MFO Repository
http://publications.mfo.de:8080
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 16 Jun 2018 16:13:04 GMT2018-06-16T16:13:04ZMFO Repositoryhttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:8080
Categorical Linearly Ordered Structures
http://publications.mfo.de/handle/mfo/1361
Categorical Linearly Ordered Structures
Downey, Rod; Melnikov, Alexander; Ng, Keng Meng
We prove that for every computable limit ordinal $\alpha$ there exists a computable linear ordering $\mathcal{A}$ which is $\Delta^0_\alpha$-categorical and $\alpha$ is smallest such, but nonetheless for every isomorphic computable copy $\mathcal{B}$ of $\mathcal{A}$ there exists a $\beta< \alpha$ such that $\mathcal{A} \cong_{\Delta^0_\beta} \mathcal{B}$. This answers a question left open in the earlier work of Downey, Igusa, and Melnikov. We also show that such examples can be found among ordered abelian groups and real-closed fields.
Thu, 26 Apr 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13612018-04-26T00:00:00ZDowney, RodMelnikov, AlexanderNg, Keng MengWe prove that for every computable limit ordinal $\alpha$ there exists a computable linear ordering $\mathcal{A}$ which is $\Delta^0_\alpha$-categorical and $\alpha$ is smallest such, but nonetheless for every isomorphic computable copy $\mathcal{B}$ of $\mathcal{A}$ there exists a $\beta< \alpha$ such that $\mathcal{A} \cong_{\Delta^0_\beta} \mathcal{B}$. This answers a question left open in the earlier work of Downey, Igusa, and Melnikov. We also show that such examples can be found among ordered abelian groups and real-closed fields.On the Gauss Algebra of Toric Algebras
http://publications.mfo.de/handle/mfo/1360
On the Gauss Algebra of Toric Algebras
Herzog, Jürgen; Jafari, Raheleh; Nasrollah Nejad, Abbas
Let $A$ be a $K$-subalgebra of the polynomial ring $S=K[x_1,\ldots,x_d]$ of dimension $d$, generated by finitely many monomials of degree $r$. Then the Gauss algebra $\mathbb{G}(A)$ of $A$ is generated by monomials of degree $(r-1)d$ in $S$. We describe the generators and the structure of $\mathbb{G}(A)$, when $A$ is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree $2$, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph $G$ with one loop, the embedding dimension of $\mathbb{G}(A)$ is bounded by the complexity of the graph $G$.
Wed, 25 Apr 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13602018-04-25T00:00:00ZHerzog, JürgenJafari, RahelehNasrollah Nejad, AbbasLet $A$ be a $K$-subalgebra of the polynomial ring $S=K[x_1,\ldots,x_d]$ of dimension $d$, generated by finitely many monomials of degree $r$. Then the Gauss algebra $\mathbb{G}(A)$ of $A$ is generated by monomials of degree $(r-1)d$ in $S$. We describe the generators and the structure of $\mathbb{G}(A)$, when $A$ is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree $2$, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph $G$ with one loop, the embedding dimension of $\mathbb{G}(A)$ is bounded by the complexity of the graph $G$.Homogenization of a nonlinear monotone problem with nonlinear Signorini boundary conditions in a domain with highly rough boundary
http://publications.mfo.de/handle/mfo/1359
Homogenization of a nonlinear monotone problem with nonlinear Signorini boundary conditions in a domain with highly rough boundary
Gaudiello, Antonio; Mel'nyk, Taras A.
We consider a domain $\Omega_\varepsilon\subset\mathbb{R}^N$, $N\geq2$, with a very rough boundary depending on~$\varepsilon$. For instance, if $N=3$ the domain $\Omega_\varepsilon$ has the form of a brush with an $\varepsilon$-periodic distribution of thin cylinders with fixed height and a small diameter of order $\varepsilon$. In $\Omega_\varepsilon$ a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on $\varepsilon$, on the lateral boundary of the cylinders is considered. We study the asymptotic behavior of this problem, as $\varepsilon$ vanishes, i.e. when the number of thin attached cylinders increases unboundedly, while their cross sections tend to zero. We identify the limit problem which is a nonstandard homogenized problem. Namely, in the region filled up by the thin cylinders the limit problem is given by a variational inequality coupled to an algebraic system.
