We consider testing regression coefficients in high dimensional generalized linear models. An investigation of the test of Goeman et al. (2011) is conducted, which reveals that if the inverse of the link function is unbounded, the high dimensionality
in the covariates can impose adverse impacts on the power of the test. We propose a test formation which can avoid the adverse impact of the high dimensionality. When the inverse of the link function is bounded such as the logistic or probit regression, the proposed test is as good as Goeman et al. (2011)s test. The proposed tests provide p-values for testing significance for gene-sets as demonstrated in a case study on an acute lymphoblastic leukemia dataset.
The authors of this paper deal with the existence and regularities of weak solutions to the homogenous $hbox{Dirichlet}$ boundary value problem for the equation $-hbox{div}(| abla u|^{p-2} abla u)+|u|^{p-2}u=frac{f(x)}{u^{alpha}}$. The authors apply
the method of regularization and $hbox{Leray-Schauder}$ fixed point theorem as well as a necessary compactness argument to prove the existence of solutions and then obtain some maximum norm estimates by constructing three suitable iterative sequences. Furthermore, we find that the critical exponent of $m$ in $|f|_{L^{m}(Omega)}$. That is, when $m$ lies in different intervals, the solutions of the problem mentioned belongs to different $hbox{Sobolev}$ spaces. Besides, we prove that the solution of this problem is not in $W^{1,p}_{0}(Omega)$ when $alpha>2$, while the solution of this problem is in $W^{1,p}_{0}(Omega)$ when $1<alpha<2$.
The authors of this paper study singular phenomena(vanishing and blowing-up in finite time) of solutions to the homogeneous $hbox{Dirichlet}$ boundary value problem of nonlinear diffusion equations involving $p(x)$-hbox{Laplacian} operator and a nonl
inear source. The authors discuss how the value of the variable exponent $p(x)$ and initial energy(data) affect the properties of solutions. At the same time, we obtain the critical extinction and blow-up exponents of solutions.
