1 - Oberwolfach Preprints (OWP)
http://publications.mfo.de/handle/mfo/19
The Oberwolfach Preprints (OWP) mainly contain research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs program and the Oberwolfach Leibniz Fellows, but this can also include an Oberwolfach Lecture, for example.Fri, 15 Feb 2019 22:46:48 GMT2019-02-15T22:46:48ZApplications 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.Cataland: 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.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.Criteria for Algebraicity of Analytic Functions
http://publications.mfo.de/handle/mfo/1392
Criteria for Algebraicity of Analytic Functions
Bochnak, Jacek; Gwoździewicz, Janusz; Kucharz, Wojciech
We consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing trough a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.
Mon, 12 Nov 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13922018-11-12T00:00:00ZBochnak, JacekGwoździewicz, JanuszKucharz, WojciechWe consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing trough a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.Global Variants of Hartogs' Theorem
http://publications.mfo.de/handle/mfo/1391
Global Variants of Hartogs' Theorem
Bochnak, Jacek; Kucharz, Wojciech
Hartogs' theorem asserts that a separately holomorphic function, defined on an open subset of $\mathbb{C}^n$, is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.
Tue, 06 Nov 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13912018-11-06T00:00:00ZBochnak, JacekKucharz, WojciechHartogs' theorem asserts that a separately holomorphic function, defined on an open subset of $\mathbb{C}^n$, is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.Real Analyticity is Concentrated in Dimension 2
http://publications.mfo.de/handle/mfo/1390
Real Analyticity is Concentrated in Dimension 2
Bochnak, Jacek; Kucharz, Wojciech
We prove that a real-valued function on a real analytic manifold is analytic whenever all its restrictions to $2$-dimensional analytic submanifolds are analytic functions. We also obtain analogous results in the framework of Nash manifolds and nonsingular real algebraic sets. These results can be regarded as substitutes in the real case for the classical theorem of Hartogs, asserting that a complex-valued function defined on an open subset of $C^n$ is holomorphic if it is holomorphic with respect to each variable separately. In the proofs we use methods of real algebraic geometry even though the initial problem is purely analytic.
Mon, 05 Nov 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13902018-11-05T00:00:00ZBochnak, JacekKucharz, WojciechWe prove that a real-valued function on a real analytic manifold is analytic whenever all its restrictions to $2$-dimensional analytic submanifolds are analytic functions. We also obtain analogous results in the framework of Nash manifolds and nonsingular real algebraic sets. These results can be regarded as substitutes in the real case for the classical theorem of Hartogs, asserting that a complex-valued function defined on an open subset of $C^n$ is holomorphic if it is holomorphic with respect to each variable separately. In the proofs we use methods of real algebraic geometry even though the initial problem is purely analytic.Computing Congruence Quotients of Zariski Dense Subgroups
http://publications.mfo.de/handle/mfo/1389
Computing Congruence Quotients of Zariski Dense Subgroups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n \geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n, \mathbb{Q})$ for $n > 2$.
Fri, 26 Oct 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13892018-10-26T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderWe obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n \geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n, \mathbb{Q})$ for $n > 2$.