Solution of the determinantal assignment problem using the Grassmann matrices
The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral...
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Gespeichert in:
| Autor*in: |
Karcanias, Nicos [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016 |
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| Rechteinformationen: |
Nutzungsrecht: © 2015 Taylor & Francis 2015 |
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| Schlagwörter: |
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| Übergeordnetes Werk: |
Enthalten in: International journal of control - London : Taylor & Francis, 1965, 89(2016), 2, Seite 352 |
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| Übergeordnetes Werk: |
volume:89 ; year:2016 ; number:2 ; pages:352 |
| Links: |
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| DOI / URN: |
10.1080/00207179.2015.1077525 |
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OLC1971624357 |
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| 520 | |a The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge-Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge-Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist. | ||
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10.1080/00207179.2015.1077525 doi PQ20160430 (DE-627)OLC1971624357 (DE-599)GBVOLC1971624357 (PRQ)i1452-b9309505c1b4d9f0db1259726b68db5d2443f491948c48ea20f80266d7f655d00 (KEY)0006630320160000089000200352solutionofthedeterminantalassignmentproblemusingth DE-627 ger DE-627 rakwb eng 620 DNB Karcanias, Nicos verfasserin aut Solution of the determinantal assignment problem using the Grassmann matrices 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge-Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge-Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist. Nutzungsrecht: © 2015 Taylor & Francis 2015 frequency assignment control theory linear and multilinear algebra Leventides, John oth Enthalten in International journal of control London : Taylor & Francis, 1965 89(2016), 2, Seite 352 (DE-627)129595780 (DE-600)240693-7 (DE-576)015088804 0020-7179 nnns volume:89 year:2016 number:2 pages:352 http://dx.doi.org/10.1080/00207179.2015.1077525 Volltext http://www.tandfonline.com/doi/abs/10.1080/00207179.2015.1077525 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC GBV_ILN_70 GBV_ILN_2020 GBV_ILN_4314 GBV_ILN_4318 GBV_ILN_4700 AR 89 2016 2 352 |
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10.1080/00207179.2015.1077525 doi PQ20160430 (DE-627)OLC1971624357 (DE-599)GBVOLC1971624357 (PRQ)i1452-b9309505c1b4d9f0db1259726b68db5d2443f491948c48ea20f80266d7f655d00 (KEY)0006630320160000089000200352solutionofthedeterminantalassignmentproblemusingth DE-627 ger DE-627 rakwb eng 620 DNB Karcanias, Nicos verfasserin aut Solution of the determinantal assignment problem using the Grassmann matrices 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge-Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge-Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist. Nutzungsrecht: © 2015 Taylor & Francis 2015 frequency assignment control theory linear and multilinear algebra Leventides, John oth Enthalten in International journal of control London : Taylor & Francis, 1965 89(2016), 2, Seite 352 (DE-627)129595780 (DE-600)240693-7 (DE-576)015088804 0020-7179 nnns volume:89 year:2016 number:2 pages:352 http://dx.doi.org/10.1080/00207179.2015.1077525 Volltext http://www.tandfonline.com/doi/abs/10.1080/00207179.2015.1077525 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC GBV_ILN_70 GBV_ILN_2020 GBV_ILN_4314 GBV_ILN_4318 GBV_ILN_4700 AR 89 2016 2 352 |
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10.1080/00207179.2015.1077525 doi PQ20160430 (DE-627)OLC1971624357 (DE-599)GBVOLC1971624357 (PRQ)i1452-b9309505c1b4d9f0db1259726b68db5d2443f491948c48ea20f80266d7f655d00 (KEY)0006630320160000089000200352solutionofthedeterminantalassignmentproblemusingth DE-627 ger DE-627 rakwb eng 620 DNB Karcanias, Nicos verfasserin aut Solution of the determinantal assignment problem using the Grassmann matrices 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge-Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge-Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist. Nutzungsrecht: © 2015 Taylor & Francis 2015 frequency assignment control theory linear and multilinear algebra Leventides, John oth Enthalten in International journal of control London : Taylor & Francis, 1965 89(2016), 2, Seite 352 (DE-627)129595780 (DE-600)240693-7 (DE-576)015088804 0020-7179 nnns volume:89 year:2016 number:2 pages:352 http://dx.doi.org/10.1080/00207179.2015.1077525 Volltext http://www.tandfonline.com/doi/abs/10.1080/00207179.2015.1077525 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC GBV_ILN_70 GBV_ILN_2020 GBV_ILN_4314 GBV_ILN_4318 GBV_ILN_4700 AR 89 2016 2 352 |
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620 DNB Solution of the determinantal assignment problem using the Grassmann matrices frequency assignment control theory linear and multilinear algebra |
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ddc 620 misc frequency assignment misc control theory misc linear and multilinear algebra |
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ddc 620 misc frequency assignment misc control theory misc linear and multilinear algebra |
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Solution of the determinantal assignment problem using the Grassmann matrices |
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Solution of the determinantal assignment problem using the Grassmann matrices |
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Karcanias, Nicos |
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International journal of control |
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10.1080/00207179.2015.1077525 |
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620 |
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solution of the determinantal assignment problem using the grassmann matrices |
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Solution of the determinantal assignment problem using the Grassmann matrices |
| abstract |
The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge-Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge-Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist. |
| abstractGer |
The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge-Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge-Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist. |
| abstract_unstemmed |
The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge-Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge-Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist. |
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Solution of the determinantal assignment problem using the Grassmann matrices |
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