3,507 Works

Swarming robots

Magnus Egerstedt
When lots of robots come together to form shapes, spread in an area, or move in one direction, their motion has to be planned carefully. We discuss how mathematicians devise strategies to help swarms of robots behave like an experienced, coordinated team.

Random sampling of domino and lozenge tilings

Éric Fusy
A grid region is (roughly speaking) a collection of ``elementary cells'' (squares, for example, or triangles) in the plane. One can ``tile'' these grid regions by arranging the cells in pairs. In this snapshot we review different strategies to generate random tilings of large grid regions in the plane. This makes it possible to observe the behaviour of large random tilings, in particular the occurrence of boundary phenomena that have been the subject of intensive...

Das Problem der Kugelpackung

Maria Dostert, Stefan Krupp & Jan Hendrik Rolfes
Wie würdest du Tennisbälle oder Orangen stapeln? Oder allgemeiner formuliert: Wie dicht lassen sich identische 3-dimensionale Objekte überschneidungsfrei anordnen? Das Problem, welches auch Anwendungen in der digitalen Kommunikation hat, hört sich einfach an, ist jedoch für Kugeln in höheren Dimensionen noch immer ungelöst. Sogar die Berechnung guter Näherungslösungen ist für die meisten Dimensionen schwierig.

High performance computing on smartphones

Anthony T. Patera & Karsten Urban
Nowadays there is a strong demand to simulate even real-world engineering problems on small computing devices with very limited capacity, such as a smartphone. We explain, using a concrete example, how we can obtain a reduction in complexity – to enable such computations – using mathematical methods.

Polyhedra and commensurability

Rafael Guglielmetti & Matthieu Jacquemet
This snapshot introduces the notion of commensurability of polyhedra. At its bottom, this concept can be developed from constructions with paper, scissors, and glue. Starting with an elementary example, we formalize it subsequently. Finally, we discuss intriguing connections with other fields of mathematics.

How to choose a winner: the mathematics of social choice

Victoria Powers
Suppose a group of individuals wish to choose among several options, for example electing one of several candidates to a political office or choosing the best contestant in a skating competition. The group might ask: what is the best method for choosing a winner, in the sense that it best reflects the individual preferences of the group members? We will see some examples showing that many voting methods in use around the world can lead...

Special values of zeta functions and areas of triangles

Jürg Kramer & Anna-Maria Von Pippich
In this snapshot we give a glimpse of the interplay of special values of zeta functions and volumes of triangles. Special values of zeta functions and their generalizations arise in the computation of volumes of moduli spaces (for example of Abelian varieties) and their universal spaces. As a first example, we compute the special value of the Riemann zeta function at s=2 and give its interpretation as the volume of the moduli space of elliptic...

Modelling the spread of brain tumours

Amanda Swan & Albert Murtha
The study of mathematical biology attempts to use mathematical models to draw useful conclusions about biological systems. Here, we consider the modelling of brain tumour spread with the ultimate goal of improving treatment outcomes.

Quantum diffusion

Antti Knowles
If you place a drop of ink into a glass of water, the ink will slowly dissipate into the surrounding water until it is perfectly mixed. If you record your experiment with a camera and play the film backwards, you will see something that is never observed in the real world. Such diffusive and irreversible behaviour is ubiquitous in nature. Nevertheless, the fundamental equations that describe the motion of individual particles – Newton's and Schrödinger's...

The ternary Goldbach problem

Harald Helfgott
Leonhard Euler (1707–1783) – one of the greatest mathematicians of the eighteenth century and of all times – often corresponded with a friend of his, Christian Goldbach (1690–1764), an amateur and polymath who lived and worked in Russia, just like Euler himself. In a letter written in June 1742, Goldbach made a conjecture – that is, an educated guess – on prime numbers: "Es scheinet wenigstens, dass eine jede Zahl, die größer ist als 2,...

Arrangements of lines

Brian Harbourne & Tomasz Szemberg
We discuss certain open problems in the context of arrangements of lines in the plane.

Drugs, herbicides, and numerical simulation

Peter Benner, Hermann Mena & René Schneider
The Colombian government sprays coca fields with herbicides in an effort to reduce drug production. Spray drifts at the Ecuador-Colombia border became an international issue. We developed a mathematical model for the herbicide aerial spray drift, enabling simulations of the phenomenon.

Wie man einen Sieger wählt: Die Mathematik der Sozialwahl

Victoria Powers
Angenommen, eine Gruppe von Einzelpersonen möchte unter verschiedenen Optionen wählen, zum Beispiel einen von mehreren Kandidaten für ein politisches Amt oder den besten Teilnehmer einer Eiskunstlaufmeisterschaft. Man könnte fragen: Was ist die beste Methode, einen Sieger in dem Sinne zu wählen, dass er die individuellen Präferenzen der Gruppenmitglieder am besten widerspiegelt? Wir werden anhand einiger Beispiele sehen, dass viele Wahlverfahren, die weltweit in Gebrauch sind, zu Paradoxa und nachgerade schlechten Ergebnissen führen können, und wir...

