@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} } @misc{KiselevWolf2010, author = {Kiselev, A. V. and Wolf, Thomas}, title = {Classification of integrable super-systems using the SsTools environment}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/nlin.SI/0609065}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11792}, number = {10-17}, year = {2010}, abstract = {A classification problem is proposed for supersymmetric \%scaling\/-\/in\-va\-ri\-ant evolutionary PDE that satisfy the assumptions of nonlinearity, nondegeneracy, and homogeneity. Four classes of nonlinear coupled boson\/-\/fermion systems are discovered under the weighting assumption \$|f|=|b|=|D_t|=\oh\$. The syntax of the \Reduce\ package \SsTools, which was used for intermediate computations, and the applicability of its procedures to the calculus of super\/-\/PDE are described.}, language = {en} } @misc{WolfEfimovskaya2010, author = {Wolf, Thomas and Efimovskaya, Olya V.}, title = {Classification of integrable quadratic Hamiltonians on e(3)}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11807}, number = {10-18}, year = {2010}, abstract = {Linear Poisson brackets on e(3) typical of rigid body dynamics are considered. All quadratic Hamiltonians of Kowalevski type having additional first integral of fourth degree are found. Quantum analogs of these Hamiltonians are listed.}, language = {en} } @misc{KiselevWolf2005, author = {Kiselev, Arthemy V. and Wolf, Thomas}, title = {On weakly non-local, nilpotent, and super-recursion operators for N=1 super-equations}, arxiv = {http://arxiv.org/abs/nlin.SI/0511056}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8850}, number = {05-52}, year = {2005}, abstract = {We consider nonlinear, scaling-invariant \$N=1\$ boson\$+\$fermion supersymmetric systems whose right-hand sides are homogeneous differential polynomials and satisfy some natural assumptions. We select the super-systems that admit infinitely many higher symmetries generated by recursion operators; we further restrict ourselves to the case when the dilaton dimensions of the bosonic and fermionic super-fields coincide and the weight of the time is half the weight of the spatial variable. We discover five systems that satisfy these assumptions; one system is transformed to the purely bosonic Burgers equation. We construct local, nilpotent, triangular, weakly non-local, and super-recursion operators for their symmetry algebras.}, language = {en} } @misc{KiselevWolf2005, author = {Kiselev, Arthemy V. and Wolf, Thomas}, title = {Supersymmetric representations and integrable super-extensions of the Burgers and Boussinesq equations}, arxiv = {http://arxiv.org/abs/math-ph/0511071}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8869}, number = {05-53}, year = {2005}, abstract = {New evolutionary supersymmetric systems whose right-hand sides are homogeneous differential polynomials and which possess infinitely many higher symmetries are constructed. Their intrinsic geometry (symmetries, conservation laws, recursion operators, Hamiltonian structures, and exact solutions) is analyzed by using algebraic methods. A supersymmetric \$N=1\$ representation of the Burgers equation is obtained. An \$N=2\$ KdV-component system that reduces to the Burgers equation in the diagonal \$N=1\$ case \$\theta^1=\theta^2\$ is found; the \$N=2\$ Burgers equation admits and \$N=2\$ modified KdV symmetry. A one\/-\/parametric family of \$N=0\$ super\/-\/systems that exte nd the Burgers equation is described; we relate the systems within this family with the Burgers equation on associative algebras. A supersymmetric boson\$+\$fermion representation of the dispersionless Boussinesq equation is investigated. We solve this equation explicitly and construct its integrable deformation that generates two infinite sequences of the Hamiltonians. The Boussinesq equation with dispersion is embedded in a one-parametric family of two-component systems with dissipation. We finally construct a three-parametric supersymmetric system that incorporates the Boussinesq equation with dispersion and dissipation but never retracts to it for any values of the parameters.}, language = {en} } @article{OdesskiiWolf2013, author = {Odesskii, Alexander and Wolf, Thomas}, title = {Compatible quadratic Poisson brackets related to a family of elliptic curves}, volume = {63}, journal = {Journal of Geometry and Physics}, number = {0}, issn = {0393-0440}, pages = {107 -- 117}, year = {2013}, language = {en} } @article{AncoMacNaughtonWolf2012, author = {Anco, Stephen and MacNaughton, Steven A. and Wolf, Thomas}, title = {Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions}, volume = {53}, journal = {Journal of Mathematical Physics}, number = {5}, doi = {10.1063/1.4711814}, year = {2012}, language = {en} } @article{WolfSchrueferWebster2012, author = {Wolf, Thomas and Schr{\"u}fer, Eberhard and Webster, Kenneth}, title = {Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs}, volume = {38}, journal = {Programming and Computer Software}, number = {2}, issn = {0361-7688}, doi = {10.1134/S0361768812020065}, pages = {73 -- 83}, year = {2012}, language = {en} } @article{EulerEulerWolf2013, author = {Euler, Marianna and Euler, Norbert and Wolf, Thomas}, title = {The Two-Component Camassa-Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws}, volume = {19}, journal = {Journal of Nonlinear Mathematical Physics}, number = {1}, doi = {10.1142/S1402925112400025}, pages = {13 -- 22}, year = {2013}, abstract = {Recently, Holm and Ivanov, proposed and studied a class of multi-component generalizations of the Camassa-Holm equations [D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys A: Math. Theor. 43 (2010) 492001 (20pp)]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete set of first-order integrating factors for the systems in Cauchy-Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).}, language = {en} } @article{AncoMohiuddinWolf2012, author = {Anco, Stephen and Mohiuddin, Mohammad and Wolf, Thomas}, title = {Traveling waves and conservation laws for complex mKdV-type equations}, volume = {219}, number = {2}, doi = {10.1016/j.amc.2012.06.061}, pages = {679 -- 698}, year = {2012}, 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} }