaller than 200 by budget constraints, then case-control
study may be a better choice because of its power.

Abstract:
The many-body space fractional quantum system is studied using the density matrix method. We give the new results of the Thomas-Fermi model, and obtain the quantum pressure of the free electron gas. We also show the validity of the Hohenberg-Kohn theory in the space fractional quantum mechanics and generalize the density functional theory to the fractional quantum mechanics.

Abstract:
The 2D space-fractional Schrodinger equation in the time-independent and time-dependent cases for the scattering problem in the fractional quantum mechanics is studied. We define and give the mathematical expression of the Green's functions for the two cases. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems.

Abstract:
The solution to the fractional Schr\"odinger equation with infinite square well is obtained in this paper, by use of the L\'evy path integral approach. We obtain the even and odd parity wave functions of this problem, which are in accordance with those given by Laskin in [Chaos 10 (2000), 780--790].

Abstract:
Integral form of the space-time-fractional Schr\"odinger equation for the scattering problem in the fractional quantum mechanics is studied in this paper. We define the fractional Green's function for the space-time fractional Schrodinger equation and express it in terms of Fox's H-function and in a computable series form. The asymptotic formula of the Green's function for large argument is also obtained, and applied to study the fractional quantum scattering problem. We get the approximate scattering wave function with correction of every order.

Abstract:
Backgrounds: Although many disease-associated common variants have been discovered through genome-wide association studies, much of the genetic effects of complex diseases have not been explained. Population-based association studies are vulnerable to population stratification. A possible solution is to use family-based tests. However, if tests only estimate the genetic effect from the within-family variation to avoid population stratification, they may ignore the useful genetic information from between-family variation and lose power. Methods: We have developed an adaptive weighted sum test for family-based association studies. The new test uses data driven weights to combine two test statistics, and the weights measure the strength of population stratification. When population stratification is strong, the proposed test will automatically put more weight on one statistic derived from within-family variation to maintain robustness against spurious positives. On the other hand, when the effect of population stratification is relatively weak, the proposed test will automatically put more weight on the other statistic derived from both within-family and between-family variation to make use of both sources of genetic variation; and at the same time, the degrees of freedom of the test will be reduced and power of the test will be increased. Results: In our study, the proposed method achieves a higher power in most scenarios of linkage disequilibrium structure as well as Hap Map data from different genes under different population structures while still keeping its robustness against population stratification.

This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows COPE's Retraction
Guidelines. The aim is to promote the circulation of scientific research by offering an ideal research publication platform with due consideration of internationally accepted standards on publication ethics. The Editorial Board would like to extend its sincere apologies for any inconvenience this retraction may have caused.

Abstract:
Studies have suggested that one volcanic eruption can influence seasonal to inter-annual climate variations. This study indicates that the Pinatubo eruption in 1991 may have actually induced the stratospheric decadal cooling recorded in the early 1990s. Using the NCEP/NCAR reanalysis and TOMS/SBUV satellite data, a decadal abrupt cooling of stratospheric tropical air temperature was found to have occurred in the early 1990s during a long-term descending trend. We generated the spatio-temporal structures of the decadal abrupt changes (DACs) for the stratosphere, and explored the relationship between the Pinatubo volcano eruption in 1991 and stratospheric DACs in the early 1990s. Our results suggest that the eruption of Pinatubo prompted a decadal decrease of ozone by the activation of nitrate and sulfate volcanic aerosols on ClO free radicals. The stratospheric heat absorbed by ozone decreased over a decadal time scale. As a result, decadal abrupt cooling of stratospheric tropical air temperatures occurred in the early 1990s, and may be attributed to the Pinatubo eruption. The results therefore indicate that one strong volcanic eruption can induce stratospheric decadal climate variation.

Abstract:
We develop a score test based on wavelet transform with empirical Bayesian thresholding. Extensive simulation studies are carried out under various LD structures as well as using HapMap data from many different chromosomes for both qualitative and quantitative traits. Simulation results show that the proposed test automatically adjusts the level of noise suppression according to LD structures, and it is able to consistently achieve higher or similar powers than many commonly used association tests including the principle component regression method (PCReg).The wavelet-based score test automatically suppresses the right amount of noise and uses the information contained in spatial ordering of SNPs to achieve higher power.In a genome-wide association study (GWAS), if a SNP has a strong LD with a disease locus, single marker methods should have more power than multiple marker methods. However, if several SNPs have moderate associations with disease genes, multiple marker methods (such as Hotelling's T2 test [1-4] or multiple logistic regression) can provide higher power [5]. One of the problems of multiple marker methods is their large number of degrees of freedom, which in turn may lead to low power. Therefore, reducing the number of degrees of freedom becomes a key issue in gaining power for a multilocus method. For example, in haplotype association studies, tests based on haplotype sharing [6,7] have fewer degrees of freedom and higher power than tests based on haplotype frequencies. Another common approach to reducing the number of degrees of freedom is PCReg [8,9]. Projections of genotype data onto a few principal component directions of the variance-covariance matrix can often capture a majority of genotypic variances, and have fewer degrees of freedom. Tests based on projected genotype data can potentially have higher power. A weighted score test based on the Fourier transform [10] was also introduced to reduce the number of degrees of freedom. The basic idea be

Abstract:
We propose a weighted selective collapsing strategy for both candidate gene studies and genome-wide association scans. The strategy considers genetic information from both common and rare variants, selectively collapses all variants in a given region by a forward selection procedure, and uses an adaptive weight to favor more likely causal rare variants. Under this strategy, two tests are proposed. One test denoted by BwSC is sensitive to the directions of genetic effects, and it separates the deleterious and protective effects into two components. Another denoted by BwSCd is robust in the directions of genetic effects, and it considers the difference of the two components. In our simulation studies, BwSC achieves a higher power when the casual variants have the same genetic effect, while BwSCd is as powerful as several existing tests when a mixed genetic effect exists. Both of the proposed tests work well with and without the existence of genetic effects from common variants.Two tests using a weighted selective collapsing strategy provide potentially powerful methods for association studies of sequencing data. The tests have a higher power when both common and rare variants contribute to the heritable variability and the effect of common variants is not strong enough to be detected by traditional methods. Our simulation studies have demonstrated a substantially higher power for both tests in all scenarios regardless whether the common SNPs are associated with the trait or not.Genome-wide association studies (GWAS) have been used successfully in detecting associations between common genetic variants and complex diseases. However, common SNPs detected by current GWAS only explain a small proportion of heritable variability [1]. These identified common SNPs usually have a relatively small to modest genetic effect, which suggests that another type of variants, rare variants, need to be considered in the current GWAS. Recent studies showed that common diseases can be cau