For every two-dimensional random walk on the square lattice$Z$^{2}having zero mean and finite variance we obtain fine asymptotic estimates of the probability that the walk hits the negative real line for the first time at a site ($s$,0), when it is started at a site far from both (0,$s$) and the origin.

Let$L$=−Δ+$V$be a Schrödinger operator on ℝ^{$d$},$d$≥3. We assume that$V$is a nonnegative, compactly supported potential that belongs to$L$^{$p$}(ℝ^{$d$}), for some$p$>$d$$/$2. Let$K$_{$t$}be the semigroup generated by −$L$. We say that an$L$^{1}(ℝ^{$d$})-function$f$belongs to the Hardy space $H^{1}_{L}$ associated with$L$if sup_{$t$>0}|$K$_{$t$}$f$| belongs to$L$^{1}(ℝ^{$d$}). We prove that $f\in H^{1}_{L}$ if and only if$R$_{$j$}$f$∈$L$^{1}(ℝ^{$d$}) for$j$=1,…,$d$, where$R$_{$j$}=($∂$/$∂$$x$_{$j$})$L$^{−1$/$2}are the Riesz transforms associated with$L$.

Let$R$be a Noetherian ring, $\mathfrak{a}$ an ideal of$R$,$M$an$R$-module and$n$a non-negative integer. In this paper we first study the finiteness properties of the kernel and the cokernel of the natural map $f\colon\operatorname{Ext}^{n}_{R}(R/\mathfrak{a},M)\to \operatorname{Hom}_{R}(R/\mathfrak{a},\mathrm{H}^{n}_{\mathfrak{a}}(M))$ , under some conditions on the previous local cohomology modules. Then we get some corollaries about the associated primes and Artinianness of local cohomology modules. Finally we will study the asymptotic behavior of the kernel and the cokernel of the natural map in the graded case.

Let$X$be a compact Hausdorff space. A kernel function on$X$×$X$, enjoying additional properties, naturally defines a semi-inner product structure on certain subspaces of all finite signed Borel measures on$X$. This paper discusses the question of completeness of such spaces.

In this note we establish some general finiteness results concerning lattices Γ in connected Lie groups$G$which possess certain “density” properties (see$Moskowitz$, M., On the density theorems of Borel and Furstenberg,$Ark. Mat.$$16$(1978), 11–27, and$Moskowitz$, M., Some results on automorphisms of bounded displacement and bounded cocycles,$Monatsh. Math.$$85$(1978), 323–336). For such groups we show that Γ always has finite index in its normalizer$N$_{$G$}(Γ). We then investigate analogous questions for the automorphism group Aut($G$) proving, under appropriate conditions, that Stab_{Aut($G$)}(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup $\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}$ , of Aut($G$) has finite index in Stab_{Aut($G$)}(Γ). We test the limits of our results with various examples and counterexamples.