2008
http://publications.mfo.de/handle/mfo/1340
Fri, 24 Sep 2021 21:58:46 GMT2021-09-24T21:58:46ZA note on k[z]-Automorphisms in Two Variables
http://publications.mfo.de/handle/mfo/1222
A note on k[z]-Automorphisms in Two Variables
Edo, Eric; Essen, Arno van den; Maubach, Stefan
We prove that for a polynomial $f \in k[x, y, z]$ equivalent are: (1)$f$ is a $k[z]$-coordinate of $k[z][x,y]$, and (2) $k[x, y, z]/(f)\cong k^[2]$ and $f(x,y,a)$ is a coordinate in $k[x,y]$ for some $a \in k$. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate $f \in k[x,y,z]$ which is also a $k(z)$-coordinate, is a $k[z]$-coordinate. We discuss a method for constructing automorphisms of $k[x, y, z]$, and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method -essentially linking Nagata with a non-tame $R$-automorphism of $R[x]$, where $R=k[z]/(z^2)$.
OWLF 2007
Tue, 01 Jan 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12222008-01-01T00:00:00ZEdo, EricEssen, Arno van denMaubach, StefanWe prove that for a polynomial $f \in k[x, y, z]$ equivalent are: (1)$f$ is a $k[z]$-coordinate of $k[z][x,y]$, and (2) $k[x, y, z]/(f)\cong k^[2]$ and $f(x,y,a)$ is a coordinate in $k[x,y]$ for some $a \in k$. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate $f \in k[x,y,z]$ which is also a $k(z)$-coordinate, is a $k[z]$-coordinate. We discuss a method for constructing automorphisms of $k[x, y, z]$, and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method -essentially linking Nagata with a non-tame $R$-automorphism of $R[x]$, where $R=k[z]/(z^2)$.On the Directionally Newton-non-degenerate Singularities of Complex Hypersurfaces
http://publications.mfo.de/handle/mfo/1221
On the Directionally Newton-non-degenerate Singularities of Complex Hypersurfaces
Kerner, Dmitry
We introduce a minimal generalization of Newton-non-degenerate singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called directionally Newton-non-degenerate if the local embedded topological singularity type can be restored from a collection of Newton diagrams. A singularity that is not directionally Newton-non-degenerate is called essentially Newton-degenerate. For plane curves we give an explicit and simple characterization of directionally Newton-non-degenerate singularities, for hypersurfaces we give some examples. Then we treat the question: is Newton-non-degenerate or directionally Newton-non-degenerate a property of singular types or of particular representatives.Namely, is the non-degeneracy preserved in an equi-singular family? This is proved for curves. For hypersurfaces we give an example of a Newton-non-degenerate hypersurface whose equi-singular deformation consists of essentially Newton-degenerate hypersurfaces. Finally, the classical formulas for the Milnor number (Kouchnirenko) and the zeta function (Varchenko) of the Newton-non-degenerate singularity are generalized to some classes of directionally Newton-non-degenerate singularities.
OWLF 2008
Tue, 01 Jan 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12212008-01-01T00:00:00ZKerner, DmitryWe introduce a minimal generalization of Newton-non-degenerate singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called directionally Newton-non-degenerate if the local embedded topological singularity type can be restored from a collection of Newton diagrams. A singularity that is not directionally Newton-non-degenerate is called essentially Newton-degenerate. For plane curves we give an explicit and simple characterization of directionally Newton-non-degenerate singularities, for hypersurfaces we give some examples. Then we treat the question: is Newton-non-degenerate or directionally Newton-non-degenerate a property of singular types or of particular representatives.Namely, is the non-degeneracy preserved in an equi-singular family? This is proved for curves. For hypersurfaces we give an example of a Newton-non-degenerate hypersurface whose equi-singular deformation consists of essentially Newton-degenerate hypersurfaces. Finally, the classical formulas for the Milnor number (Kouchnirenko) and the zeta function (Varchenko) of the Newton-non-degenerate singularity are generalized to some classes of directionally Newton-non-degenerate singularities.On the δ=const Collisions of Singularities of Complex Plane Curves
http://publications.mfo.de/handle/mfo/1220
On the δ=const Collisions of Singularities of Complex Plane Curves
Kerner, Dmitry
We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\delta$ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A new invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in collisions. We consider in details the $\delta=const$ deformations of ordinary multiple point, the deformation of a singularity into the collection of ordinary multiple points and the deformation of the type $x^p+y^{pk}$ into a collection of $A_k$'s.
