2012
http://publications.mfo.de/handle/mfo/1344
Thu, 08 Aug 2024 20:32:20 GMT2024-08-08T20:32:20ZOn the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc
http://publications.mfo.de/handle/mfo/1253
On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc
Brandenbursky, Michael; Kȩdra, Jarek
Let $D^2$ be the open unit disc in the Euclidean plane and let $G := Diff(D^2; area)$ be the group of smooth compactly supported area-preserving diffeomorphisms of $D^2$. For every natural number $k$ we construct an injective homomorphism $Z^k → G$, which is bi-Lipschitz with respect to the word metric on $Z^k$ and the autonomous metric on $G$. We also show that the space of homogeneous quasi-morphisms vanishing on all autonomous diffeomorphisms in the above group is infinite dimensional.
OWLF 2012
Sun, 01 Jan 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12532012-01-01T00:00:00ZBrandenbursky, MichaelKȩdra, JarekLet $D^2$ be the open unit disc in the Euclidean plane and let $G := Diff(D^2; area)$ be the group of smooth compactly supported area-preserving diffeomorphisms of $D^2$. For every natural number $k$ we construct an injective homomorphism $Z^k → G$, which is bi-Lipschitz with respect to the word metric on $Z^k$ and the autonomous metric on $G$. We also show that the space of homogeneous quasi-morphisms vanishing on all autonomous diffeomorphisms in the above group is infinite dimensional.Invariants of Closed Braids via Counting Surfaces
http://publications.mfo.de/handle/mfo/1252
Invariants of Closed Braids via Counting Surfaces
Brandenbursky, Michael
A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.
OWLF 2012
Sun, 01 Jan 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12522012-01-01T00:00:00ZBrandenbursky, MichaelA Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.Positive Margins and Primary Decomposition
http://publications.mfo.de/handle/mfo/1251
Positive Margins and Primary Decomposition
Kahle, Thomas; Rauh, Johannes; Sullivant, Seth
We study random walks on contingency tables with fixed marginals, corresponding to a (log-linear) hierarchical model. If the set of allowed moves is not a Markov basis, then there exist tables with the same marginals that are not connected. We study linear conditions on the values of the marginals that ensure that all tables in a given fiber are connected. We show that many graphical models have the positive margins property, which says that all fibers with strictly positive marginals are connected by the quadratic moves that correspond to conditional independence statements. The property persists under natural operations such as gluing along cliques, but we also construct examples of graphical models not enjoying this property. Our analysis of the positive margins property depends on computing the primary decomposition of the associated conditional independence ideal. The main technical results of the paper are primary decompositions of the conditional independence ideals of graphical models of the $N$-cycle and the complete bipartite graph $K_{2,N2-2}$, with various restrictions on the size of the nodes.
OWLF 2011
Sun, 01 Jan 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12512012-01-01T00:00:00ZKahle, ThomasRauh, JohannesSullivant, SethWe study random walks on contingency tables with fixed marginals, corresponding to a (log-linear) hierarchical model. If the set of allowed moves is not a Markov basis, then there exist tables with the same marginals that are not connected. We study linear conditions on the values of the marginals that ensure that all tables in a given fiber are connected. We show that many graphical models have the positive margins property, which says that all fibers with strictly positive marginals are connected by the quadratic moves that correspond to conditional independence statements. The property persists under natural operations such as gluing along cliques, but we also construct examples of graphical models not enjoying this property. Our analysis of the positive margins property depends on computing the primary decomposition of the associated conditional independence ideal. The main technical results of the paper are primary decompositions of the conditional independence ideals of graphical models of the $N$-cycle and the complete bipartite graph $K_{2,N2-2}$, with various restrictions on the size of the nodes.Rate of Convergence of the Density Estimation of Regression Residual
http://publications.mfo.de/handle/mfo/1211
Rate of Convergence of the Density Estimation of Regression Residual
Györfi, László; Walk, Harro
Consider the regression problem with a response variable $Y$ and with a $d$-dimensional feature vector $X$. For the regression function $m(x) = \mathbb{E}\{Y|X = x\}$, this paper investigates methods for estimating the density of the residual $Y -m(X)$ from independent and identically distributed data. If the density is twice differentiable and has compact support then we bound the rate of convergence of the kernel density estimate. It turns out that for $d\leq3$ and for partitioning regression estimates, the regression estimation error has no influence in the rate of convergence of the density estimate.
