Let $u$ be a harmonic function on a complete simply connected manifold $M$ whose sectional curvatures are bounded between two negative constants. It is proved here a pointwise criterion of non-tangential convergence for points of the geometric boundary: the finiteness of the density of energy, which is the geometric analogue of the density of the area integral in the Euclidean half-space.

Lewis and Vogel proved that a bounded domain whose Poisson kernel is constant and whose surface measure to the boundary has at most Euclidean growth is a ball. In this paper we show that this result is stable under small perturbations. In particular a bounded domain whose Poisson kernel is smooth and close to a constant, and whose surface measure to the boundary has at most Euclidean growth is a smooth deformation of a ball.

The construction of Freudenthal's Magic Square, which contains the exceptional simple Lie algebras of types $F_4,E_6,E_7$ and $E_8$, in terms of symmetric composition algebras is further developed here. The para-Hurwitz algebras, which form a subclass of the symmetric composition algebras, will be defined, in the split case, in terms of the natural two dimensional module for the simple Lie algebra $\mathfrak{sl}_2$. As a consequence, it will be shown how all the Lie algebras in Freudenthal's Magic Square can be constructed, in a unified way, using copies of $\mathfrak{sl}_2$ and of its natural module.

We provide an integral formula for the Poisson kernel of half-spaces for Brownian motion in real hyperbolic space $\mathbb{H}^n$. This enables us to find asymptotic properties of the kernel. We also show convergence to the Poisson kernel of the whole space $\mathbb{H}^n$. For $n=3$, $4$ or $6$ we compute explicit formulas for the Poisson kernel itself.

The aim of this paper is to classify the finite minimal non-$p$-supersoluble groups, $p$ a prime number, in the $p$-soluble universe.

We give a new proof of a Theorem of S. Mardešić, generalized by G. E. Bredon, that Čech and singular homology groups of certain locally connected spaces coincide. We use the chain complexes of integral flat chains (H. Whitney) and integral currents (H. Federer and W. H. Fleming) to define new homology groups of subsets of Euclidean space. We show these verify the axioms of Eilenberg and Steenrod, and we provide Lipschitz-flavored local connectedness conditions which guarantee these groups coincide with Čech's. Relations between these theories is relevant for the solvability and regularity of many geometric variational problems.

We consider the definition and regularity properties of convex functions in Carnot groups. We show that various notions of convexity in the subelliptic setting that have appeared in the literature are equivalent. Our point of view is based on thinking of convex functions as subsolutions of homogeneous elliptic equations.

Nontangential a.e. vanishing of the oblique derivative of a harmonic function with respect to a Hölder continuous vector field on a Lipschitz boundary is shown to imply that the harmonic function is constant.

In this article we give new conditions for the density of continuous or smooth functions in variable exponent Sobolev spaces. Our first result combines the previously known sufficient conditions, a monotony condition by Edmunds and R#x00E1;kosn#x00ED;k and a continuity condition independently due to Samko and Diening, into a single weaker condition. The second main result gives a sufficient condition in terms of the regularity of the level-sets of the variable exponent.

We present a new logarithmic Sobolev inequality adapted to a log-concave measure on $\mathbb{R}$ between the exponential and the Gaussian measure. More precisely, assume that $\Phi$ is a symmetric convex function on $\mathbb{R}$ satisfying $(1+\varepsilon)\Phi(x)\leq {x}\Phi'(x)\leq(2-\varepsilon)\Phi(x)$ for $x\geq 0$ large enough and with $\varepsilon\in ]0,1/2]$. We prove that the probability measure on $\mathbb{R}$ $\mu_\Phi(dx)=e^{-\Phi(x)}/Z_\Phi dx$ satisfies a modified and adapted logarithmic Sobolev inequality: there exist three constants $A,B,C>0$ such that for all smooth functions $f>0$, $$ \mathbf{Ent}_{\mu_\Phi}{\left(f^2\right)}\leq A\int H_{\Phi}\left(\frac{f'}{f}\right)f^2d\mu_\Phi, $$ with $$ H_{\Phi}(x)= \left\{ \begin{array}{l} x^2 \text{ if }\left|x\right|< C,\\ \Phi^*\left(Bx\right) \text{ if }\left|x\right|\geq C, \end{array} \right. $$ where $\Phi^*$ is the Legendre-Fenchel transform of $\Phi$.

We show that if a smooth function that never vanishes to infinite order, then the set of points within the distance $\delta$ from the zeroes of this function has volume $O(\delta).$

For any subset $E \subset \mathbb{R}^n$, let $C^m (E)$ denote the Banach space of restrictions to $E$ of functions $F \in C^m (\mathbb{R}^n)$. It is known that there exist bounded linear maps $T:C^m(E)\longrightarrow C^m(\mathbb{R}^n)$ such that $Tf = f$ on $E$ for any $f \in C^m (E)$. We show that $T$ can be taken to have a simple form, but cannot be taken to have an even simpler form.

We discuss questions related to the non-existence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog and Ono, and Alkan using new results about exponential sums and the distribution of $\mathfrak{B}$-free numbers.

We construct Generalized Multifractional Processes with Random Exponent (GMPREs). These processes, defined through a wavelet representation, are obtained by replacing the Hurst parameter of Fractional Brownian Motion by a sequence of continuous random processes. We show that these GMPREs can have the most general pointwise H#x00F6;lder exponent function possible, namely, a random H#x00F6;lder exponent which is a function of time and which can be expressed in the strong sense (almost surely for all $t$), as a $\liminf$ of an arbitrary sequence of continuous processes with values in $[0,1]$.

We obtain a regularity conditions for solutions of the 3D Navier-Stokes equations with fractional powers of the Laplacian, which incorporates the vorticity direction and its magnitude simultaneously. We find that regularity assumption of direction field of the vorticity compensates with the integrability condition for the magnitude of vorticity. The regularity of direction field is most naturally measured in terms of the Triebel-Lizorkin type of norms. This unifies and extends previous results in this direction of studies, where the geometric structure of the vortex stretching term is used to obtain refined regularity conditions, initiated by Constantin and Fefferman.

We show that the number of monochromatic solutions of the equation $x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_r^{\alpha_r}=g$ in a $2$-coloring of a finite group $G$, where $\alpha_1,\ldots,\alpha_r$ are permutations and $g\in G$, depends only on the cardinalities of the chromatic classes but not on their distribution. We give some applications to arithmetic Ramsey statements.