OWLF 2008
Tue, 01 Jan 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12202008-01-01T00:00:00ZKerner, DmitryWe study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\delta$ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A new invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in collisions. We consider in details the $\delta=const$ deformations of ordinary multiple point, the deformation of a singularity into the collection of ordinary multiple points and the deformation of the type $x^p+y^{pk}$ into a collection of $A_k$'s.Weyl-Titchmarsh Functions of Vector-Valued Sturm-Liouville Operators on the Unit Interval
http://publications.mfo.de/handle/mfo/1219
Weyl-Titchmarsh Functions of Vector-Valued Sturm-Liouville Operators on the Unit Interval
Chelkak, Dmitry; Korotyaev, Evgeny
The matrix-valued Weyl-Titchmarsh functions $M(\lambda)$ of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of $M(\lambda)$) and the residues of $M(\lambda)$ is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, $N \times N$ Weyl-Titchmarsh functions) corresponding to $N \times N$ self-adjoint square-integrable matrix-valued potentials is given, if all $N$ eigenvalues of the averaged potential are distinct.
OWLF 2008
Tue, 01 Jan 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12192008-01-01T00:00:00ZChelkak, DmitryKorotyaev, EvgenyThe matrix-valued Weyl-Titchmarsh functions $M(\lambda)$ of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of $M(\lambda)$) and the residues of $M(\lambda)$ is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, $N \times N$ Weyl-Titchmarsh functions) corresponding to $N \times N$ self-adjoint square-integrable matrix-valued potentials is given, if all $N$ eigenvalues of the averaged potential are distinct.Stratifying modular representations of finite groups
http://publications.mfo.de/handle/mfo/1137
Stratifying modular representations of finite groups
Benson, David J.; Iyengar, Srikanth B.; Krause, Henning
We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.
Research in Pairs 2008
Thu, 20 Mar 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11372008-03-20T00:00:00ZBenson, David J.Iyengar, Srikanth B.Krause, HenningWe classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.Epsilon-hypercyclic operators
http://publications.mfo.de/handle/mfo/1136
Epsilon-hypercyclic operators
Badea, Catalin; Grivaux, Sophie; Müller, Valdimir
For each fixed number $\varepsilon$ in $(0, 1)$ we construct a bounded linear operator on the Banach space $\ell_1$ having a certain orbit which intersects every cone of aperture $\varepsilon$, but with every orbit avoiding a certain ball of radius $d$, for every $d>0$. This answers a question from [8]. On the other hand, if $T$ is an operator on the Banach space $X$ such that for every $\varepsilon>0$ there is a point in $X$ whose orbit under the action of $T$ meets every cone of aperture $\varepsilon$, then $T$ has a dense orbit.
Research in Pairs 2008
Wed, 19 Mar 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11362008-03-19T00:00:00ZBadea, CatalinGrivaux, SophieMüller, ValdimirFor each fixed number $\varepsilon$ in $(0, 1)$ we construct a bounded linear operator on the Banach space $\ell_1$ having a certain orbit which intersects every cone of aperture $\varepsilon$, but with every orbit avoiding a certain ball of radius $d$, for every $d>0$. This answers a question from [8]. On the other hand, if $T$ is an operator on the Banach space $X$ such that for every $\varepsilon>0$ there is a point in $X$ whose orbit under the action of $T$ meets every cone of aperture $\varepsilon$, then $T$ has a dense orbit.Quantities that frequency-dependent selection maximizes
http://publications.mfo.de/handle/mfo/1135
Quantities that frequency-dependent selection maximizes
Matessi, Carlo; Schneider, Kristian
We consider a model of frequency-dependent selection, to which we refer as the Wildcard Model, that accommodates as particular cases a number of diverse models of biologically specific situations. Two very different particular models (Lessard, 1984; Bürger, 2005; Schneider, 2006), subsumed by the Wildcard Model, have been shown in the past to have a Lyapunov functions (LF) under appropriate genetic assumptions. We show that the Wildcard Model: (i) in continuous time is a generalized gradient system for one locus, multiple alleles and for multiple loci, assuming linkage equilibrium, and its potential is a Lyapunov function; (ii) the LF of the particular models are special cases of the Wildcard Model's LF; (iii) the LF of the Wildcard Model can be derived from a LF previously identified for a model of density- and frequency- dependent selection due to Lotka-Volterra competition, with one locus, multiple alleles, multiple species and continuous-time dynamics (Matessi and Jayakar, 1981). We extend the LF with density and frequency dependence to a multilocus, linkage equilibrium dynamics.
