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.Tue, 30 May 2023 13:32:10 GMT2023-05-30T13:32:10ZComputer Algebra with GAP
http://publications.mfo.de/handle/mfo/4023
Computer Algebra with GAP
Piterman, Kevin I.; Vendramin, Leandro
This monograph includes the following topics: a basic introduction to the language, basic arithmetic, permutations, matrices, polynomial rings, finite fields, finite and finitely presented groups, small groups, group representations and character theory, and simple groups. Advanced topics include testing several open conjectures and theorems. In addition, each chapter ends with an extensive list of problems. We hope the reader will find some problems challenging and exciting as they are based on outstanding research papers. Selected solutions can be found at the end of the book.
Thu, 13 Apr 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40232023-04-13T00:00:00ZPiterman, Kevin I.Vendramin, LeandroThis monograph includes the following topics: a basic introduction to the language, basic arithmetic, permutations, matrices, polynomial rings, finite fields, finite and finitely presented groups, small groups, group representations and character theory, and simple groups. Advanced topics include testing several open conjectures and theorems. In addition, each chapter ends with an extensive list of problems. We hope the reader will find some problems challenging and exciting as they are based on outstanding research papers. Selected solutions can be found at the end of the book.Real Enumerative Invariants Relative to the Anti-Canonical Divisor and their Refinement
http://publications.mfo.de/handle/mfo/4022
Real Enumerative Invariants Relative to the Anti-Canonical Divisor and their Refinement
Itenberg, Ilia; Shustin, Eugenii
We introduce new invariants of the projective plane (and, more generally, of
certain toric surfaces) that arise from the appropriate enumeration of real
elliptic curves. These invariants admit a refinement (according to the quantum
index) similar to the one introduced by Grigory Mikhalkin in the rational case.
We also construct tropical counterparts of the refined elliptic invariants
under consideration and establish a tropical algorithm allowing one to compute,
$via$ a suitable version of the correspondence theorem, the above
invariants.
Fri, 24 Mar 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40222023-03-24T00:00:00ZItenberg, IliaShustin, EugeniiWe introduce new invariants of the projective plane (and, more generally, of
certain toric surfaces) that arise from the appropriate enumeration of real
elliptic curves. These invariants admit a refinement (according to the quantum
index) similar to the one introduced by Grigory Mikhalkin in the rational case.
We also construct tropical counterparts of the refined elliptic invariants
under consideration and establish a tropical algorithm allowing one to compute,
$via$ a suitable version of the correspondence theorem, the above
invariants.Flag-Accurate Arrangements
http://publications.mfo.de/handle/mfo/4012
Flag-Accurate Arrangements
Mücksch, Paul; Röhrle, Gerhard; Tran, Tan Nhat
In [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is put forward in the present paper. More specifically, let $\mathscr A$ be a free arrangement of rank $\ell$. Suppose that for every $1\leq d \leq \ell$, the first $d$ exponents of $\mathscr A$ - when listed in increasing order - are realized as the exponents of a free restriction of $\mathscr A$ to some intersection of reflecting hyperplanes of $\mathscr A$ of dimension $d$. Following [MR21], we call such an arrangement $\mathscr A$ with this natural property accurate. If in addition the flats involved can be chosen to form a flag, we call $\mathscr A$ flag-accurate. We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate. The main result of [MR21] shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy.
Mon, 13 Feb 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40122023-02-13T00:00:00ZMücksch, PaulRöhrle, GerhardTran, Tan NhatIn [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is put forward in the present paper. More specifically, let $\mathscr A$ be a free arrangement of rank $\ell$. Suppose that for every $1\leq d \leq \ell$, the first $d$ exponents of $\mathscr A$ - when listed in increasing order - are realized as the exponents of a free restriction of $\mathscr A$ to some intersection of reflecting hyperplanes of $\mathscr A$ of dimension $d$. Following [MR21], we call such an arrangement $\mathscr A$ with this natural property accurate. If in addition the flats involved can be chosen to form a flag, we call $\mathscr A$ flag-accurate. We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate. The main result of [MR21] shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy.Convolution in Dual Cesàro Sequence Spaces
http://publications.mfo.de/handle/mfo/4002
Convolution in Dual Cesàro Sequence Spaces
Curbera, Guillermo P.; Ricker, Werner J.
