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satellite [2016/11/07 21:00] anton created |
satellite [2018/04/13 22:28] (current) anton + linearly satellite |
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- | The ''Satellite'' package contains //Maple// procedures | + | The ''Satellite'' package contains //Maple// procedures to determine //satellite// and //linearly satellite// |

- | to determine //satellite// unknowns in linear differential systems. | + | unknowns in linear differential systems. Procedures for satellite unknowns recognizing implement |

- | These procedures implement partial algorithms, so they | + | partial algorithms, so they cannot be applied to all differential systems and thus they solve the |

- | cannot be applied to all differential systems and thus | + | problem in some cases. In other cases they do not give any answer (nor positive, nor negative). |

- | they solve the problem in some cases. In other cases | + | |

- | they do not give any answer (nor positive, nor negative). | + | |

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//A<sub>r</sub>y//<sup>(//r//)</sup>+//A//<sub>//r//-1</sub>//y//<sup>(//r//-1)</sup>+ ... +//A//<sub>1</sub>//y//'+//A//<sub>0</sub>y=0 | //A<sub>r</sub>y//<sup>(//r//)</sup>+//A//<sub>//r//-1</sub>//y//<sup>(//r//-1)</sup>+ ... +//A//<sub>1</sub>//y//'+//A//<sub>0</sub>y=0 | ||

- | where A, A_0, ..., A_r are //n// ''x'' //n// matrices over //K//=**//Q//**(//x//), | + | where //A//, //A<sub>0</sub>//, ..., //A<sub>r</sub>// are //n// ''x'' //n// matrices over //K//=**//Q//**(//x//), |

//y//=(//y//<sub>1</sub>,...,//y//<sub>n</sub>)<sup>T</sup> is a vector of unknowns. | //y//=(//y//<sub>1</sub>,...,//y//<sub>n</sub>)<sup>T</sup> is a vector of unknowns. | ||

We assume that some unknowns (entries of the vector //y//) are //selected//. | We assume that some unknowns (entries of the vector //y//) are //selected//. | ||

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**Definition.** | **Definition.** | ||

- | The unselected unknown //y<sub>j</sub>// is called //satellite// unknown | + | An unselected unknown //y<sub>j</sub>// is called a //satellite// unknown |

for the set of selected unknowns //s// in //S// | for the set of selected unknowns //s// in //S// | ||

if minimal subfield of a Picard--Vessio field over //K// for //S//, | if minimal subfield of a Picard--Vessio field over //K// for //S//, | ||

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also contains the //y<sub>j</sub>// component of any solution. | also contains the //y<sub>j</sub>// component of any solution. | ||

- | The ''Satellite'' package exports two procedures: | + | **Definition.** |

+ | The unselected unknown //y<sub>j</sub>// is called a //linearly satellite// unknown for the set of selected unknowns //s// | ||

+ | in //S// if the //j//-th component of any solution to //S// can be linearly expressed only via selected | ||

+ | components of this solution and their derivatives. | ||

+ | | ||

+ | The ''Satellite'' package exports the following procedures: | ||

* ''Testing''; | * ''Testing''; | ||

- | * ''Determination''. | + | * ''Determination''; |

+ | * ''LinSatTesting''; | ||

+ | * ''LinearlySatellite''. | ||

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but the partial algorithm cannot determine this. | but the partial algorithm cannot determine this. | ||

''Testing'' procedure cannot determine | ''Testing'' procedure cannot determine | ||

- | if //y//<sub>''v''/<sub> is not a satellite. | + | if //y//<sub>''v''</sub> is not a satellite. |

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- indices of unselected unknowns for which ''Testing'' algorithm cannot determine if they are satellite; | - indices of unselected unknowns for which ''Testing'' algorithm cannot determine if they are satellite; | ||

- indices of unselected unknowns that are not satellite for sure. | - indices of unselected unknowns that are not satellite for sure. | ||

+ | |||

+ | ---- | ||

+ | |||

+ | ===== Satellite:-LinSatTesting procedure ===== | ||

+ | |||

+ | |||

+ | ==== Calling Sequence ==== | ||

+ | |||

+ | ''LinSatTesting(A, s, v)'' | ||

+ | |||

+ | ==== Parameters ==== | ||

+ | |||

+ | * A - square matrix of the normal differential system //y'=Ay// | ||

+ | * s - set of positive integers — indices of selected unknowns | ||

+ | * v - positive integer - index of the testing unknown | ||

+ | |||

+ | |||

+ | ==== Description ==== | ||

+ | |||

+ | ''LinSatTesting'' procedure determines whether the unknown of index v (//y<sub>v</sub>//) of differential system | ||

+ | //y'=Ay// is linearly satellite for the set of selected unknowns //s//. | ||

+ | ''LinSatTesting'' returns «true» if //y<sub>v</sub>// is linearly satellite for selected unknowns //s//; otherwise it | ||

+ | returns «false». | ||

+ | |||

+ | |||

+ | ---- | ||

+ | |||

+ | ===== Satellite:-LinearlySatellite procedure ===== | ||

+ | |||

+ | |||

+ | ==== Calling Sequence ==== | ||

+ | |||

+ | ''LinearlySatellite(A, s)'' | ||

+ | |||

+ | ==== Parameters ==== | ||

+ | |||

+ | * A — square matrix of the normal differential system //y'=Ay// or a list of high-order differential system matrices | ||

+ | * s — set of positive integers — indices of selected unknowns | ||

+ | |||

+ | ==== Description ==== | ||

+ | |||

+ | ''LinearlySatellite'' procedure builds and returns the set of linearly satellite unknown indices for the set of | ||

+ | selected unknowns //s//. | ||

====== Source ======= | ====== Source ======= |

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