This paper provides an implementation of a novel signal processing co-processor using a Geometric Algebra technique tailored for fast and complex geometric calculations in multiple dimensions. This is the first hardwa...This paper provides an implementation of a novel signal processing co-processor using a Geometric Algebra technique tailored for fast and complex geometric calculations in multiple dimensions. This is the first hardware implementation of Geometric Algebra to specifically address the issue of scalability to multiple (1 - 8) dimensions. This paper presents a detailed description of the implementation, with a particular focus on the techniques of optimization used to improve performance. Results are presented which demonstrate at least 3x performance improvements compared to previously published work.展开更多
If a first-order algebraic ODE is defined over a certain differential field,then the most elementary solution class,in which one can hope to find a general solution,is given by the adjunction of a single arbitrary con...If a first-order algebraic ODE is defined over a certain differential field,then the most elementary solution class,in which one can hope to find a general solution,is given by the adjunction of a single arbitrary constant to this field.Solutions of this type give rise to a particular kind of generic point—a rational parametrization—of an algebraic curve which is associated in a natural way to the ODE’s defining polynomial.As for the opposite direction,we show that a suitable rational parametrization of the associated curve can be extended to a general solution of the ODE if and only if one can find a certain automorphism of the solution field.These automorphisms are determined by linear rational functions,i.e.,Möbius transformations.Intrinsic properties of rational parametrizations,in combination with the particular shape of such automorphisms,lead to a number of necessary conditions on the existence of general solutions in this solution class.Furthermore,the desired linear rational function can be determined by solving a comparatively simple differential system over the ODE’s field of definition.These results hold for arbitrary differential fields of characteristic zero.展开更多
This paper generalizes the method of Ngo and Winkler(2010,2011)for finding rational general solutions of a first order non-autonomous algebraic ordinary differential equation(AODE)to the case of a higher order AODE,pr...This paper generalizes the method of Ngo and Winkler(2010,2011)for finding rational general solutions of a first order non-autonomous algebraic ordinary differential equation(AODE)to the case of a higher order AODE,provided a proper parametrization of its solution hypersurface.The authors reduce the problem of finding the rational general solution of a higher order AODE to finding the rational general solution of an associated system.The rational general solutions of the original AODE and its associated system are in computable 1-1 correspondence.The authors give necessary and sufficient conditions for the associated system to have a rational solution based on proper reparametrization of invariant algebraic space curves.The authors also relate invariant space curves to first integrals and characterize rationally solvable systems by rational first integrals.展开更多
The Manley-Rowe constants of motion(MRC)are conservation laws written out for a dynamical systemdescribing the time evolution of the amplitudes in resonant triad.In this paper we extend the concept of MRC to resonance...The Manley-Rowe constants of motion(MRC)are conservation laws written out for a dynamical systemdescribing the time evolution of the amplitudes in resonant triad.In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants(gMRC)and give a constructive method how to compute them.We also give details of a Mathematica implementation of this method.WhileMRC provide integrability of the underlying dynamical system,gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.展开更多
文摘This paper provides an implementation of a novel signal processing co-processor using a Geometric Algebra technique tailored for fast and complex geometric calculations in multiple dimensions. This is the first hardware implementation of Geometric Algebra to specifically address the issue of scalability to multiple (1 - 8) dimensions. This paper presents a detailed description of the implementation, with a particular focus on the techniques of optimization used to improve performance. Results are presented which demonstrate at least 3x performance improvements compared to previously published work.
文摘If a first-order algebraic ODE is defined over a certain differential field,then the most elementary solution class,in which one can hope to find a general solution,is given by the adjunction of a single arbitrary constant to this field.Solutions of this type give rise to a particular kind of generic point—a rational parametrization—of an algebraic curve which is associated in a natural way to the ODE’s defining polynomial.As for the opposite direction,we show that a suitable rational parametrization of the associated curve can be extended to a general solution of the ODE if and only if one can find a certain automorphism of the solution field.These automorphisms are determined by linear rational functions,i.e.,Möbius transformations.Intrinsic properties of rational parametrizations,in combination with the particular shape of such automorphisms,lead to a number of necessary conditions on the existence of general solutions in this solution class.Furthermore,the desired linear rational function can be determined by solving a comparatively simple differential system over the ODE’s field of definition.These results hold for arbitrary differential fields of characteristic zero.
基金supported by the Austrian Science Foundation(FWF) via the Doctoral Program "Computational Mathematics" under Grant No.W1214Project DK11,the Project DIFFOP under Grant No.P20336-N18+2 种基金the SKLSDE Open Fund SKLSDE-2011KF-02the National Natural Science Foundation of China under Grant No.61173032the Natural Science Foundation of Beijing under Grant No.1102026,and the China Scholarship Council
文摘This paper generalizes the method of Ngo and Winkler(2010,2011)for finding rational general solutions of a first order non-autonomous algebraic ordinary differential equation(AODE)to the case of a higher order AODE,provided a proper parametrization of its solution hypersurface.The authors reduce the problem of finding the rational general solution of a higher order AODE to finding the rational general solution of an associated system.The rational general solutions of the original AODE and its associated system are in computable 1-1 correspondence.The authors give necessary and sufficient conditions for the associated system to have a rational solution based on proper reparametrization of invariant algebraic space curves.The authors also relate invariant space curves to first integrals and characterize rationally solvable systems by rational first integrals.
基金support of the Austrian Science Foundation(FWF)under project P22943-N18”Nonlinear resonances of water waves”and in part–by the Project of Knowledge Innovation Program(PKIP)of Chinese Academy of Sciences,Grant No.KJCX2.YW.W10.E.K.is very much obliged to the organizing committee of the program”New Directions in Turbulence”(KITPC/ITP-CAS,2012)and the hospitality of Kavli ITP,Beijing,where part of this work has been accomplished.
文摘The Manley-Rowe constants of motion(MRC)are conservation laws written out for a dynamical systemdescribing the time evolution of the amplitudes in resonant triad.In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants(gMRC)and give a constructive method how to compute them.We also give details of a Mathematica implementation of this method.WhileMRC provide integrability of the underlying dynamical system,gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.