@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} } @article{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}, journal = {SIGMA 7}, doi = {10.3842/SIGMA.2011.066}, 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 uses 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 group-foliation reduction 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} } @article{AncoAliWolf2011, author = {Anco, Stephen and Ali, Sajid and Wolf, Thomas}, title = {Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions}, volume = {379}, journal = {Journal of Mathematical Analysis and Applications}, number = {2}, doi = {10.1016/j.jmaa.2011.01.073}, pages = {748 -- 763}, year = {2011}, language = {en} } @article{Wolf2011, author = {Wolf, Thomas}, title = {A Dynamical Systems Approach for Static Evaluation in Go}, volume = {3}, journal = {Computational Intelligence and AI in Games, IEEE Transactions on}, number = {2}, doi = {10.1109/TCIAIG.2011.2141669}, pages = {129 -- 141}, year = {2011}, abstract = {In the paper, arguments are given as to why the concept of static evaluation has the potential to be a useful extension to Monte Carlo tree search. A new concept of modeling static evaluation through a dynamical system is introduced and strengths and weaknesses are discussed. The general suitability of this approach is demonstrated.}, language = {en} }