3,540 Works

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.

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...

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.

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...

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.

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.

A study on gradient blow up for viscosity solutions of fully nonlinear, uniformly elliptic equations

Bernhard Kawohl & Nickolai Kutev
We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we derive results on local and global Lipschitz continuity of continuous viscosity solutions. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition...

Realizing Spaces as Classifying Spaces

Gregory Lupton & Samuel Bruce Smith
Which spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of $\mathbb{C}P^2$ and $S^4$ are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.

A categorical model for the virtual braid group

Louis H. Kauffman & Sofia Lambropoulou

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.

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.

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.

Categoric Aspects of Authentication

Jeroen Schillewaert & Koen Thas

Strongly Consistent Density Estimation of Regression Redidual

László Györfi & Harro Walk
Consider the regression problem with a response variable $Y$ and with a $d$-dimensional feature vector $X$. For the regression function $m(x) = \mathbb{E}\{Y|X = x\}$, this paper investigates methods for estimating the density of the residual $Y -m(X)$ from independent and identically distributed data. For heteroscedastic regression, we prove the strong universal (density-free) $L_1$-consistency of a recursive and a nonrecursive kernel density estimate based on a regression estimate.

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