@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{WolfEfimovskaya2012, author = {Wolf, Thomas and Efimovskaya, Olga}, title = {On Integrability of the Kontsevich Non-Abelian ODE System}, volume = {100}, journal = {Letters in Mathematical Physics}, number = {2}, doi = {10.1007/s11005-011-0527-4}, pages = {161 -- 170}, year = {2012}, 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} } @article{HussinKiselevKrutovetal.2010, author = {Hussin, V. and Kiselev, A. V. and Krutov, A.O. and Wolf, Thomas}, title = {N=2 Supersymmetric a=4 - Korteweg-de Vries hierarchy derived via Gardner's deformation of Kaup-Bousinesq equation}, volume = {51}, journal = {Journal of Mathematical Physics}, doi = {10.1063/1.3447731}, year = {2010}, language = {en} } @article{Wolf2009, author = {Wolf, Thomas}, title = {Positions in the Game of Go as Complex Systems}, journal = {Proceedings of 2009 IEEE Toronto International Conference (TIC-STH 2009) - Science and Technology for Humanity}, isbn = {978-1-4244-3878-5}, year = {2009}, language = {en} } @article{TsarevWolf2009, author = {Tsarev, Sergey and Wolf, Thomas}, title = {Hyperdeterminants as integrable discrete systems}, volume = {42}, journal = {Journal of Physics A: Mathematical and Theoretical}, doi = {/10.1088/1751-8113/42/45/454023}, year = {2009}, language = {en} } @misc{MarsWolf1997, author = {Mars, Marc and Wolf, Thomas}, title = {G2 Perfect-Fluid Cosmologies with a proper conformal Killing vector}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2774}, number = {SC-97-08}, year = {1997}, abstract = {We study the Einstein field equations for spacetimes admitting a maximal two-dimensional abelian group of isometries acting orthogonally transitively on spacelike surfaces and, in addition, with at least one conformal Killing vector. The three-dimensional conformal group is restricted to the case when the two-dimensional abelian isometry subalgebra is an ideal and it is also assumed to act on non-null hypersurfaces (both, spacelike and timelike cases are studied). We consider both, diagonal and non-diagonal metrics and find all the perfect-fluid solutions under these assumptions (except those already known). We find four families of solutions, each one containing arbitrary parameters for which no differential equations remain to be integrated. We write the line-elements in a simplified form and perform a detailed study for each of these solutions, giving the kinematical quantities of the fluid velocity vector, the energy-density and pressure, values of the parameters for which the energy conditions are fulfilled everywhere, the Petrov type, the singularities in the spacetimes and the Friedmann-Lema\^{\i}tre-Robertson-Walker metrics contained in each family.}, language = {en} } @misc{StephaniWolf1996, author = {Stephani, Hans and Wolf, Thomas}, title = {Spherically symmetric perfect fluids in shearfree motion - the symmetry approach}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2253}, number = {SC-96-14}, year = {1996}, abstract = {In General Relativity, the motion of expanding shearfree perfect fluids is governed by the ordinary differential equation \$y^{\prime \prime }=\$ \$\% F(x)\,y^2\$ , where \$F\$ is an arbitrary function from which the equation of state can be computed. A complete symmetry analysis of this differential equation is given; its solutions are classified according to this scheme, and in particular the relation to Wyman's Painlev\'e analysis is clarified.}, language = {en} } @misc{Wolf1995, author = {Wolf, Thomas}, title = {The program CRACK for solving PDEs in General Relativity. Lecture given at the 152. WE-Heraeus-Seminar on RELATIVITY AND SCIENTIFIC CUMPUTING: Computer Algebra, Numerics, Visualization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1881}, number = {SC-95-22}, year = {1995}, abstract = {In the introduction an approach to solving differential equations is motivated in which non-linear DEs are not attacked directly but properties like infinitesimal symmetries or the existence of an equivalent variational principle are investigated. In the course of such investigations overdetermined PDE-systems are generated which are to be solved (where the term `overdetermined' just stands for `more conditions than free functions'). In section 2.\ algorithms for simplifying and solving overdetermined PDE systems are given together with examples. References for more details of the corresponding program {\tt CRACK}, written by A.\ Brand and the author, are given. In sections 3.-05.\ applications of the program {\tt CRACK} are discussed. The first application is the investigation of symmetries of space-time metrics by solving Killing equations for Killing vectors and Killing tensors and their integrability conditions. A program {\tt CLASSYM} that formulates these equations, written by G.\ Grebot, is briefly described. In section 4.\ an example of the original application of {\tt CRACK} is discussed which is the determination of symmetries of a PDE system. The problem is to find the symmetries of an unusual unified field theory of gravitational and hadronic interactions. The application of symmetries with a program {\tt APPLYSYM} is the content of section 5.