@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} } @misc{TsuchidaWolf2005, author = {Tsuchida, Takayuki and Wolf, Thomas}, title = {Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I}, arxiv = {http://arxiv.org/abs/nlin.SI/0412003}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8391}, number = {05-05}, year = {2005}, abstract = {We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov--Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of \$2^{\mbox{\scriptsize nd }}\$order systems with a \$3^{\mbox{\scriptsize rd }}\$order or a \$4^{\mbox{\scriptsize th }}\$order symmetry and \$3^{\mbox{\scriptsize rd }}\$order systems with a \$5^{\mbox{\scriptsize th }}\$order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.}, language = {en} } @misc{Wolf2005, author = {Wolf, Thomas}, title = {Partial and complete linearization of PDEs based on conservation laws}, arxiv = {http://arxiv.org/abs/nlin.SI/0501034}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8409}, number = {05-06}, year = {2005}, abstract = {A method based on infinite parameter conservation laws is described to factor linear differential operators out of nonlinear partial differential equations (PDEs) or out of differential consequences of nonlinear PDEs. This includes a complete linearization to an equivalent linear PDE (-system) if that is possible. Infinite parameter conservation laws can be computed, for example, with the computer algebra package {\sc ConLaw}.}, language = {en} } @misc{Wolf2005, author = {Wolf, Thomas}, title = {Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket}, doi = {10.1007/s10714-006-0293-2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8414}, number = {05-07}, year = {2005}, abstract = {Quadratic Hamiltonians with a linear Lie-Poisson bracket have a number of applications in mechanics. For example, the Lie-Poisson bracket \$e(3)\$ includes the Euler-Poinsot model describing motion of a rigid body around a fixed point under gravity and the Kirchhoff model describes the motion of a rigid body in ideal fluid. Advances in computer algebra algorithms, in implementations and hardware, together allow the computation of Hamiltonians with higher degree first integrals providing new results in the search for integrable models. A computer algebra module enabling related computations in a 3-dimensional vector formalism is described.}, language = {en} } @misc{AncoWolf2005, author = {Anco, Stephen and Wolf, Thomas}, title = {Some classifications of hyperbolic vector evolution equations}, arxiv = {http://arxiv.org/abs/nlin.SI/0412015}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8429}, number = {05-08}, year = {2005}, abstract = {Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several \$O(N)\$-invariant classes of hyperbolic equations \$Utx=f(U,Ut,Ux)\$ for an \$N\$-component vector \$U(t,x)\$ are considered. In each class we find all scaling-homogeneous equations admitting a higher symmetry of least possible scaling weight. Sigma model interpretations of these equations are presented.}, language = {en} } @misc{SokolovWolf2004, author = {Sokolov, Vladimir V. and Wolf, Thomas}, title = {Integrable quadratic Hamiltonians on so(4) and so(3,1)}, arxiv = {http://arxiv.org/abs/nlin.SI/0405066}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8085}, number = {04-33}, year = {2004}, abstract = {We investigate a special class of quadratic Hamiltonians on \$so(4)\$ and \$so(3,1)\$ and describe Hamiltonians that have additional polynomial integrals. One of the main results is a new integrable case with an integral of sixth degree.}, language = {en} } @misc{EulerEulerWolf2012, 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}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16087}, year = {2012}, abstract = {Recently, Holm and Ivanov, proposed and studied a class of multi-component generalisations 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, {\it J. Phys A: Math. Theor} {\bf 43}, 492001 (20pp), 2010]. 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 sent 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} } @misc{OdesskiiWolf2012, author = {Odesskii, Alexander and Wolf, Thomas}, title = {Compatible quadratic Poisson brackets related to a family of elliptic curves}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/1204.1299}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16097}, year = {2012}, abstract = {We construct nine pairwise compatible quadratic Poisson structures such that a generic linear combination of them is associated with an elliptic algebra in \$n\$ generators. Explicit formulas for Casimir elements of this elliptic Poisson structure are obtained.}, language = {en} } @misc{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}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16102}, year = {2012}, abstract = {The paper reports on a computer algebra program {\sc LSSS} (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M.\ Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14.}, language = {en} } @misc{AncoMacNaughtonWolf2012, author = {Anco, Stephen and MacNaughton, Steven A. and Wolf, Thomas}, title = {Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/1109.1719}, doi = {10.1063/1.4711814}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16111}, year = {2012}, abstract = {Symmetries and conservation laws are studied for two classes of physically and analytically interesting radial wave equations with power nonlinearities in multi-dimensions. The results consist of two main classifications: all symmetries of point type and all conservation laws of a general energy-momentum type are explicitly determined, including those such as dilations, inversions, similarity energies and conformal energies that exist only for special powers or dimensions. In particular, all variational cases (when a Lagrangian formulation exists) and non-variational cases (when no Lagrangian exists) for these wave equations are considered. As main results, the classification yields generalized energies and radial momenta in certain non-variational cases, which are shown to arise from a new type of Morawetz dilation identity that produces conservation laws for each of the two classes of wave equations in a different way than Noether's theorem.