### Modeling communication and movement: from cells to animals and humans

Raluca Eftimie
Communication forms the basis of biological interactions. While the use of a single communication mechanism (for example visual communication) by a species is quite well understood, in nature the majority of species communicate via multiple mechanisms. Here, I review some mathematical results on the unexpected behaviors that can be observed in biological aggregations where individuals interact with each other via multiple communication mechanisms.

### Billard und ebene Flächen

Diana Davis
Billard, die Zick-Zack-Bewegungen eines Balls auf einem Tisch, ist ein reichhaltiges Feld gegenwärtiger mathematischer Forschung. In diesem Artikel diskutieren wir Fragen und Antworten zum Thema Billard, und zu dem damit verwandten Thema ebener Flächen.

### Drogen, Herbizide und numerische Simulation

Peter Benner, Hermann Mena & René Schneider
Die kolumbianische Regierung versprüht Unkrautbekämpfungsmittel (Herbizide) über Coca-Feldern, um die Drogenproduktion im Land zu reduzieren. Sprühverwehungen entlang der Grenze Kolumbiens zu Ecuador wurden zu einem internationalen Streitfall. Wir haben ein mathematisches Modell für die Ausbreitung der Chemikalien in der Luft entwickelt, das es uns ermöglicht, das Phänomen am Computer zu simulieren.

### Towards a Mathematical Theory of Turbulence in Fluids

Jacob Bedrossian
Fluid mechanics is the theory of how liquids and gases move around. For the most part, the basic physics are well understood and the mathematical models look relatively simple. Despite this, fluids display a dazzling mystery to their motion. The random-looking, chaotic behavior of fluids is known as turbulence, and it lies far beyond our mathematical understanding, despite a century of intense research.

### Cocharacter-closure and spherical buildings

Michael Bate, Sebastian Herpel, Benjamin Martin & Gerhard Röhrle
Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion of cocharacter-closed $G(k)$-orbits in $V$. In earlier work we used a rationality condition on the point stabilizer of a $G$-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponing $G(k)$-orbit in $V$. In the present paper we employ building-theoric techniques to...

### Proof mining in metric fixed point theory and ergodic theory

Laurenţiu Leuştean
In this survey we present some recent applications of proof mining to the fixed point theory of (asymptotically) nonexpansive mappings and to the metastability (in the sense of Terence Tao) of ergodic averages in uniformly convex Banach spaces.

### Rational Approximation on Products of Planar Domains

Richard M. Aron, Paul Montpetit Gauthier, Manuel Maestre, Vassili Nestoridis & Javier Falcó
We consider $A(\Omega)$, the Banach space of functions $f$ from $\overline{\Omega}=\prod_{i \in I} \overline{U_i}$ to $\mathbb{C}$ that are continuous with respect to the product topology and separately holomorphic, where $I$ is an arbitrary set and $U_i$ are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off $\overline{U_i}$ are uniformly dense in $A(\Omega)$. This generalizes previous results where $U_i=\mathbb{D}$ is...

### G-complete reducibility in non-connected groups

Michael Bate, Sebastian Herpel, Benjamin Martin & Gerhard Röhrle
In this paper we present an algorithm for determining whether a subgroup $H$ of a non-connected reductive group $G$ is $G$-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of $G^0$ is $G^0$-cr. This essentially reduces the problem of determining $G$-complete reducibility to the connected case.

### Late-Time Behaviour of Israel Particles in a FLRW Spacetime with Λ>0

Ho Lee & Ernesto Nungesser
In this paper we study the relativistic Boltzmann equation in a spatially flat FLRW space-time. We consider Israel particles, which are the relativistic counterpart of the Maxwellian particles, and obtain global-in-time existence and the asymptotic behaviour of solutions. The main argument of the paper is to use the energy method of Guo, and we observe that the method can be applied to study small solutions in a cosmological case. It is the first result of...

### Spectral triples and finite summability on cuntz-krieger algebras

Magnus Goffeng & Bram Mesland
We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space" coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are...

### On Siegel modular forms of level p and their properties mod p

Siegfried Böcherer & Shoyu Nagaoka
Using theta series we construct Siegel modular forms of level p which behave well modulo $p$ in all cusps. This construction allows us to show (under a mild condition) that all Siegel modular forms of level p and weight 2 are congruent mod $p$ to level one modular forms of weight $p+1$; in particular, this is true for Yoshidal lifts of level $p$.

### Fibonacci-like unimodal inverse limit spaces

H. Bruin & S. Štimac
We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allows us to introduce certain chains that enable a more detailed analysis of symmetric arcs within this space than is possible in the general case. We show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs. This leads to simplification of some existing results, including the Ingram Conjecture for Fibonacci-like...

### Polynomiality, wall crossings and tropical geometry of rational double hurwitz cycles

Aaron Bertram, Renzo Cavalieri & Hannah Markwig
We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and “modular” description. A main goal of this paper is to simultaneously carry out this investigation...