\ where an ODE, resulting from an attempt to generalize Weyl's class of solutions of Einsteins field equations, is solved. The final section is devoted to future work on, first, making a general PDE-solver more flexible and effective, and secondly, on applying it to more advanced applications. This section contains so far unpublished work. An example requiring the extension of {\tt CRACK} to deal with non-polynomial non-linearities results from an investigation of interior solutions of Einstein's field equations for a spherically symmetric perfect fluid in shear-free motion by H.\ Stephani. A possible future application of {\tt CRACK} is the determination of Killing tensors of higher rank. In the last sub-section an algorithm for formulating corresponding integrability conditions has been sketched. The maximal number of Killing tensors of rank \$r\$ in a \$n\$-dimensional Riemannian space has been found to be \$\frac{1}{r+1}\left( ^{n + r - 1}_{\;\;\;\;\,r} \right) \left( ^{ n+r}_{\;\;\,r} \right)\$.}, language = {en} } @misc{Wolf1995, author = {Wolf, Thomas}, title = {Programs for Applying Symmetries of PDEs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1710}, number = {SC-95-05}, year = {1995}, abstract = {In this paper the programs {\tt APPLYSYM}, {\tt QUASILINPDE} and {\tt DETRAFO} are described which aim at the utilization of infinitesimal symmetries of differential equations. The purpose of {\tt QUASILINPDE} is the general solution of quasilinear PDEs. This procedure is used by {\tt APPLYSYM} for the application of point symmetries for either \begin{itemize} \item calculating similarity variables to perform a point transformation which lowers the order of an ODE or effectively reduces the number of explicitly occuring independent variables in a PDE(-system) or for \item generalizing given special solutions of ODEs/PDEs with new constant parameters. \end{itemize} The program {\tt DETRAFO} performs arbitrary point- and contact transformations of ODEs/PDEs and is applied if similarity and symmetry variables have been found. The program {\tt APPLYSYM} is used in connection with the program {\tt LIEPDE} for formulating and solving the conditions for point- and contact symmetries which is described in LIEPDE(1992). The actual problem solving is done in all these programs through a call to the package {\tt CRACK} for solving overdetermined PDE-systems.}, language = {en} } @misc{Wolf1999, author = {Wolf, Thomas}, title = {A comparison of four approaches to the calculation of conservation laws}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3896}, number = {SC-99-01}, year = {1999}, abstract = {The paper compares computational aspects of four approaches to compute conservation laws of single differential equations or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints may be given. Examples of new conservation laws include non-polynomial expressions, an explicit variable dependence and conservation laws involving arbitrary functions. Examples involve the following equations: Ito, Liouville, Burgers, Kadomtsev-Petviashvili, Karney-Sen-Chu-Verheest, Boussinesq, Tzetzeica, Benney.}, language = {en} } @misc{SokolovWolf1999, author = {Sokolov, Vladimir V. and Wolf, Thomas}, title = {A symmetry test for quasilinear coupled systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3901}, number = {SC-99-02}, year = {1999}, abstract = {It is well known that the following class of systems of evolution equations \begin{eqnarray} \label{nsgen} \cases{ u_{t}=u_{xx}+F(u,v,u_x,v_x),\cr v_{t}=-v_{xx}+G(u,v,u_x,v_x),\cr} \end{eqnarray} is very rich in integrable cases. The complete classification problem is very difficult. Here we consider only the most interesting (from our opinion) subclass of systems (1). Namely, we consider equations linear in all derivatives of the form \begin{eqnarray} \label{kvazgen} \cases{ u_t = u_{xx} + A_{1}(u,v) u_x + A_{2}(u,v) v_x + A_{0}(u,v)\cr v_t = - v_{xx} + B_{1}(u,v) v_x + B_{2}(u,v) u_x + B_{0}(u,v). \cr} \end{eqnarray} without any restrictions on the functions \$A_{i}(u,v), B_{i}(u,v)\$.}, language = {en} } @misc{Wolf1997, author = {Wolf, Thomas}, title = {Structural equations for Killing tensors of arbitrary rank}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3310}, number = {SC-97-62}, year = {1997}, abstract = {An algorithm is given for bringing the equations of monomial first integrals of arbitrary degree of the geodesic motion in a Riemannian space \$V_n\$ into the form \$(F_A)_{;k} = \sum_B \Gamma_{kAB} F_B\$. The \$F_A\$ are the components of a Killing tensor \$K_{i_1\ldots i_r}\$ of arbitrary rank \$r\$ and its symmetrized covariant derivatives. Explicit formulas are given for rank 1,2 and 3. \%The maximal number of Killing tensors \%(reducible + non-reducible) is found to be \%\$\frac{1}{r+1}\left( ^{n + r - 1}_{\;\;\;\;\,r} \right) \% \left( ^{ n+r}_{\;\;\,r} \right)\$. Killing tensor equations in structural form allow the formulation of algebraic integrability conditions and are supposed to be well suited for integration as it is demonstrated in the case of flat space. An alternative proof of the reducibility of these Killing tensors is given which shows the correspondence to structural equations for rank 2 Killing tensors as formulated by Hauser \& Malhiot. They used tensors with different symmetry properties.