@misc{TsarevWolf2009, author = {Tsarev, Sergey and Wolf, Thomas}, title = {Hyperdeterminants as integrable discrete systems}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/0903.3864}, doi = {10.1088/1751-8113/42/45/454023}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11292}, number = {09-17}, year = {2009}, abstract = {We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A.~Cayley in 1845. We prove integrability (understood as \$4d\$-consistency) of a nonlinear difference equation defined by the \$2 \times 2 \times 2\$ - hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants of any size are integrable. We show that this hypothesis already fails in the case of the \$2\times 2\times 2\times 2\$ - hyperdeterminant.}, language = {en} } @misc{WolfNeun2008, author = {Wolf, Thomas and Neun, Winfried}, title = {About a Computer Algebra based online Mathtest}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10688}, number = {08-14}, year = {2008}, abstract = {The article describes the online mathematics test {\tt http://lie.math.brocku.ca/mathtest}, its typical applications and experiences gathered.}, language = {en} } @misc{Wolf2010, author = {Wolf, Thomas}, title = {Positions in the Game of Go as Complex Systems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11823}, number = {10-19}, year = {2010}, abstract = {The paper gathers evidence showing different dimensions of the game of Go: the continuous and discrete nature of the game and different types of relations between state variables happening on ultra local, local, regional, and global scales. Based on these observations a new continuous local model for describing a board position is introduced. This includes the identification of the basic variables describing a board position and the formulation and solution of a dynamical system for their computation. To be usable as a static evaluation function for a game playing program at least group-wide (regional)aspects will have to be incorporated.}, language = {en} } @misc{AncoAliWolf2011, author = {Anco, Stephen and Ali, Sajid and Wolf, Thomas}, title = {Exact solutions of nonlinear partial differential equations by the method of group foliation reduction}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13171}, number = {11-25}, year = {2011}, abstract = {A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method is based on group foliation reduction and employs a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation.}, language = {en} } @misc{AncoAliWolf2011, author = {Anco, Stephen and Ali, Sajid and Wolf, Thomas}, title = {Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13213}, number = {11-26}, year = {2011}, abstract = {A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in \$n\>1\$ dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.}, language = {en} } @misc{WolfEfimovskaya2011, author = {Wolf, Thomas and Efimovskaya, Olga}, title = {On integrability of the Kontsevich non-abelian ODE system}, arxiv = {http://arxiv.org/abs/1108.4208v1}, doi = {10.1007/s11005-011-0527-4}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13113}, number = {11-24}, year = {2011}, abstract = {We consider systems of ODEs with the right hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M. Kontsevich. We prove the integrability of the Kontsevich system by finding a Lax pair, corresponding first integrals and commuting flows. We also provide a pre-Hamiltonian operator which maps gradients of integrals for the Kontsevich system to symmetries.}, language = {en} } @misc{BoucharebClementChenetal.2010, author = {Bouchareb, Adel and Cl{\´e}ment, G{\´e}rard and Chen, Chiang-Mei and Wolf, Thomas and Gal'tsov, Dmitri V. and Scherbluk, Nikolai G.}, title = {G_2 generating technique for minimal D=5 supergravity and black rings}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/0708.2361}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11732}, number = {10-14}, year = {2010}, abstract = {A solution generating technique is developed for \$D=5\$ minimal supergravity with two commuting Killing vectors based on the \$G_2\$ U-duality arising in the reduction of the theory to three dimensions. The target space of the corresponding 3-dimensional sigma-model is the coset \$G_{2(2)}/(SL(2,R)\times SL(2,R))\$. Its isometries constitute the set of solution generating symmetries. These include two electric and two magnetic Harrison transformations with the corresponding two pairs of gauge transformations, three \$SL(2,R) \; S\$-duality transformations, and the three gravitational scale, gauge and Ehlers transformations (altogether 14). We construct a representation of the coset in terms of \$7\times 7\$ matrices realizing the automorphisms of split octonions. Generating a new solution amounts to transforming the coset matrices by one-parametric subgroups of \$G_{2(2)}\$ and subsequently solving the dualization equations. Using this formalism we derive a new charged black ring solution with two independent parameters of rotation.}, language = {en} } @misc{Wolf2010, author = {Wolf, Thomas}, title = {Merging solutions of non-linear algebraic systems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11769}, number = {10-13}, year = {2010}, abstract = {In solving large polynomial algebraic systems that are too big for standard Gr{\"o}bner basis techniques one way to make progress is to introduce case distinctions. This divide and conquer technique can be beneficial if the algorithms and computer programs know how to take advantage of inequalities. A further hurdle is the form of the resulting general solutions which often have unnecessarily many branches. In this paper we discuss a procedure to merge solutions by dropping inequalities which are associated with them and, if necessary, by re-parametrizing solutions. In the appendix the usefulness of the procedure is demonstrated in the classification of quadratic Hamiltonians with a Lie-Poisson bracket \$e(3)\$. This application required the solution of algebraic systems with over 200 unknowns, 450 equations and between 5000 and 9000 terms.}, language = {en} } @misc{Wolf2010, author = {Wolf, Thomas}, title = {A Dynamical Systems Approach for Static Evaluation in Go}, issn = {1438-0064}, doi = {10.1109/TCIAIG.2011.2141669}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11771}, number = {10-15}, year = {2010}, abstract = {In the paper arguments are given why the concept of static evaluation (SE) has the potential to be a useful extension to Monte Carlo tree search. A new concept of modeling SE through a dynamical system is introduced and strengths and weaknesses are discussed. The general suitability of this approach is demonstrated. A Remark: Among users of the Internet Go server KGS the abbreviation SE is used for 'Score Estimator'. Although different from 'Static Evaluation' a score estimator is easily obtained from static evaluation by adding up probabilities of chains to be alive at the end of the game or points to be owned by White or Black}, language = {en} } @misc{HussinKiselevKrutovetal.2010, author = {Hussin, V. and Kiselev, A. V. and Krutov, A.O. and Wolf, Thomas}, title = {N=2 supersymmetric a=4-KdV hierarchy derived via Gardner's deformation of Kaup-Boussinesq equation}, issn = {1438-0064}, doi = {10.1063/1.3447731}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11787}, number = {10-16}, year = {2010}, abstract = {We consider the problem of constructing Gardner's deformations for the \$N{=}2\$ supersymmetric \$a{=}4\$--\/Korteweg\/--\/de Vries equation; such deformations yield recurrence relations between the super\/-\/Hamiltonians of the hierarchy. We prove the non\/-\/existence \%P.~Mathieu's Open problem on constructing for of supersymmetry\/-\/invariant \%Gardner's deformations that \%solutions, retract to Gardner's formulas for the KdV equation \%whenever it is assumed that, under the \%respective component reduction. \% in the \$N{=}2\$ super\/-\/field. the solutions . At the same time, we propose a two\/-\/step scheme for the recursive production of the integrals of motion for the \$N{=}2\$,\ \$a{=}4\$--\/SKdV. First, we find a new Gardner's deformation of the Kaup\/--\/Boussinesq equation, which is contained in the bosonic limit of the super\/-\/\%\$N{=}2\$,\ \$a{=}4\$--\/SKdV hierarchy. This yields the recurrence relation between the Hamiltonians of the limit, whence we determine the bosonic super\/- /Hamiltonians of the full \$N{=}2\$, \$a{=}4\$--\/SKdV hierarchy.}, language = {en} }