2010
http://publications.mfo.de/handle/mfo/1342
Wed, 20 Jan 2021 17:39:04 GMT2021-01-20T17:39:04ZSupertropical Matrix Algebra III : Powers of Matrices and Generalized Eigenspaces
http://publications.mfo.de/handle/mfo/1237
Supertropical Matrix Algebra III : Powers of Matrices and Generalized Eigenspaces
Izhakian, Zur; Rowen, Louis
We investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in controlling the rank of a power of a matrix. This leads to a Jordan-type decomposition of supertropical matrices, together with a generalized eigenspace decomposition of a power of an arbitrary supertropical matrix.
OWLF 2010
Fri, 01 Jan 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12372010-01-01T00:00:00ZIzhakian, ZurRowen, LouisWe investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in controlling the rank of a power of a matrix. This leads to a Jordan-type decomposition of supertropical matrices, together with a generalized eigenspace decomposition of a power of an arbitrary supertropical matrix.Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into a projective variety intersecting hypersurfaces
http://publications.mfo.de/handle/mfo/1236
Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into a projective variety intersecting hypersurfaces
Tran, Van Tan; Vu, Van Truong
In 1985, Fujimoto established a non-integrated defect relation for meromorphic maps of complete Kähler manifolds into the complex projective space intersecting hyperplanes in general position. In this paper, we generalize the result of Fujimoto to the case of meromorphic maps into a complex projective variety intersecting hypersurfaces in general position.
OWLF 2010
Fri, 01 Jan 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12362010-01-01T00:00:00ZTran, Van TanVu, Van TruongIn 1985, Fujimoto established a non-integrated defect relation for meromorphic maps of complete Kähler manifolds into the complex projective space intersecting hyperplanes in general position. In this paper, we generalize the result of Fujimoto to the case of meromorphic maps into a complex projective variety intersecting hypersurfaces in general position.Weak-duality based adaptive finite element methods for PDE-constrained optimization with pointwise gradient state-constraints
http://publications.mfo.de/handle/mfo/1234
Weak-duality based adaptive finite element methods for PDE-constrained optimization with pointwise gradient state-constraints
Hintermüller, Michael; Hinze, Michael; Hoppe, Ronald H. W.
Adaptive finite element methods for optimization problems for second order linear elliptic partial di erential equations subject to pointwise constraints on the $\ell^2$-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint quali cation such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.
OWLF 2009
Fri, 01 Jan 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12342010-01-01T00:00:00ZHintermüller, MichaelHinze, MichaelHoppe, Ronald H. W.Adaptive finite element methods for optimization problems for second order linear elliptic partial di erential equations subject to pointwise constraints on the $\ell^2$-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint quali cation such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.Supertropical linear algebra
http://publications.mfo.de/handle/mfo/1233
Supertropical linear algebra
Izhakian, Zur; Knebusch, Manfred; Rowen, Louis
The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of "ghost surpasses." Special attention is paid to the various notions of "base," which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in a d-base may vary according to the d-base, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identifed with a set of "critical" elements. Linear functionals and the dual space are also studied, leading to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence, as well as a supertropical version of a theorem of Artin.
OWLF 2009
Fri, 01 Jan 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12332010-01-01T00:00:00ZIzhakian, ZurKnebusch, ManfredRowen, LouisThe objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of "ghost surpasses." Special attention is paid to the various notions of "base," which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in a d-base may vary according to the d-base, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identifed with a set of "critical" elements. Linear functionals and the dual space are also studied, leading to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence, as well as a supertropical version of a theorem of Artin.On the complement of the dense orbit for a quiver of type A
http://publications.mfo.de/handle/mfo/1232
On the complement of the dense orbit for a quiver of type A
Baur, Karin; Hille, Lutz
Let $\mathbb{A}_t$ be the directed quiver of type $A$ with $t$ vertices. For each dimension vector $d$ there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the irreducible components in the complement of the dense orbit. Then we compare this result with already existing ones by Knight and Zelevinsky, and by Ringel. Moreover, we compare with the fan associated to the quiver $\mathbb{A}$ and derive a new formula for the number of orbits using nilpotent classes. In the complement of the dense orbit we determine the irreducible components and their codimension. Finally, we consider several particular examples.
Research in Pairs 2009
Fri, 01 Jan 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12322010-01-01T00:00:00ZBaur, KarinHille, LutzLet $\mathbb{A}_t$ be the directed quiver of type $A$ with $t$ vertices. For each dimension vector $d$ there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the irreducible components in the complement of the dense orbit. Then we compare this result with already existing ones by Knight and Zelevinsky, and by Ringel. Moreover, we compare with the fan associated to the quiver $\mathbb{A}$ and derive a new formula for the number of orbits using nilpotent classes. In the complement of the dense orbit we determine the irreducible components and their codimension. Finally, we consider several particular examples.A series of algebras generalizing the Octonions and Hurwitz-Radon Identity
http://publications.mfo.de/handle/mfo/1231
A series of algebras generalizing the Octonions and Hurwitz-Radon Identity
Morier-Genoud, Sophie; Ovsienko, Valentin
We study non-associative twisted group algebras over $(\mathbb{Z}_2)^n$ with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: $(a_1^2+...+a_N^2)(b_1^2+...+b^2_{\rho(N)})=c_1^2+...+c_N^2$, where $c_k$ are bilinear functions of the $a_i$ and $b_j$ and where $\rho(N)$ is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop.