Research in Pairs 2012
Sun, 01 Jan 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12112012-01-01T00:00:00ZGyörfi, LászlóWalk, HarroConsider the regression problem with a response variable $Y$ and with a $d$-dimensional feature vector $X$. For the regression function $m(x) = \mathbb{E}\{Y|X = x\}$, this paper investigates methods for estimating the density of the residual $Y -m(X)$ from independent and identically distributed data. If the density is twice differentiable and has compact support then we bound the rate of convergence of the kernel density estimate. It turns out that for $d\leq3$ and for partitioning regression estimates, the regression estimation error has no influence in the rate of convergence of the density estimate.Strongly Consistent Density Estimation of Regression Redidual
http://publications.mfo.de/handle/mfo/1210
Strongly Consistent Density Estimation of Regression Redidual
Györfi, László; Walk, Harro
Consider the regression problem with a response variable $Y$ and with a $d$-dimensional feature vector $X$. For the regression function $m(x) = \mathbb{E}\{Y|X = x\}$, this paper investigates methods for estimating the density of the residual $Y -m(X)$ from independent and identically distributed data. For heteroscedastic regression, we prove the strong universal (density-free) $L_1$-consistency of a recursive and a nonrecursive kernel density estimate based on a regression estimate.
Research in Pairs 2012
Sun, 01 Jan 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12102012-01-01T00:00:00ZGyörfi, LászlóWalk, HarroConsider the regression problem with a response variable $Y$ and with a $d$-dimensional feature vector $X$. For the regression function $m(x) = \mathbb{E}\{Y|X = x\}$, this paper investigates methods for estimating the density of the residual $Y -m(X)$ from independent and identically distributed data. For heteroscedastic regression, we prove the strong universal (density-free) $L_1$-consistency of a recursive and a nonrecursive kernel density estimate based on a regression estimate.Contractive Idempotents on Locally Compact Quantum Groups
http://publications.mfo.de/handle/mfo/1051
Contractive Idempotents on Locally Compact Quantum Groups
Neufang, Matthias; Salmi, Pekka; Skalski, Adam; Spronk, Nico
A general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators. As a consequence a one-to-one correspondence between contractive idempotents and a certain class of ternary rings of operators is established.
Research in Pairs 2012
Sun, 01 Jan 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10512012-01-01T00:00:00ZNeufang, MatthiasSalmi, PekkaSkalski, AdamSpronk, NicoA general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators. As a consequence a one-to-one correspondence between contractive idempotents and a certain class of ternary rings of operators is established.A Uniform Model for Kirillov-Reshetikhin Crystals I: Lifting the Parabolic Quantum Bruhat Graph
http://publications.mfo.de/handle/mfo/1050
A Uniform Model for Kirillov-Reshetikhin Crystals I: Lifting the Parabolic Quantum Bruhat Graph
Lenart, Cristian; Naito, Satoshi; Sagaki, Daisuke; Schilling, Anne; Shimozono, Mark
We consider two lifts of the parabolic quantum Bruhat graph, one into the Bruhat order in the affine Weyl group and the other into a level-zero weight poset first considered by Littelmann. The lift into the affine Weyl group gives rise to Diamond Lemmas for the parabolic quantum Bruhat graph. Littlemann's poset is defined on Lakshmibai-Seshadri paths for arbitrary (not necessarily dominant) weights. Here we consider this poset for level-zero weights and determine its local structure (such as cover relations) in terms of the parabolic quantum Bruhat graph. Littelmann had not determined the local structure of this poset. In addition, we show a generalization of results by Deodhar, coined the tilted-Bruhat theorem, which involves the compatibility of the quantum Bruhat graph with the cosets for every parabolic subgroup of the Weyl group. We will use the results in this paper in a second paper to establish the equality between the Macdonald polynomials $P_\lambda (q,t)$ specialized at $t = 0$ and $X_\lambda (q)$ which is the graded character of a simple Lie algebra coming from tensor products of Kirillov-Reshetikhin (KR) modules.
