On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space
Let for s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 be the space of s-vectors in the Clifford algebra IR0,m+1 constructed over the quadratic vector space IR0,m+1 and let r, p, q, ∈ IN be such that 0 ≤ r ≤ m + 1, p < q and r + 2q ≤ m + 1. T...
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| Format: | Article |
| Language: | English |
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Universidad de La Frontera
2010-01-01
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| Series: | Cubo |
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| Online Access: | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000200010 |
| _version_ | 1818842341596725248 |
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| author | Richard Delanghe |
| author_facet | Richard Delanghe |
| author_sort | Richard Delanghe |
| collection | DOAJ |
| description | Let for s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 be the space of s-vectors in the Clifford algebra IR0,m+1 constructed over the quadratic vector space IR0,m+1 and let r, p, q, ∈ IN be such that 0 ≤ r ≤ m + 1, p < q and r + 2q ≤ m + 1. The associated linear system of first order partial differential equations derived from the equation ∂xW = 0 where W is IR(r,p,q)0,m+1 = ∑q j=p ⊕IR(r+2j)0,m+1 -valued and ∂x is the Dirac operator in IRm+1, is called a generalized Moisil-Théodoresco system of type (r, p, q) in IRm+1. For k ∈ N, k ≥ 1,MT+(m+ 1; k; IR(r,p,q)0,m+1), denotes the space of IR(r,p,q)0,m+1-valued homogeneous polynomials Wc of degree k in IRm+1 satisfying ∂xWx = 0. A characterization of Wk∈ MT+(m + 1; k;IR(r,p,q)0,m+1) is given in terms of a harmonic potential Hk+1 belonging to a subclass of IR(r,p,q)0,m -valued solid harmonics of degree (k + 1) in IRm+1. Furthermore, it is proved that each Wk∈ MT+(m+ 1; k; IR(r,p,q)0,m+1) admits a primitive Wk+1 ∈ MT+(m+ 1; k + 1; IR(r,p,q)0,m+1). Special attention is paid to the lower dimensional cases IR³ and IR4. In particular, a method is developed for constructing bases for the spaces MT+(4; k; IR(r,p,q)0,4), r being even.<br>Para s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 el espacio de los s-vectors en el algebra de Clifford IR0,m+1 construida sobre el espacio de vectores cuadráticos IR0,m+1 sea r, p, q, ∈ IN tal que 0 ≤ r ≤ m + 1, p < q. El sistema lineal asociado de ecuaciones diferenciales parciales de primer orden derivado de la ecuaci´on ∂xW = 0 donde W es IR(r,p,q)0,m+1 = ∑q j=p ⊕IR(r+2j)0,m+1 1-valuada y ∂x es el operador de Dirac en IRm+1, es llamado un sistema de Moisil-Théodoresco generalizado de tipo (r, p, q) en IRm+1. Para k ∈ N, k ≥ 1,MT+(m+ 1; k; IR(r,p,q)0,m+1), denota el espacio de polinomios homogéneosWk IR(r,p,q)0,m+1- valuados de grado k en IRm+1. satisfaciendo ∂xWx = 0. Una caracterización de Wk∈ MT+(m+1; k; IR(r,p,q)0,m+1) es dada en términos de un potencial armónico Hk+1 perteneciendo a una subclase de armónicos consistentes IR(r,p,q)0,m -valuados de grado (k + 1) in IRm+1. Además es probado que todo Wk∈ MT+(m + 1; k; IR(r,p,q)0,m+1) admite una primitiva Wk+1 ∈ MT+(m + 1; k + 1; IR(r,p,q)0,m+1). Una especial atención es dada a los casos de dimensión baja IR³ y IR4. En particular, un metodo es desarrollado para construir bases para espaciosMT+(4; k; IR(r,p,q)0,4 ), r siendo par. |
| first_indexed | 2024-12-19T04:40:26Z |
| format | Article |
| id | doaj.art-7366d30464dd4840a023c1b23ba9323c |
| institution | Directory Open Access Journal |
| issn | 0716-7776 0719-0646 |
| language | English |
| last_indexed | 2024-12-19T04:40:26Z |
| publishDate | 2010-01-01 |
| publisher | Universidad de La Frontera |
| record_format | Article |
| series | Cubo |