Mon, 16 Apr 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13592018-04-16T00:00:00ZGaudiello, AntonioMel'nyk, Taras A.We consider a domain $\Omega_\varepsilon\subset\mathbb{R}^N$, $N\geq2$, with a very rough boundary depending on~$\varepsilon$. For instance, if $N=3$ the domain $\Omega_\varepsilon$ has the form of a brush with an $\varepsilon$-periodic distribution of thin cylinders with fixed height and a small diameter of order $\varepsilon$. In $\Omega_\varepsilon$ a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on $\varepsilon$, on the lateral boundary of the cylinders is considered. We study the asymptotic behavior of this problem, as $\varepsilon$ vanishes, i.e. when the number of thin attached cylinders increases unboundedly, while their cross sections tend to zero. We identify the limit problem which is a nonstandard homogenized problem. Namely, in the region filled up by the thin cylinders the limit problem is given by a variational inequality coupled to an algebraic system.The Sylow Structure of Scalar Automorphism Groups
http://publications.mfo.de/handle/mfo/1358
The Sylow Structure of Scalar Automorphism Groups
Herfort, Wolfgang; Hofmann, Karl Heinrich; Kramer, Linus; Russo, Francesco G.
For any locally compact abelian periodic group A its automorphism group contains as a subgroup those automorphisms that leave invariant every closed subgroup of A, to be denoted by SAut(A). This subgroup is again a locally compact abelian periodic group in its natural topology and hence allows a decomposition into its p-primary subgroups for p the primes for which topological p-elements in this automorphism subgroup exist. The interplay between the p-primary decomposition of SAut(A) and A can be encoded in a bipartite graph, the mastergraph of A. Properties and applications of this concept are discussed.
Thu, 22 Mar 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13582018-03-22T00:00:00ZHerfort, WolfgangHofmann, Karl HeinrichKramer, LinusRusso, Francesco G.For any locally compact abelian periodic group A its automorphism group contains as a subgroup those automorphisms that leave invariant every closed subgroup of A, to be denoted by SAut(A). This subgroup is again a locally compact abelian periodic group in its natural topology and hence allows a decomposition into its p-primary subgroups for p the primes for which topological p-elements in this automorphism subgroup exist. The interplay between the p-primary decomposition of SAut(A) and A can be encoded in a bipartite graph, the mastergraph of A. Properties and applications of this concept are discussed.Exceptional Legendrian Torus Knots
http://publications.mfo.de/handle/mfo/1357
Exceptional Legendrian Torus Knots
Geiges, Hansjörg; Onaran, Sinem
We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact structures on certain Seifert fibred manifolds with boundary allow us to place upper bounds on the number of tight contact structures on the complements of torus knots; the classification of exceptional realisations of these torus knots is then achieved by exhibiting suffciently many realisations in terms of contact surgery diagrams. We also discuss a couple of general theorems about the existence of exceptional Legendrian knots.
Wed, 21 Mar 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13572018-03-21T00:00:00ZGeiges, HansjörgOnaran, SinemWe present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact structures on certain Seifert fibred manifolds with boundary allow us to place upper bounds on the number of tight contact structures on the complements of torus knots; the classification of exceptional realisations of these torus knots is then achieved by exhibiting suffciently many realisations in terms of contact surgery diagrams. We also discuss a couple of general theorems about the existence of exceptional Legendrian knots.The Martin Boundary of Relatively Hyperbolic Groups with Virtually Abelian Parabolic Subgroups
http://publications.mfo.de/handle/mfo/1356
The Martin Boundary of Relatively Hyperbolic Groups with Virtually Abelian Parabolic Subgroups
Dussaule, Matthieu; Gekhtman, Ilya; Gerasimov, Victor; Potyagailo, Leonid
Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space ${\mathcal H}^n$, we show that the Martin boundary coincides with the $CAT(0)$ boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.