Research in Pairs 2008
Tue, 18 Mar 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11352008-03-18T00:00:00ZMatessi, CarloSchneider, KristianWe consider a model of frequency-dependent selection, to which we refer as the Wildcard Model, that accommodates as particular cases a number of diverse models of biologically specific situations. Two very different particular models (Lessard, 1984; Bürger, 2005; Schneider, 2006), subsumed by the Wildcard Model, have been shown in the past to have a Lyapunov functions (LF) under appropriate genetic assumptions. We show that the Wildcard Model: (i) in continuous time is a generalized gradient system for one locus, multiple alleles and for multiple loci, assuming linkage equilibrium, and its potential is a Lyapunov function; (ii) the LF of the particular models are special cases of the Wildcard Model's LF; (iii) the LF of the Wildcard Model can be derived from a LF previously identified for a model of density- and frequency- dependent selection due to Lotka-Volterra competition, with one locus, multiple alleles, multiple species and continuous-time dynamics (Matessi and Jayakar, 1981). We extend the LF with density and frequency dependence to a multilocus, linkage equilibrium dynamics.Nonlinear Optimization for Matroid Intersection and Extensions
http://publications.mfo.de/handle/mfo/1134
Nonlinear Optimization for Matroid Intersection and Extensions
Berstein, Yael; Lee, Jon; Onn, Shmuel; Weismantel, Robert
We address optimization of nonlinear functions of the form $f(W_x)$ , where $f : \mathbb{R}^d \to \mathbb{R}$ is a
nonlinear function, $W$ is a $d \times n$ matrix, and feasible $x$ are in some large finite set $\mathcal{F}$ of integer points in $\mathbb{R}^n$ . Generally, such problems are intractable, so we obtain positive algorithmic results by looking
at broad natural classes of $f$, $W$ and $ \mathcal{F}$.
One of our main motivations is multi-objective discrete optimization, where $f$ trades off the linear functions given by the rows of $W$ . Another motivation is that we want to extend as much as possible the known results about polynomial-time linear optimization over trees, assignments, matroids, polymatroids, etc. to nonlinear optimization over such structures.
We assume that the convex hull of $\mathcal{F}$ is well-described by linear inequalities (i.e., we have an efficient separation oracle). For example, the set of characteristic vectors of common bases of a pair of matroids on a common ground set satisfies this property for $\mathcal{F}$. In this setting, the problem is already known to
be intractable (even for a single matroid), for general $f$ (given by a comparison oracle), for (i) $d = 1$ and binary-encoded $W$, and for (ii) $d = n$ and $W = I$.
Research in Pairs 2007
Mon, 17 Mar 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11342008-03-17T00:00:00ZBerstein, YaelLee, JonOnn, ShmuelWeismantel, RobertWe address optimization of nonlinear functions of the form $f(W_x)$ , where $f : \mathbb{R}^d \to \mathbb{R}$ is a
nonlinear function, $W$ is a $d \times n$ matrix, and feasible $x$ are in some large finite set $\mathcal{F}$ of integer points in $\mathbb{R}^n$ . Generally, such problems are intractable, so we obtain positive algorithmic results by looking
at broad natural classes of $f$, $W$ and $ \mathcal{F}$.
One of our main motivations is multi-objective discrete optimization, where $f$ trades off the linear functions given by the rows of $W$ . Another motivation is that we want to extend as much as possible the known results about polynomial-time linear optimization over trees, assignments, matroids, polymatroids, etc. to nonlinear optimization over such structures.
We assume that the convex hull of $\mathcal{F}$ is well-described by linear inequalities (i.e., we have an efficient separation oracle). For example, the set of characteristic vectors of common bases of a pair of matroids on a common ground set satisfies this property for $\mathcal{F}$. In this setting, the problem is already known to
be intractable (even for a single matroid), for general $f$ (given by a comparison oracle), for (i) $d = 1$ and binary-encoded $W$, and for (ii) $d = n$ and $W = I$.Noncommutative topological entropy of endomorphismus of Cuntz Algebras
http://publications.mfo.de/handle/mfo/1133
Noncommutative topological entropy of endomorphismus of Cuntz Algebras
Skalski, Adam; Zacharias, Joachim
Noncommutative topological entropy estimates are obtained for ‘finite range’ endomorphisms of Cuntz algebras,generalising known results for the canonical shift endomorphisms. Exact values are computed for a class of polynomial endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura.
Research in Pairs 2008
Sun, 16 Mar 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11332008-03-16T00:00:00ZSkalski, AdamZacharias, JoachimNoncommutative topological entropy estimates are obtained for ‘finite range’ endomorphisms of Cuntz algebras,generalising known results for the canonical shift endomorphisms. Exact values are computed for a class of polynomial endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura.A 3-local characterization of Co2
http://publications.mfo.de/handle/mfo/1132
A 3-local characterization of Co2
Parker, Christopher; Rowley, Peter
Conway’s second largest simple group, $Co_2$, is characterized
by the centralizer of an element of order 3 and certain
fusion data.
Research in Pairs 2007
Sat, 15 Mar 2008 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11322008-03-15T00:00:00ZParker, ChristopherRowley, PeterConway’s second largest simple group, $Co_2$, is characterized
by the centralizer of an element of order 3 and certain
fusion data.