Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Tue, 27 Sep 2022 07:13:16 GMT2022-09-27T07:13:16ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
Birational Rowmotion on a Rectangle over a Noncommutative Ring
http://publications.mfo.de/handle/mfo/3974
Birational Rowmotion on a Rectangle over a Noncommutative Ring
Grinberg, Darij; Roby, Tom
We extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained.
Tue, 20 Sep 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39742022-09-20T00:00:00ZGrinberg, DarijRoby, TomWe extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained.Root Cycles in Coxeter Groups
http://publications.mfo.de/handle/mfo/3973
Root Cycles in Coxeter Groups
Hart, Sarah; Kelsey, Veronica; Rowley, Peter
For an element $w$ of a Coxeter group $W$ there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on $\Phi$, the root system of $W$. This paper investigates the interaction between these two features of $w$, introducing the notion of the crossing number of $w$, $\kappa(w)$. Writing $w = c_1 \cdots c_r$ as a product of disjoint cycles we associate to each cycle $c_i$ a `crossing number' $\kappa(c_i)$, which is the number of positive roots $\alpha$ in $c_i$ for which $w\cdot \alpha$ is negative. Let Seq$_k(w)$ be the sequence of $\kappa(c_i)$ written in increasing order, and let $\kappa(w)$ = max Seq$_k(w)$. The length of $w$ can be retrieved from this sequence, but Seq$_k(w)$ provides much more information. For a conjugacy class $X$ of $W$ let $k_{\min}(X)=\min \{\kappa(w) \;|\;w \in X\}$ and let $\kappa(W)$ be the maximum value of $k_{\min}$ across all conjugacy classes of $W$. We call $\kappa(w)$ and $\kappa(W)$, respectively, the crossing numbers of $w$ and $W$. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups if $u$ and $v$ are two elements of minimal length in the same conjugacy class $X$, then Seq$_k(u)$ = Seq$_k(v)$ and $k_{\min}(X)=\kappa(u)=\kappa(v)$. Also it is shown that the crossing number of an arbitrary Coxeter group is bounded below by the crossing number of a standard parabolic subgroup. Finally, examples are given to show that crossing numbers can be arbitrarily large for finite and infinite irreducible Coxeter groups.
Thu, 15 Sep 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39732022-09-15T00:00:00ZHart, SarahKelsey, VeronicaRowley, PeterFor an element $w$ of a Coxeter group $W$ there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on $\Phi$, the root system of $W$. This paper investigates the interaction between these two features of $w$, introducing the notion of the crossing number of $w$, $\kappa(w)$. Writing $w = c_1 \cdots c_r$ as a product of disjoint cycles we associate to each cycle $c_i$ a `crossing number' $\kappa(c_i)$, which is the number of positive roots $\alpha$ in $c_i$ for which $w\cdot \alpha$ is negative. Let Seq$_k(w)$ be the sequence of $\kappa(c_i)$ written in increasing order, and let $\kappa(w)$ = max Seq$_k(w)$. The length of $w$ can be retrieved from this sequence, but Seq$_k(w)$ provides much more information. For a conjugacy class $X$ of $W$ let $k_{\min}(X)=\min \{\kappa(w) \;|\;w \in X\}$ and let $\kappa(W)$ be the maximum value of $k_{\min}$ across all conjugacy classes of $W$. We call $\kappa(w)$ and $\kappa(W)$, respectively, the crossing numbers of $w$ and $W$. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups if $u$ and $v$ are two elements of minimal length in the same conjugacy class $X$, then Seq$_k(u)$ = Seq$_k(v)$ and $k_{\min}(X)=\kappa(u)=\kappa(v)$. Also it is shown that the crossing number of an arbitrary Coxeter group is bounded below by the crossing number of a standard parabolic subgroup. Finally, examples are given to show that crossing numbers can be arbitrarily large for finite and infinite irreducible Coxeter groups.Solving inverse problems with Bayes’ theorem
http://publications.mfo.de/handle/mfo/3972
Solving inverse problems with Bayes’ theorem
Latz, Jonas; Sprungk, Björn
The goal of inverse problems is to find an unknown parameter based on noisy data. Such problems appear in a wide range of applications including geophysics, medicine, and chemistry. One method of solving them is known as the Bayesian approach. In this approach, the unknown parameter is modelled as a random variable to reflect its uncertain value. Bayes’ theorem is applied to update our knowledge given new information from noisy data.