Mon, 19 Mar 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13562018-03-19T00:00:00ZDussaule, MatthieuGekhtman, IlyaGerasimov, VictorPotyagailo, LeonidGiven a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space ${\mathcal H}^n$, we show that the Martin boundary coincides with the $CAT(0)$ boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.Computing the long term evolution of the solar system with geometric numerical integrators
http://publications.mfo.de/handle/mfo/1355
Computing the long term evolution of the solar system with geometric numerical integrators
Fiorelli Vilmart, Shaula; Vilmart, Gilles
Simulating the dynamics of the Sun–Earth–Moon system
with a standard algorithm yields a dramatically
wrong solution, predicting that the Moon is ejected
from its orbit. In contrast, a well chosen algorithm
with the same initial data yields the correct behavior.
We explain the main ideas of how the evolution of
the solar system can be computed over long times
by taking advantage of so-called geometric numerical
methods. Short sample codes are provided for the
Sun–Earth–Moon system.
Wed, 27 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13552017-12-27T00:00:00ZFiorelli Vilmart, ShaulaVilmart, GillesSimulating the dynamics of the Sun–Earth–Moon system
with a standard algorithm yields a dramatically
wrong solution, predicting that the Moon is ejected
from its orbit. In contrast, a well chosen algorithm
with the same initial data yields the correct behavior.
We explain the main ideas of how the evolution of
the solar system can be computed over long times
by taking advantage of so-called geometric numerical
methods. Short sample codes are provided for the
Sun–Earth–Moon system.Computing with symmetries
http://publications.mfo.de/handle/mfo/1354
Computing with symmetries
Roney-Dougal, Colva M.
Group theory is the study of symmetry, and has many
applications both within and outside mathematics.
In this snapshot, we give a brief introduction to symmetries,
and how to compute with them.
Tue, 06 Mar 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13542018-03-06T00:00:00ZRoney-Dougal, Colva M.Group theory is the study of symmetry, and has many
applications both within and outside mathematics.
In this snapshot, we give a brief introduction to symmetries,
and how to compute with them.Topological recursion
http://publications.mfo.de/handle/mfo/1353
Topological recursion
Sułkowski, Piotr
In this snapshot we present the concept of topological
recursion – a new, surprisingly powerful formalism
at the border of mathematics and physics, which has
been actively developed within the last decade. After
introducing necessary ingredients – expectation values,
random matrices, quantum theories, recursion
relations, and topology – we explain how they get
combined together in one unifying picture.
Mon, 05 Mar 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13532018-03-05T00:00:00ZSułkowski, PiotrIn this snapshot we present the concept of topological
recursion – a new, surprisingly powerful formalism
at the border of mathematics and physics, which has
been actively developed within the last decade. After
introducing necessary ingredients – expectation values,
random matrices, quantum theories, recursion
relations, and topology – we explain how they get
combined together in one unifying picture.Spaces of Riemannian metrics
http://publications.mfo.de/handle/mfo/1352
Spaces of Riemannian metrics
Bustamante, Mauricio; Kordaß, Jan-Bernhard
Riemannian metrics endow smooth manifolds such as
surfaces with intrinsic geometric properties, for example
with curvature. They also allow us to measure
quantities like distances, angles and volumes. These
are the notions we use to characterize the “shape” of
a manifold. The space of Riemannian metrics is a
mathematical object that encodes the many possible
ways in which we can geometrically deform the shape
of a manifold.
Thu, 28 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13522017-12-28T00:00:00ZBustamante, MauricioKordaß, Jan-BernhardRiemannian metrics endow smooth manifolds such as
surfaces with intrinsic geometric properties, for example
with curvature. They also allow us to measure
quantities like distances, angles and volumes. These
are the notions we use to characterize the “shape” of
a manifold. The space of Riemannian metrics is a
mathematical object that encodes the many possible
ways in which we can geometrically deform the shape
of a manifold.