2014
http://publications.mfo.de/handle/mfo/1346
Fri, 21 Jun 2024 01:48:12 GMT2024-06-21T01:48:12ZAlgebraic Matroids with Graph Symmetry
http://publications.mfo.de/handle/mfo/202
Algebraic Matroids with Graph Symmetry
Király, Franz J.; Rosen, Zvi; Theran, Louis
This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal polynomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely.
OWLF 2013
Wed, 01 Jan 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/2022014-01-01T00:00:00ZKirály, Franz J.Rosen, ZviTheran, LouisThis paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal polynomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely.An Explicit Formula for the Dirac Multiplicities on Lens Spaces
http://publications.mfo.de/handle/mfo/188
An Explicit Formula for the Dirac Multiplicities on Lens Spaces
Boldt, Sebastian; Lauret, Emilio A.
We present a new description of the spectrum of the (spin-) Dirac operator $D$ on lens spaces. Viewing a spin lens space $L$ as a locally symmetric space $\Gamma \setminus Spin(2m)/Spin(2m-1)$ and exploiting the representation theory of the Spin groups, we obtain explicit formulas for the multiplicities of the eigenvalues of $D$ in terms of infinitely many integer operations. As a consequence, we present conditions for lens spaces to be Dirac isospectral. Tackling classic questions of spectral geometry, we prove with the tools developed that neither spin structures nor isometry classes of lens spaces are spectrally determined by giving infinite families of Dirac isospectral lens spaces. These results are complemented by examples found with the help of a computer.
OWLF 2013
Wed, 01 Jan 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1882014-01-01T00:00:00ZBoldt, SebastianLauret, Emilio A.We present a new description of the spectrum of the (spin-) Dirac operator $D$ on lens spaces. Viewing a spin lens space $L$ as a locally symmetric space $\Gamma \setminus Spin(2m)/Spin(2m-1)$ and exploiting the representation theory of the Spin groups, we obtain explicit formulas for the multiplicities of the eigenvalues of $D$ in terms of infinitely many integer operations. As a consequence, we present conditions for lens spaces to be Dirac isospectral. Tackling classic questions of spectral geometry, we prove with the tools developed that neither spin structures nor isometry classes of lens spaces are spectrally determined by giving infinite families of Dirac isospectral lens spaces. These results are complemented by examples found with the help of a computer.Spherical Actions on Flag Varieties
http://publications.mfo.de/handle/mfo/187
Spherical Actions on Flag Varieties
Avdeev, Roman; Petukhov, Alexey
For every finite-dimensional vector space $V$ and every $V$ -flag variety $X$ we list all connected reductive subgroups in $GL(V)$ acting spherically on $X$.
OWLF 2013
Fri, 25 Apr 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1872014-04-25T00:00:00ZAvdeev, RomanPetukhov, AlexeyFor every finite-dimensional vector space $V$ and every $V$ -flag variety $X$ we list all connected reductive subgroups in $GL(V)$ acting spherically on $X$.Central Limit Theorems for the Radial Spanning Tree
http://publications.mfo.de/handle/mfo/1090
Central Limit Theorems for the Radial Spanning Tree
Schulte, Matthias; Thäle, Christoph
Consider a homogeneous Poisson point process in a compact convex set in d-dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length.
Research in Pairs 2014
Wed, 01 Jan 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10902014-01-01T00:00:00ZSchulte, MatthiasThäle, ChristophConsider a homogeneous Poisson point process in a compact convex set in d-dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length.A Generalization of the Discrete Version of Minkowski’s Fundamental Theorem
http://publications.mfo.de/handle/mfo/1089
A Generalization of the Discrete Version of Minkowski’s Fundamental Theorem
González Merino, Bernardo; Henze, Matthias
One of the most fruitful results from Minkowski’s geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of an o-symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.
Research in Pairs 2014
Wed, 01 Jan 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10892014-01-01T00:00:00ZGonzález Merino, BernardoHenze, MatthiasOne of the most fruitful results from Minkowski’s geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of an o-symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.Cocharacter-Closure and the Rational Hilbert-Mumford Theorem
http://publications.mfo.de/handle/mfo/1088
Cocharacter-Closure and the Rational Hilbert-Mumford Theorem
Bate, Michael; Herpel, Sebastian; Martin, Benjamin; Röhrle, Gerhard
For a field $k$, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. Using the notion of cocharacter-closed $G(k)$-orbits in $V$ , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.
