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This paper establishes the hyper-contractivity in L∞ (ℝd) (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the dif;fusion exponent m > 1 -2/d. The results show that for the supercritical and critical case 1 - 2/d < m ≤ 2 - 2/d, if || U0||d(2-m)/2) < Cd, m where Cd, m is a universal constant, then for any t > 0, || u (⋅, t)|| ... is bounded and decays as t goes to infinity, For the subcritical case m > 2 - 2/d, the solution u (⋅, t) ∈ L∞ (ℝd) with any initial data U0 ∈ L1+ (ℝd) for any positive time.

In this paper the existence of positive 2\pi -periodic solutions to the ordinary differential equation \begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}is studied, where 2\pi is a positive 2\pi -periodic smooth function. By virtue of a new generalized Blaschke鈥揝antal贸 inequality, we obtain a new existence result of solutions.

We classify three fold isolated quotient Gorenstein singularity C^ 3/G . These singularities are rigid, ie there is no non-trivial deformation, and we conjecture that they define 4d C^ 3/G SCFTs which do not have a Coulomb branch.