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http://publications.mfo.de:8080
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http://publications.mfo.de:8080
On radial basis functions
http://publications.mfo.de/handle/mfo/1410
On radial basis functions
Buhmann, Martin; Jäger, Janin
Many sciences and other areas of research and applications
from engineering to economics require the approximation
of functions that depend on many variables.
This can be for a variety of reasons. Sometimes
we have a discrete set of data points and we
want to find an approximating function that completes
this data; another possibility is that precise
functions are either not known or it would take too
long to compute them explicitly. In this snapshot
we want to introduce a particular method of approximation
which uses functions called radial basis functions.
This method is particularly useful when approximating
functions that depend on very many variables.
We describe the basic approach to approximation
with radial basis functions, including their computation,
give several examples of such functions and
show some applications.
Wed, 13 Mar 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14102019-03-13T00:00:00ZBuhmann, MartinJäger, JaninMany sciences and other areas of research and applications
from engineering to economics require the approximation
of functions that depend on many variables.
This can be for a variety of reasons. Sometimes
we have a discrete set of data points and we
want to find an approximating function that completes
this data; another possibility is that precise
functions are either not known or it would take too
long to compute them explicitly. In this snapshot
we want to introduce a particular method of approximation
which uses functions called radial basis functions.
This method is particularly useful when approximating
functions that depend on very many variables.
We describe the basic approach to approximation
with radial basis functions, including their computation,
give several examples of such functions and
show some applications.On a Group Functor Describing Invariants of Algebraic Surfaces
http://publications.mfo.de/handle/mfo/1409
On a Group Functor Describing Invariants of Algebraic Surfaces
Dietrich, Heiko; Moravec, Primož
Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.
Fri, 01 Mar 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14092019-03-01T00:00:00ZDietrich, HeikoMoravec, PrimožLiedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.Weighted Surface Algebras: General Version
http://publications.mfo.de/handle/mfo/1408
Weighted Surface Algebras: General Version
Erdmann, Karin; Skowroński, Andrzej
We introduce general weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular disc, triangle, tetrahedral and spherical algebras, are symmetric tame periodic algebras of period 4.
Thu, 28 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14082019-02-28T00:00:00ZErdmann, KarinSkowroński, AndrzejWe introduce general weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular disc, triangle, tetrahedral and spherical algebras, are symmetric tame periodic algebras of period 4.Group Algebras of Compact Groups. A New Way of Producing Group Hopf Algebras over Real and Complex Fields: Weakly Complete Topological Vector Spaces
http://publications.mfo.de/handle/mfo/1407
Group Algebras of Compact Groups. A New Way of Producing Group Hopf Algebras over Real and Complex Fields: Weakly Complete Topological Vector Spaces
Hofmann, Karl Heinrich; Kramer, Linus
Weakly complete real or complex associative algebras $A$ are necessarily projective limits of finite dimensional algebras. Their group of units $A^{-1}$ is a pro-Lie group with the associated topological Lie algebra $A_{\rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $\exp\colon A\to A^{-1}$ as the exponential function of $A^{-1}$. With each topological group $G$, a weakly complete group algebra $\mathbb K[G]$ is associated functorially so that the functor $G\mapsto \mathbb K[G]$ is left adjoint to $A\mapsto A^{-1}$. The group algebra $\mathbb K[G]$ is a weakly complete Hopf algebra. If $G$ is compact, then $\mathbb R[G]$ contains $G$ as the set of grouplike elements.
The category of all real weakly complete Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is equivalent to the category of compact groups. The group algebra $A=\mathbb R[G]$ of a compact group $G$ contains a copy of the Lie algebra $\mathfrak L(G)$ in $A_{\rm Lie}$; it also contains all probability measures on $G$. The dual of the group algebra $\mathbb R[G]$ is the Hopf algebra ${\cal R}(G,\mathbb R)$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis
of a duality ${\cal R}(G,\mathbb R)\leftrightarrow \mathbb R[G]$ and thus yields a new aspect of Tannaka duality. In the case of a compact abelian $G$, an alternative concrete construction of $\mathbb K[G]$ is given both for $\mathbb K=\mathbb C$ and $\mathbb K=\mathbb R$. Because of the presence of $\mathfrak L(G)$, the enveloping algebra of weakly complete Lie algebras are introduced and placed into relation with $\mathbb K[G]$.
