# Differences

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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 =======