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http://publications.mfo.de:8080
Snake graphs, perfect matchings and continued fractions
http://publications.mfo.de/handle/mfo/1405
Snake graphs, perfect matchings and continued fractions
Schiffler, Ralf
A continued fraction is a way of representing a real
number by a sequence of integers. We present a new
way to think about these continued fractions using
snake graphs, which are sequences of squares in the
plane. You start with one square, add another to
the right or to the top, then another to the right or
the top of the previous one, and so on. Each continued
fraction corresponds to a snake graph and vice
versa, via “perfect matchings” of the snake graph. We
explain what this means and why a mathematician
would call this a combinatorial realization of continued
fractions.
Wed, 13 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14052019-02-13T00:00:00ZSchiffler, RalfA continued fraction is a way of representing a real
number by a sequence of integers. We present a new
way to think about these continued fractions using
snake graphs, which are sequences of squares in the
plane. You start with one square, add another to
the right or to the top, then another to the right or
the top of the previous one, and so on. Each continued
fraction corresponds to a snake graph and vice
versa, via “perfect matchings” of the snake graph. We
explain what this means and why a mathematician
would call this a combinatorial realization of continued
fractions.Applications of BV Type Spaces
http://publications.mfo.de/handle/mfo/1403
Applications of BV Type Spaces
Appell, Jürgen; Bugajewska, Daria; Kasprzak, Piotr; Merentes, Nelson; Reinwand, Simon; Sánchez, José Luis
Wed, 13 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14032019-02-13T00:00:00ZAppell, JürgenBugajewska, DariaKasprzak, PiotrMerentes, NelsonReinwand, SimonSánchez, José LuisA Function Algebra Providing New Mergelyan Type Theorems in Several Complex Variables
http://publications.mfo.de/handle/mfo/1401
A Function Algebra Providing New Mergelyan Type Theorems in Several Complex Variables
Falcó, Javier; Gauthier, Paul Montpetit; Manolaki, Myrto; Nestoridis, Vassili
For compact sets $K\subset \mathbb C^{d}$, we introduce a subalgebra $A_{D}(K)$ of $A(K)$, which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.
Mon, 11 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14012019-02-11T00:00:00ZFalcó, JavierGauthier, Paul MontpetitManolaki, MyrtoNestoridis, VassiliFor compact sets $K\subset \mathbb C^{d}$, we introduce a subalgebra $A_{D}(K)$ of $A(K)$, which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.Mixed volumes and mixed integrals
http://publications.mfo.de/handle/mfo/1400
Mixed volumes and mixed integrals
Rotem, Liran
In recent years, mathematicians have developed new
approaches to study convex sets: instead of considering
convex sets themselves, they explore certain functions
or measures that are related to them. Problems
from convex geometry become thereby accessible to
analytic and probabilistic tools, and we can use these
tools to make progress on very difficult open problems.
We discuss in this Snapshot such a functional extension
of some “volumes” which measure how “big”
a set is. We recall the construction of “intrinsic volumes”,
discuss the fundamental inequalities between
them, and explain the functional extensions of these
results.
Sat, 29 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14002018-12-29T00:00:00ZRotem, LiranIn recent years, mathematicians have developed new
approaches to study convex sets: instead of considering
convex sets themselves, they explore certain functions
or measures that are related to them. Problems
from convex geometry become thereby accessible to
analytic and probabilistic tools, and we can use these
tools to make progress on very difficult open problems.
We discuss in this Snapshot such a functional extension
of some “volumes” which measure how “big”
a set is. We recall the construction of “intrinsic volumes”,
discuss the fundamental inequalities between
them, and explain the functional extensions of these
results.Jahresbericht | Annual Report - 2017
http://publications.mfo.de/handle/mfo/1399
Jahresbericht | Annual Report - 2017
Mathematisches Forschungsinstitut Oberwolfach
Tue, 01 Jan 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13992019-01-01T00:00:00ZMathematisches Forschungsinstitut OberwolfachCataland: Why the Fuß?
http://publications.mfo.de/handle/mfo/1398
Cataland: Why the Fuß?
Stump, Christian; Thomas, Hugh; Williams, Nathan
The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuß-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuß-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.
Mon, 21 Jan 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13982019-01-21T00:00:00ZStump, ChristianThomas, HughWilliams, NathanThe three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuß-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuß-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.Estimating the volume of a convex body
http://publications.mfo.de/handle/mfo/1396
Estimating the volume of a convex body
Baldin, Nicolai
Sometimes the volume of a convex body needs to
be estimated, if we cannot calculate it analytically.
We explain how statistics can be used not only to
approximate the volume of the convex body, but also
its shape.
Sun, 30 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13962018-12-30T00:00:00ZBaldin, NicolaiSometimes the volume of a convex body needs to
be estimated, if we cannot calculate it analytically.
We explain how statistics can be used not only to
approximate the volume of the convex body, but also
its shape.The Tutte Polynomial of Ideal Arrangements
http://publications.mfo.de/handle/mfo/1395
The Tutte Polynomial of Ideal Arrangements
Randriamaro, Hery
The Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo.
Fri, 21 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13952018-12-21T00:00:00ZRandriamaro, HeryThe Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo.Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D
http://publications.mfo.de/handle/mfo/1394
Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D
Beckus, Siegfried; Bellissard, Jean; De Nittis, Giuseppe
The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.
Mon, 17 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13942018-12-17T00:00:00ZBeckus, SiegfriedBellissard, JeanDe Nittis, GiuseppeThe existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.Sur le Minimum de la Fonction de Brjuno
http://publications.mfo.de/handle/mfo/1393
Sur le Minimum de la Fonction de Brjuno
Balazard, Michel; Martin, Bruno
The Brjuno function attains a strict global minimum at the golden section.
Tue, 11 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13932018-12-11T00:00:00ZBalazard, MichelMartin, BrunoThe Brjuno function attains a strict global minimum at the golden section.