1.
The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions - Nazarov, Fedor; Tolsa, Xavier; Volberg, Alexander
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.

2.
Embeddings of local fields in simple algebras and simplicial structures - Skodlerack, Daniel
We give a geometric interpretation of Broussous-Grabitz embedding types. We fix a central division algebra $D$ of finite index over a non-Archime\-dean 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...

3.
Comparison principle and constrained radial symmetry for the subdiffusive \boldmath$p$-Laplacian - Greco, Antonio
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...

4.
Third-power associative absolute valued algebras with a nonzero idempotent commuting with all idempotents - Mira, José Antonio Cuenca
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...

5.
Explicit minimal Scherk saddle towers of arbitrary even genera in \boldmath$\mathbb{R}^3$ - Hancco, A. J. Yucra; Lobos, G. A.; Batista, V. Ramos
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...

6.
On restricted weak-type constants of Fourier multipliers - Oseękowski, Adam
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.

7.
On separated Carleson sequences in the unit disc - Amar, Eric
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"...

8.
Smoothing properties of the discrete fractional maximal operator on Besov and Triebel-Lizorkin spaces - Heikkinen, Toni; Tuominen, Heli
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.

9.
Atomic decomposition of real-variable type for Bergman spaces in the unit ball of \boldmath$\mathbb{C}^n$ - Chen, Zeqian; Ouyang, Wei
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...

10.
Stable sampling and Fourier multipliers - Matei, Basarab; Meyer, Yves; Ortega-Cerdà, Joaquim
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.

11.
Groups with normality conditions for subgroups of infinite rank - De Falco, Maria; de Giovanni, Francesco; Musella, Carmela
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.

12.
Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field - Szpiro, Lucien; Tepper, Michael; Williams, Phillip
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...

13.
A PDE approach of inflammatory phase dynamics in diabetic wounds - Cónsul, N.; Oliva, S. M.; Pellicer, M.
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.

14.
Bautin ideals and Taylor domination - Yomdin, Y.
We consider families of analytic functions with Taylor coefficients\guio{polynomials} in the parameter $\lambda$:
$f_\lambda(z)=\sum_{k=0}^\infty a_k(\lambda) z^k$, $a_k \in {\mathbb C}[\lambda]$.
Let $R(\lambda)$ be the radius of convergence of $f_\lambda$. The "Taylor domination'' property for this family is the
inequality of the following form: for certain fixed~$N$ and $C$ and for each $k\geq N+1$ and $\lambda,
$|a_{k}(\lambda)|R^{k}(\lambda)\leq C \max_{i=0,\dotsc,N} |a_{i}(\lambda)|R^{i}(\lambda).$
¶ Taylor domination property implies a uniform in $\lambda$ bound on the number of zeroes of~$f_\lambda$. In this
paper we discuss some known and new results providing Taylor domination (usually, in a smaller disk) via the Bautin
approach. In particular, we give new conditions on $f_\lambda$ which imply...

15.
Uniform methods to establish Poincar type linearization theorems - Wu, Hao
We find a uniform method to establish Poincare type linearization theorems for regular systems including classical autonomous, random and almost periodic
ones via modified majorant norm methods.

16.
Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields - Schlomiuk, Dana
We describe the origin and evolution of ideas on topological and polynomial invariants and their interaction, in problems of classification of polynomial
vector fields. The concept of moduli space is discussed in the last section and we
indicate its value in understanding the dynamics of families of such systems. Our
interest here is in the concepts and the way they interact in the process of topologically classifying polynomial vector fields. We survey the literature giving an ample
list of references and we illustrate the ideas on the testing ground of families of quadratic vector fields. In particular, the role of polynomial invariants is illustrated...

18.
A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations - Oliva, Waldyr M.
For stably dissipative Lotka-Volterra equations the dynamics on the attractor are Hamiltonian and we argue that complex dynamics can occur. We also
present examples and properties of some infinite dimensional Volterra systems with
applications related with stably dissipative Lotka-Volterra equations. We finish by
mentioning recent contributions on the subject.

19.
On nonsmooth perturbations of nondegenerate planar centers - Novaes, Douglas D.
We provide sufficient conditions for the existence of limit cycles of non-smooth perturbed planar centers when the discontinuity set is a union of regular
curves. We introduce a mechanism which allows us to deal with such systems. The
main tool used in this paper is the averaging method. Some applications are explained
with careful details.

20.
Invariant tori in the lunar problem - Meyer, Kenneth R.; Palacian, Jesus F.; Yanguas, Patricia
Theorems on the existence of invariant KAM tori are established for perturbations of Hamiltonian systems which are circle bundle flows. By averaging the perturbation over the bundle flow one obtains a Hamiltonian
system on the orbit (quotient) space by a classical theorem of Reeb. A non-degenerate critical point of the system on the orbit space gives rise to a family of periodic solutions of the perturbed
system. Conditions on the critical points are given which insure KAM tori for the perturbed flow.
¶ These general theorems are used to show that the near circular periodic solutions
of the planar lunar problem are orbitally stable...