In another article we associated a dynamical system to a non-properly ordered Bratteli diagram. In this article we describe how to compute the $K$-group $K_0$ of the dynamical system in terms of the Bratteli diagram. In the case of properly ordered Bratteli diagrams this description coincides with what is already known, namely the so-called dimension group of the Bratteli diagram. The new group defined here is more relevant for non-properly ordered Bratteli diagrams. We use our main result to describe $K_0$ of a substitutional system.

It is shown that for $H^\infty_\mathbb{R}(\mathbb{D})$ functions $f_1$ and $f_2$ with

$\inf_{z\in\mathbb{D}}(\vert f_1(z)\vert+\vert f_2(z)\vert)\geq\delta>0$

and $f_1$ being positive on the real zeros of $f_2$, then there exists $H^\infty_\mathbb{R}(\mathbb{D})$ functions $g_2$ and $g_1$, $g_1^{-1}$ with norm controlled by a constant depending only on $\delta$ and

$g_1f_1+g_2f_2=1\quad\forall\; z\in\mathbb{D}$.

These results are connected to the computation of the stable rank of the algebra $H^\infty_\mathbb{R}(\mathbb{D})$ and to results in Control Theory.

We consider maximal operators $M_{\mathcal B}$ with respect to a basis ${\mathcal B}$. In the case when $M_{\mathcal B}$ satisfies a reversed weak type inequality, we obtain a boundedness criterion for $M_{\mathcal B}$ on an arbitrary quasi-Banach function space $X$. Being applied to specific ${\mathcal B}$ and $X$ this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function $M^{+}$ on $L^p_w$.

We introduce a many-variable version of the $f$-polynomial and $h$-polynomial associated to a finite simplicial complex. In this context the $h$-polynomial is actually a rational function. We establish connections with the $l^2$-Euler characteristic of right-angled buildings. When $L$ is a triangulation of a sphere we obtain a new formula for the $l^2$-Euler characteristic.

Gerbes are locally connected presheaves of groupoids on a small Grothendieck site $\mathcal{C}$. They are classified up to local weak equivalence by path components of a cocycle category taking values in the big $2$-groupoid $\mathbf{Iso}(\Gr(\mathcal{C}))$ consisting of all sheaves of groups on $\mathcal{C}$, their isomorphisms and homotopies. If $\mathcal{F}$ is a full subpresheaf of $\mathbf{Iso}(\Gr(\mathcal{C}))$ then the set $[\ast,B\mathcal{F}]$ of morphisms in the homotopy category of simplicial presheaves classifies gerbes locally weakly equivalent to objects of $\mathcal{F}$. If $\St(\pi \mathcal{F})$ is the stack completion of the fundamental groupoid $\pi\mathcal{F}$ of $\mathcal{F}$, if $L$ is a global section of $\St(\pi\mathcal{F})$, and if $F_{L}$ is the homotopy fibre over $L$ of the canonical map $B\mathcal{F} \to B\St(\pi\mathcal{F})$, then $[\ast,F_{L}]$ is in bijective correspondence with Giraud's non-abelian cohomology object $H^{2}(\mathcal{C},L)$ of equivalence classes of gerbes with band $L$.

We consider families of entire transcendental maps given by $F_{\lambda,m} (z) = \lambda z^m \exp(z) $ where $m \ge 2$. All these maps have a superattracting fixed point at $z=0$ and a free critical point at $z=-m$. In parameter planes we focus on the capture zones, i.e., we consider $\lambda$ values for which the free critical point belongs to the basin of attraction of $z=0$. We explain the connection between the dynamics near zero and the dynamics near infinity at the boundary of the immediate basin of attraction of the origin, thus, joining together exponential and polynomial behaviors in the same dynamical plane.

We classify noninvertible, holomorphic selfmaps of the projective plane that preserve an algebraic web. In doing so, we obtain interesting examples of critically finite maps.

Let ${\mathbb F}_p$ be the finite field of a prime order $p$. Let $F \colon {\mathbb F}_p \times {\mathbb F}_p\rightarrow {\mathbb F}_p$ be a function defined by $F(x,y)=x(f(x)+by)$, where $b \in {\mathbb F}_p^*$ and $f\colon {\mathbb F}_p \rightarrow {\mathbb F}_p$ is any function. We prove that if $A \subset {\mathbb F}_p$ and $|A|<p^{1/2}$ then

$|A+A|+|F(A,A)| \gtrapprox |A|^{\frac{13}{12}}.$

Taking $f=0$ and $b=1$, we get the well-known sum-product theorem by Bourgain, Katz and Tao, and Bourgain, Glibichuk and Konyagin, and also improve the previous known exponent from $\frac{14}{13}$ to $\frac{13}{12}$.

We prove that, in general, ${\mathbb H}$-regular surfaces in the Heisenberg group $\mathbb{H}^1$ are not bi-Lipschitz equivalent to the plane ${\mathbb R}^2$ endowed with the ``parabolic'' distance, which instead is the model space for $C^1$ surfaces without characteristic points. In Heisenberg groups $\mathbb{H}^n$, ${\mathbb H}$-regular surfaces can be seen as intrinsic graphs: we show that such parametrizations do not belong to Sobolev classes of metric-space valued maps.

In this paper we introduce the concept of the index of an implicit differential equation $F(x,y,p)=0,$ where $F$ is a smooth function, $p=\frac{dy}{dx}$, $F_{p}=0$ and $F_{pp}=0$ at an isolated singular point. We also apply the results to study the geometry of surfaces in $\mathbb{R}^{5}$.

We show an Itô's formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s \,dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in Itô's s formula as an integral over space and time with respect to local time.

We apply an effective multidimensional $\Omega$ result of Voronin in order to obtain effective universality-type theorems for the Riemann zeta-function. We further use this approach to study approximation properties of linear combinations of derivatives of the zeta-function.

We discuss two possible definitions for Sobolev spaces associated with ultraspherical expansions. These definitions depend on the notion of higher order derivative. We show that in order to have an isomorphism between Sobolev and potential spaces, the higher order derivatives to be considered are not the iteration of the first order derivatives. Some discussions about higher order Riesz transforms are involved. Also we prove that the maximal operator for the Poisson integral in the ultraspherical setting is bounded on the Sobolev spaces.

We show the uniqueness for the measure of maximal entropy for regular automorphisms of $\mathbb{C}^k$.