TY - JOUR A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction JF - SIGMA 7 N2 - 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. Y1 - 2011 U6 - https://doi.org/10.3842/SIGMA.2011.066 ER - TY - JOUR A1 - Wolf, Thomas A1 - Efimovskaya, Olga T1 - On Integrability of the Kontsevich Non-Abelian ODE System JF - Letters in Mathematical Physics Y1 - 2012 U6 - https://doi.org/10.1007/s11005-011-0527-4 VL - 100 IS - 2 SP - 161 EP - 170 ER - TY - JOUR A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions JF - Journal of Mathematical Analysis and Applications Y1 - 2011 U6 - https://doi.org/10.1016/j.jmaa.2011.01.073 VL - 379 IS - 2 SP - 748 EP - 763 ER - TY - JOUR A1 - Wolf, Thomas T1 - A Dynamical Systems Approach for Static Evaluation in Go JF - Computational Intelligence and AI in Games, IEEE Transactions on N2 - 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. Y1 - 2011 U6 - https://doi.org/10.1109/TCIAIG.2011.2141669 VL - 3 IS - 2 SP - 129 EP - 141 ER - TY - JOUR A1 - Hussin, V. A1 - Kiselev, A. V. A1 - Krutov, A.O. A1 - Wolf, Thomas T1 - N=2 Supersymmetric a=4 - Korteweg-de Vries hierarchy derived via Gardner's deformation of Kaup-Bousinesq equation JF - Journal of Mathematical Physics Y1 - 2010 U6 - https://doi.org/10.1063/1.3447731 VL - 51 ER - TY - JOUR A1 - Wolf, Thomas T1 - Positions in the Game of Go as Complex Systems JF - Proceedings of 2009 IEEE Toronto International Conference (TIC-STH 2009) - Science and Technology for Humanity Y1 - 2009 SN - 978-1-4244-3878-5 ER - TY - JOUR A1 - Tsarev, Sergey A1 - Wolf, Thomas T1 - Hyperdeterminants as integrable discrete systems JF - Journal of Physics A: Mathematical and Theoretical Y1 - 2009 U6 - https://doi.org//10.1088/1751-8113/42/45/454023 VL - 42 ER - TY - GEN A1 - Mars, Marc A1 - Wolf, Thomas T1 - G2 Perfect-Fluid Cosmologies with a proper conformal Killing vector N2 - 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. T3 - ZIB-Report - SC-97-08 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2774 ER - TY - GEN A1 - Stephani, Hans A1 - Wolf, Thomas T1 - Spherically symmetric perfect fluids in shearfree motion - the symmetry approach N2 - 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. T3 - ZIB-Report - SC-96-14 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2253 ER - TY - GEN A1 - Wolf, Thomas T1 - 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 N2 - 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 $\frac{1}{r+1}\left( ^{n + r - 1}_{\;\;\;\;\,r} \right) \left( ^{ n+r}_{\;\;\,r} \right)$. T3 - ZIB-Report - SC-95-22 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1881 ER - TY - GEN A1 - Wolf, Thomas T1 - Programs for Applying Symmetries of PDEs N2 - 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. T3 - ZIB-Report - SC-95-05 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1710 ER - TY - GEN A1 - Wolf, Thomas T1 - A comparison of four approaches to the calculation of conservation laws N2 - 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. T3 - ZIB-Report - SC-99-01 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3896 ER - TY - GEN A1 - Sokolov, Vladimir V. A1 - Wolf, Thomas T1 - A symmetry test for quasilinear coupled systems N2 - 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 $A_{i}(u,v), B_{i}(u,v)$. T3 - ZIB-Report - SC-99-02 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3901 ER - TY - GEN A1 - Wolf, Thomas T1 - Structural equations for Killing tensors of arbitrary rank N2 - 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. T3 - ZIB-Report - SC-97-62 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3310 ER - TY - JOUR A1 - Abbas, Amira A1 - Ambainis, Andris A1 - Augustino, Brandon A1 - Bärtschi, Andreas A1 - Buhrman, Harry A1 - Coffrin, Carleton A1 - Cortiana, Giorgio A1 - Dunjko, Vedran A1 - Egger, Daniel J. A1 - Elmegreen, Bruce G. A1 - Franco, Nicola A1 - Fratini, Filippo A1 - Fuller, Bryce A1 - Gacon, Julien A1 - Gonciulea, Constantin A1 - Gribling, Sander A1 - Gupta, Swati A1 - Hadfield, Stuart A1 - Heese, Raoul A1 - Kircher, Gerhard A1 - Kleinert, Thomas A1 - Koch, Thorsten A1 - Korpas, Georgios A1 - Lenk, Steve A1 - Marecek, Jakub A1 - Markov, Vanio A1 - Mazzola, Guglielmo A1 - Mensa, Stefano A1 - Mohseni, Naeimeh A1 - Nannicini, Giacomo A1 - O’Meara, Corey A1 - Tapia, Elena Peña A1 - Pokutta, Sebastian A1 - Proissl, Manuel A1 - Rebentrost, Patrick A1 - Sahin, Emre A1 - Symons, Benjamin C. B. A1 - Tornow, Sabine A1 - Valls, Víctor A1 - Woerner, Stefan A1 - Wolf-Bauwens, Mira L. A1 - Yard, Jon A1 - Yarkoni, Sheir A1 - Zechiel, Dirk A1 - Zhuk, Sergiy A1 - Zoufal, Christa T1 - Challenges and opportunities in quantum optimization JF - Nature Reviews Physics Y1 - 2024 U6 - https://doi.org/10.1038/s42254-024-00770-9 SN - 2522-5820 VL - 6 SP - 718 EP - 735 PB - Springer Science and Business Media LLC ER - TY - GEN A1 - Wolf, Thomas T1 - Applications of CRACK in the Classification of Integrable Systems N2 - 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. T3 - ZIB-Report - 02-48 KW - Integrability KW - evolution equations KW - quadratic Hamiltonians KW - Computer algebra KW - Bi-linear algebraic systems KW - conservation laws Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7159 ER - TY - GEN A1 - Wolf, Thomas T1 - The Symbolic Integration of Exact PDEs N2 - 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. T3 - ZIB-Report - 00-02 KW - symbolic integration KW - partial differential equations Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5702 ER - TY - GEN A1 - Wolf, Thomas T1 - A Study of Genetic Algorithms solving a combinatorial Puzzle N2 - 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. T3 - ZIB-Report - SC-98-01 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3445 ER - TY - GEN A1 - Wolf, Thomas A1 - Brandt, Andreas A1 - Mohammadzadeh, Majid T1 - Computer algebra algorithms and routines for the computation of conservation laws and fixing of gauge in differential expressions N2 - 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. T3 - ZIB-Report - SC-98-05 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3482 ER - TY - GEN A1 - Wolf, Thomas T1 - Killing Pairs in Flat Space N2 - In this paper the integrability conditions for Killing pairs in flat spaces are investigated and it is shown that only trivial Killing pairs exist. T3 - ZIB-Report - SC-97-58 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3275 ER -