Mon, 05 Sep 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39722022-09-05T00:00:00ZLatz, JonasSprungk, BjörnThe goal of inverse problems is to find an unknown parameter based on noisy data. Such problems appear in a wide range of applications including geophysics, medicine, and chemistry. One method of solving them is known as the Bayesian approach. In this approach, the unknown parameter is modelled as a random variable to reflect its uncertain value. Bayes’ theorem is applied to update our knowledge given new information from noisy data.Convergence and Error Analysis of Compressible Fluid Flows with Random Data: Monte Carlo Method
http://publications.mfo.de/handle/mfo/3970
Convergence and Error Analysis of Compressible Fluid Flows with Random Data: Monte Carlo Method
Feireisl, Eduard; Lukáčova-Medviďová, Mariá; She, Bangwei; Yuan, Yuhuan
The goal of this paper is to study convergence and error estimates of the Monte Carlo method for the Navier-Stokes equations with random data. To discretize in space and time, the Monte Carlo method is combined with a suitable deterministic discretization scheme, such as a fnite volume method. We assume that the initial data, force and the viscosity coefficients are random variables and study both, the statistical convergence rates as well as the approximation errors. Since the compressible Navier-Stokes equations are not known to be uniquely solvable in the class of global weak solutions, we cannot apply pathwise arguments to analyze the random Navier-Stokes equations. Instead we have to apply intrinsic stochastic compactness arguments via the Skorokhod representation theorem and the Gyöngy-Krylov method. Assuming that the numerical solutions are bounded in probability, we prove that the Monte Carlo fnite volume method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results.
Thu, 25 Aug 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39702022-08-25T00:00:00ZFeireisl, EduardLukáčova-Medviďová, MariáShe, BangweiYuan, YuhuanThe goal of this paper is to study convergence and error estimates of the Monte Carlo method for the Navier-Stokes equations with random data. To discretize in space and time, the Monte Carlo method is combined with a suitable deterministic discretization scheme, such as a fnite volume method. We assume that the initial data, force and the viscosity coefficients are random variables and study both, the statistical convergence rates as well as the approximation errors. Since the compressible Navier-Stokes equations are not known to be uniquely solvable in the class of global weak solutions, we cannot apply pathwise arguments to analyze the random Navier-Stokes equations. Instead we have to apply intrinsic stochastic compactness arguments via the Skorokhod representation theorem and the Gyöngy-Krylov method. Assuming that the numerical solutions are bounded in probability, we prove that the Monte Carlo fnite volume method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results.On a Conjecture of Khoroshkin and Tolstoy
http://publications.mfo.de/handle/mfo/3967
On a Conjecture of Khoroshkin and Tolstoy
Appel, Andrea; Gautam, Sachin; Wendlandt, Curtis
We prove a no-go theorem on the factorization of the lower triangular part in the Gaussian decomposition of the Yangian's universal $R$-matrix, yielding a negative answer to a conjecture of Khoroshkin and Tolstoy from [Lett. Math. Phys. vol. 36 1996].
Tue, 02 Aug 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39672022-08-02T00:00:00ZAppel, AndreaGautam, SachinWendlandt, CurtisWe prove a no-go theorem on the factorization of the lower triangular part in the Gaussian decomposition of the Yangian's universal $R$-matrix, yielding a negative answer to a conjecture of Khoroshkin and Tolstoy from [Lett. Math. Phys. vol. 36 1996].Embedding Spaces of Split Links
http://publications.mfo.de/handle/mfo/3966
Embedding Spaces of Split Links
Boyd, Rachael; Bregman, Corey
We study the homotopy type of the space $\mathcal{E}(L)$ of unparametrised embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$. Inspired by work of Brendle and Hatcher, we introduce a semi-simplicial space
of separating systems and show that this is homotopy equivalent to $\mathcal{E}(L)$. This combinatorial object provides a gateway to studying the homotopy type of $\mathcal{E}(L)$ via the homotopy type of the spaces $\mathcal{E}(L_i)$. We apply this tool to find a simple description of the fundamental group, or motion group, of $\mathcal{E}(L)$, and extend this to a description of the motion group of embeddings in $S^3$.
