Let $\Omega $ be a bounded open set in $\mathbb R^n$, $n \geq 2$. In a well-known paper {\it Indiana Univ. Math. J.}, 20, 1077--1092 (1971) Moser found the smallest value of $K$ such that $$ \sup \bigg \{\int _{\Omega } \exp \Big (\Big (\frac {\left |f(x)\right |}K\Big )^{n/(n-1)}\Big )\colon f\in W^{1,n}_0(\Omega ),\|\nabla f\|_{L^n}\leq 1\bigg \}<\infty . $$ We extend this result to the situation in which the underlying space $L^n$ is replaced by the generalized Zygmund space $L^n\log ^{n-1}L \log ^{\alpha }\log L$ $(\alpha <n-1)$, the corresponding space of exponential growth then being given by a Young function which behaves like $\exp (\exp (t^{n/(n-1-\alpha )}))$ for large $t$. We also discuss the case of an embedding into triple and other multiple exponential cases.
Assume that $X$, $Y$ are continuous-path martingales taking values in $\mathbb R^\nu $, $\nu \geq 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality $$ \|\sup _{t\geq 0} |Y_t| \|_1\leq 2\|\sup _{t\geq 0} |X_t| \|_1. $$ The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder's method and rests on the construction of an appropriate special function.
We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results., Lihua You, Yujie Shu, Xiao-Dong Zhang., and Obsahuje seznam literatury
The theorem about the characterization of a GS-quasigroup by means of a commutative group in which there is an automorphism which satisfies certain conditions, is proved directly.