Random self-affine multifractal Sierpinski sponges in $${\mathbb{R}^d}$$

In this paper we study the Hausdorff and packing dimensions and the Rényi dimensions of random self-affine multifractal Sierpinski sponges in \({\mathbb{R}^{d}}\).     

Separable forms of {{\mathbb{G}}_m} -ACTIONS ON {{\mathbb{A}}^3}

Let k be a field of characteristic zero. We consider k-forms of $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ and show that they are linearizable. In particular, $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ are linearizable, and k-forms of $ {\mathbb A} $ ³ that admit an effective action of an infinite reductive group are trivial.     

Heat Flow and Perimeter in \boldsymbol{{\mathbb{R}}^m}

Let Ω be an open set in Euclidean space ? ^m with finite perimeter ${\mathcal{P}}(\Omega),$ and with m-dimensional Lebesgue measure |Ω|. It was shown by M. Preunkert that if T(t) is the heat semigroup on L ²(? ^m ) then $H_{\Omega}(t):=\int_{\Omega}T(t)\textbf{1}_{\Omega}(x)dx=|\Omega|-\pi^{-1/2}{\mathcal{P}}(\Omega)t^{1/2}+o(t^{1/2}), \ t\downarrow 0$ . H _Ω(t) represents the amount of heat in Ω if Ω is at initial temperature 1 and if ? ^m ???Ω is at initial temperature 0. In this paper we will compare the quantitative behaviour of H _Ω(t) with the usual heat content Q _Ω(t) associated to the Dirichlet heat semigroup on Ω. We analyse the heat content for horn-shaped open sets of the form Ω(α, Σ)?=?{(x, x′)?∈?? ^m : x′?∈?(1?+?x)^???α Σ, x?>?0}, where α?>?0, and where Σ is an open set in ? ^m???1 with finite perimeter in ? ^m???1, which is star-shaped with respect to 0. For m?≥?3 we find that there are four regimes with very different behaviour depending on α, and a further two limiting cases where logarithmic corrections appear.     

Strong Central Limit Theorem for isotropic random walks in {\mathbb{R}^d}

We prove an optimal Gaussian upper bound for the densities of isotropic random walks on ${\mathbb{R}^d}$ in spherical case (d ?? 2) and ball case (d ?? 1). We deduce the strongest possible version of the Central Limit Theorem for the isotropic random walks: if ${\tilde S_n}$ denotes the normalized random walk and Y the limiting Gaussian vector, then ${\mathbb{E} f(\tilde S_{n}) \rightarrow \mathbb{E} f(Y)}$ for all functions f integrable with respect to the law of Y. We call such result a ??Strong CLT??. We apply our results to get strong hypercontractivity inequalities and strong Log-Sobolev inequalities.     

Trajectory of an Observer Tracking Object Motion around Convex Obstacles in $${{\mathbb{R}}^{2}}$$ and $${{\mathbb{R}}^{3}}$$

Doklady Mathematics - An autonomous object moves at a constant speed along the shortest path, while bypassing an ordered collection of pairwise disjoint convex sets. The object is tracked by an...     

Circle Geometries Modeled in Projective Lines over {{\mathbb{R}}^2} -rings

In this paper, we consider classical circle geometries and connect them with places of planar Cayley–Klein geometries. There are, in principle, only three types of $ {{\mathbb{R}}^2} $ -ring structures and, thus, only three types of corresponding circle geometries. Thus, each generalization to non-Euclidean planes turns out to be just another representation of the classical Euclidean cases. We believe that even the Euclidean cases of circle geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle geometries might also be of interest in themselves. For example, among the planar Cayley–Klein geometries, the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated similarly to the isotropic Möbius geometry by suitable projections of the Blaschke cylinder.     

On regular seven-membered loops in {{\mathbb{R}}^3} with arbitrary join angle

The problem of ring molecules and macromolecules arises in a number of contexts in physical chemistry. Perhaps the simplest example of a seven-membered loop is cycloheptane \({\rm C_{7}\rm H_{14}}\), which is a molecule where the carbon–carbon bonds form a regular seven-membered loop. However, it is possible to envisage much more complicated arrangements of proteins in chains comprising straight rigid sections linked in ways that enforce the same angle at all of the joins. In this paper, we present a coordinate system that reduces the problem to four free variables and three constraints. We then survey the solutions numerically and find that there are families of solutions for all join angles \({\theta}\) between \({\pi/7}\) and \({5\pi/7}\) with fixed planar solutions existing for \({\theta = \pi/7}\), \({3\pi/7}\) and \({5\pi/7}\). The available families of solutions undergo a major reorganisation at the join angle \({\theta = \pi/3}\), where 28 intersecting solutions form a single connected network of configurations.     

Discrete fractal dimensions of the ranges of random walks in {{\mathbb Z}^d} associate with random conductances

Let ${X= \{X_t, t \ge 0\}}$ be a continuous time random walk in an environment of i.i.d. random conductances ${\{\mu_e \in [1,\infty), e \in E_d\}}$ , where E _d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice ${\mathbb{Z}^d}$ and d ≥ 3. Let ${{\rm R} = \{x \in \mathbb{Z}^d: X_t = x {\rm \,for\, some}\,t \ge 0\}}$ be the range of X. It is proved that, for almost every realization of the environment, dim_H R = dim_P R = 2 almost surely, where dim_H and dim_P denote, respectively, the discrete Hausdorff and packing dimension. Furthermore, given any set ${A \subseteq \mathbb{Z}^d}$ , a criterion for A to be hit by X _t for arbitrarily large t > 0 is given in terms of dim_H A. Similar results for Bouchoud’s trap model in ${\mathbb{Z}^d}$ (d ≥ 3) are also proven.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

Composition Operators on Bounded Convex Domains in {{\mathbb{C}}^n}

Let \({H^{\infty}(E)}\) be a non commutative Hardy algebra, associated with a \({W^*}\)-correspondence E. In this paper we construct factorizations of inner-outer type of the elements of \({H^{\infty}(E)}\) represented via the induced representation, and of the elements of its commutant. These factorizations generalize the classical inner-outer factorization of elements of \({H^\infty(\mathbb{D})}\). Our results also generalize some results that were obtained by several authors in some special cases.     

A Liouville theorem for conformal Gaussian curvature type equations in {{\mathbb{R}}^2}

In this paper, we obtain a Liouville type theorem for a class of elliptic equations including the conformal Gaussian curvature equation $$-\Delta u=K(x)e^{2u}\quad {\rm in}\,\, {\mathbb{R}}^2,$$ where K(x) is a H?lder continuous function in ${{\mathbb{R}}^2}$ that does not have a fixed sign near infinity. The main tool in our approach is an asymptotic formula for the solution at infinity and the method of moving planes. We also show how our Liouville theorem can be used to obtain a priori bound for solutions of the prescribing Gaussian curvature equation in S ², namely $$\Delta\, u+K(x)e^{2u}=1\, {\rm in}\, S^2,$$ where K(x) is H?lder continuous and nonnegative in S ² but vanishes on a set with nonempty interior, a case left open in previous research.     

Convex hulls of indicatrices and singularities of the transitivity zone in {{\mathbb{R}}^3}

We consider a control system on two- and three-dimensional manifolds, the indicatrices of which are the images of smooth mappings, and present a survey on classification of the generic singularities of the boundary of the local transitivity set.