On Rado conditions for nonlinear Diophantine equations

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado's characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado's theorem. We verify that such a conjecture holds for the equations x2−xy+ax+by+cz=0 and x2−y2+ax+by+cz=0 for a,b,c∈Z such that abc=0 or a+b+c=0. To deal with these equations, we establish new results concerning the partition regularity of polynomial configurations in Z such as x,x+y,xy+x+y, building on the recent result on the partition regularity of x,x+y,x

Upper density problems in infinite Ramsey theory

We consider the following question in infinite Ramsey theory, introduced by Erdős and Galvin [EG93] in a particular case and by DeBiasio and McKenney [DM19] in a more general setting. Let H be a countably infinite graph. If the edges of the complete graph on the natural numbers are colored red or blue, what is the maximum value of λ such that we are guaranteed to find a monochromatic copy of H whose vertex set has upper density at least λ? We call this value the Ramsey density of H. The problem of determining the Ramsey density of the infinite path was first studied by Erdős and Galvin, and was recently solved by Corsten, DeBiasio, Lang and the author [CDLL19]. In this thesis we study the problem of determining the Ramsey density of arbitrary graphs H. On an intuitive level, we show that three properties of a graph H have an effect on the Ramsey density: the chromatic number, the number of components, and the expansion of its independent sets. We deduce the exact value of the Ramsey density for a wide variety of graphs, including all locally finite forests, bipartite factors, clique factors and odd cycle factors. We also determine the value of the Ramsey density of all locally finite graphs, up to a factor of 2. We also study a list coloring variant of the same problem. We show that there exists a way of assigning a list of size two to every edge in the complete graph on N such that, in every list coloring, there are monochromatic paths with density arbitrarily close to 1.Wir betrachten die folgende Fragestellung aus der Ramsey-Theorie, welche von Erdős und Galvin [EG93] in einem Spezialfall sowie von DeBiasio und McKenney [DM19] in einem allgemeineren Kontext formuliert wurde: Es sei H ein abzählbar unendlicher Graph. Welches ist der größtmögliche Wert λ, sodass wir, wenn die Kanten des vollständigen Graphen mit Knotenmenge N jeweils entweder rot oder blau gefärbt sind, stets eine einfarbige Kopie von H, dessen Knotenmenge eine obere asymptotische Dichte von mindestens λ besitzt, finden können? Wir nennen diesen Wert die Ramsey-Dichte von H. Das Problem, die Ramsey-Dichte des unendlichen Pfades zu bestimmen wurde erstmals von Erdős und Galvin untersucht und wurde vor kurzem von Corsten, DeBiasio, Lang und dem Autor [CDLL19] gelöst. Gegenstand der vorliegenden Dissertation ist die Bestimmung der Ramsey-Dichten von Graphen. Auf einer intuitiven Ebene zeigen wir, dass drei Parameter eines Graphen die Ramsey-Dichte beeinflussen: die chromatische Zahl, die Anzahl der Zusammenhangskomponenten sowie die Expansion seiner unabhängigen Mengen. Wir ermitteln die exakten Werte der Ramsey-Dichte für eine Vielzahl von Graphen, darunter alle lokal endlichen Wälder, bipartite Faktoren, Kr-Faktoren sowie Ck-Faktoren für ungerade k. Ferner bestimmen wir den Wert der Ramsey-Dichte aller lokal endlichen Graphen bis auf einen Faktor 2. Darüber hinaus untersuchen wir eine Variante des oben beschriebenen Problems für Listenfärbungen. Wir zeigen, dass es möglich ist, jeder Kante des vollständigen Graphen mit Knotenmenge N eine Liste der Größe Zwei zuzuweisen, sodass in jeder zugehörigen Listenfärbung monochromatische Pfade mit beliebig nah an 1 liegender Dichte existieren

Elliptic Partial Differential Equations in Geometric Analysis and the Calculus of Variations

The subject of this thesis is Elliptic PDE's that appear in the fields of Geometric Analysis and The Calculus of Variations, such as the Beltrami equation and its generalizations. The main results are the existence and uniqueness of solutions in function spaces such as the Sobolev-spaces, as well as regularity and properties of solutions. The thesis contains four scientific articles on the subject. The first two articles contain results on generalized Beltrami equations, where the solvability is investigated using functional analytic methods. New results for the corresponding singular integral operators are also found, such as finding the L^2-norm of the Beurling transform for the Dirichlet problem. The third and fourth papers cover properties of solutions to Euler-Lagrange and Hopf-Laplace equations for certain energy functionals. One of the main results is the generalization of the classic Radó-Kneser-Choquet theorem for the p-harmonic energy in the plane. The proof is based on a new subharmonicity result for the Jacobian of a solution, and similar other new subharmonicity results are also obtained in the thesis.Väitöskirja käsittelee matematiikassa esiintyviä osittaisdifferentiaaliyhtälöitä. Osittaisdifferentiaaliyhtälöt ovat matemaattisia yhtälöitä joissa esiintyy tuntematon funktio sekä sen (osittais-) derivaattoja. Motivaatio tutkia näitä yhtälöitä perustuu siihen, että niillä voi mallintaa esimerkiksi fysikaalisia suureita kuten lämpöä ja ääntä joiden suuruus usein riippuu myös niiden muutosnopeudessa. Tässä väitöskirjassa tutkitaan erilaisia ns. elliptisiä osittaisdifferentiaaliyhtälöitä jotka nousevat esille matematiikan osa-alueissa nimeltä Geometrinen Analyysi ja Variaatiolaskenta. Peruskysymyksiä johon väitöksessä vastataan on kyseisten yhtälöiden ratkaisujen olemassaolo, yksikäsitteisyys, ja niiden säännöllisyysominaisuudet kuten derivoituvuus. Väitöksessä osoitetaan esimerkiksi että tietynlaisilla yleistetyillä Beltrami-tyyppisillä yhtälöillä on aina olemassa yksikäsitteinen ratkaisu. Beltrami-tyyppiset yhtälöt ovat yleistyksiä klassisesta Beltrami-yhtälöstä, jonka avulla mallinnetaan matematiikassa kvasikonformikuvauksia: kuvauksia jotka jossain määrin venyttävät alkuperäistä kuvaa vain rajoitetussa määrin pienellä skaalalla. Lisäksi väitöksessä tutkitaan energiafunktionaaleja ja niiden minimoivia kuvauksia, jotka tyypillisesti totetuttavat myös sopivanlaisia osittaisdifferentiaaliyhtälöitä. Yksi päätuloksista on klassisen Radó-Kneser-Choquet'n lauseen yleistäminen ns. p-harmonisille kuvauksille. Lause näyttää, että p-harmonisen yhtälön ratkaisut ovat homeomorfisia tasoalueen sisällä mikäli niiden reuna-arvot ovat homeomorfisia. Tämän voi karkeasti hahmottaa tuloksena joka kertoo, että tietynlainen elastinen kappale ei voi venytessään lytistyä kasaan ellei lytistymistä tapahdu jo kappaleen reunalla

Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators

Topological crystalline insulators (TCIs) can host anomalous surface states which inherits the characteristics of crystalline symmetry that protects the bulk topology. Especially, the diversity of magnetic crystalline symmetries indicates the potential for novel magnetic TCIs with distinct surface characteristics. Here, we propose a topological magnetic Dirac insulator (TMDI), whose two-dimensional surface hosts fourfold-degenerate Dirac fermions protected by either the $p'_c4mm$ or $p4'g'm$ magnetic wallpaper group. The bulk topology of TMDIs is protected by diagonal mirror symmetries, which give chiral dispersion of surface Dirac fermions and mirror-protected hinge modes. We propose candidate materials for TMDIs including Nd$_4$Te$_8$Cl$_4$O$_{20}$ and DyB$_4$ based on first-principles calculations, and construct a general scheme for searching TMDIs using the space group of paramagnetic parent states. Our theoretical discovery of TMDIs will facilitate future research on magnetic TCIs and illustrate a distinct way to achieve anomalous surface states in magnetic crystals.Comment: 10+36 pages, 4+23 figures, published versio

The CECAM Electronic Structure Library and the modular software development paradigm

First-principles electronic structure calculations are very widely used thanks to the many successful software packages available. Their traditional coding paradigm is monolithic, i.e., regardless of how modular its internal structure may be, the code is built independently from others, from the compiler up, with the exception of linear-algebra and message-passing libraries. This model has been quite successful for decades. The rapid progress in methodology, however, has resulted in an ever increasing complexity of those programs, which implies a growing amount of replication in coding and in the recurrent re-engineering needed to adapt to evolving hardware architecture. The Electronic Structure Library (\esl) was initiated by CECAM (European Centre for Atomic and Molecular Calculations) to catalyze a paradigm shift away from the monolithic model and promote modularization, with the ambition to extract common tasks from electronic structure programs and redesign them as free, open-source libraries. They include ``heavy-duty'' ones with a high degree of parallelisation, and potential for adaptation to novel hardware within them, thereby separating the sophisticated computer science aspects of performance optimization and re-engineering from the computational science done by scientists when implementing new ideas. It is a community effort, undertaken by developers of various successful codes, now facing the challenges arising in the new model. This modular paradigm will improve overall coding efficiency and enable specialists (computer scientists or computational scientists) to use their skills more effectively. It will lead to a more sustainable and dynamic evolution of software as well as lower barriers to entry for new developers

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants

Resistively-limited current sheet implosions in planar anti-parallel (1D) and null-point containing (2D) magnetic field geometries

Implosive formation of current sheets is a fundamental plasma process. Previous studies focused on the early time evolution, while here our primary aim is to explore the longer-term evolution, which may be critical for determining the efficiency of energy release. To address this problem we investigate two closely-related problems, namely: (i) 1D, pinched anti-parallel magnetic fields and (ii) 2D, null point containing fields which are locally imbalanced ('null-collapse' or 'X-point collapse'). Within the framework of resistive MHD, we simulate the full nonlinear evolution through three distinct phases: the initial implosion, its eventual halting mechanism, and subsequent evolution post-halting. In a parameter study, we find the scaling with resistivity of current sheet properties at the halting time is in good agreement - in both geometries - with that inferred from a known 1D similarity solution. We find that the halting of the implosions occurs rapidly after reaching the diffusion scale by sudden Ohmic heating of the dense plasma within the current sheet, which provides a pressure gradient sufficient to oppose further collapse and decelerate the converging flow. This back-pressure grows to exceed that required for force balance and so the post-implosion evolution is characterised by the consequences of the current sheet `bouncing' outwards. These are: (i) the launching of propagating fast MHD waves (shocks) outwards and (ii) the width-wise expansion of the current sheet itself. The expansion is only observed to stall in the 2D case, where the pressurisation is relieved by outflow in the reconnection jets. In the 2D case, we quantify the maximum amount of current sheet expansion as it scales with resistivity, and analyse the structure of the reconnection region which forms post-expansion, replete with Petschek-type slow shocks and fast termination shocks

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..