To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number-theoretical free-energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free-energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued-fraction digits exceeding a given number which tends to infinity.

Let α₁, α₂,…, α_m be linear forms defined on ℂⁿ and $\mathcal{X}=\mathbb{C}^{n}\setminus\bigcup_{i=1}^{m}V(\alpha_{i})$ , where V(α_i)={p∈ℂⁿ:α_i(p)=0}. The coordinate ring $O_{\mathcal{X}}$ of $\mathcal{X}$ is a holonomic A_n-module, where A_n is the nth Weyl algebra and since holonomic A_n-modules have finite length, $O_{\mathcal{X}}$ has finite length. We consider a “twisted” variant of this A_n-module which is also holonomic. Define M ${}_{α}^{β}$ to be the free rank-1 ℂ[x]_α-module on the generator α^β (thought of as a multivalued function), where $\alpha^{\beta}=\alpha_{1}^{\beta_{1}},\ldots,\alpha_{m}^{\beta_{m}}$ and the multi-index β=(β₁,…, β_m)∈ℂ^m. Our main result is the computation of the number of decomposition factors of M ${}_{α}^{β}$ and their description when n=2.

In [Rong, F., Quasi-parabolic analytic transformations of Cⁿ, J. Math. Anal. Appl. 343 (2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of Cⁿ. Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.

We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L₂ Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.

In this note, we characterize maximal invariant subspaces for a class of operators. Let T be a Fredholm operator and $1-TT^{*}\in\mathcal{S}_{p}$ for some p≥1. It is shown that if M is an invariant subspace for T such that dim M⊖TM<∞, then every maximal invariant subspace of M is of codimension 1 in M. As an immediate consequence, we obtain that if M is a shift invariant subspace of the Bergman space and dim M⊖zM<∞, then every maximal invariant subspace of M is of codimension 1 in M. We also apply the result to translation operators and their invariant subspaces.

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.

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.

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 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 L¹(ℝ^d)-function f belongs to the Hardy space $H^{1}_{L}$ associated with L if sup _t>0|K_tf| belongs to L¹(ℝ^d). We prove that $f\in H^{1}_{L}$ if and only if R_jf∈L¹(ℝ^d) for j=1,…, d, where R_j=(∂/∂x_j)L^−1/2 are the Riesz transforms associated with L.

For every two-dimensional random walk on the square lattice Z² 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.

We study bijections between algebras of smooth functions preserving certain parts of its structure. In particular, we show that multiplicative bijections are implemented by diffeomorphisms and they are automatically algebra isomorphisms. This confirms a conjecture by Mrčun and Šemrl.

We give a sufficient condition on a Lévy measure μ which ensures that the generator L of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., $\mathop {\mathrm {sing\,supp}}u\subseteq \mathop {\mathrm {sing\,supp}}Lu$ for all admissible u. In particular, we assume that $\mu|_{\mathbb {R}^{d}\setminus \{0\}}\in C^{\infty}(\mathbb {R}^{d}\setminus \{0\})$ . We also show that this condition is necessary provided that $\mathop {\mathrm {supp}}\mu$ is compact.