}, language = {en} } @misc{AncoMohiuddinWolf2012, author = {Anco, Stephen and Mohiuddin, Mohammad and Wolf, Thomas}, title = {Travelling waves and conservation laws for complex mKdV-type equations}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/1110.2403}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16124}, year = {2012}, abstract = {Travelling waves and conservation laws are studied for a wide class of \$U(1)\$-invariant complex mKdV equations containing the two known integrable generalizations of the ordinary (real) mKdV equation. The main results on travelling waves include deriving new complex solitary waves and kinks that generalize the well-known mKdV \$\sech\$ and \$\tanh\$ solutions. The main results on conservation laws consist of explicitly finding all 1st order conserved densities that yield phase-invariant counterparts of the well-known mKdV conserved densities for momentum, energy, and Galilean energy, and a new conserved density describing the angular twist of complex kink solutions.}, language = {en} } @misc{WolfAmadi2017, author = {Wolf, Thomas and Amadi, Chimaobi}, title = {Rational Solutions of Underdetermined Polynomial Equations}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64488}, year = {2017}, abstract = {In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our work provides a method to compute rational solutions of single polynomial equations that are typically large with 10^2 ... 10^5 terms and that are heavily underdetermined. It was possible to obtain this functionality by adding a number of new modules for a new type of splitting of equations to the existing package CRACK that is normally used to solve polynomial algebraic and differential systems of equations.}, language = {en} } @misc{SokolovWolf2001, author = {Sokolov, Vladimir V. and Wolf, Thomas}, title = {Classification of integrable polynomial vector evolution equations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6601}, number = {01-34}, year = {2001}, abstract = {Several classes of systems of evolution equations with one or two vector unknowns are considered. We investigate also systems with one vector and one scalar unknown. For these classes all equations having the simplest higher symmetry are listed.}, language = {en} } @misc{Wolf2002, author = {Wolf, Thomas}, title = {The integration of systems of linear PDEs using conservation laws of syzygies}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6751}, number = {02-08}, year = {2002}, abstract = {The purpose of the paper is to formulate and use syzygies for systems of linear PDEs. The computation of an equivalent of a GCD for linear partial differential operators will save us their factorization which is otherwise only possible algorithmically in special cases. After showing the computation with the new and the traditional method and comparing both in the next three sections, the algorithm is explained in general and an overview is given.}, language = {en} } @misc{AncoBlumanWolf2008, author = {Anco, Stephen and Bluman, George and Wolf, Thomas}, title = {Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Admitted Conservation Laws}, organization = {Department of Mathematics, Brock University, St. Catharines, ON Canada L2S 3A1}, issn = {1438-0064}, doi = {10.1007/s10440-008-9205-7}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10652}, number = {08-11}, year = {2008}, abstract = {An algorithmic method using conservation law multipliers is introduced that yields necessary and sufficient conditions to find invertible mappings of a given nonlinear PDE to some linear PDE and to construct such a mapping when it exists. Previous methods yielded such conditions from admitted point or contact symmetries of the nonlinear PDE. Through examples, these two linearization approaches are contrasted.}, language = {en} } @misc{Wolf2008, author = {Wolf, Thomas}, title = {On solving large systems of polynomial equations appearing in Discrete Differential Geometry}, issn = {1438-0064}, doi = {10.1134/S0361768808020047}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10646}, number = {08-10}, year = {2008}, abstract = {The paper describes a method for solution of very large overdetermined algebraic polynomial systems on an example that appears from a classification of all integrable 3-dimensional scalar discrete quasilinear equations \$Q_3=0\$ on an elementary cubic cell of the lattice \${\mathbb Z}^3\$. The overdetermined polynomial algebraic system that has to be solved is far too large to be formulated. A probing' technique which replaces independent variables by random integers or zero allows to formulate subsets of this system. An automatic alteration of equation formulating steps and equation solving steps leads to an iteration process that solves the computational problem.}, language = {en} } @misc{Wolf2008, author = {Wolf, Thomas}, title = {The Parametric Solution of Underdetermined linear ODEs}, issn = {1438-0064}, doi = {10.1134/S0361768811020113}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10693}, number = {08-15}, year = {2008}, abstract = {The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are arbitrary analytic functions in the independent variable. A second conceptual aim is to present an algorithm that is in some sense dual to the fundamental Euclids algorithm, and thus an alternative to the special case of a Gr\"{o}bner basis algorithm as it is used for solving linear ODE-systems. In the paper Euclids algorithm and the new dual version' are compared and their complementary strengths are analysed on the task of solving underdetermined ODEs. An implementation of the described algorithm is interactively accessible at http://lie.math.brocku.ca/crack/uode.}, language = {en} } @misc{TsarevWolf2008, author = {Tsarev, Sergey and Wolf, Thomas}, title = {Classification of 3-dimensional integrable scalar discrete equations}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/0706.2464}, doi = {10.1007/s11005-008-0230-2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10667}, number = {08-13}, year = {2008}, abstract = {We classify all integrable 3-dimensional scalar discrete affine linear equations \$Q_3=0\$ on an elementary cubic cell of the lattice \${\mathbb Z}^3\$. An equation \$Q_3=0\$ \%of such form is called integrable if it may be consistently imposed on all \$3\$-dimensional elementary faces of the lattice \${\mathbb Z}^4\$. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only ontrivial(non-linearizable) integrable equation from this class is the well-known dBKP-system.}, language = {en} }