Mon, 01 Aug 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39662022-08-01T00:00:00ZBoyd, RachaelBregman, CoreyWe study the homotopy type of the space $\mathcal{E}(L)$ of unparametrised embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$. Inspired by work of Brendle and Hatcher, we introduce a semi-simplicial space
of separating systems and show that this is homotopy equivalent to $\mathcal{E}(L)$. This combinatorial object provides a gateway to studying the homotopy type of $\mathcal{E}(L)$ via the homotopy type of the spaces $\mathcal{E}(L_i)$. We apply this tool to find a simple description of the fundamental group, or motion group, of $\mathcal{E}(L)$, and extend this to a description of the motion group of embeddings in $S^3$.Shock-avoiding Slicing Conditions: Tests and Calibrations
http://publications.mfo.de/handle/mfo/3963
Shock-avoiding Slicing Conditions: Tests and Calibrations
Baumgarte, Thomas W.; Hilditch, David
While the 1+log slicing condition has been extremely successful in numerous numerical relativity simulations, it is also known to develop "gauge-shocks" in some examples. Alternative "shockavoiding" slicing conditions suggested by Alcubierre prevent these pathologies in those examples, but have not yet been explored and tested very broadly. In this paper we compare the performance of shock-avoiding slicing conditions with those of 1+log slicing for a number of "text-book" problems, including black holes and relativistic stars. While, in some simulations, the shock-avoiding slicing conditions feature some unusual properties and lead to more "gauge-dynamics" than the 1+log slicing condition, we find that they perform quite similarly in terms of stability and accuracy, and hence provide a very viable alternative to 1+log slicing.
Tue, 19 Jul 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39632022-07-19T00:00:00ZBaumgarte, Thomas W.Hilditch, DavidWhile the 1+log slicing condition has been extremely successful in numerous numerical relativity simulations, it is also known to develop "gauge-shocks" in some examples. Alternative "shockavoiding" slicing conditions suggested by Alcubierre prevent these pathologies in those examples, but have not yet been explored and tested very broadly. In this paper we compare the performance of shock-avoiding slicing conditions with those of 1+log slicing for a number of "text-book" problems, including black holes and relativistic stars. While, in some simulations, the shock-avoiding slicing conditions feature some unusual properties and lead to more "gauge-dynamics" than the 1+log slicing condition, we find that they perform quite similarly in terms of stability and accuracy, and hence provide a very viable alternative to 1+log slicing.On the Enumeration of Finite $L$-Algebras
http://publications.mfo.de/handle/mfo/3961
On the Enumeration of Finite $L$-Algebras
Dietzel, Carsten; Menchón, Paula; Vendramin, Leandro
We use Constraint Satisfaction Methods to construct and enumerate finite L-algebras up to isomorphism. These objects were recently introduced by Rump and appear in Garside theory, algebraic logic, and the study of the combinatorial Yang-Baxter equation. There are 377322225 isomorphism classes of $L$-algebras of size eight. The database constructed suggest the existence of bijections between certain classes of $L$-algebras and well-known combinatorial objects. On the one hand, we prove that Bell numbers enumerate isomorphism classes of finite linear $L$-algebras. On the other hand, we also prove that finite regular $L$-algebras are in bijective correspondence with infinite-dimensional Young diagrams.
Wed, 29 Jun 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39612022-06-29T00:00:00ZDietzel, CarstenMenchón, PaulaVendramin, LeandroWe use Constraint Satisfaction Methods to construct and enumerate finite L-algebras up to isomorphism. These objects were recently introduced by Rump and appear in Garside theory, algebraic logic, and the study of the combinatorial Yang-Baxter equation. There are 377322225 isomorphism classes of $L$-algebras of size eight. The database constructed suggest the existence of bijections between certain classes of $L$-algebras and well-known combinatorial objects. On the one hand, we prove that Bell numbers enumerate isomorphism classes of finite linear $L$-algebras. On the other hand, we also prove that finite regular $L$-algebras are in bijective correspondence with infinite-dimensional Young diagrams.Discretization of Inherent ODEs and the Geometric Integration of DAEs with Symmetries
http://publications.mfo.de/handle/mfo/3953
Discretization of Inherent ODEs and the Geometric Integration of DAEs with Symmetries
Kunkel, Peter; Mehrmann, Volker
Discretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties.
Wed, 08 Jun 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39532022-06-08T00:00:00ZKunkel, PeterMehrmann, VolkerDiscretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties.Jewellery from tessellations of hyperbolic space
http://publications.mfo.de/handle/mfo/3952
Jewellery from tessellations of hyperbolic space
Gangl, Herbert
In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional hyperbolic space. Many of the building blocks of these tessellations exhibit beautiful symmetry and have inspired the design of 3D printed jewellery.
Thu, 02 Jun 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39522022-06-02T00:00:00ZGangl, HerbertIn this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional hyperbolic space. Many of the building blocks of these tessellations exhibit beautiful symmetry and have inspired the design of 3D printed jewellery.