In this short paper we prove that, for $3 le N le 9$, the problem $ -Delta u = e^u$ on the entire Euclidean space $mathbb {R}^N$ does not admit any solution stable outside a compact set of $mathbb {R}^N$. This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems. [ABSTRACT FROM AUTHOR]
In this paper, we give an estimate from below of the bounding genera for homology $3$-spheres defined by Y. Matsumoto in terms of $w$-invariants. In particular, combining with Matsumoto's estimates we determine the values of the bounding genera for several infinite families of Brieskorn homology $3$-spheres. [ABSTRACT FROM AUTHOR]
*RINGS of integers, *UNIFORM distribution (Probability theory), *STOCHASTIC processes, *FUNCTIONS of bounded variation, *MATHEMATICAL constants, *MATHEMATICS
Abstract
Chung, Diaconis, and Graham considered random processes of the form $X_{n+1}=a_nX_n+b_npmod p$ where $p$ is odd, $X_0=0$, $a_n=2$ always, and $b_n$ are i.i.d. for $n=0,1,2,dots $. In this paper, we show that if $P(b_n=-1)=P(b_n=0)=P(b_n=1)=1/3$, then there exists a constant $c>1$ such that $clog _2p$ steps are not enough to make $X_n$ get close to being uniformly distributed on the integers mod $p$. [ABSTRACT FROM AUTHOR]