Random walk on the random connection model

We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like ${|x?y|}^{?\alpha }$ with $\alpha >d$, where $d$ denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits similar properties as classical discrete long-range percolation models studied by Berger (2002) with regard to recurrence and transience of the random walk. Moreover, we address a question which is related to a conjecture by Heydenreich, Hulshof and Jorritsma (2017) for this graph.     

The lifespan of classical solutions for the inviscid Surface Quasi-geostrophic equation

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation ϵ from a radial stationary solution $\theta =|x|$. We use a modified energy method to prove the existence time of classical solutions from $\frac{1}{ϵ}$ to a time scale of $\frac{1}{{ϵ}^4}$. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space.     

Words that almost commute

The Hamming distance $\mathrm{ham}(u,v)$ between two equal-length words u, v is the number of positions where u and v differ. The words u and v are said to be conjugates if there exist non-empty words $x,y$ such that $u=xy$ and $v=yx$. The smallest value $\mathrm{ham}(xy,yx)$ can take on is 0, when x and y commute. But, interestingly, the next smallest value $\mathrm{ham}(xy,yx)$ can take on is 2 and not 1. In this paper, we consider conjugates $u=xy$ and $v=yx$ where $\mathrm{ham}(xy,yx)=2$. More specifically, we provide an efficient formula to count the number $h(n)$ of length-n words $u=xy$ over a k-letter alphabet that have a conjugate $v=yx$ such that $\mathrm{ham}(xy,yx)=2$. We also provide efficient formulae for other quantities closely related to $h(n)$. Finally, we show that $h(n)$ grows erratically: cubically for n prime, but exponentially for n even.     

Anisotropic torsional creep problems involving rapidly growing differential operators

The asymptotic behaviour of the sequence of positive solutions for a family of torsional creep-type problems involving anisotropic rapidly growing differential operators is studied in a bounded domain from the Euclidian space ${\mathbb{R}}^N$. We prove that the sequence of solutions converges uniformly on the domain to a certain distance function defined in accordance with the anisotropy of the problem.     

On the universal completion of pointfree function spaces

This paper approaches the construction of the universal completion of the Riesz space $\mathrm{C}(L)$ of continuous real functions on a completely regular frame L in two different ways. Firstly as the space of continuous real functions on the Booleanization of L. Secondly as the space of nearly finite Hausdorff continuous functions on L. The former has no counterpart in the classical theory, as the Booleanization of a spatial frame is not spatial in general, and it offers a lucid way of representing the universal completion as a space of continuous real functions. As a corollary we obtain that $\mathrm{C}(L)$ and $\mathrm{C}(M)$ have isomorphic universal completions if and only if the Booleanization of L and M are isomorphic and we characterize frames L such that $\mathrm{C}(L)$ is universally complete as almost Boolean frames. The application of this last result to the classical case $\mathrm{C}(X)$ of the space of continuous real functions on a topological space X characterizes those spaces X for which $\mathrm{C}(X)$ is universally complete. Finally, we present a pointfree version of the Maeda-Ogasawara-Vulikh representation theorem and use it to represent the universal completion of an Archimedean Riesz space with weak unit as a space of continuous real functions on a Boolean frame.     

Representation and factorization theorems for almost-Lp-spaces

We extend the notions of $p$-convexity and $p$-concavity for Banach ideals of measurable functions following an asymptotic procedure. We prove a representation theorem for the spaces satisfying both properties as the one that works for the classical case: each almost $p$-convex and almost $p$-concave space is order isomorphic to an almost-$L^p$-space. The class of almost-$L^p$-spaces contains, in particular, direct sums of (infinitely many) $L^p$-spaces with different norms, that are not in general $p$-convex – nor $p$-concave –. We also analyze in this context the extension of the Maurey–Rosenthal factorization theorem that works for $p$-concave operators acting in $p$-convex spaces. In this way we provide factorization results that allow to deal with more general factorization spaces than $L^p$-spaces.     

Asymptotic behavior of spreading fronts in the anisotropic Allen–Cahn equation on Rn

We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on ${\mathbb{R}}^n$ with $n\ge 2$ and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow $V_{\gamma }={\kappa }_{\gamma }+c$ with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen–Cahn equation plays a key role.     

Singular p-Laplacian equations with superlinear perturbation

We consider a nonlinear Dirichlet problem driven by the p-Laplace operator and with a right-hand side which has a singular term and a parametric superlinear perturbation. We are interested in positive solutions and prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter $\lambda >0$ varies. In addition, we show that for every admissible parameter $\lambda >0$ the problem has a smallest positive solution ${\stackrel{￣}{u}}_{\lambda }$ and we establish the monotonicity and continuity properties of the map $\lambda \to {\stackrel{￣}{u}}_{\lambda }$.     

Mappings preserving B-orthogonality

It is well known that a linear mapping $T:X\to Y$ preserving the Birkhoff orthogonality (i.e. ${?}_{x,y\in X}x{\perp }_{\text{B}}y?Tx{\perp }_{\text{B}}Ty$), has to be a similarity. For real spaces it has been proved by Koldobsky (1993); a proof including both real and complex spaces has been given by Blanco and Turn?ek (2006). In the present paper the author would like to present a somewhat simpler proof of this nice theorem. Moreover, we extend the Koldobsky theorem; more precisely, we show that the linearity assumption may be replaced by additivity.     

Limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes describe the stochastic dynamics of interacting particle systems of Calogero–Moser–Sutherland type and are related with $\beta$-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and Miyashita that for fixed starting points, these processes admit interesting limit laws when the multiplicities $k$ tend to $\infty$, where in some cases the limits are described by the zeros of classical Hermite and Laguerre polynomials. In this paper we use SDEs to derive corresponding limit laws for starting points of the form $\sqrt{k}?x$ for $k\to \infty$ with $x$ in the interior of the corresponding Weyl chambers. Our limit results are a.s. locally uniform in time. Moreover, in some cases we present associated central limit theorems.