| spelling | doaj.art-7366d30464dd4840a023c1b23ba9323c2022-12-21T20:35:37ZengUniversidad de La FronteraCubo0716-77760719-06462010-01-01122145167On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean spaceRichard DelangheLet for s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 be the space of s-vectors in the Clifford algebra IR0,m+1 constructed over the quadratic vector space IR0,m+1 and let r, p, q, ∈ IN be such that 0 ≤ r ≤ m + 1, p < q and r + 2q ≤ m + 1. The associated linear system of first order partial differential equations derived from the equation ∂xW = 0 where W is IR(r,p,q)0,m+1 = ∑q j=p ⊕IR(r+2j)0,m+1 -valued and ∂x is the Dirac operator in IRm+1, is called a generalized Moisil-Théodoresco system of type (r, p, q) in IRm+1. For k ∈ N, k ≥ 1,MT+(m+ 1; k; IR(r,p,q)0,m+1), denotes the space of IR(r,p,q)0,m+1-valued homogeneous polynomials Wc of degree k in IRm+1 satisfying ∂xWx = 0. A characterization of Wk∈ MT+(m + 1; k;IR(r,p,q)0,m+1) is given in terms of a harmonic potential Hk+1 belonging to a subclass of IR(r,p,q)0,m -valued solid harmonics of degree (k + 1) in IRm+1. Furthermore, it is proved that each Wk∈ MT+(m+ 1; k; IR(r,p,q)0,m+1) admits a primitive Wk+1 ∈ MT+(m+ 1; k + 1; IR(r,p,q)0,m+1). Special attention is paid to the lower dimensional cases IR³ and IR4. In particular, a method is developed for constructing bases for the spaces MT+(4; k; IR(r,p,q)0,4), r being even.<br>Para s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 el espacio de los s-vectors en el algebra de Clifford IR0,m+1 construida sobre el espacio de vectores cuadráticos IR0,m+1 sea r, p, q, ∈ IN tal que 0 ≤ r ≤ m + 1, p < q. El sistema lineal asociado de ecuaciones diferenciales parciales de primer orden derivado de la ecuaci´on ∂xW = 0 donde W es IR(r,p,q)0,m+1 = ∑q j=p ⊕IR(r+2j)0,m+1 1-valuada y ∂x es el operador de Dirac en IRm+1, es llamado un sistema de Moisil-Théodoresco generalizado de tipo (r, p, q) en IRm+1. Para k ∈ N, k ≥ 1,MT+(m+ 1; k; IR(r,p,q)0,m+1), denota el espacio de polinomios homogéneosWk IR(r,p,q)0,m+1- valuados de grado k en IRm+1. satisfaciendo ∂xWx = 0. Una caracterización de Wk∈ MT+(m+1; k; IR(r,p,q)0,m+1) es dada en términos de un potencial armónico Hk+1 perteneciendo a una subclase de armónicos consistentes IR(r,p,q)0,m -valuados de grado (k + 1) in IRm+1. Además es probado que todo Wk∈ MT+(m + 1; k; IR(r,p,q)0,m+1) admite una primitiva Wk+1 ∈ MT+(m + 1; k + 1; IR(r,p,q)0,m+1). Una especial atención es dada a los casos de dimensión baja IR³ y IR4. En particular, un metodo es desarrollado para construir bases para espaciosMT+(4; k; IR(r,p,q)0,4 ), r siendo par.http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000200010Clifford analysisMoisil-Théodoresco systemsconjugate harmonic funtionsharmonic potentialspolynomial bases |
| spellingShingle | Richard Delanghe On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space Cubo Clifford analysis Moisil-Théodoresco systems conjugate harmonic funtions harmonic potentials polynomial bases |
| title | On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space |
| title_full | On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space |
| title_fullStr | On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space |
| title_full_unstemmed | On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space |
| title_short | On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space |
| title_sort | on homogeneous polynomial solutions of generalized moisil theodoresco systems in euclidean space |
| topic | Clifford analysis Moisil-Théodoresco systems conjugate harmonic funtions harmonic potentials polynomial bases |
| url | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000200010 |
| work_keys_str_mv | AT richarddelanghe onhomogeneouspolynomialsolutionsofgeneralizedmoisiltheodorescosystemsineuclideanspace |