OWLF 2010
Fri, 01 Jan 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12312010-01-01T00:00:00ZMorier-Genoud, SophieOvsienko, ValentinWe study non-associative twisted group algebras over $(\mathbb{Z}_2)^n$ with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: $(a_1^2+...+a_N^2)(b_1^2+...+b^2_{\rho(N)})=c_1^2+...+c_N^2$, where $c_k$ are bilinear functions of the $a_i$ and $b_j$ and where $\rho(N)$ is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop.Supertropical semirings and supervaluations
http://publications.mfo.de/handle/mfo/1230
Supertropical semirings and supervaluations
Izhakian, Zur; Knebusch, Manfred; Rowen, Louis
We interpret a valuation $\upsilon$ on a ring $R$ as a map $\upsilon:R \rightarrow M$ into a so called bipotent semiring $M$ (the usual max-plus setting), and then define a supervaluation $\varphi$ as a suitable map into a supertropical semiring $U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $\upsilon$ via the ghost map $U \rightarrow M$. The set Cov($\upsilon$) of all supervaluations covering $\upsilon$ has a natural ordering which makes it a complete lattice. In the case that $R$ is a field, hence for $\upsilon$ a Krull valuation, we give a complete explicit description of Cov($\upsilon$). The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual max-plus setting. We illustrate this by giving a supertropical version of Kapranov's lemma.
OWLF 2009
Fri, 01 Jan 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12302010-01-01T00:00:00ZIzhakian, ZurKnebusch, ManfredRowen, LouisWe interpret a valuation $\upsilon$ on a ring $R$ as a map $\upsilon:R \rightarrow M$ into a so called bipotent semiring $M$ (the usual max-plus setting), and then define a supervaluation $\varphi$ as a suitable map into a supertropical semiring $U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $\upsilon$ via the ghost map $U \rightarrow M$. The set Cov($\upsilon$) of all supervaluations covering $\upsilon$ has a natural ordering which makes it a complete lattice. In the case that $R$ is a field, hence for $\upsilon$ a Krull valuation, we give a complete explicit description of Cov($\upsilon$). The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual max-plus setting. We illustrate this by giving a supertropical version of Kapranov's lemma.A construction of hyperbolic Coxeter groups
http://publications.mfo.de/handle/mfo/1229
A construction of hyperbolic Coxeter groups
Osajda, Damian
We give a simple construction of Gromov hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension. Our construction provides new examples of such groups. Using this one can construct e.g. new groups having some interesting asphericity properties.
OWLF 2010
Fri, 01 Jan 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12292010-01-01T00:00:00ZOsajda, DamianWe give a simple construction of Gromov hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension. Our construction provides new examples of such groups. Using this one can construct e.g. new groups having some interesting asphericity properties.There is a unique real tight contact 3-ball
http://publications.mfo.de/handle/mfo/1228
There is a unique real tight contact 3-ball
Öztürk, Ferit; Salepci, Nermin
We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional direction. For the proofs, we use the real versions of contact neighborhood theorems and the invariant convex surface theory in real contact manifolds.
OWLF 2009
Fri, 01 Jan 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12282010-01-01T00:00:00ZÖztürk, FeritSalepci, NerminWe prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional direction. For the proofs, we use the real versions of contact neighborhood theorems and the invariant convex surface theory in real contact manifolds.Stochastic mean payoff game: smoothed analysis and approximation schemes
http://publications.mfo.de/handle/mfo/1176
Stochastic mean payoff game: smoothed analysis and approximation schemes
Boros, Endre; Elbassioni, Khaled; Fouz, Mahmoud; Gurvich, Vladimir; Manthey, Bodo
We consider two-person zero-sum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWR-games games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games, stochastic parity games, and Markov decision processes. They can also be used to model parlor games such as Chess or Backgammon. It is a long-standing open question whether a polynomial algorithm exists that solves BWR-games. In fact, a pseudo-polynomial algorithm for these games with an arbitrary number of random nodes would already imply their polynomial solvability. Currently, only two classes are known to have such a pseudo-polynomial algorithm: BW-games (the case with no random nodes) and ergodic BWR-games (i.e., in which the game's value does not depend on the initial position) with constant number of random nodes. In this paper, we show that the existence of a pseudo-polynomial algorithm for BWR-games with constant number of random vertices implies smoothed polynomial time complexity and the existence of absolute and relative polynomial-time approximation schemes. In particular, we obtain smoothed polynomial time complexity and derive absolute and relative approximation schemes for the above two classes.
Research in Pairs 2010
Sat, 20 Mar 2010 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11762010-03-20T00:00:00ZBoros, EndreElbassioni, KhaledFouz, MahmoudGurvich, VladimirManthey, BodoWe consider two-person zero-sum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWR-games games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games, stochastic parity games, and Markov decision processes. They can also be used to model parlor games such as Chess or Backgammon. It is a long-standing open question whether a polynomial algorithm exists that solves BWR-games. In fact, a pseudo-polynomial algorithm for these games with an arbitrary number of random nodes would already imply their polynomial solvability. Currently, only two classes are known to have such a pseudo-polynomial algorithm: BW-games (the case with no random nodes) and ergodic BWR-games (i.e., in which the game's value does not depend on the initial position) with constant number of random nodes. In this paper, we show that the existence of a pseudo-polynomial algorithm for BWR-games with constant number of random vertices implies smoothed polynomial time complexity and the existence of absolute and relative polynomial-time approximation schemes. In particular, we obtain smoothed polynomial time complexity and derive absolute and relative approximation schemes for the above two classes.