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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 //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 =======
satellite.txt · Last modified: 2018/04/13 22:28 by anton
 
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