Profinite groups

Laurent Bartholdi
Profinite objects are mathematical constructions used to collect, in a uniform manner, facts about infinitely many finite objects. We shall review recent progress in the theory of profinite groups, due to Nikolov and Segal, and its implications for finite groups.

Eine visuelle Analyse der Sterblichkeit männlicher Spanier

J.S. Marron
Die statistische Visualisierung benutzt graphische Methoden um Erkenntnisse aus Daten zu gewinnen. Wir zeigen wie mit dem Verfahren der Hauptkomponentenanalyse die Sterblichkeit in Spanien im Laufe der letzten hundert Jahre analysiert werden kann. Diese Datenzerlegung zeigt sowohl erwartete geschichtliche Ereignisse auf, als auch einige, teilweise überraschende Entwicklungen der Sterblichkeit im Laufe der Zeit.

Holomorphic automorphic forms and cohomology

Roelof W. Bruggeman, Yŏng-Ju Ch'oe & Nikolaos Diamantis
We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. We use Knopp’s generalization of this integral to real weights, and apply it to complex weights that are not an integer at least 2. We show that for these weights the generalized Eichler integral gives an injection into the first cohomology group with...

A construction of hyperbolic coxeter groups

Damian Osajda
We give a simple construction of Gromov hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension. Our construction provides new examples of such groups. Using this one can construct e.g. new groups having some interesting asphericity properties.

Quantities that frequency-dependent selection maximizes

Carlo Matessi & Kristian Schneider
We consider a model of frequency-dependent selection, to which we refer as the Wildcard Model, that accommodates as particular cases a number of diverse models of biologically specific situations. Two very different particular models (Lessard, 1984; Bürger, 2005; Schneider, 2006), subsumed by the Wildcard Model, have been shown in the past to have a Lyapunov functions (LF) under appropriate genetic assumptions. We show that the Wildcard Model: (i) in continuous time is a generalized gradient...

Simple graded commutative algebras

Sophie Morier-Genoud & Valentin Ovsienko
We study the notion of $\Gamma$-graded commutative algebra for an arbitrary abelian group $\Gamma$. The main examples are the Clifford algebras already treated in [2]. We prove that the Clifford algebras are the only simple finite-dimensional associative graded commutative algebras over $\mathbb{R}$ or $\mathbb{C}$. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.

Some Combinatorial Identities Related to Commuting Varieties and Hilbert Schemes

Gwyn Bellamy & Victor Ginzburg
In this article we explore some of the combinatorial consequences of recent results relating the isospectral commuting variety and the Hilbert scheme of points in the plane.

Cryptanalysis of Public-key Cryptosystems Based on Algebraic Geometry Codes

Irene Márquez-Corbella, Edgar Martínez-Moro & Ruud Pellikaan
This paper addresses the question of retrieving the triple $(\mathcal{X},\mathcal{P},\mathcal{E})$ from the algebraic geometry code $\mathcal{C}_L(\mathcal{X},\mathcal{P},\mathcal{E})$, where $\mathcal{X}$ is an algebraic curve over the finite field $\mathbb{F}_q, \mathcal{P}$ is an $n$-tuple of $\mathbb{F}_q$-rational points on $\mathcal{X}$ and $E$ is a divisor on $\mathcal{X}$. If deg($E$) $\geq 2g + 1$ where $g$ is the genus of $\mathcal{X}$, then there is an embedding of $\mathcal{X}$ onto $\mathcal{Y}$ in the projective space of the linear series of the...

Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, Adrian R. D. Mathias & Jiří Rosický
We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $\mathcal{S}$ of morphisms in an accessible category $\mathcal{C}$, the orthogonal class of objects...

Composition of Irreducible Morphisms in Quasi-Tubes

Claudia Chaio & Piotr Malicki
We study the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of the Auslander-Reiten quivers of artin algebras $A$ in relation with the powers of the radical of their module category mod $A$.

The Index of Singular Zeros of Harmonic Mappings of Anti-Analytic Degree One

Robet Luce & Olivier Sète
We study harmonic mappings of the form $f(z) = h(z) - \overline{z}$, where $h$ is an analytic function. In particular we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of $f$, where the Jacobian of $f$ is non-vanishing, it is known that this index has similar properties as the classical multiplicity of zeros of analytic functions. Little is known about the index of zeros on...

Virtual Polytopes

Gaiane Panina & Ileana Streinu
Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as the Grothendick group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. A selection of applications demonstrates their versatility.

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