Research in Pairs 2012
Sat, 20 Dec 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10882014-12-20T00:00:00ZBate, MichaelHerpel, SebastianMartin, BenjaminRöhrle, GerhardFor a field $k$, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. Using the notion of cocharacter-closed $G(k)$-orbits in $V$ , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.Nonlinear Multi-Parameter Eigenvalue Problems for Systems of Nonlinear Ordinary Differential Equations Arising in Electromagnetics
http://publications.mfo.de/handle/mfo/1087
Nonlinear Multi-Parameter Eigenvalue Problems for Systems of Nonlinear Ordinary Differential Equations Arising in Electromagnetics
Angermann, Lutz; Shestopalov, Yury V.; Smirnov, Yury G.; Yatsyk, Vasyl V.
We investigate a generalization of one-parameter eigenvalue problems arising in the theory of nonlinear waveguides to a more general nonlinear multiparameter eigenvalue problem for a nonlinear operator. Using an integral equation approach, we derive functional dispersion equations whose roots yield the desired eigenvalues. The existence and distribution of roots are verified.
Research in Pairs 2014
Sat, 20 Dec 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10872014-12-20T00:00:00ZAngermann, LutzShestopalov, Yury V.Smirnov, Yury G.Yatsyk, Vasyl V.We investigate a generalization of one-parameter eigenvalue problems arising in the theory of nonlinear waveguides to a more general nonlinear multiparameter eigenvalue problem for a nonlinear operator. Using an integral equation approach, we derive functional dispersion equations whose roots yield the desired eigenvalues. The existence and distribution of roots are verified.Abundance of 3-Planes on Real Projective Hypersurfaces
http://publications.mfo.de/handle/mfo/1086
Abundance of 3-Planes on Real Projective Hypersurfaces
Finashin, Sergey; Kharlamov, Viatcheslav
We show that a generic real projective n-dimensional hypersurface of odd degree $d$, such that $4(n-2)=\binom{d+3}{3}$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3$ log d, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.
Research in Pairs 2014
Tue, 11 Nov 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10862014-11-11T00:00:00ZFinashin, SergeyKharlamov, ViatcheslavWe show that a generic real projective n-dimensional hypersurface of odd degree $d$, such that $4(n-2)=\binom{d+3}{3}$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3$ log d, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.Equidistribution of Elements of Norm 1 in Cyclic Extensions
http://publications.mfo.de/handle/mfo/1085
Equidistribution of Elements of Norm 1 in Cyclic Extensions
Petersen, Kathleen L.; Sinclair, Christopher D.
Upon quotienting by units, the elements of norm 1 in a number field $K$ form a countable subset of a torus of dimension $r_1+r_2-1$ where $r_1$ are the numbers of real and pairs of complex embeddings. When $K$ is Galois with cyclic Galois group we demonstrate that this countable set is equidistributed in this torus with respect to a natural partial ordering.
Research in Pairs 2013
Wed, 20 Aug 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10852014-08-20T00:00:00ZPetersen, Kathleen L.Sinclair, Christopher D.Upon quotienting by units, the elements of norm 1 in a number field $K$ form a countable subset of a torus of dimension $r_1+r_2-1$ where $r_1$ are the numbers of real and pairs of complex embeddings. When $K$ is Galois with cyclic Galois group we demonstrate that this countable set is equidistributed in this torus with respect to a natural partial ordering.Random dynamics of transcendental functions
http://publications.mfo.de/handle/mfo/1084
Random dynamics of transcendental functions
Mayer, Volker; Urbański, Mariusz
This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order and whose derivative satisfies some growth condition at infinity. This class contains most of the classical families of transcendental functions and goes much beyond. Based on uniform versions of Nevanlinna’s value distribution theory we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with exponential decay of correlations and a central limit theorem are shown. This part relies on our construction of new positive invariant cones that are adapted to the setting of unbounded phase spaces. This setting rules out the use of Hilbert’s metric along with the usual contraction principle. However these cones allow us to apply a contraction argument stemming from Bowen’s initial approach.
Research in Pairs 2013
Wed, 20 Aug 2014 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10842014-08-20T00:00:00ZMayer, VolkerUrbański, MariuszThis work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order and whose derivative satisfies some growth condition at infinity. This class contains most of the classical families of transcendental functions and goes much beyond. Based on uniform versions of Nevanlinna’s value distribution theory we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with exponential decay of correlations and a central limit theorem are shown. This part relies on our construction of new positive invariant cones that are adapted to the setting of unbounded phase spaces. This setting rules out the use of Hilbert’s metric along with the usual contraction principle. However these cones allow us to apply a contraction argument stemming from Bowen’s initial approach.