We investigate convolution operators in the sequence spaces $d_p$, for 1 $\leqslant p<\infty$. These spaces, for $p$ > 1, arise as dual spaces of the Cesàro sequence spaces $ces_p$ thoroughly investigated by G. Bennett. A detailed study is also made of the algebra of those sequences which convolve $d_p$ into $d_p$. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of $l^{p}$.
Fri, 16 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40022022-12-16T00:00:00ZCurbera, Guillermo P.Ricker, Werner J.We investigate convolution operators in the sequence spaces $d_p$, for 1 $\leqslant p<\infty$. These spaces, for $p$ > 1, arise as dual spaces of the Cesàro sequence spaces $ces_p$ thoroughly investigated by G. Bennett. A detailed study is also made of the algebra of those sequences which convolve $d_p$ into $d_p$. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of $l^{p}$.Hutchinson's Intervals and Entire Functions from the Laguerre-Pólya Class
http://publications.mfo.de/handle/mfo/4001
Hutchinson's Intervals and Entire Functions from the Laguerre-Pólya Class
Nguyen, Thu Hien; Vishnyakova, Anna
We find the intervals $[\alpha, \beta (\alpha)]$ such that if a univariate real polynomial or entire function $f(z) = a_0 + a_1 z + a_2 z^2 + \cdots $ with positive coefficients satisfy the conditions $ \frac{a_{k-1}^2}{a_{k-2}a_{k}} \in [\alpha, \beta(\alpha)]$ for all $k \geq 2,$ then $f$ belongs to the Laguerre-Pólya class. For instance, from J.I. Hutchinson's theorem, one can observe that $f$ belongs to the Laguerre-Pólya class (has only real zeros) when $q_k(f) \in [4, + \infty).$ We are interested in finding those intervals which are not subsets of $[4, + \infty).$
Mon, 12 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40012022-12-12T00:00:00ZNguyen, Thu HienVishnyakova, AnnaWe find the intervals $[\alpha, \beta (\alpha)]$ such that if a univariate real polynomial or entire function $f(z) = a_0 + a_1 z + a_2 z^2 + \cdots $ with positive coefficients satisfy the conditions $ \frac{a_{k-1}^2}{a_{k-2}a_{k}} \in [\alpha, \beta(\alpha)]$ for all $k \geq 2,$ then $f$ belongs to the Laguerre-Pólya class. For instance, from J.I. Hutchinson's theorem, one can observe that $f$ belongs to the Laguerre-Pólya class (has only real zeros) when $q_k(f) \in [4, + \infty).$ We are interested in finding those intervals which are not subsets of $[4, + \infty).$Quasi-Equilibria and Click Times for a Variant of Muller's Ratchet
http://publications.mfo.de/handle/mfo/3993
Quasi-Equilibria and Click Times for a Variant of Muller's Ratchet
González Casanova, Adrian; Smadi, Charline; Wakolbinger, Anton
Consider a population of $N$ individuals, each of them carrying a type in $\mathbb N_0$. The population evolves according to a Moran dynamics with selection and mutation, where an individual of type $k$ has the same selective advantage over all individuals with type $k' > k$, and type $k$ mutates to type
$k+1$ at a constant rate. This model is thus a variation of the classical Muller's ratchet: there the selective advantage is proportional to $k' - k$. For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the click times of the ratchet (i.e. the times at which the hitherto minimal ('best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a "dual" hierarchical multitype logistic system, and also via the distribution of the final minimal
displacement in a branching random walk with one-sided steps. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations.
Wed, 30 Nov 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39932022-11-30T00:00:00ZGonzález Casanova, AdrianSmadi, CharlineWakolbinger, AntonConsider a population of $N$ individuals, each of them carrying a type in $\mathbb N_0$. The population evolves according to a Moran dynamics with selection and mutation, where an individual of type $k$ has the same selective advantage over all individuals with type $k' > k$, and type $k$ mutates to type
$k+1$ at a constant rate. This model is thus a variation of the classical Muller's ratchet: there the selective advantage is proportional to $k' - k$. For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the click times of the ratchet (i.e. the times at which the hitherto minimal ('best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a "dual" hierarchical multitype logistic system, and also via the distribution of the final minimal
displacement in a branching random walk with one-sided steps. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations.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.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].