### Torsion-free Covers of Solvable Minimax Groups

Peter H. Kropholler & Karl Lorensen
We prove that every finitely generated solvable minimax group can be realized as a quotient of a torsion-free solvable minimax group. This result has an application to the investigation of random walks on finitely generated solvable minimax groups. Our methods also allow us to completely characterize the solvable minimax groups that are homomorphic images of torsion-free solvable minimax groups.

### The Initial and Terminal Cluster Sets of an Analytic Curve

Paul Montpetit Gauthier
For an analytic curve $\gamma : (a,b) \to \mathbb{C}$, the set of values approaches by $\gamma(t)$, as $t ↘a$ and as $t↗b$ can be any two continuua of $\mathbb{C} \cup \{\infty\}$.

### An Identification Therorem for PSU6(2) and its Automorphism Groups

Chris Parker & Gernot Stroth
We identify the groups PSU6(2), PSU6(2):2, PSU6(2):3 and Aut(PSU6(2)) from the structure of the centralizer of an element of order 3.

### Random dynamics of transcendental functions

Volker Mayer & Mariusz Urbański
This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order and whose derivative satisfies some growth condition at infinity. This class contains most of the classical families of transcendental functions and goes much beyond. Based on uniform versions of Nevanlinna’s value distribution theory we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with...

### Mesh Ratios for Best-Packing and Limits of Minimal Energy Configurations

A. V. Bondarenko, Douglas P. Hardin & Edward B. Saff
For $N$-point best-packing configurations $\omega_N$ on a compact metric space $(A, \rho)$, we obtain estimates for the mesh-separation ratio $\gamma(\rho_N , A)$, which is the quotient of the covering radius of $\omega_N$ relative to $A$ and the minimum pairwise distance between points in $\omega_N$ . For best-packing configurations $\omega_N$ that arise as limits of minimal Riesz $s$-energy configurations as $s \to \infty$, we prove that $\gamma(\omega_N , A) ≤ 1$ and this bound can be...

### Shape Theory and Extensions of C*-Algebras

Vladimir M. Manujlov & Klaus Thomsen
Let A, A' be separable $C^*$-algebras, $B$ a stable $\sigma$-unital $C^*$-algebra. Our main result is the construction of the pairing $[[A', A]] \times Ext^{-1/2}(A,B) \to Ext^{-1/2}(A',B)$, where $[[A', A]]$ denotes the set of homotopy classes of asymptotic homomorphisms from $A'$ to $A$ and $Ext^{-1/2}(A,B)$ is the group of semi-invertible extensions of $A$ by $B$. Assume that all extensions of $A$ by $B$ are semi-invertible. Then this pairing allows us to give a condition on $A'$...

### Boundary Representations of Operator Spaces, and Compact Rectangular Matrix Convex Sets

Adam H. Fuller, Michael Hartz & Martino Lupini
We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed...

Stefan Suhr & Kai Zehmisch
We show that the Lagrangian of classical mechanics on a Riemannian manifold of bounded geometry carries a periodic solution of motion with prescribed energy, provided the potential satis es an asymptotic growth condition, changes sign, and the negative set of the potential is non-trivial in the relative homology.

### Steinberg groups for Jordan pairs

Ottmar Loos & Erhard Neher
We introduce categories of groups with commutator relations with respect to root groups and Weyl elements, permuting the root groups. This allows us to view the classical Steinberg groups, for example the Steinberg group of a ring, as an initial object in an appropriate category. The general framework is then specialized to groups associated to Jordan pairs, first for arbitrary Jordan pairs and then later for Jordan pairs with Peirce gradings or more general gradings...

### Classification of idempotent states on the compact quantum groups Uq(2), SUq(2) and SOq(3)

Uwe Franz, Adam Skalski & Reiji Tomatsu
We give a simple characterisation of those idempotent states on compact quantum groups which arise as Haar states on quantum subgroups, show that all idempotent states on quantum groups $U_q(2)$, $SU_q(2)$, and $SO_q(3) (q \in (-1,0)\cup (0,1])$ arise in this manner and list the idempotent states on compact quantum semigroups $U_o(2)$, $SU_o(2)$, and $SO_o(3)$. In the Appendix we provide a simple proof of coamenability of the deformations of classical compact Lie groups.

### A new counting function for the zeros of holomorphic curves

J. M. Anderson & Aimo Hinkkanen
Let $f_1,..., f_p$ be entire functions that do not all vanish at any point, so that $(f_1,..., f_p)$ is a holomorphic curve in $\mathbb{CP}^{p-1}$. We introduce a new and more careful notion of counting the order of the zero of a linear combination of the functions $f_1,..., f_p$ at any point where such a linear combination vanishes, and, if all the $f_1,..., f_p$ are polynomials, also at infinity. This enables us to formulate an inequality,...

### New representations of matroids and generalizations

Zur Izhakian & John L. Rhodes
We extend the notion of matroid representations by matrices over fields by considering new representations of matroids by matrices over finite semirings, more precisely over the boolean and the superboolean semirings. This idea of representations is naturally generalized to include hereditary collections (also known as abstract simplicial complexes). We show that a matroid that can be directly decomposed as matroids, each of which is representable over a field, has a boolean representation, and more generally...

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