http://publications.mfo.de/handle/mfo/1340 Wed, 08 Apr 2026 15:44:47 GMT 2026-04-08T15:44:47Z 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 GMT http://publications.mfo.de/handle/mfo/1222 2008-01-01T00:00:00Z 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 $R=k[z]/(z^2)$. 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 GMT http://publications.mfo.de/handle/mfo/1221 2008-01-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1220 2008-01-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1219 2008-01-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1137 2008-03-20T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1136 2008-03-19T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1135 2008-03-18T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1134 2008-03-17T00:00:00Z 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 $W = I$. 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 GMT http://publications.mfo.de/handle/mfo/1133 2008-03-16T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1132 2008-03-15T00:00:00Z 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.