Wed, 27 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14072019-02-27T00:00:00ZHofmann, Karl HeinrichKramer, LinusWeakly complete real or complex associative algebras $A$ are necessarily projective limits of finite dimensional algebras. Their group of units $A^{-1}$ is a pro-Lie group with the associated topological Lie algebra $A_{\rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $\exp\colon A\to A^{-1}$ as the exponential function of $A^{-1}$. With each topological group $G$, a weakly complete group algebra $\mathbb K[G]$ is associated functorially so that the functor $G\mapsto \mathbb K[G]$ is left adjoint to $A\mapsto A^{-1}$. The group algebra $\mathbb K[G]$ is a weakly complete Hopf algebra. If $G$ is compact, then $\mathbb R[G]$ contains $G$ as the set of grouplike elements.
The category of all real weakly complete Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is equivalent to the category of compact groups. The group algebra $A=\mathbb R[G]$ of a compact group $G$ contains a copy of the Lie algebra $\mathfrak L(G)$ in $A_{\rm Lie}$; it also contains all probability measures on $G$. The dual of the group algebra $\mathbb R[G]$ is the Hopf algebra ${\cal R}(G,\mathbb R)$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis
of a duality ${\cal R}(G,\mathbb R)\leftrightarrow \mathbb R[G]$ and thus yields a new aspect of Tannaka duality. In the case of a compact abelian $G$, an alternative concrete construction of $\mathbb K[G]$ is given both for $\mathbb K=\mathbb C$ and $\mathbb K=\mathbb R$. Because of the presence of $\mathfrak L(G)$, the enveloping algebra of weakly complete Lie algebras are introduced and placed into relation with $\mathbb K[G]$.Hölder Continuity of the Spectra for Aperiodic Hamiltonians
http://publications.mfo.de/handle/mfo/1406
Hölder Continuity of the Spectra for Aperiodic Hamiltonians
Beckus, Siegfried; Bellissard, Jean; Cornean, Horia
We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.
Tue, 26 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14062019-02-26T00:00:00ZBeckus, SiegfriedBellissard, JeanCornean, HoriaWe study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.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 LuisTime Discretization Schemes for Hyperbolic Systems on Networks by ε-Expansion
http://publications.mfo.de/handle/mfo/1402
Time Discretization Schemes for Hyperbolic Systems on Networks by ε-Expansion
Altmann, Robert; Zimmer, Christoph
We consider partial differential equations on networks with a small parameter $\epsilon$, which are hyperbolic for $\epsilon>0$ and parabolic for $\epsilon=0$. With a combination of an $\epsilon$-expansion and Runge-Kutta schemes for constrained systems of parabolic type, we derive a new class of time discretization schemes for hyperbolic systems on networks, which are constrained due to interconnection conditions. For the analysis we consider the coupled system equations as partial differential-algebraic equations based on the variational formulation of the problem. We discuss well-posedness of the resulting systems and estimate the error caused by the $\epsilon$-expansion.
Tue, 12 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14022019-02-12T00:00:00ZAltmann, RobertZimmer, ChristophWe consider partial differential equations on networks with a small parameter $\epsilon$, which are hyperbolic for $\epsilon>0$ and parabolic for $\epsilon=0$. With a combination of an $\epsilon$-expansion and Runge-Kutta schemes for constrained systems of parabolic type, we derive a new class of time discretization schemes for hyperbolic systems on networks, which are constrained due to interconnection conditions. For the analysis we consider the coupled system equations as partial differential-algebraic equations based on the variational formulation of the problem. We discuss well-posedness of the resulting systems and estimate the error caused by the $\epsilon$-expansion.A 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.