The objective of the present paper is the modeling and analysis of the dynamics of macrophages and certain growth factors in the inflammatory phase, the first one of the wound healing process. It is the phase where there exists a major difference between diabetic and non-diabetic wound healing, an effect that we will consider in this paper.

We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. The main theorems of this paper (5.5 and 5.6) establish that, for a degree 2 map, semi-stability in the Geometric Invariant Theory sense on the space of self maps, implies minimality of the resultant. We prove the singular reduction of a semi-stable presentation coincides with the simple bad reduction (Theorem 4.1). Given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattès map has unstable bad reduction (Proposition 4.6). Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample (Example 3.1) to a simple dynamical analog to Theorem 1.1.

A well-known theorem of B. H. Neumann states that a group has finite conjugacy classes of subgroups if and only if it is central-by-finite. It is proved here that if $G$ is a generalized radical group of infinite rank in which the conjugacy classes of subgroups of infinite rank are finite, then every subgroup of $G$ has finitely many conjugates, and so $G/Z(G)$ is finite. Corresponding results are proved for groups in which every subgroup of infinite rank has finite index in its normal closure, and for those in which every subgroup of infinite rank is finite over its core.

We study the relationship between stable sampling sequences for band-limited functions in $L^p(\mathbb{R}^n)$ and the Fourier multipliers in $L^p$. In the case that the sequence is a lattice and the spectrum is a fundamental domain for the lattice the connection is complete. In the case of irregular sequences there is still a partial relationship.

In this paper we show that, for any $0 < p \le 1$ and $\alpha > -1$, every (weighted) Bergman space $\mathcal{A}^p_{\alpha} (\mathbb{B}_n)$ admits an atomic decomposition of real-variable type. More precisely, for each $f \in \mathcal{A}^p_{\alpha} (\mathbb{B}_n)$ there exist a sequence of $(p, \infty)_{\alpha}$-atoms $a_k$ with compact support and a scalar sequence $\{\lambda_k \}$ such that $f = \sum_k \lambda_k a_k$ in the sense of distribution and $\sum_k | \lambda_k |^p \lesssim \| f \|^p_{p, \alpha};$ and moreover, $f = \sum_k \lambda_k P_{\alpha} ( a_k)$ in $\mathcal{A}^p_{\alpha} (\mathbb{B}_n),$ where $P_{\alpha}$ is the orthogonal projection from $L^2_{\alpha} (\mathbb{B}_n)$ onto $\mathcal{A}^2_{\alpha} (\mathbb{B}_n).$ The proof is constructive and our construction is based on analysis inside the unit ball $\mathbb{B}_n$ associated with a quasimetric.

Motivated by the results of Korry, and Kinnunen and Saksman, we study the behaviour of the discrete fractional maximal operator on fractional Hajłasz spaces, Hajłasz-Besov, and Hajłasz-Triebel-Lizorkin spaces on metric measure spaces. We show that the discrete fractional maximal operator maps these spaces to the spaces of the same type with higher smoothness. Our results extend and unify aforementioned results. We present our results in a general setting, but they are new already in the Euclidean case.

The interpolating sequences $S$ for $H^{\infty }(\mathbb{D})$, the bounded holomorphic functions in the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$, were characterized by L. Carleson using metric conditions on $S$. Alternatively, to characterize interpolating sequences we can use the existence in $H^{\infty }(\mathbb{D})$ of an infinity of functions $\lbrace \rho _{a}\rbrace _{a\in S}$, uniformly bounded in $\mathbb{D}$, the function $\rho _{a}$ being $1$ at the point $a\in S$ and $0$ at any $b\in S\setminus \lbrace a\rbrace$. A. Hartmann recently proved that just one function in $H^{\infty }(\mathbb{D})$ was enough to characterize interpolating sequences for $H^{\infty }(\mathbb{D})$. In this work we use the "hard" part of Carleson's proof of the corona theorem to extend Hartmann's result and to answer a question he asked in his paper.