Research in Pairs 2012
Sun, 01 Jan 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10502012-01-01T00:00:00ZLenart, CristianNaito, SatoshiSagaki, DaisukeSchilling, AnneShimozono, MarkWe consider two lifts of the parabolic quantum Bruhat graph, one into the Bruhat order in the affine Weyl group and the other into a level-zero weight poset first considered by Littelmann. The lift into the affine Weyl group gives rise to Diamond Lemmas for the parabolic quantum Bruhat graph. Littlemann's poset is defined on Lakshmibai-Seshadri paths for arbitrary (not necessarily dominant) weights. Here we consider this poset for level-zero weights and determine its local structure (such as cover relations) in terms of the parabolic quantum Bruhat graph. Littelmann had not determined the local structure of this poset. In addition, we show a generalization of results by Deodhar, coined the tilted-Bruhat theorem, which involves the compatibility of the quantum Bruhat graph with the cosets for every parabolic subgroup of the Weyl group. We will use the results in this paper in a second paper to establish the equality between the Macdonald polynomials $P_\lambda (q,t)$ specialized at $t = 0$ and $X_\lambda (q)$ which is the graded character of a simple Lie algebra coming from tensor products of Kirillov-Reshetikhin (KR) modules.K-Triviality, Oberwolfach randomness, and differentiability
http://publications.mfo.de/handle/mfo/1049
K-Triviality, Oberwolfach randomness, and differentiability
Bienvenu, Laurent; Greenberg, Noam; Kučera, Antonin; Nies, André; Turetsky, Dan
We show that a Martin-Lof random set for which the effective version of the Lebesgue density theorem fails computes every $K$-trivial set. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem (Question 4.6 in Randomness and computability: Open questions. Bull. Symbolic Logic, 12(3):390{410, 2006). On the other hand, we settle stronger variants of the covering problem in the negative. We show that any witness for the solution of the covering problem, namely an incomplete random set which computes all $K$-trivial sets, must be very close to being Turing complete. For example, such a random set must be LR-hard. Similarly, not every $K$-trivial set is computed by the two halves of a random set. The work passes through a notion of randomness which characterises computing K-trivial sets by random sets. This gives a "smart" $K$-trivial set, all randoms from whom this set is computed have to compute all $K$-trivial sets.
Research in Pairs 2012
Fri, 21 Dec 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10492012-12-21T00:00:00ZBienvenu, LaurentGreenberg, NoamKučera, AntoninNies, AndréTuretsky, DanWe show that a Martin-Lof random set for which the effective version of the Lebesgue density theorem fails computes every $K$-trivial set. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem (Question 4.6 in Randomness and computability: Open questions. Bull. Symbolic Logic, 12(3):390{410, 2006). On the other hand, we settle stronger variants of the covering problem in the negative. We show that any witness for the solution of the covering problem, namely an incomplete random set which computes all $K$-trivial sets, must be very close to being Turing complete. For example, such a random set must be LR-hard. Similarly, not every $K$-trivial set is computed by the two halves of a random set. The work passes through a notion of randomness which characterises computing K-trivial sets by random sets. This gives a "smart" $K$-trivial set, all randoms from whom this set is computed have to compute all $K$-trivial sets.On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras
http://publications.mfo.de/handle/mfo/1217
On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras
Goodwin, Simon M.; Röhrle, Gerhard
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $\mathbb{k}$ of characteristic zero. We consider the commuting variety $\mathcal{C}(\mathfrak{u})$ of the nilradical $\mathfrak{u}$ of the Lie algebra $\mathfrak{b}$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\mathfrak{u}$ with only a finite number of orbits, we verify that $\mathcal{C}(\mathfrak{u})$ is equidimensional and that the irreducible components are in correspondence with the distinguished $B$-orbits in $\mathfrak{u}$. We observe that in general $\mathcal{C}(\mathfrak{u})$ is not equidimensional, and determine the irreducible components of $\mathcal{C}(\mathfrak{u})$ in the minimal cases where there are infinitely many $B$-orbits in $\mathfrak{u}$.
Research in Pairs 2011
Tue, 04 Dec 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12172012-12-04T00:00:00ZGoodwin, Simon M.Röhrle, GerhardLet $G$ be a connected reductive algebraic group defined over an algebraically closed field $\mathbb{k}$ of characteristic zero. We consider the commuting variety $\mathcal{C}(\mathfrak{u})$ of the nilradical $\mathfrak{u}$ of the Lie algebra $\mathfrak{b}$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\mathfrak{u}$ with only a finite number of orbits, we verify that $\mathcal{C}(\mathfrak{u})$ is equidimensional and that the irreducible components are in correspondence with the distinguished $B$-orbits in $\mathfrak{u}$. We observe that in general $\mathcal{C}(\mathfrak{u})$ is not equidimensional, and determine the irreducible components of $\mathcal{C}(\mathfrak{u})$ in the minimal cases where there are infinitely many $B$-orbits in $\mathfrak{u}$.Polynomiality, wall crossings and tropical geometry of rational double hurwitz cycles
http://publications.mfo.de/handle/mfo/1216
Polynomiality, wall crossings and tropical geometry of rational double hurwitz cycles
Bertram, Aaron; Cavalieri, Renzo; Markwig, Hannah
We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and “modular” description. A main goal of this paper is to simultaneously carry out this investigation for the corresponding objects in tropical geometry, underlining a precise combinatorial duality between classical and tropical Hurwitz theory.
Research in Pairs 2012
Tue, 04 Dec 2012 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12162012-12-04T00:00:00ZBertram, AaronCavalieri, RenzoMarkwig, HannahWe study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and “modular” description. A main goal of this paper is to simultaneously carry out this investigation for the corresponding objects in tropical geometry, underlining a precise combinatorial duality between classical and tropical Hurwitz theory.