}, language = {en} } @article{AbbasAmbainisAugustinoetal.2024, author = {Abbas, Amira and Ambainis, Andris and Augustino, Brandon and B{\"a}rtschi, Andreas and Buhrman, Harry and Coffrin, Carleton and Cortiana, Giorgio and Dunjko, Vedran and Egger, Daniel J. and Elmegreen, Bruce G. and Franco, Nicola and Fratini, Filippo and Fuller, Bryce and Gacon, Julien and Gonciulea, Constantin and Gribling, Sander and Gupta, Swati and Hadfield, Stuart and Heese, Raoul and Kircher, Gerhard and Kleinert, Thomas and Koch, Thorsten and Korpas, Georgios and Lenk, Steve and Marecek, Jakub and Markov, Vanio and Mazzola, Guglielmo and Mensa, Stefano and Mohseni, Naeimeh and Nannicini, Giacomo and O'Meara, Corey and Tapia, Elena Pe{\~n}a and Pokutta, Sebastian and Proissl, Manuel and Rebentrost, Patrick and Sahin, Emre and Symons, Benjamin C. B. and Tornow, Sabine and Valls, V{\´i}ctor and Woerner, Stefan and Wolf-Bauwens, Mira L. and Yard, Jon and Yarkoni, Sheir and Zechiel, Dirk and Zhuk, Sergiy and Zoufal, Christa}, title = {Challenges and opportunities in quantum optimization}, volume = {6}, journal = {Nature Reviews Physics}, publisher = {Springer Science and Business Media LLC}, issn = {2522-5820}, arxiv = {http://arxiv.org/abs/2312.02279}, doi = {10.1038/s42254-024-00770-9}, pages = {718 -- 735}, year = {2024}, language = {en} } @misc{Wolf2002, author = {Wolf, Thomas}, title = {Applications of CRACK in the Classification of Integrable Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7159}, number = {02-48}, year = {2002}, abstract = {The talk given by the author at the CRM workshop on Superintegrability in Sep.\ 2002 and this related paper report on work in two subjects. One is the collaboration with Vladimir Sokolov and Takayuki Tsuchida in an effort to classify polynomial integrable vector evolution equations. The other is the computer algebra package {\sc Crack} which did the main computations in solving large bi-linear algebraic systems. Although originally designed to solve over-determined systems of partial differential equations a number of extensions made {\sc Crack} a powerful tool for solving systems of bi-linear algebraic equations. Such systems turn up in many different classification problems some of which were investigated by other participants of this workshop. Two additional applications are outlined. In the talk on which this article is based a method to reduce the length of equations was presented which proved to be useful in solving the bi-linear algebraic systems. Due to numerous asked questions about the computer program, a more complete overview is given in the appendix.}, language = {en} } @misc{Wolf2000, author = {Wolf, Thomas}, title = {The Symbolic Integration of Exact PDEs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5702}, number = {00-02}, year = {2000}, abstract = {An algorithm is described to decide if a given polynomial differential expression \$\Delta\$ of multivariate functions is exact, i.e. whether there exists a first integral \$P\$ such that \$D_xP = \Delta\$ for any one of a set of variables \$x\$ and to provide the integral \$P\$. A generalization is given to allow integration in the case that the exactness is prohibited by terms which contain only functions of not all the independent variables.}, language = {en} } @misc{Wolf1998, author = {Wolf, Thomas}, title = {A Study of Genetic Algorithms solving a combinatorial Puzzle}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3445}, number = {SC-98-01}, year = {1998}, abstract = {The suitability of Genetic Algorithms (GAs) to solve a combinatorial problem with only one solution is investigated. The dependence of the performance is studied for GA-hard and GA-soft fitness functions, both with a range of different parameter values and different encodings.}, language = {en} } @misc{WolfBrandtMohammadzadeh1998, author = {Wolf, Thomas and Brandt, Andreas and Mohammadzadeh, Majid}, title = {Computer algebra algorithms and routines for the computation of conservation laws and fixing of gauge in differential expressions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3482}, number = {SC-98-05}, year = {1998}, abstract = {Three different approaches for the determination of conservation laws of differential equations are presented. For three corresponding REDUCE computer algebra programs CONLAW1/2/3 the necessary subroutines are discribed. One of them simplifies general solutions of overdetermined PDE systems so that all remaining free functions and constants correspond to independent conservation laws. It determines redundant functions and constants in differential expressions and is equally useful for the determination of symmetries or the fixing of gauge freedom in differential expressions.}, language = {en} } @misc{Wolf1997, author = {Wolf, Thomas}, title = {Killing Pairs in Flat Space}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3275}, number = {SC-97-58}, year = {1997}, abstract = {In this paper the integrability conditions for Killing pairs in flat spaces are investigated and it is shown that only trivial Killing pairs exist.}, language = {en} }