We exhibit a large class of symbols $m\colon \mathbb{R}^d\to \mathbb{C}$ for which the corresponding Fourier multipliers $T_m$ satisfy the following restricted weak-type estimates: if $A\subset \mathbb{R}^d$ has finite Lebesgue measure, then

$$||T_m\chi_A||_{p,\infty}\leq \frac{p}{2}e^{(2-p)/p}||\chi_A||_p,\quad p\geq 2.$$

In particular, this leads to novel sharp estimates for the real and imaginary part of the Beurling-Ahlfors operator on $\mathbb{C}$. The proof rests on probabilistic methods: we exploit a stochastic representation of the multipliers in terms of Lévy processes and appropriate sharp inequalities for differentially subordinated martingales.

Starting from works by Scherk (1835) and by Enneper-Weierstrass (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher. In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996. However, Traizet's construction is implicit and excludes towers, namely the desingularisation of more than two concurrent planes. Then, new explicit towers were found only in 2006 by Martín and Ramos Batista, all of them with genus one. For genus two, the first such towers were constructed in 2010. Back to 2009, implicit towers of arbitrary genera were found in An end-to-end construction for singly periodic minimal surfaces. In our present work we obtain explicit minimal Scherk saddle towers, for any given genus $2k$, $k\ge3$.

This paper deals with the determination of the absolute valued algebras with a nonzero idempotent commuting with the remaining idempotents and satisfying $x^2 x = x x^2 $ for every $x$. We prove that, in addition to the absolute valued algebras $\mathbb R $, $\mathbb C $, $\mathbb H $, or $\mathbb O $ of the reals, complexes, division real quaternions or division real octonions, one such absolute valued algebra $A$ can also be isometrically isomorphic to some of the absolute valued algebras $\overset{\star}{\mathbb C}$, $\overset{\star}{\mathbb H}$, or $\overset{\star}{\mathbb O}$, obtained from $\mathbb C $, $\mathbb H$, and $\mathbb O $ by imposing a new product defined by multiplying the conjugates of the elements. In particular, every absolute valued algebra having the above properties is finite-dimensional. This generalizes some well known theorems of Albert, Urbanik and Wright, and El-Mallah.

A comparison principle for the subdiffusive $p$-Laplacian in a possibly non-smooth and unbounded open set is proved. The result requires that the involved sub and supersolution are positive, and the ratio of the former to the latter is bounded. As an application, constrained radial symmetry for overdetermined problems is obtained. More precisely, both Dirichlet and Neumann conditions are prescribed on the boundary of a bounded open set, and the Neumann condition depends on the distance from the origin. The domain of the problem, unknown at the beginning, turns out to be a ball centered at the origin if a positive solution exists. Counterexamples are also discussed.

We give a geometric interpretation of Broussous-Grabitz embedding types. We fix a central division algebra $D$ of finite index over a non-Archimedean local field $F$ and a positive integer $m$. Further we fix a hereditary order $\mathfrak{a}$ of $\operatorname{M}_m(D)$ and an unramified field extension $E|F$ in $\operatorname{M}_m(D)$ which is embeddable in $D$ and which normalizes $\mathfrak{a}$. Such a pair $(E,\mathfrak{a})$ is called an embedding. The embedding types classify the $\operatorname{GL}_m(D)$-conjugation classes of these embeddings. Such a type is a class of matrices with non-negative integer entries. We give a formula which allows us to recover the embedding type of $(E,\mathfrak{a})$ from the simplicial type of the image of the barycenter of $\mathfrak{a}$ under the canonical isomorphism, from the set of $E^\times$-fixed points of the reduced building of $\operatorname{GL}_m(D)$ to the reduced building of the centralizer of $E^\times$ in $\operatorname{GL}_m(D)$. Conversely the formula allows to calculate the simplicial type up to cyclic permutation of the Coxeter diagram.

We show that, given a set $E\subset{\mathbb R}^{n+1}$ with finite $n$-Hausdorff measure${\mathcal H}^n$, if the $n$-dimensional Riesz transform

$$R_{{\mathcal H}^n{\lfloor} E} f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}}\,f(y)\,{\mathcal H}^n(y)$$

is bounded in $L^2({\mathcal H}^n{\lfloor} E)$, then $E$ is $n$-rectifiable. From this result we deduce that a compact set $E\subset{\mathbb R}^{n+1}$ with ${\mathcal H}^n(E)<\infty$ is removable for Lipschitz harmonic functions if and only if it is purely $n$-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions.